NASA/TMm2001-209981

Small-Scale Spatial Fluctuations

in the Soft X-Ray Background

Kip D. Kuntz

NASA Goddard Space Flight Center, Greenbelt, Maryland

National Aeronautics and

Space Administration

Goddard Space Flight Center

Greenbelt, Maryland 20771

February 2001

https://ntrs.nasa.gov/search.jsp?R=20020009013 2018-05-15T10:45:47+00:00Z
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ABSTRACT

Title of Dissertation: SMALL-SCALE SPATIAL FLUCTUATIONS IN THE SOFT

X-RAY BACKGROUND

K. D. Kuntz, Doctor of Philosophy, 2000

Dissertation Directed by: Prof. Stuart N. Vogel,

Department of Astronomy

In order to isolate the diffuse extragalactic component of the soft X-ray background, we have

used a combination of ROSAT All-Sky Suvey and IRAS 100/_m data to separate the soft X-ray

background into five components.

We find a Local Hot Bubble similar to that described by Snowden et al (1998).

We make a first calculation of the contribution by unresolved Galactic stars to the diffuse

background.

We constrain the normalization of the Extragalactic Power Law (the contribution of the unresolved extragalactic point sources such as AGN, QSO's, and normal galaxies) to 9.5 -4-0.9 keV

cm -2 s-1 sr -1 keV -1, assuming a power-law index of 1.46.

We show that the remaining emission, which is some combination of Galactic halo emission

and the putative diffuse extragalactic emission, must be composed of at least two components

which we have characterized by thermal spectra. The softer component has log T -,- 6.08 and a

patchy distribution; thus it is most probably part of the Galactic halo. The harder component has

log T -,_ 6.46 and is nearly isotropic; some portion may be due to the Galactic halo and some portion

may be due to the diffuse extragalactic emission. The maximum upper limit to the strength of the

emission by the diffuse extragalactic component is the total of the hard component, --* 7.4 :i= 1.0

keV cm -_ s-1 st-1 keV-1 in the 43-keV band.

We have made the first direct measure of the fluctuations due to the diffuse extragalactic emission

in the 3 keV band. Physical arguments suggest that small angular scale (_ 10') fluctuations in

the Local Hot Bubble or the Galactic halo will have very short dissipation times (--, l0 s years).

Therefore, the fluctuation spectrum of the soft X-ray background should measure the distribution

of the diffuse extragalactic emission. Using mosaics of deep, overlapping PSPC pointings, we find

an autocorrelation function value of _ 0.0025 for 10' < _ < 20', and a value consistent with zero

on larger scales. Measurement of the fluctuations with a tiI/I method produces consistent results.

SMALL-SCALE SPATIAL FLUCTUATIONS IN THE SOFT X-RAY

BACKGROUND

by

K. D. Kuntz

Dissertation submitted to the Faculty of the Graduate School of the

University of Maryland, College Park in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

2000

Advisory Committee:

Prof. Stuart Vogel, Chairman

Prof. Daniel I. Fivel

Prof. Richard Mushotzky

Prof. Eve Ostriker

Dr. Steven L. Snowden

Prof. Sylvain Veilleux
© Copyright by

K. D. Kuntz

2000
PREFACE

Meae culpae

First, I apologize profusely to my readers; my style is prolix and baroque, and the dissertation is

long. Second, I apologize for the format. I know that most readers expect to receive the dissertation

in the official format. I have turned to the two-column, single-spaced, ten-point journal format,

not only because it looks better and is easier to read, but it decimates the fewest forests. Third, I

apologize for insufficient gravitas.

I would also like to apologize for a matter of notation, about which I was very careful in the

earlier chapters, and quite sloppy in the later chapters. The 0.1-0.284 keV energy band is sometimes

referred to as band R1L2, and sometimes as band R82. Both of these bands are the same. Band

R12 is slightly different from band R82, but not sufficiently so as to make a difference for most

purposes. Sometimes these bands are referred to collectively as the ,,_1keV" band, but I have tried

to eliminate most of those instances. The 0.44-1.21 keV band is referred to as the R45 band, but

occasionally I slip and refer to is as the ,,3 keV" band. The 1.21-2.04 keV band is sometimes called

the "1.5 keV" band. Other works using the ROSATPSPC usually use these same bands, but often

refer to them slightly differently, For example, the 0.44-2.04 keV band often is called the 0.5-2.0

keV band. Then, of course, there are a few individuals who have created non-standard bands; in

those cases I have tried to refer to them in a way that will make their relation to the standard

bands apparent.

I could also apologize for the diffuse nature of the term "Soft X-ray Background", as that term

sometimes means 0.1-0.5 keV, and sometimes E < 2 keV. However, that terminology has been

mixed for a long time, so I decline to take the blame.

Some Administrative Matters

Chapter 3, together with Appendix D, is contained in Kuntz & Snowden (1998), a manual for

use of the mosaicking software. At the beginning of the project it was thought that the existing

mosaicking software was sufficient, but repeated tinkering led to its wholesale reconstruction, and

thus the need for a new manual.

Chapter 4, on the tracers of absorption, began as an attempt to understand the absorption of

X-rays by the ISM. I was led to examine a number of results in the literature, and at times the

examination got out of hand; this chapter was the result. There are a number of interesting issues

buried in this chapter that I hope to revisit in the future, such as curious results obtained by

Arabadjis & Bregman (1999).

Chapter 5 and Chapter 6, on the TAE, are in press (set for the 1 November issue of the

Astrophysical Journal) as "The Deconstruction of the Spectrum of the Soft X-ray Background". I

did need to determine a good spectrum for the SXRB in order to do a good deabsorption (and to

pursue some other topics, such as X), but the treatment here is a bit of an indulgence, though a

profitable one. Appendix A and Appendix E cover related issues that were raised during seminar

presentations about the TAE.

Chapter 7 (with Appendix F), on the X-ray contribution due to stars, was originally projected

due to the need for a measure of the variance due to unresolved stellar X-ray sources. The question

of how much of the TAE could be due to stars provided a more immediate motive. It was thought

that a simple galactic model, together with the Schmitt, Fleming, & Giampapa (1995) and Schmitt

(1997) data would be adequate. The Guillout et al. (1996) paper led me to believe that my problem

had already been solved. However, the difference in energy bands modeled and the concentration

of resolved sources rather than the unresolved sources, did not allow application of the results to

my problem. Contacting the author was of little use: his proprietary software was too complex and

too directed to other problems to be easily applied to mine, and the author became unavailable due
to the XMM launch. So I took a month and built my own. After another round of improvements,

this chapter will be submitted to the Astrophysical Journal.

Chapter 8, on deabsorption, is necessary and (almost) completely uninteresting. Clearly, the

absorption of X-rays by the ISM is more complex than we would like to admit, and many of the

required data do not exist. Those data will be very hard to obtain, but will have major implications

for the general study of the ISM.

Chapter 10, on fluctuations, is the ostensible reason for the dissertation, as supported by an

ADP grant. A number of difficulties make the results somewhat less useful than one might have

hoped (as might be implied by the introductory quote). Nevertheless, the results may be a useful

base for future study of a clumpy diffuse IGM. The essential results, together with bits of Chapter

8 and Chapter 2, and together with some projected work described in Chapter 11, are expected

to turn into a paper over the next two years, during which I will be funded by an ADP grant to

pursue this work.

This dissertation was formatted with _using a the aastex style file as modified by M. Pound,

M. Regan, and, very likely, others. The style file was further modified to better handle figure and

table numbering, reduce the wasted space in references, and to include a number of other tweaks.

REFERENCES

Arabadjis, J. S., & Bregman, J. N. 1999, ApJ, 510, 806, submitted

Guillout, P., Haywood, M., Motch, C., & Robin, A. C. 1996, A&A, 316, 89

Kuntz, K. D., & Snowden, S. L. 1998, Cookbook for analysis procedures for rosat xrt observations of

extended objects and the diffuse background, part ii: Mosaics, Technical report, NASA/GSFC

Schmitt, J. H. M. M. 1997, A&A, 318, 215

$chmitt, J. H. M. M., Fleming, T. A., & Giampapa, M. S. 1995, ApJ, 450, 392

ii
DEDICATION

Tohewhomadeit possible,

Dr. Dr. Prof.L.M.A.F.P.

iii
ACKNOWLEDGEMENTS

Aswith anyworkof this length, there are many individuals and institutions who have contributed to its nascence.

My greatest thanks go to Steve Snowden, who made the mistake of inviting me to work with

him for the summer of 1994, and then was foolish enough to invite me back to work on a thesis. He

has shown patience approaching infinite with my (sometimes egregious) mistakes, with my pursuit

of the inconsequential, and with the all too frequent absences in which I worked to renovate either

the house at 2329 North Calvert Street, or the neighborhood in which it stands.

Richard Mushotzky fronted for this project within the Laboratory for High Energy Astrophysics.

I have not been the most flexible graduate student with which he has worked, I've been slow and

methodical when what was really wanted was results. And even then I was off by a factor of 4rrl06.

Stuart Vogel agreed to front for this project within the Department of Astronomy. More importantly, he found a source of funding for the first several months of this project.

John Wang, before his untimely death, had agreed to be a member of my disseration committee.

In the early stages he provided acute insights about the thesis process, and I had looked forward

to his guidance on the fluctuation analysis. I miss his warm and witty presence even more than I

miss his advice.

The remainder of my committee, Sylvain Veilleux, Eve Ostriker, and James J. Kelly displayed

remarkable restraint and patience. Dr. Kelly, in particular, took the responsibility to act as the

Dean's Representative at, literally, the last moment.

Keith Jahada practiced great forebearance in providing guidance in what I suspect he (rightly)

thought a fool's errand. Many times he has set aside his own work at length and at short notice

to restrain my galloping misconceptions.

Robin Shelton has provided many stimulating conversations in which we attempted to determine

the extent of knowledge about the halo and LHB, and how that knowledge had come to be. I am

also thankful for the several collaborative projects in which she included me long before I had

proved myself capable.

Rick Schafer helped me to understand some of the background for Chapter 10 during the early

parts of this project, and provided many interesting discussions about propagating uncertainties

through Fourier transforms.

Fran(tic) Verter introduced me to a number of new problems in the ISM, and encouraged me

to further study of the X-ray method of determining X.

Eli Dwek, and Tom Sodroski helped me understand the ISM as seen by IRAS, and provided

moral support when I was the organizer of the Building 2-Building 21 ISM bag-lunch series.

Peter Teuben often inquired after my progress and made a number of very useful suggestions

for finding a way to analyse the ACFs.

Richard Rand provided a sounding board for some of the discussion about the halo, and its

relation to those of other galaxies, in Chapter 6.

At the University of Leicester, Bob Warwick provided the thesis topic, who was perplexed by

excess variations in his 1⁄4 keV data, and who posed the question as to their nature. (I suppose I

should also thank Colin Barber for not pursuing this course of investigation.)

At the Evangelische Studienwerk, Villigst, Dr. Eberhard Mfiller provided great hospitality

during my visit to Well der Stadt. There I spent several pleasant afternoons drafting parts of

Chapter 8 while sitting under the gaze of Kepler.

My path through the sometimes labyrinthine University bureaucracy was smoothed by Thelma

Bublitz, Maryann Philips, and Ms. Zita. Though it is tradition for graduate students to complain

about University administration, the rapaciousness of the bursar's office, etc., I must acknowledge

that the University administration has done much to support me, transparently, without my request,

iv
such as deferring my student loans.

At Goddard, Maybelline Burrell and the GSRP office, who handled the funding of the first

several months of this project, and have been helpful ever since.

Finally, I must also thank that person who set into motion the whole curious chain of events

that led me not only to start a thesis, but to start one dealing with the ISM, a topic that I had

learned to loathe under different masters. Without Laura Danly, this thesis, 2329 Calvert Street,

and L.M.A.F.P. would not have happened, and I surely would not have become the person I am

today, let alone be as happy as I am.

Other Acknowledgements

This research was supported in part by the NASA GSRP program through the GSFC/LHEA.

This research was also supported in part by the NASA ADP program.

This research has made use of data obtained through the High Energy Astrophysics Science

Archive Research Center Online Service (HEASARC), provided by the NASA/Goddard Space

Flight Center.

This research has made use of data obtained through NED.

This research has made use of data obtained through SIMBAD.

This research has made use of the Schlegal, Finkbeiner, & Davis maps of the IRAS 100 ttm

emission obtained from Douglas Finkbeiner in advance of publication.

Table of Contents

Chapter 1

1

2

Introduction 1

Introduction ........................................ 1

Motivation ........................................ 2

2.1 Missing Baryons - Why To Look at the 0.1-1.2 keV Background ...... 2

2.2 Observations of 0.1-1.2 keV Fluctuations .................... 3

2.3 The Observational Difficulties Below 0.284 keV ................. 4

2.4 ...And Why They May Not Be So Horrible .................. 4

Outline .......................................... 5

Chapter 2

1 Data

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Data, its Selection and Reduction 7

............................................

ROSAT Pointed PSPC Observations ...................... 7

ROSAT All-Sky Survey ............................. 7

ROSAT Bright Source Catalogue ........................ 7

HI ......................................... 9

DIRBE-IRAS 100 pm Emission ......................... 9

Abell Cluster Catalogue ............................. 9

Pseudo-Data ................................... 9

1.7.1 Thermal Models of X-ray Emission ................. 9

1.7.2 Absorption Cross-sections ....................... 10

Data Selection ...................................... 10

2.1 High-b Mosaic Criteria .............................. 10

2.1.1 LHB Mosaic Criteria ......................... 10

2.2 Pointing Criteria ................................. 16

Data Reduction ...................................... 16

3.1 The Instrument .................................. 17

3.2 Detector Efficiency ................................ 18

3.3 The Non-Cosmic Backgrounds ......................... 18

3.3.1 The Particle Background ....................... 18

3.3.2 The After-Pulse Contamination ................... 21

3.3.3 The SXX, STE, and LTE Backgrounds ............... 21

3.3.3.1 Descriptions ......................... 21

3.3.3.2 Measurement ......................... 22

3.3.3.3 Spatial Distribution ..................... 23

3.4 Point Source Removal .............................. 23

3.5 Output Products ................................. 23

3.6 Execution ..................................... 23

Uncertainties ....................................... 24

4.1 Detector Map & Effective Exposure Map ................... 24

4.2 Spatial and Temporal Gain Variation ..................... 24

vii
4.3 ParticleBackgroundMap ............................ 25

Chapter 3

1

2

Mosaicking 29

Introduction ........................................ 29

The LTE Offset Rate ................................... 29

2.1 Direct Comparison to the RASS ........................ 30

2.1.1 Point Sources .............................. 30

2.1.2 Uncertainties .............................. 30

2.1.3 Use of the R1 vs. the R1L Band ................... 33

2.1.4 Weighting Change ........................... 34

2.2 Self-Consistent Solution ............................. 34

2.3 Synthesis ..................................... 36

2.4 Calculating Offsets in Overlap Regions ..................... 37

2.4.1 Background Gradients ......................... 39

2.4.2 The Exclusion Radius ......................... 40

2.5 Point Source Detection ............................. 41

2.5.1 Recalcitrant Point Sources ...................... 42

2.5.2 Point Source Detection Limits .................... 44

2.6 An HRI Note ................................... 44

2.7 "What have I done wrong?" . .......................... 44

Chapter 4

1

2

Tracers of Absorption 47

Statement of the Problem ................................ 47

Dust Emission and ISM ................................. 48

2.1 Review of SFD .................................. 48

2.2 H I ......................................... 50

2.3 H II ........................................ 55

2.4 H_ ......................................... 56

Implications........................................ 57

Chapter 5

1

2

The Spectrum of the SXRB 61

Introduction ........................................ 61

Analysis of Previous Results ............................... 61

2.1 Spectral Fitting .................................. 63

2.2 All-Sky Deabsorption .............................. 66

2.2.1 1⁄4 keV (The R12 Band) ........................ 66

2.2.2 43-keV .................................. 67

2.2.3 Synthesis ................................ 68

2.3 Summary ..................................... 68

Chapter 6

1

2

The Spectrum of the TAE 71

Introduction ........................................ 71

Data ............................................ 7l

2.1 The RASS ..................................... 71

2.2 The Absorbing Column ............................. 72

Limits ........................................... 73

Vl|l
4 External Constraints ................................... 73

4.1 Properties of the EPL .............................. 73

3 keV Emission from the LHB 75 4.2 _ .........................

5 The Iterative Algorithm ................................. 75

5.1 Details of Local/Distant Separation ...................... 75

5.2 Details of the R4-R7 SED Fitting ........................ 76

6 Analysis of Current Results ............................... 77

6.1 The TAE ..................................... 77

6.1.1 The Soft Component .......................... 77

6.1.2 The Hard Component ......................... 80

6.1.3 Comparisons .............................. 80

6.1.4 The Diffuse Background Budget ................... 82

6.1.5 Spectral Fitting Redux ........................ 82

6.2 The LHB ..................................... 86

7 Discussion ......................................... 87

Chapter 7 Stellar Contribution to the SXRB 93

1 Introduction ........................................ 93

1.1 Motivation .............. - ...................... 93

1.2 Previous Work .................................. 94

2 Data ............................................ 94

2.1 Model Inputs ................................... 96

2.2 Method ...................................... 98

3 Comparisons ....................................... 99

3.1 The Guillout et al. Model ............................ 99

3.2 Point Source Counts ............................... 100

4 Model Description .................................... 102

5 Results ........................................... 102

6 Future Improvements ................................... 104

Chapter 8 Deabsorption 107

1 Introduction ........................................ 107

1.1 Method ...................................... 107

1.2 Errors ....................................... 108

2 lo ............................................. 108

2.1 Point Source Correction ............................. 108

2.2 Ntt Binning .................................... 110

3 _re1! ............................................ 110

3.1 The Uncertainty due to o'e1! .......................... 111

4 NH ............................................. 111

4.1 Some Considerations ............................... 111

4.2 Overview of Method ............................... 113

4.3 Details ....................................... 113

4.3.1 Scaling ................................. 115

4.3.2 The Iloo-NH Relation ......................... 116

ix
6

4.3.3 The fo-NH Relation .......................... ll9

4.3.4 Another Structure/Scaling Method ................. 119

4.4 Uncertainties in NH ............................... 121

4.4.1 NH Scaling Errors ........................... 121

4.4.1.1 NH Baseline - An Offset Error ............... 121

4.4.1.2 Error in the I10o - NH Relation - A Scaling Error .... 122

4.4.2 NH Structure Errors .......................... 122

4.4.2.1 Systematic Variations in H l/I100 ............. 124

4.4.2.2 H II .............................. 124

4.4.3 NH Errors ............................... 125

Deabsorption ....................................... 125

5.1 Uncertainties ................................... 127

Summary ......................................... 128

6.1 Prescription .................................... 128

6.2 Uncertainties ................................... 129

Chapter 9 ID Revealed 131

1 Boring Stuff ........................................ 131

2 Not-So-Boring Stuff ................................... 132

2.1 Quality of the Deabsorption. ........................... 132

2.2 Band-Band Cross-Correlations ......................... 144

Chapter i0 Fluctuation Analysis 147

1 Introduction........................................ 147

1.1 The Angular CorrelationFunction ....................... 148

1.2 A Problem .................................... 148

2 The AutocorrelationFunction - W(O) ......................... 149

2.1 EquivalentFormulations ............................. 149

2.2 Algorithm ..................................... 149

2.3 Uncertainty .................................... 150

2.4 Data Preparation ................................. 150

2.5 Application .................................... 151

2.5.1 Averaging the ACFs .......................... 152

2.5.2 The PSF ................................ 152

3 Jl/1- A Statistical Method ............................... 155

3.1 Relation to w(O) ................................. 155

3.2 Algorithm ..................................... 155

3.2.1 Counting Statistics .......................... 155

3.2.2 Point Source Contributions ...................... 156

3.2.3 The relative contribution ....................... 157

3.2.4 The distribution of values within a single pixel ........... 157

3.2.5 The variance of a collection of pixels ................. 157

3.3 Uncertainty .................................... 157

3.4 Data Preparation ................................. 158

3.5 Application .................................... 158

X
3.5.1 The Effect of Offsets .......................... 158

3.5.2 Interpretive Difficulties ........................ 160

3.5.3 Averaging ................................ 160

3.5.4 _I/I Summary ............................. 161

4 Results ........................................... 161

4.1 Some Considerations ............................... 161

4.1.1 The LHB ................................ 161

4.1.2 The Halo ................................ 161

4.2 Comparison with Previous Results ....................... 163

5 Summary ......................................... 164

Chapter 11 Summary 167

1 Cosmological Considerations ............................... 167

1.1 The Status of the Problem ........................... 167

1.2 WHIMsical Detections .............................. 168

1.2.1 Current Claims ............................. 168

1.2.2 Future Claims ............................. 168

2 The Current Work .................................... 168

2.1 Total Flux ..................................... 168

2.1.1 The EPL . ............................... 171

2.2 Spectrum ..................................... 171

2.3 Angular Distribution ............................... 171

3 Improvements ....................................... 172

3.1 Building Upon The Current Data ........................ 172

3.2 Without Mosaics? ................................ 172

3.3 Deconstructing the Background ......................... 173

4 Some Unfinished Business ................................ 173

5 ArrivederLa ........................................ 174

Appendix A Model Temperatures 175

Appendix B Observations 177

Appendix C

C.1

C.2

C.3

Reduction Scripts 185

UNIX Scripts for Data Reduction ............................ 185

Prescription for Use ................................... 185

Scripts ........................................... 186

C.3.1 initial ....................................... 186

C.3.2 rate.fit.4 ...................................... 186

C.3.3 middle ....................................... 186

C.3.4 final ........................................ 188

C.3.5 cleanup ...................................... 188

C.3.6 clearout ...................................... 188

Appendix D Mosaicking 189

D.1 Introduction ........................................ 189

xi
D.2

D.3

D.4

D.5

D.6

Appendix E

Appendix F

Appendix G

Appendix H

Appendix I

Glossary

References

Accessing and Compiling ................................ 189

D.2.1 Retrieving Files .................................. 189

D.2.2 Compiling ..................................... 189

Prescription ........................................ 189

D.3.1 Further Refinements ............................... 193

A Final Word ....................................... 194

Program Descriptiogs .................................. 194

UNIX Scripts ....................................... 224

The TAE in the Southern Galactic Hemisphere

A Note on the Lx to Count Rate Conversions

The Uncertainty in W(O)

Fluctuation Figures

Calculating the Suppression Factor

227

229

233

237

247

249

253

ySi
2.1

2.2

2.3

2.4

2..5

2.6

2.7

3.1

3.2

3.3

4.1

4.2

5.1

6.1

6.2

6.3

7.1

7.2

7.3

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

9.1

9.2

9.3

9.4

List of Tables

Data Summary .................................... 8

PSPC PSF-Field Radius Relation .......................... 8

Fields ......................................... 11

Broad Band Definitions ............................... 19

Detector Map Counts ................................ 25

Effect of Gain Reduction on Broad Bands ..................... 26

Particle Background Distribution .......................... 26

RASS Point Source Exclusion Limits ........................ 33

R1L/R1 Ratios for the LHB ............................. 35

LTE Offset (R2) ................................... 38

Fits to Nx = A + B(X)lloo ............................. 59

Uncertainties in Equivalent Absorbing Columns .................. 59

Compendium . ..... 62

Diffuse Background Budget ............................. 83

Pointed Observations Used ............................. 83

Spectral Fits ..................................... 83

Stellar Data ...................................... 97

Mid-Plane Stellar Densities ............................. 98

Predictions ...................................... 104

Effect of Spectral Shape ............................... 112

Fitting with Scaled NH ............................... 117

ll00 to H IScaling ................................... 119

I100 Fitting to the X-ray Mosaic .......................... 120

Point-by-Point Scaling/Fitting ........................... 121

Effect of AN_ .................................... 123

Effect of Ab = 0.09 .................................. 123

Effect of H I1 ...................................... 126

Error Summary .................................... 126

Image Display Parameters .............................. 133

Deabsorption Parameters I ............................. 133

Deabsorption Parameters II ............................. 143

Pearson r0 Values ................................... 143

10.1 PSPC PSF-Field Radius Relation .......................... 150

10.2 Maximum Number of Pixel Pairs .......................... 153

xiii
10.3

10.4

B.1

B.2

B.3

B.4

B.5

B.6

B.7

B.8

B.9

C.1

C.2

D.1

D.2

D.3

D.4

D.5

D.6

F.1

Summary . ...................................... 156

Comparison of ACFs ................................. 165

035+78 Observations

086+74 Observations

138+52 Observations

161+51 Observations

171+50 Observations

220+84 Observations

295-89 Observations

045+00 Observations

178-22 Observations

................................. 178

................................. 178

................................. 179

................................. 180

................................. 181

................................. 182

................................. 182

................................. 183

................................. 184

Reduction Shell Parameters ............................. 187

Log Files and Their Origins ............................. 187

Suggested DETECT/MASK parameters ...................... 193

Mosaicking Programs ................................. 195

Files Produced by Mosaicking Programs ...................... 196

Columns in DETECT, MERGE_DETECT, and GET_RASS_BSC Output . . . 210

Columns in OFFSET(_XX) Output ......................... 215

Mosaicking Shell Parameters ............................ 225

Fits to the HR- Lx Relation ............................ 230

xiv
2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

3.2

3.1

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

3.14

3.15

3.16

4.1

4.2

4.3

4.4

4.5

4.6

4.7

List of Figures

Maps of the PSPC pointings at high Galactic latitude ............... 11

The 1⁄4 keV emission for each mosaic region ...................... 12

The IRAS 100 pm emission for each mosaic region ................. 13

The H ! spectrum in the direction of each mosaic .................. 14

The distribution of exposure and IRAS 100/am emission within each mosaic... 15

Map of PSPC pointings in the Galactic disk ..................... 16

Map of PSPC pointings in the Galactic disk ..................... 17

The broad energy bands ................................ 19

The distribution of Uncertainty/Rate for a single panel of the RASS in the direction of (2, b) -- (180, 0) ............................... 30

A mosaic of R2 band pointings where the LTE offset has been determined by

three different methods ................................. 31

A demonstration of the hazards of comparing PSPC pointings to the RASS. . . 32

The distribution of the RASS pixel values for each individual pointing (band R12). 33

The behavior of the R1L/R1 ratio as a function of halo emission fraction and

absorbing column density ............................... 34

A graphic portrayal of the determination of the LTE offset ............. 35

A demonstration of the change in LTE offset solution as a function of the relative

weights of the self-consistent and direct-comparison equations ........... 37

A demonstration of point source problems in the region of overlap ........ 39

A demonstration of the effect of a gradient in the cosmic background on the count

rates measured in overlap regions ........................... 40

A measure of the effect of a gradient on the count rates measured in overlap regions. 40

A graphical representation of the uncertainty in the count rate for the overlap

region as a function of field separation and exclusion radius ............ 41

Point source detection limits .............................. 42

Comparison of "rejected" sources to better exposures of the same regions .... 43

An unmasked hot-spot ................................. 43

An example of a "raw" mosaic where the count rates are continuous over exposure

boundaries, but scaling of "sparse" images reveals the exposure edges ...... 45

One dimensional demonstration of the effect of scaling on sparse images ..... 46

The relative elemental contributions to the absorption cross-section ........ 47

A schematic depiction of the source of the structure in the Iloo-Ngl relation. 49

The ratio of X-ray detected gas to atomic gas plotted as a function of the atomic

gas ............................................ 50

The angular variation of the Iloo/NHI ratio for the northern Galactic hemisphere. 51

The angular variation of the Iloo/NHI ratio for the northern Galactic hemisphere. 52

The angular variation of the Iloo/NHI ratio for the southern Galactic hemisphere. 53

The anugular variation of the Iloo/NH1 ratio for the southern Galactic hemisphere. 54

XV
5.1

5.2

5.3

5.4

5.5

6.1

6.2

6.3

6.5

6.4

6.6

6.7

6.8

6.9

6.10

6.11

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.9

7.8

The amount of gas detected by X-ray absorption versus the SFD I100 measure of

the dust column ..................................... 57

The amount of gas detected by X-ray absorption versus the SFD XT I100 measure

of the dust column ................................... 58

A comparison of the spectra of thermal components with the available independent spectral elements for the ROS,4T PSPC .................... 63

The degeneracy in the thermal component fit parameters .............. 64

The R2/R1 ratio as a function of temperature .................... 65

Checking spectral fit parameters using absorption .................. 65

A demonstration of the bias due to the method of Snowden et al. (1997) ..... 68

The distribution of the statistical significance of individual 12_ × 12t pixels in

individual and combined bands of the RASS ..................... 72

The column densities at various energies necessary to produce a given optical

depth .......................................... 74

Flowchart for the bicyclic iterative fitting of two thermal components to the SED

of the TAE ....................................... 76

The distribution of the R2/R1 band ratios for the local and distant components. 77

The spatial distribution of measured quantities ................... 79

The derived R2/R1 ratio as a function of NH if NH were underestimated .... 79

The distribution of the temperatures produced by fitting Raymond & Smith models to the R4-R7 SED after the emission due to the LHB and the soft component

of the TAE have been removed ............................ 80

Correlation of the R12 band emission of the TAE and the R45 band emission of

the TAE ......................................... 80

Fitting one- and two-component TAE models to the R1-R7 band SED ...... 81

Simultaneous fits of three PSPC spectra from relatively long pointings ...... 84

The 1⁄4 keV hardness ratio as a function of Galactic longitude for the region within

5° of the Galactic plane ................................ 87

The luminosity function, N(S)dlogS, for band R17 shown with the fractional

contributions of the different age classes and spectral types ............ 95

The relative surface number densities for each age group as a function of z-height,

compared to the absorbing column through which they are observed ....... 99

The effect of uncertainties of the input parameters on the unresolved flux and

the cumulative luminosity function shown as the ratio between the luminosity

functions ......................................... 99

The model calculations for the north Galactic pole subdivided by age and spectral

class ........................................... I00

Comparison between the current model and that of Guillout et al. (1996) .... 101

The band R47 predicted cumulative luminosity function with measured values.. 101

The luminosity function for different bands in the direction of the north Galactic

pole ........................................... 102

The luminosity function for main-sequence stars, white dwarfs, and extragalactic

objects, for band R12 and band R47 ......................... 102

The ratio of the luminosity function at b = 30 ° to the luminosity function at

b = 90 ° ......................................... 103

xvi
7.10

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

8.10

8.11

8.12

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

9.10

9.11

9.12

10.1

10.2

10.3

10.4

10.6

10.5

10.7

10.8

10.9

10.10

ThecontributionofunresolvedstellarsourcestotheRASS as a function of Galactic latitude ....................................... 103

Typical correction for variable point source detection limits ............ 109

Comparison of effective cross-sections for a reasonable emission model and some

single equilibrium temperature models ........................ 111

Images of the 138+52 soft X-ray mosaic and tracers of the absorbing column. 114

Comparison of the I_o-NH relations for mosaic region 138+52 ........... 115

Example IRAS 100 tam emission to Dwingeloo H I21 cm emission scaling ..... 117

Example IRAS 100 tam emission to Lockman H I21 cm emission scaling ..... 118

Example fits of I100 to the X-ray absorption, and the residuals to the fits ..... 120

Comparison of the Dwingeloo and Lockman data sets ................ 121

The fractional amplitude of spurious fluctuations produced by uncertainties in

the absorbing column .................................. 124

The fractional amplitude of spurious fluctuations produced by fluctuations in the

column density of H II ................................. 125

Demonstration of the effects of the multiplicative and additive methods of deabsorption ......................................... 127

Comparison of the systematic uncertainties for the multiplicative and additive

methods of deabsorption ................................ 128

ll00-H I relations for the mosaics ........................... 132

Distribution of point source detection limits ..................... 134

I_ - NH relations for the mosaics ........................... 135

The R1L2 band images after deabsorption ...................... 136

The R45 band images after deabsorption ....................... 137

The R1L2 band images after deabsorption smoothed by a 121 t boxcar ...... 138

The R45 band images after deabsorption smoothed by a 121' boxcar ....... 139

The R1L2 band images after deabsorption high-pass filtered with a 121' threshold. 140

The R45 band images after deabsorption high-pass filtered with a 121' threshold. 141

The relative areas of mosaics with different maximum PSF sizes .......... 142

Pearson r, values for the cross-correlation of the deabsorbed images with the

maps of the absorbing column ............................. 145

Pearson re values for the cross-correlation of the R1L2 deabsorbed images with

the R45 deabsorbed images .............................. 145

Comparison of PSF profiles .............................. 149

The ACF of a typical mosaic ............................. 152

Mean ACF for high-pass-filtered mosaics ....................... 154

The unremoved point sources that dominate the ACF ............... 154

The _I/I measure of a typical mosaic ........................ 159

Comparison of the erB and fib measures ........................ 159

The effect of offset errors on aB(¢) and o'b(¢) .................... 160

Mean (O'b(¢)/I) 2 for High-Pass filtered mosaics ................... 161

Band R1L2 ACF for the LHB ............................. 162

Comparison of ACFs .................................. 163

xvii
11.1 GraphicaldecompositionofthesoftX-raybackground: removingtheunresolved

Galacticstars...................................... 169

11.2 Graphicaldecompositionf thesoftX-raybackground: removingtheunresolved

extragalacticsources.................................. 169

11.3 Graphicaldecompositionf thesoftX-raybackground: removingtheLHBcomponent.......................................... 170

11.4 Graphicaldecompositionofthe soft X-ray background: fitting the TAE components ........................................... 170

A.1

C.1

C.2

D.1

D.2

D.3

D.4

D.5

D.6

D.7

E.1

F.1

F.2

F.3

F.4

G.1

H.1

H.2

H.3

H.4

H.5

H.6

H.7

H.8

1.1

Comparison of the R2/R1 band ratios as a function of temperature for equilibrium

and non-equilibrium cooling ..............................

The header text .....................................

Output from fit_ap ...................................

A sample directory tree for a mosaic of the Rll and R2 bands made from two

exposures ........................................

A comparison of measured count ratesand the calculatedpoint sourcedetection

limitforBETTER_LIMIT ...............................

Detected pointsourcesin a mosaic compared to pointsourcesin the BrightSource

Catalogue ........................................

Detected point sourcesin a mosaic..........................

Comparison of "rejected" sources to better exposures of the same regions ....

A comparison of measured count rates and the calculated point source detection

limit for PS_LIMIT ...................................

Two of the images output by PS_LIMIT .......................

The distribution of the R2/R1 band ratios for the local and distant components.

Fit of HR vs. log Lx for each spectral type .....................

Fit of HR vs. log Lx for all spectral classes together using the iterative fitting

technique described in the text. Data from URBs are included ..........

Fit of HR vs. log Lx for all spectral classes together using the iterative fitting

technique described in the text. Data from URBs are not included ........

Comparison of the "Universal Slope" fit to the data for each spectral type ....

Comparison of the uncertainty in W(O) determined by the variance and shuffle

methods .........................................

ACFs for

ACFs for

ACFs for

ACFs for

Measures

Measures

Measures

Measures

175

186

186

191

198

2OO

208

209

220

222

228

229

231

231

231

235

band R1L2 .................................. 238

band R45 .................................. 239

high-pass filtered images of band R1L2 .................. 240

high-pass filtered images of band R45 .................. 241

of ab(¢}/l for band R1L2 .......................... 242

of Crb(_b)/I for band R45 .......................... 243

ofab(¢)/l for band R1L2 .......................... 244

of ab(¢)/I for band R45 .......................... 245

Cartoon demonstrating values used in calculating the suppression factor ..... 247

xviii
Chapter 1: Introduction

Parturient montes, nascetur ridiculus mus, attamen labor quam perutilis.

-not Horace, Ars Poetica, line 168

Abstract

This chapter introduces the astrophysical problem that the dissertation was intended to address: to place limits on the existence of the "missing" baryons in the

local universe in the form of warm-hot filaments of gas in the intergalactic medium.

We argue that the soft X-ray band (E < 1 keV) is the most likely energy band in which

to detect these filaments, and that the observational difficulties in this band, though

grave, are surmountable.

1. Introduction

Observations have shown the baryon density at high redshift to be consistent with that

predicted by big-bang nucleosynthesis models

(Burles & Tytler 1998), but at low redshifts,

the best accounting of stars, stellar remnants,

and hot gas (iogT > 7.0) reveals an embarrassing deficit of baryons (Fukagita, Hogan, &

Peebles 1998). The cosmological models of Cen

et al. (1995) suggest that, depending upon the

nature of the dark matter and the form of the

universe, as much as 30% to 50% of the lowredshift baryonic matter may be in the form

of warm-hot gas filaments in the intergalactic

medium (IGM) with 5.0 < logT < 7.0, and

should emit strongly in the ROSAT bandpass

< 2 keV. However, some 70-80% of the X-ray

background in the 0.5-2.0 keV band has been resolved into point sources (Hasinger et al. 1998).

If the missing baryons are in the form of hot

gas, that gas must not emit too strongly above

-_ 0.75 keV, and thus must have temperatures

,,- 106 K, and should show strong emission in

the 0.1-0.284 keV or 1⁄4 keV band.

Extragalactic studies are difficult at 0.1-0.5

keV band due both to the greater absorption

by the Galactic ISM than at higher energies and

to the addition of a number of Galactic emission

components whose emission is minimal at higher

energies. Nonetheless, the study of the spatial

fluctuations in the 0.1-0.284 keV and 0.44-1.21

keV bands, especially when coupled to a similar

study at higher energies, can yield valuable constraints on the distribution of the extragalactic

gas that may contain the missing baryons, the

distribution of the Galactic halo gas, and upon

the population of point sources emitting in the

0.1-1.0 keV bands.

We proposed to study these spatial fluctuations using large (-_ 10 square degree) mosaics

of overlapping ROSAT PSPC pointings, which

would provide the largest product of collecting

area, field of view, and cumulative exposure for

any mission of the foreseeable future. The mosaics were to be created in two different energy bands, the 0.1-0.284 keV band (also called

the 1⁄4 keV, R12, or R82 band), and the 0.44-

1.21 keV band (also called the 3 keV, or R45

band). (Due to the carbon edge, the ROSAT

PSPC has virtually no response in the 0.284-0.44

keV band.) Cen & Ostriker (1999) estimated

that some 7 keV cm -_ s -1 keV -1 of the soft

X-ray background at 0.7 keV was due to emission from large scale filamentary structure in the

intergalactic medium. Initial calculations made

the detection of such filaments hopeful. On the

scale of the largest point spread function in a

ROSAT PSPC pointing, 10', given a modest exposure, 10 ks, the relative uncertainty in SXRB

(just the root of the total number of counts)

is _, 31(-_ ll)xl0 -6 counts s -1 arcmin -_, in

the R82(R45) band mosaic which translates into

,-, 3(--, 1) keV cm -2 s-1 keV -1 at 0.7 keV.

Of course, there are difficulties with such a

calculation; any single filament will not have

so large a flux, (at the time of the proposal,

the emission of any single filament had not yet
beenpredictedin print)andtherewouldlikely

bemanyof thesestructuresof variousbrightnessesalongthelineof sight. Thus,although

onemightnotexpecto detectanysinglesuch

object,one would expect to be able to measure

the power spectrum the fluctuations in the background caused by such objects.

Initial calculations showed that the uncertainty in the autocorrelation function, the standard method for measuring the power spectrum

of fluctuations, would, once the information for

all of the mosaics were combined be about a factor of two lower than the best yet measured in

the 0.5-2.0 keV energy band (Chen et al. 1994).

Once the conversion is made of their ACF from

the broad 0.5-2.0 keV band to the narrower 0.44-

1.21 keV band that we use, we expected to do

about a factor of eight better; in reality, we have

done about a factor of seven better.

Studies on large scales are essential to the

detection of fluctuations for more than an increase in the detection sensitivity. If, as suggested by the cosmological models of Bryan &

Norman (1998), the large scale structure has a

low filling factor, then one needs a very large region of study to ensure a reasonable probability

of including at least one such structure. Such

a large-scale study has not been previously possible; only by searching the ROSAT archive for

large numbers of overlapping PSPC exposures,

and the application of our unique mosaicking

software has this study become possible. At the

time the project was begun, the actual filling

factor for any of the models was poorly characterized, and remains so.

In the following section we will flesh-out the

recent theoretical innovations that motivate the

study of fluctuations at 1⁄4 keV and _ keV and

describe the difficulties of such study. In Section 3 we will describe the necessary parts of

the research program and provide a chapter-bychapter description of the following dissertation.

2. Motivation

2.1. Missing Baryons - Why To Look at

the 0.1-1.2 keY Background

The baryonic density calculated from standard big-bang nucleosynthesis using measurements of the D/H ratio in Lyman limit systems

is _bh_o "- 0.0394 4-0.00286 (Buries & Tytler

1998). A number of investigators have matched

cosmological models to spectra of the Lyman alpha forest to find similar values of the baryonic

density at z --, 3 (Zhang et al. 1997; Hernqu_st

et al. 1996). Despite the large amount of matter shown to exist in the hot (log T --, 7.0) intergalactic medium of galaxy clusters, a complete census of matter in stars and gas in galaxies, groups, and clusters in the low-redshift universe accounts for only about 1/3 of the baryon

density predicted by nucleosynthesis (Fukagita,

Hogan, & Peebles 1998; Cen _: Ostriker 1999).

Where is the remaining 2/3? If it emits, it must

emit in some region of the spectrum either hidden from us, such as the extreme UV, where

Galactic absorption obscures all but the brightest sources, or in some spectral region that has

been poorly explored.

Hydrodynamical simulations of the universe,

such as those by Cen et al. (1995), have shown

that gravitational accumulations of matter lead

not only to the well known and observed hot

shocked gas in galaxy clusters (T = 10_'°-zs),

but also to substantial sheets and filaments of

cooler (T = 106"°-v'3) gas at low redshift. This

cooler gas, forming large-scale structures, should

dominate X-ray emission at energies below 1

keV. Ostriker & Cen (1996) have shown that a

wide range of models (CDM, HDM, and COBE

normalized models) require between 30% and

50% of the baryons in the current epoch to be in

this cooler gas with 105.0 < T < 10 v°. It should

be noted that the COBE limits on intergalactic

gas (Mather et al. 1994) apply only to gas with

T ,_ l0 s.

If the gas that forms the large-scale structure

is hotter than T ,,_ 106.5 K, then the gas should

have been detected in the ROSAT _ keV band.

However, it is common knowledge that there is

a distinct lack of diffuse emission at energies

above 0.5 keV; Hasinger et al. (1998) has resolved some 80% of the X-ray background at this

energy into point sources (mostly AGN). This

understanding requires qualification. Hasinger

et al. (1998} has resolved some 80% of the X-ray

background in the 0.5-2.0 keV band; the primary

uncertainty being the exact level of the SXRB.

The lower portion of this band, the 0.5-1.0 keV is

known to be contaminated by emission from the

Galaxy, but the absolute value of that contamination can not be determined at present. All one

can measure is the excess above an extrapola-
tionoftheextragalacticbackgroundwhosespectral shapeandemissionstrengthismeasuredat

higherenergies(E > 2 keV).Theremayeven

besomecontaminationithe1.0-2.0keVband

(seeChapter6). Of thatcontamination,there

is nocurrentlyavailabletechniquefor separatingtheGalacticcomponentfromanyputative

extragalacticcomponent.Thus,it issuggested

thatalthoughHasingeretal. (1998)havenearly

completelyresolvedtheSXRBintopointsources

in the 1.0-2.0keVband,thereis still "wiggle

room"for adiffuseextragalacticcomponent at

E < 1 keV.

Measurements of the autocorrelation function

in a combined 0.5-2.0 keV band at scales from

2_ to several degrees are consistent with unresolved point sources just under a point source

detection limit of 10-14 ergs cm -2 s -1 (Chen

et al. 1994), but, the measured ACFs in that

band are very strongly limited by noise (compared in particular with the ACF in the 1.0-2.0

band measured by Soltan & Hasinger 1994) and

the claims based upon the ACF are derived by

fitting a functional form to the measured ACFs.

If, however, instead of being at T = 106.6 K,

the gas is as cool as T = 106.0 K, then it will be

virtually undetectable in the ROSAT 0.44-1.21

keV band, but will dominate the ROSAT 0.1-

0.284 band. Due to the greater observational

difficulties in the 0.1-0.284 keV band than in

the 0.44-1.21 band, the 0.1-0.284 band has not

recieved the same attention given to higher energies. For example, the X-ray logN - logs

curve was determined only for energies greater

than 0.5 keV, and fluctuation analyses are rarely

performed at energies as low as 0.5 keV.

At the time this project was begun it appeared that the cosmo-hydrodynamical models

made very different predictions about the distribution of gas in large-scale structures, depending sensitively upon the unknown cosmological

parameters. Some models produced sparsely

distributed, small, high surface brightness clumps,

while others produced very densely distributed,

low surface brightness filaments. Although direct comparisons of the clumping due to different models of the universe have not yet been

published, one can get a good flavor of their

difference by comparing the X-ray simulations

in Bryan & Norman (1998) and Cen & Ostriker (1999). The spatial distribution of hot

gas, and thus the fluctuations it produces in the

soft X-ray background, could depend strongly

on cosmological parameters, lending greater impetus to this project. More recent work by

Pierre, Bryan, & Gastaud (2000) suggests that

the strength of the filaments does depend upon

cosmological parameters, though somewhat perhaps more weakly than previously thought, and

Dav_ et al. (2000) states categorically that cosmological parameters have a non-negligible effect on the warm-hot IGM, but that the differences have not yet been fully explored.

In summary, cosmological hydrodynamic models suggest that there may be a large reservoir

of baryons in warm-hot gas, in filaments in the

IGM, and observations suggests that the gas

does not reside in the hotter phases, but, if it

exists, should be most apparent in the 0.1-1.0

keV band.

2.2. Observations of 0.1-1.2 keY

Fluctuations

Soltan et al. (1996) measured the autocorrelation function (ACF) in the R6 band (E ,_

0.73-1.56 keV) as well as in the R5 band (E ,_

0.56-1.21 keV). The two were very different (see

their Figure 2a), the R6 ACF dropping to indetectability for scales greater than -_ 6° , while the

R5 ACF has a significant value at 6° and a slow

decline to zero only at ,-, 300 . The difference was

attributed to the higher Galactic contamination

in the lower energy band, and great consideration was given to using the ACF in the R5 band

to remove this contamination from the R6 band.

On large scales the P_6 ACF appears to match

well ACF's measured at higher energies (2 - 10

keV) and provides a good measurement down to

12t. However, the dismissal of the R5 ACF as

being unduly contaminated with Galactic emission may have been a bit hasty, it may, rather,

have been of sign of more interesting phenomena

at lower energies, such as large scale structures

of 10 e K gas.

The Leicester group, studying the UK Medium

Deep Survey has been attempting to model fluctuations from R12 band (E -,- 0.1-0_284 keV)

ROSATdata. Assuming that the emission of the

Galactic foreground is unabsorbed, and assuming that extragalactic and Galactic halo emission is absorbed by the total H l column in a

given direction, they noticed that after remov-
ing the emission due to the Local Hot Bubble (LHB) and deabsorbing the remaining extragalactic flux, that there remained greater

than anticipated residuals (Barber, Warwick, &

Snowden 1995; Warwick et al. 1997). These

fluctuations had sizes 15' - 30' and amplitudes

of---20%. These fluctuations are not related to

known clusters and have too large an angular

extent for their flux to be due to previously unknown clusters.

Thus, there is some evidence that there may

be fluctuations in the soft X-ray background at

energies for which the X-ray background has entirely resolved into point sources. However, at

these energies, interpretation of X-ray images

becomes much more difficult than at higher energies.

2.3. The Observational Difficulties

Below 0.284 keY ...

Galactic Absorption: Measurement of fluctuations at 1⁄4 keV is complicated by stronger

Galactic absorption than at higher energies; a

typical Galactic column density toward a highGalactic-latitude field is 10 _° cm -2. This column absorbs _65% of the Galactic halo/extragalactic flux at -_ keV, but only _5% at _ keV.

This absorption not only reduces the signal but,

due to the uncertainty in the column density,

can produce spurious structure.

The Multiplicity of Emission Components: In

the 0.1-0.284 keV band there are many more

emission components than there are at higher

energies. In the 2-10 keV energy range, the spectrum is dominated by the extragalactic powerlaw (EPL) which is due to the many types of

unresolved extragalactic point sources (AGN,

QSOs, etc.). When the extragalactic power-law

spectrum, measured at energies > 1.5 keV is extrapolated into the lower energy bands, there

is a clear excess that is well modeled by two

thermal components with T = 108"°-es (Chen,

Fabian, & Gendreau 1997; Miyaji et al. 1998)

that are loosely called "Galactic components".

These components are the Local Hot Bubble

(LHB, a region of T = 106.0 plasma surrounding the sun), and the Galactic halo, which would

include any diffuse extragalactic emission. In

Chapter 5 we will see that there are problems

with previous decompositions of the spectrum

of the SXRB. There are, in fact (see Chapter 6),

an even larger number of components, but the

increased number of components does not add

any significant complexity.

From shadowing experiments using the disks

of spiral galaxies, the extragalactic flux in this

energy band is thought to be "_30 keV cm -2 s-1

sr -1 keV -l (Barber & Warwick 1994; Snowden

Pietsch 1995; Barber, Roberts, & Warwick

1996; Cui et al. 1996), a value that matches that

expected from the extrapolation of the higher

energy power law, which is thought to be due to

the superposition of extragalactic point sources.

However, the value is derived from a few very

small areas on the sky, and although all of

the determinations agree fairly well, the limited sampling cannot adequately constrain the

flux in other directions, where it may be significantly higher. From the modeling of 0.1-

0.284 keV RASS data (ignoring regions where

the model is known to have difficulties) Snowden et al. (1998) found that somewhat more

than half of the received 0.1-0.284 keV emission

is due to the LHB. Of the remainder, one quarter

to perhaps all could be extragalactic. Tighter

limits on the diffuse extragalactic emission are

placed in Chapter 6 and Chapter 7.

2.4 .... And Why They May Not Be So

Horrible

Despite the Galactic signal being comparable to or greater than the extragalactic signal,

nature has been forgiving. Fluctuations due to

foreground emission (the sum of the LHB and

Galactic halo emission) are likely to be small,

and the fluctuations due to the Local Hot Bubble in the 0.1-0.284 keV band can be measured

in mosaics made in carefully chosen parts of the

Galactic plane where Galactic absorption blocks

the emission not due to the LHB. Preliminary

work has shown these fluctuations to be small.

In the 0.44-1.21 keV band, the LHB has a very

much smaller emission (Snowden, McCammon,

&Verter 1993; Kuntz, Snowden, & Verter 1997),

and thus does not produce a substantial 0.44-

1.21 keV foreground.

The Galactic halo is also likely to be smooth

on the scales of interest. The crossing time of a

clump capable of producing the Warwick 1⁄4 keV

fluctuations, if they are in the Galactic halo, is a
mere106years,suggestingthat fluctuationson

this scaleareunlikelyto persistunlessreplenishedbyanasyetunknownmechanism.Reasonableheatingmechanismsarelikelyto affect

onlythelargescales.Halosupernovae, which

havefluxescomparableto that of theGalactic

haloevenasoldremnants,wouldaffecthe5° -

10 ° angular scales (Shelton 1998). Nearby galactic fountains have not been conclusively detected

with either X-ray or UV absorption line surveys,

and therefore should not effect the high latitude

sky. Measuring fluctuations at 1⁄4 keV may place

interesting constraints on the existence of halo

supernovae. Thus, although one can argue that

the halo can be expected to be smooth, any measurement of structure in the 0.1-1.21 keV band

will be of interest for understanding the Galactic

halo.

3. Outline

To study the spatial fluctuations one must

find the appropriate data; regions where there

are many overlapping long exposures. The noncosmic backgrounds must be removed from the

individual exposures, and the exposures mosaicked together. The foreground due to the

LHB must be removed, and the remaining image

corrected for absorption by the Galactic ISM.

The fluctuations in the resultant image can then

be studied using autocorrelation or other methods.

Chapter 2 describes all of the different types

of data used, from the X-ray data themselves to

the H I and IRAS 100/Jm data used in correcting for the absorption by the Galactic ISM. Also

included is a discussion of the models which are

used to convert count-rates to physical quantities. (This discussion is continued in Appendix

A.) The second section of this chapter reviews

the data selection criteria and briefly describes

the selected data. Tables containing a complete

list of the selected X-ray data are given in Appendix B. The third section describes the reduction of the X-ray data; the formation of images,

the removal of non-cosmic backgrounds, and the

correction for instrument response. Some minor

reduction tools are presented in Appendix C

Chapter 3 describes the process by which individual pointings can be mosaicked together

to form images covering several square degrees.

The most significant problem is correcting for

long-term variation in the non-cosmic backgrounds.

This chapter describes the algorithm by which

the relative offsets between observations can be

determined. A step-by-step description of the

process and the software by which it is accomplished can be found in Appendix D

Correcting the X-ray images for absorption

due to the Galactic ISM, the process of "deabsorption", requires knowledge of the absorbing

column and the spectrum of the radiation being

absorbed. The spectrum of the absorbed radiation is not known ab initio, but must be determined after the absorbing column is found. The

process of deabsorption is discussed in Chapter

4 through Chapter 8.

Chapter 4 describes the current state of understanding of the X-ray absorption cross-sections and abundances of the various components

of the ISM. The absorption is produced by metals associated with the molecular, neutral, and

ionized components of the ISM. Maps of these

components do not exist at the resolution of the

X-ray data. This chapter explores the extent to

which, on multidegree scales, the total absorbing

column is traced by IRAS 100 pm emission.

Chapter 5 describes the difficulties with previous measures of the spectrum of the SXRB,

and the following chapter solves several of those

problems. Chapter 6 describes a large-scale separation of the diffuse soft X-ray background into

the foreground emission due to the Local Hot

Bubble (LHB) and the background emission due

to the Galactic halo and the putative diffuse

extragalactic emission, a "deabsorption" of the

northern Galactic hemisphere. The chapter contalns a determination of the spectrum (actually

the broad-band Spectral Energy Distribution or

SED) of the background emission, and provides

a budget for the SXPd3. A short consideration

of the southern Galactic hemisphere is provided

in Appendix E

Chapter 7 is essential to several different

chapters. This chapter calculates the contribution of unresolved Galactic stars to the diffuse

X-ray background. The model that is described

here is used to calculate the fluctuations due to

unresolved point sources in Chapter 10. Ancillary calculations used by this chapter are found

in Appendix F.

Chapter 8 considers the computational task
of correcting mosaics for absorption. This chap- this dissertation to more concrete results.

ter also considers the extent to which the IRAS

100 gm emission traces absorbtion on arcminute

scales, and the extent to which the as-yet-unmeasured REFERENCES

H II component will contribute to the uncertainty. This chapter displays, sometimes in excruciating detail, the mechanics of deabsorption,

and the uncertainties involved. The superfluity

of detail is meant to convince the reader that we

really have done the deabsorption as well as possible, and minimized the spurious fluctuations

that can be introduced by this process.

Chapter 9 contains pictures of the deabsorbed

data in a number (surfeit?) of different mannets. This chapter also includes a small amount

of analysis to demonstrate the degree to which

the deabsorption was successful.

Chapter 10 contains the fluctuation analysis.

We consider two measures of fluctuations, the

autocorrelation function and the 6I/I measure.

Because the data to which we are applying these

methods are very different from the usual types

of data to which they are applied (our data have

very strongly varying exposure, noise, and point

source detection limits) we devote considerable

discussion to data preparation. We also develop

a new version of the JI/I measure that is more

useful than those used in the past. We then

compare our ACFs to previous work. Some of

the more tedious derivations, as well as a large

amount of graphical output are relegated to Appendix G through Appendix I.

We find that although our ACF in the 0.44-

1.21 keV band is better than any previously

measured in the broader 0.44-2.04 keV band,

there appears to be structure in the ACF of

somewhat greater amplitude than the uncertainty which may be due to unremoved point

sources and the windowing function. The modeling required to remove these features (or at

least place limits upon their reality) has not yet

been done. We also show that contributions to

the 0.1-0.284 keV band ACF by the LHB are

very small, and thus that the contributions in

the 0.44-1.21 keV band will be even smaller.

Chapter 11 discusses the progress made in

this dissertation towards detecting the "missing"

baryons. We show that our ACFs could be improved by further modeling, but that it is not

yet known what order of magnitude signal is expected from the warm-hot intergalactic medium.

We suggest a number of projects that could tie

Barber, C., Warwick, R. S., & Snowden, S. L. 1995,

in Rbntgenstrahlung from the Universe, ed.

H. Zimmermann, J. Trtimper, & H. Yorke,

MPE Report 263, 319

Barber, C. R., Roberts, T. P., & Warwick, R. S.

1996, MNRAS, 282, 157

Barber, C. R., & Warwick, R. S. 1994, MNRAS,

267, 270

Bryan, G. L., & Norman, M. L. 1998, ApJ, 495, 80

Buries, S., & Tytler, D. 1998, ApJ, 499, 699

Cen, R., Kang, H., Ostriker, J. P., & Ryu, D. 1995,

ApJ, 451,436

Cen, R., & Ostriker, J. P. 1999, ApJ, 514, 1

Chen, L.-W., Fabian, A. C., & Gendreau, K. C.

1997, MNRAS, 285, 449

Chen, L.-W., Fabian, A. C., Warwick, R. S.,

Branduardi-Raymont, G., & Barber, C. R.

1994, MNRAS, 266, 846

Cui, W., Sanders, W. T., McCammon, D., Snowden,

S. L., & Womble, D. S. 1996, ApJ, 468, 117

Dav_, R., et al. 2000, ApJ, submitted

Fukagita, M., Hogan, C. J., & Peebles, P. J. E. 1998,

ApJ, 503, 518

Hasinger, G., Burg, R., Giacconi, R., Schmidt, M.,

Triimper, J., & Zamorani, G. 1998, A&A,

329, 482

Hernquist, L., Katz, N., Weinberg, D. H., &

Miralda-Escud_, J. 1996, ApJ, 457, LS1

Kuntz, K. D., Snowden, S. L., & Verter, F. 1997,

ApJ, 484, 245

Mather, J. C., et al. 1994, ApJ, 420, 439

Miyaji, T., Ishisaki, Y., Ogasaka, Y., Ueda, Y.,

Freyberg, M. J., Hasinger, G., & Tanaka, Y.

1998, A&A, 334, L13

Ostriker, J. P., & Cen, R. 1996, ApJ, 464, 27

Pierre, M., Bryan, G., & Gastaud, R. 2000, A&A,

submitted

Shelton, R. L. 1998, ApJ, 504, 785

Snowden, $. L., Egger, R., Finkbeiner, D., F-Yeyberg,

M. J., & Plucinsky, P. P. 1998, ApJ, 493, 715

Snowden, $. L., McCammon, D., & Verter, F. 1993,

ApJ, 409, L21

Snowden, S. L., & Pietsch, W. 1995, ApJ, 452,627

Soltan, A., & Hasinger, G. 1994, A&A, 288, 77

Soltan, A. M., Hasinger, G., Egger, R., Snowden, S.,

& "l¥iimper, .]. 1996, A&A, 305, 17

Warwick, R., Hutchinson, I., Willingale, R., Kuntz,

K., & Snowden, S. 1997, in The Local Bubble, ed. D. Breitschwerdt & M. Freyberg, 321

Zhang, Y., Anninos, P., Norman, M. L., & Meiksin,

A. 1997, ApJ, 485, 496
Chapter 2: Data, its Selection and Reduction

Abstract

This chapter presents an overview of the many types of data and pseudo-data used

in this analysis. The second second section describes the criteria by which PSPC mosaics were chosen for analysis, and the criteria by which pointings within those mosaics

were accepted or rejected. The third section describes the reduction of individual PSPC

pointings. The mosaicking of PSPC pointings is described in the following chapter.

1. Data 1.2. ROSAT All-Sky Survey

This section provides a description of the

principal data set used, the PSPC observations,

as well as the subsidiary data sets necessary

for the analysis. We also describe the various "pseudo-data" sets, such as the Raymond

Smith plasma models, that are used in the

analysis.

1.1. ROSAT Pointed PSPC

Observations

The PSPC pointed observations cover regions

with a radius of ,-_ 1°. The angular resolution

for a given pointing varies from _ 15" on the

optical axis to ,-_ 10' at a radius of _-- 1°. At

the edge of the pointing where the resolution

is degrading rapidly, the field is also strongly

vignetted, so each pointing is restricted to the

central 53' radius. The PSPC response covers

the energy range 0.1 keV < E < 2.04 keV. The

energy resolution is given by

AE/E _ 0.43_/E. (2.1)

Due to the relatively poor energy resolution,

the energy range is divided into seven standard

broad bands, R1 through R7.

The data were retrieved from the High Energy Astrophysics Archive at the Goddard Space

Flight Center. The bulk of the data retrieved

used the "Revision 2" reprocessing. On average, 22% of the exposure time for a given pointing was discarded for having either overly large

or unquantifiable backgrounds.

The data reduction of PSPC pointed observations is discussed in §3, while the formation of

mosaics is discussed in Chapter 3.

The ROSAT AII-Sky Survey (the RASS), was

taken with the prime PSPC camera over --_ 200

days (Snowden et al. 1995; Snowden et al.

1997). Although the angular resolution of the

survey was ,-_ 3', the effective exposure for any

position, -_ 600 seconds, does not provide a statistically meaningful number of photons on those

scales. The latest release of the RASS (Snowden

et al. 1997) binned the data to 12' pixels.

Because some sources of non-cosmic background contamination vary over long periods of

time (particularly the so-called Long Term Enhancements or LTEs, see §3.3.3 below) and the

RASS covers a much longer time-line than any

single PSPC pointing, the RASS background

contamination removal is on average superior to

that for any single PSPC pointing.

The point source removal limits for the RASS

were nearly uniform and set to the level obtainable for 90% of the survey region; 0.025 counts

s-1 in the combined R12 band, 0.020 counts s-1

in the combined R45 band, and 0.020 counts s- 1

in the combined R67 band.

1.3. ROSAT Bright Source Catalogue

The ROSAT Bright Source Catalogue (Voges et al. 1996) is a catalogue of the sources

detected in the RASS. Selecting objects from

the catalogue with count rates greater than the

RASS point source detection limits will produce

a (nearly-correct) list of objects removed from

the RASS. This is useful for comparing individual PSPC pointings to the RASS to remove

residual background contamination.
TABLE 2.1

DATA SUMMARY

Data Resolution FOV

(,) (,2)

RASS 12 All-Sky

PSPC 1-5 53' Radius

Dwingeloo H I 30 6 > -37 °

Lockman H I 15 -,_15° x 15 °

DIRAS 5 All-Sky

TABLE 2.2

PSPC PSF-FIELD RADIUS RELATION

PSF" Field Radius

Diameter R1L2 R45 R67

(m_min) (m_cmin)

2 12.7 15.7 14.7

3 20.7 21.7 20.7

4 26.7 27.2 25.7

5 32.1 31.6 30.6

6 36.6 36.6 35.1

7 41.6 41.1 39.6

8 45.6 45.1 43.6

9 49.6 49.1 47.6

10 53.6 53.1 51.6

• This is the diameter encircling 90%

of the energy.
1.4. HI

The bulk of the H I 21 cm data used is

from the Dwingeloo survey (Hartmann & Burton 1997) which has complete coverage for _ >

-37. The velocity sampling was 1.03 km s -1

The beam size was 36' and the beam spacing

was 30 _, somewhat less than fully sampled. Although the data were collected on a regular 30 _

grid, the data were interpolated to points spaced

(At, Ab) = (30', 30'). To form images in the coordinates of the X-ray mosaics requires a second

interpolation, a problem that could have been

avoided if the data had been released as a list of

spectra on the original pointing centers.

One key advantage of the Dwingeloo survey is the sidelobe decontamination (Hartmann

et al. 1996). The sidelobes were removed using a bootstrapped model of the Dwingeloo response and a self-referential algorithm first developed by Kalberla (1978). The uncertainty after the removal of the sidelobe contamination is

Ta "_ 0.084 or Nxl _ 1.6 x 1017

perature. They used the DIRBE 240 pm/100

pm ratio to derive the dust temperature. From

the dust temperature they then calculated a correction factor X, that, when applied to the 100

#m flux, produces the total dust column density. At high Galactic latitudes dust emission is

small, so the uncertainties in the derived dust

temperature are large, even when the DIRBE

data are smoothed.

The efficacy of using the 100 pm emission to

trace the total absorbing column and the value

of the X correction factor for dust temperature

are discussed in a general way in Chapter 4, and

are discussed for small fields in Chapter 8.

1.6. Abell Cluster Catalogue

The Abell cluster catalogue used in Chapter

9 to remove known clusters from the mosaics was

that of Abell, Corwin, & Olowin (1989) (readily

available through the HEASARC) with subsequent corrections.

1.5. DIRBF_,-IRAS 100 pm Emission

We have used the de-striped DIRBE-corrected

IRAS 100 pm all-sky survey as presented by

Schlegel, Finkbeiner, & Davis (1998) as a surrogate tracer of the total absorbing column density. The IRAS data have a high resolution,

_- 5r, but a poor absolute calibration, having

a variable gain and offset. The DIRBE data,

conversely, have poor resolution, 0°.7, but a precise absolute calibration. Schlegel, Finkbeiner,

& Davis (1998) used Fourier analysis to de-stripe

the IRAS 100 pm data, and used the DIRBE 100

pm data to correct the IRAS calibration.

The systematic uncertainty in the resultant

data product is a vexed question, depending as

it does on the removal of the zodiacal dust emission, etc.. However, the systematic uncertainty

will play a very small rSle in our analysis since

we will always be scaling the 100 pm emission to

H I emission. Where we have needed the pixelto-pixel uncertainty we have typically taken the

variance over a region 15' x 15', about 3 x 3 resolution elements.

The main objective of Schlegel, Finkbeiner, &

Davis (1998) was to derive a measure of the total

dust column from the 100 pm emission. The 100

pm dust emissivity depends upon the dust tem1.7. Pseudo-Data

Pseudo-data are those large collections of

model calculations whose results are often so

taken-for-granted that one forgets that they are

subject to uncertainty. The model calculations

used are not unique but are those commonly

used by X-ray astronomers, and were readily

available for use.

1.7.1. Thermal Models of X-ray

Emission

The connection between band ratios and the

physical state of the emitting plasma is made

through model spectra. Due to their ready availability, we have chosen to use the 1991 release of

the Raymond _ Smith (1977) equilibrium model

spectra. The applicability of equilibrium models

to astronomical diffuse X-ray emission is debatable. However, except in Appendix A, where the

problem is discussed more fully, the spectrum

models provide no more than a low-resolution

spectral shape for use in deabsorption. The

equilibrium model spectra provide a convenient

set of "basis-vectors" for describing the broadband emission.
1.7.2. AbsorptionCross-sections

Wehaveused the Balucinska-Church & McCammon (1992) cross-sections for all elements

save He, for which we have used the cross-section

from Yah, Sadeghpour, & Dalgarno (1998). The

Balucinska-Church & McCammon (1992) crosssections are, save for He, the same used in Morrison & McCammon (1983). The BalucinskaChurch & McCammon (1992) He cross-section is

very different from other determinations of the

He cross-section. Following Arabadjis & Bregman (1999), we have chosen to use a He crosssection that is more similar to others in the literature.

The elemental cross-sections are, of course,

not valid for the same elements when in molecular form. For this reason, among others, we

avoid regions with molecular gas for the bulk of

the analysis. The elemental cross-sections are

also not valid for ionized gas. The few crosssections quoted for ionized gas may be out of

date and should be reconsidered before the controversies mentioned in Chapter 4 can be resolved.

2. Data Selection

The data selection criteria set forth here are

those that were applied when the project was

started. Little change could be applied to the

criteria for the high-b mosaics. However, the

LHB mosaic criteria could be improved, and

there are several other potential mosaics that

may be analysed in the future.

2.1. High-b Mosaic Criteria

Figure 2.1 shows the positions of individual

PSPC pointings over high Galactic latitudes.

There are a number of regions containing large

aggregates of overlapping (or nearly overlapping) pointings which are potential mosaics. For

the study of background fluctuations we have

chosen those aggregates that meet the following

criteria. 1.) The region must contain a large

number of deep exposures and cover a significant area, _ 10 square degrees. 2.) The region

must not contain large angular extent clusters of

galaxies, (i.e., avoid the Coma and Virgo clusters). 3.) The region must not contain large angular extent X-ray features due to the Galaxy

(i.e., Loop I). 4.) The region must not contain strong high or intermediate velocity clouds.

Since the surrogate tracer, the IRAS 100 pm

emission, is known not to trace high and intermediate velocity clouds, we have excluded these

regions where we have no adequate tracer of the

total absorbing column. 5.) The region must not

contain any molecular gas, as the conversion to

total absorbing column density is uncertain in

such regions. This condition was considered to

be met if the H I column was less than 3 x 1020

cm -2, or, equivalently, if the IRAS 100 pm emission was generally _ 2.5 MJy sr -1 . Small areas

within a mosaic containing suspected molecular

gas could be removed later.

The regions meeting these criteria are marked

in Figure 2.1, are shown in Figure 2.2 with respect to the 1⁄4 keV background, and are listed in

Table 2.3. Figure 2.3 shows the regions with respect to the IRAS 100 gm emission, which can

be considered equivalent to the amount of Xray absorption, while Figure 2.5 shows the distribution of the strength of the IRAS 100 #m

emission. The distribution of the exposure times

within each mosaic is also shown in Figure 2.5.

Figure 2.4 shows the velocity distribution of the

H l within the fields. These figures demonstrate

that all of the criteria are reasonably well met

for the seven chosen mosaics.

2.1.1. LHB Mosaic Criteria

Mosaics for the study of the variationof the

Local Hot Bubble were chosen with the following

criteria.I.)The regionmust not contain bright

extended sources at any X-ray energy. Because

the R2 band has a small amount of response

at energiesgreaterthan the Carbon edge (0.284

keV), hard sources,even ifnot readily visible

to the eye in the R1 and R2 band maps, may

cause considerablecontamination. 2.) The region must contain a sufficientcolumn densityof

H I to block most of the 1⁄4 keV band emission

originatingfrom beyond the Local Hot Bubble.

Figure2.6shows the positionsof PSPC pointings in the Galacticplane with referenceto the

Galactic column density as measured by the

IRAS 100 #m emission. Of course, with this

measure of the column density,there isno way

ofdetermining whether the absorbing column is

immediately outside the LHB, or whether the

I0
30 0 330 150 180 210

1 keY Fig. 2.1.-- Maps of the PSPC pointings at high Galactic latitude. The grey-scale is the

X-ray background. Quadrangles mark the locations of mosaics. Longitudes are marked every 30 °,

latitudes every 15°. Left." The north Galactic pole. Right: The south Galactic pole.

TABLE 2.3

FIELDS

Moniker Field R.A. Dec. g b Area a <Exposure>

deg. deg. deg. deg. sq. deg. ks

NGPD 035+78 206.3343 26.9168 35.00 78.00 13.8(10.9) 15. 5

NGPE 086+74 205.2314 39.3763 86.00 74.00 10.7(8.84) 25. 1

LockA 138+52 171.8789 62.5172 138.50 52.00 29.2(25.1) 8.3

LockB 161+51 153.4315 52.5778 161.00 51.25 29.4(23.7) 7.9

NGPF 171+50 149.0283 46.8985 171.00 50.50 21.1(19.5) 7.4

NGPA 220+84 186.0158 26.1826 220.50 84.00 28.8(26.3) 9.2

SGPA 295-89 13.0158 -28.1186 295.00 -89.00 14.3(12.1) 13. 7

Hyades 178-22 66.0426 17.2034 178.50 -22.00

Plane 045+00 288.4274 10.7237 45.00 00.00 22.6(21.4) 3.2

aUnmasked area(masked area)

11
mlO

035+78

161+51

295-89

086+74

171+50

Fig. 2.2.-- Tile x keV emission for each mosaic region. The dark lines show the mosaic regions

after source exclusion. The scale bar is given at the bottom of the first panel.

12
035+78

mlO

161+51

295-89

086+74

171+50

Fig. 2.3.-- The IRAS 100 /_m emission for each mosaic region. The dark lines show the mosaic

regions after source exclusion. The scale bar is given at the bottom of the first panel.

13
3.0

2.5

_, 2.0

1.5

1.0

0.5

0.0

3.0

2.5

2.0

1.5

1.0

0.5

0.0

3.0

2.5

2.o

1.5

1.0

0.5

0.0

3.0

2.5

2.0

1.5

t-. 1.0

0.5

=

035+78

086+74

138+52 2

- 161+51

0.0 -

3.0 ='-

2.5 ---=-

_. 2.0-

_ 1.5

1.0-

0.5

0.0

3.0

2.5 L

_ e.o L

_ 1.5-

1.0

OA

0.0

3.0

2.5

2.0

1.5

E-. 1.0

0.5 k

0.0.

-200

171+50

220+84

k 295-89

k

-150 -100 -50 0 50 100

v (km/s)

Fig. 2.4.-- Tile H [ spectrum in the direction of each mosaic. The light line is the spectrum scaled

by a factor of 10.

14
o16

lo'_I .J, 086+74]

z.

0.4

oO:o

1.o _ 138+52

_ 0.8

_ 0.6

"g 0.4

0.2

0.0 .....

_ 0.8

z. o.s

_ 0.4

0.2

0.0, . .......

;':_,. 171+50

_ 0.6

_ 0.4

0.2

0.0

0" 84

0.6

_ 0.4

0.2

0.0 , -

_ 0.6

_ 0.,1

02

olo

0 10 20 30 40

Exposure (ks)

i 0.8

z 0.6

_. 0.4

0.2

0.0 , -

035+78

1.0

0.8

0.6

_ 6.4

0.2

0.0

1.0

_ 0.8

z_ 0.6

_ 0.4

0.2

O.Oi

1"0 I

_ 0.8

_ 0.6

_ 0.4

0.2

0,0 ,

1.0

_ 0.8

_ 0.6

"_ 0.4

r_

0.2

0.0

0.0 0.5

086+74

0.8

o.6

0.4

02

010. :

_ 0.8

z. o.e

_ 0.4

0.2

0.0 + - -

171+50

. i i

_ 220+84

_.._.. :95- 89

1.0 1.5 2.0 2.5 3.0

I_

Fig. 2.5.-- The distribution of exposure (left) and IRAS i00 ym emission (right) within each

mosaic.

15
o.o 0.5 1.o 1.5 2.o 2.5

Fig. 2.6.-- Map of PSPC pointings in the Galactic disk. The greyscale is the logarithm of the

4! keV optical depth calculated from the IRAS 100 Inn emission, g = 1800 is to the right, ( = 360 °

is to the left.

column density is distributed along the entire

lille of sight. However, there are only two large

potential mosaics; one in the direction of the

Hyades cluster (e,b) = (181 ° ,-22°), and one

towards a rather anonynaous region at ((', b) =

(45°,0°). The absorption in the direction of

the Hyades is predominately due to the TaurusAuriga molecular clouds that are just external

to the LHB. The absorption in the direction of

g = 45 ° is less understood, but the maps of Sfeir

et al. (1999) suggest a significant amount of absorption near the LHB boundary.

Although the Hyades mosaic was reduced and

analysed (Kuntz, Snowden, & Warwick 1998) it

was not. included in the dissertation; the large

point source population is particularly difficult

to mode[, so fluctuation measurements using

this field would be very uncertain.

2.2. Pointing Criteria

Not all pointings overlapping a mosaic of

other pointings were used. Typically, pointings

with less than 3 ks were removed, as were other

pointings for which the background removal was

uncertain. Short exposures (--. 3 ks) were sometimes included if they had significant (_. 75%)

overlap, in which case the region that did not

overlap any other pointing was masked out. Regions with less than 0.01 tile mean exposure,

typically the unoverlapped edges of lower exposures and regions near point sources that had

been removed, were masked out. A complete

list of the pointings used in each mosaic can be

found in Appendix A.

We were able to use a larger number of pointings than originally expected; the original mosaicking software required all of the pointings

to have a substantial overlap with at least one

other pointing. The version of the mosaicking

software developed by Kuntz & Snowden (1998)

allowed pointings with little (or no) overlap to

be included as well.

3. Data Reduction

This section provides an overview of ROSAT

PSPC data reduction. The purpose of data

reduction is to remove the non-cosmic backgrounds and to correct for the temporal and spatial variation in the detector efficiency. These

topics have been discussed elsewhere at length;

this chapter provides a summary of previous

work keyed to the appropriate sources. Since

we are primarily interested in the diffuse background, the detection and removal of point sources

are also included under the rubric of data reduction. The first sub-section discusses the instrument, §1.3.2 discusses the "fiat-field" correction, §1.3.3 describes the non-cosmic back16
0.0 0.s 1.0 _.5 2.0 2.5

Fig. 2.7.-- MapofPSPCpointingsin theGalacticdisk. Thegreyscaleis thelogarithmof the

t keVopticaldepthcalculatedfromtheIRAS 100 tim emission, g = 180 ° is to the left, g = 0° is 4

to the right.

grounds and their measurement, §1.3.4 describes

the point source removal algorithm, and §1.3.5

describes the output data products.

3.1. The Instrument

The ROSAT satellite carried a pair of Position Sensitive Proportional Counters (PSPC).

The first, PSPC C, was used for the ROSAT

All-Sky Survey and a few pointed observations

before 25 January 1991, when, due to loss of

attitude control, the spacecraft tumbled and

scanned the detector over the sun. The second,

PSPC B, was nearly identical to the first, and

was used from January 1991 to the end of the

ROSAT mission, when the detector's gas was

depleted. All of the PSPC pointed observations

used here were taken with PSPC B.

PSPC B was operated in two different gain

states. Before 11 October 1991 it was operated

at the same gain as PSPC C. After that date

the gain was reduced by _ 30% to eliminate a

"warm spot" (low-level leakage events) that appeared as the detector degraded.

For each detected photon, the instrument

produced a time, an (x, y) detector position, and

an 8-bit pulse height that is photon-energy dependent. The Standard Analysis Software System (SASS) (Gruber 1992; Voges et al. 1992)

corrected the pulse height for temporal and spatim variation in the gain to produce a pulseheight invariant (PI) channel that is independent of the gain state of the detector. The SASS

also coverts the detector coordinates to sky coordinates.

The full 0.1-2.4 keV energy range of the PSPC

is covered by the 256 PI channels. However,

the energy resolution is much lower than the PI

channel spacing,

AE 0.93

0.43_ (2.2) E

so that the AE/E at 0.1 keV is _ 1.3 and the

AE/E at 1.0 keV is -,_ 0.41. Given the low

energy resolution, and that the count rate for

the diffuse cosmic background in any given PI

channel is small, for studies of the diffuse background the PSPC energy range is broken into

six standard bands, R1 through R7, excluding

R31. The PI channels and approximate energy

range for each band are given in Table 2.4, and

the band response functions are shown in Figure 2.8. Although the energy coverage of the

bands is often redundant (i.e., the R1 and R2

bands cover much of the same energy range) the

1The R3 band exists, but it has a small response on either

side of the carbon absorption edge. It served primarily

as a gap to more distinctly separate the 1⁄4 keV and the

3

keV data.

17
bandshavebeenformedfromdifferentPI channelrangesandthusarestatistically independent.

Given the statistical independence and tile difference in energy centroids, even the R1 and R2

bands can be used to form a hardness ratio, or an

X-ray "color", that is capable of distinguishing,

for example, between different tltermal models.

(For an extensive set of band ratios as a function of emission model temperature and absorbing column density see Snowden et al. (1997).)

Ill tile low gain mode, the R1L band (sometimes also called the R8 band) nmst be substituted for the RI band. In tire high gain state,

channels 7 and below are set to zero by" the pulse

hight discriminator; ira the low gain state, channels 1O and below contain no usable data.

3.2. Detector Efficiency

Map Formation: (See Snowden et al. (1994),

§§3.1, 3.2) The X-ray equivalent of tile "flatfield" is the detector efficiency map. The detector efficiency maps for PSPC B data were made

differently for the R1 and R2 bands than the),

were for the R4-R7 bands. The R1 and R2 band

detector efficiency maps were created from the

--_ 11 days of scanning mode data taken with the

PSPC B to finish the ROSAT All-Sky Survey

(RASS). Point sources, bright extended objects,

and periods with high non-cosmic background

were removed from the data, and all the remaining photon detections were mapped ira detector

coordinates. From this image was subtracted

the modelled particle background and the aherpulse background 2. The remainder is the detector efficiency map.

Since the cosmic background in the hard

bands is approximately a factor of five lower

than in the soft bands, the statistics do not allow the employ of the same method for the R4-

R7 band maps. Instead, the detector efficiency

maps were created from the _ 180 days of scanning mode data accumulated with the PSPC C.

Tile PSPC C detector efficiency map was then

shifted and normalized to the data accumulated

with the PSPC B.

['se: (See Snowden et al. (1994), §§3.1, 3.3)

Over the course of an exposure, which may be

divided into seperate segments spread over days

if not weeks, the roll angle of the detector on

the sky will change. There is also a 400 second

period "wobble" intended to reduce obscuration

by the window support structure. To create an

effective exposure map in sky coordinates, one

divides an exposure into short observation intervals (typically ,_ 1 second long) and for each interval shifts, rotates and sums an appropriately

scaled copy of the detector efficiency map.

3.3. The Non-Cosmic Backgrounds

There are five different non-cosmic backgrounds

that must be removed; the particle background

(PB), the after-pulse contamination (AP), shortterm enhancements (STE), long-term enhancements (LTE), and scattered solar X-rays (SSX).

For each, one nmst know the spectrum, tile

spatial distribution, and the strength of the

background in order to effect the background

subtraction. The particle background and the

after-pulse contamination can be separated easily from the other backgrounds and are described individually Tile STE, LTE, and SSX

backgrounds all enter through the detector window; their detection and measurement is executed sinmltaueously, and their spatial distributions are all derived from the exposure map, so

they will discussed together. The spatial and

spectral distribution of the STE cannot be determined ab initio, so periods containing STEs

are merely removed.

For each of the backgrounds there is a map

of the spatial distribution of that background

over the detector. The method of generating

these maps is described below; except for the

SSX background, these maps are temporally invariant and were produced by the ROSAT calibration team. Each of the maps is normalized

so that the integral over the detector is unity,

representing the probability of a single event as

a function of location; the map need only be

nmltiplied by the number of events to create (ira

detector coordinates) a map of the background.

2No other non-cosmic backgrounds need be subtracted

as, with the exception of the after-pulse background, all

the others will be vignetted in the same manner as the

cosmic photons,

3.3.1. The Particle Background

The particle background is discussed extensively in Snowden et al. (1992) and Plucinsky

18
200 '

.-. 150 _--

o

-< 100

o

_ 50 -

0 ,

1.8

I i i

/

/

/

/

/ x

/

/

/

/

/

2.0 2.2 2.4

i i i i

A

R2 / \

R611 II AR_

.-II/ I I

R5.1"" I1"_'\ / It I

..-Z.. / ",\ / /

2.6 2.8 3.0 3.2

Log,, Energy (eV)

Fig. 2.8.-- The broad energy bands.

TABLE 2.4

BROAD BAND DEFINITIONS

Band PI Channel Energy a

(keV)

R1 8-19 0.11-0.284

RIL=R8 11-19 0.11-0.284

R2 20-41 0.14-0.284

R3 42-51 0.20-0.83

R4 52-69 0.44-1.01

R5 70-90 0.56-1.21

R6 91-131 0.73-1.56

R7 132-201 1.05-2.04

alO% of peak response.

19
et al. (1993) as well as Snowden et al. (1994),

§4.1. Some 99.83% of the particle background

events are rejected by the five-sided vetoing and

logic discrimination of the instrument, and so

are not included in the data. Still, given the

high incidence rate, the particle background remains significant, especially at higher energies

where the X-ray rate decreases as E -_, where

1.4 < a < 2.0. Typically the particle background provides 10% of the counts in the R7

band.

There are two different particle backgrounds

that must be measured. The internal particle

background is observed when the detector is in

(or out) of the focal plane and the filter wheel

is closed. The internal background is composed

primarily of energetic charged particles. The external particle background is most clearly observed when either the detector is out of the

focal plane and the filter wheel is open or the

detector is in the focal plane, the filter wheel

is open, and the instrument is pointed at the

unilluminated earth. In neither of these cases is

the external particle background observed independently from the internal particle background.

The external background is composed of particle

induced AI hic_ X-rays, and a component that is

thought to be electrons with energies of a few

keV.

The clearest evidence for the external background is obtained when the detector is out of

the focal plane but the filter wheel is open. The

detector itself is square, but is masked by the

window mounting plate to a circular opening.

When the filter wheel is closed, the rates in the

corners of the detector, under the mask, are similar to the values in the center of the detector.

When the filter wheel is opened, the count rate

in the corners does not increase, though the rate

in the unmasked region of the detector does.

The A1 Kc_ component is only detected where

the detector is unmasked. Further, the external background bears the faint imprint of the

shadow of the window support structure.

There is no concrete way of knowing whether

the external background increases or decreases

when the detector is moved into the focal plane,

but the brief observation of the unilluminated

earth (178 s, immediately before the detector

was pointed at the sun) suggests that the external background does not increase significantly.

Measurement: The Master Veto Rate (MV) is

the rate at which the instrument rejects particle

background events using five-sided vetoing and

logic selection. The mean MV rate is a function

of the geocentric position of the satellite; below

a threshold MV rate of about 220 counts s-1,

the MV rate as a function of geocentric position is relatively constant. MV rates over this

threshold (which occur less than about one percent of the time) are very erratic and typically

occur in proximity to the particle belts or the

South Atlantic Anomaly.

Both the internal and the external particle

backgrounds are well correlated with the MV

rate for MV rates below about 220 counts s -1.

Thus, the total particle background for any observation can be determined from the MV rate

so long as the MV rate is below the threshold.

Spatial distribution: The spatial distribution

of the external particle background was measured while the detector was out of the focal

plane and the filter wheel was closed. The only

spatial variation within the "accepted" detector

region 3 is a slight increase with distance from

the detector center. The external particle background map used for data reduction is a constant

with the slight radial dependence.

The spatial distribution of the internal particle background was measured while the detector was out of the focal plane and the shutter

was open. To within the uncertainties, the spatial variation was limited to the shadow of the

window support structure. The external particle background map used for data reduction is

a de-vignetted detector map, which imitates a

constant background shadowed by the window

support structure.

Spectrum: The spectrum of the internal particle background is

sp(m) = al(m + 0.5)-°_-+a3 (2.3)

where PI is the pulse-height invariant channel.

The external particle background has the same

form, with the addition of the AI Ka line. There

is no detectable MV dependence of a2, while the

MV dependence of ax and a3 depends upon the

epoch of the data.

3The accepted detector region is that area which is not

masked and excludes regions near the edge of the physical detector where the vetoing is inefficient due to the

variable electric field near the attachment points of the

high-voltage anodes.

2O
3.3.2. The After-Pulse Contamination

The after-pulse (AP) contamination is discussed by Plucinsky et al. (1993), §§1, 3.1, 3.3.2,

and 3.4. The after-pulse contamination was first

observed as an increase in the count rates of the

lowest PI channels. The events that produced

this increase occurred within 0.35 ms and within

a few arcminutes of a previous event. These

events are thought to have been due to negative

ion formation in the counter gas.

The AP contamination is reduced by accepting only those events which occur more than 0.35

ms after a previous event. Since many of the

events that produce the AP counts are particle

background events that are rejected by the vetoing (and thus not recorded as events), rejecting all events that occur less than 0.35 ms after

a previous event will not remove all of the AP

counts. The remaining AP contamination can

contribute ,,- 15% of R1 band counts, and thus

is a significant background. The AP contamination is time-variable and need not be correlated

with the MV rate

Spectrum: Although the spectrum of the AP

contamination is given by

-,2w,+0._1 [ P/+ 0.5] PAp(PI) = ele -3 (2.4)

e3

where PI is the pulse-height invariant channel,

el, e2, and e3 can rarely be determined from the

data. The AP spectrum is generally modeled in

a more empirical manner.

Measurement: The AP contamination is determined in the following manner. A PI channel spectrum is formed from all events occurring greater than 0.35 ms after a previous event.

This is taken as a first order approximation of

the true spectrum. A PI channel spectrum is

formed from all events occurring less than 0.35

ms after a previous event. This "AP" spectrum

will contain predominately AP events, but will

also contain photon events and particle background events. If CR is the total count rate

from our first our "true" spectrum, then the exposure represented by the AP spectrum will be

cn * (0.35 - 0.25)/1000s (2.5)

since no events are recorded in the 0.25 ms after

a previous event. To first order, one can remove

the photon and PB events from the AP spectrum by subtracting from it the "true" spectrum

scaled to the exposure time of the AP spectrum.

To determine the number of AP counts, NAP,

one fits the "true" spectrum by

NApAP + NTHS(T) + SSX + PB (2.6)

where AP is the corrected AP spectrum derived above, ArT is the normalization of a Raymond & Smith thermal spectrum, RS(T), folded

through the detector response, SSX is the measured scattered solar X-ray background, and PB

is the measured particle background. The thermal spectrum models a combination of the cosmic background and the LTE. The temperature

of the model is varied to provide the best fit.

Spatial dzstribution: The spatial distribution

of the AP contamination was determined from

data taken during the All-Sky Survey completion as well as several long pointed exposures

with high AP rates.

3.3.3. The SXX, STE, and LTE

Backgrounds

3.3.3.1. Descriptions SSX: Solar X-rays are

scattered off of the earth's atmosphere along the

observation line of sight. The reduction software uses a standard model atmosphere evaluated for the observation time and position and

an empirical two-component solar X-ray spectrum to model Thompson scattering and fluorescent scattering by the principal species (Snowden & Freyberg 1993).

STE: A short-term enhancement is a period

of increased background with time scales from

several minutes to a few hours, though STE's of

the longer periods are unusual. The enhancements are typically soft, affecting the 1⁄4 keV

bands more than the _ keV bands. There are

several suggested origins for the STE's: aurorae,

solar flares momentarily increasing the SSX, and

enhanced charged-particle precipitation during

periods of increased geomagnetic field activity.

Given the difficulty of determining the true origin of any single STE, data segments that are

contaminated are simply removed; there is no a

priori means of determining the spatial distribution of the contamination.

LTE: Long-term enhancements (LTEs in the

American literature, LXEs in the German literature) are discussed at length in Snowden et al.

(1994) §4.5 and Snowden et al. (1995) §3.1.4.

Suggestions about their physical origin, which is

21
unknown,arediscussedin Freyberg(1998)and

Cox(1998).LTEsareperiodsof increasedbackgroundwith timescalesfromseveralhoursto

severaldays4. Theemissionisvignettedto the

samedegreeasthediffusebackground, indicatingthat theLTEisdueto photonsratherthan

particles(Snowdenet al. 1994)Theemissionis

soft,but thehardnessgenerallyincreaseswith

emissionstrength.

TheabsolutestrengthoftheLTEemissionis

unknown.TheR.ASSI (Snowdenet al. 1995)

hasmadethebestmeasureto date(< 30×10-6

countss-1 arcmin-2 to > 300×10-6countss-1

arcmin-2) overbaselinesof severaldays.However,suchameasurecannotproducemorethan

a localminimum;whethertheLTEevergoesto

zero,or hassomefinitevaluecannotbemeasuredin thisway.

There is evidence that the minimum LTE

measured during the RASS is the true minimum LTE as the RASS count ratesagree well

with other all-skysurveysexecuted over the last

twenty years.There issome evidence that suggeststhat the minimum LTE value isnear zero

(Snowden et al. 1995); a ROSAT observation

of the moon (Schmitt et al. 1991) revealsthat

the X-ray fluxfrom the unilluminatedportionof

the moon was comparable to the LTE emission

rate.(As pointed out by Snowden et al. (1995),

ifthatX-ray emissionwere produced on the surface of the moon, then the equivalencemay be

a misleadingserendipity.)

The lunar observationssuggest that the LTE

originates within cis-lunar space, a suggestion

that can be better understood in the light of

the surprising X-ray emission by comets such

as those of Lisse et al. (1996). It has been

suggested that the X-ray emission from comets

is the result of solar wind colliding with neutral atoms (Cravens 1996), a process that could

also work with the geocorona (Cox 1998). This

mechanism is supported by the very weak correlation between the LTE rate and various measures of geomagnetic storms and the solar wind 5.

4Freyberg (1998) makes the curious claim that enhancements last "for typically up to eight hours (i.e. a couple

of orbits)". However, a cursory glance at Figure 7 of

Snowden et al. (1994) shows that enhancements often

last up to several days, and there may be variation on

the order of several weeks.

5For the wide disparity in opinion about the correlation

of the LTE rate and either geomagnetic activity or solar

Cox (1998), in a notorious back-of-the-envelope

calculation based on "flaming comets" has suggested that the solar wind interaction with the

exospheric hydrogen of the geocorona (through

the subsequent recombination) might produce

the correct emission rates for the LTE, but that

the calculations depend critically on the recombination spectrum.

3.3.3.2. Measurement Allofthe above phenomena are identifiedby their temporal signatures. In the case of the STE's, observationsegments with anomalously high count rates

are merely removed. For the SSX and the

LTE backgrounds, the lightcurve isfitto both

backgrounds simultaneously to determine the

strength of the backgrounds.

SSX: The lightcurve of the SSX background

depends upon the change of the zenith angle,

which can be calculatedfrom the spacecraftattitude,and the column density along the line

of sight.The scalingof the SSX background is

determined from fittingthe model to the light

curve. (The current reductionsoftware alsoallows the scalingto have a lineartemporal variation.However, except for some of the longest

continuous data sets,such variationisgenerally

not needed.)

Time segments with an SSX background rate

over all bands exceding 5 counts s-I are discarded; those time periods are very short,so

discardingthem removes a largeamount ofnoise

while removing only a small amount of signal.

The 1⁄4 keV SSX background isdominated by

Thompson scattering,while the higher energy

bands are dominated by the O Ka fiourescence

lineat 0.53 keV. The two sources do tend to

change independently. Thus, the contribution

to the R4-R7 bands can be determined from

the contributionto the R4 band (sinceallthose

bands are dominated by the O Ka line),but

the contributionsto the R2 band (which contains both the Thompson scatteringand the

0 Ka line)and the RI band (containingonly

the Thompson scattering)must be determined

seperately.

LTE: The LTE lightcurve isassumed to be

reasonablyfitby a low order (lessthan fifthorder) polynomial. From any given observation,

activity compare Snowden et ai. (1994) §4.5, Snowden

et al. (1995) §3.1.4, and Freyberg (1998).

22
one can only remove the LTE emission greater

than the minimum LTE emission during that observation. Removing the LTE to an absolute

level requires comparison of the observation to

the equivalent RASS field (which is assumed to

be free of any LTE emission, but note the caveat

several paragraphs above). This comparison is

discussed in the following chapter.

B(x, y), and the effective exposure, E(x, y). There

is also a mask, M(x, y), which is zero for regions

containing point sources, and regions with very

low effective exposure (i.e., under the wagon

wheel of the window support structure).

3.6. Execution

3.3.3.3.

is vignetted in the same manner as the cosmic

emission. The spatial distribution of the LTE is

just a scaling of the effective exposure map.

SSX: The SSX is vignetted in the same manner as the cosmic emission, but the SSX emission is also a strong function of zenith angle.

Over the diameter of the PSPC field (< 2°)

the change in SSX with zenith angle is approximately linear. The introduction of this gradient

need not be done for every observation interval

since the sum of a series of images with different count rates and different gradients will be

the sum of the gradients weighted by the count

rates. Thus, the gradient can be applied to the

sum of the SSX backgrounds from each observation interval.

The data were reduced following the proceSpatial Distribution LTE: The LTEdure described in the Cookbook for Analysis Pro3.4. Point Source Removal

Point source removal was accomplished with

an enhanced sliding box method, where the size

of a circular box is scaled to the local PSF. The

background was determined from the raw counts

in an annulus from the 90% encircled energy radius to 2.5 times that radius. If the raw counts

within the 90% encircled energy radius form a 4o"

or greater deviation from the background, then a

point source was deemed to have been detected.

Since the size of the PSF increases with radius,

the point source detection limit will be spatially

variant (see §3.2.5 of the following chapter).

Given the limited count rate in any single

band and the overlap of the energy response,

point source detection was executed on the combined R12 band and the combined R47 band.

3.5. Output Products

The output products consist of three images

for each band, the raw counts, C(x,y), the

sum of the modeled non-cosmic backgrounds,

cedures for ROSAT XRT Observations of Extended Objects and the Diffuse Background, Part

I: Individual Observations (Snowden & Kuntz

1998) which describes the Extended Source Analysis Software (ESAS). The ESAS implements

the calibration and procedures described in a series of papers about the PSPC (Snowden et al.

1992; Snowden & Freyberg 1993; Plucinsky et al.

1993; Snowden et al. 1994).

The ESAS was developed by S.L. Snowden as

a suite of about a dozen individual FORTRAN

routines, each of which queried the user for data

and control parameters. When faced with the

task of reducing over 280 observations, it became necessary to reduce the amount of human

interaction necessary for the reduction of each

observation. K.D. Kuntz modified the FORTRAN routines so parameters could be passed

from one routine to another through data files.

These modifications have been incorporated into

the current ESAS release, which is available

to the entire community through the ROSAT

Guest Observer Facility at NASA/GSFC. A set

of UNIX shell scripts that invoke the FORTRAN

routines in the correct order are described in Appendix C.

The steps of the reduction may be summarized as follows.

Initial Time Selection: Time periods when

the MV rate exceded the threshold are removed.

All events occuring less than 0.35 ms after a previous event are removed.

Subsequent Time Selection: The remaining

data are used to create a light curve. All times

for which the SSX> 5 counts s -1 are removed.

On average, time selection removes ~ 20%

of the exposure. This loss is unavoidable; the

data that are removed either are uncalibratable

or have a very low signal-to-noise ratio.

Light Curve Fitting: The strength of the SSX

and LTE backgrounds are determined by fit23
tingthemodelsto thelightcurve.Observation

segmentsthataresignificanlyabovethefit are

removedunderthesuspicionthat theycontain

STEs.

Mapping (also called "casting"): The events

remaining after time selection (raw counts) are

mapped into sky coordinates, as are the effective

exposure, the particle backgrounds (as measured

from the MV rate), the SSX, and the LTE.

AP Fitting: Once the particle background

and SSX count rates have been determined, the

AP count rate can be determined. As described

above, and the distribution of the AP counts

mapped into sky coordinated.

Background Summation: The PB, SSX, LTE,

and AP backgrounds are summed to produce a

single background image.

Point Source Removal: The point source detection is done using the raw count image and

exposure map, and the mask is created.

The final, reduced image is given by

raw counts modeled background (2.7)

effective ezposure

The reduced images were created only to check

for reduction problems; the mosaicking software

to be discussed in the next chapter does not use

reduced images, but rather the raw counts image, the background image, the effective exposure image, and the mask.

4. Uncertainties

4.1. Detector Map _ Effective

Exposure Map

To produce an effective exposure map the

observation is split into observation segments

that are sufficiently small that the shift or rotation within any single segment is less than

a detector pixel, 14_/957 x 14_/957. A typical

observation contains 200 different shift/rotation

combinations. A pixel of the exposure map

(14_/957 x 14_/957 in sky coordinates) will be the

weighted sum of all of the detector map pixels that overlaid that position in each of the

shift/rotation combinations.

The detector map used for this work is the

image, in detector coordinates, of all of the photons that were detected in the -,, 11 days that the

PSPC B was used to finish the All-Sky Survey.

The uncertainty in a given pixel of the detector map will be the uncertainty in the number

of photons detected in that pixel, and will be

given by Poisson statistics. If we consider a case

where all the obervation segments have the same

weight, then the uncertainty ofa 14(J957× 14'.'957

pixel in the exposure map will be the root of the

sum of the counts in all of the detector map pixels that overlaid that position. Table 2.5 contains the typical number of counts at both the

center and the periphery of the detector map.

Assuming 200 observation segments, there are

200 different detector map pixels that overlay a

given exposure map pixel; the percentage uncertainty for a pixel of the effective exposure map

will range from -_ 11.5% to ,,- 6.8%.

For a 1' x 1_ pixel (the size typically used

in mosaics) the calculation is similar. However,

in this case, one no longer has 200 independent

samples; although the number of detector map

pixels that overlay a given exposure map pixel

will increase a small amount, the number of independent samples will decrease 6. The uncertainty per pixel will decrease only slightly; for

1_ pixels the uncertainty ranges from 9.5% to

5.6%. In all cases the uncertainty in the exposure map is much less than the uncertainty

due to the counting statistics of individual exposures.

4.2. Spatial and Temporal Gain

Variation

Snowden, Turner, & Freyberg (1999) have recently completed a study of the spatial and temporal variation of the PSPC. After the temporal

gain corrections of Prieto, Hasinger, & Snowden

(1996), they found that there were variations of

up to -_ 4% in the gain around the nominal PHA

to PI conversion. In particular, the gain in the

central 15' region of the PSPC was reduced by

up to ,_ 4%, depending upon the epoch.

The implications of the gain variation are not

clear for broad band studies. A change in gain

SAn approximatecalculation can be made as follows. Assume that the detector map pixels used to calculate a single 14t/957 x 14_957 exposure map pixel form a square

with sides of _ pixels. If one is seeking the uncertainty for an exposure map pixel that is n x n detector

pixels in size, the square of detector pixels that will contribute will be _+ (n - 1) on a side. However, there

will be only (_+ (n - 1))2 independent contributions.

24
TABLE2.5

DETECTORMAPCOUNTS

Band Counts a % Uncer. b

Edge Center Edge Center

R1 0.39 1.09 11.3 6.96

RIL 0.38 1.09 11.5 6.77

R2 0.62 0.99 8.96 7.10

R4 0.49 0.96 10.1 7.22

R5 0.49 0.98 10.1 7.14

R6 0.50 0.94 9.98 7.30

R7 0.38 0.96 11.5 7.22

_Average for a single detector pixel measured

over a 16 x 16 pixel box. The "edge" region was

selected to be [455:470,225:270]; the "center" region was selected to be [235:250,225:240]

bThe percent uncertainty was calculated for a

single detector pixel using the method described

in the text.

changes the PI channel into which a given event

is binned. For a broad energy band such as those

used here, the effect of the gain reduction depends upon the spectral shape; what is the balance between counts shifted out of one end of

the band and those shifted in at the other?

To explore this effect we have taken several

Raymond & Smith model spectra of different

temperatures, determined their expected PI distribution, and reduced the gain by 4.5% (the

counts in each channel are reduced by 4.5% and

then increased by 4.5% of the next higher channel.) The results are shown in Table 2.6, where

it can be seen that, due to the low spectral resolution of the PSPC, a small change in the gain

does little to change the total number of counts.

Note also that in most cases the wobble smears

the effect of the gain variation so modeling a

4.5% gain shift is conservative.

4.3. Particle Background Map

The total number of counts per detector pixei

in the particle background maps measured by

Plucinsky et al. (1993) are on the order of 1.02

counts arcmin -2 for the internal PB, and 0.40

counts arcmin -2 for the external PB. (Here I

have combined their data sets 2, 3, and 5 for the

internal PB and data sets 1 and 4 for the external PB.) Given these low count values, only direct measurement of large scale structure in the

PB map is possible. For bands R4-R7, the variance within the measured PB maps was consistent with a Poisson distribution. In band R1L2,

there is a strong residual grid pattern with a 5_

scale (Plucinsky et al. 1993).

Although one cannot measure the grid pattern in the particle background, it is unlikely

to cause significant spurious structure. First,

the "dither" smears the structure. Second, the

mean particle background rate for the sum of

the internal and external particle background in

the data used by Plucinsky et al. (1993) are on

the order of 13x10 -6 counts s -1 arcmin -2 for

all bands combined. From Table 2.7, one can

see that the contribution to any single band will

be significantly smaller; in band R1L2 the PB

rate will be _. 1% of the detected cosmic rate,

while in band R45 the PB rate will be -_ 4% of

the detected cosmic rate 7.

REFERENCES

Abel], G. O., Corwin, H. G., & O|owin, R. P. 1989,

ApJS, 70, 1

7I have used the detected cosmic rate here rather loosely.

The detected cosmic rate is the total number of non-LHB

photons detected. This is the total number of photons

that actually enter the detector that cannot be attributed

to the LHB.

25
TABLE 2.6

EFFECT OF GAIN REDUCTION ON BROAD BANDS

log T a Percent Change

R1 R1L R2 R4 R5 R6 R7

5.8 0.04 0.14 0.54 0.84 0.52 0.62 0.64

6.0 -0.12 -0.03 0.42 0.41 0.40 0.46 0.46

6.2 -0.22 -0.15 0.33 0.14 0.30 0.34 0.34

6.4 -0.24 -0.16 0.34 -0.06 0.13 0.28 0.28

6.6 -0.19 -0.12 0.32 -0.22 0.04 0.24 0.25

6.8 -0.01 0.04 0.33 -0.33 -0.02 0.21 0.23

_Raymond & Smith model temperature.

TABLE 2.7

PARTICLE BACKGROUND DISTRIBUTION

Band Internal External

Continuum AI Ka

RIL 0.185 0.044 0.000

R2 0.148 0.105 0.000

R3 0.049 0.047 0.000

R4 0.081 0.084 0.000

R5 0.O9O 0.O96 0.000

R6 0.168 0.182 0.189

R7 0.270 0.293 0.804

RIL2 0.333 0.149 0.000

R45 0.171 0.180 0.000

R67 0.438 0.475 0.993

The quantities listed are the percentage of

each component in each band for the PSPC B

in low gain mode.

26
Arabadjis, J. S., & Bregman, J. N. 1999, ApJ, 510,

806, submitted

Balucinska-Chureh, M., & McCammon, D. 1992,

ApJ, 400, 699

Cox, D. P. 1998, in The Local Bubble, ed. D. Breitschwerdt & M. Freyberg, 121

Cravens, T. E. 1996, Geophysical Research Letters,

24, 105

Freyberg, M. J. 1998, in The Local Bubble, ed.

D. Breitschwerdt & M. Freyberg, 113

Gruber, R. 1992, in Data Analysis in Astronomy IV,

ed. V. D. Gesia, 153

Hartmann, D., & Burton, W. B. 1997, Atlas of

Galactic Neutral Hydrogen (Cambridge University)

Hartmann, D., Kalberla, P. M. W., Burton, W. B.,

& Mebold, U. 1996, A&A, 119, 115

Kalberla, P. M. W. 1978, Ph.D. thesis, University of

Bonn

Kuntz, K. D., & Snowden, S. L. 1998, Cookbook

for analysis procedures for rosat xrt observations of extended objects and the diffuse

background, part ii: Mosaics, Technical report, NASA/GSFC

Kuntz, K. D., Snowden, S. L., & Warwick, R. S.

1998, in The Local Bubble, ed. D. Breitschwerdt & M. Freyberg, 329

Lisse, C. M., et al. 1996, Science, 274, 205

Morrison, R., & McCammon, D. 1983, ApJ, 270,

119

Plucinsky, P. P., Snowden, S. L., Briel, U. G.,

Hasinger, G., & PfetTermann, E. 1993, ApJ,

418, 519

Prieto, M. A., Hasinger, G., & Snowden, S. L. 1996,

A&A, 120, 187

Raymond, J. C., & Smith, B. W. 1977, ApJS, 35

Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998,

ApJ, 500, 525

Schmitt, J. H. M. M., Snowden, S. L., Aschenbach,

B., Hasinger, G., Pfeffermann, E., Predehl,

P., & Triimper, J. 1991, Nature, 349, 583

Sfeir, D., Lallement, R., Crifo, F., & Welsh, B. Y.

1999, A&A, submitted

Snowden, S. L., et al. 1997, ApJ, 485, 125

Snowden, S. L., & Freyberg, M. J. 1993, ApJ, 404,

403

Snowden, S. L., et al. 1995, ApJ, 454, 643

Snowden, S. L., & Kuntz, K. D. 1998, Cookbook

for analysis procdures for rosat xrt observations of extended objects and the diffuse

background, part i: Individual observations,

Technical report, NASA/GSFC

Snowden, S. L., McCammon, D., Burrows, D. N., &

Mendenhall, J. A. 1994, ApJ, 424,714

Snowden, S. L., Plucinsky, P. P., Briel, U., Ha.singer,

G., & Pfeffermann, E. 1992, ApJ, 393, 819

Snowden, S. L., Turner, T. J., & Freyberg, M. J.

1999, A&A

Voges, W., et al. 1992, in Proceedings of Satellite

Symposium 3, Space Science with Particular

Emphasis on High-Energy Astrophysics, ed.

T. D. Guyenne & J. J. Hunt, 223

Voges, W., et al. 1996, http://www.rosat.mpegarching.mpg.de/survey/rass-bsc/

Yan, M., Sadeghpour, H. R., & Dalgarno, A. 1998,

ApJ, 496, 1044

27

Chapter 3: Mosaicking

Abstract

The following cookbook describes the process of creating a mosaic from a number

of overlapping ROSAT PSPC or HR/pointings, whose reduction was described in the

previous chapter. The mosaicking process described here is a substantial improvement

over previous versions; we have implemented a more complex process that, now that

the high resolution maps from the ROSAT All-Sky Survey have become publically

available, will produce a mosaic with better LTE offset removal especially on large

scales, and automatic normalization to the All-Sky Survey. Although this chapter is

primarily concerned with describing this new method of mosaicking, the method not

utilizing the ROSAT All-Sky Survey is described as well.

1. Introduction

Mosaicking 1 is a potentially powerful tool for

producing large field-of-view images (several degrees by several degrees) that are much more

sensitive than the ROSAT All-Sky Survey (the

RASS). As with any mosaicking technique, determining and removing the zero-point offsets

between overlapping pointings is of prime concern. Unlike those who mosaic optical images,

we have little astrometric concern, the pixels are

large (usually several times the 14.957" pointed

image pixels) compared to the pointing accuracy

of the instrument (10").

As in the previous chapter, it is important

to distinguish between cosmic and non-cosmzc

backgrounds. The cosmic background is the sum

of the counts received from diffuse or unresolved

distant sources. The non-cosmic background,

usually referred to simply as the "background",

is the sum of the counts received from local

sources, such as scattered solar X-rays and the

Long Term Enhancement (LTE), or produced in

the instrument, such as the particle background

and the "after-pulse" counts. See Snowden et al.

(1994) for an extensive discussion of the PSPC

background, Snowden (1998) for the HRI particle background. In general, the cosmic background is what we want, and the background is

the nuisance we want to eliminate, though its

t A number of commentators have expressed displeasure

at this spelling of the gerund form of "mosaic". Though

it may look a bit odd, this form is recognized by dictionaries, following the model of picnicking and panicking.

A "c" followed by an 'T' should be soft, leading to something sounding like "mS - sA - _ - s_ng", like "icing".

components may, in themselves, be of interest.

2. The LTE Offset Rate

The principal task in forming a mosaic is to

remove the effect of any residual offsets between

fields: to eliminate any variation in the zero level

across the fields. The residual offsets are primarily due to the different LTE offset rate of

each exposure, but may also include any over or

undersubtraction of the other background components. As was described in Part I, for each

individual exposure, the light-curve can be described roughly by a low order polynomial. The

global minimum of that polynomial is the true

cosmic background signal plus any component

constant (over the period of the observation) of

the LTE, while the flux above that minimum is

clearly due to the LTE. Due to the limited time

over which an exposure is made, one cannot be

sure that a global minimum (LTE = 0) has been

reached. Therefore, although one can take the

true cosmic background rate to be the minimum

rate observed during the exposure, what remains

is still the true cosmic background plus some unknown LTE rate, the LTE offset.

Clearly, since the LTE is by definition a time

variable phenomenon, the residual offset will be

different for different exposures. The following sections describe three methods by which

one can determine the LTE offsets for all of

the pointings within a mosaic. The first two

methods, though implemented in the mosaicking

software, are primarily for illustration, as both

have potentially serious drawbacks. The third

29
methodis a synthesisof theprevioustwoand

istherecommendedmethod.Theresultsof the

threemethodscanbeseenin Figure3.1.

2.1. Direct Comparison to the RASS

The first method, direct comparison (DC),

is rather simple-minded; one can compare the

mean count rate in each pointing to the mean

count rate from the matching position in the

ROSAT All-Sky Survey (the RASS);

LTE = R_oi,,ed - RRASS, (3.1)

where LTE is the residual LTE count rate for

a given pointed exposure, Rpointed is the mean

count rate in that exposure, and RRASS is the

mean RASS count rate for the area of the sky

covered by the pointed observation. It is assumed that the LTE has been correctly removed

from the RASS, an assumption discussed at

length in Snowden et al. (1994) and Snowden

et al. (1995).

2.1.1. Point Sources

The above equation also assumes that the

point sources have been removed to the same

level for both the RASS and the pointed observation. In general, the point sources can be

removed from the pointed observation to much

smaller count rates than in the RASS due to the

limiting source count rate used for the RASS

(see Table 3.1) and the typically much longer

exposure of the pointed observations. As can be

seen in Table 3.3, the total contribution from

point sources that are detected in individual

pointings but which are fainter than the RASS

exclusion threshold can be comparable to the

LTE offset. To make the comparison properly,

one should make a point source mask based upon

the ROSAT Bright Source Catalogue or BSC

(Voges et al. 1996), the catalogue used to make

the point source masks for the RASS, using the

count-rate thresholds listed in Table 3.1.

2.1.2. Uncertainties

For a given pointed exposure there are -,- 90

12' x 12' RASS pixels, each of which has an uncertainty of,-- 25 - 75%. The distribution of the

uncertainty of the RASS pixels can be judged

from Figure 3.2. The uncertainty in the difference between the mean count rate of a pointed

observation and the mean count rate for the

same region of the RASS is _ 5- 7 x 10 -6

counts s -1 arcmin -2 for a single band (see Table 3.3). Figure 3.3 shows a typical RASS field

and the regions corresponding to the individual

pointings in a mosaic. Several of the regions appear to contain "hot" pixels, i.e., those which are

either statistically high or contain unremoved

point sources. This impression is reinforced by

Figure 3.4, where the "hot" pixels appear as outliers. As implemented, these outliers, both positive and negative, are removed before the difference between the pointed observation and the

RASS is calculated.

u

iJl

$.| .

u

lul

u lu iJ |J _ _ s,lLI _ |J

R1 R2

R6 R7 0t_

Fig. 3.2.-- The distribution of Uncertainty/Rate for a single panel of the RASS in

the direction of (2, b) = {180, 0).

The difficulty with this method is that there

is no pointing-to-pointing consistency. Given

the uncertainties in the RASS, adjacent pointings may be normalized to different values of

the diffuse cosmic background; causing obvious

"seams" between pointings. The seams can only

be seen with difficulty in Figure 3.1 due, in part,

to the inherent smoothness of this field.

3O
SC

W

DC

W-SC

DC-SC

W-DC

Fig. 3.1.-- A mosaic of R2 band pointings where the LTE offset has been determined by three

different methods. Top Left: Self-consistent solution (SC) adjusted by the difference between the

entire mosaic and the RASS. Top Right: Difference of the equal weighting and self-consistent

solutions. Middle Left: Equal weighting (W) of self-consistent and direct comparison solutions.

Middle Ri9ht: Difference between the direct comparison and self-consistent solutions. Bottom

Left: Direct comparison (with the RASS) solution (DC). Bottom Right: Difference between the

equal weighting and the direct-comparison solution. The image mosaics (left-hand column) were

binned to 6' × 6' pixels and were further smoothed by a 30' radius tophat function. The mean

value of the images is ,,, 200 × 10 -6 counts s-1 arcmin -2, with a range from 140 to 230 ×10 -6

counts s -1 arcmin -2. The difference images (right-hand column) were binned to 1._5 × 1(5 pixels.

The difference have a range from -30 (white) to +30 (black) ×10 -6 counts s -1 arcmin -2. The

{min,max} range for each of the difference images is W-DC: {-7.95,15.68}, DC-SC: {-27.24,21.22},

and W-SC: {-19.07,14.92} in the same units as above. One should not take the uncertainty in the

LTE offset to be so large since much of the difference is accounted for by the systematic trend

across the image, not pointing-to-pointing variation. Scale: Each pointing has a radius of 53'.

31
17

25

@

18

@26

19

®

20 Ill

14

22 23 24

Fig. 3.3.-- A demonstration of the hazards of comparing PSPC pointings to the RASS. Top Left:

A typical RASS field, in this case centered on (l,b) = (180,0) for band R2. Darker pixels have

higher X-ray intensity. Shown is the outline of a mosaic for which the LTE offset is to be found.

(The dot in the center is a region missing from the mosaic_ not a point source.) Top Right: The

RASS at the location of the mosaic. Bottom: The RASS at the location of each individual pointing.

The size scale is that same as for the panel in the upper right. Note the existence of "hot" pixels

in some of these pointings.

32
TABLE 3.1

RASS POINT SOURCE EXCLUSION LIMITS

Band Limit (counts s -l)

RI2 0.025

R45 0.02

R67 0.02

R47 a 0.04

*This is an approximation. A

point source with a steep spectral index might be detected in

R45 but not in R67, meaning

that 0.02 counts s -1 would be

removed from the BASS, rather

than 0.04 counts s -1 .

-4 -2 0 2 --4 -t 0 t 4

(h_-XeQn)/o (xate-_u)/a

Fig. 3.4.-- The distribution of the RASS pixel values for each individual pointing (band R12).

The horizontal axis is the pixel value minus the mean, divided by the standard deviation of the

distribution. Note that the hot pixels in Figure 3.3 are the outliers above 3a.

2.1.3. Use of the R1 vs. the R1L Band

The RASS was executed with the R1 band,

while most mosaics will be formed with the RIL

band. To convert from one to the other requires

some assumptions about the temperature of the

emitting gas and the amount of intervening absorption. These assumptions should, of course,

be consistent with the R1/R2 band ratio determined from the RAS5 q.

sit should also be remembered that the BASS was not

executed with the same detector used for most of the

If the bulk of the 1⁄4 keV emission in the field

of view is produced by the Local Hot Bubble

(LHB), then the appropriate R1L/R1 band ratio

can be read from Table 3.2. This table contains

the mean R2/RI band ratio for the LHB, as well

as the values at the half-maxima of the distribution (Snowden et al. 1998), the corresponding

Raymond & Smith model temperatures, (Raymond & Smith 1977) and the resulting R1L/R1

pointed observations; the cross-calibration of the two detectors is described in Snowden et al. (1994).

33
band ratios. These values have been calculated

assuming both no absorption and an absorption

due to the Local Fluff, ,,- 1.5 x 1018 cm -2.

If an object other than the LHB fills most of

the field of view, one will need to determine the

R1L/R1 ratio intrinsic to the object through independent spectral modeling. As a first cut, one

could determine the Raymond & Smith model

temperature from the R2/R1 band ratio, and

use that to determine the R1L/R1 band ratio.

The user will have to determine whether such a

solution is appropriate.

One other interesting situation is a highGalactic-latitude pointing containing a contribution from the LHB as well as a contribution

from the Galactic halo. From (Snowden et al.

1998), we find that the halo has a lower model

temperature (lOgl0T = 6.06) than the LHB

(logl0T = 6.14). An unabsorbed contribution

would increase R1L/R1, since the lower temperature gas would be softer, but the halo emission is absorbed, which hardens the halo contribution to an R1L/R1 ratio closer that of the

LHB. When the absorption makes the halo component very hard, it is sufficiently attenuated,

that it contributes very little to the total emission. This behavior can be seen in Figure 3.5.

As can be seen, for most fields containing the

Galactic halo, taking the R1L/R1 ratio of the

LHB will be sufficient.

2.1.4. Weighting Change

Since the initial release of this software, we

have changed the weighting scheme used to compare the RASS to the individual pointings. Originally, each pixel in a pointing was weighted by

its uncertainty. For low surface brightness objects, where there are only a few counts per

pixel, the uncertainty will be correlated with the

pixel value, thus significantly biasing the mean.

(This will not be a problem with most observations of diffuse but recognizeable objects, only

with observations of the diffuse background.)

To correct this problem, we removed the

weighting. Without the weighting, pixels at the

edge of a pointing where the exposure is low,

introduce significant uncertainty into the value

of the mean. To reduce this uncertainty we introduced an iterative discard. In each iteration,

the mean and variance are calculated. Any pix0.850;

0,840

0.830

0.820

0.810

18

......... i ......... i ......... i ......... i ......... _...... ,,

19 20 21 22 23 24

Log., Column Density)

Fig. 3.5.-- The behavior of the R1L/R1 ratio

as a function of halo emission fraction and absorbing column density. The horizontal line is

the value of R1L/R1 for the LHB. Each curve

deviating from that line is the value of R1L/R1

as a function of the column density absorbing

the halo component. The curves progressively

farther from the horizontal show an unabsorbed

halo to LHB contribution of 10%, 20%, 30%, to

80%.

els that fall outside 3.5a are removed, and the

mean and variance are calculated again. The

entire set of pixels, including pixels that have

been previously discarded, is then compared to

the new standard deviation. The process is repeated until the number of discarded pixels converges, typically three or four iterations.

2.2. Self-Consistent Solution

The self-consistent solution determines the

relative LTE offsets between pointings, and then

compares the entire mosaic to the RASS.

The effect of the different offsets between two

overlapping exposures can be removed by comparing the total count rates in the region of overlap. The total count rate is defined as

total counts - background counts Rexposure time

(3.2)

where each quantity is summed over the region of overlap. Here, as above, "background"

is shorthand for the sum of the particle background, the scattered solar X-rays, the after34
TABLE 3.2

R1L/RI RATIOS FOR THE LHB

R2/R1 a log T R1L/R1

Foreground N H 0 cm -2b 1.5 × 1018 cm -2b 0 cm -2b 1.5 X 1018 cm -2b

-HWHM 1.02 6.093 6.088 0.8088 0.8095

Mean 1.12 6.143 6.138 0.8149 0.8153

+HWHM 1.23 6.214 6.207 0.8234 0.8239

aFrom Snowden et al. 1997.

bColumn density of absorbtion in front of the LHB emission due to the Local Fluff.

pulse counts, and the portion of the LTE greater

than the minimum LTE (during the exposure).

Assuming that there is no cosmic source of variation, (e.g., a variable star) the difference in the

count rates in the region of overlap is the difference in LTE offset rates between the two exposures. In other words, if Sis is the count rate

due to cosmic sources in the region of overlap of

exposure1 and exposures, LTE1 and LTE2 are

the LTE offsets for exposure1 and exposures, respectively, and R1 and R_ are the total count

rates in exposure1, and exposures, respectively,

then

Sis + LTE1 = R1, (3.3)

SIS "_- LTE2 = Rs, (3.4)

and thus

ALTE =_ LTE1 - LTE2 = R1 - R2. (3.5)

At this point, one could subtract the difference

in LTE rates (ALTE) from the exposure with

the larger rate, could add it to the exposure

with the smaller rate, or, pardoning the pun,

one could split the difference between the two.

Although the first option might seem the best,

it really does not matter, as the absolute LTE

offset is yet to be determined.

The method for multiple exposures is similar.

For an example of three exposures,

LTE1 - LTE2 = R1 - Rs (3.6)

LTEs - LTEa = Rs - R3 (3.7)

where the P_ - Rm are measured quantities.

(See Figure 3.6 for a graphic portrayaL.) To

these equations can be added the constraint

LTE1 + LTEs + LTE3 = O, (3.8)

Fig. 3.6.-- A graphic portrayal of the determination of the LTE offset. Here there are three exposures (1, 2, and 3) and two regions of overlap

(a and b). We thus have the system of equations:

Rla - LTE1 = R2a - LTE2

Rsb - LTEs = l_3b -- LTEa

LTEI + LTEs + LTEa = 0

which can be easilysolved.

which closes the system and sets the condition

that the LTE offset rates are as small as possible.

This condition ensures a reasonable solution and

essentially "splits the difference".

The advantage of this method is that adjacent pointings are normalized to the same cosmic background rate, and thus visible "seams"

can be avoided. However, small errors between

pairs of overlapping pointings can accumulate,

causing a "drift" in the measured cosmic back35
groundlevel(relativeto thetruecosmicbackgroundlevel)fromonepartof themosaictoanother. "Drift" is especiallya problemforlinearchainsofoverlappingexposures, butcanoccuranyplacetherelativeLTEoffsetsaredeterminedby a limitednumberof pointings,or

pointingswithlowexposure.

2.3. Synthesis

The best method combines the self-consistent

method, and its ability to avoid "seams", with

the direct comparison to the RASS, which avoids

the problem of "drift". For a two-pointing mosaic, we wish to combine

LTE1 - LTE2 = R1 - R_ (3.9)

from the self-consistent method with

LTE, = R, - RASSx (3.10)

LTE2 = R_ - RASS2 (3.11)

where RASS, and RASS_ are the mean count

rates determined from the RASS at the positions

of the two pointings.

The equations described in the previous section, as implemented, are solved with a singular value decomposition (SVD) algorithm, which

is very stable for over-determined sets of equations. The algorithm is sufficiently complicated

that most introductory texts merely describe its

properties without describing the algorithm itself (e.g., Strang 1976) or provide source code

without explanation (Press et ai. 1986). Understanding the algorithm itself, however, is not

necessary to determine the proper way to combine the above three equations.

Assuming, for the moment, that there are no

uncertainties, the above system of equations can

be written as

[ [R1R2] 1 -1 LTE1 = R1 - RASSa

l 0 LTE2

0 1 R2 - RASS2

(3.12)

For example, take

R1

R2

RASS1

RASS_.

= 110

= 120

= 90

= 110.

(3.13)

The self-consistent solution (after comparing the

entire mosaic to the RASS) would yield

LTEsc = LTE2 = 20 "

while direct comparison with the RASS would

yield

LTEDc = LTE2 = 10

The SVD solution of the three equations together yields

LTEws = LTE_ 16.6666

which is 1⁄2 the self-consistent solution (LTEsc)

(which is represented by 1 equation) plus _ the

direct comparison solution (LTEDc) (which is

represented by two equations).

The two solutions should be weighted by their

uncertainties, and not by the number of equations representing each. For the simple example above, the weighting can be accomplished

by multiplying each direct comparison equation

by a factor W,

[W1

0

-10

W

LTE_LTE11[= (RI

(R2 -

-RI-R RASS1)W

RASS_)W ]

(3.17)

which yields the solution

LTEws = LTEsc + 2W2LTEDc (3.18)

or, to weight the solutions equally

w=v/_. (3.19)

In this case, each equation should then be further weighted by its uncertainty.

Increasing the number of pointings substantially increases the complexity, and the result is

no longer the weighted sum of the two solutions.

Figure 3.7 demonstrates this complexity; each

curve shows the LTE offset for a given pointing

as a function of the weighting applied to the direct comparison equations. The left edge of the

plot is the modified 3 self-consistent solution; the

right edge is the direct comparison solution; between is the result for various weightings. The

36
400 ""''"'I ....T, "I ...... "i '''''_''

300

0

200

100

/L

f_

J

O_ f

-2 -1 0 1 2

Logm(Weight)

Fig. 3.7.-- A demonstration of the change in

LTE offset solution as a function of the relative weights of the self-consistent and directcomparison equations. The left edge is the modified (see text) self-consistent solution for the

LTE offsets of the mosaic shown in Figure 3.1.

There are twenty-six pointings in this mosaic.

The right edge is .the direct-comparison solution. Plotted between the vertical bars are the

solutions obtained as the relative weight of the

direct-comparison equations is increased from

1/100 to 100. Note that not all of the LTE

values change monotonically as the weighting is

changed.

change from one solution to the other is not

monotonic.

It should be noted that the X2 will be higher

for the weighted solution than for either the selfconsistent and direct-comparison solutions, simply because there are more conflicting conditions

to be met. However, it should also be remembered that X2 loses significance if W is more than

one 4 .

Once the LTE offset rates are determined for

each pointing, the spatial distribution of the

LTE offset counts for each pointing is taken to

be the distribution of the effective exposure, i.e.,

that which is put into the mosaic is a scaled exposure map.

2.4. Calculating Offsets in Overlap

Regions

Before the total count rate in the overlap region is calculated, the image must be carefully

cleaned. First, the heavily vignetted edges must

be removed; the low effective exposure, and the

resultant low source-detection rates, result in a

high uncertainty in the count rates. Excluding

regions more than 53" from the center of the

field removes the most heavily vignetted regions.

Second, point sources and bright small-scale

extended sources must be removed. As has been

noted before, the PSF at the edge of the field is

8The self-consistent solution was modified by adding a

constant. The self-consistent solution produces the relative LTE values; the absolute LTE values had to be obtained by comparing the mosaic to the RASS. As soon as

one combines the self-consistent and direct comparison

equations, no matter how small the weight of the direct

comparison equations, they set the absolute LTE value.

This LTE value assumes a different cosmic background

than does the self-consistent solution. The direct comparison solution v-_sumes a cosmic background that is the

mean of the RASS values of individual pointings, which

is not at all the same as the mean of the RASS in the

region of the mosaic; even though both cover the same

area, they weight it differently.

4If n is the number of self-consistent equations, m

the number of pointings, W the weighting the directcomparison equations, LTEn the fit value of the LTE

offset, and using the nomenclature above, then

X2 = _.((R, - R_) - (LTEi - LTE])) 2 (3.20)

m+Wn

"t _'_m(W(R' - RASSi - LTE,)) 2 (3.21)

m+Wn

37
TABLE 3.3

LTE OFFSET (R2)

Pointing 1 b PSC" SC b WS -0.3 ¢ WS 1.0 c WS 0,3 ¢ DC d

rp200020/ 180,74 -22.61 21.27 -16.154.5.45 -7.964.5,30 -2.95-I-3.66 0.354-4.99 3.21-I-6.98

rp200083/ 179,79 -20.73 7.75 -4.47J,-6.89 -4.154-6,96 -4.464-5.39 -4.624-7.56 -4,734-9.76

rp200441n00/ 180.48 -23.54 12.04 9.03-I-6.51 17.444-6.10 21.864-4.30 23.044-5,81 20.27-I-8.42

rp200442nO0/ 178.12 -21.73 12.82 -12.01+6.47 -17.354-6.22 -19.664-4.49 -20.88+6.03 -23,07+8.11

rp200443nO0/ 179.95 -19,65 10.80 92.744-7.79 86,054-7.13 79,78+4.94 74.634-6,40 71.274-9,02

rp200444nO0/ 182.33 -21,40 10.35 87.114-7.67 96.764-6.73 101.484-4,78 103.944-6.24 106.424-8.08

rp200776nO0/ 181.14 -24.13 12.90 42.274.7.66 51.614.7.03 57,194-5.08 60.474-6.61 63.50+8.26

rp200777nO0/ 180.16 -21.94 6.81 74.054-5.06 81.054-5,16 85.574-3.61 88.704-5.03 90,174-7.30

rp200942/ 183,14 -23.12 357 202,954-12.51 208.154-8.22 207.424-5.37 206.634-6,71 206.154-8.05

rp200981nO0/ 177.96 -23.01 3.66 46.704-8.58 42.124-8.00 36.954-5,80 30.974-7.55 21,274-10.26

rp200982n00/ 182,40 -22,74 5.32 123.364-10.38 130.484-7.50 132.244-5,17 133.324-6,50 134.024-7.56

rp201504nO0/ 174.91 -19.69 5.83 27.76+13.37 16.914-8.79 17.324-5.52 19.914-6.42 22.674-7.24

rp700044/ 176,42 -20.03 8,40 4,934-9,48 -7,864-7.19 -14.144-4.96 -18,924-6.34 -22.314-7.70

rp900353nO0/ 179.42 -19.59 13,38 61.284-7,65 54,334-6.99 48.064-4.80 42,754-6,18 37.514-8.83

rp2OOOO2sum/ 184.12 -23.03 7.26 92.504-13.99 96,354-8.92 93.974-5.55 92,004-6.33 90.934-6.88

rp200912sum/ 177.87 -23.33 5.42 56.964-8.69 53.044-8.05 49.314-5.77 47.474-7.30 51.51-I-9.32

wp200775nO0/ 179.46 -23.21 6.12 31.664-6,88 37.504-6.75 39.864-5.13 39.754-7.13 38.634-9.14

wp201313/ 179.50 -19.38 8.60 388,504-8,36 381.384-7.65 375.234-5.50 371.124-7.40 370.704-10,33

wp201316n00/ 175.03 -20.40 3.98 51,604-13.02 39.914-8,73 38.274-5.69 37.524-7,02 36,01±8.40

wp201369n00/ 179,54 -21.21 17.32 22.754-5,18 26.964-5,52 30.214-4.05 33.344-5,57 36.314-7,31

wp201747nO0/ 181.98 -20,52 17,00 24.974-7.72 34.374-6.87 38.414-4.94 38.644-6.55 35.03+8.79

wp2OOOOlsum/ 181,15 -21.28 13.80 87.044-6.30 95.964-5.89 100.994-4.21 104,094-5,77 106,764-7.90

wp201314sum/ 178,06 -19,85 9.93 62.854-8.15 56.364-7.38 56,314-5.35 60.104-6.75 64,264-7.70

wp201315sum/ 176.61 -20.28 3.97 31.814-9.08 19.584-7,21 13.374-5.10 8.854-6.66 5,814-8.30

wp201370sum/ 177,44 -21.I0 6,85 88.814-6.76 82.124"6.21 80.054-4.40 80.504"5.76 82.834"7.45

*The point source contamination in the RASS for the combined band RIL+R2. The units axe I0 -s counts s-l

bThe self-consistent solution sdjusted by the LTE offset determined from the entire mo4aic region (67.16 x I0 -s counts s-l

aJrcmin-2). The units ace I0 -s counts s -j arcmin -_

¢The solution of the weighted combination of the self-consistent and direct comparison equations, The number is the logx0

of the weight (W) applied to the direct-comparison equations. The units are 10 -s counts s-x arcmin -_.

dThe direct-comparison solution, The units are 10 -s counts s-I arcmin -_.

38
much larger than at the center of the field. Thus,

when the edge of one field overlaps the center of

another, the same point source, just outside the

region of overlap, may enter the region of overlap for one exposure but not for another (See

Figure 3.8). Even if the point source lies completely within the overlap region for both exposures, the Poisson variation in point source

counts between the two exposures, differential

vignetting, as well as possible intrinsic variability of the point source can both cause miscalculation of the background offsets.

2.4.1. Background Gradients

The above treatment of uncertainty in the

count rate in the overlap region assumes that

the background is fiat. The calculation of the

LTE offsets can stumble if there is a strong gradient across the overlap region. Consider the

following one-dimensional demonstration, which

is shown schematicMly in Figure 3.9. In the region of Overlap, let the cosmic background be

represented by a linear function (solid line, top

panel). Let the effective exposure be another

simple function of position for two overlapping

pointings, pointing 1 and pointing 2 (dotted and

dashed lines, top panel). The detected counts,

as shown as a function of position, are shown for

the two exposures in the lower panel.

The overlap routine calculates the count rate

in the region of overlap as

CR = _ counts - y]_ background (3.22)

effective exposure

In this simplified case, _-_(effective exposure)

is the same for both pointings. Assume also,

for simplicity, that the background is the same

for both pointings. As can be seen in the lower

panel, _counts will be very different for the

two pointings.

Given this extreme example, what is likely

in typical fields? Consider a 30 _ overlap in a

region where the cosmic background has a linear gradient, decreasing by n% from one side of

the overlap region to the other. One can calculate the count rate in the overlap region that

one would determine using both of the exposures. Figure 3.10 plots the difference between

the count rates divided by the true count rate as

a function of n. Given two pointings, 1 and 2,

,.°°°

,°°°-'°

Fig. 3.8.-- Top: A demonstration of point

source problems in the region of overlap. The

thick lines represent the edges of two overlapping exposures; the crosses show the exposure

centers. The thin lines show the PSF size for

a single point source near, but not in, a region

of overlap. Although the point source is not in

the region of overlap, it contributes counts to

the overlap region for one exposure, but not the

other. Bottom: A mosaic of two observations in

the LMC. Both observations have a bright point

source near their center which lies just outside

of the processed area of the other observation.

Note the "fans" produced by not removing the

PSF region of a source outside the field of view.

39
1.2

0.6_

0+4_

0.2 _

"to.o

1.0

o.s

u= 0-8 I

=

O.4

11.2

0.0 ,

-20 0 20 40 60 SO I00 120

Position

+I '1

1.0 • r

i

i

, I

o., ' i

,'

_ 0.4 ..... t

O.2 . ". I

0,0 , , , i . , , , , , , , , "b ,

-20 0 |0 40 60 80 I00 120

PoeiUon

Fig. 3.9.-- A demonstration of the effect of a

gradient in the cosmic background on the count

rates measured in overlap regions. Top: The

solid line is the cosmic background as a function of position on a one-dimensional sky. The

dotted line and the dashed line are the effective exposures as a function of position for two

overlapping exposures. Bottom: The measured

count rates as a function of position for the two

exposures, shown only in the region where the

two exposures overlap. (No vignetting correction has been applied.) Note that the sum under

the curves will not be the same.

and taking all of the following sums to be over

the region of overlap, then the percentage difference is

_ COtlnt 8 l _ COUnt$3

true count rate

(3.23)

The cosmic background has been taken to be a

linear function across the region of overlap. The

percent decline is defined as

100 x M az(backgr°und) - M in(background)

Max(background)

(3.24)

where the Max(background) is at one edge of

the region of overlap and Min(background) at

the other.

As can be seen, even for a very steep slope,

the difference in counts between the two exposures is only -,-10% of the true number of counts.

Thus, although gradients can lead to serious

n_

100

8o

60

40

20

0 ,..i ........ , ......... p ......... i .........

60 70 SO 90 1 O0

Separation (arcmin)

Fig. 3.10.-- A measure of the effect of a gradient on the count rates measured in overlap regions. The percentage difference between the

count rates determined from two overlapping

pointings as a function of the separation of the

centers of the pointings and the steepness of the

slope of the cosmic background over the region

of overlap. The structure in the plot is due to

the spoke-and-ring structure of the PSPC window. See text for an complete explanation of the

plotted quantities.

problems, for most circumstances the effects will

be less than a percent.

2.4.2. The Exclusion Radius

It is tempting to use the exclusion radius to

minimize the uncertainty in the count rate in

the overlap region. It would appear that increasing the exclusion radius would increase the

uncertainty by including an ever larger number

of low-signal-to-noise pixels in the more heavily vignetted region. However, for a given separation of two overlapping fields, increasing the

exclusion radius includes an ever larger number

of high-signal-to-noise pixels near the field center. Thus, it can be seen that the uncertainty in

the count rate in the overlap region is formally

minimized by maximizing the exclusion radius

(i.e. excluding as little as possible). The scaled

uncertainty per pixel 5 is shown graphically in

SThe scaled uncertainty per pixel is defined as follows:

if RT {the rate multiplied by the exposure time) is the

4O
6O

58

== 56

54

52

5O

70 SO 90 100 1 tO

Separation (aremin)

Fig. 3.11.-- A graphical representation of the

uncertainty in the count rate for the overlap region as a function of field separation (center of

field to center of field) and exclusion radius. The

plotted quantity is the scaled uncertainty per

pixel, the plotted contours are logarithmic from

-5.5 to 0 in steps of 0.5. The empty region to

the right denotes the combination of exclusion

radius and separation for which there is no overlap. The scaling of the uncertainty per pixel is

described in the text.

Figure 3.11.

We have not found a criterion which, from

first principles, determines the best exclusion ranumber of counts in an unvignetted pixel and V,,3 is the

vignetting function in pixel {i,j}, the the total number

of counts in the overlap region will be

Z RTV,,_ (3.25)

i,j

while the count rate in the overlap region will be

2_-,i,._ RTV,,j

(3.26)

_"_i,_ TV,,j

with the uncertainty

(3.27)

_'_i,j TV,,j

The scaled uncertainty per pixel is thus

i /r_ 1

(3.28)

N V _ _ V, °

where N is the number of pixels in the over]ap region.

dius to use. However, experience has shown that

using a 53' exclusion radius produces reasonable

solutions.

2.5. Point Source Detection

Detecting and removing point sources is a

multi-step process, given that the same point

source may (or may not) be detected several

times in overlapping exposures, and false sources

may be detected in a short exposure and not

seen in longer, overlapping exposures. The problem is further complicated by the fact that not

only will the detection limit vary from exposure

to exposure, but, due to the variation of the size

of the PSF 6, will change drastically from the

center to the edge of a single exposure. (See

Figure 3.12.)

Once one has detected all of the sources in

all of the exposures, the easiest way of sorting

through the multiple detections and the possible

false detections is to accept only the "best" detections. The "best" detections are those made

in the exposure which, at the location of the

detection, has the best point source detection

limit. That is, for every pixel in the mosaic, one

determines which exposure has the best point

source detection limit at that point. Then, taking a list of all of the point source detections, and

the fields in which each detection was made, one

discards all of those detections which were made

in the parts of exposures with non-optimal detection limits. The remainder is a "master list"

of detections which can then be used to mask

out the point sources•

The actual method implemented is slightly

more sophisticated. It compares the discarded

point source detections with the list of "good"

detections to determine which are duplicates,

i.e., the same objects as the "good" detections,

barely detected in sub-optimal pointings. The

rejects that are not duplicates are selected for

closer scrutiny; they may be unmasked hot-spots

or transient sources, both of which should be

excluded. The bottom-left panel of Figure 3.13

shows an example of an unmasked hot-spot detected as a point source, but placed in the reject

6Unless explicitely defined otherwise, the point spread

function radius is taken to be the radius at which the

point spread function contains ninety percent of the energy for the PSPC, fifty percent for the HRI.

41
0.040

0.030

2 0.020

0.010

O.OO_

1.2xlO =

l.OxlO 4

A

m

8.OxlO 3

oo

"_ 6.0x I03

_ 4.0xlO 3

2.0xlO 3

0.008

= 0.006

2

o

0.0O4

0.002

0.000

0

........ , ......... r ......... i ......... i ......... i,.-

........ , ......... i ......... i ......... i ......... i...

........ , ......... , ......... , ......... , ......... h,,

I0 20 30 40 50

Radius(arcminutes)

Fig. 3.12.-- Point source detection limits.

These plots were made from an azimuthally averaged exposure map det_20__l_b.fits. Top: The

point source detection limit for an unvignetted

exposure with a uniform background count rate

of 450× 10 -6 counts s -x arcmin -_ and an exposure of 10 ks. The source of the variation with

radius is only the change in the PSF size with

radius. Middle: The vignetting function. The

sharp dip at ,_ 21 t is due to the window support

structure. Bottom: The point source detection

limit for a vignetted exposure with a uniform

count rate of 450x10 -6 counts s -x arcmin -2.

The variation with radius is due to both the

change in the size of the PSF and the decrease

in effective exposure.

list. This hot-spot is reproduced in Figure 3.14

at several different contrasts to show its structure. The other rejected detections do not appear to be significant.

Occasionally, when a point source falls right

on the boundary between being accepted from

one pointing and being accepted from another,

there can be problems; each pointing's detection

can fall in the other pointing's "accept" region.

Such problems are readily found by a brief examination of the "rejected" point sources.

The above method produces a detection limit

that varies greatly across the mosaic. Why not,

one might ask, use a uniform detection limit that

is as bad or worse than the worst detection limit

within the mosaic? The difficulty is that the

worst detection limit is set by the outer parts

of the exposures (typically outside of the inner

support ring at -,-22' ) which provide the bulk of

the area of the mosaic. The resulting detection

limits are usually worse, sometimes, much worse,

than the detection limits of the RASS.

2.5.1. Recalcitrant Point Sources

Once the mosaic has been formed, enhancements may exist which appear to be point sources

that have not been adequately removed. This

may or may not be a real problem. Consider

an area where the edge of one field (where the

PSF is large) overlaps the center of another field

(where the PSF is small). If a point source has

been removed, then a larger area of one field

will have been removed than has been removed

from the other. As a result, there may be a

small region with no exposure, there will certainly be a surrounding annulus with low exposure, and a surrounding region with higher

exposure. If the "low exposure" is sufficiently

small that not all the pixels contain counts and

the "higher exposure" is significantly greater

than the "low exposure", then, in the mosaic,

the pixels in the "low exposure" region will be

scaled to have the same average count rate as the

higher exposure region which means that those

individual pixels in the "low exposure" region

which have counts, will have much higher values

than the surrounding area, while those pixels

without counts (but which have modeled noncosmic background counts) will be more negative. Thus, though all of the averages will be

42
©

©

O

©

O

i

O .

©

• O "

0

• !

• ! 0

s

o

Fig. 3.13.-- Comparison of "rejected" sources to

better exposures of the same regions. Left two

panels: ten "rejected detections". The leftmost

panel is overplotted with the local PSF. Right

two panels: the same positions (R.A., dec.) in

the pointings with the best point source detection limit at those positions. The right-most

panel is overpiotted with the local PSF. Each

panel has been scaled by the extremes within

that panel• Note the strong source in the lowest

panel.

!

%

|" . _%

| . : _,

I

Fig. 3.14.-- An unmasked hot-spot. The region

is shown at several different contrasts to demonstrate its structure. Note that at low levels it

looks vaguely point-like, but that the brightest

pixels reveal the spacecraft wobble.

correct, a point source will seem to exist, as the

eye more readily perceives positive excursions

than the negative excursions. The solutions are

to use a larger pixel size, or to mask out the offending region "by hand", or to use an exposure

threshold.

Although many residual point sources are of

this spurious nature, real residual point sources

may occur in mosaics formed from bands R1

or R1L. In this case, the PSF is poorly characterized even at small detector radii, due to

the ghost-image problem z. The theoretical PSF

radii are too small, a problem that is especially

notable towards the center of the field. This can

be corrected by setting a minimum removal radius of eight 15" pixels (a 2' radius).

Reszdual Point Sources will also appear in

rWhen an event occurs in the PSPC it creates a signal

in both sets of crossed cathode wires. The x-position of

the event is the centroid of the signals produced by the

cathode wires running in the y-direction. The y-position

of the event is the centroid of the signals produced by the

cathode wires running in the x-direction. If the signal

from a set of wires is too small to allow centroiding, (i.e.,

the signal is above the threshold on only one wire) the

position of that event is taken to be the iposition of that

wire with the greatest signal. This causes faint sources

to be spread along the closest set of crossed wires. The

faint wings of stronger sources will assume a "screen"

pattern•

43
regionswheremanyshortexposuresoverlap.

Thesepointsourceswereindetectablein any

singleexposure.At thistime,nosoftwarehas

beendevelopedto detectandremovethesepoint

sources.If thesesourcesmustberemoved,one

mustdeterminetheirpositionmanuallyandproceedasdescribedaboveforextendedsources.

2.5.2. Point Source Detection Limits

If the smallest allowable statistical significance for a detected point source is set to S,

then, given Poisson statistics,

S = P(C, Cback)dC, (3.29)

JC=O

Ohm

s = e-c-o, c£ok/c! (3.30)

C--.=O

where Cback is the number of background counts

expected in the area covered by a point source

(i.e., one PSF radius) and C#i,_ is the smallest

number of counts necessary for the statistical

significance of S. At a pixel position [m,n] a radius r from the field center (the optical axis)

Cback[m, n]-_ R[m, n] rr PSF(r) _ E[m,n]

(3.31)

where R[m,n] is the background count rate (counts

s-1 arcmin -2) expected in pixel [m,n], E[m,n] is

the exposure time (sec), and PSF(r) is the PSF

radius (arcmin) at that position. Given a constant exposure and background count rate, the

point source detection limit as a function of radius is shown in Figure 3.12a. The slight sawtooth pattern is due to the Poisson probability

being an integral function. Given the exposure

map, which shows strong vignetting at the edges

of the field (Figure 3.12b) one can derive the a

typical point-source detection limit as a function of radius (Figure 3.12c). It should be noted

that, due to the shape of this function and the

increase of PSF size with radius, the center of a

shorter exposure can have a better point source

detection limit than the edge of a much longer

exposure.

2.6. An HRI Note

The above discussion has been primarily targeted towards the PSPC and is not completely

applicable to the HRI. First, although the HRI

also is subject to the LTE, the greater part of

the offsets between pointings is due to the large

uncertainty in the HRI particle background calibration (Snowden 1998). However, the difference in the source of the offset does not change

the method of removal. Second, the method

of determining offsets is different because there

is no RASS equivalent for the HRI. The difference in bandpass between the HRI and PSPC

is sufficiently large that it sould be unwise to

attempt the type of band ratio correction we

have employed to scale the RASS to the PSPC

band R1L. Therefore one can only use the selfconsistent method of determining relative offsets.

2.7. "What have I done wrong?"

Once the mosaic has been produced, unless

all the exposures were of similar length, or the

pixel size was chosen to be quite large, the end

result will not look very nice; the boundaries

between different exposures will be readily apparent. This type of result can be seen in Figure

3.15. The reason for this ugliness is not incorrect background subtraction or a problem with

the LTE offset rate, rather, the problem is with

the eye.

In the typical ROSA T exposure the majority

of the pixels do not have detected photons. A

longer exposure has more photon-bearing pixels

than a shorter exposure. When scaling a shorter

exposure to a longer exposure, one increases the

values in the few pixels containing detections,

rather than increasing the number of pixels with

detections. Thus, even when a shorter exposure

is scaled to a longer exposure, the two will still

look very different. This is graphically demonstrated for a one-dimensional detector in Figure

3.16.

The solution to this problem is to increase the

size of the pixels. If, on the basis of aesthetics,

one needs to increase the pixei size, then one

probably needs to increase the pixel size for scientific reasons, namely, in order to gain enough

counts per pixel to have reasonable statistics.

REFERENCES

Press, W. H., Flannery, B. P., Teukolsky, S. A., &

44
Fig. 3.15.--Anexampleof a "raw"mosaicwhere the count rates are continuous over exposure

boundaries, but scaling of "sparse" images reveals the exposure edges. Top Left: A map of the

logarithm of the exposure; higher exposure regions are darker. Top Right: The "raw" mosaic, Note

that the lower exposure regions appear "rougher" than higher exposure regions. Bottom Left: The

mosaic formed with a sufficiently large pixel size (4 × 4 the size of the "raw" mosaic) that every

pixel has nearly the same uncertainty. Note that there is structure in this field, but it is not related

to exposure boundaries. Bottom Right: The uncertainty for the previous mosaic.

45
_9

0.5

0.0 . . . : .... : , :

_ 1.0

0.5

0.0 ....

0 2 4 6 8 I00 2 4 6 8 I0

Pixel Pixel

Fig. 3.16.-- One dimensional demonstration of

the effectof scaling on sparse images. Top: Two

one-dimensional "images" with the same count

rate; the left image has twice the exposure as

the right image. Bottom: The two images scaled

to the same exposure time. Note that the two

images will look very different despite the fact

they have the same count rates.

Vetterling,W. T. 1986, Numerical Recipes

(Cambridge University Press)

Raymond, J. C., & Smith, B. W. 1977, ApJS, 35

Snowden, S. L. 1998, ApJS, 117, 223

Snowden, S. L.,Egger, R., Finkbeiner, D., Freyberg,

M. J.,& Plucinsky, P. P. 1998, ApJ, 493, 715

Snowden, S. L., et al. 1995, ApJ, 454,643

Snowden, S. L., McCammon, D., Burrows, D. N., &

MendenhaU, J. A. 1994, ApJ, 424, 714

Strang, G. 1976, Linear Algebra and itsApplications

(Academic Press)

Voges, W., et al. 1996, http://www.rosat.mpegarching.mpg.de/survey/rass-bsc/

46
Chapter 4: Tracers of Absorption

Abstract

This chapter explores the problem of finding an adequate tracer of the entire Xray absorbing column. X-ray absorption is produced by H and He (for X-rays with

E < 1⁄4 keV) and metals (for X-rays with E > _ keV) distributed through all the

phases of the ISM. As dust is likely distributed th_rc igh'all but hottest phases of the

ISM, and a high-resolution all-sky survey exists, 100 pm emission by dust has become a

commonly used tracer of the X-ray absorbing column. The IRAS 100 pm emission has

become even more attractive as a tracer since the destriping, recalibration, and dust

temperature correction by Schlegel, Finkbeiner, & Davis (1998). However, there are a

number of well known (though not always well understood) systematic effects that make

100 pm emission less than an ideal tracer, including unresolved temperature effects in

neutral gas, and very uncertain contributions by H I1 to either 100 /Jm emission or

X-ray absorption. We review the current understanding of the problem and attempt

to quantify the uncertainties.

1. Statement of the Problem

When observing X-ray emission from objects

beyond the Local Cavity one typically wants to

correct for the effects of the absorption. At

!4 keV absorption is primarily due to hydrogen and helium, with smaller contributions from

oxygen and iron. The relative contributions of

the various elements in a completely neutral

atomic gas is shown for three ROSAT bands

in Figure 4.1. The amount of absorption is, of

course, quite sensitive to the ionization states,

but much less sensitive to chemical state, i.e., C

and O versus CO. It is generally assumed that

the abundance of these absorbers is traced by

the abundance of H in its three phases, H II, H I,

and H2. Ideally, then, determining the absorbing column would be accomplished by making

a map of H in each phase, and correcting those

maps for the change in absorption cross-section

due to the mean ionization or chemical state of

the absorbing species related to each phase of H.

Two problems are immediately apparent; first,

metallicity varies (and thus the number of X-ray

absorbers per H atom), and second, although H I

is well traced by 21 cm emission, no all-sky highresolution maps (better than 30') exist of any of

the phases of H.

For such reasons, use of the IRAS 100 pm

emission is particularly appealing. The IRAS

maps, particularly after the destriping and crosscalibration work of Sch]egel Finkbeiner, & Davis

(1998), provide a high resolution (6') all-sky, map

of a quantity that might trace all three states of

H. Relying on the dubious assumption that the

dust to gas ratio is nearly constant , one is left

with the complication of finding the dust emir,-

0.70_¢¢'¢¢¢¢¢¢¢_¢

0.60

0.50

._ 0,40

_ o.ao

-II ee 0.20

0,10

0.00 L

H He

i"i

!-!

, !

I I

i i

i i

i

! ! ,...,

N 0 NeNaMgAI Si S CI ArCaCrFeNi

Element

Fig. 4.1.-- The relative elemental contributions

to the absorption cross-section, solid: The relative contributions at 1⁄4 keV. dotted: The relative contributions at _ keV. dashed: The relative

contributions at 1.5 keV. Note that the bands

are not dominated by single elements that differ from band to band. Thus, the broad-band

ROSAT data is not useful for diagnosing elemental abundances.

47
sivityperH atomfor eachof thethreestates,

aquantitydependinguponavarietydusttemperaturesalongthelineofsight.Theproblem

iscomplicatedbythefactthat,unliketheH I

andH11data,theIRAS data is devoid of velocity information, thus complicating the task of

identifying the contributions from the different

components. Despite the seemingly Sisyphaen

nature of this task, the following sections explore

the extent to which the IRAS 100 pm emission

can be used to trace the X-ray absorbing column.

We will first consider the extent to which the

IRAS 100 pm emission tracks H I in regions that

ought to be free of contamination by H2. As that

correlation is quite strong, we then consider the

extent to which the H II may or may not be

traced. We then consider IRAS 100 pm emission

in regions with contributions from H2, and in

particular, the extent to which the temperature

dust correction of Schlegel, Finkbeiner, & Davis

(1998) may be deemed reliable for the purpose

of tracing the X-ray absorption column.

2. Dust Emission and ISM

Consider the components to the 100 pm signal, measured over some limited region, as a

function of neutral hydrogen column, NHI, as

schematically depicted in Figure 4.2. There is

a constant component due to the Cosmic Infrared Background (CIB) which will not vary

with NHI. To this component can be added the

contribution due to H I, which, over most of the

high Galactic latitude sky, is the largest contribution. The slope of the relation will be the

dust emissivity per H atom, et. This slope will

depend upon the dust temperature, but for the

moment we will assume that the 100 pm emission has been corrected for dust temperature. Of

the H !1, some fraction, f ,_30%, is correlated to

the H I (Reynolds et al. 1995), and the remainder is not. If the H II has an emissivity of eii,

then the relation between NH and I100 will have

a slope el + NHl-_mf_ll, while there will be

the added offset, (1 -f)NnHeH, and increased

scatter in the Iloo-Nnl relation. Finally, above

some threshold, there will be the contribution

of H2, with an emissivity e2. The threshold over

which H2 may begin to appear will depend upon

the local H I density and the radiation field, and

estimates based upon the column density range

from 3 x 1020 for high Galactic latitude fields

(Reach, Wall, & Odegard 1998) to 5 x 1020 in

general (Savage et al. 1977). The amount of H2

on a given line of sight is, of course, but poorly

correlated with H I, so points may fall within a

wide region.

This pedantic exercise demonstrates the only

tool (currently existing) for diagnosing the IRAS

100 pm emission: deviations in the Ixoo-NHl relation. In the following sections we will make use

of this schematic to understand limits placed on

emissivities, column densities, and temperature

deviations.

2.1. Review of SFD

Schlegel, Finkbeiner, & Davis (1998) (henceforth, SFD) created an optical extinction map

based upon the IRASand DIRBE 100 pm maps.

The DIRBE map is well calibrated, but has poor

resolution (,-, 0.75°), while the IRAS map has

calibration problems, but much higher resolution (--, 6'). Beyond destriping the IRAS maps,

SFD adjusted the IRASmap by both a scale factor and an offset so that the IRAS map, when

convolved to the DIRBE beam, would have the

same value as the DIRBE 100 pm map. Thus,

they obtained a 100 pm map with the IRAS resolution, and a calibration nearly as good as the

DIRBE map.

To determine the relative dust column, they

determined the 100pm/240pm band ratio for

each pixel in the DIRBE map and used a u s

emission model to determine the dust temperature from that ratio. They then determined a

scaling factor XT between the 100 pm emission

at 18.2 K and the 100/_m emission the temperature of the dust. The value XTIloo is the amount

of 100 pm emission that would be produced if all

the dust were at 18.2 K, i.e., a quantity proportional to the total dust column.

There are two important caveats to be observed. First, SFD did their own Inter-Planetary

Dust (IPD) removal from the DIRBE data set

which is not quite so sophisticated as that produced by Kelsa]l et al. (1998). The DIRBE

maps produced by Kelsall et al. (1998) used a

sophisticated model of the IPD that relied upon

the movement of the spacecraft to produce a

time-varying IPD signal. The time-varying portion of the signal in any band could then be re48
12

1o CIB+HI+HII+H,

8

CIB+HI+HII

6 CIB+HI+Hn(c)

CIB+HI

4

0 2 4 6 8

N./lo" (era")

Fig. 4.2.-- A schematic depiction of the source of the structure in the lloo-NH1 relation. We have

assumed no temperature variation in the dust associated with H I. The emissivities and relative

offsets are reasonably accurate. The shaded region contains the expected distribution of observed

points.

moved with minimal effect upon the absolute

zero-point. As determining the CIB was the

object of this mission, this method was ideal.

SFD, conversely, were interested in isolating the

contribution due to the Galactic ISM, and their

method explicitly removes the CIB. They assume that the spatial distribution of the IPD

is well represented by the 25 pm emission, and

the spatial distribution of the Galactic emission

(in regions without contributions from H2) is assumed to be well tracked by the H I, as measured

by Leiden-Dwingeloo survey (Hartmann & Burton 1997). The magnitude of the IPD emission

to be removed from the emission map at a band

b is given by that scaling of the 25 pm emission

which will minimize the scatter in the relation

between band b and the H l emission. As can

be seen from SFD Equation 3, a constant offset

is also removed in this process 1. The resulting

map will then include only that emission which

has the same spatial distribution as the Galactic

HI.

l Equation 3 from SFD is

D_ = Db - {Ab + QbD2s(/3))D25 - Bb (4.1)

where bold face indicates images, and subscripts, bands.

The quantity D25 (_3) is the mean 25 /zm flux as a function of _, ecliptic latitude. The quantity Db is the original DIRBE image in band b, and D Q is the quantity

remaining after the IPD removal process.

The offset removed from DIRBE 100 pm image, B100 = 0.62 + 0.04 MJy sr -1, compares

favourably with the CIB value found by Arendt

et al. (1998), 0.66 4- 0.017 MJy sr -_. This offset contains (at least) three components; 1.) the

CIB which SFD measure 2 as < 0.5 MJy sr -1,

2.) the constant term of the 25 #m emission

map, though this is likely small 3, < 0.12 MJy

sr -1, and 3.) the contribution of the uncorrelated H II. This last contribution should be quite

small as, overall, the H 11 should have a cscb

distribution similar to the overall distribution of

HI.

The second caveat is that the dust temperature correction is calculated from the ratio of two

differently weighted means 4 over a given line of

2SFD use a linear regression (as opposed to the quasiquadratic regression in their Equation 3) to measure the

CIB from all data fulfilling the condition that f_ > 300

and Nit < 200 K km s -1.

3The following quantity is derived by assuming that D25

is formed of two components, the IPD and a constant

background. The constant flux is taken to be 192 MJy

sr -1, that measured by Hauser et al. (1998) in "High

Quality Region B". This quantity was then propagated

through SFD Equation 3.

4That is, emission in a given band will be

o °_ B_,(T(l))d[ (4.2)

where Bv is the Planck function. Thus the emission

is a given band will weighted by the emission from re49
1.5

M

:/

z

0.5

+

0.0 . . * _ I . . I , i , , I i , i

5.0xlO zO I.OxlO II t.6xiO 21 _*.Ox 102]

N{H.m)

Fig. 4.3.-- The ratio of X-ray detected gas to

atomic gas plotted as a function of the atomic

gas. N(H,X) is the amount of gas setected by

X-ray absorption; N(H,HI) is the amount of gas

detected by 21 cm radio observations. Note that

the data points are "paired" due to there being two X-ray measures per background source.

This is a recreation of the Figure 2 in AB97,

whose data were kindly provided by the authors.

sight. Any single line of sight will sample several

different dust temperature environments. Dust

near the disk will sample its local environment,

which will change rapidly with position due both

to the localized nature of the heating sources and

the patchy extinction. Dust at higher z will be

subject to a more uniform radiation field, and

large-scale variation is expected due to, for example, the Galactic bulge. (See Sodroski et al.

(1997) for a detailed three-dimensional model of

dust temperature.) It is not clear to what extent

sampling different dust environments along the

line of sight will effect the dust temperature corrections (a subject far beyond this work) but it

should be taken that the dust temperature correction will, by its nature, be no more than an

estimate.

2.2. H I

In order to isolate as much as possible the

contribution of H I to 1100, we restrict ourgions with the temperature at which dust will emit most

strongly in that band.

selves to column densities below the threshold

above which H2 contamination might be expected. The value of that threshold is rather

an open question. Savage et al. (1977), using Copernicus absorption line spectra of H2,

noted that at columns of 5 x 1020 that the molecular fraction went from negligible to 0.1. Of

course, their lines of sight were primarily in the

disk, so one might expect a contribution to the

H I column density by gas not at all associated with the H2 clouds, thus raising the threshold. Boulanger et al. (1996), when plotting

NHI versus I100 noted that below T = 250 K

km s-1 (,,_ 4.5 x 1020 cm-2), that the relation

was fairly tight, but over that value, the dispersion increased, presumably due to contribution

from H2. Arabadjis & Bregman (1999), in a

study comparing the X-ray absorbing column,

Nx, with NHI in high b lines of sight noted a

fairly tight correlation between Nx and NHI below ,_ 5 x 10-20 cm -u, but an immediate and

substantial increase in the dispersion over that

value (see Figure 4.3). Reach, Wall, &: Odegard (1998), using the formula in Reach, Koo,

& Heiles (1994), suggest that at high b that the

threshold is more ,-, 3 x 102° cm -2.

Restricting ourselves for the moment to H l

below the most restrictive threshold, we turn to

Figure 4.4. Here we have plotted Iloo/N_z in

the top panel for that part of the northern hemisphere with b > 40 °. The 21 cm data were taken

from the Leiden-Dwingeloo survey (Hartmann &

Burton 1997) and has a 30' resolution. The/100

data was taken from SFD and was smoothed to

approximately the same resolution.

If we disregard those regions with NHZ >

3 x 1020 cm -2 (the strong white contours) we

see that there is still a great deal of variation

in the ratio. Reach, Wall, & Odegard (1998),

using DIRBE data binned to 10 ° cells, found

such variation to be greater than the statistical

uncertainty, and, on the basis of 100 pm/240

pm colors, suggest that the scatter in the ratio is due to temperature variation. The strong

patterning on many scales supports this conclusion. Comparison of Figure 4.4 with the bottom

right panel of Figure 4.5, which incorporates the

SFD dust temperature correction, suggests that

whereas temperature has some effect, it does not

account for the spatial pattern of the variation

(unless, of course, there is some serious problem with the SFD dust temperature correction,

5O
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Fig. 4.4.-- Theangularvariationof theIloo/NHt ratio for the northern Galactic hemisphere.

Iloo/NHI in Galactic coordinates for tile north Galactic pole. 0° is at bottom with g increasing

counter-clockwise; Ab = 15 °. The highest gray-scale value is 1.4 MJy sr-1/102° cm -2. The NH1

is from the Leiden-Dwingeloo survey, and the column is summed over -100 km s -1 < v < 100 km

s -1 and represents "disk" gas. The thick white contours mark NH1 ---- 3 x 10-'° cm-2; the light

white contours mark NHI = 2 × 1020 cm-2; the black contours mark the Intermediate Velocity

Arch (Kuntz _z Danly 1996) in -100 km s-1 < v < -70 km s -I at NH = 1019 cn1-2. Tile bright

region at (t', b)=(316 °, 51 °) is the H II region surrounding Spica.

51
0.0 0.2 0.4 0.6 0.8 ! .0

Fig. 4.5.-- The angular variation of the Iloo/NHi ratio for the northern Galactic hemisphere. Top

Left: The "disk" gas. The highest gray-scale value is 10 -°1cm --_. The contours are as in the previous

figure. Top Right: The Intermediate Velocity Arch. The highest gray-scale value is 1.5 × 10-_°em --°.

The contours are as in the previous figure. Bottom Left: The SFD dust temperature correction

factor -_'T. The highest gray-scale value is 1.4. Bottom Right: The Ilo0/NHl(disk) multiplied by

the SFD dust temperature correction factor. The gray-scale is the same as for the previous figure.

52
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Fig. 4.6.-- The angular variation of the Iloo/NHI ratio for the northern Galactic hemisphere. The

panels and scales are as in the previous figures.

53
0.0 0.2 0.4 0.6 0.8 1.0

Fig.4.7.--TheangularvariationoftheIloo/NHi ratio for the southern Galactic hemisphere. The

panels and scales are as in the previous figure.

54
whichit wouldbefaroutsideofthescopeofthis

chapterto attempttodiagnose).

FromFigure4.4,onecan see that the Iloo/NHI

ratio has some dependence upon NHI, though

this is suggested in only some regions and does

not explain, for example, the low values in the

fourth quadrant, or the low "trough" between

1700 and 180 °. The low "trough" can, in fact

be explained by the presence of intermediate velocity gas 5 between -100 km s-1 < v < -70 km

s-1. Removing this intermediate velocity gas

from the NHI column improves the Iloo-NHt

relation in those very limited regions with intermediate velocity gas (see the lower right panel

of Figure 4.5), but does not account for much of

the variation.

2.3. H 11

In the canonical model of the H 11 distribution (Reynolds 1991), the scale-height (910 pc)

and planar density (0.025 cm -s) of the H II has

been determined from pulsar dispersion measures, which give a direct measure of n_ along

the line of sight. These measures imply that

NttH/Nxl "" 0.33. In the canonical model,

NH_tl/NHe "_ 0.5 (DomgSrgen & Mathis 1994),

thus reducing the absorption cross-section (with

respect to that of neutral gas) to 0.62. Thus,

H II accounts for about 20% of the X-ray absorbing column. Using the canonical model, Arabadjis & Bregman (1999) calculate C =_ Nx /NH! ",

1.23.

There have been two recent claims that the

H ll columns are much smaller. Both Laor et ai.

(1997) and Arabadjis & Bregman (1999) claim

that the H I accounts for all of the absorption seen in the spectra of extragalactic objects.

SThis particular part of the IV gas defines the backbone

of the Intermediate Velocity Arch (Kuntz & Danly 1996),

and is morphologically related to both lower and high

velocity gas. The HVC gas has not been detected at

100 #m despite careful searches (Wakker & Boulanger

1986). Thus, it is not surprising that the intermediate

velocity gas might be deficient as well. It is far from clear

whether the high velocity clouds are deficient in dust due

to grain destruction or whether the temperature of the

dust is such that the emission at 100 _m is too small to be

teased out of the much larger signal due to the disk gas.

The deficiency is not a deficiency in the materials from

which to make dust; in detecting high velocity complex M

in absorption with IUE, Danly, Albert, & Kuntz (1993)

showed that at least some of the high velocity clouds are

not metal deficient.

Arabadjis & Bregman (1999) in particular, measure the ratio Nx/NHI (see Figure 4.3) for thirteen lines of sight with Nt-tl < 5 x 102° cm -2

and find the mean Nx/NHI = 0.972 + 0.022;

their 99% confidence interval upper limit to C

is 1.023. There are two reasons for suspecting

this result. First, if there were no H II the physical lower limit of Nx/Nnt, C, must be unity. In

order for the uncertainties in Nx/Nnt to be consistent with this physical limit, they must be increased by a factor or 2.7. Increasing the uncertainty by this factor expands the upper limit of

C to 1.11, allowing for Nntl/Nnt "-' .18, which

is still low, but not so shockingly so. The published results of Laor et al. (1997) do not allow

a similar re-analysis.

Both Laor et al. (1997) and Arabadjis &

Bregman (1999) determined the X-ray absorbing

column through spectral fits; Laor et al. (1997)

using quasar spectra and Arabadjis & Bregman

(1999} using galaxy cluster spectra. Given that

these fits were made on objects with different

spectral types it is difficult to ascribe the results

to systematic problems with the fitting. Both

groups propose that their results suggest that

the He associated with the H II must appear

mostly in the form of He Ill in order to avoid

significant X-ray absorption. It is also suggested

that the H 11 observed by the Reynolds group is

not the same H I1 that produces the pulsar dispersion measures.

A full discussion of this possibility is beyond the scope of this work, but a few obvious

considerations can be made. Reynolds (1985)

measured a temperature of ._ 8000 K for local

(r < 1 kpc, b < 30 °) H II using the forbidden

line ratio IS Ill/Ha. For one not-so-local field

(r < 3.8 kpc, b -,- 20 °) (Reynolds et al. 1995)

a similar temperature was determined from Hc_

line widths. This field would suggest that the

spectroscopically observed H II is the same as

that seen in the pulsar dispersion measures, but

no significant conclusion can be drawn. Conversely, if the H II observed by pulsar dispersion measures has He Ill/He _ 0.5, it must have

iogT > 4.8 (collisional ionization equilibrium)

or logT > 4.1 {isochorically cooling gas) according to the calculations of Schmutzler & Tscharnuter (1993). The latter temperature is suggestive given that the cooling curve for an isochoric

gas has a minimum at logT = 4.0. In either

case, the gas is entering the regime where it may

55
beproducing X-rays that would "fill in" the expected absorption.

For a worst-case scenario, we will continue to

assume that the H [1 is described by the canonical model. For the canonical model some fraction of the H II, as measured by Ha, is correlated

with the H I. Thus, the amount of spatial variation in the component of the ISM that is traced

only by H II will contain only ,-, 12% of the Xray absorbing column. Of than portion, some

portion may be spatially variant, the remainder

lost in the uncertainty of the offset.

How well is this component traced by I100?

Boulanger et al. (1996), after removing the contribution to I100 by H I, searched for any csc b

dependence in the residuals (such as one might

expect from the uniform plane-parallel distribution of H If) and found none. Fixsen et al.

(1998) attempted to correlate the II00 with both

H 121 cm and [C Ill 158/Jm (as a tracer ofH I1),

and found little emission correlated with [C II]

(Arendt et al. 1998). The COBE team concentrating on the Galactic emission attempted to

determine the emissivity of the H II related gas

by fitting

11oo = _INHI + e11NHH + CIB (4.3)

where NHII was determined from some combination of Ha and dispersionmeasures for a

slectednumber of linesof sight.Their 3a upper

limitto EZl/EI= 0.75.This valuesuggeststhat

the totalcontributionby H IIto If00 is_<20_,

and the component that isuncorrelatedwith H I

willcontribute< 13%.

Thus, the upper limitto the contributionby

H IIto X-ray absorptionisof the same order as

the upper limitto itscontributionto If00,due to

the fact that, by chance, _11/_! *" all/_I, where

ao is the absorption cross-section of component

a. However, since these are upper limits, Ii00

may trace H 11 much more poorly than does its

X-ray absorption, or vice versa.

2.4. H2

Given all of the difficulties encountered below

the molecular threshold, it might seem foolish to

even attempt deab_orption for regions over the

threshold However, X-ray absorption by molecular clouds holds great promise for determining total molecular content (Snowden. McCammon, & Verter 1993; Herbstmeier, Heithausen,

& Mebold 1993; Mortiz et al. 1998; Kuntz,

Snowden, & Verter 1997), and such studies often use 100 pm emission as a tracer of the total

gas content. Thus, there are some valid reasons

for pushing into the realm of the absurdly uncertain.

The molecular gas can be detected at some

threshold in plots of I100 versus NHI as a spray

of points above the linear Iloo-Nnl relation established at even lower column densities (see

Figure 4.2). This effect can be seen in the plot

of Nx/Nnl versus Nnl from Arabadjis & Bregman (1999) (Figure 4.3 and Figure 4.8), where

the molecular column is seen as an excess of

Nx for NH -_ 4 x 102° cm -2. Using Nx as

a measure of the total column density, we plot

Nx versus Iloo in Figure 4.8. The dotted line

represents the best fit to the Iloo-Nx relation

for data points without molecular contamination; the points with substantial molecular components fall far above the line and have a substantial scatter around a fit to all of the data

(dotted line). It is well known that this is due

to 1.) a substantially different dust temperature

between neutral and molecular regions and 2.) a

substantial spread in temperatures between different molecular regions. SFD attempted to correct the Ii00 for dust temperature; just how well

did they do?

Our reanalysis uses the Nx data published in

Arabadjis & Bregman (1999). To make a consistent comparison between Ii00 and Xlloo, we reextracted the values of Ii00 from the SFD maps

and extracted the values of XT from the same

regions. A comparison of our value of I100 and

that used by Arabadjis & Bregman (1999) (and

kindly provided by Dr. Arabadjis) shows only

some minor differences, presumably due to different extraction/averaging procedures. The uncertainties used are those published for the Nx

and, for the I100, the RMS of the pixels within

the extraction region.

The fits to the Nx-Iloo and Nx-XTIIo0 relations were made using the BCES(Nx l(XT)ll00)

estimator ofAkritas & Bershady (1996), and the

uncertainties were estimated using the bootstrap

method 6. The results are shown in Figure 4.9.

When fitting to XTIloo, we do not use the valeThe BCES(X[Y) estimator finds the best fit to Y given

X under the assumption that both variables have uncertaintes and that both variables have intrinsic scatter.

56
40

3O

20

Z

lo

o

o

.... ....! I-

--,-....,..,_- , .......

1o 20 30

Iioo

Fig. 4.8.-- The amount of gas detected by X-ray absorption versus the SFD 1100 measure of the

dust column. The dashed line is the fit to points with N(H,X) < 4 x 102°, the dotted line is the fit

to all points. We have plotted boxes around the points to aid the eye in distinguishing those with

the smallest uncertainties.

ues directly extracted from the SFD data sets.

SFD warn that at high Galactic latitudes, where

the I100 emission is low (and by extension, the

I140 and 124o emission is small) the dust temperature correction factor, XT will be poorly determined. This is not disastrous, as these regions,

having low dust columns, are unlikely to have

a substantial molecular component, and so the

correction is likely close to unity. Therefore, for

the purposes of fitting, we have taken the mean

of the XT values for all points where XT < 1.3

and divided all the XT values by this mean. For

XT < 1.3 we have set XT "- 1.0. This removes

the large uncertainties for the XT values near

unity in a consistent manner. Since our mean

XT for XT < 1.3 was slightly greater than unity,

we will be slightly underestimating the XT at

higher values.

When fitting XTIlOO (in opposition to fitting just ll00), the difference in emissivity per H

atom between the fits using all of the data and

fits using data from below the NH threshold becomes less, and the 1 a uncertainty regions for

the slopes come closer to overlapping. Clearly,

the XT correction is an improvement, and one

can see that the dispersion of the data over the

molecular threshold has decreased. However,

the XT correction, despite the fact that we have

decreased its magnitude slightly to improve the

The bootstrap method of determining the uncertainty

in the fit parameters ta&es an arbitrarily large number

of subsamples of the data, fits each sub-sample, and

determines the statistical properties of the resulting fit

parameters.

signal-to-noise, is a bit too large. Clearly, increasing the data set will be necessary before

determining the correctness of XT.

The results of the fits are given in Table 4.1 to

demonstrate the above trend in a more quantitative manner. The parameters for fitting Nx =

A + BIloo using only data with Nx < 4 × 1020

cm -2 are particularly useful and will be used

in the summary to calculate the uncertainties

in equivalent X-ray absorbing columns from 100

gm emission.

3. Implications

A summary of the uncertainties in tracing the

X-ray absorbing column is given in Table 4.2.

1.) The 100 pm emission traces the H 1 very

well over many degree scales. However, dust

temperature effects limit its usefulness over area._ less than a few degrees, as there is clear

structure on scales down to the resolution limit

in maps of Iloo/NHz in regions with no molecular component. This structure may also be due

to H !i.

2.) In regions with high or intermediate velocity gas, the Ii00 measure will underestimate

th_ total column due to the well known dust

emission deficiency of high velocity gas. This

deficiency seems to exist for the higher velocity

parts of the Intermediate Velocity Arch, and furthor work will be necessary to see if it is true for

the" -.- -60 km s-1 gas as well.

3.) H Il is poorly traced by I100 as the emis57
+

0

0 I0 20

X_|ioo

30

Fig. 4.9.-- The amount of gas detected by X-ray absorption versus the SFD XT/'100 measure of

the dust column. The dashed line is the fit to points with N(H,X) < 4 x 102°, the dotted line is the

fit to all points.

sivity per H atom appears to be much smaller

in H II than in H I. The impact of this difficulty

may be ameliorated by substantial decrease in

the absorption cross-section of gas related to

HII.

4.) The SFD dust temperature correction

offers a substantial improvement in obtaining

rough absorption measures for regions with molecular components. The SFD dust temperature

correction does not adequately correct I100 from

diffuse neutral gas, perhaps due to the low emission values of such gas.

5.) Given that there is no all-sky H I survey

with resolution smaller than 30', the best projected survey of H II has a 1° resolution, and

surveys of H_ (through its CO surrogate) have a

1 ° sampling, the IRAS 100 pm survey remains

the only tracer on scales between 5' and 30'.

We will explore the efficacy of using the IRAS

100/zm survey on sub-degree scales in the following chapter.

REFERENCES

Akritas, M. G., & Bershady, M. A. 1996, ApJ, 470,

7O6

Arabadjis, J. S.. & Bregman, J. N. 1999, ApJ, 510,

806, submitted

Arendt, R. G., et al. 1998, ApJ, 508, 74

Boulanger. F., Abergel, A., Bernard, J.-P., Burton, W. B., D6sert, F.-X., Hartmaan, D., Lagache, G., & Puget, J.-L. 1996, A&A, 312,

256

Danly, L., Albert, C. E., & Kuntz, K. D. 1993, ApJ,

416, L29

Domg6rgen, H., & Mathis, J. S. 1994, ApJ, 428, 647

Fixsen, D. J., Dwek, E., Mather, J. C., Bennett,

C. L., & Sharer, R. A. 1998, ApJ, 508, 123

Hartmann, D., & Burton, W. B. 1997, Atlas of

Galactic Neutral Hydrogen (Cambridge University)

Hauser, M. G., et al. 1998, ApJ, 508, 25

Herbstmeier, U., Heithausen, A., & Mebold, U.

1993, A&A, 272, 514

Kelsall, T., et al. 1998, ApJ, 508, 44

Kuntz, K. D., & Danly, L. 1996, ApJ, 457, 703

Ktmtz, K. D., Snowden, S. L., & Verter, F. 1997,

ApJ, 484, 245

Laor, A., Fiore, F., Elvis, M., Wilkes, B. J., & McDowel], J. C. 1997, ApJ, 477, 93

Mortiz, P., Wennmacher, A., Herbstmeier, U.,

Mebold, U., Egger, R., & Snowden, S. 1998,

A&A, 336, 682

Reach, W. T., Koo, B.-C., & Heiles, C. 1994, ApJ,

429, 672

Reach, W. T., Wall, W. F., & Odegard, N. 1998,

ApJ, 507, 507

Reynolds, R. J. 1985, ApJ, 333, 341

Reynolds, R. J. 1991, in The Interstellar Disk-Halo

Connection in Galaxies, ed. H. Bioernen, 67

Reynolds, R. J., Tufte, S. L., Kung, D. T., McCullough, P. R., & Heiles, C. 1995, ApJ, 448,

715

Savage, B. D., Bohlin, R. C., Drake, J. F., & Budich,

W. 1977, ApJ, 216, 291

Schleget, D. J., Finkbeiner, D. P.. & Davis, M. 1998,

ApJ, 50O, 525

Schmutzler, T., & Tscharnuter. W. M. 1993, A&A,

273, 318

Snowden, S. L., McCammon, D., & Verter, F. 1993,

ApJ, 409, L21

Sodroski, T. J., Odegard, N., Arendt, R. G., Dwek,

E., Weiland, J. L., Hauser, IVI. G., & Kelsall,

T. 1997, ApJ, 480, 173

Walcker, B. P., & Boulanger, F. 1986, A&A, 170, 84

58
TABLE 4,1

FITS TO Nx = A + B(X)Itoo

Fit" A b B _ CIB a

Fit Fit

(Bootstrap)" (Bootstrap)

/too

NH <_ 4e20 -0.019 • 0,098 1.477 4- 0.079 -0.003 -t,- 0.139

(0.016 • 0.128) (1,490 • 0.096) (0.022 i 0.149)

NH < 5e20 -0,192 + 0,119 1,755 • 0,086 0.312 i 0,121

(0.072 • 0.259) (1.769 • 0.I00) (0.111 4- 0.316)

All -0.473 • 0.299 2.161 • 0.094 0.298 • 0.096

(0.355 • 0.481) (2.164 • 0.105) (0.102 • 0.304)

Xfl0o

NH ,_ 4e20 -0.019 + 0.098 1,477 • 0.079 -0.003 • 0,139

(0.012 • 0.109) (1.493 + 0.077) (0.034 -f 0.181)

]VH <_ 5820 -0.192 • 0.119 1755 • 0.086 0_312 • 0.121

(0.049 • 0.216) (1.758 + 0.093) (0.117 • 0.323)

All 0.825 • 0,242 1.318 • 0.039 -0.773 • 0.174

(0.708 4- 0.457) (1.329 • 0.054) (0.583 • 0.496)

aThis column lists which portion of the data was used for the fit.

b1020 cm -2

¢1020 cm -_ (MJy st-t) -l

dRes|dualCIBin the SFD 100pro data set MJy sr -l

eResult of the bootstrap analysis.

TABLE 4.2

UNCERTAINTIES IN EQUIVALENT ABSORBING COLUMNS

Source Full Field" Variation within Field b

cm -2 cm -_

(o_ginal units) (original units)

• 6.35 x 10 Is None

(•043 MJy sr -l)c

lloo Of[set

Columns not trtu:ed by /too

H H d

H I[¢

IV gas

t : Variation

H2

(I -- f)NHllol'ttl/ol.t!

= 0.30 X 10_° cscb

0.16 x 10a°cscb

Poeition Dependent

None as a local e I

c_ be determined from the data

None for lzoo < 2 MJy sr -l

(1 -/)NHuomdom

= 0.30 x 102°cscb

016 x 10Z°cscb

Calculable from NHt

TBD

Suspect areas cam be removed

"On the order of 10 ° x I0 °.

bScales 2 ° to 4 °.

<Calculated from the SFD uncertainty in the offset using the Nx/lloo rat,o 1 49 determined by the

fit to the Arabax:ljis & Bregma.n (1999) data listed in 'l_ble 4.1

dA_uming the canonical model of the H H layer

"Assuming the low H H absorption suggested by Arabadjis and Bregman (1999)

59

Chapter 5: The Spectrum of the SXRB

Abstract

ThesoftX-raybackgroundinthe0.1-1.0keVbandisknownto beproducedbyat

leastthreesources;theLocalHotBubble(LHB),theextragalacticpowerlaw(EPL),

andaseeminglyGalacticcomponent thatliesoutsidethebulkof theabsorptionthat

isdueto theISMoftheGalacticdisk.Thislastcomponent, whichwecalltheTransAbsorptionEmission(TAE),hasbeenmodeledbyanumberofgroupswhohavederived

disparatemeasuresof its temperature.Thedifferenceshavearisenfromdifferingassumptionsaboutthestructureof theemittinggasandunrecognizedmethodological

difficulties.

Wefindthatspectralfitting methodsdonotuniquelyseparatetheTAEfromthe

foregroundemissionthat isduetheLHB,buttheangularvariationof theabsorption

canbeusedto breakthisdegeneracy. However,large-scaledeabsorptionstudiesthat

haveusedthis variationto determinethestrengthof the TAE have all had serious

flaws, most notably, the lack of sufficient energy coverage.

I. Introduction

The canonical model of the soft X-ray background (0.1-1.0 keV) posits a Local Cavity (in

the H 1 of the Galactic disk) that is filled, at

least in part, by hot gas emitting mostly in the

i

keV band. This region of X-ray emitting gas

is known as the Local Hot Bubble (LHB). The

LHB emission is absorbed by the nearby very

tenuous Local Interstellar Cloud (LIC), otherwise known as the Local Fluff. Beyond the Local Cavity there are thicker layers of absorbing gas which partially absorb emission from an

isotropic extragalactic component and emission

from at least one other component. This latter component we will call the "trans-absorption

emission" (TAE) or the "distant component".

The local component and the distant component are usually modeled as thermal plasmas.

The isotropic extragalactic component, which is

most clearly seen at E > 1 keV, and is presumably due to the accumulation of unresolved

point sources, is modeled as a power-law, and

will be referred to as the Extragalactic Power

Law (EPL). The distant component is required

to contribute to the observed excess over the extrapolation of the EPL in both the 1⁄4 keV and

a keV bands.

Table 5.1 contains a compendium of measures

of the temperature of the thermal components

using various techniques. The ROSAT mensures alone show a wide dispersion. In particular, PSPC spectral fitting produces lower LHB

temperatures and higher halo temperatures than

do early broad-band measures and PSPC broadband deabsorption measures. In order to understand the source of this difference, the next section assumes the canonical model and explores

the limitations of the methods used for determining the parameters of that model.

We find that although a portion of the difference is due to inadequacies in the spectral fitting

method, the bulk of the difference is due to assumptions about the nature of the distant component. Most investigators have assumed that

the TAE is due to a single-temperature Galactic

halo. We find that none of the existing singlecomponent models satisfies all of the constraints

placed upon the data by the angular variation of

the absorption of the TAE.

2. Analysis of Previous Results

Using the canonical model described above,

several efforts have been made to determine the

spectrum of the soft X-ray background. We divide these methods into two groups; spectral fitting of individual PSPC pointings, and all-sky

deabsorption of broad-band images. In both of

these cases, the data were fit to the function

fO : IL e-(N1)a(v'N1's)

61
TABLE 5.1

COMPENDIUM

Source TL TDI TD2 Method"

keY log T keV log T keV log T

Pre-ROSAT Results

Bloch eta]. (1986) 0.077-0.097 5.95-6.05 ...... BB

3uda et ai.(1991) 0.054-0.136 5.8-6.2 ...... BB

Snowden et al, (1990) 0.086 6.0 ...... BB

ROSAT Era Results

Chen, Fabian 0.05 5.76 0.12 6.14 SF

& Gendreau (1997)

Kerp(1994) n nF;I +°'°16 5.85 4- 0.1 n 179 +0.045 6.3 4- 0.I

..... --0,015 ..... --0.035

Miyaji et al.(1998) .057-.09 5.82-6.02 0.142-0.148 6.217-6.233

Peitz et al,(1998) b ...... 0.135 6,195+0.05 b

Sidher(1996) ....... 0.137(0.216) 6.2 (6.4) c

Snowden et al.(1998) d N 119 +0.034 fi l&+O.1lef& n lOi_ '+0.029 fi 0_ ÷O'Ose ..... --0.01 ") .... --0.08 ..... --0.0IS .... --0,08

Snowden et al.(2000) b n 110 +0.034 _ ld ÷0"tier8 n lf_N +0'033 _ N7 "I'0"120

..... --0.024 .... --0. I0 ..... --0,030 .... --0.16

DA

SF

SP/DA

SF/DA

DA/BB

DA/BB

ROSAT Results - This Work

Hyades n 1ha +°.°°2 _ n_+o.o,i ...... BB ..... -0.002 .... -o,o]

n INR +O'00s /_ 1| +O'03ei Galactic Plane ..... -0.007 .... -o,os ...... BB

Chap. 6 §6.1 n |IR +0"Oss _ 11 +0"lEer n n_R +0"Oss K n/_ +0lee N 9Kn +0 o79 _ 4K+°'120 DA/BB ..... --0.023 .... --0.07 ..... --O.OSY .... --O20 ..... --0 045 .... --O.0S

Chap 6 §6.1.4 0.108 4- 0,04 6.10 4- 0.02 0.087 4- 0.003 6.00 4- 0.02 0 180 4. 0.008 6.32 4. 0.02 SF

Chap. 6. §61.4 0.1274.0.010 6,174-0.03 0.3134-0.055 6.564-0.07 SF

Post-ROSAT Results

Psrmar et al.(1999) k 0 137 +0"0]I fi _)÷0.o3 /_ 7+0"4 _ gt+0 20k SF ..... --0.022 --" ---0.07 --'----0.3 .... --0.94

"BB - Broad-Bared Ratic_, SF - Spectral Fitting, DA - De-Absorption

bTemperature derived from the Mewe-Kanstra (Mewe, Gronenschild, & van den Oord 1985) code. A fit of a Raymond ,t, Smith

(1977) spectra to the Mewe-Kaastra spectrum abeorbed by 1.5 x 1090 cm -a and folded through the ROSAT PSPC response yielded a

Raymond & Smith plasma temperature of log T = 6.18. If a Mewe-Kanstr_-Leidah] spectrum were used rather than a Mewe-Ksnstra

spectrum, the Raymond & Smith plnsma temperature would be logT = 6.17.

eTemperatures derived from the Lsndini Jr, Monaignori-Foasi (Lsndini & Monsignori-Fo_i 1990) code Sidher et al. note that

Raymond & Smith code would produce a log TD = 6.4.

dA second, hotter component is implied, but not explicitly characterized in this paper. The values given here are not quite the

values given in the original paper Those v_lues had been calculated from band ratios using an older detector response matrix; we

have used the current detector response matrix.

"Cited range is the FWHM of the distribution. These distributions ere dominated by fit and/or data uncertmnties; the distribution

of temperatures is very narrow. See text §4 for a fuller explanation.

rThe upper limit is not given by the upper FWHM of the distribution but by the highest R2/RI ratio permitted by an unabsorbed

Raymond & Smith spectra

|These LHB temperatures have not taken into account atmorption within the LHB Auummg an absorbing column ,_ 4 x 10 Is

cm -2, the temperatures drop by AlogT _ 0.01

_Thls paper mumed a second, hotter component with logTD3 _ 6.4.

'This temperature was derived in the anti-center. Snowden, Schmitt, & Edwards (1990) showed that the temperature of the LHB

depends upon Galactic longitude, with Galactic center directions having slightly higher temperatures than the Galactic anti-center

"Result obtaxned with the BeppoSAX LECS. The temperatures were derived using the Mewe-Kaastra-Liehdal code. These results

should not be compared to the current work because the Parmar et al. (1999) data selection does not well sample the Galactic

halo and the fit methodology does not consider some known physical constraints. The fitted spectrum was the sum of spectra from

many Iocat,ons on the sky All of the spectra were from relatively low Galactic latitude, most have absorbing columns > 3 x 109°,

making the detection of the soft TAE component difficult When the spectra were fitted ind,vtdually, the TAE component was fixed

to I%gT -- 69, rather than being fitted in a simultaneous fit The foreground absorption used for the LHB component, 7 x l0 ts

cm--, is unreasonably high

62
+[De -( IV6)a(v'Nb'8) (5.1)

_{_IEe--(N6)a(t_,Nb,S)

where Io is the observed X-ray flux, IL is the

local (LHB) X-ray emission which may be absorbed by an intervening layer of column N/,

ID is the distant emission (TAE) and IE is the

contribution due to the EPL. Both the distant

and extragalactic components are absorbed by

a column Nb. The quantity a is the absorption cross-section. In the case of spectral fitting, a is dependent only upon energy (v), but

for broad-band deabsorption we use the effective

cross-section for a given band which is dependent both upon the magnitude of the absorbing

column (N), and upon the spectral shape of the

background emission (S).

2.1. Spectral Fitting

Several groups (e.g., Chen, Fabian, & Gendreau 1997 and Miyaji et al. 1998) have attempted to fit the spectrum of the X-ray background as observed in individual PSPC pointings. Attention has focused upon determining

the index of the EPL; determination of the temperatures and normalizations of the two thermal components has been incidental. Due to

this focus, the ROSAT PSPC data have usually

been supplemented with ASCA observations of

the same field in order to gain a greater lever arm

for fitting the EPL. Fitting the thermal components is more problematic.

The thermal components occupy only a very

limited spectral region in the ROSA T range; the

emission lies mostly within 0.1-0.4 keV, which is

completely outside of the ASCA range. At these

energies, absorption, and the uncertainty due to

its correction, is much greater. Also greater are

the non-cosmic contaminants; the Long Term

Enhancement (LTE) and the Scattered Solar Xrays (SSX). The LTE is a particular problem

for PSPC pointed observations as adequate removal requires a light curve extending over (at

minimum) several weeks to months (Snowden

etal. 1995; Snowden et M. 1994). Much of the

disagreement between different spectral fits has

been attributed to inadequate non-cosmic background removal (Miyaji et al. 1998).

Inadequate spectral resolution is also a problem. There are only about five independent

spectral resolution elements in the ROSA T PSPC

250 L

200 ER IR2 R4 R5 R6 R7

100 _/_I

0.s[ i R_s lDgr = 5.s5

0sI [i

0.4 ::

0.2 :

0.0 I m

x 1.0' ! '

"q o.a i R/kS log T -- 6.05

_o._: i

_ ._,t ): i : i

i I i! RkS io_ r = s._s

0.6 _i 0.,I 0.2 i I

o.o ..J _ _.. :.. i

0.5 1.0 1.5 2.0

Enerly (keY)

Fig. 5.1.-- A comparison of the spectra of thermal components with the available independent

spectral elements for the ROSAT PSPC. Top:

the ROSA T PSPC band response functions. Below: Three thermal spectra spanning the range

of temperatures found in the literature. The vertical lines show the divisions between independent spectral resolution elements, i.e., elements

with non-overlapping FWHM. The bulk of the

emission from Galactic thermal components falls

almost entirely within two independent spectral

resolution elements.

energy band, and only three spanning the energies emitted by the thermal components. Thus,

even given some a pr/or/model of the EPL, the

four fit parameters necessary to describe two

thermal components are under-determined. In

practice, the situation is even worse. Figure 5.1

shows the Raymond & Smith (1977) spectra for

three representative thermal components, and a

maximal set of independent resolution elements.

(As with all future references to the Raymond &

Smith spectra in this paper, except in the context of XSPEC, we are using the 1991 release

with cosmic abundances.) The bulk of the emission is contained in ,,- 2 resolution elements.

At the risk of appearing to labour the point,

consider the following. The six standard ROSAT

63
6.25 _5 o "

6.20 t =_

6.15 _ >

6:a5 R4,R5,R6 .?_..',' 1

1

,5.70 ,5.80 5.90 6.00 6.10 6.20 6.30

1o£ 1",

Fig. 5.2.-- The degeneracy in the thermal

component fit parameters. We have used a

"mean" high Galactic latitude background spectrum with a mean absorbing column (Nb) of

1.21 x 1020 cm -2 (and N! = 0), have assumed a Chen, Fabian & Gendreau (1997) EPL

(their Model A), and have assumed a singletemperature distant component. Top: The

range of normalizations required for the local

emission component, in units of 1000 nenpL (he

and nh in cm -3 and L in pc), as a function of

the local and distant component temperatures.

Middle: The same as above for the distant component. Bottom: The region of parameter space

allowed by the R4. RS, and R6 band fluxes,

allowing a +Sx 10-6 counts s-z arcmin -_ uncertainty in the fluxes. Sohd: R4, Dotted: I_5,

and Dashed. R6. Results are similar for other

EPL models, though no EPL model can force

the three regions to have significant overlap.

PSPC bands RI-R7 (the RI and P,2 bands forming the 1⁄4 keV band. the R4 and R5 bands forming the a_ keV band. and excluding R3 as usual,

Snowden et al 1995) contain all of the information in the PSPC spectrum. The bulk of the information about the thermal components is contained in the R1 and R2 bands. Using only the

R1 and R2 bands, given the absorbing column

and the EPL, one can choose (within reason)

any pair of local and distant temperatures, TL

and TD, and find some pair of normalizations

NL and ND that provide the observed R1 and

R.2 band values. Figure 5.2 shows the ranges of

normalizations required over a reasonable range

of TL and TD z.

Also shown in Figure 5.2 is the region allowed

by the R4, RS, and Ft6 band values. The horizontal branch of the allowed parameter space

represents a "minimal LHB" model, where the

1 keV flux is dominated by the distant emission;

the LHB component is then poorly constrained

and can assume a broad range of values. The

vertical branch of the allowed parameter space

represents a "maximal LHB" model, where the

z keV flux is dominated by the LHB; the distant

emission is then poorly constrained and can assume a broad range of values. It should also be

noted that the vertical branch falls at the TL at

which the R2/R1 ratio for the local component

is a maximum (see Figure 5.3.)

Most of the published parameters of the local and distant thermal components fall in the

region allowed by the R4 flux.

The "degeneracy" in the fit parameters can

be broken with the addition of angular information; if Nb varies over scales smaller than the

angular scales of the variation in the emission,

then the local and distant normalizations can be

determined, and from them, the temperatures.

It is not our desire to single out any particular

practitioner of spectral fitting for criticism, especially since the thermal parameters were not

their prime objective. However, one can easily

show that the thermal components fit, for example, by Chen, Fabian, & Gendreau (1997) are

incorrect. Taking a small strip of the RASS in

the southern Galactic hemisphere that is free of

strong 1⁄4 keV structures (t = 165 °, -55 ° > b >

-75°), and assuming that the temperatures of

the local and distant emission do not vary with

position, one can solve for the normalizations as

a function of position. A plot of the normalization as a function of absorbing column density,

Figure 5.4, shows a strong correlation between

the absorbing column density and the normalization of the distant component. One should

ZThe data used in the construction of Figure 5.2 were

regions with b > 45 °, 45 ° < ! < 270 °. and 1.1 x 102° <

NH < 1.3 x 10 2° cm -_.

64
1.5

1.0

0.5

,,i ...... J......... i ......... i ......... i ......... t ......... i ........

5.70 5.80 5.90 6.00 8.10 6.20 6.30 6.40

LOg T

Fig. 5.3.-- The R2/R1 ratio as a function of

temperature. The top curve is the R2/R1 ratio for the distant component after absorption

by a "mean" high Galactic latitude column of

1.21 x 10_° cm-2; the bottom curve is the ratio for the local component. The horizontal line

is the ratio for the "mean" high Galactic latitude background. Note that since the local

component has an R2/R1 ratio less than the observed R2/R1 ratio in all cases, the distant component must have a temperature greater than

log T ,-, 5.95.

not expect the emission from beyond the absorbing layer to be correlated with the intervening

column density of H I. Therefore, the normalization must be wrong.

Indeed, repeating the exercise using the parameters determined by Snowden et al. (1998),

shows no correlation between the absorbing column density and the normalization for the distant component. The uncertainties for the Snowden et al. {1998) parameters are larger than for

the Chen, Fabian, &: Gendreau (1997) parameters because a softer distant component, absorbed by the intervening column, will require

a greater change in Nz) before effecting the observed flux.

Consider a spatially invariant background absorbed by a spatially varying column of H l, to

which is added a spatially invariant foreground.

If one assumes that the background component

has a higher temperature than the true background, then the assumed effective absorption

cross-sections will be lower than the true effective absorption cross-sections. Thus the expected variation due to absorption (in the value

of the transmitted flux) will be smaller than the

true variation, and that part of the true variation that is not accounted for by the assumed

a

z

0.014

0.012

0.010

0.008

0.008

0.006

0.004

0.002

1.0

CFG []

D

[]

0 D

D 0 _3n

Sea

[] o

%o

[]

[]

o f3

o el3

P [] D

1.5 2.0 2.5 3.0 3.5

N. (lO" em"l

Fig. 5.4.-- Checking spectral fit parameters using absorption. Top: We have used the temperatures of Chen, Fabian & Gendreau (1997)

(iogTL = 5.76 and IogTo = 6.14) to demonstrate how the angular variation of the absorption can be used to check the temperature

and normalizations derived from spectral fitting.

Since one does not expect the true magnitude

of the TAE to be correlated with the absorbing column, the fact that the derived TAE normalization is correlated with NH indicates that

the assumed component temperatures are incorrect. Bottom: We have used the temperatures

of Snowden et al. (1998) {logTL = 6.12 and

IogTp = 6.09). There is no visible correlation

between the TAE normalization and the absorbing column.

65
variation will be attributed (by whatever fitting

mechanism one is using) to a variation in the

strength of the background component. Thus,

if the assumed temperature of the distant component is wrong, or if the magnitude of the distant component is wrong for any reason, the fitted normalization will be correlated with the absorbing column.

2.2. All-Sky Deabsorption

The second approach to determining the Xray background spectrum utilizes the Galactic

absorption as a tool to separate the local emission from the distant emission. One assumes

that at some angular scale the foreground and

background emissions are constant at some IL

and ID, and that the variation in observed Xray flux is due to absorption by the intervening

column. Given a tracer of the absorbing column (e.g., either an H ! map or an IRAS 100

/am map) one fits the relation in Equation 5.1 to

find IL and ID.

Since we have very little knowledge of the distribution of NI, the absorption due to clouds

within the LHB, one fits I_ and It) in

1o - IEe -(NDa(v'l%'s) - I'L + IDe -(Nb)°(''_c_'s)

(5.2)

where

fL = IL e-(NI)a(_''NI's)" (5.3)

When using the I[ determined from several

bands to solve for the temperature and emission measure of the local component, a uniform

N! is generally assumed (see § 6.2). N I is small

so. at worst, it can contribute only a few percent

error.

!

2.2.1. _ keY (The R12 Band)

Snowden et al. (1998) used the singleband

RASS maps of Snowden et al. (1997) to separate the 1⁄4 keV (RI2 band) emission intoforeground and background components over the entiresky. Although these maps have a mean exposure much lower than pointed observations,

they have the advantage of having the best possibleLTE and SSX removal (Snowden et al.

1995),far betterthan can be accomplished for

any single typical PSPC pointing.

Snowden et al. (1998) assumed that both the

local emission and the distant emission were fiat

on -_ 4?8 scales, and they fit Equation 5.1 assuming Nf = O. The fit for each 4?8 x 4?8 region was

then applied to the center ninth of that region.

They used the IRAS 100 pm map of Schlegel,

Finkbeiner, & Davis (1998) as a tracer of the

absorbing column. This map was scaled by the

mean value of H I/I100 where the 21 cm data

(-74 km s-1 < V < 24 km s -l) was taken from

the Dwingeloo survey of Hartmann & Burton

(1997). The EPL was assumed to have a F of

1.96 (as per Hasinger et al. 1993) but was normalized to be consistent with the observed upper limit to the extragalactic background in the

R12 band (Barber & Warwick 1994; Snowden

& Pietsch 1995; Barber, Roberts, & Warwick

1996; Cui et al. 1996); although an inconsistent

procedure, the inconsistency is inconsequential,

as shown by Snowden et al. (2000). The decomposition of the emission into foreground and

background components was done separately for

the R1 and R2 bands, and the R2/R1 band ratio

was then used to determine the temperatures of

the local and distant components.

This method works well if the emission and

absorption is well described by the canonical

model. If, however, the emission and the absorption are interleaved, then significant overestimates of the distant component will occur. Conversely,ifthe absorption tracerunderestimates

the totalabsorbing column, then the distant

emission willbe underestimated. Both problems

pertain to regionswith high and intermediate

velocitycomplexes. (SeeHerbstmeier etal. 1995

and Kerp et al. 1999 for preliminarystudiesof

thisproblem.)

As can be seen in Snowden et al. (1998),Figure 8, (and again in our Figure 6.4)the distant

emission appears to be strongly mottled, even

away from the problematic high velocityclouds,

and there isevidence of temperature variation.

Surprisingly,none of the structureobserved in

the 1⁄4 keV TAE isseen in the _ keV band, which

issmooth (seeFigure4 in Snowden et al.(1997)

or our Figure 6.4).This differenceled Snowden

et al. (1998) to conclude that the component

responsibleforthe structurein the 1⁄4 keV band

must not have significantemission in the _ keV

band, and must be differentfrom the component producing the very smooth 3 keV emission.Accordingly,they feltjustifiedin measur66
ing the temperature of the component producing

the 1⁄4 keV structure from the R2/R1 band ratio

alone, leaving the measure of the hotter component to groups concentrating on the _ keV band.

Note that almost all other measures of the

temperature of the TAE assumed a single-temperature component, and therefore measure a significantly higher temperature than Snowden et al.

(1998). In this case, visible structure in the

1_.keV (R12) band but relative smoothness in 4

the _ keV (R45) band is understandable; for

logT = 6.14, R45/R12 .-. 0.026, so much of the

structure visible in the 1⁄4 keV band will be reduced to the level of the noise in the _ keV band.

Since the effective cross-section for a given energy band depends upon both the column density of the absorbing gas and the spectral shape

of the background emission, one must assume a

spectral shape (or temperature) for the distant

emission before separating the observed emission

into local and distant components. After the

local/distant separation one can take the band

ratios of the distant component to determine a

new spectral shape for the distant emission, and

repeat the process.

Did the assumption of the initial temperature

of the distant emission bias the subsequent measure of the temperature of the distant emission?

For a range of local and distant temperatures,

we assumed a fixed single-temperature distant

component, and determined the normalizations

required to produce the total observed R1 and

R2 band emission. Using those temperatures

and normalizations we then created vectors of

flux vs. absorbing column for both bands. Using

a method similar to that used in Snowden et al.

(1998), those vectors were then fit to derive the

local and distant emission values, assuming the

cross-sections appropriate for a distant component with Iog TD = 6.09. We used the R2/R1

band ratio from the resulting components to

compute a "derived" temperature. Figure 5.5

shows the "derived" temperature for a range of

true (or input) local and distant temperatures.

One can see that for many temperatures, the

Snowden et al. (1998) method (without iteration} could produce a biased measure of the

temperatures. However, for temperatures near

the assumed temperature, the bias is small and

does not preclude finding the true temperature

through iteration. For all reasonable cases, the

biases are A logTL < 0.02 and AlogTD < 0.04,

and so are on the order of the systematics of the

analysis (e.g., choice of plasma emission model).

2.2.2. 43-keV

A large-scale deabsorption technique was used

in Snowden et al. (1997) to demonstrate the existence of the X-ray emitting bulge with a temperature of log T = 6.6 which dominates the

Galactic center and the Loop I region, but does

not affect most of the high Galactic latitude sky.

Pietz et al. (1998) took an interesting, though

Procrustean, approach to determining the TAE

in the 43-keV band. They applied simultaneous

spectral fits to three deep PSPC images from

the same region of the sky, each with a different absorbing column, to determine the temperature of the local and distant components. This

method minimizes the problems with inadequate

non-cosmic background subtraction and avoids

the degeneracy described in §2.1.

They then used the fitted temperature for

the TAE to fit a single-temperature hydrostatic

equilibrium model of the Galactic halo to the

observed _ keV distribution. Given 1) that the

high Galactic latitude _ keV emission is smooth,

2) that one does not expect the canonical model

to work well near the plane, where the classical

"infill" problem s remains unsolved, 3) that the

flattened halo model produces very little variation at high Galactic latitude, and 4) that the

flattened halo model has its greatest variation in

the direction of Galactic center where the multiple emission components are very confused, it is

not surprising that the Pietz et al. (1998) model

is a "good fit" at _ keV. Their fitting has found

the high Galactic latitude normalization, but is

not good at constraining the variation with latitude, since the region with the greatest lever arm

is also the region that must be excluded due to

Galactic absorption and confusion with Loop I.

Pietz et al. (1998) used a very limited number

_The "infill" problem is as follows: at 1⁄4 keV and 1.5

keV the average emission level in the Galactic plane is

roughly the same as the average emission at high GalacLic

latitude, despite the fact that one would expect Galactic disk to "shadow" the extragalactic flux. See, for example, Sanders et al. (1983). Thus, there must be an

additional Galactic source of diffuse 1⁄4 keV and 1.5 keV

emission that almost exactly compensates for the emission that is absorbed by the Galactic disk. However, that

source of emission remains unkown.

67
tJ

om

M

o

m

6.20

6.10

1_:_8

6.30

6.20

6.10

6.00

5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40

10, T,

Fig. 5.5.--A demonstration of the bias due to

the method of Snowden et al. (1998). Input

temperatures forboth localand distantcomponents were used to calculatea model of the observed flux as a function of absorbing column

density.Those model data were then analyzed

using the method of Snowden et al. (1998) to

find "derived" temperatures for the local and

distant components. The difference between the

input and "derived" temperatures measures the

bias in the Snowden et al. (1998) analysis. The

axes are the input temperatures, the contours

are the "derived" temperature. Top: The "derived" temperature of the distant component as

a function of the true local and distant temperatures. The shaded region produces measured

R2/R1 ratios that do not correspond to any temperature. Bottom: The "derived" temperature

of the local component as a function of the true

local and distant temperatures. Making a bad

initial assumption will produce an incorrect (biased) measure of the distant temperature, but iteration will produce the correct distant temeperature.

of PSPC pointings towards a fairly atypical region (Draco) to make a local/distant separation

at 1⁄4 keV. They then claim that the distant component of the 1⁄4 keV flux (excluding regions with

High Velocity Clouds or known structures such

as Loop I) is produced solely by the extrapolation to 1⁄4 keV of the 43-keV emission produced by

their Galactic halo model. Their evaluation of

the degree to which this extrapolation accounts

for the angular structure of the 1⁄4 keV emission

is overly optimistic; as they have no LHB model,

they can always attribute poorly fitting regions

to the effects of the LHB.

2.2.3. Synthesis

In Snowden et al. (2000), a large (but by no

means comprehensive) catalogue of 378 ROSA T

1 keV band shadows 3 was used to determine the

values of the foreground and background emission in the R1 and R2 bands for many very

localized regions at high Galactic latitude for

both the northern and southern Galactic hemispheres. The ] keV band emission was determined for the same regions by subtracting the

absorbed EPL from the observed emission and

deabsorbing the remainder. It was found that although both the distant 1⁄4 keY and ] key band

emission varied by a factor of five, there was

no clear correlation between the distant 1⁄4 keV

emission and the distant _ keV emission (see

the Figure 15 from Snowden et al. 2000). The

lack of correlation was used to support the argument made in Snowden et al. (1998) that the

distant component producing the bulk of the

_ keV emission is not the same thermal component producing the _ keV emission.

2.3. Summary

Spectral fitting methods are incapable of determining the parameters of the thermal components due to absorption uncertainty, poor noncosmic background subtraction, and inadequate

spectral resolution. The two large-scale deab3An X-ray shadow is traditionally defined a.s a region

that has a deficit of X-rays and the region of the deficit

is spatially matched to an H 1 21 cm or 100 pm emission

feature. Snowden et al. (2000) expand that definition

to regions with an excess of X-rays, where the excess is

matched to the decrease in some measure of the absorbing column.

68
sorption studies, executed in different bands, are

each blind to the analysis in the other. Snowden et al. (1998) posited two components to the

TAE, but for historical reasons deferred the necessary analysis in the 3 keV band to expand the

proof of their case. Pietz et al. (1998) attempted

1 keV to extrapolate their _ keV results into the

band, but without taking into account the structure of the LHB, or the possibility of additional

halo components.

The Snowden et al. (1998) self-consistent solution for the temperature of the distant component derives a low temperature distant component (logT ,,_ 6.07) 4 that produces little 3 keV

emission. Assuming that both the 1⁄4 keV and the

3

keV distant emission are produced by the same

thermal component requires a higher temperature (IogT > 6.14) but, as shown in Figure 5.4

and § 2.1, the higher temperature is inconsistent

with the observed angular variation of the emission with absorbing column. Further, the extensive shadowing study by Snowden et al. (2000)

shows that the distant _ keV emission is not

correlated with the distant 1⁄4 keV emission.

Therefore, the distant emission, the TAE,

must have more than a single thermal component and, in fact, may have multiple thermal

components. The temperature of those components may be spatially variable. The intensity is

undoubtedly variable. Due to the limitations of

the data, no more than an incremental improvement can be made by extending the Snowden

et al. (1998) analysis to the R4-R7 bands. In

the following chapter, we will assume the TAE

to be composed of two thermal components, a

soft thermal component with temperature Ts

dominating the 1⁄4 keV band, and a hard thermal component with temperature Tn dominats keV band. We will derive model tem- ing the

peratures for those components, and will evaluate the extent to which further thermal components may exist.

4The temperatures quoted here from Snowden et al.

(1998) and Snowden et al. (2000) are not quite the temperatures found in those papers. The values here have

been recalculated from the R2/RI band ratios using an

updated detector response matrix, while the Snowden et

al. papers used earlier versions to remain consistent with

previous papers.

REFERENCES

Barber, C. R., Roberts, T. P., & Warwick, R. S.

1996, MNRAS, 282, 157

Barber, C. R., & Warwick, R. S. 1994, MNRAS,

267, 270

Bloch, J. J., Jahoda, K., Juda, M., McCammon, D.,

Sanders, W. T., & Snowden, S. L. 1986, ApJ,

308, L59

Chen, L.-W., Fabian, A. C., & Gendreau, K. C.

1997, MNRAS, 285, 449

Cui, W., Sanders, W. T., McCammon, D., Snowden,

S. L., & Womble, D. S. 1996, ApJ, 468, 117

Hartmann, D., & Burton, W. B. 1997, Atlas of

Galactic Neutral Hydrogen (Cambridge University)

Hasinger, G., Burg, R., Giacconi, R., Hartner, G.,

Schmidt, M., Triimper, .I., & Zamorani, G.

1993, A&A, 275, I

Herbstmeier, U., Mebold, U., Snowden, S. L.,

Hartmann, D., Burton, W. B., Moritz, P.,

Kalberla, P. M. W., & Egger, R. 1995, A&A,

298, 606

Juda, M., Bloch, J. J., Edwards, B. C., McCammon,

D., Sanders, W. T., Snowden, S. L., & Zhang,

J. 1991, ApJ, 367, 182

Kerp, J. 1994, A&A, 289, 597

Kerp, J., Burton, W. B., Egger, R., Freyberg, M. J.,

Hartmann, D., Kalberla, P. M. W., Mebold,

U., & Pietz, J. 1999, A&A, 342, 213

Landini, M., & Monsignori-Fossi, B. C. 1990,

A&AS, 82,229

Mewe, R., Gronenschild, E., & van den Oord, G.

H. J. 1985, A&AS, 62, 197

Miyaji, T., lshisaki, Y., Ogasaka, Y., Ueda, Y.,

Freyberg, M. J., Hasinger, G., & Tanaka, Y.

1998, A&A, 334, L13

Parmar, A. N., Guainazzi, M., Oosterbroek, T., Orr,

A., Favata, F., Lumb, D., & Malizia, A. 1999,

A&A, submitted

Pietz, J., Kelp, J., Kalberla, P., Burton, W. B.,

Hartmann, D., & Mebold, U. 1998, A&A,

332, 55

Raymond, J. C., & Smith, B. W. 1977, ApJS, 35

Sanders, W. T., Burrows, D. N., McCammon,

D., & Kraushaar, W. L. 1983, in Supernova Remnants and their X-ray Emission, ed.

J. Dartziger & P. Gorenstien, 361

Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998,

ApJ, 500, 525

Sidher, S. D., Summer, T. J., Quenby, J. J., &

Garnbhir, M. 1996, A&A, 305, 308

Snowden, S. L., Cox, D. P., McCammon, D., &

Sanders, W. T. 1990, ApJ, 354, 211

Snowden, S. L., Egger, R., Finkbeiner, D., Freyberg,

M. J., & Plucinsky, P. P. 1998, ApJ, 493, 715

Snowden, S. L., et al. 1997, ApJ, 485, 125

Snowden, S. L., Freyberg, M. J., Kuntz, K. D., &

Sanders, W. T. 2000, ApJS, in press

Snowden, S. L., et al. 1995, ApJ, 454,643

69
Snowden, S.L., McCammon, D., Burrows, D. N., &

MendenhaU, J. A. 1994, ApJ, 424, 714

Snowden, S. L., & Pietsch, W. 1995, ApJ, 452, 627

Snowden, S. L., Schmitt, J. H. M. M., & Edwards,

B. C. 1990, ApJ, 364, 118

7O
Chapter 6: The Spectrum of the TAE

Abstract

We show that the TAE cannot be characterized by a single thermal component; no

single-component model can be consistent with both the spectral energy distribution

of the TAE emission and the angular variation due to absorption by the Galactic disk.

Under the assumption that the LHB does not vary on scales smaller than _ 5°, we use

the angular anticorrelation of the ROSAT All-Sky Survey with the Galactic absorption

to separate local from distant emission components, and to fit the spectral energy

distribution of the resulting distant emission.

We find that the emission is best described by a two-thermal-component model

with logTs = a....n_+-020°-19 and logTH = fl

.....

aa+°l_0.08" The soft component has a patchy

distribution, while the hard component appears nearly constant over the high Galactic latitude sky. This two-thermal-component TAE fits the ROSAT spectral energy

distribution significantly better than single-component models, and is consistent with

both angular variation and spectral constraints. This two-thermal-component model is

similar to the halo emission seen in external spiral galaxies.

I. Introduction

We attempt to determine the thermal structure of the TAE by combining the all-sky deabsorption method of Snowden et al. (1998) with

six-band Spectral Energy Distribution (SED)

fitting. Ideally, to determine the thermal and

spatial structure of the TAE, one would use the

angular structure of the absorbing column to

separate the emission in each band into local

and distant components, and then one would

determine what combination of thermal components would produce the spectrum (or SED) of

the distant emission. Unfortunately, the separation of the emission of any band into local

and distant components requires knowledge of

the spectrum of the distant component in order to calculate the appropriate effective crosssections. Thus, any determination of the nature

of the distant emission requires iterative solution of both the separation into local and distant

components and the modeling of the thermal

component structure. The extent to which such

a process can be successful is strongly limited by

the interplay of the response of the instrument,

the spectrum of the emission, the energy dependence of the absorption cross-sections, and the

available range of absorption column density.

In the following sections, we will briefly describe the available X-ray data and some considerations concerning the measurement of the

absorbing column. Section 3 describes the limitations to the separation of the X-ray data into

local and distant components that were alluded

to above. Section 4 describes the external constraints that can be used to set the non-TAE

parameters in Equation 5.2. Finally, we detail

the iterative method used to separate the local

and distant components and to determine the

thermal structure of the distant emission.

2. Data

2.1. The RASS

The X-ray data are the single-band RASS

maps of Snowden et al. (1997), which have

12' x 12_ pixels. Of these band maps, the combined R1 and R2 bands (the R12 band) map

forms a "1⁄4 keV" map, the combined R4 and

R5 bands (the R45 band) map forms a "_ keV"

map, and the combined R6 and R7 bands (the

R67 band) map forms a "1.5 keV" map. Histograms of the statistical significance of the pixels in each band are given in Figure 6.1. Given

the long temporal baseline possible with the

RASS, the LTE and SSX removal were far better

than is possible with any single pointed PSPC

71
4.0x10 4 :

3.0xlO 4

_ 2.0xl 04

Z

1.0xl0 4-

i

i'1

I.

lit"

li" 4.

I: I:,

I:.

: t':

: IL:"

4000

R45

n

3000 - J _ ii

/

II

mHI _

v ,_i ,"1

if" 3"l't

I:' 'I..

le:_ _Jri..

I,: "I.L_j ': I

1000 i : "_i' - II/ "":'".

: "-i ....... ..

o ._ .L,:__--_ 4000: "

: R67

3000 i

i!. /i

% ::

1000 "- :_'. t

":_::'i. -.it

0.0 0.2 0.4 0.6 0.0 l.O 1.2

a../Rate

Fig. 6.1.-- Histograms of the statistical significance of individual 12' × 12' pixels in individual

and combined bands of the RASS. For bands R4-

R7, only pixels in the north polar cap (b > 55 °,

450 < t < 270 °) were used. In each panel, the

solid line is the combined band, the dotted line

is the lower energy band, and the dashed line is

the higher energy band.

L

a_ 2000

Z

k,,

,.D

2000

Z

observation I. Some systematics remain. Even

in seemingly uniform regions, a high contrast

stretch of the raw RASS images will reveal the

scan pattern, more strongly at _ keV than at

1 keV

The R1 and R2 band maps have had point

sources removed to a level of (R,l+R2)=0.025

counts s -1 while the R4-Ft7 band maps have had

point sources removed to a level of (R4+R,5)=0.2

counts s-I and (R6+R7)=0.02 counts s-1. Correction of the RASS bands for the band-variable

point source removal is discussed in §4.1.

For reasons that will become apparent shortly,

the analysis was restricted to b > 55 °, 45 ° < t <

270°; the selection in Galactic longitude was set

to exclude the Galactic center/Loop I emission.

Within this region we also removed pixels contaminated by extended sources, Coma, Virgo,

and A1367.

2.2. The Absorbing Column

We have taken as a tracer of the absorbing column the IRAS 100 pm maps of Schlegel,

Finkbeiner, & Davis (1998), as no other tracer

of the absorbing column has either all-sky coverage, or angular resolution comparable to that

of the RASS. As has been discussed in Arendt

et al. (1998) 100 pm emission traces, to varying degrees, the molecular, neutral atomic, and

ionized atomic components of the ISM. (For a

review see Chapters 3 & 5 of Kuntz 2000.)

Low velocity disk H I appears to be well

traced by the 100/_m emission, but the 100 pm

emissivity of the H I does appear to vary (Reach,

Wall, & Odegard 1998). The high and intermediate velocity H I appears to have very low

100 pm emissivity, and is essentially invisible at

100 pro, meaning that the IRAS 100 pm maps

in regions with intermediate (-100 < v < -75

km s -l) or high (v < -100 km s -1) velocity

clouds will underestimate the absorbing column

(Wakker & Boulanger 1986).

Only upper limits to the 100 pm emissivity of

the H li have been determined, so the 100/_m

emission should be considered a poor tracer of

]Removal of residual background contamination for individual PSPC pointings after reduction with the ESAS

software (Snowden & Kuntz 1998) can only be done

by reference to the RASS. The method is described by

Kuntz & Snowden (1908).

72
the ionized gas (Arendt et al. 1998).

The emissivity of the molecular gas is a very

strong function of the cloud temperature. The

work of Schlegel, Finkbeiner, & Davis (1998) attempted to correct for the cloud temperature

in order to produce maps of the true dust column density, but the accuracy of this correction, denoted by X, is not well understood.

Therefore the absorbing column of molecular

gas (even if one assumes a constant dust-tohydrogen-nucleon ratio) may be poorly determined by the IRAS 100 pm emission. However,

molecular gas covers only a small portion of the

high Galactic latitude sky, appearing only where

the H I column is greater than 3 x 102o cm -2

(Reach, Wall, & Odegard 1998), and the H 1 21

cm/IRAS 100 pm ratio is quite sensitive to the

presence of molecular material.

The dominant tracer of absorption is H I. (At

1 keV He provides roughly half of the physical

absorption, while at energies > 0.5 keV metals provide nearly all of the absorption.) The

amount of absorption due to H II is a matter of

debate (Arabadjis _ Bregman 1999; Laor et al.

1997); the cumulative absorbing column due to

H I! appears to be less than 0.2 that of the absorbing column due to H I at 1⁄4 keV and may be

significantly smaller.

Since the 100 pm emission is very good at

tracing the H I, and the H I 21 cm/100 pm ratio is very sensitive to the presence of H2, the

100/Jm emission is the single best tracer of the

total absorbing column. That the 100 pm emission is not sensitive to H II is offset by the X-ray

absorption being less sensitive to H I[. To convert from the native units of dust emission (MJy

sr -1) to absorbing column, we use the ratio of

N{H I)/Xl_oo "-, 1.475 determined from the

northern Galactic hemisphere (b > 45 °) where

N(H l)< 3 x 102o cm -_, a value which varies by

--- 5%, depending upon the latitude and column

density limits used.

We have removed the regions where NH with

v < -75 km s-1 has column densities above 1 x

l019 cm -_. This selection removes the strongest

intermediate and high velocity clouds, which are

known to be deficient in IRAS 100 pm emission.

3. Limits

Optical Depths: In order to separate the distant emission from the local emission, one must

have absorption over a wide range of optical

depths within a limited angular region. (The region must be small in order for the assumption

of a spatially invariant local and distant X-ray

emission made by Equation 5.1 to be valid.) In

practice, the smallest r must be _ 1, in order

to produce a good measure of ID, the distant

emission, and the largest r must be _ 2 in order to produce a good measure of 1L, the local

emission. Either of these limits can be relaxed if

the other end of the optical depth range extends

well beyond its limit. If the first condition is not

met, but the latter is, Io may be estimated from

Io = (Io- IL )e T, where Io is the observed flux.

Since the absoption cross section, a o¢ E -s/a,

regions fulfilling the 7" requirements at 1⁄4 keV

generally do not fulfi]l the r requirements at

keV or 1.5 keV (see Figure 6.2). Separation at

1 keV is usually possible if N-H'H< 4 x 102° cm -2,

whereas separation at 43-keV is generally only

possible if N----H>> 4 x 10_° cm -2. Thus, while

1⁄4 keV separation into lo and lt. is usually possible at high Galactic latitude, _ keV separation

is generally only possible at low latitude.

Count Rates." The difficulty in separating It)

from IL at _ keV is exacerbated by the _ keV

count rates, which are typically one-tenth of the

count rates at 1⁄4 keV (note the lower statistical significance of higher energy band pixels in

Figure 6.1). Thus, separation of the RASS is

generally possible for the R1 and 1%2bands, individually, but at higher energy, it is only possible

in the combined R45 and R67 bands.

Regions with sufficiently high column densities to show good _ keV shadows are typically near the Galactic plane, where there are

likely to be many emission components (and

absorbing screens) along a given line of sight.

Thus, although there are regions for which the

local/distant separation is mathematically possible, those regions are unlikely to produce a

meaningful measure of either the local or the

distant emission.

4. External Constraints

4.1. Properties of the EPL

The EPL is the aggregate of unresolved sources

It must be the result of a complex convolution

of source populations with different spectra, the

73
o

m

E

m

e_

o.5

E-

, p ......... E T r

, , I ,

15 .. 3'0 4b 6'o o8o9o,oo zooR6 

" ' ' ' ' ' ' ' ' R45:

I0 20 30 40 50 60 70 80 90 I()0

"__ 3 4 5 6 7 8

0.0

0 1 2 3 4 5

Optical Depth

Fig. 6.2.-- The column densities at various energies necessary to produce a given optical depth.

The column density marked on each bar is in units of 10 =° cm-L We have assumed a logT = 6.05

thermal background in band R12, and a logT = 6.35 thermal background for both R45 and R67.

distribution of redshifts, and the distribution of

intrinsic absorption. None of the existing populations of point sources thought to compose the

EPL has spectra as hard as that of the EPL.

Given the cosmological significance of the EPL,

a number of groups have attempted to measure

the photon index (r) and normalization (Nr)

of the EPL (i.e., the spectrum of the aggregate

of unresolved sources). Indices measured from

ROSA T data alone have been higher than those

measured from ASCA, which provides a significantly longer, absorption free, lever arm (e.g.

Chen, Fabian, & Gendreau 1997). It has been

suggested by a number of authors that this discrepancy is due to calibration uncertainties in

the ROSA T PSPC. Fits using both ROSA T and

ASCA data tend to indices more similar to the

ASCA values. We have chosen the values of

Chen, Fabian, & Gendreau (1997), r = 1.46,

Nr = 10.5 keV cm -_ s-z sr -1 keV -l, as being

representative of a number of recent measures of

the EPL from joint ROSAT/ASCA fits.

The Chen, Fabian, & Gendreau (1997) point

source removal limit for the region from which

the r = 1.46, Nr = 10.5 EPL was derived was

0.00524 counts s-1 for the R47 band. To correct to the _-,mt source detection limit of the

RASY,, nearly an order of magnitude larger, we

used the log N - log S relation of Hasinger et al.

(1998) and have assumed a photon index for the

point sources between the point source removal

limits of Chen, Fabian, & Gendrean (1997) and

the RASS . We have assumed that those point

sources can be characterized by the point sources

detected by Chen, Fabian, & Gendreau (1997)

at somewhat lower fluxes (_ 0.00524 counts s -z

but _ 0.0003 counts s- 1). A F = 2.54 index was

derived by Chen, Fabian, & Gendreau (1997)

for a sample of confirmed QSO's detected in this

flux interval. However, the confirmed QSO sample is softer than the complete sample of objects

in this flux interval, so a harder index may be

more apropriate. We have performed the analysis using both F -- 2.00 (the value used by

Hasinger et al. (1998) in their construction of

a log N-log S relation) and F = 2.54 for the

point source correction.

Caveat: Although much effort has been expended to determine the spectra of individual

sources, the spectrum of the unresolved extragalactic background, the EPL, is unknown in the

I keV band. We have assumed that the power

law that describes the extragalactic background

for E > 1.0 keV extends into the 1⁄4 keY band.

It has been suggested that the spectrum of

the extragalactic background turns up below 1

keV. However, that turn-up can not be too great;

Snowden et al. (2000) show that the isotropic

74
extragalacticbackgroundat 1⁄4keVcannotbe

morethan_ 325x 10-6 counts s-1 arcmin-:.

If the spectrum breaks at 1 keV, then F < 1.64;

if it breaks at 0.75 keV, then F < 1.67. Thus,

even if there is a turn-up in the extragalactic

spectrum, the effect is small.

3 keV Emission from the LHB 4.2.

Given all of the caveats placed on _ keV local/distant separation in _3, any external constraint on the magnitude of the local 43-keV emission is useful. Lines of sight useful for measuring

the local component of the 43-keV emission must

be chosen carefully. The highest signal-to-noise

measure of the local _ keV emission remains

that of Snowden, McCammon,/_ Verter (1993)

towards MBM 12, (a molecular cloud situated

just inside the LHB) which yields 2q upper limits of 23 × 10 -6 counts s -1 arcmin -:_ (_ keV)

and 39 x 10 -6 counts s-1 arcmin -2 (1.5 keV).

In this direction, the 1⁄4 keV flux is 385 x 10-6

counts s- 1 arcmin-2.

If the temperature of the LHB is ,-, log TL =

6.10, then the R45/R12 ratio is 0.017, and the

3

keV count rate due to the LHB in the direction of MBM 12 is 7.1 × 10 -6 counts s -1

arcmin-2, well below the upper limit and similar to the measured value of ,_ 5 x 10-6 counts

s-1 arcmin-2. At high Galactic latitudes, where

the LHB flux is on the order of 400 - 600 x 10 -6

counts s -1 arcmin-:, the _ keV counts due to

the LHB will be 8.75- 13.25 x 10 -6 counts

s -1 arcmin -2, which is a substantial fraction of

the high Galactic latitude 3 keV flux, typically

35 x 10-6 counts s-1 arcmin -_. Thus, ignoring the LHB contribution at _ keV in the

manner of Pietz et al. (1998) is unwise.

5. The Iterative Algorithm

We use a bicyclic iterative scheme to solve

for the distant component in each band and, simultaneously, fit two thermal components to the

resulting SED. The process is described graphically in Figure 6.3. Given the difficulty of performing a local/distant separation in the R4-R7

bands, we have treated those bands somewhat

differently than the R1 and R2 bands; the R1

and R2 bands are separated into local and distant components with a method similar to that

used by Snowden et al. (1998), while the R4-

R7 bands are corrected only for Galactic absorption after the contribution of the LHB has

been removed. The lack of correlation between

3

the 1⁄4 keV and _ keV bands, as well as the low

model temperature found for the 1⁄4 keV emission

by Snowden et al. (1998) suggested that there is

little contribution from the 1⁄4 keV TAE emission

to the 3 keV TAE emission (when compared to

the much greater flux at 1⁄4 keV ) and vice versa.

The fitting algorithm begins with the assumption, but the assumption is not reified by the

fitting algorithm.

The first cycle (the left side of Figure 6.3) iteratively determines the temperature of the softer

component from the bands R1 and R2, after

separating the local from the distant emission.

The second cycle (the right side of Figure 6.3)

iteratively determines the temperature of the

harder component from the bands R4 through

R7. Each cycle passes to the other the extrapolation into the complementary band range of the

component it has fit. The first cycle finds the

single best thermal component fit to the R1 and

R2 bands, and extrapolates that component into

the R4-R7 bands. That extrapolation is then

subtracted from the R4-R7 data, and a single

thermal component is fit to the remainder. That

thermal component is then extrapolated to the

R1-R2 band range, where it is subtracted from

the R1-R2 data. The process is then repeated.

5.1. Details of Local/Distant Separation

The process by which the distant emission is

separated from the local emission in bands R1

and R2 is similar to that used by Snowden et al.

(1998). Both the local and distant emissions are

assumed to be spatially invariant over 4°.8 x 4°.8

regions. The bulk of the distant emission in the

R12 band is assumed to come from a single thermal component. From both RASS single band

images (R] and R2) is subtracted the contribution of the EPL, after absorption by the Galactic

ISM, and the contribution to that band from the

hard thermal component, also after absorption

by the Galactic ISM. The X-ray data from each

40.8 x 40.8 region are then binned by the absorbing column density into 1 x 1019 cm -2 wide bins.

The binned data are then fit by Equation 5.2, assuming some log T for the distant emission. The

Lampton, Margon, & Bowyer (1976) criteria are

75
Splitter

Parsmeters

AdjusL6ble: T S

Noa-Adjmstable: F,N F

for a=l,_ do

R_ = R. -- pae--lupC)

Fit R_ to C _ l_l " ,I_D

DSt_c

RZ ,R2

El ,E2

I

¢

I

__ D6ta

C

R4,Rs,RG,R 7

E 4 .E 5 ,E G .E7

I

Fitter

Parameterg

Adjuatable: T H

Non-Adjustable: r,N F

I

for Iz--_4,? do

I. = (Ra -- I.L - P.e--(uP C) - E.z-(°Ts C))e(OTH C)

SE,D_ = Z In X _ASK

¢

i

Fit RkS models to SE_D ¢_ T H

for Itll,2 do E m = (Bl,/B46) X (I 4 + 15}

Fig. 6.3.-- Flowchart for the bicyclic iterative fitting of two thermal components to the SED of the

TAE. Ts is the model temperature of the softer component (which is fit in the left iterative cycle)

and Tn is the model temperature of the harder component (which is fit in the right iterative cycle).

P_ is an image of the observed emission in band n, En is an image of the emission extrapolated from

other energies into band n, In is an image of the TAE emission in band n, corrected for absorption,

Pn is the value of the extragalactic power law in band n, and C is the absorbing column. The

subscripts L and D denote the local and &stant components. Bn is the band value derived from

a Raymond & Smith spectrum. The "T = T'?" is a consistency check for the temperature of the

component.

used to determine the uncertainties in the fit parameters. The fitting is done for each 4?8 x 4?8

region for which the N-'-H"< 4 x 10_° cm -2. The

4?8 x 4?8 regions overlap by 176, so the solutions

are applied only to the center ninth of the region

(1762).

In their work, Snowden et al. (1998) found

that the uncertainty in the distant component

could be substantially reduced by taking the

quantity

I_, = (;o - 7,.)_" (6.])

rather than the value produced by the fit We

have followed this procedure, although, owing

to slight differences in the fitting procedure, we

find that the improvement is less significant.

Once the separation is accomplished for both

the R1 and the R2 band images, the hardness

ratio is determined for the distant component.

This leads to a new estimation of the log Ts for

the distant emission, and the process is repeated

until the assumed temperature and the temperature recovered from the hardness ratio agree to

A logTs < 0.005. Convergence typically occurs

in three iterations.

5.2. Details of the R4-R7 SED Fitting

From each RASS single band image is subtracted the contribution from the EPL, absorbed

by the Galactic ISM, the contribution from the

soft thermal component that dominates the R1

and R2 bands, absorbed by the Galactic ISM,

and the contribution by the LHB, without absorption. The image of the remainder in each

band is spatially smoothed by a three pixel boxcar and is then deabsorbed (i.e., multiplied by e _

where r is the effective cross-section multiplied

by the column density) assuming some log TH

76
forthedistantemission.

Afterallfourbandshavebeendeabsorbedin

this manner,the meandeabsorbedSEDfor a

regionwithb > 550 and 450 < if < 2700 is calculated. A thermal model is fitted to this SED

to determine the new log TH, and the process is

repeated until AlogTH < 0.005. Convergence

typically occurs in three iterations. Convergence

between the two halves of the iteration usually

occurs in three or four iterations.

Unlike Snowden et al. (1998), we have not

smoothed the distant emission images before calculating the R2/R1 ratios, so our uncertainties

are somewhat higher.

In order to minimize the effects of the tail we

have chosen to characterize the R2/R1 ratio by

the mode. The mode is 0.950; the corresponding model temperature is log Ts = 6.056. Snowden et al. (1998) found logT = 6.09 (FWHM

= 6.01 - 6.17) for the soft distant component,

6. Analysis of Current Results

The results of the local/distant separation

and the SED fitting are shown in Figure 6.4.

The top images show the soft and hard components in the bands in which they dominate, and

below that are their uncertainties (3cr for the

soft, la for the hard). The third row contains

the spatial distribution of the temperature of the

two components. For the soft band the temperature is in the form of the R2/R1 band ratio,

while for the hard component the temperature

is the fitted Raymond & Smith (1977) equilibrium temperature. The fourth row contains the

absorbing column, and the raw R47 band image.

6.1. The TAE

6.1.1. The Soft Component

The temperature of the soft component was

derived from the R2/R1 ratio over the north polar cap region (b > 45 °, 450 < e < 270 °, excluding regions with strong X-ray clusters of galaxies or strong (NH > 1019 em -2) high and intermediate velocity clouds). The histogram of

the R2/R1 band ratio over the north polar cap

region is shown in Figure 6.5, along with the

conversion to Raymond _ Smith model temperatures (assuming cosmic abundances). The

FWHM of the R2/R1 ratio is 0.610-1.290, and

the distribution is asymmetric, with a strong

high R2/R1 tail. The values in the high tail do

not correspond to a Raymond _ Smith model

temperature, and thus indicate that the bulk of

the dispersion is due not to an intrinsic spread

of temperatures, but to the uncertainty of the

local/distant split. The mean uncertainty ill

R2/R1, obtained by propagating the uncertainties in the distant components of R2 and R1 is

sufficient to produce the entire observed FWHM.

70 .... i .... i .... i .... .

LHB

6.2

40 ___

"_ 5

z_ a0

20

10

0 : '. :'.'",'_._: : I : ...: ,'_L_.:

I

5_ 6.2

,, 20

Z

10

0 I _, ml

0.0 0.5 1.0 1.5 2.0

R2/m

Fig. 6.5.-- The distribution of the R2/R1 band

ratios for the LHB and TAE. The squiggle above

the histogram shows the conversion to Raymond

& Smith model temperature. For the TAE, both

the R2/R1 ratio and Raymond & Smith conversion are after deabsorption. For the LHB the

histogram is of the R2/R1 ratio derived from

[}_ values; the temperature conversion assumes

a 4 x l0 is cm -2 absorption. Note that only the

lower branch of the temperature conversion is of

interest as the upper branch would over-predict

the R47 flux.

{I
30 0 330 30 0 330

78
Fig. 6.4.-- Prevwus page: The spatial distribution of measured quantities. All images are of the

northern Galactic hemisphere. Galactic longitude 0 ° is down, 90 o is on the left. The longitude

spacing is 30 °, the latitude spacing is 15°. Note that the pixelization is 1?6. a: The soft TAE

component as seen in the R12 band. Greyscale: minimum 0, maximum 5 x 10 -3 counts s -I

arcmin -2. b: The hard TAE component as seen in the R47 band. Greyscale: minimum 0,

maximum 5 × 10 -4 counts s-1 arcmin -2. c.' The 3_r uncertainty of the soft TAE component.

Greyscale as in a. d: The la uncertainty of the hard TAE component. Greyscale as in b. e: The

R2/R1 band ratio for the soft component. Greyscale: minimum 0, maximum 3. f: The log of the

fitted temperature of the hard TAE component. Greyscale: minimum 6.0, maximum 7.0. g: The

absorbing column density. Greyscale: minimum 0, maximum 4 x 102° cm -2. h: The observed R47

band flux. Greyscale as in b. Note that the bulk of the observed structure is due to the scan paths.

without an extrapolation of a hard thermal component. If we follow Snowden et al. (1998), and

ignore the extrapolation of the hard component,

we find an R2/R1 ratio of 0.92, with a corresponding model temperature of logT = 6.040.

Since we used a much more restricted portion

of the sky, and a different fitting algorithm, the

difference between the value from Snowden et al.

(1998) and our value (without the extrapolation

of the hard component) is a good measure of the

systematic uncertainties.

Systematics: Most of the systematics in the

local/distant separation have been discussed in

Snowden et al. (1998). The magnitude of the

distant component is not correlated with the absorbing column, though the uncertainty will be.

The value of the R2/R1 ratio for the distant

component is not correlated with the value of

R2, R1, or the magnitude of the absorbing column. Snowden et al. (2000) discuss the variation of the R2/R1 ratio of the distant component

with Galactic latitude and longitude and they

find none. We are not so sensitive to the variation of R2/RI with position as were Snowden

et al. (2000) because, although this study has

a greater number of pixels, the Snowden et al.

(2000) study carefully chose regions to have a

large dynamic range in absorption.

Although the R2/R1 ratio of the distant component is not correlated with NH, the derived

temperature of the distant component may still

be systematically affected by the absorbing column. If the absorbing column is systematically underestimated, one might overestimate

the model temperature, and that overestimate

may depend upon the absorbing column. As

can be seen in Figure 6.6, for the approximate

background temperature and range of columns

used (NH < 4.0 x 1020 cm-2), the overestimate has little dependence on absorbing column. If Nntl/NHI "_ 0.33 (Reynolds 1991) and

e_ttl/aHI "" 0.62 (assuming that Nn_it/Nne "_

0.5 as per DomgSrgen & Mathis 1994) then the

absorbing column due to H II will be --, 0.2 times

that due to H 1. Assuming a distant component

temperature of logT = 6.15, if one does not account for the 20% extra absorption due to the

H II, one would overestimate the distant component temperature by A log T ,-- 0.05.

One might also expect to find that the lack

of a sufficient dynamic range in the absorption

1.5

1.0

0._ ......... i ......... i .........

19 20 21 22

Iog..(Nm)

Fig. 6.6.-- The derived R2/R1 ratio as a function of Nn if Nil were underestimated. We assumed the distant emission to have logT = 6.15

and measured the R2/R1 ratio of the deabsorbed emission when the absorbing column had

been underestimated by (from bottom to top)

0% to 30% in steps of 5%.

79
mightproducea systematicallyhigh(or low)

estimateof thedistantcomponent.Werecalculated the R2/R1 ratio using only those 1762

pixels with "good" fits, defining "good" fits to

be those for which the lo" lower limit was greater

than zero. The difference between this value of

R2/R1 and that calculated from all of the pixels

was insignificant.

6.1.2. The Hard Component

The temperature of the hard component was

derived from fitting the R4-R7 SED The uncertainty of the value was determined from the dispersion of the temperatures found from fitting

the SED of each 176: region. The histogram

of the distribution of temperatures is shown in

Figure 6.7. The mode of the temperature distribution was log TH = 6.46 and the FWHM of

the distribution was 6.38-6.55.

To within the uncertainties, the spatial distribution of the temperature of the hard component is uniform. The gradient from bottom

to top in Figure 6.4 matches the scan pattern

seen in the RASS images; the pattern has an

amplitude of A log T --, 0.1.

50i • . , , , . . , . . . , . . . , . .

: DH 4O

._ 3O

2o,o O;

O, D r_ .

6.0 6.2 6.4 6.6 6.8 7.0

log T,

Fig. 6.7.-- The distribution of the temperatures

produced by fitting Raymond & Smith models

to the R4-R7 SED after the emission due to the

LHB and the soft component of the TAE have

been removed.

6.1.3. Comparisons

Comparison of the components: If the TAE

were produced by a single component, then one

would expect the R12 and R45 bands for the

TAE to be correlated. The lack of correlation is

shown in Figure 6.8.

Comparison of one- and two-component models: Given the R1 and R2 measures of the TAE

(from the local/distant separation), and the R4-

R7 measures of the TAE (from direct RASS observation), we can construct the SED for each

1".62 region on the sky. We can then measure

the relative goodness of fit for both the one- and

two-component models.

To minimize the introduction of a bias through

deabsorption, we created an absorbed SED from

that part ofthe sky with 0.95 < Nu/102° < 1.15

cm -_. For all bands we have taken the SED elements to be Io - (IL + 1E) where IE is the

contribution due to the absorbed EPL We have

then fit the two component model to each 1'.62

pixel using the previously derived tempe___ratures,

and determined the mean X" per fit. _,_ = 3.43

c

100

50

0

-1000

........ n ......... *......... , ........

0 !000 2000 3000

RI2 (cn_ s" arcmin _)

Fig. 6.8.--Correlationof the _ keV (RI2) band

emissionof the TAE and the -_keV (R45) band

emissionof theTAE. The pointsareshown without errorbars forclarity.

8O
e.

o

t_

r_

t,3

{-

o

M

t_

250

200

ISO

100

5O

0

250

200

150

100

50

o!

.... r......... u......... n......... T......... u......... U''"

Y

I 2 4 5 6 ?

Bands

Fig. 6.9-- Fitting one- and two-component

TAE models to the RI-R7 band SED. Both

panels use a "mean" SED calculated from all

regions where 0.95 < N///102° < 1.15 and

(45 ° < _ < 270°,b > 55°). The uncertainties are representative of those of the individual

1762 pixeis for which the fits were made. Top:

The data are the x, the solid line is the best

fit two-component TAE model, the dotted line

is the hard component, the dashed line is the

soft component. Bottom: The data are the x,

the solid line is the best fit one-component TAE

model. Note that the SED used for the onecomponent model is slightly different from that

used for the two-component model. See text for

explanation. Note that the relative X 2 values

given in the text were derived from all of the

pixels within (45 ° < t < 270°,b > 55 °) while

the fits shown are to the mean of a subsample

of the pixels used.

(v = 4).

The temperature of the single-component model

was determined iteratively in a manner similar

to that used for the two-component model. We

fit a sing]e-component model to the SED used

for the two component fit. That temperature

was then used to do the R1 and R2 band local/distant separation, and to redetermine the

SED. The single-component model was refit to

the R1-R7 SED, and the procedure was repeated

until sucessive fits produced the same value. We

then found the temperature for which a singleco___mponent model had the best mean X2 per fit,

X_ = 6.79 (v = 5) at log T = 6.22. If the singlecomponent model were correct, one would expect that the temperature produced by fitting

the SED should be consistent with that found

from the R2/R1 ratio. Using iogT = 6.22 as

the TAE temperature in the local/distant separation, we find the unabsorbed R2/R1 band ratio to be 1.00, equivalent to a Raymond _: Smith

model temperature of log T = 6.08. Therefore, although our procedure attempts to make

the temperatures found by SED fitting and by

local/distant separation consistent, no singlecomponent model can be made consistent.

Comparison of the X2 show that the twocomponent TAE model is clearly a significantly

better fit. Results for a "mean" SED derived

from a limited range of absorbing column density (0.95 < NH/IO _° < 1.15) can be seen

for both models in Figure 6.9. The singlecomponent temperature derived here is consistent with other single-component analyses. The

relatively poor fit of the single-component model

is due to forcing the SED fit to be consistent

with the local/distant separation.

More Components?: The data are clearly insufficient to determine the existence of further

components; any combination of thermal components that mimic the two derived here will

work as well. The spatial distribution of the

temperature of the hard component (Figure 6.4)

suggests that that temperature may vary with

position. Whether there is a single component

with a spatially variable temperature, or several

components with different temperatures must,

at this point, remain a semantic distinction.

Conversely, one can ask if the hard thermal

component is actually due to some component of

the extragalactic background. If it is due to the

unresolved extragalactic point sources, then the

81
spectrumof the background must turn up at _ 1

keV, but turn back down before the _ keV band;

possible, but contrived. If it is due to a diffuse

extragalactic background, then that background

resembles a thermal plasma.

It is likely that some portion of the hard thermal component is due to unresolved Galactic

stars. Stellar X-ray emission is usually modeled

by thermal plasmas with log T -,, 6.3-6.5. A calculation of the X-ray flux due to stars, similar to

that made by Guiliout et al. (1996), shows that

no more than -_ 17% of the hard component flux

can be due to unresolved stellar point sources.

At high Galactic latitudes, in directions of nominal diffuse background levels, only ,-* 14% of the

3

keV and 1.5 keV band flux which is not due to

the LHB or the EPL can be due to unresolved

stars.

Comparison of point source spectrum models: As noted in §4.1, one must correct the

RASS for point sources below the RASS removal

limit but brighter than those discarded by Chen,

Fabian, & Gendreau (1997) when measuring the

spectrum of the EPL. The spectrum of those

sources is poorly understood in the 1⁄4 keV and

3

keV bands. The results presented above use

F = 2.00 for the point source correction. If

a F = 2.54 spectrum were used instead, the

changes to the derived temperatures is minimal;

/x logT < 0.001 for all of the components.

Comparison of plasma models: From Table 5.1 it will be remembered that the model

temperature derived depended, to some degree,

upon the plasma model used. To some extent. this disagreement is immaterial; no plasma

model has been able to reproduce the line spectrum of the diffuse X-ray background (Sanders

et al. 1998; Vallerga _: Slavin 1998; Deiker et al.

1997). Therefore, the model temperatures derived here may be nothing more than convenient

markers on an artificial but universally accessible scale. However, these markers certainly allow comparison of some measure of the relative

temperatures of the gas.

l ising XSPEC (Arnaud 1996) to fit MeweKaastra- Leidahl models to unabsorbed Raymond

& Smith models convolved with the ROSAT response, both models using cosmic abundances,

we find that

log TMKL "" 0.92 + 0.83 log Tns. (6.2)

Thus. the equivalent MKL temperatures will be

systematically lower than the Raymond & Smith

values quoted here.

6.1.4. The Diffuse Background Budget

Table 6.1 presents the budget for the diffuse soft X-ray background toward the north

Galactic pole (b > 85 ° , 45 ° < l < 270 ° ,

with the region around the Coma Cluster excluded). The line marked "Remainder" is the

total observed flux from which has been subtracted the contributions from the LHB, the

EPL, and the point source correction necessary

for point sources not excluded from the RASS,

but too bright to have been included in the

Chen, Fabian, & Gendreau (1997) formulation

of the EPL. Hard and soft thermal components

with log Tn = 6.46 and log Ts = 6.06 were

fit to the remainder to determine their normalizations. The final line shows what portion of

the remainder is due to unresolved stellar X-ray

sources. It should also be noted from the work of

Snowden et al. (2000) that of the remainder no

more than 80x 10 -6 counts s -1 arcmin -2 in the

! keV band can be truly extragalactic. (That is, 4

after the removal of the EPL, the minimum flux

observed originating from beyond the Galactic

H I layer is 80x 10 -_ counts s -1 arcmin-2.)

6.1.5. Spectral Fitting Redux

Using the knowledge gained above, one can

return to the spectral fitting method. We have

used three deep pointed PSPC exposures to test

the consistency of our broad-band KASS results

with more detailed spectral fitting. Whereas

Pietz et al. (1998) selected pointings that were

close together but had different absorbing columns, we used maps of the distant 1⁄4 keV emission to choose directions with different TAE intensities. We then searched the ROSAT archive

at the HEASARC for pointed observations longer

than 10 ks in those directions, and which did

not include clusters of galaxies or particularly

bright point sources. Table 6.2 details these observations. The three observations, one exposure of the Lockman Hole deep survey, one of

an AGN, and one of a star, were then prepared

for spectral fitting. We first examined the light

curves of the observations and eliminated times

of bright scattered solar X-ray contamination,

82
TABLE 6.1

DIFFUSE BACKGROUND BUDGET

Component Observed Fluxes _ Fluxes Before Absorption b

10 -_ counts s -1 arcmin -2 ]0 -6 countss -I arcmin -3

R12 R45 R67 R]2 R45 R67

Total Observed" 1036.6 122.3 1226

LHB 622.0 12.2 06

EPL 83.6 57.1 I05.4 245.6 60.5 107.4

PSC 19.5 8.9 11 4

Remainder 311.5 44. i 4 7

Best Fit Soft a 251.5 5.8 02 649.3 67 0.2

Best Fit Hard" 43.8 395 12.6 97.0 41.8 13.0

Stars ! 87 5.5 1 7

=FtASS data taken from b > 85 °, 45 ° < t < 270 °

bMean absorbing column was 132 x 10 _° cm -2

CThe best fit emiasion measure was 4.7 x 10 -3 cm -s pc.

aThe best fit emission measure was 2.1 x 10 -3 crn -6 pc

¢Unreaolved stellar X-ray sources which contribute to the hard thermal component.

TABLE 6.2

POINTED OBSERVATIONS USED

Observation NH t b Exposure(ks)

Total Useful

rp900029s04 0.56 x 1020 149.51 53 16 46.7 46.7

rpT00996n00 2.16 x 102° 217.00 55 45 19.6 14.4

rp200721n00 1.88 x I0 _° 8655 4091 46.9 28.0

TABLE 6.3

SPECTRAL FITS

Model log TL N/. • log TD t ND, * log TD2 ND2 • X _ v

2 Comp 6.10 4- 0.02 1.28 -4- 0.53 6.00 -6 0.02 < 0 77 6.32 4" 0.02 0.64 4- 0.24 1,22 450

0.734-0.14 1 72:[:060

2,61 4- 0.04 4 14 :k () 89

1 Comp b 6.17+ 0.03 1.764- 0.04 6.564- 0.07 < 0 05 1.39 452

1.28 4- 0.04 0 20 d: n 09

2.61 4- 0.04 < 0 o5

1 Comp c 6.10 4- 0.02 1.28 4- 0.24 6.31 4- 0.23 0 64 4- 0 35 1,26 450

0.094-0.14 6.24 4- 0.02 1 254- 042

1.584-0.26 6.134-0.01 3 01-t- 0 7l

=Units are I0 -a photons cm -_ s-l keV -t in a solid angle of 8 54 x I0 -s sr

bThe temperature of the absorbed thermal component was constrained to be the same for all data sets. The

normalizat,on of the extragalactic power law was fixed at the fitted value from the two-component model

¢The temperature of the al:_orbed thermal component was fitted independently for aI) the data sets.

83
10.0

x 1.0

m

O.l

4

2

0

-2

tjj_ "1

2 TAE Components " '_

x 1.0 " I

_=

O.1

1 TAE Component - Simultaneous T fits t_

!

-2

1.0 M

2

b

0.1

I TAE Component - Individual T fits

2 I,++ ++I+I 1

-4

0.2

} 0.0

c_ 10.2 -

O. 1.0

Energy (keV)

Fig. 6.10.-- Simultaneous fitsof three PSPC spectra from deep pointings. Top. Fit by a model

containing the EPL, LHB and a two-component TAE. _nd Panel: The residuals to the twocomponent fit.3rd Panel: Fit by a model containing the EPL, LHB, and a one-component TAE;

all the spectra are fitted with the same temperature TAE. ,lthPanel." The residuals to the one

component fit. 5th Panel: Fit by a model containing the EPL, LHB, and a one-component TAE;

each spectrum is fitted with its own TAE temperature. 6th Panel: The residuals to the onecomponent fit. Bottom: The best fittwo-component models minus the best fitone-component

models, where the one-component models were required to have the same TAE temperature, i.e.

the models in the third panel.

84
short-term enhancements (most likely a background due to auroral X-rays), and long-term

enhancements (the observations were relatively

unaffected by the last). We examined images of

the data and created an extraction region which

included most of the area inside of the ring of the

window-support structure but excluded the targets of the AGN and stellar observations. The

same extraction region was used on all observation data sets. We binned the data into pulseheight channels with a minimum of 25 counts

and created ancillary region files (ARFs).

XSPEC V.10 was used to fit the three observations simultaneously with several different

models. We first fit the data with the twocomponent TAE model: an unabsorbed thermal

spectrum for the LHB, two absorbed thermal

spectra for the TAE, and the absorbed EPL.

For the fit we fixed the power-law slope to the

Chen, Fabian, & Gendreau (1997) value of 1.46

but allowed the normalization to vary. (The fitted normalization was 12.1 keV cm -2 s-1 sr -1

keV-1.) The temperatures of the three thermal components were allowed to vary but they

were constrained to be the same for all three

data sets. The normalizations of the local component and the soft absorbed component were

fit independently for the three data sets but the

normalizations of the hard thermal component

and power law were constrained to be the same.

The data were fit in the 0.12-1.90 keV band. Table 6.3 lists the results, which show a reasonable

goodness of fit {:_ = 1.22 with v = 450). The

fitted values of the temperatures are consistent

with our broad-band analysis.

We also fit the data with the one-component

TAE model: an unabsorbed thermal spectrum

for the LHB, one absorbed thermal component

for the TAE. and the absorbed EPL. The temperatures of the thermal components and the

normalization of the power law were constrained

to be the same for all data sets but the normalizations for the thermal components were fit

individually. With two fewer fitted parameters

the reduced )2 value increased from 1.22 to 1.38.

This rather significant reduction in the quality of

the fit relative to the two-component model supports our claim that the one-component model

cannot, in general, be fit simultaneously to multiple directions on the sky. Examination of the

fitted parameters (Table 6.3) reveals a much

higher temperature for the absorbed thermal

component (logT ,-, 6.56), and lower values for

the normalizations. The fitting algorithm forced

the local emission to accept a portion of the distant emission.

To examine the situation further, we allowed

the temperature of the absorbed thermal component in the one-component model to be fit

individually to each data set. This produced

a considerable improvement in the X2 value, to

1.26, now with the same number of fitted parameters as the two-component model. However, the fitted values for the temperatures (Table 6.3) were completely inconsistent with a constant value (as expected with the large change

in the quality of the fit). The brighter the 1⁄4 keV

TAE the cooler the fitted temperature. This is

exactly the expected result if the _ keV intensity

arises from a different emission component than

the _ keV intensity. Thus the two-component

model is completely consistent with the detailed

spectral fits of the three observations while the

one-component model is not.

The results listed in Table 6.3 also demonstrate the ease with which the fitting algorithm

can shift intensities and spectral parameters between one component and another. The choice

of details used for the spectral model strongly

affects the results which are obtained. The two

separate fits to the one-component model provide an excellent example. In the first case,

when the temperature of the TAE is constrained

to be the same for all three data sets, the fitted

temperature is very high and the fitted intensities are very low; such low intensities are inconsistent with the variation of the halo emission

which has been determined by shadowing experiments. This is clearly the case with the observation in the high-intensity direction (towards

the Draco enhancement) where the fit provides

a low value for the upper limit; flux from the

TAE has been shunted to the LHB.

The difference between the temperature derived for the hot component by spectral fitting

and that derived by SED fitting should not be

considered too seriously; the two methods have

very different systematics. Whereas, in the bicyclic iteration, the normalization and temperature of the LHB was very strongly constrained

by the anticorrelation with the absorbing column, in the spectral fitting, the properties of

the LHB are less strongly constrained, and so

the fitting routine can distribute flux among the

85
variouscomponentsmorefreely,producingthe

bestfit forthosethreeparticularspectra.

6.2. The LHB

Overview: The local/distant split that produces the R1 and R2 measures of the TAE

also provides a measure of the emission due

to the LHB. We find an LHB temperature of

logTL = 6.11 (FWHM = 6.04-6.26) from the

R2/R1 band ratio 1.06 (FWHM = 0.920-1.220).

Although Snowden et al. (1998) and Snowden

et al. (2000) contain detailed discussions of a

similar analysis, as applied to the LHB, several

points are worth noting. First, the temperature of the LHB, as derived by this analysis or

by spectral fitting, depends sensitively upon the

temperature found for the TAE. The total R1

and R2 band emission must be conserved, and

the imposition of a given TAE temperature will

determine the temperature to be found for the

LHB. However, there are some checks upon the

temperature of the LHB. Since, in the Galactic

plane, the column density in many directions is

sufficient to block background 1⁄4 keV emission,

one can determine the R2/R1 ratio for the LI'IB

without TAE contamination. After marshaling

some essential data about the LHB, we will compare our measure of the LHB temperature to

those in the literature.

Shape and Size: The shape of the LHB was

determined by Snowden et al. (1998) from the

all-sky deabsorption, but the size can not be set

without knowing the emission measure within

the LHB. The standard scaling of the LI-IB is

given by the observation of the molecular cloud

MBM 12, which produces a barely significant

14 keV shadow, implying that it is just barely

within the LHB emission. The distance to MBM

12 has recently be redetermined to be -_ 90 pc

by Sfeir et al. (1999). Thus, the high scaling of

Snowden et ai. (1998), 0.23 pc (10 -6 counts s-1

arcmin-_) -l, should be used.

Internal Absorption: Sfeir et al. (1999) has

compiled a homogeneous data set of 456 lines of

sight with Na 1 D absorption spectra with which

they have mapped the Local Cavity. Their comparison of the Snowden et al. (1998) LHB contours to the Local Cavity shows that the bulk of

the LHB fits inside the dislance at which the cumulativecolumn density i.-significantly less than

2 × 1019 cm -2 (in the Na I D line, Wx = 10m,_);

only some of the more peripheral regions of the

LHB overlap directions/distances with such high

column densities.

Absorption of the LHB emission is produced

by the Local Interstellar Cloud (LIC) and several similar clouds within _ 5 pc. A mean absorbing column due to these features can be derived from Hutchinson, Warwick, & Willingale

(1998) who showed that the mean absorbing column within -_ 50 pc was about 4 x 10 is cm -2,

about a factor of two greater than the highest

LIC column densities (Lallement et al. 1995).

We will use this value for the mean absorption of

the LHB. It corresponds to approximately 3.6%

absorption of the 1⁄4 keV emission. Other investigators have used greater columns; Sidher et al.

(1996) used 6.6 x I0 Is cm -2 (-_ 5.9% absorption), Kerp (1994) and Pietz et al. (1998) used

1 x 1019 cm -2 (--8.7% absorption), and Parmar

et al. (1999) used 7 x 1019 cm -2 (-_ 43% absorption). The last of these values is completely

incompatible with optical absorption line studies.

Temperature: The model temperature of the

LHB can be determined from the R2/RI (or

R2/RIL) ratio in a direction in which absorption

by the Galactic ISM blocks 1⁄4 keV emission from

more distant sources. Figure 6.11 demonstrates

the difficulty of this problem. Numerous strong

features visible in the a keV band, mostly supernovae and super-bubbles, penetrate the Galactic absorption to contaminate the local emission,

and increase the R2/R1 ratio. One can calculate

the hardness ratio in a large window towards the

anticenter in between known sources of contamination, and one finds that the temperature is

-- logT = 6.10 (R2/R1 = 1.057). The temperature of the LHB towards the Galactic center

does seem to be higher (Snowden, Schmitt, &

Edwards 1990; Snowden et al. 2000), but the

contamination by the Galactic bulge "leaking"

through the absorption does not allow a determination with this data set.

One might think to avoid most sources of contamination by sampling just off of the Galactic

plane The only region that is near the Galactic plane and has a substantial absorbing column (1021 <Ntt < 2.0 x 1021 cm -2) is the

region of the Taurus-Auriga molecular clouds.

Absorption line data place these clouds at --. 120

pc, so there is little room for extra emission

86
p

1.8 

. 1.4

1.0

0.8- Scan 2

0 100 200 300

Galactic Longitude

Fig. 6.11.-- The _ keV hardness ratio as a function of Galactic longitude for the region within

5° of the Galactic plane. Brighter, hotter, absorbed features that "leak" into the R12 band are

labeled. "Scan" refers to region with poor exposure due to a missing scan. The horizontal bar

overlaying the data shows the mean hardness ratio over the given ranges of Galactic longitude.

components between the LHB (radius _ 100 pc

in this direction) and the absorbing clouds. A

sample of the RASS of -_ 360 square degrees

yields a R2/R1 = 1.02 + 0.01, or an equivalent Raymond & Smith model temperature of

log T = 6.09 -t- 0.01

Thus, the current analysis produces an LHB

temperature through broad-band ratios that is

consistent with independent measures also made

with broad-band ratios. The significantly lower

temperatures found by the spectral fitting of

past analyses are due to the need for a much

higher temperature TAE, when one assumes

that the TAE is due to a single thermal component.

7. Discussion

Hzstorical Background: The discovery of the

soft X-ray emission that we have come to associate with the LHB occurred as the result of

attempts to measure the soft extrapolation of

the extragalactic power law (Bowyer, Field, &

Mack 1968; Bunner et al. 1969; Davidsen et al.

1972). The ubiquity of the 1⁄4 keV emission was

a mystery that was not to be solved by the discovery of instrumental effects or incorrect modeling of the particle background, though these

problems remain. (See, for example, the discussion of the Long Term Enhancements and

the earth as a "flaming comet" in Cox 1998.)

The 1⁄4 keV emission anti-correlation with the total neutral hydrogen column led to two main

schools of thought. The first, the absorption

model (Bowyer, Field, & Mack 1968; Bunner

et al. 1969; Davidsen et al. 1972), was that the

ubiquitous emission was extragalactic and absorbed, though it was found that the absorption

cross-sections could not produce the observed

dynamic ranges (McCammon et al. 1976; Long,

Agrawal, & Garmire 1976; Burrows et al. 1984)

or band ratios (Bloch et al. 1986; Burrows 1989;

Juda et al. 1991). The second, the displacement model (Sanders et al. 1977; Cox _: Snowden 1986; Cox & Reynolds 1987; Snowden et al.

1990), was that the emission was proportional to

the unabsorbed path length, which was roughly

anti-correlated to the total column. There was a

third model, the embedded cloud model, Jakobsen & Kahn (1986), based somewhat on the McKee & Ostricker (1977) model of the ISM, that

suffered from many of the same problems as did

the absorption model (Burrows 1989). The advent of X-ray shadows (Snowden et al. 1991;

Burrows & Mendenhall 1991), observed early in

the ROSATera, showed that both models were,

to some extent, correct.

It was clear that the soft X-ray background

was composed of at least three components, the

LHB, the EPL, and some component beyond

the Galactic absorption, which we have called

the TAE. Although the Draco shadows (g =

890 -91 ° , b = 38 ° -40 °) were the first glimpse of

a hot Galactic halo, it was not the long-theorized

Galactic corona (Spitzer 1956); the 1⁄4 keV TAE

was too patchy (Burrows & Mendenhall 1991;

Snowden et al 1991; Snowden et al. 1994). It

i keV and 3 was suggested that the _ _ keV TAE

were due to different components (Burrows &

Mendenhall 1991) as the _ keV emission was

87
muchsmootherthanthe 1⁄4keVemission.Like

theinfamousLeiden-Groningen H 121cmdata

splitthat producedtheartificialseparationbetweenhighandintermediatevelocityclouds,the

RASS was split between the 1⁄4 keV and _ keV

bands and analysed by different groups.

Summary of the Argument: The current work

attempts to effect the obvious synthesis of the

two bands. Our results have shown that the

TAE is composed of at least two thermal components, perhaps more. The result is based on five

arguments: 1) That the TAE of the R12 and

R45 bands are not correlated, 2) That no single component can produce both the observed

anti-correlation of emission and absorbing column and a consistent SED, 3) That the twocomponent models fit the SED better than any

single-component model, 4) That single component models are not acceptable because they do

not allow the LHB to have a temperature that is

consistent with that measured by the B,2/R,1 ratio in the Galactic plane where there is no "leaking" background emission, and 5) That the two

component model is more consistent with detailed spectral fitting than are the single component models. The argument that the 43-keV

emission is much smoother than the 1⁄4 keV emission is no more than suggestive. With the exception of such strong features as the Draco enhancement, a single-temperature model, in order to get the proper R45/R12, will necessarily

scale down the structures seen in the R12 band.

Since much of the emission in the R45 band is

due to the EPL, the small variations due to the

TAE would become difficult to detect in the R45

band.

Although we posit two (or more) components

to the TAE, the components are not neatly segregated into the 1⁄4 keV and 33-keV bands; -,,15%

of the _ keV band flux (before absorption) is

produced by the soft component, while ,,-10%

of the 1⁄4 keY band flux (also before absorption)

is produced by the hard component. Because

the soft component does not have a significant

trend with Galactic longitude, global analyses

that utilize only the R4-R7 bands, such as Wang

(1997), will not be significantly affected by our

analysis.

Global Propertzes of the TAE: The mean surface brightness of the soft TAE component is

~ 642 counts s -1 arcmin -_ or -,, 2.6 x 10-s ergs

cm -_ s -l sr -I over the 0.1-2.0 keV range; for

the hard component, _ 110 counts s-x arcmin -2

or _ 8.5 x 10 -9 ergs cm -z s-1 sr -x over the

0.1-2.0 keV range. Even away from Loop I

(g = 330 ° + 60 °) and the Draco enhancement

(g _- 90°), the soft component is quite mottled,

though the azimuthally averaged distribution is

quite constant. The hard component, away from

Loop I, is smooth (see Figure 6.4).

These characteristics suggest that the hard

component might be identified with the Galactic corona or a Galactic halo in hydrostatic equilibrium (Wang 1997; Pietz et al. 1998), though

that identification is by no means certain. If the

hard component is the corona, with what is the

soft component to be identified?

The missing piece in halo studies is often the

distance to the emission, and this study is no

exception. Several studies have set the 1⁄4 keV

terminus a quo. The _ keV shadowing clouds

in Draco have distances z > 180 pc (Lilienthal

et al. 1991). Benjamin et al. (1996) found

distances of z ~ 285:1:75 pc to an infrared

cirrus cloud in the Lockman hole that shadows the distant X-ray emission. The determination of the terminus ad quem is more difficult. The attempts to use high and intermediate velocity clouds are difficult as the disk absorption is much stronger, and there have been

suggestions that the high velocity clouds may

have X-ray emitting bow-shocks. A search for

1⁄4 keV shadows due to high velocity complex M

(1.5 < z < 4.4 kpc) was equivocal (Herbstmeier

et al. 1995). A trivial upper limit is set by

the shadowing of the extragalactic background

by several galaxies (Barber & Warwick 1994;

Snowden & Pietsch 1995; Barber, Roberts, &

Warwick 1996; Cui et al. 1996). Distance determinations for the 43-keV emission by shadowing

are even more difficult as high and intermediate velocity clouds do not have sufficient column

density to produce a measurable shadow.

External Galames: Observations of external

galaxies provide further clews to the distribution of the TAE. The 1⁄4 keV flux due to the

TAE (both components) is ~ 706 counts s-_

3 keV flux is --- 47 counts arcmin -2, while the

s -_ arcmin -_. Thus, despite the absorption

by our own galaxy, the closer external galaxies

should have detectable halos.

The X-ray halos of face-on galaxies are contaminated by the point sources in the disk, but

this contamination has diminished as resolu88
tionhasimproved.Thebeststudyto dateis

thatof theclosestface-onMilkyWayanalogue,

M101,whichwasobservedbytheROSA T PSPC

(Snowden & Pietsch 1995). If we use the relative D25 values for the two galaxies to scale

to the equivalent of the galactocentric radius of

the sun, a radius of -_ 8!5, and if we assume

that the entire M101 flux, after point source removal, is due to the halo, we find 1570 counts s-I

arcmin -2 at 1⁄4 keV, while the flux at 43-keV at

that radius is indistinguishable from the background. Significant 1⁄4 keV flux (,-, 350 counts

s -l arcmin -2) extends to -_ 20 _. (The scale

length of the 1⁄4 keV emission is ,-- 0.4D25/2, and

the scale length of the _ keV emission would appear to be similar.) Thus, both the strength and

extent of the soft component of the TAE is consistent with that observed in M101. The temperature determined from the R2/R1 ratio for

the M101 halo is, however, cooler, logT = 5.86.

Edge-on spiral galaxies for which extra-planar

X-ray emission has been detected are often strong

star-burst galaxies. Three non-starburst edgeon spiral galaxies have ROSAT-detected X-ray

halos: NGC 891, NGC 4631, and NGC 4565.

Of these, NGC 4565 has the lowest stellar formation rate (SFR, as measured by LFtR/D_5 )

and has no detected extended diffuse ionized gas

(DIG). Vogler, Pietsch, & Kahabka (1996) detected some extraplanar diffuse X-ray emission,

but were unable to determine a scale-height,

and could only say that the temperature was

consistent with a few times 106 K gas. NGC

4631, which has a higher SFR, is a disturbed

galaxy that may be interacting (Weliachew, Sancisi, & Gudlin 1978; Rand, Kulkarni, & Hester

1992; Rand & Stone 1996; Donahue, Aldering,

& Stocke 1995). It has a strong nuclear star formation, and nuclear Ha filaments, which may be

related to a 1⁄4 keV emission region. The _ keV

emission has a HWHM z-extent of almost 3 kpc,

a radial extent of about 3 kpc (0.25 D25/2),

and a temperature logTRs = 6.48 (Wang et al.

1995) or iOgTMKL = 6.31 (Dahlem, Weaver, &

Hackman 1998). NGC 891, which has an even

higher SFR, has a more uniformly extended Xray halo; the _ keV emission has a HWHM zextent of 2.1 kpc, a radial scale length of 4.1 kpc

(0.3 D2_/2), and a temperature of logT = 6.54

(Bregman & Houk 1997).

The only extragalactic detection (in an edgeon spiral) of a component that might be similar

to the soft component of the TAE is that by

Wang et al. (1995) in NGC 4631, where they

detect a component with logT < 5.78. Dahlem,

Weaver, & Hackman (1998) also detect a component with logT = 5.76, but Read, Ponman,

& Strickland (1997) do not. Such a soft component could not be detected in NGC 891 due

to the higher Galactic absorption. Lack of detection of the soft TAE component in external

galaxies is not surprising; it is subject to strong

absorption by both the Milky Way and the host

galaxy. If the soft component has a lower scaleheight than the hard component, then not only

will the absorption by the host galaxy be important, but the ROSATresolution would not allow

separation of the soft TAE from disk emission.

Pure Speculation: Both NGC 4631 and NGC

891 have SFRs, based upon LFtR/D_5, that are

similar to that of the Milky Way (Rand 1998).

Both NGC 4631 and NGC 891 have halo temperatures that are consistent with the logT = 6.46

that we have found for the hot component of the

TAE. Given the similarities of halo temperature,

and that the extent of the extra-planar diffuse

ionized gas is roughly dependent upon the SFR,

it seems likely that the smooth hard component

of the TAE has z-scale heights ,-- 2 kpc, somewhat smaller than the 4 kpc exponential scale

height fitted by Pietz et al. (1998). Both M101

and NGC 891 imply that the hard component

should have a radial scale length ,_ 0.35D25/2.

or ~ 4 kpc, which is much smaller than the 15

kpc scale length fitted by Pietz et al. (1998),

but implies a much more disk-like distribution

than the model of Wang (1997).

The _ keV emission in the Milky Way is quite

symmetric, the mean flux in the south being very

similar to that of the north, the emission is quite

symmetric in NGC 891, and the asymmetry of

NGC 4631 is thought to be an inclination effect

(Wang et al. 1995). With a disk-like distribution, this symmetry would argue that the hard

component is hot gas in hydrostatic equilibrium,

where the symmetry across the disk is enforced

by the gravitational potential.

The soft component, however, is very different. The soft component is patchy, and there is

a distinct north-south asymmetry in the 1⁄4 keV

TAE; the northern hemisphere is about 1.9 times

stronger than the southern hemisphere (Snowden et al. 1998), and the temperature of the

soft component in the southern hemisphere may

89
belowerthanthatof thenorth(Snowdenetal.

1998)asis impliedby Figure11of Snowden

et al. (2000).Thesepropertiessuggestthat the

softcomponent istheresultoflocalphenomena.

Perhapsthesoftcomponent iscomposedof the

plumesof hotgasejectedfromGalacticchimneys.Snowdenetal. (1998)suggestedthatthe

Dracoenhancement wastheextraplanarbreakoutof theCygnussuperbubble, thoughtheinterveningmid-latitudeabsorptionmakestesting

this modeldifficult. (It is interestingto note

thatsuchamodelsuggestshattheDracoemissionoccurstoz _ 2 kpc.) However, as the 1⁄4 keV

TAE is not restricted to lower latitudes any local phenomenon must be fairly uniformly distributed or must send gas to fairly high z. Shelton (1998) has suggested that supernovae remnants in the halo might dominate the soft X-ray

emission and produce the patchy structure.

As has been pointed out by Gehrels & Williams

(1993) in the context of stellar coronae, the cooling curve for a hot thin plasma has two local

minima at temperatures that would emit in the

soft X-ray range. Of course, the exact temperatures of those minima (IogT = 5.9 and 6.6)

do not agree particularly well with the observed

temperatures (log T = 6.06 and 6.46). Reference to Schmutzler & Tscharnuter (1993) shows

that the location of the minima of the cooling

curve are sensitive to the metallicity and previous ionization history of the gas. It is only suggestive that the minima of the cooling curve may

correspond to the temperatures of the dominant

components. In this case, the ubiquity of the

soft component suggests that the soft component may represent some part of the hard component at a more advanced age.

Future Work: Determining the true thermal

structure of the Galactic halo will be difficult.

Next generation X-ray instruments will have

lower background contamination, but typically

will have small fields-of-view, so using angular

variations caused by absorption to separate local and distant components will be more difficult. Although spectral resolution will be good

at 3 keV, the spectral resolution below the carbon edge, where the LHB and the soft component of the TAE has most of their emission,

remains poor. Experience with the DXS has

shown that given high spectral resolution, understanding the spectrum of the LHB is still difficult, and the parameter space allowed by nonequilibrium plasma codes is large.

Attention will also need to be given to incorporating the X-ray TAE into our understanding of the halo optical/UV emission/absorption

line ratios. Many line ratios are adequately explained by photoionization due to massive disk

stars, whose ionizing radiation is absorbed and

hardened by passage through the gaseous disk

(see, for example, the review by Rand 1998).

However, there are still outstanding difficulties,

such as the run of line ratios with z-height, in

which the X-ray halo may play a part.

REFERENCES

Arabadjis, J. S., & Bregman, J. N. 1999, ApJ, 510,

806, submitted

Arendt, R. G., et al. 1998, ApJ, 508, 74

Arnand, K. A. 1996, in Astronomical Data Analysis

Software and Systems V, ed. G. Jacoby &

J. Barnes, 17

Barber, C. R., Roberts, T. P., & Warwick, R. S.

1996, MNRAS, 282, 157

Barber, C. R., & Warwick, R. S. 1994, MNRAS,

267, 270

Benjamin, R. A., Venn, K. A., Hiltgen, D. D., &

Sneden, C. 1996, ApJ, 464, 836

Bloc.h, J. J., Jahoda, K., Juda, M., McCammon, D.,

Sanders, W. T., & Snowden, S. L. 1986, ApJ,

308, L59

Bowyer, C. S., Field, G. B., & Mack, J. E. 1968,

Nature, 217, 32

Bregman, J. N., & Houk, J. C. 1997, ApJ, 485, 159

Bunner, A. N., Coleman, P. L., Kraushaar, W. L.,

McCammon, D., Palmieri, T. M., Shilepsky,

A., & Ulmer, M. 1969, Nature, 233, 1222

Burrows, D. N. 1989, ApJ, 340, 775

Burrows, D. N., McCammon, D., Sanders, W. T., &

Kraushaar, W. L. 1984, ApJ, 287, 208

Burrows, D. N., & Mendenhall, J. A. 1991, Nature,

351,629

Chert, L.-W., Fabian, A. C., & Gendreau, K. C.

1997, MNRAS, 285, 449

Cox, D. P. 1998, in The Local Bubble, ed. D. Breitschwerdt & M. Freyberg, 121

Cox, D. P., & Reynolds, R. J. 1987, ARA&A, 25,

3O3

Cox, D. P., & Snowden, S. L. 1986, Advances in

Space Research, 6(2), 97

Curl, W., Sanders, W. T., McCamrnon, D., Snowden,

S. L., & Womble, D. S. 1996, ApJ, 468, 117

Dahlem, M., Weaver, K. A., & Heckman, T. M.

1998, ApJS, 118, 401

Davidsen, A., Shulman, S., Fritz, G., Meekins, J. F.,

Henry, Ft. C., & Friedman, H. 1972. ApJ, 177,

629

9O
Deiker, S., Kelley, R. L., Lesser, A., McCammon,

D., Porter, F. S., Stahle, W. T. S. C. K., &

Szymkowiak, A. E. 1997, in Proceedings of

the Seventh International Workshop on Low

Temperature Detectors, 113

DomgSrgen, H., & Mathis, J. S. 1994, ApJ, 428, 647

Donahue, M., Aldering, G., & Stocke, J. T. 1995,

ApJ, 450, L45

Gehrels, N., & Williams, E. D. 1993, ApJ, 418, L25

Guillout, P., Haywood, M., Motch, C., & Robin,

A. C. 1996, A&A, 316, 89

Hasinger, G., Burg, R., Giacconi, R., Schmidt, M.,

Triimper, J., & Zamorani, G. 1998, A&A,

329, 482

Herbstmeier, U., Mebold, U., Snowden, S. L.,

Hartmarm, D., Burton, W. B., Moritz, P.,

Kalberla, P. M. W., & Egger, R. 1995, A&A,

298, 606

Hutchinson, I. B., Warwick, R. S., & Willingale, R.

1998, in The Local Bubble, ed. D. Breitschwerdt & M. Freyberg, 283

Jakobsen, P., & Kahn, S. M. 1986, ApJ, 309, 682

Juda, M., Bloc.h, J. J., Edwards, B. C., McCammon,

D., Sanders, W. T., Snowden, S. L., & Zhang,

J. 1991, ApJ, 367, 182

Kerp, J. 1994, A&A, 289, 597

Kuntz, K. D. 2000, Ph.D. thesis, University of Maryland

Kuntz, K. D., & Snowden, S. L. 1998, Cookbook

for analysis procedures for rosat x.rt observations of extended objects and the diffuse

background, part ii: Mosaics, Technical report, NASA/GSFC

Lallement, R., Ferlet, R., Lagrange, A. M., Lemoine,

M., & Vidal-Madjar, A. 1995, A&A, 304, 461

Lampton, M., Margon, B., & Bowyer, S. 1976, ApJ,

208, 177

l_,aor, A., Fiore, F., Elvis, M., Wilkes, B. J., & McDowel], J. C. 1997, ApJ, 477, 93

Lilienthal, D.. Wermmacher, A., Herbstmeier, U., &

Mebold, U. 1991, A&A, 250, 150

Long, K. S., Agrawal, P. C., & Garmire, G. P. 1976,

ApJ, 206, 411

McCammon, D., Meyer, S. S., Sanders, W. T,, &

Williamson, F. O. 1976, ApJ, 209, 46

McKee. C. F., & Ostricker, J. P. 1977, ApJ, 218,

148

Parmar, A N., Gualnazzi, M., Oosterbroek, T., Orr,

A., Favata, F., Lumb, D., & Malizia, A. 1999,

A&A, submitted

Pietz, J., Kerp, J., Kalberla, P., Burton, W. B.,

Hartmann, D., & Mebold, U. 1998, A&A,

332, 55

Rand, R. J. 1998, Proc. Astron. Soc. Aust., 15, 106

Rand, R. J., Kulkarm, S. R., & Hester, J. J. 1992,

ApJ, 396, 97

Rand, R. J., & Stone, J. M. 1996, A J, 111, 190

Raymond, J. C., & Smith, B. W. 1977, ApJS, 35

Reach, W. T., Wall, W. F., & Odegard, N. 1998,

ApJ, 507, 507

Read, A. M., Ponman, T. J.. & Stricldand, D. K.

1997, MNRAS, 286, 626,668

Reynolds, R. J. 1991, in The Interstellar Disk-Halo

Connection in Galaxies, ed. H. Bloemen, 67

Sanders, W. T., Edgar, R. J., Liedahl, D. A., & Morgenthaler, J. P. 1998, in The Local Bubble,

ed. D. Breitschwerdt & M. Freyberg, 83

Sanders, W. T., Kraushaar, W. L., Nousek, J. A., &

Fried, P. M. 1977, ApJ, 217, L87

Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998,

ApJ, 50O, 525

Schmutzler, T., & Tscharnuter, W. M. 1993, A&A,

273, 318

Sfeir, D., Lallement, R., Crifo, F., & Welsh, B. Y.

1999, A&A, submitted

Shelton, R. L. 1998, ApJ, 504, 785

Sidher, S. D., Summer, T. J., Quenby, J. J., &

Gambhir, M. 1996, A&A, 305,308

Snowden, S. L., Cox, D. P., McCammon, D., &

Sanders, W. T. 1990, ApJ, 354, 211

Snowden, S. L., Egger, R., Finkbeiner, D., Freyberg,

M. J., & Plucinsky, P. P. 1998, ApJ, 493, 715

Snowden, S. L., et al. 1997, ApJ, 485, 125

Snowden, S. L., Freyberg, M. J., Kuntz, K. D., &

Sanders, W. T. 2000, ApJS, in press

Snowden, S. L., Hasinger, G., Jahoda, K., Lockman,

F. J., McCammon, D., & Sanders, W. T.

1994, ApJ, 430, 601

Snowden, S. L., & Kuntz, K. D. 1998, Cookbook

for analysis procdures for rosat xrt observations of extended objects and the diffuse

background, part i: Individual observations,

Technical report, NASA/GSFC

Snowden, S. L., McCammon, D., & Verter, F. 1993,

ApJ, 409, L21

Snowden, S. L., Mebold, U., Hirth, W., Herbstmeier, U., & Schmitt, J. H. M. M. 1991, Science, 252, 1529

Snowden, S. L., & Pietsch, W. 1995, ApJ, 452,627

Snowden, S. L., Schmitt, J. H. M. M., & Edwards,

]3. C. 1990, ApJ, 364, 118

Spitzer, L. 1956, ApJ, 124. 20

Vallerga, J., & Slavin, J. 1998, in The Local Bubble,

ed. D. Breitschwerdt & M. Freyberg, 79

Vogler, A., Pietsch, W., & Kahabka, P. 1996, A&A,

305, 74

Wakker, B. P., & Boulanger, F. 1986, A&A, 170, 84

Wang, Q. D. 1997, in The Local Bubble, ed. D. Breitsdawerdt & M. Freyberg, 503

Wang, Q. D., Walterbos, R. A. M., Stealdey, M. F.,

Norman, C. A., & Braun, R. 1995, ApJ, 439,

176

Weliachew, L., Sancisi, R., & Gu_lin, M. 1978, A&A,

65, 37

91

Chapter 7: The Stellar Contribution to the SXRB

Abstract

Using stellar luminosity functions derived from ROSAT data, the contributions of

Galactic stars to the diffuse X-ray background are calculated for ROSAT PSPC passbands. The model follows that of Guillout et al. (1996), but uses ROSA T rather than

Einstein data to determine the intrinsic luminosity distributions. The model provides

nearly the correct number of stellar sources in deep ROSAT surveys, though it may

underestimate the number of stars at low fluxes.

The contribution of unresolved stellar sources to the ROSAT All-Sky Survey at

the Galactic poles is 8.28, 5.46, and 1.66 x 10 -6 counts s -1 arcmin -_ in bands R12

(1⁄4 keV), R45(_ keV) and R67(1.5 keV), respectively, which is equivalent to 1.8, 1.2

and 0.35 × 10-13 ergs cm -_ s -1 deg -:. These values correspond to 2%, 19%, and 37%

of the TAE emission. The uncertainty in the normalization could increase this quantity

by a factor of _ 1.6.

1. Introduction

1.1. Motivation

Recent advances in the study of the diffuse

softX-ray background have attempted to place

limitson the amount of emission from a diffuse extragalacticbackground. Cen & Ostriker

{1999) suggested that such a background might

exist in the 1⁄4 keV and _ keV range as a result of gas accretion on sub-cluster size objects

forming filamentary structures on cosmological

scales, the exact form of these structures being

very sensitive to the cosmological model. At

higher energies (above 1 keV) truly diffuse X-ray

extragalactic emission is ruled out as such gas

would produce unacceptably large perturbations

in the microwave background. Such considerations do not hold at 1⁄4 keV, but the multiplicity of backgrounds (at least two diffuse Galactic components, as well as unresolved Galactic point sources, and unresolved extragalactic

sources) make the detection of a diffuse extragalactic background difficult. ROSAT observations have been used to attack the problem on

several different fronts.

The extragalactic background due to unresolved point sources can be characterized as a

power-law at E>I keV, and a variety of results,

including recent Chandra observations The work

of (Mushotzky et al. 2000) suggests that this

power-law extends, without a break, to lower

energies. Snowden et al. (2000) and Kuntz &

Snowden (2000), following the work of Snowden

et al. (1998), have studied the emission that

is not due to unresolved extragalactic sources.

Snowden et al. (1998) separated the 1⁄4 keV emission originating from beyond the Galactic absorption from that due to the Local Hot Bubble and this work was extended by Kuntz &

Snowden (2000) to the remainder of the ROSAT

band. Snowden et al. (2000) placed an upper limitto the diffuseextragalacticemission

at 1⁄4 keV. Although Kuntz & Snowden (2000)

characterizedthe emission originatingbeyond

the Galacticabsorptionin the _ keV band, they

were not able to effecta separationof the emission from beyond the Galactic absorption into

itsvarious components, the Galactic halo,the

diffuseextragalacticemission, and unresolved

Galactic point sources. Thus, although there

isalmost no doubt that thereisa Galactichalo,

the upper limitto the diffuseextragalacticemission in the _ keV band remains the total emission originating beyond the Galactic absorption.

Included in the measure of the emission originating beyond the Galactic absorption was the

contribution due to unresolved Galactic point

sources. It is thus necessary to calculate that

contribution to further tighten the limits on the

diffuse extragalactic background.

From the model of Cen &: Ostriker (1999),

the diffuse extragalactic background will not be

smooth, and the clumpiness is a function of

q3
thecosmological model.Thisdissertationseeks

to placelimitsonthespatialfluctuationsthat

mightbedueto thediffuseextragalacticbackground,thusit isnecessaryto knowthedegree

of fluctuationdueto unresolvedGalacticpoint

sources.

It is thus particularly interesting for the study

of the diffuse extragalactic background to study

the luminosity function of Galactic point sources,

that isto say,stars.

levels, but will turn over and become constant

below some limiting flux. The published relations do not reach the "turn-over" point, so it is

difficult to know how far to extrapolate in order

to determine the total point source contribution.

The following model calculation uses the best

current data to derive the luminosity function at

high Galactic latitude, which will be of greatest

interest to those studying the diffuse extragalactic background. Rather than the CLF, we calculate the more useful form,

1.2. Previous Work

A stellar log N-log S relation in the form of a

cumulative luminosity function (CLF, the number of sources, N, with fluxes greater than S) has

been calculated by Guiliout et al. (1996) from

a mixture of Einstein IPC and ROSAT PSPC

data, using the Bienaymd, Robin, & Crdzd (1987)

model of the spatial distribution of stars. The

N(S)dlogS, (7.1)

which allows the calculation of

os' S N ( S)d log S, (7.2)

the flux in unresolved sources. The model is,

in principle, capable of calculating the stellar

contribution at low Galactic latitudes, but some

Guillout et .al.(1996) study concentrated on poorly determined input parameters greatlyincomparing themodel distributionofstellarsourcescreasethe uncertaintiesnear the Galacticplane.

with observed count rates as a function of position, the complement of the information required for studies of the diffuse X-ray background.

There are a number of difficulties in applying

the published Guiilout et al. (1996) CLF to the

problem of unresolved stellar sources. First, the

Guillout et al. (1996) study published the CLF

for the entire ROSA T PSPC band-pass. Studies

of the diffuse X-ray background are generally interested in only some portion of that band-pass.

Given the strong differences in the absorption

by the ISM from one end of the band-pass to

the other, no simple scaling can be applied to

the published relations to obtain, for example,

the relation for the 1⁄4 keV band. Second, as can

be seen in Guillout et al (1996), and again in

our Figure 7.1, although older stars have lower

X-ray luminosities, their population is considerably larger and they dominate the luminosity

function, especially at lower flux levels. However, this is the population that is the least well

characterized in the Guillout et al. (1996)study;

they were forced to take data from Einstein IPC

studies of field stars. Far superior ROSATstudies have since become available (Schmitt, Fleming, & Giampapa 1995: Schmitt 1997) Third,

because the source population has a finite distance limit, the CLF will increase to lower flux

2. Data

In order to calculate the contribution of stellar sources, the stellar populations were binned

by spectral type 87. and age .A. We used the

spectral-type divisions of Schmitt, Fleming, &

Giampapa (1995) and Schmitt (1997) and the

age divisions of GuiUout et al. (1996), all of

which are listed in Table 7.1.

The division by age follows Guillout et al.

(1996): the 0-0.15 Gyr class is represented by the

Pleiades (171 ROSAT detected members, _- 0.1

Gyr, 115 pc), the 0.15-1.0 Gyr class is represented by the Hyades (191 detected members,

"--0.6 Gyr, 45 pc), and the 1.0-10.0 Gyr class is

represented by the field population.

The data are insufficient to determine the Xray luminosity functions for subsamples of the

1-10 Gyr field population, which, especially for

the later type stars, will be nearly uniformly distributed through the age range. However, stars

in the 1-2 Gyr age range will have a very different z-height distribution than stars in the 5-10

Gyr age range. Thus, for calculation, the stars

in the 1-10 Gyr age class are subdivided into the

age ranges used by Bienaymd, Robin, & Crdzd

(1987) (see Table 7.2) although the same X-ray

94
-8 -6 -4 -'2 0 -fl -6 -4 -2 0

Log,,(S/counts s-') Log,,(S/counts s")

Fig. 7.1.-- The luminosity function, N(S) d log S, for band R17 shown with the fractional contributions of the different age classes (left) and spectral types (right).

luminosity function for all the age sub-divisions

in the 1-10 Gyr age class.

A number of other well-studied clusters could

be included with the Hyades for a sample of

stars in the 0.15-1.0 Gyr group; Coma Berenices

(42 detected members, ,-, 0.5 Gyr, 80 pc) and

Praesepe (56 detected members, _ 0.6 Gyr, 160

pc). X-ray emission from Coma Berenices members is similar to that from Hyades members

(Randich, Schmitt, & Prosser 1996), while that

from Praesepe members is lower than that from

(coeval) Hyades members (Randich & Schmitt

1995). Given that the Hyades would seem to

be representative of its age class, and that there

is a relatively larger sample available from the

Hyades cluster than from other clusters, the

other clusters have not been included in this age

class. However, it should be borne in mind that

the Hyades luminosities may need to be adjusted

downwards to truly represent its age class.

Although X-ray emission has been detected

from nearly every spectral type, the dominant

contribution by stellar sources is from low-mass

main sequence dwarfs, due both to intrinsic luminosity and the number of sources.

Stars of spectral type A have been excluded

from the model. The intrinsic X-ray luminosity probability distribution is poorly determined

for the younger age classes, and is currently undetermined for the 1-10 Gyr age class, which

would dominate the emission from the A spectral type 1. The intrinsic X-ray luminosities,

where measurable, are about an order of magnitude below those of F stars, and the mid-plane

number density is lower than F stars, by a factor of 0.005 for the 1-10 Gyr age class. Since the

younger age classes will have low z-heights, the

error introduced will be minimal for high Galactic latitudes. Guiliout et al. (1996) estimate the

maximum contribution from A stars to be ,_ 2%.

White dwarfs have also been excluded from

the calculation as their contribution is typically

two orders of magnitude below that of mainsequence stars in band R12, and significantly

lower in band R47.

RS CVn systems have been excluded from the

calculation. Although they have high luminosities, the median intrinsic luminosity, p -,_ 30.3

(Dempsey et al. 1993), their number density

is quite low, probably less than 1 x 10 -4 pc -3

(Ottmann & Schmitt 1992) as is their scale

height (,-- 325 pc).

Giants and super giants have been excluded

from the calculation as their proper calculation would require extensive analysis beyond the

ITo determine the intrinsic luminosity function for the

old disk A stars, one would need access to the ROSAT

All-Sky Survey data from which the Bright Source Catalogue (Voges et al. 1999) was formed in order to determine the upper limits for undetected stars.

95
scope of this paper. Nearly all giants are Xray sources with logLx .-- 27.3, similar to solar

type stars (SchrSder, Hiinsch, & Schmitt 1998;

HiJnsch et al. 1998) but the source population

is much smaller, and more closely restricted to

the plane.

We have also excluded Population II stars

from the calculation. If a comparison between

disk RS CVn systems and Population II binaries is any indication of the relative emission

from single Population I and Population II stars,

then the Population II stars should have emission that is more than an order of magnitude below that of Population I stars (Ottmann, Fleming, & Pasquini 1997).

2.1. Model Inputs

For each spectral-type/age-classbin,the contributionto the luminosity function is determined by four parameters: /Jand or,the mean

and dispersionof the intrinsicluminosityprobabilityfunction,c,which describesthe z distributionof the stars,and no, the mid-plane (z = 0)

density of stars.

Following Schmitt, Fleming, & Giampapa

(1995) and Schmitt (1997), we have assumed

that the intrinsicluminosityfunctionfor a particularspectral-type/age-classbin is given by

the Iog-normaidistribution,

P(logL.) = _--_exp-_ 1 1 [logL_-/J] 2

(7.3)

The parameters p(ST, -4) and a(Sy,.4) were derived from the ROSAT data sets listed in Table 7.1.

Since these data sets contain points with

upper limits, the parameters were determined

by fitting the Kaplan-Meier estimator (Schmitt

1985) of the cumulative distribution function,

where the uncertainty in the Kaplan-Meier estimator was taken from Fiegeison & Nelson

(1985). The uncertainties were derived using

a bootstrap method (Babu & Feigelson 1996).

We corrected for binaries in the same manner

as Schmitt, Fleming, & Giampapa (1995): we

have divided the flux equally between members

of binary systems. G uillou t et al. ( 1g96) sh owed

that correction for binaries made little difference

to the luminosity function.

Since there were no data available for late M

stars of the 0-0.15 Gyr and 0.15-1.0 Gyr age

classes, the /_ for the 1-10 Gyr age class was

extrapolated to the younger age classes using

the general trend observed in the other spectral classes; a A log Lx = 0.92 decrease between

the 0-0.15 Gyr and 0.15-1.0 Gyr classes, and a

A logLx = 0.98 decrease between the 0.15-1.0

Gyr and 1.0-10.0 Gyr classes. For the late M

stars the same value of _r was used for each age

class, that of the oldest age class; this tends to

increase the number of sources with high fluxes.

To convert from X-ray luminosity to countrates, we used the inverse of the conversion used

by the original authors. Schmitt, Fleming, &

Giampapa (1995) used the conversion factor e

(5.30HR + 8.31) x 10-12 ergs cm -2 counts -1,

where HR, the hardness ratio, is defined by

(R47-R82)/(R47+R82) and R_z is the count

rate in band zx. For each spectral type, HR

is a linear function of the log of the luminosity. (See Appendix F.) Typically, that linear

relation was determined for a span of Iog Lx

smaller than used in the model calculation and

unreasonable values of I-IR could be obtained;

HR was limited to the range -1 to 0. Stern,

Schmitt, & Kahabka (1995) used the conversion

6.0 x 10 -12 ergs cm -2 counts -1, which is somewhat softer than that used by Schmitt, Fleming,

& Giampapa (1995). Stauffer et al. (1994) used

a conversion of _ 8.4 x 10-12 ergs cm- 2 counts- 1

(for Ntt= 0), characteristic of logT = 7.06.

Due to the shape of the conversion curve, this is

also the conversion factor for log T = 6.88.

The spatial distribution for each age-class was

assumed to be given by equations A1 and A2 of

Bienaym_, Robin, & Crdzd (1987), where c(.4),

the axial ratio for ellipsoids describing the disk

populations, were taken from the "best fit" column of their Table 1 and are listed in our Table 7.2.

To maintain some consistency between the

initial mass function (IMF), the stellar formation rate (SFR), and the az-Age relation, the

mid-plane density of each spectral-type/age-class

bin, no(ST,A), was derived in a manner similar

to that used by Guillout et al. (1996). The disk

was assumed to have an age of 10 Gyr, a constant SFR, and an invariant IMF. The IMF was

2This relation was derived for a Raymond & Smith model

two component corona fitted to a small number of high

S/N spectra. See Fleming et al. (1995) for a complete

description.

96
I"...

.I

nn

,<

,<

f.,Q

E-,

o3

o

i

_o

0

z

_o

I.

0

Z

0

a

0

Z

>

p..

6-

rn

E

• QD _'_ I'_

V_A

_666

_o_AAA

A

E

0

rm_

"_,_.

._-VV

,_._._ -_

m_

m

Q,

e,

E

-7

c_

e_

0'_ 0")

"T._I

i

r. r.

I

Q

ct_

Q.

r_

1 m I ,-

x _ _ _

_, -_ _ _ .__ -

.E R'_ ._ _ = = o

-_ : v _._

o , _ =..

_-_= = .= _ __

.I: _''_ t; .- _._

i. llO_ ° "-

c_

0
TABLE 7.2

MID-PLANE STELLAR DENSITIES

Type Age Group (Gyr)

0-0.15 0.15-1 I-2 2-3 3-5 5-I0

A 2.16 × 10 -4 3.93 x 10 -4 1.01 x 10 -4 2.55 x 10 -7 0 0

F 2.93 x 10 -4 6.14 x 10 -4 4.40 x 10 -4 2.59 x 10 -4 2.29 × 10 -4 6.99 x I0 -s

G 2.78 x 10 -4 5.82 x 10 -4 4.17 x 10 -4 2.88 x 10 -4 4.39 × 10 -4 9.15 x lO -4

K 3.31 x 10 -4 6.93 x 10 -4 4.96 x 10 -4 3.42 x 10 -4 5.22 x 10 -4 1.18 x 10 -3

M(ear]y) 4.28 x 10 -3 8.96 × 10 -3 6.42 × 10 -3 4.43 × 10 -3 6.75 x 10 -3 1.53 x 10 -2

M(late) 2.45 x 10 -3 5.12 × 10 -3 3.67 x 10 -3 2.53 x 10 -3 3.86 x 10 -3 8.73 x 10 -3

c a 0.0140 0.0279 0.0457 0.0662 0.0867 0.0958

aParameter describing the ellipticity of the stellar distribution. See equations A 1 and A2 of Bienaym6,

Robin, & CrUz6 (1987).

taken from Haywood, Robin, _z Crdzd (1997):

dN = m-(l+_)dm (7.4)

where

z = 0.7 for rn < Ira®

x = 1.5 for lm®< rn < 3m® (7.5)

z = 2.0 for m > 3rn®.

The total number of stars was normalized by

comparing the number produced by the above

formula with r < 7 pc in the 8 < Mv <

15 range (i.e., the K and early M stars) with

the number actually observed (79). The fraction of the number of stars in each age-class

was calculated using the main-sequence lifetimes

listed in Scalo (1986} and the mass-absolute

magnitude-spectral type relations of Kroupa,

Tout, _z Gilmore (1993).

2.2. Method

For each spectral-type/age-class bin, the intrinisic luminosity distribution was evaluated for

A log Lx = 0.1 intervals. For each distance interval, Ar = lpc, from 0 to 3000 pc, the value

of log Lx was adjusted for r -_ diminution and

absorption by the ISM. The vertical distribution of the ISM was taken from Lockman (1984),

and was assumed to be plane-parallel. The absorption cross-sections were taken from Morrison & McCammon (1983). The absorption was

calculated assuming that the stellar emission is

well described by an equilibrium plasma model

with log T = 6.4. Although the bulk of the stars

lie beyond the bulk of the absorption (see Figure 7.2), only the R12 band reaches column densities for which r = 1. (The next most sensitive

band is R17, for which r _, 1 at .-- 5.5 x 10_°

cm-_.) Thus, the luminosity function is not too

sensitive to absorption by the ISM.

The intrinsic luminosity function was then interpolated to a standard A log (S/counts s-1) =

0.1 spacing, and scaled by the size of the volume element and the relative stellar density

for that z-height. The luminosity function for

each spectral-type/age-class was summed over

z-height, and scaled by the total number of stars

in that spectral-type/age-class.

The aggregate luminosity function for band

B.17 is shown in Figure 7.1 for the north Galactic pole. As might be expected, the luminosity

function is dominated by the older age classes

because, although their intrinsic luminosities

may be an order of magnitude smaller than the

younger age classes, they account for the bulk of

the number of sources. For similar reasons, the

luminosity function is dominated by the longerlived spectral classes. The high-count-rate end

of the luminosity function is strongly influenced

by the younger age-classes as they have larger

intrinsic luminosities and smaller scale-heights

The lower half of Figure 7.3 shows the effect of

the changes of parameters on the cumulative luminosity function of a single spectral-type/ageclass. The increments for each of the parameters

are their typical uncertainties. As can be seen,

except for the uncertainty in no, whmh would

98
2.5

_" 2.0

E

1

0

,_ 1.0

e_

2

o 0.5

O.0

0

'_ -1

-4

500 zooo zSoo 2000 2500 3000

R(pc)

&p.=0.15

..... Ao=O.15

- -&N,=2X

.... 5c=0.003

.... Mog,.T=O,2

--&p.=0.15

....... &o=0.15

- - -&N.=2X

.... _,c=0.O03

.... &tog.,T= 0.2

/

/

/

-tO -8 -6 -4 -2 0 2

Iog.(S/count= a")

Fig. 7.2.-- The relative surface number densities

for each age group as a function of z-height, compared to the absorbing column through which

they are observed. Top: The total absorbing

column through which and object of z-height R

is observed. The r = 1 point for band R12 is

marked; to achieve the same r for band R47 requires a much greater column density. Bottom:

The relative surface densities for ._tars of a given

age group at a z-height R. Each curve has a midplane number density of unity.

cause increases at all count rates, uncertainties

in _, Tej;, and/_ only cause significant changes

for count-rates abov, 10-2 counts s-1. Uncertainties in c and N_ cause very little change at

any count rate.

3. Comparisons

3.1. The Guillout et a/. Model

It is difficult to check our model against that

Fig. 7.3.-- The effect of uncertainties of the

input parameters on the unresolved flux (top)

and the cumulative luminosity function (bottom)

shown as the ratio between the luminosity functions. For the M(early), 1-2 Gyr age class we

show the luminosity function calculated with a

single parameter increased by A, where the A

is the typical uncertainty divided by the fiducia]

luminosity function.

ofGuillout et a.]. (1996) because, although their

c values can be reconstructed from their Figure

3, their p values recovered from their references,

and their no values approximated from their description of the IMF and SFR, the c, values are

provided in neither the references nor the paper proper. Figure 7.4 shows the results of our

model broken into the same age/spectral classes

as Guillout et al. (1996) and can be compared

directly to their Figure 8.

The two models agree fairly well except for

the "late M" class. For this class we find a factor of-_ 10 more sources than Guillout et al

99
1

o

g

-2

-3

-4

A

-3 -2 -1

lol.(s)

A 0 "'_"

1

-_'%

-4 -3 -2 -I

Iog.(s)

Z

-2

iI......................... 0

0 "<'\_

._ %

\x

"\\ "-,

-, -3 -2 -I 0

Iog,.(s)

0

2 ......... ' ......... • ......... f .........

K

!

iiiiiiiii•\\ "..

_3 ........ , ......... , ......... • ......

-4 -3 -2 -! 0

ioz.(s)

m 0

o"-t

-2

o

;¢

_o"-1

......... , ......... , ......... ¢ .........

M,

'\

• <\

........... N"_.

"\ "<..

"% %"..

"\ %'..

\ \

-3 -2 -1

loi.(s)

-2

-3 -3

-4 0 -4

......... t ......... i ......... , ........

M,

% •

"% %

.. "\ \

-3 -2 -1

Iol.(s)

Fig. 7.4.--The model calculationsfor the north Galactic pole subdivided by age and spectral

class.Solid:the totalluminosityfunctionforthe spectralclass,Dotted: the contributionfrom the

0-0.15Gyr age class,Dashed: the contributionfrom the 0.15-iGyr age class,and Dot-Dashed:

the contributionfrom the 1-10Gyr age clasess.Me and Mt indicateforthe earlyand lateM stars

respectively.

(1996);itisnot clearwhether thisdue to a different IMF, a differentlow-mass cut-off(0.08M@

was used here),or merely a differentma.ss-tospectral-typeconversion.

The current model uses the same c valuesas

Guillout et al. (1996). The values of no can

be checked forthosespectral-type/ogeclassbins

for which Guilloutet al.(1996)provide plotsof

the cumulative luminosityfunctionthat extend

below the "turn-over".The no agree fairlywell

(to within a factorof _ 2).

Since the valuesof p used by Guilloutet al.

(1996) are known, one could, in principle,reconstructthe valuesof crused from theirFigure

8. However, the computational effortoutweighs

the usefulnessof thistest.The valuesof crused

for the currentmodel are clearlysuperiorgiven

the greatercompleteness of the data from which

they are drawn. Overall,the values of _r used

here appear to be somewhat largerthan those

used by Guillout et al. (1996),causing a horizontalshiftof the cumulative luminosityfunction to higher count-rates.The current cumulativeluminosityfunctionmatches the Guillout

et al. (1996) values between logS = -4.2 and

logs = -2.7. At higherluminosities,our band

RI7 cumulative luminosityfunctiongrows to 1.6

that of Guillout et al. (1996) at logs = -0.6,

while at lower luminosities,our cumulative luminosity functiongrows to 1.3that of Guillout

et al. (1996)at logs = -4.8. The overallcomparison of the two models can be seen in Figure 7.5.

3.2. Point Source Counts

We have checked our model against the number of stellar point sources found in surveys and

against the stellar log N-log S relation of Kraut100
1

-_ ._....

•- .', :

-3 ....... _......... , ......... , ......... , ....... ":-'!

-5 -4 -3 -2 -I 0

log,.(S/counts s")

2.S f"I T_

2.0

1.5

0

0.5

0.0

-3 -2 -I 0

log,.(S/counts s")

Fig. 7.5.-- Comparison between the current

model and that of Guillout et al. (1996). Solid."

the total CLF for the current model, Dotted: the

current model CLF for various age groups, o:

the total Guillout CLF, D: the Guillout CLF

for 1-10 Gyr, _-: for 0.15-1 Gyr, +: for 0-0.15

Gyr.

ter et al. (1999). The comparison is listed in

Table 7.3 and shown for the R47 band in Figure 7.6.

As can be seen, the model somewhat underestimates the number of stellar sources for deep

ROSAT surveys in band R47, but the predictions are within Poisson uncertainties, with the

exception of the survey of Carballo et al. (1995),

which has a low identification fraction, and the

survey of Boyle, Wilkes, _ Elvis (1997). The

predwted number listed in the Table is the number of sources with flux above the survey limit at

the (/, b) position of the survey, in deg -2, multipli,,d by the survey area and the fraction of

sour,es for which identifications have been made

(the "'ID completeness").

qhe bright end of the model can be compared

to the Krautter et al. (1999) log N-log S relation for band R17 (shown for three values of S)

and matches reasonably well for logs < 0. The

diffi,'ulty with this comparison is that Krautter

et at (1999) do not note whether this relation

is for b = 90* or for some lower Galactic latitud,' Predictions for both b = 90* and b = 30 °

are g_ven in the table. Transforming the Krautter ,.r al. (1999) log N-log S function to the

R47 hand using their meal_ stellar hardness raFig. 7.6.-- The band R47 predicted cumulative

luminosity function with measured values, o:

deep surveys listed in Table 7.3, O: values from

the Krautter et al (1999) log N-log S relation.

tio, HR = 0.11 - 0.12, we find that our model

slightly underestimates the observed function.

There are two potential methods for correcting the model. The first is to adjust the normalization, that is, scaling the whole model. This

will increase the number counts at all countrates by a factor of _- 1.6. This would increase

the number of K and early M stars in the solar neighborhood to 118. which is well outside

the Poisson uncertainty..Such an increase might

be accomodated if the IM F/SFR were changed 3.

The second method would be to change the

shape of the cumulative luminosity function; increasing p or a for an3 of the young populations

would preferentially incr,.ase number counts at

higher count-rates wit ho_,t increasing the number counts at lower count-rates.

3However, it should be noted t hat the most likely change

necessary for the IMF/SFR. o. the grounds of stellar observations, is to decrease the low-mass end of the IMF

at early epochs t_, account f,,r the seeming de_ciency

of old late M stars (Bes-_,.ll ,_. Stringfellow 1993). The

normalization used only h a.d early M stars, and it is

not clear whether t he obs,-rva, ional evidence can tolerate

any similar reduction in the ,,umber of early M stars. If

so, the relative contributi,,n of young stars would be increased, thus increasing th_ c.mulative luminosity function at higher count-rate

101
4. Model Description

It is instructive to see how the luminosity

function changes with detection band, Galactic

latitude, and Galactic longitude.

Figure 7.7 shows the stellar luminosity function at the north Galactic pole for several ROSAT

bands. The R12 band is substantially different

from the R45 band due both to the different instrumental response {which produces a horizontal shift in the curves) and the differing effects of

absorption (which increases the number counts

below the peak and decreases them above the

peak, though not at the highest fluxes).

Figure 7.8 shows the ratio of the stellar luminosity function at b = 30 ° to that at the Galactic

pole for the R12 and R47 bands. We have assumed a plane-parallel absorption model. For

typical deep ROSAT surveys, there is a factor

of ,,- 2.5 greater number of sources at b = 30 °

than at b = 90 ° in band R47. The increase in

number counts for band R12 is less severe due to

absorption. As can be seen from the Figure, at

b = 30 ° the effect of the Galactic radial density

gradient is negligible.

5. Results

Figure 7.9 shows the luminosity functions in

3 o °

2 g"

i

-2[ --V\

-8 -s -4 -2 o

I.og,.(S/counta s _)

Fig. 7.7.-- The luminosity function for different bands in the direction of the north Galactic

pole. .s'ohd hne: band R17 +: band RI2, D:

band B45. O: band R67, Dotted line: band R47

(almosl indistinguishable from band R45)

2

z

u

_q

3

2

1

0

-1

-2

,

-6 -4 -2

Log,.(s)

o

Fig. 7.9.-- The luminosity function for mainsequence stars, white dwarfs, and extragalactic

objects, for band R12 (dotted) and band R47

(solid). The relations are shown for a total

NH = 5 x 10 z9 cm -2 (further to the right) and

for NH = 2.1 x 10_° (further to the left). For

band R47, the relations for the different values

of NH are indistinguishable. The band R47 relation for white dwarfs is not shown, but would be

significantly below the R12 relation. The point

source detection limit for the ROSAT All-Sky

Survey is also shown.

PSPC bands RI2 (dotted) and R47 (solid)for

main-sequence stars,white dwarfs, and the extragalacticluminosityfunctionderivedby Hasinger

et am. (1998)4. For band R47, above the RASS

detectionlimits,stellarsources account forone

thirdto one halfof pointsources,in agreement

with a number of studies,while for band RI2,

the fractiondepends upon the totalabsorbing

column.

Figure 7.10 shows the contribution to the

RA,.9,.9of unresolved stellarsources as a function of Galactic latitude. The model predicts

the unresolvedstellarcontributionto the diffuse

X-ray background measured by the ROSA T AllSky Survey to be

8.28 x 10-6 counts s-z arcmin --_(band RI2),

4The white dwarf luminosity function is shown to demonstrate its inconsequentiality. It was calculated using the

data of Fleming et al. (19q6) with the a._sumption that

intrinsic X-ray luminosity ,d white dwarfs extends down

to at least IogLx -- 27.5

102
3.0 r , i , ,

2.5

.a 2.0

O

,,,1.5

e,,.,

1.0

0.5

, . r I ' ' ' _ " " " _ '

R12

......... t=O

t=90

--- t=lSO

FL , J ' ' "' I ' '

"iI R4 7

I I k ",,

-8 -6 -4 -2 0 -8 -6 -4 -2 0

Log,.(S/counts s") Log,.(S/counts s-')

Fig. 7.8.-- The ratio of the luminosity function at b = 30 ° to the luminosity function at b = 90 °.

Solzd: The ratio for t = 90 °, Dotted." the ratio for f = 0°, Dashed. • the ratio for t = 180 °.

5.46 x 10-s counts s-1 arcmin -_ (band R45),

and

1.66 x 10-6 counts s-z arcmin -2 (band R67)

for the north Galactic pole. Using the standard

stellar conversion factor of 6 x 10-12 erg cm -_

count -t , the unresolved stellar contribution is

1.80 x 10-la ergs cm -_ s-t deg -_ (band R12),

1.18 × 10-is ergs cm -_ s-I deg -2 (band R45),

and

3.58 × 10-14 ergs cm -= s-t deg -2 (band R67).

It should be noted that the Fleming et al. (1995)

conversion ranges from 3 to 8.3 x 10 -12 erg cm-2

count- I for hardness ratios in the range -1 to 0.

in bands 11.12,R45, and R67, the unresolved

stellar sources can account for, at most, 2%, 19%

and 37% of the flux that is thought not to be due

to the Local Hot Bubble or the unresolved extragalactic point sources (essentially the component that Kuntz & Snowden (2000) attribute to

the Galactic halo) The variation of the contribution from unresolved stellar sources with Galacti, latitude would be unnoticeable in the RA.',;S

given Ihe uncertainty in the RASS and, more

importantly, the variable effects of Galactic absorpl ion.

F"

e-

'm

e,

o

t._

i

e20J

15

10

• R47

RI2

R45 +

R67 x

+ O

0 D 0

qd- + +

×

X X X X X

20 40 80 80 1oo

Gilictic IAtitude

Fig. 7.10.-- The contribution of unresolved stellar sources to the RASS as a function of Galactic

latitude. It is assumed that the point source removal limits were 0.025 counts s-l in band R12

and 0.02 counts s- 1 in band R45 and band R67.

If the normalization suggested by the deep

surveys discussed in §3.2 is correct, then these

values would increase by a factor of ,-, 1.6.

103
TABLE 7.3

PREDICTIONS

Survey (t, b) Band Area ID Source

2000 deg 2 Comp. Limit

I0 -3 c/s

Pred. Obs.

Bower et al. (1996) ( 96.4, 29.8) 47 0.21 0.90 .763

Schmidt et al. (1998) (149.5, 53.5) 47 0.28 1.00 .459

Zamorani et al. (1999) (270.2,-51.8) 47 0.20 0.84 .309

McHardy et al. (1998) ( 85.0, 75.9) 47 0.17 1.00 .175

Carballo et al. (1995) (162.6, 51.1) 47 2.20 0.57 1.42

Boyle, Wilkes, & Elvis (1997) various 47 3.92 0.89 1.7

1.9 3

2.8 3

2.4 4

3.0 5

4.6 9

11.9 20

Comparison to measured log N-log S function

Krautter et al. (1999) various 47 9.52

47 95.2

47 952

Krautter et al. (1999) various 17 17

17 170

17 1700

0.40-0.72 0.6

0.030-0.038 0.04

0.0014-0.0016 0.00 4

0.6-1.01 0.6

0.058-0.073 0.04

0.0032-0.0036 0.00 4

6. Future Improvements

In the future, this nmdel will be applied to

the problem of the X-ray "infilr' of the Galactic

plane. A number of improvements will be made.

1.) The My-mass relation has been improved

by (Henry & McCarthy 1993), and there is likely

to be a better low-mass IMF available. (Bessell

& Stringfellow (1993), for example, suggest that

there is a deficit of old late M stars.) Since the

M stars dominate the main-sequence luminosity function, changes in the number of M stars

could cau_ significant changes in the luminosity

function.

2.) Schmitt, Fleming. & Giampapa (1995)

and Schmitt (1997) ha_e shown that the hardnes_ ratio of a star depeuds upon its luminosity;

fainter stars are softer/, ooler. This change can

amount t,, a change of a factor of two in the

flux-to-counts ,'onversio, factor from one end of

the probability distribulion to the other. Since

the intrin_icallv more I,minous stars have less

negative hardness rat 0c_. the conversion factor is

higher for higher luminosity stars; the high luminosity tail is at tenualed and the low luminosity

tail is augmented Alth,_ugh the hardness ratio

dependent conversnon was used for the 1-10 Gyr

populatiou, the luminoqty-hardness-ratio relatioiD has not yel been del,rmined for the younger

populatio.s

3.) The field A star luminosity distribution

should be determined. This will be important

for working close to the plane.

4.) Similarly, the RS CVn luminosity function should be included for work near the Galactic plane.

5.) The Hyades luminosity function could be

improved by the point source data from pointed

PSPC observations. These data are in- i land, but

have not been examined

6.) The contribution from OB stars and giants of all spectral classes needs to be r,'assessed.

REFERENCES

Babu, G. J., & Feigelson, E. D. 1996, Astr,,statistics

(Chapman & Hall)

B_ll, M. S., & Stringfellow, G. S. 1993 ARA&A,

31, 433

Bienaym_, O., Robin, A. C., & CrUz& M. 1987,

A&A, 180, 94

Bower, R. G., et al. 1996, MNRAS, 281 -,'_

Boyle, B. J., Wilkes, B. J. & Eivis, M I'_97, MNRAS, 285, 511

Carballo, R., Warwick, R. S., Bar, ol_, X.,

Gonzalez-Serrano, J. I., Barbe_ C. R.,

Martine_Gonzalez, E., Peres-Foun .on, I., &

Burgos, J. 1995, MNRAS, 277, l:_l:

Cen, R., & Ostriker, J. P. 1999, ApJ, 514 1

Dempsey, R. C., Linsky, ,1. L., Fleming V. A., &

Schmitt, J. H. M. lkl. 1993. ApJS. _6. 599

Fiegelson, E. D., & Nelson. P. I. 1985, Apl 293, 192

104
Fleming, T. A., Molendi, S., Maccacaro, T., &

Woher, A. 1995, ApJS, 99, 701

Fleming, T. A., Snowden, S. L., Pfeffermann, E.,

Briel, U., & Greiner, J. 1996, A&A, 316, 147

Guillout, P., Haywood, M., Motch, C., & Robin,

A. C. 1996, A&A, 316, 89

Ha.singer, G., Burg, R., Giacconi, R., Schmidt, M.,

Triimper, J.) & Zamorani, G. 1998, A&A,

329, 482

Haywood, M., Robin, A. C., & Cr4z4, M. 1997,

A&A, 320, 428

Henry, T. J., & McCarthy, D. W. 1993, A J, 106, 773

Hiinsch, M., Schmitt, J. H. M. M., SchrSder, K.-P.,

& Zickgraf, F.-J. 1998, A&A, 330, 225

H/insch, M., Schmitt, J. H. M. M., Sterzik, M. F.,

& Voges, W. 1999, A&AS, 135, 319

Krautter, J., et al. 1999, A&A, 350, 743

Kroupa, P., Tout, C. A., & Gilmore, G. 1993, MNRAS, 262, 545

Kuntz, K. D., & Snowden, S. L. 2000, ApJ

Lockman, F. J. 1984, ApJ, 283, 9(}

McHardy, 1. M., et al. 1998, MNRAS, 295, 641

Morrison, R., & McCammon, D. 1983, ApJ, 270,

119

Mushotzky, R. F., Cowie, L. L., Barger, A. J., &

Arnaud, K. A. 2000, Nature, in press

Ottmann, R., Fleming, T. A., & Paaquini, L. 1997,

A&A, 322, 785

Ottmann, R., & Schmitt, J. H. M. M. 1992, A&A,

256, 421

Randich, S., & Schmitt, J. H. M. M. 1995, A&A,

298, 115

Randich, S., Schmitt, J. H. M. M., & Prosser, C.

1996, A&A, 313, 815

Scalo, J. M. 1986, Fundamentals of Cosmic Physics,

11, 1

Schmidt, M., et al. 1998, A&A, 329, 495

Schmitt, J. H. M. M. 1985, ApJ, 293, 178

Schmitt, J. H. M. M. 1997, A&A, 318, 215

Schmitt, J. H. M. M., Fleming, T. A., & Giampapa,

M. S. 1995, ApJ, 450, 392

SchrSder, K.-P., H/insch, M., & Schmitt, J. H. M. M.

1998, A&A, 335, 591

Snowden, S. L., Egger, R., Finkbeiner, D., Freyberg,

M. J., & Plucinsky, P. P. 1998, ApJ, 493, 715

Snowden, S. L., Freyberg, M. J., Ktmtz, K. D., &

Sanders, W. T. 2000, ApJS, in press

Stauffer, J. R., CaiUauh, J.-P., Gagn6, M., Prosser,

C. F., & Hartman, L. W. 1994, ApJS, 91,625

Stern, R. A., Schmitt, J. H. M. M., & Kahabka,

P. T. 1995, ApJ, 448, 683

Voges, W., et aJ 1999, A&A, 349, 389

Zamorani, G., et aJ. 1999, A&A, 346, 731

105

Chapter 8: Deabsorption

Abstract

Thischapterdiscussestheoptimalmannerbywhichto correctX-rayimagesfor

absorptiondueto theGalacticISMin theabsenceof high-resolution(_ 1_)surveys

of tracersof the total absorbingcolumn. Thischaptertreatsthreeissues.First,

althoughthe IRAS 100 pm survey is adopted as the best currently available highresolution (_ 5 _) "surrogate" tracer of the absorbing column, it is necessary to scale

the "surrogate" tracer to approximate most closely the true absorbing column, without

having a measure of that absorbing column other than the absorbed X-ray images.

Second is the purely technical issue of the optimal algorithm for calculating the emission

from beyond the absorbing column using the observed absorbed emission and the map of

the surrogate tracer. Third is the uncertainty in the deabsorbed emission, particularly

the uncertainties that produce spurious spatial variation in the measured deabsorbed

emission.

We remove regions with significant molecular contribution by removing regions with

H l column densities greater than 3 × 10 _° cm -2. We find that the IRAS 100 pm

emission is a better tracer of the total X-ray absorbing column than the H 1 21 cm

emission. The 100 pm emission may be a better tracer of the absorbing column because

1.) it traces dust (i.e., absorbers) rather than the gas containing the absorbers or 2.) it

traces all phases of H (H2, H I, and H If), rather than just the neutral component. The

best agreement between X-ray absorbing column and 100/Jm emission occurs when the

100 _m emission is scaled to the column density of H ! in the velocity range 4-75 km

s -1 Although we have little knowledge of the distribution of H II, and H II is poorly

traced by 100 pm emission, we show that the error produced by absorption by H II is

< 3% in band R12 and < 1% in band R45.

1. Iatroduction

Deabsorption is the process by which one uses

a map of the absorbing column density to determine the amount of emission before and behind

the screen of absorption. The goal of the operation is either to calculate the mean emission

from beyond the absorbing screen, as in Chapter

5, or to determine the structure of the emission

from beyond the absorbing screen, as in the following chapter.

1.1. Method

In order to correct for the effects of Galactic absorption, we have assumed the "standard"

model for the distribution of the X-ray emission

and absorption; the solar system is surrounded

by a Local Hot Bubble (LHB) beyond which the

bulk of the absorption is due to diffusely distributed material in the Galactic disk. Exterior

to the disk is X-ray emission due to the Galactic halo as well as the X-ray emission due to the

extragalactic background. The observed X-ray

flux, Io, is given by

Io = IL + ]De -(NH)a(v'Ntt's°)

"_-IE e-(NH}a(v'N_''qB) (8.1)

where IL is the local (LHB) X-ray emission,

Iv is the contribution due to the extragalactic background by unresolved point sources, and

ID is the distant component of the X-ray emission including the Galactic halo and the putative emission from a diffuse extragalactic background. Note that we are assuming that the

local emission is "fiat" on the scale of region

being deabsorbed. The e(v,N//,3) is the effective cross-section, which is a function of the

energy band, v, the total absorbing column, NH,

and the shape of the background spectrum, _q.

Throughout the following chapter, NH is the total absorbing column density, expressed in terms

of the total column density of hydrogen nucleons. In order to simpli_' the deabsorption pro107
cess, we produce

I'o = Io - IEe -Cub)°(_'reH's)E (8.2)

so that we fit

l_o = IL '[" ]D e-(Nb)a(v'NH'SD). (8.3)

In order to accomplish the deabsorption, we

must first produce an Io image by removing the

flux due to the unresolved extragalactic point

sources, for which we have assumed the functional form given by Chen, Fabian, & Gendreau

(1997). One must also correct the X-ray mosaics

for the effects of spatially varying point source

removal (§2.1). In order to find IL and ID we

must bin the 15 image by the absorbing column

density, NH (§2.2), to produce an I' - NH relation that can be fitted by Equation 8.3.

Before fitting, we must determine SD, the

shape of the spectrum of the X-ray background.

This background will include emission by the

Galactic halo and the putative extragalactic

background, as well as other, _s yet unidentified components. The shape of the background

spectrum is discussed in §3.

Before creating the I'-NH relation one must

also construct a map of the total absorbing column No single tracer will follow all of the possible states of the absorbing gas. The bulk of the

absorption will be due to gas associated with

the neutral hydrogen, but the spatial resolution

of existing neutral hydrogen surveys is insufficient (,-- 30') to resolve the fine-scale structure

of the H I. The IRAS I00 pm survey is the best

surrogate tracer, as it has much higher angular

resolution (,,- 5'), but it may not track the entire

H I column, and it traces other components as

well (i.e., H2 and. perhaps, H II). The manner

in which we can use the primary tracer (H 1 21

cm emission) and the surrogate tracer (the I00

pm emission, If00) to form the best currentlypossible tracer of the total absorbing column is

discussed in §4.

Once the I'-Nn relation has been fit, and the

quantities It. and lo have been determined, two

different methods can be used to calculate the

image of the distant emission. These methods

are compared in §5.

1.2. Errors

In each section we will estimate the magnitude of the errors introduced by the deabsorption process. There are three different types of

errors to be considered. The first type of error

is in the fit coefficients. While the error in the

fit parameters is the prime concern if one is attempting to determine the magnitude of the distant emission, when one is interested primarily

in the structure of the distant emission, the uncertainty in the fit parameters enters only when

one determines the relative size of the fluctuations compared to the background. We will see

that the uncertainty in the fit parameters is reduced in band R47 as the foreground emission

is negligible and is treated as a fixed parameter.

The second type of error is the deviation of

the fit from the mean measured I_) - N// relation. The uncertainty in this function contributes to the uncertainty in the residual to the

fit. The uncertainty in the value of the fit at a

given value of Nn is typically small as the function must fit the observed points. Changing the

scaling of the absorbing column, for example,

may change the fit parameters, but will do little to change the fitted relation itself. Errors of

this type will be dependent on Nn, and should

be visible as a correlation of emission with Nn

within the image of the distant emission.

The third type of error is in the absorbing

column. This is the most serious type of uncertainty forstudiesof the structureof the distant

emission as systematic errorsin the absorbing

column could causespuriousdeviationsfrom the

mean I_)-NH relation.This type of uncertainty

isalsothe most difficultto determine as the variation in the column densityof H Ifis virtually

unknown at small angular scales.

The relativeimpacts of varioustypesoferrors

issummarized at the end of §4.

2. Io

2.1. Point Source Correction

The X-ray mosaic must be correctedfor the

position-dependentpoint source detectionlimit

(PSDL) and pointsource removal. When creating the mosaics,the pointsourceswere removed

to the localPSDL. This means that,on average, pixelsin regions with lower PSDLs (i.e.,

I08
better PSDLs) will have more flux removed due

to point source removal than will pixels with

higher PSDLs. To correct for the spatially variable flux removal, we set a global PSDL which is

higher than the PSDL over the bulk of the mosaic. Regions with higher PSDL are removed.

To each mosaic pixel is then added the average

flux that would be contributed by point sources

whose brightnesses fall between the local PSDL

and the global PSDL.

There are two classes of point sources for

which we must correct. For extragalactic objects, we have used the Hasinger et al. (1998)

log N - log S relation 1.

One must also correct for unresolved Galactic stars. The model for creating the luminosity

function as seen from earth, including the effects of absorption, is described in Chapter 7_.

We calculated that model for the Galactic longitude and latitude for each field, for total absorbing columns of 0.5, 1.5, 2.5, and 3.5 x 102°

cm -2. For each image pixel, the correction was

obtained by interpolating to the correct column

density.

The magnitude of the corrections, and the relative amounts due to Galactic and extragalactic point sources can be seen in Figure 8.1 for

the field 138+52. As expected, because of the

relatively low column densities of our selected

] Computational}y, this process is not s trivial one. The

PSDL is in counts s -1 after absorption by the local N/4,

in a given band. The log N - log S relation is in ergs

cm -2 s -t after absorption by NH = 5 X 1019 cm -2,

in the R47 band. One must convert both the local and

global PSDLs from the count-rate after absorption by

NH to the count-rate after absorption by 5 x 101_ cm -2

by multiplying by

e-- N H °( N H ,S) e5 X lOZS °(5 x lOte,$)

e(5 x 101e--NH )O'(5 xIOtg--NH,S) ,

where the spectral shape S is taken ¢o be a power law

of photon index 2.0 for consistency with Hxsinger et al.

(1998). The count-rates are then converted to energies

using the conversions of Hasinger et al. (1998) and their

log N-log S relation used to determine the average flux

per pixel produced by point sources with brightness between the local and global limits. The flux is then convetted b_ck to a count-rate in the proper band after absorption by Nil by the reverse of the above process. Note

that the correction is dependent upon NH.

2Since the model already includes the effects of absorption, and changing the absorption by factors of two

causes little change in the observed luminosity functions.

one need not execute the same computational mess required for the extragalactic luminosity function.

too

8O

2

o eO

_ 4o

.... T .... i r , • r .... i .... q ....

R1L2

0

60 R47

_- 50

_ 30

i,o

I0

ol i': ..........

0.000 0.005 0.010 0.015 0.020 0.025 0.030

PSDL (counts s°')

Fig. 8.1.-- Typical correction for variable point

source detection limits. For both bands R12

and R47 the correction has been made to 0.02

counts s-1. Solid: correction due to extragalactic objects, Dotte& correction due to Galactic

stars. The corrections are shown for three column densities: 102° cm -_ (the greatest correction), 2 x 1020 cm -2, and 3 x 10_° cm -2. Typical

PSDL are 0.001-0.03 counts s -1 in band R1L2

and are 0.0005-0.02 in band R45.

fields, the correction is nearly independent of

colunm density in band R47. Note that the

correction due to Galactic stars is only about

10_. of the correction due to extragalactic point

sources where the corrections are the greatest,

though a significantly higher fraction where the

correction is smaller. Because the total flux of

unresolved Galactic stars is small for all point

source detection limits, the uncertainty in the

observed luminosity function for Galactic stars

will have minimal impact upon the point source

corre,'tion.

109
2.2. NH Binning

In order to determine the mean I'o - NH

relation, once the X-ray mosaic has been corrected for the variable PSDL, the data must

be binned by the absorbing column. There are

two ways of executing this binning. The crude

method is to use the rate images, and to sum

the uncertainties in quadrature 3. A more refined method is to return to the raw count, background count, and effective exposure images, bin

each quantity separately, and then determine

the X-ray rate for each NH bin 4. The crude

method weights all pixels evenly and can produce unreasonably large uncertainties in cases

where the effective exposure varies greatly (e.g.

edge pixels where the effective exposure can be

near zero). The more refined method provides

more accurate count ratesand uncertaintiesand

effectivelyweights more heavilythe higherexposure regions.

Itshould be noted thatin both cases,the uncertaintyfor the X-ray flux in any Nn bin will

be the uncertaintyin the mean of that bin. AISlf i is the index of the n pixels having h < N/4 <

h + Ah, then the X-ray rate, R, its uncertainty, e, and

the absorbing column, NH for a given bin will be

R = _ R,I.,

t

o = ,_/n, and

NH = _ NH,/n

!

respectively, where the subscribed quantities are the

pixei values.

¢]f i is the index of the n pixels having h < NH <

h + Ah, then the X-ray rate, R, its uncertainty, o, and

the &baorbing column, N H for a given bin will be

*

|

respectively, where C,, B,, and E, are the raw counts,

background, And effective exposure in pixel i. The uncertainty in the background, B,, is eussumed to be much

smaller than the Poissonian uncertainty in (',.

ternately, one could calculate as the uncertainty

the dispersion of the X-ray values in a given bin.

However, this latter method produces very large

values of the uncertainty per pixel and very poor

fits. Since the fits are attempting to determine

the mean/i, and the mean Io, use of the uncertainty of the mean X-ray emission in a given bin

is justified.

With the exception of Figure 8.4 and Table 8.2, the refined binning method has been

used throughout this paper.

3. _re/f

As can be seen in Equation 8.3, the effective cross-sections are a function of the shape

of the background X-ray spectrum, _. We have

taken as the background spectrum that found

by Kuntz & Snowden (2000). This spectrum is

composed of two components, each of which is

modeled by thermal equilibrium spectra, with

log Ts = 6.056 and log TH = 6.462. The relative strengths of the two components, Azs and

¢V'H, varies by a factor of ,,. 4, and is dominated by the variation in .h/'s. We have calculated effective cross-sections for the mean highGalactic ]t/'H/.N's, as well as a factor of two

higher and lower. Figure 8.2 compares these

effective cross-sections with those derived from

single-component thermal equilibrium models.

As might be expected, the R1L2 band effective cro_-sections are dominated by the soft

thermal component, while the R67 band effective croas-sections are dominated by the hard

thermal component. The R45 band effective

cross-sections, however, are not well represented

by either single component within the column

density range encountered in the deabsorptions

presented here (5 x 10 z9 cm -2 to 3 x 10 _° cm -_,

see the following section.).

In order to determine the correct AfH/Azs for

a given field, we first fit Equation 8.3 using the

RASS data for the field. We then select those

parts of the field within a narrow range of column densities for which there are a large number of image pixels, usually 0.9 x 1020 < NH <_

1.0 x 10_° cm -2, and determine the mean absorbing column for that region. Using the Il)

determined for bands Rl and R2, reabsorbed

by the mean column, as well as the data from

bands R4-R7, in the selected parts of the field,

110
weforma spectralenergydistribution.WedeterminethevaluesofHH andN's by fitting the

two-component spectrum to the spectral energy

distribution in a manner similar to that used in 1.0x10-20

Kuntz & Snowden (2000). Using the values of

AfH and Afs we can then create a two-component s.0x_0-21

Raymond & Smith model spectrum from which

_J

6.0xlO -21

to calculate the o'eff.

4.0xlO -21

3.1. The Uncertainty due to as//

2.0xlO -21

One can calculate the effect of using a _el!

derived from a one-component model spectrum 0

when the "true" background is produced by the sxt°-2z

two-component spectrum described above. For

column densities from 5 x 1019 cm -2 to 3 x 1020 6_10-2z

cm -2 we have calculated

Io = 1 + e -NHc'(lvn's) (8.4) _ 4×to -2_

|

b

at 5 x 10is cm -2 intervals where the background

spectrum was the two component model. We fit 2_to -2_

the Io-NH relation where _e!! was calculated

using a single thermal component model of the o

background spectrum. We then calculated the 4.0xto-ZZ

fractional difference in the 1L and 1/9 fit parameters and determined the maximum fractional

deviation between the true and fitted relations, a.o_to -_

The results are given in Table 8.1. The greatest =..

effect is upon the fit parameters, 1L and 119, with

fractional differences up to 10%. However, the "2 z'°xl°-_

fractional difference between the true lo-Ntt relation and the fit to that relation are very small, t.o_lo__

< 0.9%. This source of uncertainty will not directly contribute to the production of spurious

fluctuations, o

18

Errors due to an uncertainty in .hftt/Afs will

be substantially smaller than those listed in Table 8.1.

4. NH

4.1. Some Considerations

Deabsorption requires a tracer of the total absorbing column, NH. The material responsible

for the absorption exists in at least three phases

of the ISM, the H l, the H II, and H_. High

resolution maps of these three phases do not exist. However, the dominant phase is H l, which

is traced by the IRAS 100 /_m emission, I100,

for which there is a high resolution (5') all-sky

|.2x10 20 :..., ...... , ......... , ......... , ......... , ......... ,._ ......

R82

_:_:_:_:::

5

........ , ......... , ......... l ......... t ......... , ........

........ l ......... i ......... T......... , ......... i ........

R67

!iiiiiiiiiiiiiiiiiiiiii iii::--

19 20 21 22 23 24

tot./NI

Fig. 8.2.-- Comparison of effective crosssections. In each panel the three solid lines represent two-thermal component emission models

for the Galactic halo taken from Kuntz & Snowden (2000). The AfH/Afs for the three lines are

from top to bottom, 1.13, 0.56, and 0.28. The

dotted lines represent single-component thermal

equilibrium models: Top: logT = 6.0 to 6.2,

AlogT = 0.05, Middle Row: iogT = 6.2 to 6.5,

AlogT = 0.05, Bottom Row: logT = 6.3 to 6.6,

A log T = 0.05. Lower temperature models have

higher effective cross-sections. Band R82 is the

same as band RIL2.

111
TABLE 8.1

EFFECT OF SPECTRAL SHAPE

Band .Iv'H/Ms Iz, Io Residual b

ram. max

xl0 -2 xlO-_ xlO-4 x10-4

RIL2 1.14 -2.72 1.00 -333 2.67

RIL2 0.56 -5.90 5.44 -4.45 5.79

RIL2 0.28 -7.90 2.53 -6.29 9.23

R45 1.14 9.32 -9.35 -1.94 4.73

R45 0.56 4.29 -4.45 -1.34 2.89

R45 0.28 -3.92 3.55 -3.26 6.86

R67 0.56 4.16 -4.16 -.003 .006

All quantities are fr_ctional differences. The caJculations were

made by ca]culatin 8 [o = I + • -NPI''2 for the a_ calculated from

the Kuntz & Snowden (2000) model, and then fit the relation

]0 "_ .[L + .[De -NH¢'I , where the al were calculated from a onecomponent thermal equilibrium model.

aThe B.IL2 band comparison wM a Raymond & Smith model

with log2" = 6.1, for R45, IogT = 6.35, for R67, logT = 6.45.

bThe residual is measured over 5 x I0 t_ < NH < 3 x I0 a°.

survey. Thus, we attempt to produce a map of

the total absorbing column, N#, by scaling 11oo

to the column density of the 1,1 I, that is, we

assume that the magnitude of the total absorbing column density Is well appro:mmated by the

Hi 21 cm column density, but that the structure

is better appro_mated by the higher resolution

maps of the dust emission. However, 1100 traces

the other phases as well, so the problem is not

just forming a tracer of NH, but determining the

extent to which the 1100 traces the dominant

phase, and determining whether the deviation

of the 11oo from the H 1 is due to other absorbing components. We begin by considering the

relation between H i and 1100, followed by the

relation of a bootstrapped N_v to the observed

X-ray flux, 1o.

The relation between 11oo and H 1 is an interesting topic of its own (Arendt et al. 1998;

Reach, Koo, & Heiles 1994; Reach, Wall, & Odegard 1998). Regions with H 1columns > 3x 1020

usually contain H2, which is well traced by 11oo

(Reach, Wall. & Odegard 1998). However, in

molecular regions, 1100 will depend sensitively

on the dust temperature which is determined

at resolution., no better than ,_ 1°. Given that

there are no high resolution maps of 1-12 (or its

surrogate, CO) for most regions, and the conversion between /100 and NH will be made uncertain due to t he dust temperature, regions possibly containing H2 are best avoided.

In regions not including molecular gas, the

H 1/1100 ratio will depend upon the metallicity,

dust to gas ratio, and dust temperature, as well

as the column densities of H II. The H I/I100

ratio varies on all scales of interest (see Chapter 4), but the dominant source of the variation is unknown. It is understood that 11oo pre_

dominately traces the "warm disk gas"; H I in

high or intermediate velocity clouds (HVCs or

IVCs) is not well traced by 1100 either due to

decreased metallicity or decreased dust temperatures (Wakker & Boulanger 1986). However,

even excluding regions with HVCs and IVCs,

the dust temperature correction proposed by

Schlegel, Finkbeiner, & Davis (1998) does not

remove the H 1/1100 variation.

The variation of the H 1/1100 ratio might be

due to the column density of H II, which is not

yet known for more than small parts of the sky.

However, emissivity of dust associated with H I!

is thought to be much smaller than that associated with H I (Hauser et ai. 1998). Thus, it

should be kept in mind that since 1100 may trace

the H II as well as the H 1, it may be a better

tracer of N_ than H I alone.

It should be noted that even if a perfect correlation could be found between the H I and

1100, it would still not be clear which component of the H I contribul_, to the absorption of

112
X-rays. Tile averageH I spectrumis characterizedbya broadfeaturecenteredon,-- 0 km

s -1 with a turbulent velocity width of _ 20 km

s -1 and a scale height of _ 120 pc (Lockman

8,: Gehman 1991). There are broader components, with velocity widths of 40 km s-1 and

80 kin s-1 and scale heights of --- 280 pc and

,-_ 715 pc (Lockman & Gehman 1991). There

may be an even broader component with an

even greater scale height (Kalberla et al. 1997).

There are also components centered at intermediate (50 < v < 100 km s -x) and high (v _> 100

km s -x) velocities whose z-heights tend to increase with their velocity.

The extragalactic background will clearly be

absorbed by all of these components. However,

the Galactic halo emission may or may not be

absorbed by high-z components; as the z-height

of the Galactic halo emission is unknown and

the z-heights of the absorbers are only poorly

known. The problem is compounded as preliminary investigations have suggested that some

high velocity clouds may produce X-ray emission (Kerp et al. 1999). Thus, it is not clear

whether including H 1 with velocities _ 75 km

s-x will improve the fits to Equation 8.3.

4.2. Overview of Method

The tracer of NH must meet two requirements. The tracer must have the correct spatial distribution of absorbers, that is, the correct "structure", and the tracer must have the

correct column density of absorbers, that is, the

correct "scaling". Since we are interested only

in small fields, on the order of 15o x 15°, the

problem is somewhat simplified.

The method used to create a map of NH from

/100 is to first remove regions affected by molecular gas, that is, regions with H I > 3 x 10"-°

cm --_. \Ve then find the velocity range of H I

that most closely matches the structure seen in

the I100 map. We then remove all regions with

significant H ! emission outside of this velocity

range. The I100 image is then scaled by the best

linear relation between II00 and H I.

In general, the I100 is best matched by H I in

the +75 km s-1 range, as determined by minimizing the scatter in a linear H I to I100 relation. Removing the regions with significant

colunm densities of high velocity gas (HVG, i.e.,

H I with ]v] > 75 km s -1) does improve the

fits to Equation 8.3, and removal of the HVG to

colunm densities of 1.5 x 1019 cm -2 is sufficient.

4.3. Details

This section discusses the experiments that

were performed to reach the method and parameters described above. The experiments were

performed to answer a series of questions: What

velocity range of H I best mimics the I100 structure? Should the strong high velocity gas structures be removed, and if so, to what column

density must they be removed? Are there significant absorption components besides the H I,

or is the I1o0 tracing some other component as

well? This last question can be reformulated as

an inquiry whether the H I/I100 variation traces

a variation in the column density of absorbers

or merely variation in dust emissivity within the

HI.

To explore these questions we used a single

X-ray mosaic, 138+52, as it covers a large area,

has a large range of absorbing columns, and for

which, in addition to the Dwingeloo data, we

have a higher resolution H l data set provided

by J. Lockman. This mosaic is located in the direction of the Lockman Hole, which has been discussed by Snowden et al. (1994). Images of the

X-ray and tracer data for this region are shown

in Figure 8.3.

No part of the area shown in Figure 8.3 has

column densities, as measured by H 1 21 cm,

great enough to contain molecular gas. Almost

a quarter of the mosaic is covered by H I with

Iv[ > 75 km s-1, but very little is covered by

significant column densities of high velocity gas.

The H I/I100 ratio has a _ 30% variation. Although a portion of that variation does seem to

have the same structure as the H I, both low

and high colunm density regions can be found

having the same H I/ll00 ratio.

Both the RASS image and the mosaic show

structure due to absorption. The IIo-Ng relations are shown in Figure 8.4 for both the RASS

and the mosaic for different measures of NH.

The structure of the NH is set by our choice

of I100 as a tracer. It can be modified, presumably to make the tracer more closely conform to

the true Nu, by masking out regions where the

smoothed I100 structure does not match that of

113
Fig. 8.3.--Imagesof the138+52softX-raymosaicandtracersof theabsorbingcolumn.Each

imageis 12°.8onaside,theindividualPSPCpointingshave53t radii. Top Row: from left, the

Dwinge|oo H I data for the high velocity gas (Iv I > 75 km s-1) greyscale from 0 to 2.5 × 102o cm --_,

white contour at 1.5 × 1019 cm-'_; the Dwinge|oo H 1 data for the "disk" gas (It,[ < 75 km s-l),

greyscale from 3 × 1019 cm --_ to 2.8 x 1020 cm-2; the Lockman H I data for the "disk" gas (Ivl < ?5

kin s-l), greyscale from 0 to 3.0 × 10 -_° cm -2. Middle Row." from left, the Schlegel, Finkbeiner &

Davis (1998) temperature correction factor, X, greyscale from 0.8 to 1.3; the Schlegel, Finkbeiner

& Davis (1998) destriped 1RAS 100 pm data, greyscale from 0.1 MJy sr -1 to 4.3 MJy sr-1; the

H 1/1100 ratio, greyscale from 0.5 to 3.5. Bottom Row: from left, the RASS, greyscale from 4 to

14 xl0 -4 counts s -I aremm -; the R1L2 X-ray mosaic, greyscale from 4 to 10 xl0 -4 counts s-1

arcmln• ,) -; point source correction, greyscale from 0 to 1 x 10 -4 counts s -_ The scale bar is shown

in the lower right-hand corner.

114
tooo

8oo

6oo

o

o

400

IOOO

80O

600

400

IOOO

, . Dwing

Dwing

Lock

600800 ',_.,,, I

400 : : I ..... _..... I _: .... ,t:,,::::

10o0 I DIRAS

I

o0 l o

600

400

1 2 3 4 5 6

N,/10" (era")

Fig. 8.4.-- Comparison of the I_-NH relations

for mosaic region 138+52. Top: the RASS data

vs. the Dwingeloo 21 cm data, Upper Middle:

the band R1L2 mosaic vs. the Dwingeloo data,

Lower" Middle: the band R1L2 mosaic vs. the

Lockman 21 cm data. and Bottom: the band

R1L2 mosaic vs. the IRAS 100/lm data scaled

to the Dwingeloo 21 cm column density.

the much more coarsely sampled NH. Changing

the structure of the NH tracer has the effect,

after the data have been binned by N_u, of moving individual image pixels from one NH bin to

another; the mismatch between the tracer and

the true NH increases the scatter in the Ito-NH

relation. The scaling of tile NH is set by the H l

velocity range to which we fit the lx00-H I relation, and by the "H I scaling" process. Changing the scaling of the NH stretches the NH axis,

changing the curvature in the I_-NH relation,

but does not change the scatter in that relation.

Finally, one could attempt to use the variation

in the H I/I100 ratio to employ a position dependent scaling.

How can one test the general prescription

given in the previous section? One might try

to find the method of scaling the I100 to the

H I that minimizes the X" when fitting Equation 8.3 to the ]_)-N_, relation. As noted above,

changing the structure will change the scatter in

the I'o-NH relation, so finding the method that

gives the best approximation of the structure of

the NH (as measured by the I'o-NH relation) is

easy.

Conversely, determining the method that gives

the best scaling is difficult. If one fits

I_o = IL + IDe -(pcNÈ)°(")cNH's) (8.5)

where 35 is the scaling of the NH, then if one

increases/C, In will decrease and IL will increase

to compensate, producing a nearly identical fit s

The X2 surface is very poorly structured, having

many local minima. Thus, although IL and ID,

the parameters in which one is really interested,

are very sensitive to 35, there is no way of testing

35.

4.3.1. Scaling

An example of the difficulty of testing 35 is

given by an attempt to determine whether H II

might trace significant absorption. Some _ 33%

of the H II is thought to be correlated with H I

(Reynolds 1991) Thus, if one attempted to fit

Equation 8.5 using the H I as a measure of NH,

one ought to find L7 > 1. (The best estimate is

35 should be 1.26) However, besides any con5Fitting Equation 8.5 is non-trivial. There are two possibilities, and choosing the algorithmically simpler leads

to disaster. Equatio_ _.,q was fit using a non-llnear routine, the down-hill simplex or amoeba method. The simplest way of fitting the equation is to bin the X-ray data

by NH only once. then for each evaluation, scaling the

NH by K:, determining the new aelI, and calculating the

model Io. The better way of fitting the equation is to

bin the X-ray data by I(N H for each evaluation. The

simpler method, for some reason that is not clear, tends

to run away, producing very small K:. It appears that it

may be producing the largest K: consistent with [L = O.

The better method produces a very poorly behaved X 2

surface because slight changes in K: will change the bin

into which a given pixel is placed. If there are two pixels

with similar IV H both of which have anomalously high

X-ray fluxes, the go.dness of fit can be strongly influenced by whether those two pixels end up in the same

NH bin or not.

6We have assumed that

115
siderationof the X2 surface, there is another

problem. Since the emission by the NH tracer

is linear with column but absorption is exponential with column, and the emission data is

much lower resolution than the absorption data,

any clumping in the absorbing column should

lower K below unity. To explore this problem

we fit Equation 8.5 using the X-ray mosaic for

band R1L2, the Dwingeloo data (_ 30' resolution) and the Lockman data (--_ 9 - 15' resolution). The results are shown in Table 8.2. Although A/C _ 0.004 was sufficient to produce a

l(r change in ,(2 (using the Lampton, Margon,

& Bowyer (1976) criteria) yet we find other local

minima with nearly the same X2 but a A/C of up

to 0.2, as can be seen in the two results given for

the Dwingeloo data in Table 8.2.

Given the quality of the data, that the fitted

value of/C _ 1, and that we have little knowledge of the distribution of H II, we have set/C to

unity. At worst, this assumption will introduce

a small error in Io and IL; it will not effect the

scatter of the data around the mean I'o - N H

relation.

4.3.2. The I;oo-NH Relation

We have assumed that the magnitude of the

absorbing column density is best approximated

by the H 121 cm column density, while the structure is best approximated by the IRAS 100 #m

dust emission. We must now determine the optimal way to scale the I100 image to the H 1 21

cm image.

We know that the 100 #m emission data predominately trace that which is known as "disk"

gas. If the H I velocity limits are too broad, one

will include st ructures ,lot traced by I_00. Conversely, restri_'ting the H I velocity limits too

much will exclude structures that are traced by

the I100. To <ompare the column of H I to I100

we binned the IRAS data to the same pixel areas

contained by ,he Dwmgeloo data set. We then

K.NH! _'HI "= NHIaHI if" NHIIaHII

which is equivalt'nt to

N a

NII l a H I + iV,,, £L2fJ..Ln" . , ° H I I

1.11 NH I _nl _HI

= VHIOHI(I + NHttaHttl

NHIo.HI l"

Using the values _,iven in ('hapter 4 we find that K: _ 1.2.

fit the linear relation 7

b HI

Itoo = a q- 1---0_o. (8.6)

An example fit is shown in Figure 8.5.

As can be seen in Figure 8.5, it is not entirely

clear that a simple linear fit reflects reality. If,

for example, above a threshold of _ 1.5 x 1020

cm -2 there is some additional component of I100

that does not contribute to the H l, one would

be inclined to make a fit with a much shallower slope, as shown in the right panel of Figure 8.5. This is similar to the behavior produced

by molecular gas, which on average increases the

I100 emission at column densities greater than

3 x 1020 cm -_'. Given the low apparent threshold of the effect, ,-, 1.5 x 10-°° cm -'_, molecular

gas should not be causing this effect. H II should

not depend upon column density in this manner.

If we look at I100 vs. H I using the higher resolution Lockman data (Figure 8.6), we see that

most of the low column density structure seen in

the Dwingeloo H I to/100 relation is removed by

higher resolution. The high I100 values at high

H I values are due to dense cloud cores that are

not sufficiently resolved by the Dwingeloo data.

A simple linear fit is consistent with the Lockman data, and the best fit to the Lockman data

is consistent with the best fit to the Dwingeloo

data. Thus, the simple linear fit is sufficient in

both cases.

The spray of points to higher 11t_0in the righthand panel of Figure 8.6 raises another question. If small high-column-density regions are

not detected by H I due to the low resolution

Zlt is well known that since least-squares fitting generally

minimizes in only one direction, there is a difference when

fitting between "what is H t given ll00". (H l[ll00) and

"what is I100 given H l?", (ll0olH 0. Since we will be

taking Ilo0 images and converting them to images that

are the equivalent of H 1, our formulatitm v*'ould appear

to be backwards. We justify our backwardness in the

following manner.

We expect H I to be close to the total absorbing column.

We know that at a given value of H t, ll00 will have a

range of values due to the IR emission of other components, such as molecular gas. Fitting usillg equation 8.6

will minimize this scatter. Pixels with additional molecular gas will bias the fit to somewhat larger value of b.

Fitting the relation

HI = a' + b'Iioo

will minimize in the direction perpendicular to the known

scatter. Points with additional molecular gas will produce an even stronger bias in b.

116
TABLE 8.2

FITTING WITH SCALED N H

NH IL ID K a X2v

DwingeIoo 538.4 393.4 1.037 12.46 35

Dwingeloo 493.8 403.0 0,820 12.47 35

Lockman b 510.3 492.9 1,027 4.188 35

Lockman c 492.2 460.7 1.039 4.904 35

Regions including 1.5 x 10 a9 cm -'_ in Iv I > 75 km

s -l were excluded from the fit. The uncertainties are

discussed in the text. However, it should be noted

that with K fixed, the uncertainties in Iz and ID are

8.1 and _ 14.5 respectively. With /C free to vary, the

uncertainties are somewhat larger_ _ 8.5 and _ 18.1

respectively.

aFormally uncertainties are _ -t-0.004. Varying the

region used in the fit can change /C by up to _ 0.2, as

can be seen by comparing the two Dwinge[oo data fits.

bBinned to the Dwingeloo pixet areas.

COriginal _ 9 - 15 _ resolution.

b

2

v

g

0

0

, ... "%..,, '..

1 2 3

HI (10 _ cm")

• i ..""

. ." .._;._ .,.::"

...... ,2'i,1 .........

1 2

HI (I0 '° cm-')

.,'"

J .....

3

Fig. 8.5.-- Example IRAS 100 #m emission to Dwingeloo H I 21 cm emission scaling. Left: Best

linear fit. Rzght: An alternative, the solid line is a "by-eye" fit to the lower bound of the H I to I100

relation, the dotted line shows a possible region of avoidence due to a component that contributes

to the I100 but not the H I. See text for discussion.

117
'k,

fD

v

3

2

0

0

, i .............. I i .... ' .... J .....

• -_'"_" ._'_i:'.:

2_" _,_!_: ..... "...... "'"

..... i ......... t ........ .liL,_

1 2 3 0 1 2 3

HI (10 `° cm-') HI (lO '° cm -')

Fig. 8.6.-- Example IRAS 100 #m emission to Lockman H I 21 cm emission scaling. Left: Both

the Lockman data and the It00 data were rebinned to the Dwingeloo pixel areas. The line is the

fit to the Dwingeloo data shown in the previous figure. Right: The same data set. The [too data

has been smoothed to the Lockman resolution (_ 9 - 15') and both data sets have been binned to

15'. The solid line is the same fit as in the previous figure. The dotted line is the best fit to the

I100 - Lockman H I relation.

of the Dwingeloo survey, how can we be sure

that regions containing molecular gas have been

removed? We find that although at no place

in the field is the H I column density greater

than 3 x 10_-°, there are regions where the scaled

I100 does reach those column densities (see Figure 8.4). Those regions are noted by contours on

the [100 panel of Figure 8.3. Comparison of these

regions with the study of Reach, Koo, & Heiles

(1994) shows that all IRAS clouds with detected

CO emission fall within the NH > 3 x 10 -_°

cm --_ contours, where the NH is derived from

I100. Thus, excluding points with NH > 3 x 10 2o

cm--', where the NH is derived from I100, will

be reasonably efficient at removing regions with

molecular gas.

Since, as seen in Chapter 4, and again in Figure 8.3, the H I/I100 ratio is spatially variable,

the mean H I/I100 will vary with fit region. We

were interested in knowing how the fit values

changed with fit region.

Table 8.3 contains the fit results. Although

the velocity resolution of the data was much

finer, the velocity regions tested were limited to

+50 km s -_, =E75 km s-z, +100 km s-1, and the

entire Dwingeloo velocity range (-450 < z, <

400 km s-l) s. Fits were made for a square region surrounding the X-ray mosaic, for the same

region but with the I100 poor high velocity gas

(HVG) removed to various column density limits, and for the region covered by the mosaic.

(In the last case, the HVG was removed to the

1.5 x 1019 cm -_ level.) It has been assumed that

the random uncertainties in the IRAS data are

all the same, and they have been set to unity

8Given the broad velocity distribution of the H I, one

can assume that the structure seen withH, the chosen

velocity range will extend to more extreme velocities. To

compensate for having removed the high velocity wings

one can scale the H i column within the chosen velocity

range using the factor

(HI(-4._0 km s-l<v<400 km s-*)t

f = (HI(chosen range)) "

where the means are taken over regions not excluded

due to structure at more extreme velocities (i.e., regions

without HVCs). This scaling factor is typically on the

order of 1.05 to 1.20 when comparing the 5=100 km s-l

and -1"50km s-I velocity ranges to the entire velocity

range.

However, since this factor only scales the H i image, it

can not effect the structure, and thus it can not change

the goodness of fit for either the Iioo - Nit relation or

the 1_ - N H relation. It will only change the final fit

parameters, [L and ID.

118
TABLE8.3

/I00TOH I SCALING

Maska Hi I1oo X lloo

Av a b X_b a b kzb

All All 0.0452=t=0.0204 0.51875=0.0189 0.0359 0.10745=0.0217 0.51815=0.0202 0.0402

5=100 -0.06575=0.0176 0.69265=0.0175 0.0190 -0.01575=0.0186 0.70285=0.0186 0.0207

5=75 -0.04805=0.0168 0.70534-0.0174 0.0186 0.00065=0.0178 0.71735=0.0186 0.0200

4-50 0.06085=0.0177 0.76905=0.0226 0.0318 0.08705=0.0178 0.81095=0.0228 0.0280

HVG

> 3.5 x 1019

All -0.06655=0.0219 0.64405=0.0215 0.0259 -0.03284-0.0224 0.67325=0.0219 0.0281

5=100 -0.05935=0.0194 0.69055=0.0205 0.0196 -0.02275=0.0198 0.71955=0.0210 0.0218

5=75 -0.04645=0.0185 0.70185=0.0203 0.0187 -0.00955=0.0190 0.73145=0.0209 0.0208

5=50 0.03835=0.0186 0.7755+0.0247 0.0269 0.06765=0.0185 0.82185=0.0247 0.0268

HVG

> 1.5 X 1019

All -0.07175=0.0212 0.69574-0.0230 0.0168 -0.03145=0.0214 0.71745=0.0231 0.0184

5=100 -0.03785=0.0195 0.69835=0.0224 0.0154 0.00365=0.0197 0.72015=0.0226 0.0169

5=75 -0.02745=0.0189 0.70475=0.0223 0.0148 0.01465=0.0191 0.72635=0.0224 0.0164

5=50 0.00405=0.0180 0.83915=0.0260 0.0165 0.04505=0.0181 0.86765=0.0262 0.0175

Mosaic All -0.07795=0.0294 0.71175=0.0316 0.0528 -0.03555=0.0311 0.73915=0.0335 0.0586

5=100 -0.04745=0.0269 0.71945=0.0310 0.0484 -0.00425=0.0284 0.74755=0.0327 0.0539

d=75 -0.03215=0.0259 0.71965=0.0306 0.0465 0.01244-0.0276 0.74685=0.0325 0.0521

5=50 -0.01545=0.0238 0.84515=0.0331 0.0515 0.02845=0.0258 0.87875=0.0358 0.0547

_The high velocity gas, "HVG", is defined as HI outside 5=75 km s-1. These regions are excluded from the fit.

The mosaic mask excluded the HVG at the 1.5 × 1019 cm -2 level.

bThe uncertainty used in the calculation of X2 was assumed to be the same for all pixels. The uncertainty was

taken to be unity, so this quantity is not a "real" X2 but is the second moment with respect to the fit.

for the calculation of a modified X "_. The best

agreement between H I and I100, as determined

by the modified X 2, occurs for the -I-75 km s-x

velocity interval. Removing HVG (here defined

as emission in velocities outside +75 km s -1) to

a level of 1.5 × 1019 cm -'_ produced the best fits

in terms of X_. Removing the HVG to the level

of 1.5 × 1019 cm -2 also brings the fitted scale

factor b for all the velocity ranges (save for -t-50

km s-x) more closely into agreement.

Although there are 30% spatial variations in

the H I/I100 ratio, restricting the region of the fit

to just that of the mosaic, still excluding HVG

to the level of 1.5 x 1019 cm -_-, did not make a

significant change.

The same fits were made using XIxoo, the

temperature corrected IRAS 100 pm data, in

place of Ix00. The XIloo fits were always slightly

worse than the I100 fits. Thus, by our criteria, the temperature correction proposed by

Schlegel, Finkbeiner, & Davis (1998) does not

help in regions where H I < 3 x 10 "_°cm --_. This

is not surprising as the resolution of the temperature correction is low (_ 0.7 °) and the correction is derived from the ratio of the 100/ma and

240 pm emission, both of which will have small

values and large uncertainties in regions of such

low column density.

4.3.3. The I'o-NH Relation

For each of the H l(Iv t < 75 km s -1) to

(X)Iioo fits in Table 8.3, the total absorbing column, NH was taken to be

[(X)Iloo - a] x 10-_° N. = (8.7)

b

and the NH was fit to the X-ray emission using Equation 8.3. The results are shown in

Table 84. Removing the HVG to the level of

1.5 × 1019 cm -2 does improve the fits to the I_-

NH relation. Using XIlo9 did not produce as

good a fit as I100. Figure 8.7 shows the fit and

an inaage of the residual to the fit.

4.3.4. Another Structure/Scaling

Method

The H I/I100 ratio is spatially variable. It

is not known whether this variation is due to a

119
TABLE 8.4

11oo FITTING TO THE X-RAY _,'IOSAIC

Mask a Range 11 oo X I1 oo

IL b ID b :k_ IL b Im b X="

All 4-75 533.580 417.391 3,807 552.208 389364 3.447

HVG> 3.5 × 10 _ 4-75 535.791 406.744 2.937 545.375 392.695 3.160

HVG> 15 × 1019 4-75 525.447 405.643 2.318 535.095 391.735 2.144

Mosaic 4-75 521.861 407.830 2.191 529.036 394.793 2.526

_The high velocity gas, "H\fG ", is defined as HI outside 4-75 km s-t. These regions are

excluded from the fit_ The mosaic mask excluded the HVG at the 1.5 × 10 _a cm -2 level.

bTypical uncertainties are lc = 4.8.10 and ID = 4.14.5.

1000

E

8OO

600

400

i000

800

o

v

o

600-

400

0 1 2 3 4

N,/I 0" (era-')

-300 -200 -100 0 100 200 300

Fig. 8.7.-- Example fits of 1100 to the X-ray absorption, and the residuals to the fits. In all cases

regions with HVG> 1.5 x 1019 have been removed for the 110o to H I scaling, the :t:75 km s -1

velocity interval of the H I has been used, and the H I column density in that velocity interval has

been scaled to the total H ! column density. Left: The I_o-NH relation. Right: The residual to the

fits. The grey-scale is the Io - fit. Top: Fit used a linear relation between 11oo and H I. Bottom:

Fit used the point-by-point scaling.

120
TABLE 8.5

POINT-BY-POINT SCALING/FITTING

Ma_k Range 11oo XI=oo

IL" ID I X"_ IL _ ID 8 X"

All -I-75 529.469 411.191 4613 530.490 408,720 4.619

HVG> 3.5 x 1019 4-=75 522.592 413.826 3.456 523.229 412.006 3,375

HVG> 1.5 x 1019 -1-75 52].943 403.406 2.583 521.677 403.275 2.260

Mosaic +75 519.316 408,243 2.648 519.053 408.150 2.588

=Typical uncertainties are It, = 4-8.10 and ID = 4-14.5.

change in dust temperature, a change in dust

to gas ratio, or, perhaps, the addition of ionized 3.0

hydrogen containing dust. If the variation is due 2.5

to the presence of an absorber not correlated

with the H I, then 1100 is a better tracer of the 2.0

total absorbing column, and the variation should =

be preserved, as in the fits above. If, however, o

v 1.5

the variation in the H l/I100 ratio is due to a

variation in the dust temperature, that variation _-

1.0

ought to be minimized. 3

One can retain the high resolution structure 0.5

of the lz00 image while removing the large scale

H I/lz00 variation by smoothing the H I/I100

image by a large amount (say 30' since that is

the H I resolution) and setting the absorbing

column to be the /zoo image multiplied by the

smoothed ratio image. This is essentially a highpass filter.

The results can be seen in Table 8.5. For

the cases tested, the fits using this "point-by- age.

point" scaling are significantly worse than the

linear scaling used in the previous fits, suggesting that the H I/Ii00 variation is not due merely

to variation of the dust emissivity within the H I

or a variation in the dust-to-gas ratio, but rather

to emission from dust that traces some other gas

component. From Figure 8.7 (bottom) one can

see that the residuals to the fit are not considerably different in structure than those found with

the linear scaling.

4.4. Uncertainties in NH

There are two forms of errors in the NH, scaling errors which, as we have seen above, effect

IL and It), but do very little to the value of the

fitted value of l_), and structure errors, which

produce spurious variations in the output ]D ira0.0

0.0

/

:9.?-

. :...',

.:2." •

/

/

_, , ,l, , ,11 .... i .... i .... J ....

0.5 1.0 1.5 2.0 2.5 3.0

I_tngeloo (1_ cm")

Fig. 8.8.-- Comparison of the Dwingeloo and

Lockman data sets.

4.4.1. NH Scaling Errors

There are two forms of scaling errors. The

first is a simple offset in the value of the column density. In this case, the error will be the

same at all column densities. (As a result, the

effect of the error will be smaller at higher column densities.) The second is an error in the

scaling between some observed quantity and the

value of the column density. In this case, the

error will become large as the column density

increases.

4.4.1.1. NH Baseline- An Offset Error

Although we use the Dwingeloo data set as the

primary tracer of the total absorbing column

121
towardstheX-raymosaics,in thedirectionof

the mosaic 038+52, we have a higher resolution

data set (--, 9 - 15') provided by J. Lockman.

(The same data set was used in Snowden et al.

1994). The two data sets are compared in Figure 8.8, where the Lockman data were binned

to the same pixel areas as the Dwingeloo data.

The line is the best fit linear relation between

the two:

Lockman = (0.1208 :t: 0.0084)

+(0.9036 =l=0.0086) * Dwingeloo. (8.8)

The Dwingeloo data appear to be systematically

lower than the Lockman data at lower column

densities.

The Dwingeloo data were cleaned of sidelobe

contamination by a sophisticated self-referential

algorithm. The Lockman data were cleaned

of sidelobe contamination by reference to the

Bell Laboratories Survey (Stark et ai. 1992),

whose sidelobe contamination has been extensively studied and shown to be small (Lockman,

Jahoda, & McCammon 1986; Kuntz & Danly

1992). Kalberla et al. (1997) suggested that

the baselines of the Bell Laboratories Survey

were over-subtracted, meaning that the Lockman data should be lower than the Dwingeloo

data. Where the problems lie is not clear. The

systematic difference is less than ,,. 7 × 10 is

within the 0.5- 2.0 x 1020 cm -2 range. The systematic uncertainty quoted by Snowden et al.

(1994), 5 x l0 is cm -_, is in reasonable agreement.

The impact of a systematic uncertainty in the

H ! column density can be calculated. For column densities from 5 x 1019 cm -2 to 3 x 102°

cm -2 we have calculated

Io = 1 + e -NHa(NH's) (8.9)

at 5 x I0 Is cm -2 intervals. We then fit Io-NH

relation with an Nx vector that was either increased or decreased from the true amount by

5 x 10 is cm -2. We then calculated the fractional difference in the I£ and ID fit parameters

and determined the maximum fractional deviation between the true and fitted relations. The

results are given in Table 8.6. The effect on either the fit parameters IL and ID or on the value

of the value of the Io - NH relation are negligeable.

We have calculated the same quantity for a

ANH = 5 × 1019 , the equivalent H I column

density of H If, given the canonical distribution

of H II, 7 x 1019 cscb, and the canonical H ll

absorption cross-section, 0.62 that of H I. Significant errors occur for the fit parameters, but

the difference between the fitted and measured

IIo-NH relations are still insignificant. This calculation, of course, has assumed that the H II is

smoothly distributed. A patchy distribution is

discussed below.

4.4.1.2. Error in the ll00 - NH Relation

- A Scaling Error As can be seen from Table 8.3, the spread in the values of b in

HI

11oo = a + b-- (8.10) 102o

is -_ 0.09. The change in b causes a change in

NH that is dependent on the derived NH. We

have calculated the effect of Ab = 0.09 in a manner similar to that used in the previous section.

As we have assumed the same value of a in both

cases, the calculated change is an upper limit.

Note that the change in IL and ID is larger at

higher energies; due to the lack of curvature in

the 15 - NH relation, the fit can shift flux from

the foreground to the background more easily

than at lower energies. However, it should be

noted that since, in practice, the foreground in

band R47 is fixed, the uncertainty in the R47

background flux will be minimized. The uncertainties in the residual to the fit are again negligeable.

4.4.2. NH Structure Errors

Of the sourcesof uncertaintyconsidered,this

isthe only one that contributesto the production of spurious fluctuations.NH structureerrors are errorsin counting the number of hydrogen nucleons along the lineof sight.These

errors arise because I100 is a measure of the dust

emissivity, which may not be proportional to the

number of hydrogen nucleons. We consider two

cases. In the first, we assume that the H I traces

the bulk of the absorption and the variations in

the H I/ll00 are due to dust emissivity effects.

In the second, we assume that the 1100 traces

both the H I and H !1, and then, assuming that

the variation in H l/Iloo are due to variation in

the column density of H ii, caculate the effect of

the H II.

122
TABLE 8.6

EFFECT OF ANH

Band If, ID Residual"

ram. max.

ANH = 5 x 10 is

RIL2 -3.51xI0 -_ 4.13xID -2 -4 59xi0 -4 8.04x I0 -4

R45 -1.81xi0 -3 -4.66x 10 -3 -8.29x 10 -s l.llxl0 -s

R67 -1.95x10 -4 1.68x 10 -3 -3.17x 10 -'r 3.95x I0 -r

AN H ----- 5 X 1019

RIL2 -3.24X 10 -2 4.59x 10 -l -3.62x 10 -_ 7.38x 10 -s

R45 -1.68XI0 -2 4.58x I0 -2 -5.44x I0 -5 1.10xl0 -<

R67 -I,96xID -s 1.69xi0 -2 -7.91x 10 -7 1.27x10 -s

ANH = 5 x 10 TM, I/. fixed

R45 2.68x 10 -2 -5.03x10 -s 6.76x10 -s

R67 1,48x 10 -2 -3.77x 10 -s 3.72x10 -s

All quwmtities are fractional differences.

aThe residual is meuured over 5 x 10 ]9 < NH < 3 x 10 a°.

TABLE 8.7'

EFFECT OF Ab = 0.09

Band IL ]D P,.csid u al"

rain, m_.

RIL2 2.70x10 -_ 1,62x10 -2 -4.30xi0 -a 2.67x10 -4

R45 9.72x10 -2 -9.65x I0 -2 -I.15x10 -4 6,09xi0 -s

R67 1.07x10 -a -l.07x i0 -_ -3.23X10 -5 1.69xi0 -a

IL Fixed

R45 1.15xl0 -a -3,85x i0 -s 3.75x I0 -_

R67 -6.50x 10 -_ -2.27x 10 -_' 2.29x 10 -_

All quantities are fractional differences

"The residual is me_ured over 5 x 10 ]9 < NH < 3 x 10 2°.

123
0.10

0.05

0.00

-0.05

-0.10

R1L21

0.5 [.0 1.5 2.0 2.5 3.0

N, (10= cm")

R47

0.5 1.0 t.5 2.0 2.5 3.0

N, 110= cm _)

Fig. 8.9._ The fractionalamplitude of spuriousfluctuationsin band RIL2 produced by uncertaintiesin the absorbing column. An uncertaintyof 1.5x 1019 cm -2 was assumed. The plotted value

is the change due to thisuncertaintyin the predicted observed flux,divided by the true distant

flux.See textfor discussion.

4.4.2.1. Systematic Variations in H i//100

As can be seen in Figure 8.3, the H I/I100 ratio

varies on 30 _ scales, and higher resolution data

show variation on smaller scales. In the 138+52

field, on 30' scales, for a given value of 1100, the

r.m.s, variation in H I is --- 1.6 x 1019 cm -2. If we

assume that this dispersion is a measure of the

error in the absorbing column, we can calculate

the error in the transmitted X-ray flux (i.e., the

residual to the fit to the I_) - NH relation). This

residual to the fit, Alo, as a fraction of the true

distant flux, ID, is shown in Figure 8.9.

If we assume that the variation in H 1/11o0

ratio does not reflect variations in the absorbing

column, then the variation in 1100 can produce

-,,10%(',- 1%) uncertaintiesin the measured

fluctuationsin band RIL2(R47). The spurious

fluctuationssuggested by Figure 8.9 are likely

to be a grossexaggeration;§4.3.4suggeststhat,

at leaston the 30' scales,the variationin the

H 1/1100ratio is more likelyto be due to the

If00tracingadditionalcomponents of the total

absorbingcolumn than changes inIR emissivity.

4.4.2.2. H II As has been repeatedly mentioned, the spatial distribution of H 11 is unknown and its mean column density as a function of Galactic latitude is poorly known The

Wisconsin Ha Mapper (WHAM) survey (Reynolds

et al. 1998) is currently mapping Ha emission

at an angular resolution of 1° over that part of

the sky visible from Wisconsin. Although maps

have not yet been produced, the variation of Ha

emission on scales of a few degrees is 30%, or an

equivalent column density of-_ 3 x 1019 cm -2

(Reynolds 1999), assuming that Ha is a direct

measure of the column density of H II.

If the variation in the H 1/1100 ratio were due

to emission from dust related to H II, and the

variation in H I1 were comparable to that observed by Reynolds, then the emissivity of dust

related to H If, ell, would be 0.88 times the

emissivity of dust related to H 19, s/ This is

9The measured emissivity, that is, the value ofb in Equation 8.6 -- eM, is given by Equation 8.13. The r.m.s of

[100 at a given value of H i is assumed to be the scatter

introduced by the emission of dust related to the H H. If,

for a given value of H h we assume that - I_ value of 1100

represents 1100 emission exclusively from H i (see Figure

I of Chapter 4), then the r.rn.s va_ue, Allo0, represents

the variation in the contribution to 11oo due to dust related to H H, Assuming that the 3 x I019 cm -2 found

by Reynolds is twice the characteristic r.m.s, of the H H

column density for 1 ° pixels (i.e. that 3 x I0 l° cm -2

represents the peak-to-peak value), then then _II "_ 0.50

and t! ,_ 0.57. The measured r.m.s, of 1100 for values of

H I in the 1.0-2.0x10 _° cm -2 range is0.15, when calcu*

lated from AH i bins that are 5 x 10 TM cm -2 wide. We

124
not too unreasonable given that the best 3rr upper limit to ctt/'-I _ 0.75 (Arendt et al. 1998).

Thus attributing the variation in the H l/ll00

ratio to emission by dust related to the variable

column of H II is not unreasonable.

The potential problem is that while both the

I100 emission and the X-ray absorption are due

to the weighted sum of the contributions from

each H component (H l, H II, and H2), the

weights for absorption and emission are different. We can calculate the effect of the differing

weights in the 138+52 field. The true absorbing

have assumed that the H n column density in this region

is about a quarter of the mean relation (7 x 10lg cscb)

as per Snowden et al. (1994).

0.04

0.02

0.00

o

o

#. -0.02

-0.04

0.004

o" 0.002

r_

J

a

o.ooo

o

_. -0.002

-0.004

-._ .... i .... i .... i .... i .... ,..

0.05

"--....

.....

0.50

"---. .... " ............. ..

°40 ............................ill

0.40 .. .

u.ou ............................................

°.,5........................................ //i

............ ....................... ...

0.00..-'" .................

... ..... •.....

0.55...""

R1L2

::::::::::::::::::::::::::::::::::

0.55

0.60

0.45 ...........................................

0.40 ..........................................

G.4G ..........................................

0.40 ..........................................

0.50 ............................................

0.65 ..........................................

R45

0.5 1.0 l.S 2.0 2.5 3.0

Measured N, (cm _)

Fig. 8.10.-- The fractional amplitude of spurious fluctuations produced by fluctuations in the

column density of H If. It was assumed that the

mean H II column density was 2.2 × 1019 cm -2

and that the fluctuations have an amplitude of

1.5x 1019cm -2.

column is

NH(true) = H I + 0.62H !I. (8.11)

The measured absorbing column is

etH 1 + 8,H II

NH (measured) = , (8.12)

8M

where

H II

8M -- 81 + "_8II. (8.13)

One can calculate the I_9 using the true NH

and fit using the measured NH. The results are

shown in the fit column of Table 8.8 for an assumed uniform H II column density of 2.2 × 1019,

the value used by Snowden et al. (1994) for

this field. One can also calculate the effect of

a variation of the H II column density on the

measured deabsorbed flux. For a -I-1.5 x 1019

cm -2 column density of H II, we find an error of

+1.5%(4-0.2%) in the deabsorbed flux in band

R1L2(R45) given 8II = 0.5 (see the "Deabs."

column of Table 8.8). However, one should note

the uncertainty of the many values that entered

this calculation.

4.4.3. Nn Errors

Table 8.9 summarizes the uncertainties due

to errors discussed in this section. As the deviation of fits from the mean I_o -Ntl relation are

universally small, they have not been tabulated.

The tabulated value for the amplitude of the angular variation in ]D due to variation in the error

in NH, A/D, was calculated on 30' scales. The

error attributed to variations in H 1/I100 is an

extreme upper limit, while that error attributed

to H II is likely to be a more accurate representation of the true error.

5. Deabsorption

Once an adequate map of the absorbing column has been obtained, and the mean ]L and ID

have been determined, there are two methods by

which to create a map of the distant emission.

Both, of course, assume that the local emission

is fiat.

The obvious method is multiplicative; the

map of the distant emission is given by

ID(x, y) = (Z_o(X, y) - (IL))e _', (8.14)

125
TABLE 8.8

EFFECT OF H II

tl: ctl/tl R1L2 R45

Fit j Deabs. D Fit* Deabs. b

IL ID ID Io ID

0.40 0.67 -1.10 x 10 -2 --7.04 X 10 -3 2.45 x 10 -3 -3.33 x 10 -_ 1.99 x 10 -4

0.45 0,77 --1.48 X 10-" -3.33 x 10 -3 1.30 x 10 -_' -3.39 x 10 -_ 1.04 x 10 -3

0.50 0.88 -1,79 x 10 -2 4.45 x 10 -2 2.40 x 10 -2 -3.47 x 10 -3 1.93 x 10 -s

0.55 0.99 -2.10 x 10 -2 4.31 x 10 -2 3.53 x 10 -_ -3.54 x 10 -2 2.85 x 10 -_

*ThiJ is the fractional error in the fit parameters caused by the addition of a uniform H Ix column

density of 2.2 x 10 l_ cm -_.

bThis is the fr_ction*l error in the deLbtorbed flux due to an additional -t-1.5 x 1019 cm -2 of H II

assuming the fit including the uniform H xl column denlity of 2.2 x 10 z9 cm -_

TABLE 8.9

ERROR SUMMARY

Source Scale I D AID

R1L2 R45 R1L2 R45

0".!! ... <_( 1% _( 1% ......

NH Scaling (offer) ... < .3% < .1% .....

N_r Scaling (*caled) ... < 2% < 1%

A(H _/Ixoo) 30' . < S% < 1%

H 11 30' < "T% < 3% < 3% < 3%

126
200 j

"_ I00 I

5O

0

200

150

--" IOO

5o

2o_

x,(Mum)

I. (Add.)

150

tO0

5O

0.0 0.5 1.0 |.5 2.0 2.5 3.0

NJtO" (era")

Fig. 8.11.-- Demonstration of the effects of

the multiplicative and additive methods of deabsorption. Top: The observed I_ - NH relation, the dotted line being the mean fitted relation, Middle: ID as derived by the multiplicative

method, Bottom: ID as derived by the additive

method.

where r = NH(X, y)a(NH (z, y), S). This method

assumes that all of the deviation from the mean

I'o-NH relation is real, and it amplifies all of

the uncertainties. A fluctuation analysis performed on a field deabsorbed in this manner

will produce an extreme upper limit on the size

of fluctuations. Ignoring the systematic uncertainties (i.e., the uncertainies introduced by the

deabsorption process), and considering only the

statistical uncertainty (i.e., the propagated measurement uncertainty), we find that

Crl_(x,tt) = o'ise'. (8.15)

The systematic uncertainty will be spatially variant;

a1_<,,v) = e2r V/O.2L + o'.,,,(I' 2 o,- 1I.) 2. (8.16)

However, this may be more conveaiently recast

as:

0-2 e2r V (8.17)

A more subtle method used by Barber, Warwick. g: Snowden (I995) is additive; the map of

the distant emission is given by:

rv(x, v) = rb(x, v) - <ZL)+ <ID)(1-- e-').

(8.18)

This equation is essentially the residual to the

fit to which is added the mean distant emission,

that is, if IF is the fitted value of I_), then

ID(Z,y) = l_(x,y) - IF + (ID). (8.19)

This method assumes that the distant emission

is smooth (equivalent to assuming that any variation is due to the foreground), and minimizes

the effect of the uncertainties. If the uncertainties in Io and NH were small, a fluctuation analysis performed on a field deabsorbed in this manner would produce a lower limit on the size of

fluctuations. However, in the presence of noise

this method produces neither limit to the size of

fluctuations.

Again considering only the statistical uncertainty, we find that

'y1_,(_,_) = o"15. (8.20)

The systematic uncertainty will be

= V/ r + (8.21)

Figure 8.11 demonstrates the different effects

of these two methods. Neither method is ideal,

though the multiplicative method can be improved by deabsorbing, not to zero column density, but to a small column density, such as

5 x 1019 cm -2.

5.1. Uncertainties

Given the understanding (or lack thereof) of

the uncertainties, one may reevaluate the systematic uncertainties of the two methods of deabsorption. The systematic uncertainty for the

multiplicative method is given by Equation 8.17;

the eT_IL term dominates by an order of magnitude. The systematic uncertainty for the additive method is given by Equation 8.21, but

it should be noted that the uncertainty of the

fit value, aF is usually two orders of magnitude smaller than the uncertainty in the fitted

lb. Therefore, the systematic uncertainty for

127
i '°i M

+'iI

_'°I

i+! 1.0

I  ̧^ .....

i o._, M

O.Ol ,

o.s I .o

RIL2

R45

1.5 Z.O |+6 $.0

(iO" *'m +}

Fig. 8.12.-- Comparison of the systematic uncertainties for the multiplicative and additive

methods of deabsorption. The systematic uncertainty for the multiplicative method is shown

by solid lines; for the additive method, by dotted lines. The uncertainties were calculated for

field 138+52 and are in units of 10 -e counts s -1

arcmin- 2

the additive method is approximately air. Figure 8.12 shows the systematic errors as a function of N/_ for both methods applied to the

138+52 field. The two methods yield similar

systematic uncertainties in band R45, but the

additive method yields systematic uncertainties

that are three times smaller that those of the

multiplicative method in band R1L2.

Despite the seeming advantage of the additive method, we use the multiplicative method

because our interest is the structure of the distant component, not its absolute value. The difference in systematic uncertainty is of little consequence as, despite some dependence on Nx,

it is primarily a measure of the uncertainty in

the mean value of I/). More important is the

pixel-to-pixel uncertainty, which will be smaller

by e * for the additive method, but this advantage is outweighed by the necessity to assume

that there are no fluctuations in I1).

6. Summary

The greatest difficulty in deabsorbing an Xray image is finding a tracer of the absorbing column that has a spatial resolution comparable to

that of the X-ray image. The IRAS 100 #m AllSky Survey provides a tracer with a resolution

of--. 5', but it is not clear to what extent the 100

pm emission traces the absorbing column related

to H I1, nor is it clear to what extent the 100 pm

emission is influenced by factors other than the

total column of hydrogen nucleons (i.e., dust to

gas ratio, dust temperature, etc.). However, the

analysis shows that 100 pm emission is a better

tracer of the total absorbing column than the 21

cm emission of H l; the 100 pm emission does

seem to trace variations in the total absorbing

column that are not traced by H I.

6.1. Prescription

Using the above information, we can produce

a prescription for deabsorbing X-ray mosaics.

First, calculate the correction to the X-ray

mosaic for the point source detection limit variation. One should not correct images to a largepoint source detection limit: correcting the band

R1L2(R47) band image to 0.04(0.02)counts s-1

rather than 0.02(0.01) counts s -l, for example,

will introduce a 2% variation in the deabsorbed

image, as the point source correction is N_ dependent. Although this quantity is much smaller

than the uncertainty in any single mosaic pixel,

it is comparable to the uncertainty over larger

regions.

Second, remove regions with an

H I(i_i<_5 km s-_) > 3 x 1020 cm -2,

thus removing regions possibly contaminated

with molecular gas. Remove also regions for

which the column density for

H I(ivl>7 s km s-') > 1.5 x 1019 cm -2,

thus removing 1100 deficient high and intermediate velocity gas.

Third, determine the parameters for the linear ll00-H 1 relation,

HI

I100 = a + b 102------6, (8.22)

using the region of the mosaic that satisfy the

above conditions. Use the H I(ivl<Tsk m s-') for

deriving a and b. For some of the mosaics, the

dynamic range in the mosaic region will be insufficient to produce good fits; for these mosaics

128
usea largerregion to derive a and b The difference between fit parameters derived for the

larger region and those derived from the mosaic

region will be small when both are derived over

a reasonable dynamic range. From the linear

I100-H I relation calculate the total absorbing

column as

NH = [1100--a] x 1020 b (8.23)

Fourth, determine the {rough) shape of the

background spectrum. Using the RAgS for a

larger region including the mosaic, determine

the IL and Ix) for bands R1 and R2. Given

the low exposure of the RASS one may need

to select a limited range of NR to execute the

I_o-NR fitting. Using all of the pixels from the

region that fall within some narrow range of absorbing column (ANH -_ 1 × 1019) determine

the mean absorbed SED after the extragalactic power law has been removed. From that

SED determine the normalizations of the soft

(logT = 6.056} and hard (logT = 6.462) halo

components. Using these normalizations, construct the halo spectrum and calculate the effective cross-sections.

Fifth, apply the more refined binning method

to the mosaic data to create the mean I_ - N//

relation, and determine the mean IL and ID

by fitting Equation 8.3. In principle, one could

then check to see if the background spectrum

determined from the RASS data is appropriate.

If not, one could construct a new background

spectrum and repeat the process. However, determining the background spectrum from mosaic data is far more difficult than determining

the spectrum from the RASS; one must correct

the mosaic data for the variable point source

detection limits, and the band-to-band correction which will depend upon the assumed source

spectrum. Using the RASS which has a nearly

uniform point source removal is much easier.

Sixth, derive the map of the It) component.

Although the additive method may produce a

lower systematic uncertainty, the multiplicative

method is to be preferred as it yields an identifiable upper limit to the strength of the variation

in ID.

For band R45, one cannot determine IL by

fitting the I_ - NH relation because r changes

little for NH < 3.0 x 1020 cm -2. Therefore, one

must use the IL determined for the R1L2 band,

and scale that value to the R45 band using the

apropriate emission model, l,Ve use a Raymond

& Smith thermal model with logT = 6.11, leading to R45/R1L2=0.0185. Given this small sca/-

ing factor, the uncertainty in the R45 IL will be

quite small, as will be the uncertainty in 1D.

6.2. Uncertainties

Of the many uncertainties in the deabsorption process, most uncertainties effect the fitted

values of IL and //9. This uncertainty is important in so far as one is interested in the relative amplitude of fluctuations, AID/ID. More

troubling uncertainty is the angular variation of

the error in NH. Such variation can produce

spurious fluctuations. We have placed an upper

limit on the spurious fluctuation by comparing

I100 with H I, which is known not to trace all

the components of NH. In the R47 band these

spurious fluctuations are _ 1%, while at R1L2

they are _ 8%, and could be substantially less.

Very rough calculations equating the variation in

H 1/11oo with the variation in the column density of H II suggest spurious fluctuations _ 3%

at R1L2 and _ 0.3% at R45.

REFERENCES

Arendt, R. G., et al. 1998, ApJ, 508, 74

Barber, C., Warwick, R. S., & Snowden, S. L. 1995,

in R6ntgenstrahlung from the Universe, ed.

H. Zimmermarm, .I. Triimper, & H. Yorke,

MPE Report 263, 319

Chen, L.-W., Fabian, A. C.. & Gendreau, K. C.

1997, MNRAS, 285, 449

Hasinger, G., Burg, R., Giaccom, R., Schmidt, M.,

Triimper, J., & Zamoram, G. 1998, A&A,

329, 482

Hauser, M. G., et al. 1998, ApJ, 508, 25

Kalberla, P. M. W., Westphalen, G., Mebold, U.,

Haxtmann, D., & Burton, W. B. 1997, in

The Local Bubble, ed. D. Breitschwerdt &

M. Freyberg, 479

Kerp, J., Burton, W. B., Egger, R., Freyberg, M. J.,

Hartmann, D., Kalberla, P. M. W., Mebold,

U., & Pietz, J. 1999, A&A, 342, 213

Kuntz, K. D., & Danly, L. 1992, PASP, 104, 1256

Kuntz, K. D., & Snowden, S. L. 2000, ApJ

Lampton, M., Margon, B., & Bowyer, S. 1976, ApJ,

208, 177

Lockman, F. J., & Gehman, C. S. 1991, ApJ, 382,

182

Lockman, F. J., Jahoda, K., & McCammon, D.

1986, ApJ, 302, 432

129
Reach, W. T., Koo, B.-C., & Heiles, C. 1994, ApJ,

429, 672

Reach, W. T., Wall, W. F., & Odegard, N. 1998,

ApJ, 507, 507

Reynolds, R. J. 1991, in The Interstellar Disk-Halo

Connection in Galaxies, ed. H. Bloemen, 67

Reynolds, R. J. 1999, private communication

Reynolds, R. J., Tufte, S. L., Haffner, L. M., Jaenig,

K., & Percival, J. W. 1998, Proc. Astron. Soc.

Aust., 15, 14

Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998,

ApJ, 5OO, 525

Snowden, S. L., Hasinger, G., Jahoda, K., Lockman,

F. J., McCammon, D., & Sanders, W. T.

1994, ApJ, 430, 601

Stark, A. A., Gammie, C. F., Wilson, R. W., Bally,

J., Linke, R. A., Heiles, C., & Hur_tz, M.

1992, ApJS, 79, 77

Wakker, B. P., & Boulanger, F. 1986, A&A, 170, 84

130
Chapter 9: ID Revealed

Abstract

This chapter describes the details of the application of the deabsorption method

to the seven high Galactic latitude mosaics. The cross-correlation of the images of the

deabsorbed distant emission and the absorbing column suggests that the deabsorption

process produces errors in the image of the distant emission that are small compared

to the amplitude of the intrinsic angular variation of the emission.

1. Boring Stuff

The parameters used in and produced by the

deabsorption process are listed in Tables 9.2

and 9.3. Much of the information in the tables is

depicted in graphic form in the following figures.

The I100- NH relations are shown in Figure 9.1 both for the region of the mosaic and for

a more extended 512' × 512' region. (In the case

of mosaics with 1_5 × lf5 pixels a 768' × 768' region was used.) In only two cases (086+74 and

295-89) do the two measures of the I100 - Nil

relations diverge; in these cases (indicated by an

"E" in the sixth column of Table 9.2), the relation from the more extended region was used.

The distribution of point source detection

limits (PSDL) for each mosaic is shown in Figure 9.2. Also shown is the global PSDL to which

each mosaic has been corrected (see section 2.1

in the previous chapter). The global PSDL to

which a given mosaic was corrected was chosen

to require a minimum of correction yet allow

> 97% of the mosaic to be corrected (regions

with PSDLs greater than the global value were

discarded).

The shape of the background spectrum was

determined from the RA,qS using the "extended

regions". The local-distant separation was done

in bands R1 and R2, the local component extrapolated to bands R4-R7 assuming a Raymond

& Smith model with logT = 6.12. The SED

of the distant component was constructed from

all six bands, and the normalizations for the

soft (logT = 6.056) and hard (logT = 6.42)

components were calculated 1 We used the resul1This method could not be employed with the 295-89

mosaic. The Io - NH relation determined from the

RASS image showed no anticorretation. We therefore

assumed the ,N'bl/l_fS from the 220+84 field to calculate

tant IL values to calculate the six-band SED of

the distant component. We then recalculated

.h/'H/A/'s and repeated the process until the values of A/'H/N's converged. The resulting ratio

of the normalizations, hard-to-soft, Azil/A/'s is

given in Table 9.2.

The I'o - NH relations are shown in Figure 9.3, where the dotted brackets indicate the

range of Nil used for fitting the relations. The

results of the fits are given in Table 9.3.

Most mosaics show some divergence from the

fitted I'o - NH relation at the lowest column densities due to the small number of pixels with such

low column densities. The absorbing column

density rises above the 3 x 109-0 cm -2 threshold in three fields, though those regions were removed from the fits. Above this threshold one

might not expect the I'o - NH relation to follow the extrapolation of the fit due to the presence of molecular gas, as well as the temperature and emissivity variations likely in molecular gas. However, as one can see from mosaics

138+52 and 295-989, at these column densities,

the change in I_ due to a change in NIf is very

small. Thus, errors in the I100 to N/_ conversion

will propagate into very small errors in I_. One

should also note (from Figures 9.4 and 9.5) that

the regions with Nil > 3 × 109-0 cm -2 are quite

small, so the measurement error is large. Thus,

the goodness of the fit over the column density

threshold should not be unexpected.

Images of the distant emission, absorbed by a

uniform 5 x 1019 cm -2 are shown in Figures 9.4

and 9.5. Also shown are the regions removed

from the fit due to NH > 3.0 x 109-0cm-9- (solid

the ae[ [. We then made a local-distant separation using

the 15 - N/./ relation from the mosaic (where there is a

st rong anti-correlation).

131
7"

k

J

v

J

3.0

2.5

2.01

1.5

1.0

0.5i

0.0

3.0

2.5

2.0

t.5

1.0

9.5

0.0

3.0

7` z.5

2.0

t..S

,! 1.0

0.5

0.0

3.0

_ 2.5

2.01.5

v

j t.o

O.S

0.0

3.0

_ 2.5

• 2.0

a,b

_, ,.e

1.0

0.5

0.0

3.0

_ z.s

_ 2.0

_ 1.5

j l.o

O.S

0.0

=

v

J

3.0

2.5

2.0

i.5

1.0

0,5

0.0

0.0

035+78

086+74

138+52

161+51

171+50

220+84

295-89

0.5 i.O 1.5 2.0 2..5 3.0

HI (|0" era")

Fig. 9.1.--1100-H I relationsfor the mosaics.

Points are from the region of the mosaic only.

The dotted lineisthe linearrelationdetermined

from the mosaic region;the solidlineis determined from the extended field.The excursions

in the 295-89 fieldare due to the pixelizationof

the Dwingeloo data being much smallerthan the

resolutionof the Dwingeloo data.

contours) or NH(ivf>Tskm s-s) > 1.5 x 10]9

cm -2 (dashed contours) In order to make these

images most useful for understanding the variation of the emission, the plotted quantity is

It) - (It)) (9.1)

O'ID

where azD is calculated from a Gaussian fit to

the core of the distribution of pixel values. The

values of (ID) and aid are given in Table 9.1.

Both the R45 and R1L2 images show variation on many angular scales, from the size of the

PSF to the size of the mosaic (i.e., a gradient).

Figures 9.6 and 9.7 show the mosaics smoothed

by a 121' sliding box; Figures 9.8 and 9.9 are the

high-pass filtered images: the mosaics after the

smoothed mosaics has been subtracted.

The mosaics smoothed by 121' show a variety of structure. Some is clearlydue to problems in the mosaicking process; the left-hand

"ear"of the RIL2 mosaic of 171+50, the upper

"ear"of the R45 mosaic of the same region,and

the end-to-endgradientsof 035+78 and 086+74.

Even ifthe mosaicking algorithm were perfect,

many of these structureswould stillexistdue to

largescalevariationin background removal in

the RASS. High-contraststretchesof the RASS,

particularlyin bands R4 through R7, show features along the scan direction.These features

willbe replicatedin the mosaics. Some structure, such as the central enhancement in the

RIL2 band mosaic of 220+84, appears to due

to structurein the softX-ray background.

2. Not-So-Boring Stuff

2.1. Quality of the Deabsorption

It is not trivial to determine the amount of

distortion in the images of the distant emission

due to improper deabsorption. The distortion

can not be too gross, else it would be apparent

in the images or in the I_o - NH relations, but

distortions on a given angular scale, whose amplitude is on the order of the uncertainty in the

mean flux on that scale, would be difficult to

detect.

To gain some insight on the amplitude of such

distortions (but, alas, not providing insight on

the angular scale of such distortions) we caicu132
TABLE 9.1

IMAGE DISPLAY PARAMETERS

Field RIL2 R45

(ZD) ((ID - (ID)) 2) _ID" {_'D) {(-¢D-- (ID))") °tD"

I0 -e Counts s-i arcmin -_ i0 -s Counts s -l arcmin -2

035+78 445.0 620.1 477.9 59.9 159.8 138.9

086+74 338.0 528,4 364.0 63.8 161.8 123.2

138,1,52 315.0 605.5 486.7 24.2 159.4 116.7

161,1,51 405.3 680.3 579.8 224 189.0 170.4

171@50 414.4 934,0 770.9 20.6 282.7 197.4

220-1-84 559.4 1065. 733.1 51.9 203.1 181.2

295--89 290.8 982.6 722.3 60.7 218.2 146,8

"This value determined by fitting a single Gaussian to the distribution of pixe|s

with values greater tha_ zero. Since not al| pixels in the mosaics actually contain

recorded X-ray photons, the distribution of pixels with v_lues less than zero will be

non-sts, tistical; the values will be negative of the modeled background. Thus, they

were excluded from the fits. It can not be emphasized too greatll/ that because the

pt.cei size is so area|l, many pLCelS do not contain counts, and the o are dominated

b_l counting statistics. These values were only used to dean the images of "hot"

pt.cels.

TABLE 9.2

DEABSORPTION PARAMETERS I

Field PSI,_ a b Reg" A/H/Ms

RIL2 R47

(counts s -l ) (MJy sr -l ) (MJy sr -l cm 2)

035,1,78 0.020 0.010 +0.019610.0319 0.5147-I"00283 M 1.17

086,1,74 0.030 0.013 -0.1273+0.0200 0.6127'4-0.0200 E 0.48

138,1,52 0.030 0.018 -0,04804-0.0232 0,69364.00256 M 0.25

161,1,51 0.030 0.014 -0.10284-0.0248 0.70474-00305 M 0.16

171,1,50 0.040 0.021 -0.0324-I'0.0215 0.57264-0.0209 M 0.08

220,1,84 0.040 0.021 .1.0.07744-0.0103 0 54864-00066 M 0.74

295-89 b 0.025 0.014 +0.09814-0.0106 0.43814-0.0063 E 1,00

• "M" means that the HI-to-|to0 relation wu derived only from the mosaic region. "E"

means thltt the relation was derived from a larger area due to the lack of dynamic rtunge

within the moaalc region.

bSee text for the explanation of the method used to determine .A/'H/A/s for this field.

133
1.0

• 0.8

E

= 0.6

z.

0.4

0.2

0.0

1.0

• 0.8

E

_. 0.0

___ 0.4

0,2

0.0

1.0

• 0.0

g

_ 0.6

Z 0.4

0.2

0.0

1.0

• 0.8

e

_ O.e

_ 0.4

0.2

0.0

1.0

• O.S

_ 0.0

_ 0.4

0.2

0.0

;.0

• 0.8

e

= o.0

z

0.4

0.2

0.0

1.0

• 0.8

z= 0.6

_ 0.4

0.2

0.0

• ,, ,,, :

_ _ 035+78

086+74

Ia8+52

ii% '°'+°'

171+50

| t J ] I I I I I I

_,_ . __ 295 -89

0.000 0.010 0.020 0.030 0.0400.000 0.010 0.020 0.030 0.040

Detect/on Limit (counts s") Detract/on ldmit (counts s4)

Fig. 9.2.-- Distribution of point source detection limits, the relative number of pixels with each

detection limit. The arrows indicate the point source detection limits to which each field was

corrected in each band. Left: Band R1L2, Right: Band R45.

134
?

0

v

o

1200

1000

8OO

6O0

40O

2OO

0

1200

I000

8OO

6OO

4OO

200

0

1200

1000

800

600

400

200

0

1200

1000

800

600

400

200

0

1200

1000

800

600

400

200

0

1200

I000

8OO

600

4OO

200

0

1200_

1000_

800 _

600

400_

2°81

0

RIL2

035+78

i ,

RIL2

086+?4

R1L2 !

2

138+52

2 R1L2

171_5o ! ! ,

200

150

I00

50

0

-50

2OO

150

I00

5O

0

-50

200

150

I00

50

?

0

-50

o 200

150

7

I00

.... i *

R45

i

R45

)86_+ 4

R45 i1

138+52 '-,I I r,It_ll,_l_l],l 1

,, , i , i i

- i

I R45

0

g

o

50 ,, '-,.".,,,',!I,:111[

500 1161+51 ' iIIIll_

200 ...... 1 " ': R45

i -

I i

100 I,I :

5o _ ::

0 1_'1+50 "1'i i

-50, , ,I ,, ,

200150_ _:] ]R451

1oo[ , ,, ,[i

5oI iii'' " _ ,,i,,i,, ! III

O} 220+84 iV, I I

-5o_............. _L,. ; .

2ooI

150_

lOOp

-5°

0

.......... ', .', i)IL ,,nh. [L,,,

295 _ ILIi

1 2 3 4 5 1 2 3 4 5

N. (10 'o era-') N. (I0 '° era-')

Fig. 9.3.-- Fo - NH relations for the mosaics. The dashed brackets indicate the range of JYH over

which the fit was made.

135
035+78 086+74

--i o

161+51 171+50

295-89

I.__v... ,....::.': 0

- I .o -o.$ o,o o.5 1.o

Fig. 9.4.-- The R1L2 band images after deabsorption, smoothed by a 10' boxcar function. Contours denote regions with 2¥H > 3 X 1020 cn] -2. Tile dashed contours denote regions with IV

and HV gas with column density greater than 1.5 x 1015 cm -2. The boxes are locations of Abel[

clusters. The grey scale is from -le to +hr, where the _ was measured for the Gaussian core of

the distribution of pixel values. The values of (r are given in Table 9.1. The scale is shown in the

lower part of the upper left-hand panel.

136
035+78

)

,+ t

--1 o

161+51

295-89

'"'..: ,_, 0

086+74

171+50 I,I, =o+=

-I.o -o,5 o,o o.5 l.o

Fig. 9.5.-- The R45 band images after deabsorption, smoothed by a 101 boxcar function. Contours

denote regions with NH > 3 x 1020 cm --+. The do+shed contours denote regions with IV and HV gas

with column density greater than 1.5 x 1015 cm -2. The boxes are locations of Abell clusters. The

grey scale is from -hr to +lc_, where the c_ was measured for the Gaussian core of the distribution

of pixel values. The values of _r are given in Table 9.1. The scale is shown in the lower part of the

upper left-hand panel.

137
035+78

__io

161+51

295-89

086+74

171+50

138+52

220+84

-I.0 -0.5 0.0 0.5 L0

Fig. 9.6.-- The R1L2 band images alter deabsorption smoothed by a 12V boxcar. The greyscale

is the mean +200 × 10 -6 counts s -1 " -_ arcmm -. Some variation is clearly due to problems with the

mosaicking, as tile left-hand "ear" on 171+50, but other variation, such as the central portion of

220+84, appear to be due to true variation in the background. The scale is shown in the lower

part of the upper left-hand panel.

138
035+78

mlO

161+51

295-89

086+74

171+50

;:- 138+52

220+84

-i.o -o.5 o,o o.5 1.o

Fig. 9.7.-- The R45 band images after deabsorption smoothed by a 121' boxcar. The greyscale is

the mean +50 x 10 -6 counts s-1 arcmm " -_-. The variation in the upper "ear" pf 171+50, and the

end-to-end variation of 035-1-78 and 086+74 are likely due to mosaicking problems. The scale is

shown in the lower part of the upper left-hand panel.

139
035+78

__I o

086+74

161+51 171+50

295-89

#

138+52

220+84

° 1.0 -O.S @,0 O.5 LO

Fig. 9.8.-- The R1L2 band images after deabsorption, high-pass filtered with a 121' threshold,

and smoothed with _ 10' boxcar function. The greyscale is as in Figure 9.4. The scale is shown in

the lower part of the upper left-hand panel.

140
035+78

t

__1o

161+51

295-89

086+74 138_52

171+50 220+84

-1.0 -0+5 0.0 O.S 1.0

Fig. 9.9.-- The R45 band images after deabsorption, high-pass filtered with a 121' threshold, and

smoothed by a 10' boxcar function. The greyscale is as in Figure 9.5. The scale is shown in the

lower part of the upper left-hand panel.

t41
035+?8

--i o

161+51

295-89

4

086+74

171+50

138+52

220+84

Fig. 9.10.-- The relative areas of mosaics with maximum PSF sizes of 10 _ (the largest), 5/, and 21

(the smallest).

142
TABLE 9.3

DEABSORPTION PARAMETERS II

Field IL a ID* X_ v NH b Area

(10 -6 counts s -t arcmin -z) (1020 cm -2) (degree ;t)

RIL2

035+78 568.6-}-20.4 584.14-37.5 5.564 28 0.50-2.0 10.8

086+74 806.94-31.8 432.84-49.5 5.582 17 0,35-1.2 8.68

138+52 530,34-8.1 410.14-14.5 2.713 54 0.00-3.0 28.4

161+51 508.14-11.4 564.144419.7 4.488 40 0.00-2.4 21,4

171+50 472.744421.9 511.444441.0 2.335 36 0.00-3.0 17.0

220+84 523.544410.5 659.74-26.0 7.881 45 0.70-3.0 17.3

295-89 388.64-10.5 399.44-34.5 1.924 32 1.40-3.0 10.1

R45

035+78 10.5 61.2-1" 0.6 4.109 28 0.50-2.0 10.8

086+74 14.9 66.04-0.5 3.317 17 0,35-1.2 8.68

138+52 9.7 23,34-0.5 1,288 54 0,00-3,0 28.4

161+51 9.4 25.74-0.6 1,850 40 0.00-3.0 21.4

171+50 8.7 22,84440.7 2.141 31 0.60-3,0 17.0

220+84 9.7 54,84440.5 2.504 45 0.70-3,0 17.3

295-89 7.2 63.24440.6 2.020 32 1.40-3.0 10.1

"These 10 uncertainties were derived using the LMB criteria.

bThe range in NH over which the fit wss msde.

TABLE 9.4

PEARSON r s VALUES

Field RI L2-NH R45-NH RI L2-R45

ii" (ij)b ii" (ij)b UF c HPF c

035+78 0.0453071 0.00679737 -0.00070099 0.0105800 0.0404990 0.0512406

086+74 0.0302972 0.00011539 0,00350230 -0.0087495 0.1424650 0.1500020

138+52 0.0046646 -0.00902083 -0.00212031 -0.0148158 0,0787122 0.0709099

161+51 -0.0074366 -0.00649844 -0.00438955 -0.0088455 0.0557257 0.0536091

171+50 0.0164653 0.00156977 -0.00376389 0.0150840 0,0622010 0.0670470

220+84 0,0156349 -0,00390056 0.00578401 -0.0450213 0.0640094 0.0685017

295-89 0,0162464 0,00036065 0,00827481 0.0034546 0.0538589 0.0530195

"The Pearson r value for the X-ray image correlated with the NH image for the same field.

bThe mean Pearson r value for the X-ray image correlated with the NH images for the other

fields.

CThe Pearson r value for the RIL2 X-ray image correlated with the R45 X-ray image. HPF:

the images were high-pus filtered with a cutoff of 120 i, UP: the images were not filtered.

143
late the Pearson r value s, r,, for the image of the

distant emission and the image of the absorbing

column:

_']_( ID - 7--DD)( N H -- "_n )

(9.2)

If Nn and ID were completely correlated (or

anti-correlated) then the value of r, would be

1 (or -1); values of rs near zero would indicate

lack of correlation.

Since neither ID nor NH are distributed in a

statisticalsense,we can not use the valueof r,

to derivea probabilityof correlation.However,

we can use thisvalue to provide some sense of

how smallthe correlationmay be. In Figure 9.11

and Table 9.4 we show the valuesof r, foreach

field(the iicolumn of the tableand the boxes in

the figure).We then calculatethe valueof r, for

mismatched pairs,an ID image pairedwith the

Nx image from another mosaic. (theij column

of the table,in which case the measure is the

mean of allthe mismatched pairsusing a given

ID image, and the "x" in the figure).

For both the R45 and RIL2 bands, the value

of 7",for an ID image paired with itsNn image tends to be greater than (i.e.,more correlatedthan) the valueof r, formismatched pairs,

though there are usuallymismatched pairswith

strongercorrelations.The R45 band images appear to be less correlatedwith Nn than the

RIL2 band images. This resultsuggests that

the deabsorption process has over-correctedfor

the absorption by NH (under-correctionshould

produce an anti-correlation)and this effectis

strongerin band RIL2 than in band R45 (asone

might expect any effectto be). However, given

that the valuesof r, for matched pairsusually

lieswithin the distributionof the valuesof r,

for mismatched pairs,and in no case does the

value of r, fallvery far from the distributionof

r, formismatched pairs,the amount of correlation between ID and NH must be small,at least

compared to the dynamic range of the image.

calculated the value of r, unfiltered and highpass-filtered images. In both cases, the value of

r, is low, _ 0.15, suggesting that there is little

correlation between the two bands. Given the

results of Chapter 6, such a result might have

been expected at least on multi-degree scales.

There is very little difference between the

r, calculated from high-pass-filtered images and

that calculated from the filtered images (see Table 9.4 and Figure 9.12), suggesting that the

bulk of the correlation between the R45 and

R1L2 band images is due to small-angular-scale

sources.

2.2. Band-Band Cross-Correlations

We have alsoconsideredthe cross-correlation

of the RIL2 and R45 band images. We have

2This value is the equivalent to the zero-offset crosscorrelation.

144
e.

o

¢

o

g_

0.04

0.02

0.00

-0.02

-0.04

0.04

0.02

0.00

-0.02

-0.04

............................. ".............. Riik ....

X

X

X X

× × _ _

×

X +

_ _ ×

x x x _ x

X X X

X

X X X

•--i........ _ ........ i......... J....... J....... i....... ,--

× × R45

X

X x

X

x

+ X x [] ®

x _m x

[]

xB _ _

+ x _ x x x

, x

×

x

x

X +

i..k ......... i......... [........ , ......... _......... J......... i_

035 086 138 161 171 220 295

Field

Fig. gal.-- Pearson r, values for the crosscorrelation of the deabsorbed images with the

maps of the absorbing column. Symbols: D: r8

for a deabsorbed image and the map of the absorbing column, x: r, for a deabsorbed image

and maps of the absorbing column for other mosaics, +: the mean of the r, for a deabsorbed image and the maps of the absorbing column for

other mosaics.

0.10

0.14

i

_ 0.12

N 0.10

_ 0.00

ae

_ 0.00

0.04

0.02

0.02

E) m

..................... i

0.04 0.06 0.08 0.10 0.12 0.14 0.16

Pearson r(RIL2-R45)

Fig. 9.12.-- Pearson ra values for the crosscorrelation of the R1L2 deabsorbed images with

the R45 deabsorbed images; abscissa: r, for unfiltered images, ordinate: rs for filtered images.

145

Chapter 10: Fluctuation Analysis

Abstract

Thischapterpresentstwocommonmeasuresof theangularcorrelationfunction,

theautocorrelationfunction(ACF)andthe$I/I measure.Wediscussherethemeans

bywhichthesemeasuresareappliedto thesevenmosaickedfields,analyzetheextent

to whichthesemeasuresmaybeaffectedbysystematics, andcomparetheseresultsto

earlierresultsto determineto whatextentthesedataareuseful.

Weassumethat the observedfluctuationsarenot dueto eitherthe LocalHot

Bubbleor theGalactichalo,asflucutationsonscalesmallerthan30_wouldbevery

shortlivedin eitherenvironment.Indeed,anACFof emissiondueonlyto theLHB

showednoappreciablesignalat scalesgreaterthan3',thelowerlimit towhichit could

bemeasured.Largescalefluctuations due to calibration difficulties in the RASS or

emission variation due to the LHB and the Galactic halo are removed by high-pass

filtering.

The ACF results remain contaminated by residual point sources, and there may be

other systematic effects not yet removed. After removing our estimate for the maximum

contribution due to residual point sources we find the ACF; W(5' < 0 < 35') ,-, 0.01 in

the R1L2 band and W(10' < 8 < 25 _) -,_ 0.0025 in the R45 band, with no significant

signal at larger scales. Our measurement in the R45 band is consistent with the upper

limit measured by Chen et al. (1994) in the R47 band.

We have developed a new form of the $I/I measure of fluctuations. The results are

consistent with the ACF, showing no significant fluctuations at scales greater than 35 _

and 25' in the 1⁄4 keV and _ keV bands respectively.

We find a large variation in the fluctuation spectra determined from different mosaics.

1. Introduction

A cursory examination of the images in the

previous chapter shows that there are variations

on many scales, from the size of the PSF to the

size of the mosaic. The images clearly show the

effect of unremoved point sources, point sources

that were below the detection threshold of the

individual pointings. There may be sources with

greater angular extent, similar to the Abell clusters that have been removed, but the summation of many different PSFs at any given location make analysis difficult. The smoothed images show structure on degree scales, and the eye

leads one to believe (from the high-pass filtered

images) that there may be structure on intermediate scales. How much of that structure is due

to overlapping 10' PSFs remains to be seen.

Structure on large scales is not surprising; the

images of the distant component of the diffuse

emission in Chapter 6 (Figure 6.4) show structure on degree scales. Most of that structure is

in the soft halo component (log T = 6.056) while

the harder halo component (log T = 6.426) appears to be much smoother. Of the images studied in this chapter, those in the R1L2 band will

be dominated by the soft component. The two

components contribute comparable amounts to

the R45 band images. Structures on small scales

has not been studied but is not expected to be

strong; small scale structures tend to dissipate

on short time scales in hot plasmas.

In stellar astronomy, a prime difficulty is

connecting a theoretically determined HR diagram in (Tell, Mbot) with the measured HR diagram in (B- V,V). In this work, the difficulty is connecting some measure of fluctuations in two-dimensional images to the threedimensional structure of the universe. The first

147
link in thisconnectionis thetwo-pointangular

correlationfunction,w(t_), which is described in

§1.1. The following subsection, §1.2, explores

problems introduced by the data that are common to all the methods of measuring fluctuations.

We then consider two methods for measuring

fluctuations, the autocorrelation function (ACF)

and the M/I measure. The ACF is the most direct measure of w(O), but is sensitive to the effects of the PSF, and the uncertainty is inversely

proportional to the square of the number of pixels. The 6I/I measure is a direct calculation of

the pixel-to-pixel variance at a given scale. It is

somewhat easier to apply, but does not provide

a useful measure of w(O). The M/I is limited

to scales somewhat smaller than the size of the

mosaic, but the uncertainty is proportional only

to the the inverse of the number of pixels. The

ACF is discussed in §2, the JI/I measure in §3.

Comparison of our results to previous results,

(and with our picture of the universe) is discussed in §4. Making the final link to the structure of the universe will, of course, be model

dependent. Although the models discussed in

Chapter 1 have recently become available, determining whether the range of fluctuations (or lack

thereof) seen here is consistent with the model

(or vice versa) is beyond the scope of this dissertation.

1.1. The Angular Correlation Function

Of the ways to characterize the amount of

structure as a function of scale, many are related to the two-point angular correlation function, w(8). Following Peebles (1993) one can

define w(O) in the following manner in terms of

point sources. If there is a point source at a

given point, then the probability that there is

a point source in a solid angle d_ at angular

distance 0 is

dP(O) = N'[1 + w(O)]df_ (10.1)

where N" is the angular density of point sources.

The probability that two sources exist separated

by 0 is thus

dP(O) = Af_[1 + w(O)]dftdf_ (10.2)

Although this measure can be applied to catalogues of discrete objects, we are more interested in fluxes due to collections of objects. Peebles (1993, §7, Equations 7.62-7.67) develops the

formalism for this transformation from counting

objects to measuring flux. Consider two differential units of solid angle, Jf_i and _j separated

by Ois, each differential unit so small that it can

contain at most one source. If the expectation

value of the number of sources in the differential

Jfti is (ni) = A/'6ft, then we find that

(ninj) = A/'2512,(fftj[1 + w(Oij)] (10.3)

This formalism can be expanded to the expectation values for the number of objects in extended

fields. The extension to fluxes is artificial; one

can consider the flux in a pixel to be the number

of objects in the pixel, each object having a flux

Jr. The relation between w(O) and the ACF or

JI/I will derived in the sections below.

One can define a similar two point correlation

function such that, given a source at a given

point, the probability that there is a point source

in a volume dV at a distance r is

dP(r) = n[1 + _(r)ldV (10.4)

where n isthe source density. The relationship

between _(r) and a measured w(O) is complex,

relyingon the spatialdensityand luminositydistributionas a function of redshift.If the w(O)

ismeasured the fluxof an aggregate of sources,

then relationis even more complex, as it must

include the distributionsof allof the different

types of objectscontributingto the flux.

The general relation between a measured w(O)

and the distribution and size of diffuse emission

is likewise complex and, although it will be discussed in the following chapter, is beyond the

scope of this dissertation.

1.2. A Problem

There are a number of ways of characterizing

fluctuations, each fraught with its own particular dangers. Each of these methods has a common difficulty; the change in PSF radius with

distance from the optical axis. This change is

demonstrated in Figure 10.1 and tabulated in

Table 10.1. To analyse fluctuations on a given

scale, one must include in the mosaic only the

central parts of the individual pointings, where

the PSF diameter is comparable or smaller than

the scale of interest. Otherwise, one would need

148
1.0 '- ' ' '_--= ' _ '_-_-'.-',..:.,_--:-:-_--" "

i/12 f

0.8

°i0.6 J/

'i/// "_ 0.4

0.2_

0.0

0 2 4 6 8 10 12 14

Dismeter(arcmin)

Fig. 10.1.-- Comparison of PSF profiles for different bands at different distances from the optical axis. The labels indicate the distance from

the optical axis. Solid: band R1L2, Dotted:

band R45, Dashed: band R67.

to model the shape of the PSF as a function of

position within the mosaic; a difficult task since

at any given pixel there may be contributions

from several different pointings, each with a different PSF size at that pixel. Reducing each

pointing to only the central region significantly

reduces the area available for analysis.

In order to maximize the available region for

analysis, yet still avoid the PSF, ("threading our

way twixt one and the other") we have created

three mosaics for each field, each mosaic having

a different exclusion radius. The three mosaics

have an exclusion radii of 12', 32', and 53', allowing a maximum PSFs of ,-- 2', --, 5', and _, 10'

respectively The largest exclusion radius is set

by the vignetting function.

2. The Autoeorrelation Function = W(0)

2.1. Equivalent Formulations

The autocorrelation function (ACF) is generally defined as

E(I, - (O)(b - (i))

w(o) =

(i)2 (10.5)

where the sum is over all pairs of pixels / and j

that are 0 apart. Using the formalism developed

by Peebles (1993). one can show that W(0) is directly equivalent to u(O), the two-point angular

correlation function•

The ACF has an advantage over the M/I

measure; if f(u) is the image, then the ACF is

equivalent to

i f'(u)f(u + x)du= f lF(s)12e2i_°ds,

-- 00 --00

(lO.6)

where F(s) is the power spectrum of the fluctuations.

2.2. Algorithm

There are several different formulations of

the autocorrelation function in common use, the

most general form of which is

W(O) = Ei'i'(R - (R})(R' - (R')) wvrw'-'_

(10.7)

where the sum is over all pairs of pixels seperated

by 0, R is the count rate in those pixels, and w

is the statistical weight given to those pixels.

Depending upon the formulation

{ ,

R (10.8)

x= (ER+En')/2

i i'

We have calculated the ACF using this last form

of X to remain consistent with other calculations

of the ACF (Soltan & Hasinger 1994; Chen et al.

1994).

For the weights, w, w', one could take the uncertainties calculated in the previous chapter.

However, there are difficulties using these uncertainties. As many pixels in the image have

only a small number of counts, assuming Poisson statistics, the calculated uncertainties will

be strongly correlated with the value of the pixel;

in a region of uniform exposure (and thus, uniform true uncertainty) the variation in the number of counts per pixel will cause one incorrectly

to weight more strongly the pixels with larger

number of counts.

Rather than using the measured uncertainty,

we have followed the method of Chen et al.

(1994) with modifications. The number of counts

149
TABLEI0.I

PSPCPSF-FIELDRADIUSRELATION

PSF• FieldRadius

DiameterR1L2 R45 R67

2 12.7 15.7 14.7

3 20.7 21.7 20.7

4 26.7 27.2 25.7

5 32.1 31.6 30.6

6 36.6 36.6 35.1

7 41.6 41.1 39.6

8 45.6 45.1 43.6

9 49.6 49.I 47.6

I0 53.6 53.1 51.6

• This is the diameter encircling 90%

of the energy.

in pixel (i, j) is given by

C(ij) = EV(ij)'g(ij) (10.9)

where E is the length of the exposure, V(ij) is

the vignetting function, and "g(ij) is the cosmic

flux. Therefore, assuming Poisson statistics, and

assuming that the true variation in the cosmic

flux is small, the true uncertainty of pixel (i, j)

is

(10.10)

where, for the purposes of Equation 10.7, the

proportionality constant is immaterial and the

instrumental background is assumed to be much

smaller than the cosmic background. This is the

weighting scheme used by Chen et al. (1994).

We have modified the weights to account for the

relative uncertainties due to deabsorption:

W(iz) = EV(ij)e -r('J) (10.11)

where e_(,,i) is the factor by which pixel (i, j) is

deabsorbed.

2.3. Uncertainty

The variance in the ACF is given by

E .(w(e,,oy)-w(oO)) 2

2 sf,*,lS.'+o,_=so'

_w(s) =

(1o.12)

where n is the number of different measures of

W(# = 00). The uncertainty in the mean ACF

will thus be given by

"_(e) = "_(e) (10.13)

n

This measure of the uncertainty will be unreliable at small 0 as there will only be 7rOdO/2

different measures of the ACF. The commonly

used "shume" method for determining the uncertainty in the ACF, which operates by calculating the ACF of the image after all its pixels

have been randomly rearranged, is not applicable to these fields due to the presence of structures with amplitudes greater than that of the

noise (See Appendix G).

2.4. Data Preparation

The images used for calculating the ACF are

the deabsorbed mosaics that have been further

masked to reduce spurious fluctuations. Point

sources were detected in the individual images

and removed before the mosaicking. Regions

with N_ > 3 × 1020 cm -2 or IVC/HVC column

densities greater than 1.5 x 1019 cm -2 have been

removed. Regions with a 8' radius have been removed around every known Abell cluster; larger

radii were removed if visual inspection suggested

that the cluster emission had a greater extent.

A few regions that had X-ray emission similar to

that of clusters (but for which there was no identified cluster) were removed, as were a few point

sources that had not been detected in individual

150
pointings.Afewregionswherethedeabsorption

didnotappeartobetoosucessflll havealsobeen

removed.

Theimageswerealsocleanedof individual

"spurious"pixels;isolatedpixe]sthathavecount

ratesgreaterthan 4trlauzabovethe mean1.

Thesepixelsarepresumablydueto single(or

double)photoneventsinregionswithverysmall

effectivexposure,andareoftenontheedgeofa

maskedregion.Removingthese"spuriouspixels"reducestheamplitudeof thepeakof the

ACF at 0 = 0, and reduces very slightly the

variance of the ACF at all other values of 0.

The deabsorbed images used for calculations

include only the flux that could not be attributed to the Local Hot Bubble (LHB) or to

the Extragalactic Power Law (EPL, the contribution to cosmic background by unresolved

point sources), that is, the images only included

the "Galactic halo" and the putative diffuse extragalactic emission. (Those two components

together we have called the TAE (Chapter 6),

the Trans-Absorptive Emission.) In order to be

consistent with previous measures, rise display

ACFs were corrected to include the EPL component as well;

WEPL+TAE -- WTAE (_,,RTA-_-REPL RTA )5

(10.14)

All further ACF values quoted before §4 are for

the combined TAE-EPL ACF. The ACF for the

R1L2 band is calculated in the same manner as

the ACF for the R45 band.

2.5. Application

We have measured the ACFs of each mosaic

in each band. The results for a typical mosaic

are shown in Figure 10.2. The effect of unremoved point sources can be seen at 0 £ 10', and

the effects of large-scale structure can be seen

at 0 _ 100'. However, it should be noticed that

the large-scale structure will have a significant

tCalculating the variance of the pixel values is trivial,

but calculating a meaningful variance for this purpose

is a bit more difficult. A histogram of the pixel values

reveals a "well behaved" Gaussian core, and extensive

wings. The distribution is better fit with two overlapping

Gaussians, the one with lower amplitude having a larger

variance. The ajau= used for calculating the cleaning

limit was the variance of this broader component.

effect on the measurement of the ACF due to

small-scale structure.

Consider. for a moment, a not unreasonable

model of the sky: assume that there are uncorrelated point sources and assume that the Galactic

halo has variations on the scale of a degree. The

contribution of the large-scale structure to the

ACF will have its greatest amplitude at 0 = 0,

and will decline as 0 increases. Without knowledge of the large-scale structure, one could interpret the same ACF as due to a collection of

much smaller scale variations. Thus, if one were

attempting to place an upperlimit on the contribution of small-scale fluctuations the unrecognized existence of large-scale structure would

cause the measured upper limit to be significantly too large.

To achieve the tightest possible constraints

on the ACF at 0 < 1° one must remove the influence of the large-scale structure of the halo

as much as possible. We have chosen to minimize the impact of the halo by high-pass filtering the mosaics with a low frequency cut-off of

121', slightly larger than the size of an individual pointing. The high-pass filtering will also

ameliorate (but not entirely remove) the effects

of offset errors in the mosaicking process. Since

we expect the halo emission to have structure on

degree scales, high-pass filtering with a cut-off of

121' will, at worst, remove non-halo structures

which, were they detected, would not be separable from halo phenomena.

ACFs are shown both for the mosaic and

for the mosaic after high-pass filtering. Similar plots for the other mosaics are given in Appendix H. The ACFs from the high-pass-filtered

band R45 mosaics are fairly consistent. Most

the ACFs from the high-pass-filtered band R1L2

mosaics are consistent, with the exception of

the 086+74 mosaic. Although this mosaic is arguably the best of the seven, as the individual

pointings having very high exposure, the bulk

of the pointings having substantial overlap, and

the entire field having very smooth absorption,

the ACF is very different from the others in band

R1L2. The image appears to have structure on

the -.. 25' scale. Visual inspection of the highpass-filtered mosaic reveals no peculiarity with

respect to the other mosaics _. We have removed

2There are a few point-source like objects remaining in

this mosaic, as there are in all of the other mosaics. There

151
0.10

0.05

- 0.00

-0.05

-O. lO

010

0.05

0.00

-0.05

-0.t0

0.10

0.05

, OJO

-O.OS

-0.10

0.2

_ 0,!

-0.!

-0.2

0.2

A

":I0'

I _ "T _IF 71

10'

O.l

0.0

-0.1

-0.2 5'

o.2

o.1

o.o

-o.!

-o.2 2'

0.0

" "I "-" .... • -"y"q

lli'l{itil

0.5 |.0 I .S 2.0 2.5

i,_.,,[0 (ercm, lnuMs))

0.04 :

0.02 i

0.00

-0.02

-0.04

0.04!

0.02

0.00

-0.02

-0.O4

0.O4

0.02

0.00

-0.02

-0.04

0.10

0.05

0.00

-0.00

-0.10

0.10

0,00

0.00

-0.0'5

-0.10 .5'

0.10

0.00

0,00

-0.05

-0.10 .2'

0.0

"RI]2____ __ _ _

lO'

i0'

1:l lilllH Ill Iiill " ill

0.5 i .0 1.5 2.0 2.5

bb,(# |.cminut.))

Fig. 10.2.-- The ACF of a typical mosaic (161+51). The solid line is the ACF, the shaded region

is the uncertainty. The ACF is shown for mosaics where the maximum radius for each pointing

has been chosen to produce maximal PSFs of 10', 5', and 2'. Left: ACFs of the images. Right:

ACFs of the high-pass filtered images. Note the changes of scales.

this mosaic from further consideration in band

R1L2, but its presence should continue to pose

a caveat on our conclusions.

2.5.1. Averaging the ACFs

We have found the mean ACF for the seven

mosaics by summing the ACF for each mosaic,

weighted by the average exposure for the mosaic

and by the uncertainty. The effective mean flux

(R) for the coadded ACFs is 527.8(110.2) xl0 -s

counts s-I arcmin -2 for band R1L2(R,45).

In order to reduce the uncertainty at large 8,

we arbitrarily clipped individual ACF at the 0

at which sequential values of the ACF began to

diverge. The exposure weights and the 9ms= for

each ACF is given in Table 10.3. A number of

different weighting schemes were tried, but they

produced little difference. Restricting the sum

to the four best mosaics also changed the mean

ACF very little.

The uncertainties for ACFs measured from

are also a few abnormally low patches, as there are in

several other mosaics. Masking out these regions in the

next iteration may lead to improvement, but such is not

likely.

mosaics with smaller maximum PSFs are significantly worse; the number available pixels

is sharply reduced by the demand for a better PSF (see Table 10.2). One would expect

W(5' < 0 < 10') to be much lower for the ACF

measured from a mosaic with a maximum PSF

diameter of 5' than from a mosaic with a maximum PSF diameter of 10'. However, as can be

seen from Figure 10.3, the significantly higher

noise in the ACF measured from mosaics with

5' PSFs completely masks this effect. Therefore,

we must not expect to improve upon the ACFs

measured from mosaics with 10' PSFs; any information in the ACF for O < 10' will be accessible

only by modeling the ACF due to the PSF a.

2.5.2. The PSF

How much do unremoved point sources contribute to the ACF for 0 near the size of the

maximum PSF diameter? Figure 10.3 shows

the summed ACFs compared to the ACF of the

maximum PSF. The'ACF of the maximum PSF

has been scaled to match the measured ACF

31t is for this reasonthat we have not been more careful

in the summation of the 5' ACFs

152
"]'ABLE 10.2

MAXIMUM NUMBER OF PIXEL PAIRS

mosaic

I0'

R1L2 R45

5' 21 10' 5'

number of pixel pairs/106

21

035+78 16.23 4.56 0.31 12.81 4.00 0.34

086+74 19.02 7.86 0.51 19.23 8.36 0.64

138+52 26.34 7.78 0.17 23.18 7.01 0.17

161+51 47.23 14.16 0.25 44.18 13.21 0.31

171+50 49.22 25.98 8.63 33.30 22.14 8.10

220+84 42.08 14.22 0.38 37.64 13.40 0.53

295-89 15.37 4.23 0.05 11.81 3.33 0.06

at the PSF diameter, thus demonstrating the

maximum contribution that the PSF could have.

The extent of the bulk of the ACF is similar

to the 90% encircled energy diameter, but it is

clear that the ACF of the PSF constains nonnegligible signal to much larger scales.

In band R1L2, there is clearly signal above

that expected from the PSF on scales 10' < 0 <

25', and the ACF made from the 5' image suggests that the excess may extend to 0 = 5'. The

situation in band R45 is more equivocal; the 10'

PSF image suggests an excess for 10' < 0 < 25',

which is supported to nearly the same certainty

by the 5' PSF image. However, the excess signal in band R45 is quite small, comparable to

the size of the uncertainty.

Comparison of the measured ACF with the

ACF of the PSF is valid to the following extent:

the brightest remaining point sources will be in

the portion of the mosaic with the worst point

source detection limit, which is the region with

the largest PSF. Thus, one expects the portion

of the ACF due to unremoved point sources to

be dominated by objects whose profiles are well

described by the maximum PSF. This simple description is somewhat modified by the fact that

the number of sources drops sharply as the flux

is increased, so the flux (and PSF size) of the

sources that dominate the ACF will be the result

of a competition between the reduction of the

PSF size (and the subsequent reduction of both

the point source detection limit and the area

over which the point sources could be detected),

and the increase in the number of sources per

unit solid angle at the point source detection

limit. The result of this competition is shown

roughly in Figure 10.4, where one can see that

the flux contributed by the unremoved point

sources is nearly constant with PSF size. Thus,

the wings will be due to the largest PSF, but the

remainder to the PSF ACF will be much more

sharply peaked. Our scaling of the maximum

PSF is thus likely to overestimate the contribution from the PSF ACF.

The strongestpoint sources could be correlatedto some extent, as they willbe confined

to the regionsof the image with the worst PSF,

and not scatteredrandomly acrossthe mosaic.

For typical point source detectionlimits,0.01-

0.03 counts s-I, the point source population is

dominated by the extragalacticpointsources.If

we take the "worst PSF" regionto be a 10_wide

annulus, then we expect ,_ 0.5-5point sources

per pointing4. However, since pointings overlap, some fractionof that annulus willhave a

4We have calculated the number of sources as the number

of unremoved sources in the 43_-53 ' annulus that would

have been removed given the PSDL at 43'. This is a

crude measure of the "bright" sources in the annulus.

We find the following numbers for a single pointing of a

given exposure:

Exposure R1L2 R45

PSDL Number PSDL Number

(s) (cts/s) (cts/s)

1000 0.017 0.573 0.044 0.964

2000 0.007 1.57 0.017 2.94

10000 0.005 2.59 0.013 8.90

These results suggest that many mosaics will not have

correlated sources due to this effect because the mosaic coverage is dominated by low exposure pointings.

Most of the mosaics that have a large number of deep

(10ks) exposures also have a large amount of overlap

(035+78,086+75,161+51), which will reduce the correlation. A possible exception is 295-89.

153
O.OIO

_' 0.000

D

-0.010

0.010

0.000

-0.010

0.0

i0'

"5'

i

0.5 1.0 1.5 Z.O Z.5

Lol,.{O{irt.mtlu_))

0.004

A 0.002 ]

, 0.000

-0.002

A

-0,004

0.004

0.002

0.000

-0,002

-O.O0,i

10'

0.0 0.5 1.0 1.5 Z.O 4L5

Loi,Je(areminuteJ))

Fig. 10.3.-- Mean ACF for high-pass-filtered

mosaics. The solid line is the ACF. The shaded

region is the uncertainty. The ACF of the maximal PSF is shown as a dashed line. For band

R1L2, the addition of the 086+74 mosaic is

shown by a dotted line. Note that the highpass filtering should remove fluctuations above

log 0 = 2.08.

25 30 35 40 45 SO

Radius(arcmin)

Fig. 10.4.-- The unremoved point sources that

dominate the ACF: the plotted quantity is the

flux of the point sources just below the detection

limit; the number of excess sources in an annulus

of width 1' multiplied by the point source detection limit at the radius of that annulus. The

number of excess sources is the number of unremoved "bright" sources in an annulus; the number of unremoved sources that would have been

removed had the PSDL been that of an annulus whose radius is smaller by 10'. Solid: 10 ks

exposure, Dashed: 5 ks exposure, Dotted: 1 ks

exposure.

154
substantiallybetterpointsourcedetectionlimit;

the numberof unremovedpointsourcesin the

regionof theworstPSFwill becorrespondingly

decreasedto lessthan--- 1 per pointing. Therefore, one should not expect correlation from the

brightest point sources, but from the fainter

point sources with the better PSFs.

The contribution of unremoved point sources

to the ACF can be calculated more carefully, but

only at great expense. The ACF of unremoved,

individual point sources would be the ACF of

a complexely weighted sum of all the PSFs in

a given mosaic 5. To address the issue of correlation produced by the variable point source

detection limit, one should measure the ACF

from simulated mosaics rather than an equivalent model PSF.

It should be further noted that for point

sources on a constant background, W(0) < 0

for large 0. The negative excursions in the measured ACFs may be somewhat removed by more

proper modeling of the PSF, increasing the amplitude measured fluctuations in the 10' < 0 <

30' range.

3. 5I/I - A Statistical Method

This method begins with the seemingly simple question; "Could the observed fluctuations

be caused by the variation due merely to counting statistics and the unremoved point sources?"

For pixels of a given size, one compares their

distribution about the mean with that expected

from a "flat" background; excess variation may

sIf t is a given pixel, and j is the index of the different

pointings (observations) that contribute to the mosaic

image at pixel i, then the image whose ACF we seek is

PSDL_

_ PSFi,j f sN(s)ds

, 3 ?

where PSF is the image of the PSF at that sky location

for each pointing, and PSDLi is the point source detection limit at that pixel (the best of the PSDLi,_.), and

N(s) is the number of point sources of flux a expected

per pixel. The lower limit of the integration is a vexed

issue. One certainly would not want to extend the integral to 0. as the very faint sources will be so numerous

as to provide a uniform background when measured over

individual pixels. Setting the limit to PSDL1 will miss

the contribution by sources slightly fainter than PSDLI,

that are also slightly more numerous. Perhaps one should

set this limit to the source flux for which there is one

source per pixel.

imply fluctuation by the background on the scale

of the pixel size. A small note on language:

by fluctuation at scale ¢ we mean that the

background has intrinsic fluctuations on scale _.

Conversely, "tack of fluctuation" on scale V does

not mean that the excess variance in ¢ x ¢ sized

pixels is zero, but that the variance is produced

by fluctuations whose intrinsic scale is not ¢.

The _I/I measure is defined as

(?)_- ((1-(1))2)(-1-_ " (10.15)

This quantity is calculated over a range of pixel

sizes, ¢ x ¢, from the smallest possible to the

largest for which reasonable statistics are possible. The smallest pixel size computationally

possible for our mosaics we will call the "native"

pixel size, and is either 1' or 1._5.

3.1. Relation to w(O)

Using the formalism of Peebles (1993), one

can show that this quantity is equivalent to

1 l [/df_idf_i,w($i,,) ] (10.16) (i-7+

where the integral is over scales smaller than the

sizes of the pixels whose variance is being calculated. This is not a particularly interesting formulation but is included here for completeness.

3.2. Algorithm

There are complications in the application of

this calculation to mosaicked data; the expected

variance in the value of a pixel is spatially variable.

3.2.1. Counting Statistics

The vignetting due to the telescope optics reduces the number of counts with distance from

the optical axis, (as well as severely reducing the

counts over a limited area due to the window

support structure). Thus, the amount of variation due to counting statistics will change from

the center of the pointing to the edge. We have

used as the variance due to counting statistics

the number of counts in a mosaic pixel, and have

applied standard propagation of errors through

155
TABLE 10.3

SUMMARY

mosaic 0ma:r_ Weight ¢,n_ b

RIL2 R45 R1L2 R45 R1L2 R45

arcmin arcmin

035+78 160 160 25.27 22.07 64 60

086+74 250 250 40.35 39.01 61 53

138+52 355 320 9.09 8.03 109 106

161+51 250 160 13.09 11.92 85 85

171+50 200 100 12.20 10.54 68 68

220+84 320 250 9.82 8.43 126 117

295-89 100 100 22.42 18.81 61 59

aThe largest scale probed by the ACF.

bThe largest scale probed by the _I/I measure.

the deabsorption process. As, in most mosalcs,

t2 •

there are a large number of 1 plxelsthat do

not contain counts, thus the value of the variance must be used carefullys. Binning by a few

(where "a few" < 10 for images with 10tPSFs)

removes this problem. As we willshortly see,

thisbinning isnot wholly responsiblefor losing

informationon scalessmallerthan the PSF size.

Note that the variancedue to counting statisticsdoes not include the uncertaintiesdue to

the background subtraction. These willbe addressed later.

3.2.2. Point Source Contributions

Due to the change of the size of the PSF with

distance from the optical axis, the point source

detection limit changes with distance from the

optical axis, and thus the contribution of unresolved point sources will also change. One generally assumes the point sources to be uncorreSOne might think to alleviate this problem in the following manner. The expected variance for a pixel due

to counting statistios is the expected number of counts

in that pixel, which is given by Equation 10.9, where

R(,,j) is given by

_LI'IB + _TAE e-'t'I'AE + _EPL e-r£PL •

One could then propagate this expression through the

deabsorption process to determine the uncertainty of every pixel. However, as this expremion ignores the structure in the field, so the measure of the variance will he

systematically wrong.

lated, (though it is well known that extragalactic X-ray sources will be correlated) and so the

number of sources of a given flux in a given pixel

is given by counting statistics; the variance in

the number of point sources is the mean number of point sources. If the number of sources

in a unit area in the flux interval S, S + AS, is

given by n(S), and if the minimum detectable

point source flux is Sd, then the contribution to

the flux in a given pixel is

cps = f_ fs" Sn(S)dS (10.17)

m

and the variance in that flux is

¢_s = £t is"S=n(S)dS (10.18)

nl

where 12 is the area of the pixel. The value Srn

is either the flux of the faintest source, or flux

of sources that are so numerous as not to contribute significantly to the variance.

The point sources that contribute to fluctuations of the soft X-ray background are either

Galactic (mostly young main sequence stars), or

cosmic, typically AGN. The form of the n(S) relations to be used for Galactic point sources is

discussed in Chapter 7. For the n(S) relation of

the extragalactic sources we use that of Hasinger

et al. (1998),

n(S)=n0s -_° for s>18

n(S) = nls -_' for 18>s>2.66

n(S) = n_s -z2 for 2.66 > s (10.19)

t56
wheres is in units of 10-14 erg cm -_ S -1 in the

0.5-2.0 keV band _, and

no = 111.0

nx = 238.1

n2 - 91.0

/30 = 1.94

B1 = 2.72

_ = 2.3 (10.20)

where nn is in units of degree -_. These values are quite similar to that determined by

Mushotzky et al. (2000) for the 0.5-2.0 keV

band.

Note that these calculations do not include

the effect of the PSF. If the pixel size is smaller

than the PSF, then the PSF spreads the flux due

to unresolved point sources over several of the

surrounding pixels, and effectively decreases the

variance due to point sources. Therefore, one

should not attempt to apply the $I/I measure

to scales smaller than the PSF.

3.2.3. The relative contribution

The contribution to the variance by both

the counting statistics and the unremoved point

sources are exposure dependent. The contribution by unremoved point sources is typically

comparable to, to ,_ 3 times smaller than, that

of counting statistics. Of the contribution by unremoved point sources, less than about a third

is due to Galactic sources.

3.2.4. The distribution of values within

a single pixel

If _r_ is the variance of the distribution of

"true" background sky values, _s is the variance due to the counting statistics, and a_'s is

the variance due to unremoved point sources,

then, since the three quantities are independent,

the variance of the measured value in a given

pixel will be the sum of the variances. Therefore, the distribution from which the measured

value is drawn has a variance

_4 = a_s + a_'s + o'_. (10.21)

7These are the values presented by Hasinger et al.

(1998). If one has reMon to apply these equations (or

is obsessive about checking them) one will notice that

given the normalizations in the following lines, the equations produce a discontinuity in n(s) at s = 18. The

value for the split between _0 and _1 should probably be

9.87, not 18.

The _M is measured from the mosaics, the

_rcs can also be estimated from the mosaics, and

the crps can be calculated from the point source

models. The desired quantity is the orB, which

is sometimes referred to as the excess variance

as it is in excess of that expected from counting

statistics and the point source models.

For convenience (computational and otherwise) we will take _ to be the variance in counts

s-1 within a pixel of size ¢ × ¢. The $I used in

Equation 10.15 will thus be crB/¢ 2.

3.2.5. The variance of a collection of

pixels

Although every pixel is drawn from a different distribution one must combine them in some

way to calculate the background fluctuations.

The procedure for combining the pixels is simple; calculate the X2 statistic for pixels that have

been binned ¢ x ¢,

(C- (C)) _ (10.22) 1Z 2--if'---- 2

(i,j )

where v is the number of ¢_ sized pixels in the

sum and C is the count rate within the pixel (in

units of counts s- 1). If n is the number of native

pixels in a ¢ × V sized region, na_ = a_. If the

expected variances of the individual pixels was

under(over)estimated, then X_ statistic for the

ensemble of all the pixels will be greater(less)

than one. This same operation can be thought

of as normalizing all of the distributions from

which individual pixels are drawn to have a variance of one). One may thus determine the excess

variance (aB) by iteration.

We have departed from standard practice

in calculating the quantity o"b rather than _rB,

though it is easy enough to determine one from

the other. If n is the number of pixels in a given

¢2 sized pixel that are non-zero, then we can

include binned pixels that are not "completely

filled" into the calculation, with an apropriate

weighting. This practice improves one's statistics at all ¢. As we will see, the quantity _rb can

be more useful than aB.

3.3. Uncertainty

The value of X_ will, naturally enough, have

the properties of the "X2 distribution", which is

157
awell-behaved, thoughasymmetric, probability

distribution.Wehavedefinedthe uncertainty

intervalin X2 so that

+a

/ P(x2)dx2 = / P(x2)dx2 =.3415 (10.23)

that is, the interval contains the same integrated

probability as the standard le interval and is

centered on the mean. For _, < 20 this interval

is asymmetric due to a large difference between

the mean and the mode. The uncertainty in _rb

is just the range of values of ab that fulfills this

condition.

3.4. Data Preparation

The same data preparation used for the ACFs

were used for this calculation.

3.5. Application

Figure 10.5 shows (aB(¢)/I) z for a typical

mosaic. This plot is not too satisfactory. If the

excess fluctuations were due only to uncorrelated

fluctuations at or smaller than the native pixel

scale, then crs(¢) o¢ ¢-_; very nearly the observed behavior. One is really interested in the

deviations from the 1/_b2 behavior. For this reason we find it more useful to consider ab(¢)/I.

The quantity o'_ is the excess variance at the native pixel scale required to produce the observed

excess variance at the scale of _b. This quantity

has the useful property that if there were no

fluctuations on scales greater than that of the

native pixel, o'b would be constant. If, on the

other hand, the dominate fluctuation scale were

_I,, then, roughly, O'b(_) CX .(¢/(I_) 2 for ¢ < @.

Figure 10.5 also shows (#b(¢)/I) 2 for the same

mosaic; the effect of the large-scale fluctuations

become immediately apparent. For this mosaic

there appear to be fluctuations with ¢ ,_ 30' as

o'b is constant (to within the uncertainties) for

greater ¢.

Note that we display the quantity (O'b(¢)/I) _

rather than ab(¢)/l. If one overestimates the

contribution to the variance due to counting

statistics or unremoved point sources, then the

quantity (_b(¢)/l) _ will be less than zero. In order to handle this possibility in the most graceful

manner, we do not take the root.

Figure 10.6 shows (c,(¢)/I) -_for a typical mosaic (161+51). Here, as with all the plots of

(ab(¢)/I) _ given in Appendix H, the value of

ab has been normalized to a native pixel size of

l' x 1'. The quantity I is the mean flux in counts

s-1 arcmin -2. The upper limit on ¢ is set by the

imposed requirement that the number of ¢ x ¢

pixels be greater than ten.

The generally good agreement between ab(_b)

for images where the maximum PSF size was 10'

and for images where the maximum PSF size

was 5' suggests that the characterization of the

change in the variance as a function of exposure

must be reasonably correct, as the 10' PSF images (exclusion radius, 53') sample regions with

much larger variance than the 5' PSF images

(exclusion radius, 32'). There are, of course exceptions. For mosaic 171+50 in band R1L2, the

reason would appear to be that the 5' PSF image

excludes some regions with poor deabsorption.

The discrepancy for mosaic 138+52 in band R45

is nearly within the uncertainties, but its source

is unknown.

3.5.1. The Effect of Offsets

In the mosaicking process, each individual

pointing is corrected for the non-cosmic background to the flux level of the RASS by a uniform offset, Oi. We have not included this uncertainty in the formulation of JI/I because the

bulk of the uncertainty in the background does

not enter as a pixel-to-pixel variation, but rather

as an offset s. Of course, using a mosaic of pointings means that the offsets will be different for

different regions of the mosaic.

We could derive formal uncertainties for the

offsets of each of the pointings from the RASS,

and thus derive a formal uncertainty for the relSThe lack of pixel-to-pixel variation may not be clear

given the treatment in Chapter 2, where we stated that

the uncertainty in the exposure mask (from which the

non-cosmic background is modeled) is on the order of

10%. Due to the "dither", each 147957 x 147957 pixel

in a single PSPC pointing samples ,-, 200 adjacent pixels

in the exposure mask. Thus, the difference between to

adjacent pixels in a single PSPC pointing will be small

because they will have sampled nearly the same set of

pixels in the exposure mask. Thus the relative uncertainty for adjacent 14°.'957 × 14_._957 pixets will be less

than 1.5% Since these pixels are then summed into 1'

or 1 ._5 pixels, which are then summed with pixels from

other pointings, the relative uncertainty is very small

compared to the other variations.

158
3O

ZO

_ _o

0

30

_ 2o

0

3O

LO

0

3O

_ _°

i0

0

30

I0

0

30

_ '°

10

0

_R1L2 ,- ,

1o' IL

6_

4

il_j.e

10''

4

2

0

-2

-4

4.

2

0

-2

-4

4

2

0

-2

-4

0

5 p

2'

20 40 60 80 100 i20 20 40 60 80 |00

2'

120

Fig. 10.6.-- The 5I/I measure of a typical mosaic (161+51). The solid line is ab(¢)/l, the shaded

region is the uncertainty. The value of ab(¢)/I is shown for mosaics where the maximum radius

for each pointing has been chosen to produce maximal PSFs of 10', 5', and 2'. Left: _rb(¢)/I for

the images. Right: ab(¢)/I for the high-pass filtered images. Note the changes of scales.

" o.fl

i 10 ---_

"I °:1

0 20 40 60 BO 100 120

_,(arc,,"- )

Fig. 10.5.-- Comparison of the ve and ab measures. Top: aB. Bottom: erb.

ative offsets of any pair of pointings. Since a

large fraction of the uncertainty in the offset of

a given pointing is due to the uncertainty in the

RASS flux, such an error will be a gross overestimate of the true uncertainty of the relative

offsets.

We can not determine the _rb(_b) contributed

by the errors in the offsets, but we can determine

the behavior ab would have as a function of ¢ if

the error in the offsets were the sole source of

excess fluctuations.

If each pointing in a mosaic has an offset

Oi and an offset uncertainty gi, we can create

simulations of offset errors. For each pointing

we draw a normally distributed random number

with a mean of zero and a variance of unity, and

multiply that random number by gi to get the

error in the offset, AOi. We create an mosaic

of the offset errors; the distribution of counts in

the mosaic due to the erroneous offsets are modeled in the same way as the original offsets. The

value of o'b(¢) can then be calculated for each

simulated mosaic.

Figure 10.7 shows the results for a few trials for a typical mosaic. The quantity aB(¢) is

159
2

0 ::::::::::::::::::::::::::::::::::::::::::::::::

5 O. b

4

3

g

i

1

o

I0 20 30 40

¢(aremtn)

Fig. 10.7.-- The effect of offset errors on top:

aB(¢) and bottom: (Tb(_). Plotted are the values of aB(¢) and ab(¢) for a few simulations of

the offset errors in a typical mosaic (086+74).

nearly linear, and the slope is proportional to

the overall amplitude of the excess fluctuations.

Thus. _b(_) is nearly proportional to 1/_b. This

is not the behavior of the excess fluctuations in

the real mosaics, so the contribution of the offset

errors to the excess fluctuations must be small.

3.5.2. Interpretive Difficulties

As with the ACF, there is a difficulty due to

"scale-mixing". As was noted in the discussion

of _Bvs.ab, the measure of abe(@) is not so much

a measure of the excess fluctuations at @, but

is a measure of excess fluctuations at _b _<

(for which _'b(@) "_ sigmab(¢)) and ¢ > 0i (for

which ab(4_) 0¢ (e_/cb)ab(¢_)); it is the shape of

the function that tells one at which scales fluctuations exist. As with the ACF, one might expect

to understand better the excess fluctuations at

¢ << ¢ by high-pass filtering the images with a

threshold at ¢. Figure 10.6 contains results for

a typical mosaic.

High-pass filtering an image removes all fluctuations at scales greater than the threshold,

and decreases the amplitude of the fluctuations

at all smaller scales. The variance measured at

some scale ¢ is/the linear sum of the intrinsic

variances at all scales. The variance at some

scale of a high-pass-filtered image is the linear

sum of the intrinsic variances, each multiplied by

a "suppression factor", [1 - (¢F/¢S) 2] (see Appendix I), where CF is the scale of the fluctuation (not the scale upon which one is measuring) and Cs is the smoothing scale. Since the

smoothing scale is 121', the change in _ (and

thus ab2) from the image to the high-pass-filtered

image (as expressed as a multiplier) should be

0.99993 for fluctuations on the scale of 1' (i.e.,

not much), and 0.75 for fluctuations on the scale

of 60'. When calculating the excess fluctuation

at scale _b, one must reduce the variance expected from counting statistics and unremoved

point sources, but in this case the scale of the

fluctuations is the pixel scale. We have also multiplied the calculated ab2(¢) by the inverse of the

suppression factor before plotting.

The fact that the a_(¢) relation in the highpass-filtered images seems to level off (where it

did not in the unfiltered images) would suggest

we have isolated the scales of fluctuation. A brief

perusal of the figures in Appendix H shows a

variety of behaviors for the mosaics in a given

band; some level off before reaching the largest

measured scale, and some continue to rise.

3.5.3. Averaging

Due to the variety of behaviors, it is difficult

to coadd the (a6(¢)/I) _ relations in a reasonable manner. We have averaged the relations

over those scales which are covered by all mosaics (1_-60_). For band R1L2, the largest-scale

fluctuations appear to have ¢ ,-_ 35_-45 _, while

for band R45, the largest-scale fluctuations appear to have ¢ -,, 2U-35', though there is a great

deal of variation from mosaic to mosaic.

160
6

_- 4

_2

0

6

_-- 4

¢,

_2

0

R1L2

t-e

0 I0 20 30 40 50 60

_(arcmin)

0

0 I0 20 30 40 50 60

¢(arcmin)

Fig. 10.8.-- Mean (o'b(¢)/l) 2 for high-passfiltered mosaics. The shaded region is the uncertainty.

3.5.4. (_I/I Summary

The cr_(¢) relations in band RIL2 (for unfiltered images) shows that most mosaics have fluctuations on scales up to the size of the mosaics.

Although the same is true for some mosaics in

band R45, several mosaics (086+74, 220+84,

and 295-89) appear to show minimal fluctuations on scales greater than _ 30/. The large

differences in behavior between mosaics for this

measure of fluctuations suggests that the similarity seen in the ACFs may be misleading.

4. Results

4.1. Some Considerations

4.1.1. The LHB

It is important to know which component of

the SXRB is producing the fluctuations, whether

it be the LHB, the Galactic halo, or some extragalactic component. In principle, one can measure the fluctuations due to the LHB; observe

a region with a substantial path-length through

the LHB (almost any direction) for which there

is a sufficient column density of material just

outside the LHB to remove the contribution due

to other components (not so many directions).

This is technically possible in the R1L2 band as

e-_ = 0.1 at 3.4 x 1020 cm -2, but is difficult

(impossible?) in the R45 band where e-" = 0.1

at 5.0 x 1021 cm -2.

We have performed an ACF on the R1L2

band image of a mosaic at (e, b) = (45 °, 0°) with

(Sf, (ib) = (106 _, 960'). In this direction the LHB

wall is _ 107 pc away (assuming that all of the

R1L2 flux in the Galactic plane is due to the

LHB, and assuming n_ = 0.005 cm-3). From

the absorption line study of Sfeir et al. (1999),

the Na I equivalent width (W_(D2)) due to absorption by the ISM is very small for stars with

distances less than 115 pc. At greater distances

the equivalent width rises to 80 m_t; an equivalent width of 50 m,_ corresponds to NH = l02°,

or e -_ _ .3. Thus, at distances slightly greater

than that expected for the LHB wall, the ISM

becomes optically thick to band R1L2 photons.

The ACF and the high-pass-filtered band

R1L2 image is shown in Figure 10.9. There is

very little structure on scales larger than 10'.

There is also very little structure for 31 < 0 <; 10 I

as there will not be many point sources within

the LHB. (This will not be true in all directions, consider, for example, the direction of the

Hyades.) Thus, the LHB is unlikely to contribute to fluctuations on scales 101 < 0 < 120'.

4.1.2. The Halo

The contributions due to the halo can only

be surmised from what little we know about the

Galactic halo. It is still not known whether the

X-ray emission in the Galactic halo is due to

gas in equilibrium (i.e., virialized in the Galactic

potential) or whether it is some non-equilibrium

quasi-steady state, such as a cooling flow. Chapter 6 suggests that both may be true, as there

is more than one component to the hot Galactic halo. A number of mechanisms have been

suggested for heating halo gas, either in situ

through supernovae or magnetic reconnection,

or heating the gas in the disk and ejecting it

into the halo through galactic chimneys.

Simulations of supernovae in the Galactic

halo by She[ton (1999) show that a SNR at

161
0.04

0.02

0.0 0.5 1.0 1.5 2.0 2.5 47 45 43

Log,° O(arcminutes)

Fig. 10.9.-- A band R1L2 ACF for the LHB. Left." The ACF for a band R1L2 image towards

(£, b) -- (45 °, 0°). The shaded region is the uncertainty. Right." The high-pass filtered image of the

mosaic.

z = 1 kpc grows to diameter of--_ 6° . A SNR

at z = 0.3 kpc, while not expanding to the

same physical dimension due to the higher ambient density, will have a diameter of ,_ 10°. A

substantial portion (,-- 400 × 10-6 counts s -1

arcmin -'_) of the R12 band flux from the Galactic halo could be accounted for by overlapping

halo SNR, but the flux in band R45 is expected

to be almost two orders of magnitude smaller.

Heating by magnetic reconnection, the microflare model of Raymond (1992) for exampie, is perennially mentioned in this context,

likely exists but little work has been done to extend these models to the point where appropriate scales can be determined, perhaps because

the physical conditions of the halo are so poorly

known.

It has been suggested on the basis of observational evidence that the effluvia from Galactic

chimneys is patchy on the scale of several degrees, but at smaller scales, one must resort to

physical reasoning. A 10j radius fluctuation at

z = 2 kpc in the halo has a radius of _ 6 pc.

Consider first an isothermal fluctuation. Assuming a gas temperature of 106K, the sound crossing time will be --_3.5 × 104 years, implying that

fluctuations in the emission due to fluctuations

in the density of the emitting plasma, will be

short lived.

Consider also an isobaric fluctuation. The

emissivity per volume, A _ T_p 2, under the

condition of pressure equilibrium, will be proportional to T _-2. For the Gehrels & Williams

(1993) cooling curve, a ,_ 0.2 for 5.89 <: logT <

6.14 and a _ 0.3 for 6.65 < logT < 6.88;

between those regions a < 0. Thus, for the

range of X-ray emitting plasma temperatures

observed (6.06 < logT < 6.46), one must depress the temperature to increase the emissivity (so bright spots will be cooler than the surrounding medium). Such fluctuations, however,

are removed by the high conductivity of the surrounding plasma (Field 1965).

Thus, while the contribution of the halo to

fluctuations smaller than a few degrees is not

known per se, and could profit from further

work, it seems likely that the halo is not contributing fluctuations on the < 30' scale. Thus,

once the contamination due to residual point

sources is removed, the remaining fluctuations

should be due solely to the diffuse extragalactic

emission.

162
4.2. Comparison with Previous Results

Compared to the extensive literature on ACFs

at E > 2 keV, the literature below 2 keV is

quite restrained. The difficulties due to Galactic absorption have been well appreciated and

that, together with a host of other issues (the

smaller number of counts due to the extragalactic background, the increasing contribution by

the Galactic halo and the LHB, lack of information about the point source spectra below 0.5

keV, ghost images in band R1L2), have impeded

efforts. Studies have typically concentrated either on small scales, where the ACF can be derived from a large number of individual deep exposures, or on very large scales, where the RASS

can be used at energies not effected by Galactic absorption or contamination (E > 1 keV).

A summary of pertinent ACFs is given in Table 10.4. The remainder of this section will be

concerned with the band R45 ACF as there is

no comparison to be made for the band R1L2

ACF.

Soltan et al. (1996) calculated the ACF for

0 < 40 ° from the R5 and R6 band RASS. They

showed that although the band R6 ACF decreased rapidly from 6 = 0° to 0 = 5°and was

constant and smaller than the uncertainty at

larger scales, the band 1t5 ACF declined more

slowly, to 0.002 at I0 °, and was well described by

the linear function W(0) = 0.003 - 0.000140 at

larger scales. The band R5 ACF was clearly contaminated by Galactic emission at many scales.

At much smaller scales Soltan & Hasinger

(1994), concentrating on the bands least affected

by Galactic contamination, measured the ACF

in band R67 from 46 deep (> 8 ks) PSPC pointings. They detected a signal at 0 < 3:, but at

larger 6) the signal was much smaller than their

small uncertainty. The detected signal was attributed to clusters of galaxies (further discussed

in Soltan et al. 1996), and the lack of signal at

0 > 3: was used to place strong constraints on

the degree to which X-ray sources could be clustered in the manner of QSOs.

Chen et al. (1994) measured the ACF in the

R47 band using a single deep (_ 70 ks) PSPC

pointing and a mosaic of six not-so-deep (13_--20

ks) pointings. Their ACF is noise limited on

all scales (2:-300'), and their tightest constraints

were reached by fitting a(0/0o) -'y to the 2o" upper limits of the data. They again found a very

0.000

-0.005

2O 40 60 8O I00

8(arcminutes)

Fig. 10.10.-- Comparison of ACFs. The histogrammed data with lightly shaded error bars

is the ACF for band R45 after the removal of

the maximum contribution from the PSF, scaled

as if the fluctuations were due solely to the extragalactic component of the SXRB. (The PSF

makes no visible contribution for 0 > 20'.} The

dark boxes are the ACF calculated by Soltan &

Hasinger (1994} for band R67. The lower dashed

line is the fitted ACF in band R47 from Chen

et al. (1994). The upper dashed line is the same

ACF for band R45 if all of the fluctuations in

band R47 were due to fluctuations in band R45.

The thin solid line is the ACF in band R45 using

the model of Vikhlinin & Forman (1995).

small value of W(O) with which to constrain

clustering.

Direct comparison of our data with these previous measures (Figure 10.10) is initially disheartening. However, it should be realized,

first, that the Chen et al. (1994) and Soltan

& Hasinger (1994) ACFs have been very carefully cleaned of spurious signals by extensive

PSF modeling. Second, by restricting ourselves

to the R45 band, we have about half the counts

available in the R47 band. For this we have compensated by increasing the area over which the

ACF is measured. Third, by measuring the ACF

in band R47, Chen et al. (1994) diluted the

effects of Galactic contamination that we must

possess in full measure.

In Figure 10.10 we show our band R45 ACF

after the removal of the maximal PSF contribution under the assumptions that the fluctua163
tionsarisesolelyintheextragalacticbackground

i (i.e.,theGalactichaloisassumedtobesmooth

overthescalesof interest).Theupperdashed

lineistheACFthat wouldbeexpectedif allof

thefluctuationsmeasuredby(Chenet al. 1994)

in thebandR47weredueto fluctuationsin the

bandR45;giventheverysmallmeasureof the

fluctuationsinbandR67measuredbySoltan&

Hasinger(1994)(shownasthe dark squares),

suchan assumptionseemsreasonable.If our

removalof themaximalPSFiscorrect,thenour

measuredACFisconsistent withthefittedvalue

of theChenet ai. (1994)ACF,butit shouldhe

notedthattheChenet al. (1994)fittedACFis

afactorof _ 7smallerthantheactual 2_r upper

limits on the data (see their Figure 5b).

The thin solid line is an extrapolation of the

the results of Vikhlinin & Forman (1995) to

the band R45 ACF. Vikhlinin & Forman (1995)

measured the ACF for point sources brighter

than some limit. Since both their PSDL and

ours was spatiallyvariable,the comparison is

difficult;we have taken our PSDL to be the

mean PSDL for our mosaics, weighted by average exposure, and have taken for theirsthe

PSDL for which they have "significantcoverage". We then assumed that theirACF characterizedallthe point sourcesthat they measured

but which were not removed in our mosaics,and

that there was no other sourceof fluctuations.

5. Summary

We have measured fluctuationsforseven mosaicsin bands RIL2 and R45 by two methods;

the ACF and the 51/1 measure. Due to the PSF,

the ACF analysismostly is limitedto 0 > 10_,

and I0_< 0 < 25' may stillbe contaminated by

the PSF. Direct measurements of fluctuations

arisingin the LHB, and physical state of the

Galactic halo argue strongly that the observed

fluctuationsare extragalactic.

For the mean ACF, there are clearlyfluctuations in band RIL2 for 5_ < 0 < 30_,where

W(0) ,--0.1,while the signalisconsistentwith

zero for # > 30'. In band R45 there may be

fluctuationswith scalesI0' < # < 25_,where

W(0) _ 0.0025,and thosefluctuationsare consistentwith previousupper limitsdetermined in

the R47 band. The 51/] measure of fluctuations

isconsistentwith that produced by the ACF.

The 51/I measure, which is in some ways a

more direct measure of fluctuation, suggests that

the mosaics have very different scales and amplitudes of fluctuations (even after fluctuations

with scales > 12V are removed), leading one to

question the validity of studying the mean ACF.

We have shown that the band RIL2 fluctuations due to the LHB are small, and thus the

fluctuations contributed by the LHB to the R45

band must be ever smaller. However, it should

be noted that due to our choice of energy bands,

our data are unabashedly sensitive to fluctuations in the Galactic halo. We do not understand

what sources for those fluctuations would be,

and physicalconsiderationssuggest that smallscalehalo fluctuationswillbe quicklydissipated.

Thus, with suitableimprovements (suchas modeling the ACF produced by unremoved point

sources),and a more carefulcalculationof the

halo fluctuations,we willbe able to place more

stringentlimitson the extragalacticfluctuations.

REFERENCES

Chen, L.-W., Fabian, A. C., Warwick, R. S.,

Branduardi-Raymont, G., & Barber, C. R.

1994. MNRAS, 266, 846

Field, G. B 1965, ApJ, 142, 531

Gehrels, N. & Williams, E. D. 1993, ApJ, 418, L25

Ha.singer, C., Burg, R., Giacconi, R., Schmidt, M.,

Triimper, J., & Zamorani, G. 1998, A&A,

329.482

Mushotzky. R. F., Cowie, L. L., Barger, A. J., &

Arnaud, K. A. 2000, Nature, in press

Peebles, P. J. E. 1993, Principles of Physical Cosmology (Princeton University Press)

Raymond. I. C. 1992, ApJ, 384, 502

Sfeir, D., I._llement, R., Crifo, F., & Welsh, B. Y.

1990 A&A, submitted

Shelton, R L. 1999, ApJ, 521, 217

Soltan, A.. ,_ Hasinger, G. 1994, A&A, 288, 77

Soltan, A. M., Hasinger, G., Egger, R., Snowden, S.,

& Trdmper, J. 1996, A&A, 305, 17

Vikhlinin, \., & Forman, W. 1995, ApJ, 455, L109

164
TABLE 10.4

COMPARISON OF ACFs

Source _ Energy PSDL b N c F 0 W(8)

keV

Cea94 0.4-2.4(R47) 0.5 x 10 -14 10,9 1.626 2'-30' 3.1 x 10-3(8/1') -°'sd

0.4-2.4(R47) 1.0 x 10 -14 11.9 1.626 2'-300' 3.8 x 10-3(8/1') -°sd

SH94 0.9-2.4(Ft67) 1.04 x I0 -14e 13.4 1.12 IC3' (6.8 4- 2.5) x 10-3

3'-11' (0.3-'}-0.9) x 10 -2

VF95 0.6-2.0 4 x 10 -Isf 0:5-I0 _ 0.257(0/I0') -°sK

Sea96 0.56-1.21(R5) 20-3 ° 3.8 x 10 -3

I0°-14 ° 1.6 x 10 -3

0.73-1.56(R6) 20-3 ° 1.4 x 10 -3

10°-14 ° 5 x 10 -s

K2K 0.4-0.9(Ft45) 5.80 x 10 -14h 10.5 1.42 10'-15' 0.0031 _- 0,0008 i

15'-20' 0.0028 4- 0.0007

20'-25' 0.0015 4- 0.0007

25'-100' -0.0006 4- 0.00014

"Cea94: Chen et al. (1994), SH94: Soltan & Hasinger (1994), VF95: Vikhlinin & Forman (1995),

Sea96: Soltan et al. (1996), K2K: this work.

bThe units are ergs cm -2 s -1 in the 0.5-2.0 keV band.

CThe units are keV cm -2 s -1 sr -] keV -l. Care must be taken as the different models of the EPL take

into account varying contributions from resolved point sources.

dThese relations are not those given by Chen et al. (1994). Their text states that W(8) = aO'_ °'s (where

we presume the 01, implies 0/1') and that for the 2'-30' region a = (-4.86 4- 3.98) x 10 -_, aJnd that for the

2'-300' region a = (+0.50 4- 1.68) x 10 -3. The values given here were derived from their Figure 5b. These

are the 95% upper limits of the fits of the functional form shown to the data. The data themselves have 2a

upper limits that are a factor of ,,, 7 higher than the 95% upper limits of the fits.

eAs with the current study, the point source limit is variable. The quoted value is the mean PSDL at 5'

off-axis; the study limited itself to the central 15' of each pointing.

fThe value quoted is for the PSDL "for a significant region". The maximum PSDL in the best and worst

pointings are 5 x 10 -_4 and 1.2 x 10 -ts counts s -z .

gThe scaling factor was measured from their Figure 2 for the ACF corrected for amplification bias.

hThe PSDL is variable, the value quoted is the mean over the mosaics, weighted by the mean exposure of

the mosaics.

iThe values are the mean for the given intervals; the quoted uncertainties are the uncertainty of the mean.

The r.m.s, of the 25'-100' interval is 0.002.

165

Chapter 11" Promiscuous Observations

And a mouse is miracle enough to stagger sextillions of infidels.

-Walt Whitman, Leaves of Grass

Abstract

This chapter summarizes the results of the previous chapters in terms of an interest

in the "missing baryons", and the warm-hot IGM (WHIM) that has been predicted

to contain them. We review the current knowledge about the WHIM. The fluctuation

analyses of Chapter 10 will be applicable after a theoretical prediction of the amplitude

and scales of the ACF is made.

The spectral energy distribution analysis of Chapter 6 and Chapter 7 place constraints on the total contribution of the WHIM to the soft X-ray background. We

determine the upper limit of the diffuse extragalactic emission to be < 7.4 :t: 1.0 keV

cm -2 s -1 sr -1 keV -1 at energies less than -_ 0.6 keV, and some portion of this value

is due to the Galactic halo. We also show that the normalization of the extragalactic

power law (F = 1.46) is 9.5-/- 0.9 keV cm -_ s -1 sr -1 keV -1.

1. Cosmological Considerations

1.1. The Status of the Problem

When Chapter 1 was written, a number of

different groups had just finished their hydrocosmological simulations, and several of those

simulations were available only as pictures on

the web. It was clear that all of the simulations

put the missing baryons into what has since

come to be called the WHIM, the warm-hot intergalactic medium, and it was clear that those

baryons were not related to large structures, but

the dependencies upon the modelling codes and

cosmologies were not clear. There was a clear

need for the various groups to compare results

to elucidate these dependencies. This synthesis

has recently appeared (Davd et al. 2000), and

the results, which are nearly independent of the

modelling code, can be summarized as follows.

The missing baryons are to be found in the

WHIM, at temperatures from l0 s to 107 K. The

WHIM is predominately in filamentary structures, though a small percentage is to be found

in the coronae of galaxies and virialized structures. The WHIM is found in regions with low

over-densities, (p - _)/_, peaking at 10 to 30.

(The over-density of the local group is _ 1500}.

The WHIM is not virialized. The heating is due

to shocks as gas accretes onto larger structures,

and the gas does not cool due to the low density.

Heating by supernovae appears to be of only minor importance for the bulk of the WHIM. From

an observational perspective, it is interesting to

note that the temperature distribution of the

WHIM is broadly peaked and strongly evolving,

around logT ,_ 6.45 for z = 0, and logT~ 6.15

for z = 1. The WHIM with the highest densities

is also the gas with the highest temperatures,

and represents only a tiny fraction of the whole.

Dav_ et al. (2000) suggest two possible "reality checks". The first is the total amount of flux

provided by the WHIM. Davd et al. (2000) attempt to compare a suite of models to the limits

at 1⁄4 keV (Warwick & Roberts 1998) and 43-keV

(Wu, Fabian, & Nulsen 1999), and find them

roughly consistent, but more careful calculation

of the flux from the simulations should be done.

The second reality check is the fluctuations due

to the WHIM. These calculations have not yet

been done: Davd et al. (2000) do produce a _(r),

a two-point correlation function, for the simulation, but it is difficult to transform that quantity

into w(O), an autocorrelation function.

167
1.2. WHIMsical Detections

1.2.1. Current Claims

Leaving aside detections of extended X-ray

emitting gas between close, rich clusters (e.g.

Kull & BShringer 1999) there are two published

studies that approach the question of X-ray

emission from the low-density WHIM filaments.

(Briel &: Henry 1995) attempted to detect filaments between pairs of clusters. They found

a 2_ upper limitof 3.9 x 10-16 erg 8 -1 cm -_

arcrnin-2 in band R6 (given the assumptions

z = 0.14, logT = 7.06 and metallicity0.3 solar)and a 2_ upper limitof 3.1 x 10-16 erg s-I

cm -_ arcmin-_ in band similarto R45 (given

the assumptions z = 0.14, logT = 6.76 and

metallicity0.3 solar). This is clearlyan ideal

way to look forfilaments,so long as the clusters

are sufficientlyseparated that one is not seeing

overlapping clusterhalos where one might expect gas with a similartemperature, though of

differentorigin.

Scharf et al. (2000) make a cleardetection

of a "filamentary"objectin band R47, at a flux

levelof 1.6 x 10-16 ergs s-I cm -_ arcmin -_,

but it is not clearwhether the object is a filament of the type discussedby Davd et al.(2000).

This X-ray filamentisalignedwith a filamentary

region of enhanced galacticdensity,and there

are three candidate clustersalong the filament.

This isthe best candidate to date,but itwould

be interestingto compare the mass overdensity

within thisfilamentto those proposed by Davd

et al.(2000) to see what fractionof the WHIM

is contained in structuressuch as these;isthis

a typicalfilament,or is thisa higher density

object? A note in passing: ifthe Scharf et al.

(2000) detectionis a realfilament,the factone

filament was found in 22 3ff x 30_ fieldssuggests,with the usual caveatabout small-number

statistics,that such filamentsshould be commonly detectable.

1.2.2. Future Claims

Three types of proposals have been put forth

for the direct detection of these filaments. M.

Pierre and collaborators have proposed (Pierre,

Bryan, &: Gastaud 2000) direct imaging with

XMM-Newton In opposition to the work of

Dav_ et al. (2000), they make much of the importance of the correct modeling of supernovae.

Careful perusalof theirproposal shows thatone

can detectonly the _ 10% of the WHIM that is

peripheralto virializedclumps (and has very differentpropertiesthan the restof the WHIM) by

directimaging inthe X-ray,not the 90% that actuallyresidesinfilaments.Detecting the WHIM

by thismethod islikedetectinga dog on the basisof a few cat hairs.

The second and third types of proposal are

spectroscopic. The Constellation-X proposal

suggeststhat the bulk of the WHIM willbe detectablewith that instrument afteronly 200 ks

given a sufficientlybrightbackground QSO. Alternatively,a similarlylong "blank-sky" spectroscopicobservation willdetect the WHIM in

emission to at leastz - 0.04. Both spectroscopicobservationsrelyon the strength of the

O VII (E = 0.654keV) lineforgas in the WHIM

range. Given that the new generationof detectorshave resolutionon the orderof a few eV at I

keV, the IGM lineswillbe easilydistinguishable

from thoseof the Galactichalo for z > 0.01.

2. The Current Work

There are three predictions about the WHIM

that hydrodynamical simulations of the universe

ought to make in order to allow a comparison to

the universe itself; the total flux, the spectrum,

and the angular distribution. The predictions

have not yet been published for most hydrodynamical simulation. To what extent does this

work place limits on these values?

2.1. Total Flux

The upper limit on the WHIM emission is

the total trans-absorptive emission (TAE), that

emission that is from beyond the Galactic absorption, but not part of the extragalactic power

law (EPL). What is the magnitude of the TAE?

To answer this question we must revisit the analysis of the RASS data in Chapter 5 and Chapter 6. The primary reason for embarking on a

study of the TAE was the need to characterize the spectral shape of the TAE for deabsorption calculations. The primary scientific focus

of those chapters was the thermal structure of

the Galactic halo. Given this focus, it was most

reasonable to characterize the uncertainties in

the temperatures as the dispersion of the val168
7

7

¢D

?

-1

X -2

,, -3

0

2 Total

0 "

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

Log,, E (eV)

3

2

k.

; !

,?

0

¢J

--1

o

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

Log,, E (eV)

Fig. 11.1.-- Graphical decomposition of the

soft X-ray background. The RASS broad-band

measurements (boxes) and the model of the total SXRB spectrum at 50 eV resolution (solid).

Dashed: the contribution of unresolved Galactic stars. See text for further description. Note

that the spectral resolution of the model is 50

eV, much smaller than the ROSAT resolution.

Fig. 11.2.-- Graphical decomposition of the soft

X-ray background. The RASS broad-band measurements (boxes) and the model spectrum at

50 eV resolution (solid) after the removal of the

unresolved Galactic stars. Dashed: the contribution of unresolved extragalactic sources after

absorption by the Galactic disk. See text for

further description.

ues of the temperatures measured in individual

173 x 1°3 image pixels, and to pay little heed

to the absolute magnitude of the emission. In

any case, the strength of the soft component

varies greatly with position, though the strength

of the hard component is reasonably constant.

If, however, one is interested in the strength of

the TAE in order to place an upper limit on

the strength of the emission due to diffuse extragalactic emission, then a somewhat different

analysis is needed.

For simplicity, we have chosen to analyse the

region used for the %udget" calculations in Table 6.1 (450 < _ < 270°,b > 85°). The steps

to deconstruct the total soft X-ray background

into its components are depicted graphically in

four figures beginning with Figure 11.1. The

first panel shows the SED of the total diffuse

background (squares) after correction for bandto-band difference in point source detection limit._. The small vertical lines within the symbols

are the propagated uncertainties. The solid line

is our model of the total diffuse background at a

resolution of 50 eV, far smaller than the PSPC

re_olution. The dotted line is the spectrum due

to the unresolved Galactic stars, as calculated

by an improved version of the model in Chapter 7. As the stars are distributed throughout

the absorbing medium, the displayed spectrum

is the stellar spectrum appropriately convolved

with the absorption.

The second panel (Figure 11.2) shows the

SED after the contribution of the unresolved

Galactic stars has been removed. The dashed

line is the absorbed spectrum of the extragalactic power law (EPL) due to unresolved individual extragalactic point sources. Here we have

used the formulation of (Chen, Fabian, &: Gendreau 1997) (F = 1.46, Nr = 10.5 keV cm -_ s -]

st-1 keV-1), however, unlike our previous treatment, we have renormalized the EPL so that the

sum of all of the known components produce observed band R7 flux 1 Without this renormalization, the EPL produces too much flux in band

R7.

1Since the bulk of the emission in the R7 band is due to

the EPL, this method essentially sets the band R7 flux

of the EPL to the observed flux (,,, 40 x 10 -s counts s -I

arcmin -2) minus 0.9 x 10 -6 counts s-t arcmin-2 (for

the unresolved Galactic stars) and 0.6 x 10 -6 counts s -1

arcmin -2 for the TAE. This process is dearly iterative.

However, the subtraction of the TAE contribution has

little effect on the normalization of the EPL.

169
0

• -3 \

o

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

Log,, E (eV)

Fig. 11.3.-- Graphical decomposition of the soft

X-ray background. The RASS broad-band measurements (boxes) and the model spectrum at

50 eV resolution (solid) after the removal of the

unresolved extragalactic sources. Dashed: the

contribution of the Local Hot Bubble. See text

for further description.

The thirdpanel (Figure 11.3)shows the background afterthe subtraction of the EPL. The

dotted linesare the contributiondue to the Local Hot Bubble (LHB); the middle lineis the

best fitand the othertwo arethe plusand minus

one sigma. The uncertaintyin the LHB value

has be added to the measurement uncertainties

in quadrature. The model of the background

was calculatedfor each of these values of the

LHB, each of which isplotted.

The fourth figure(Figure 11.4)is the background aftersubtractionof the LHB and after

correctionfor Galactic absorption. Following

the analysisof Chapter 6, we assumed a two

component TAE; foreach valueof the LHB we

have calculated:1:1¢extremes. We have plotted

the extremes for the totalTAE (solid),the soft

component (dotted),and the hard component

(dashed). Both the temperatures and normalizationsof the components were fit.The temperaturesare very similarto those found forthe

entireNorth Galactic hemisphere in Chapter 6;

logTs = 6.08+0.03 and logTH = 6.43+0.01.

So how much of this need be included in the

upper limit to the diffuse extragalactic emission?

First, note that the band R12 emission is patchy,

and thus likely to be related to the Galactic halo.

T .

xe TAE

-31 ".I".

_j ...... ...

100

o 50 .. _ .

; ,'

r.t,

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

Log,, E (eV)

Fig. 11.4.-- Graphical decomposition of the soft

X-ray background. The RASS broad-band measurements (boxes) and the model spectrum at

50 eV resolution (solid) after the removal of the

LHB contribution and after correction for Galactic absorption. Dotted: the contribution by the

soft TAE component. Dashed: the contribution

by the hard TAE component.A Heavy Bar: the

maximum uniform 1⁄4 keV flux allowed by the

measurements of Snowden et al. (2000). See

text for further description.

170
IstheresomepartofthebandR12emissionthat

is constantoverthesky? Therehavebeena

numberofattemptsto measurethisvalueusing

theneutralhydrogencolumnsofexternalgalaxiesto shadowtheextragalacticbandR12background.Thevaluefoundby (Barber,Roberts,

& Warwick1996),usingNGC55asashadowing

target,was29.44-7.2keVcm-_ s-1 sr-1 keV-1

(228=[=90 x 10-6countss-1 arcmin-2).TheEPL

usedin themodelproduces231x 10-6 counts

s-1arcmin-_inbandR12,sothereislittleroom

for additioncomponents.

Asecondmeasureof thebackgroundinband

R12is providedby Snowdenet al. (2000).

Measurements of the"distantemission"beyond

manyhighGalacticlatitudeabsorptionfeatures

areconsistentwith zero. Thetwosigmaupperlimits in thesedirectionsaretypically,-_

80 x 10-6 counts s-I arcmin-2. This value is

shown in Figure 11.4 as a heavy bar. It is interestingthat itisconsistentwith the band RI2

emission of the hard TAE component, suggesting that the hard TAE component, more or less

constant over the high latitudesky,providesthe

minimal band R12 emission.

The hard TAE component can only be a loose

upper limitupon the diffuseextragalacticemission;from the arguments in Chapter 6,no doubt

thereissome contributionby the Galactichalo.

The flux due to the hard TAE component is

,--7.4 4- 1.0 keV cm -2 s-I sr-I keV -z in the

_34 keV band, and has a similarvalueinthe 1⁄4 keV

band.

2,1.1. The EPL

An important point was elided above, the

(Chen, Fabian, & Gendreau 1997) EPL formulation produces too much band R7 emission. The

mean band R7 emission from the north Galactic polar cap (450 < g < 270 ° ,b > 850 ) afler

correcting the RASS to the (Chen, Fabian, &

Gendreau 1997) point source detection limit is

40.134- 0.91 × 10 -6 counts s -1 arcmin -2. The

uncertainty cited is the uncertainty in the mean

due to counting statistics. However, there is a]so

a systematic uncertainty due to the subtraction

of the non-cosmic backgrounds. Since the noncosmic backgrounds were calculated for each individual scan, the uncertainty due to the noi_-

cosmic backgrounds appears as flux zero-point

shifts between adjacent scans; a high-contrast

stretch of otherwise smooth regions will reveal

these features. To determine the systematic uncertainty in band R7, we measured the difference

between two of the most obvious scan features;

--- 7 x 10 -6 counts s -1 arcmin -2 peak-to-peak.

The mean band R7 emission from the north

Galactic polar cap is thus 40.13 4- 3.62 x 10 -6

counts s-1 arcmin -2. The (Chen, Fabian, &:

Gendreau 1997} EPL (normalization 10.5 keV

cm -2 s-1 sr -1 keV -1) produces 43.57 x 10-6

counts s-1 arcmin -2 after absorption. Therefore, the RASS band R7 is produced by the EPL

with minor contributions by the other components (-,- 1.5 x 10 -6 counts s -1 arcmin-2), the

normalization must be 9.54- 0.87 keV cm -2 s-1

sr-I keV-l.

2.2. Spectrum

Davd et al. (2000) plots the fraction of

baryons in the WHIM as a function of the temperature. For the z = 0 WHIM, the distribution

is peaked at. logT _ 6.45, but not sharply so; the

peak fraction is 0.024, and the distribution declines to 0.02 at [ogT = 6 and 0.018 at logT "- 7.

The fraction of emission due to each temperature, the baryon fraction weighted by the emission will be much more sharply peaked. Thus,

it is of interest that the hard TAE component is

characterized by a temperature of logT ,-, 6.45.

2.3. Angular Distribution

(Soltan &: Hasinger 1994) measured the autocorrelation function in band R67 and showed

that the signal is consistent with zero for O > 5'.

This result is not surprising; in our formulation,

only _ 12W, of the band R67 emission is not due

to the EPL.

We have placed limitson the angular distribution of the diffuse extragalactic emission by measuring the autocorrelation function in band R45,

where a larger portion of the emission may be

due to the diffuse extragalactic component, and

where _he WHIM is thought to have its greatest

emission. Y_'e show that there are indeed fluctuations on the scale 10' < 0 < 20 j in band R45,

and the. signal in the ACF_is ,-, 0.0025. This is

2As is cus_,,mary the result is quoted for an ira171
adirectmeasure:thebestpreviousresult(Chen

et al. 1994) found a similar amplitude by taking the 2_ upper limit of a fit of a very noisy

autocorrelation function in band R47. (By measuring in band R47, they had twice the count

rate, but the half of that count rate due to band

R67 had already been shown to have a vanishingly small signal for 0 > 5'.)

Given that Galactic absorption is small in this

band, and that fluctuations in hot plasmas such

as the LHB and the Galactic halo are very short

lived (_. l0 s years), this signal should due to the

diffuse extragalactic emission, not to one of the

foregrounds. However, it is not clear how interesting this limit is; the hydrodynamieal models

have not yet been queried for this value.

Clearly, the hydro-cosmological simulations

exist. It is now necessary to measure the A CF of

the simulations to see what order of magnitude

we should expect. The process is, in principle,

straight forward. Given the density, temperature, and metallicity of every location within a

model cube, one can calculate the observed field

for a given instrument. This process should be

done both for the PSPC and the current generation of instruments. (Given the simulations of

Pierre, Bryan, g: Gastaud (2000), it seems unlikely that any small collection of XMM-Newton

pointings would detect a filament. What these

simulations would determine is just how often

one would be able to detect a filament, and what

properties it should have.)

3. Improvements

3.1. Building Upon The Current Data

Can these ACFs be improved? Until highresolution H !I maps become available, it would

be difficult to improve the deabsorption process.

Better point source detection algorithms exist

(see the discussion in Hasinger et al. 1998) than

the scaled-sliding-box method that we have used

on the individual pointings, though it is not clear

how much better these algorithms would be in

regions witil large PSFs. One could design a

age that contains emission both from the Galactic

halo/extragalactic diffuse component and the EPL. If the

ACF is corrected to an image containing only the Galactic halo/extragalactic diffuse component, assuming that

the EPL is smooth at these scales, the ACF signal would

be a factor of "-. 4.7 larger

sliding-box method to detect point sources in

the mosaic, but the extent of the improvement

is not clear.

One might also be able to do a better job of

removing clusters of galaxies. We used the Abell

catalogue because it was convenient. A number

of new compilations from ROSAT PSPC data

have been created, but are typically derived only

from individual deep exposures; our individual

exposures are typically not deep, merely overlapping other pointings. The signal due to clusters,

however, is expected at the scale of 3t, which is

below the size of the PSF.

The A CF could be improved by modeling the

contributions due to the unremoved point sources.

The amplitude of the ACFs of Soltan & Hasinger

(1994) were greatly reduced by modeling the

point sources. Such a process is significantly

more complicated for mosaics; multiple PSFs

must be modelled at each point. Given that the

ACF for a typical mosaic required about eight

hours on the machine with which this project

was started, calculating the ACF for 200 simulations per mosaic was not practicable. Newer

mar.hines have reduced the time to an hour; simulation may be practicable.

However, it should be noted that the variance is dominated by counting statistics, not

the point sources. Thus, some of the large scale

structure in the ACF may be removable by modeling of the point sources (which will also help remove problems due to the windowing function),

the uncertainties will not be greatly reduced.

One could place tighter limits on the systematic problems by repeating thzs analysis in band

R67 where the signal is known to be quite small.

3.2. Without Mosaics?

Mosaics have a great advantage over the RASS

having exposures that are up to a factor of 40

greater than the RASS, but still covering solid

angles of a few tens of square degrees. However,

mosaics are not without problems; one can introduce features on the scale of individual pointings, ~ 110'. Since the mosaics depend upon the

RASS for their normalization, the uncertainty

at these scales is determined by the systematic

problems in the RASS . However, these larger

scales are not of much interest for the problem

at hand. Nor can mosaics address t he probl,_m of

172
thefluctuationsatthesmallestscales,_ 5'down

totheresolution of the PSPC, due to large PSF

at the edge of the PSPC field. Restricting ourselves to the centers of the PSPC paintings reduces the available area substantially. For these

scales mosaics are not the ideal data; collections

of deep paintings in low column density regions

are much better. The data Collection of Soltan

Hasinger (199_J, perhaps augmented by further

data sets, should be reconstructed to examine the

A CF in band R_5. However, without mosaicking the data, one cannot fully remove the effects

of absorption.

3.3. Deconstructing the Background

Mushotzky has suggested that this dissertation has approached the problem of the WHIM

in yet another, somewhat different, manner.

When Chapter 6 was written, the general behavior of the spectrum of the extragalactic point

sources below 0.5 keV was not known. However, the faint sources in the deep field studied

by Mushotzky et al. {2000) do not, as a class,

show a spectral break below 0.5 keV (as, indeed,

our analysis using the known limits of the 1⁄4 keV

background showed that they could not). Chapter 6 parameterizes the flux that is not due to

the extragalactic point sources and is not due to

the LHB, that is, the combination of Galactic

halo emission and putatative WHIM.

Of course, that. parameterization is, without

doubt, wrong. We know that the LHB is not

in equilibrium, and it is not likely that the halo

is in equilibrium either. There are a number

of projects afoot to study the spectrum of the

LHB and the Galactic halo with XMM-Newton

using the lines of O VII and O VII1 and modeling them with non-equilibrium spectra. Thus,

despite the notorious non-success in modelling

the high-resolution DXS spectrua of the LHB,

there is some hope for progress in determining

just how wrong our parameterization is.

Mushotzky has suggested that one can test

the existence of a strong WHIM by comparing our model to the measure of the oxygen

lines made by (as yet unpublished) rocket flights

(see Deiker et al. [1997) for a description of

the flights). Since ,,quilibrium models produce

stronger oxygen lin,._ than the non-equilibrium

models, scaling our models to produce the ohserved oxygen line strengths will produce the

minimum possible broad-band flux due to the

halo, and thus a measure of the maximum possible contribution due to the WHIM. The addition of the non-equilibrium results will then

allow a stronger upper limit on the flux due to

the WHIM.

4. Some Unfinished Business

When this dissertation was begun, one of the

motivating factors was the excessive fluctuation

found in the 1⁄4 keV background by Barber, Warwick, & Snowden (1995). We suggested in the

opening that these "blobs" might be the fluctuations predicted by Cen & Ostriker (1999).

The ACF has clearly pointed to excess fluctuations on the scale of the Barber, Warwick, &

Snowden (1995) fluctuations (_- 20'), but are we

still observing the same phenomena? To some

extent, yes. It is difficult to compare our images directly with those of Barber, Warwick, &

Snowden (1995); although the data are nearly

the same, deabsorption methods have improved

greatly and the point sources have been handled

in a very different manner. However, comparison shows that we find generally the same pattern of positive and negative fluctuations, and

the $I/I analysis does suggest that the fluctuations have an intrinsic scale of --- 20'.

Have we made any progress in determining

the origin of these features? We have not identified the features with any particular phenomena, and we have not entirely eliminated as

their source systematic problems with deabsorption. Given the initial results of (Reynolds 1999)

(3.0 x 1019 cm -2 beam-to-beam variation in

NItlI), we expect --, 5% errors in the band R82

flux (-_ 0.5% errors in band R45). At 20' in

band R82{R45), high-pass filtered, we find that

(_rb/I) 2 ,,, 1.75(1.0), meaning that the variation

on 20' scales is ,,- 7%(5%). Thus, in band R82

we may still be victim to the deabsorption errors.

The standard caveat should also be given

that the deabsorption model may be inherently

wrong; the X-ray emission cou]d be interleaved

with the absorption. In this case. the deviations

from the mean I'o - NH relation say interesting

things about the degree to which the emission is

17:1
interleaved, rather than v_ce versa 3.

Barber, Warwick, _ Snowden (1995) suggest

that the band R82 variation is due to the Galactic halo. Simple physics suggests that fluctuations do not last long in the halo (Field 1965),

so if the fluctuations seen are due to the halo,

the halo is a much more active place than usually imagined. Then again, the "halo" has become a very active place, with supernovae explosions and galactic chimneys, and with multiple components from 104 K to several times

106 K. Perhaps the region with all this activity should be better described as the "diskhalo" interface, while the term "halo" is left for

the smoother, larger-scale-length, quasi-static,

higher-temperature component. In any case,

this active region, which must contain interface

layers, is a more likely place to find the necessary fluctuations, but the physics is much more

complicated.

5. ArrivederLa

There are still a few changes to be rung in

the analysis described in this chapter. A new

and more robust calculation of the stellar contribution is one of the projects being finished

as this is written, the area of the sky covered

by the budget calculations could be expanded,

and the TAE emission could be calculated a bit

more carefully. However, the bulk of the RASS

contribution to the problem of the diffuse extragalactic emission has probably been wrung out.

Similarly, there are still a few projects to be

pursued using individual PSPC pointings. Most

important is determining the systematics in the

band R45 ACF, both by modeling the effect of

point sources interacting with the window function and by executing the same analysis in band

R67. Similarly, a Soltan & Hasinger (1994) analImagine a simple model with two sheets of abs.rption

with optical depths _1 and _, the closer sheet having

subscript 2. If there is emission beyond both ..heels,

/t, _nd between the sheets, I_, but none in fron_ of the

sheets, the true observed emission will be

Ile-'rle-*'_ + 12e -_'_

whereas our simple model will produce l_e l't • -s'_.

Therefore, given the fits to the mean I_ - N H relation,

it must be that I_ .., lz + l_e" over a wide range of

, -T1 e-'_. So, either 11 is small, in which case • must

also be small or Is is small. In neither case is reahty too

different than our simple model.

ysis in band R45, though difficult, is certainly

possible, and would illuminate the smallest accessible scales.

Perhaps the most important project in the

near future is querying the simulations for observable parameters. We have no clew if our

ACF is a useful limit, or whether the expected

value is an order of magnitude smaller than our

limit. Such analysis is also important for the

current generation of instruments; Chandra and

XMM-Newton.

No doubt, the existance of the diffuse extragalactic emission will plague at least another

generation of graduate students before it is somewhat resolved; this dissertation is but the first

step.

REFERENCES

Barber, C., Warwick, R. S., & Snowden, S. L. 1995,

in R6ntgenstrahlung from the Universe, ed.

H. Zimmermaxm, J. Triimper, _ H. Yorke,

MPE Report 263, 319

Barber, C. R., Roberts, T. P., & Warwick, R. S.

1996, MNRAS, 282, 157

Briel, U. G., & Henry, J. P. 1995, A&A, 302, L9

Cen, R, 8z Ostriker, J. P. 1999, ApJ, 514, 1

Chen, L.-W., Fabian, A. C., & Gendreau, K. C.

1997, MNRAS, 285, 449

Chen, L.-W., Fabian, A. C., Warwick, R. S.,

Branduardi-Raymont, G., & Barber, C. R.

1994, MNRAS, 266, 846

Davd. R., et a]. 2000, ApJ, submitted

Deiker. S., Kelley, R. L., Lesser, A., McCammon,

D., Porter, F. S., Stalde, W. T. S. C. K., &

Szymkowiak, A. E. 1997, in Proceedings of

the Seventh International Workshop on Low

Temperature Detectors, 113

Field, G. B. 1965, ApJ, 142, 531

Hasinger, G., Burg, R., Giacconi, R., Schmidt, M.,

T_-iimper, J., & Zamorani, G. 1998, A&A,

329, 482

Kull, A., & BShringer, H. 1999, A&A, 341, 23

Mushotzky, R. F., Cowie, L. L., Barger, A. J., &

Arnaud, K. A. 2000, Nature, in press

Pierre. M., Bryan, G., & Gastaud, R. 2000, A&A,

submitted

Reynolds, R. J. 1999, private communication

Scharf. C., Donahue, M., Volt, G. M., Rosati, P., &

Postman, M. 2000, ApJ, 528, L73

Snowd_.n, S. L., Freyberg, M. 2, Kuntz, K. D., &

Sanders, W. T. 2000, ApJS, in press

Soltan. A., & Hasinger, G. 1994, A&A, 288, 77

Warwick, R. S., & Roberts, T. P. 1998, Astronomische Nachrichten, 319, 59

Wu, K K.S., Fabian, A. C., & Nulsen, P. E. J. 1999,

MNRAS, in press

174
Appendix A: Model Temperatures

Abstract

We find that the thermal history of likely halo plasmas has little effect on the

ROSA T broad-band ratios.

It is well known that high resolution spectra

of the LHB have shown that the gas is not in

equilibrium (Sanders et al. 1998; Deiker et al.

1997), but the true thermal state of the gas remains a mystery. The question has been raised

whether it is valid to characterize a plasma temperature by an equilibrium temperature derived

from the R2/R1 band ratio observed by the

ROSA T PSPC

The standard response has been that equilibrium temperatures provide a universally accessible relative measure of the plasma temperature. The bulk of the X-ray emission from the

plamsa is in lines. Even though the relative line

strengths will be very different between equilibrium and non-equilibrium plasmas of the same

temperature, one expects that mean energy of

the lines to increase with temperature in both

cases. Thus, one would expect the very broad

band fluxes to behave in similar manners. However, there are some very strong lines, such as

the O VlI and O VIII lines, that dominate the

emission. One might expect that these lines to

have very different behaviors in the equilibrium

and non-equilibium cases, and thus have a significant impact on the general run of R2/R1 with

temperature.

To explore this problem, we obtained a set

of non-equilibrium spectra from R. L. Shelton

to compare with equilibrium spectra. The nonequilibrium spectra were derived from a plasma

that had been shock-heated to 10s K and allowed to cool. The atomic models used by the

non-equilibrium models are the same used by

the Raymond & Smith equilibrium models. The

run of R2/R1 with plasma temperature is shown

in Figure A.1, where it can be seen that for

log T > 5.5 there are only minor differences between equilibrium and non-equilibrium models.

REFERENCES

Deiker, S., Kelley, R. L., Lesser, A., McCammon,

D., Porter, F. S., Stahle, W. T. S. C. K., &

Szymkowiak, A. E. 1997, in Proceedings of

the Seventh International Workshop on Low

Temperature Detectors, 113

Raymond, J. C., & Smith, B. W. 1977, ApJS, 35

Sanders, W. T., Edgar, R. 3., Liedahl, D. A., & Morgenthaler, J. P. 1998, in The Local Bubble,

ed. D. Breitschwerdt & M. Freyberg, 83

Shelton, R. L. 1999, private communication

o/

g_

1.5

l.O ++_

0.5

++++

0.0 .............................

5 6 7 8

log,, T

Fig. A.I.-- Comparison of the R2/R1 band ratios as a function of temperature for equilibrium

and non-equilibrium cooling. The solid line is for

a plasma cooling from 108 K (Sheiton 1999), the

crosses are the Raymond _: Smith (1977) equilibrium values.

175

Appendix B: Observations

Thisappendixpresentslistsof allof thepointedPSPCobservationsused.Eachmosaicis listed

in aseparatetable.

177
TABLEB.I

035+78OBSERVATIONS

ROR R.A. Dec. 1 b Exp Date Target

(32000) (Tot) (Eel.)

701066 204,89 26.35 31.52 79.23 9040 8619 14/07/92 CHFT FIELD A/5

201037 209.04 25.92 31.83 75.48 15533 12333 19/07/92-22/07/92 ZZ BOO

700284 207.15 26.37 32.78 77.22 2025 1894 30/06/91,01/07/91 E1346+266

701064 204.40 26.63 32.79 79,70 8356 8091 12/07/92-13/07/92 CFHT FIELD A/3

701068 205.38 26.63 33.23 78.83 8669 7821 16/07/92.17/07/92 CFHT FIELD A/7

700145 207.15 26.52 33.45 77.24 18205 16702 01/07/91 1E1346+26

1909 1544 06/01/92 1E1346+26

800105 207.23 26.59 33.80 77.18 36273 30157 04/01/92-22/01/92 ABELL 1795

800055 207.22 26.62 33.93 77,19 25803 24846 01/07/91-02/07/91 ABELL 1795

701062 204,07 27.08 35.22 80.02 9457 9073 10/07/92o14/07/92 CFHT FIELD A/1

701459 204.89 27.08 35,42 79,29 7157 6358 06/07/92-09/07/92 CFHT A/13

701063 205.71 27.08 35.63 78.57 10032 9392 15/07/92-22/07/92 CFHT FIELD A/2

701065 204.40 27.53 37.82 79.74 8971 8777 11/07/92 CFHT FIELD A/4

701069 205.37 27,53 37.87 78,88 9912 9572 13/07/92-14/07/92 CHFT FIELD A/8

701067 204.88 27.82 39.41 79.32 8209 7648 14/07/92-15/07/92 CHFT FIELD A/6

300333 205.54 28.38 42,23 78,71 13083 11904 06/06/93-22/07/93 NGC 5272

44599 12683 02/07/94-04/07/94 NGC 5272

TABLE B.2

086+74 OBSERVATIONS

ROR R.A. Dec. I b Exp. Exp. Date Target

(J2000) (J2000) (TotaJ) (Sel.)

900626 203.65 37.91 8556 7591 37658 30311 19/06/93-16/07/93 DEEP SURVEY

700283 203.65 37.91 8556 75.91 71803 71788 23/06/91-26/06/91 DEEP SURVEY

900218 203.86 38.20 8588 75.59 6410 5173 27/12/91-02/01/92 SURVEY POSN. I

900219 203.96 38,35 86 06 75 43 6080 4309 27/12/91-02/01/92 SURVEY POSN, 2

900220 204.07 38.50 8621 7526 1650 1062 26/12/91 SURVEY POSN. 3

6026 5386 03/06/92 SURVEY POSN. 3

900221 204.17 38,64 8636 75,10 7030 5328 27/12/91-30/12/91 SURVEY POSN. 4

900222 204.27 38.79 86 53 7494 4792 3443 27/12/91-03/01/92 SURVEY POSN. 5

3231 2751 02/06/92-03/06/92 SURVEY POSN. 5

900223 204.38 38.93 8665 74 78 4745 3966 29/12/91-14/01/92 SURVEY POSN 6

1663 1314 02/06/92 SURVEY POSN. 6

900232 204.48 39.08 86.82 7461 5345 4841 25/12/91-26/12/91 SURVEY POSN 7

900233 204.58 39.23 86.99 74.45 2134 1246 30/12/91 SURVEY POSN 8

4075 3956 02/06/92 SURVEY POSN. 8

900234 204.69 39.37 87 I0 74.29 5891 2162 15/12/91-16/12/91 SURVEY POSN. 9

900235 204.79 39.52 87 26 7413 6202 4140 28/12/91-29/12/91 SURVEY POSN. 10

900236 204.89 39.67 87 42 73.96 6776 5512 29/12/91-16/01/92 SURVEY POSN. 1]

900237 205.00 39.81 87 54 73.80 6536 5687 05/01/92-15/01/92 SURVEY POSN. 12

900226 205,10 39.96 87.69 73 64 5839 5101 07/01/92-15/01/91 SURVEY POSN 13

900227 205.20 40,10 87 82 73.49 6996 5508 30/12/91-14/01/92 SURVEY POSN 14

900228 205.31 40.25 8795 73.32 1574 1386 06/01/92-14/01/92 SURVEY POSN 15

4692 3954 01/06/92-02/06/92 SURVEY POSN 15

900229 205,41 40.40 88 11 7316 6723 4995 31/12/91-05/01/92 SURVEY POSN 16

900405 205,51 40.54 88 23 7300 8663 7501 08/06/92 SURVEY POSN 17

900406 205.62 40.69 88 36 72.83 7469 4713 07/07/92 SURVEY POSN 18

900407 205.72 40.83 88 48 72.68 7641 5364 03/07/92 SURVEY POSN 19

900408 205.82 40.98 8863 72 52 7911 6289 09/07/92 SURVEY POSN. 20

800252 203,84 41.00 92 64 7344 15659 12297 23/06/92-29/06/92 A1763

178
TABLE B.3

138+52 OBSERVATIONS

ROR R.A. Dec_ i b Exp. Exp. Date Target

(J2000) (J2000) (Total) (Sel.)

600280 16164

201235 161.80

701122 163.01

900331 164,38

900330 164.39

900329 164,55

900328 165.66

63.24 143.69 48.58 10473 7080 12/05/93-22/05/93 NGC 3359

63.59 143d4 48.37 4760 3784 17/11/92-19/11/92 P,,X 31047.1@6336.3

61.42 144.83 50.33 9349 7806 17/11/92-28/11/92 ABLS QSO JI0524,612

7804 6956 21/10/93 ABLS QSO 310524-612

6274 4871 15/10/93-17/10/93 ABLS QSO 310524,6125

62.27 143.19 50.14 9004 5569 26/05/93-27/05/93 UMA7

63.40 142.03 49,29 7292 5045 23/05/93-27/05/93 UMA6

64.49 140.88 48.50 10089 7604 11/05/93-24/04/93 UMA5

65.62 139.28 47.91 6793 5157 10/11/92-28/11/92 UMA4

5448 3697 05/04/93-06/04/93 UMA4

3174 2503 11/10/93 UMA4

60.89 143.86 51.60 20242 15957 18/05/93-22/05/93 G290.1-0.8

64.54 139.27 49.26 6192 4465 11/04/93-14/04/93 UMA8

9249 7493 11/10/93-13/10/93 UMA8

3126 1986 04/05/94 UMA8

83.01 140,40 50.62 6617 3864 19/10/93 A 1180

58.79 143.53 54.46 4606 3740 30/11/92-04/12/92 NGC 3610

6178 4913 10/05/93-11/05/93 NGC 3610

59.08 142.54 54.52 7804 6956 21/10/93 NGC 3642

64.00 138,03 50.44 5886 4171 30/11/92-01/12/92 UMA9

3636 3077 14/10/93 UMA9

4530 3467 04/06/94-06/05/94 UMA9

1466 1018 23/04/92 NGC 3690

3532 3442 23/04/93 NGC 3690

63.84 136.76 51.10 16663 9794 29/10/91-31/10/91 UMA CLOUD 2

66.41 134.67 48.90 2972 2151 29/11/92 A1302

59.72 13953 54.89 10315 7293 29/05/93-30/05/93 UMA12

64.53 135.54 50.72 5823 3658 01/12/92-04/12/92 UMA11

4397 3432 10/05/93-21/05/93 UMA11

753 509 04/05/94 HS 11334--6753

5893 4371 15/10/93-16/I0/93 HS 11334-6753

18289 10636 19/11/91-25/11/91 UMA CLOUD 3

6000 4721 12/05/93-27/05/93 UMAI0

1769 1696 20/11/91-22/11/91 3C263

5018 3512 30/05/93 4C58.22

500157 166.72

900332 167.58

800540 168.04

600250 169.61

701412 170.58

900333 170.63

700442 172.13 58.56 141.91 55.41

900148 ]72 96

800260 173.34

900336 ]73.84

900335 174 11

900521 174.12 67.62 13345 47.95

900150 174 44 61.22 137.83 53.73

900334 ]74 83 62.37 136,66 52.79

700068 174.99 65.80 134,16 49.74

800402 17571 68.13 13943 56,75

179
TABLEB.4

16I+51 OBSERVATIONS

ROR R.A. Dec. I b Exp. Exp. Date "l_arget

(32000) (J2000) (Total) (Sel.)

900214 152.55 50.75 164.03 51.49 15664 13320 12/II/91-13/11/91 SURVEY FIELD 5

700265 152.56 51.75 ]62.56 51.11 3587 3191 16/04/91-09/05/91 SURVEY FIELD 4

20122 13774 17/11/91-20/11/91 SURVEY FIELD 4

700263 152.56 52.75 161.11 50.70 13905 11132 14/04/91-10/05/91 SURVEY FIELD 3

700264 152.57 53.75 159.68 50.28 14758 13558 13/04/91-14/04/91 SURVEY FIELD 2

900213 152.57 54.75 158.29 49.83 16614 11670 10/11/91-12/11/91 SURVEY FIELD 4

900215 152.58 55.75 156.91 49.38 18824 14878 11/04/92-15/04/92 SURVEY FIELD 6

900400 154.82 52.75 160.15 51.93 2544 2065 20/11/92-21/11/92 LOCKMAN SPUR A

13256 11760 24/04/93-03/05/93 LOCKMAN SPUR A

900401 156.81 53.49 158.15 52.62 796 727 17/11/92 LOCKMAN SPUR B

16070 14576 25/04/93-01/05/93 LOCKMAN SPUR B

Additional Pointings

701401 149.41 55.38 158.60 47.93 4465 3343 08/11/93-09/11/93 0954+556

700319 15049 55.68 157.81 48.36 18836 11024 14/11/91-15/11/91 NGC 3079

700220 153.77 49.43 165.54 52.72 5151 4688 11/05/91 GB 1011+498

600624 155.55 51.40 161.79 52.94 8427 7443 21/10/93-22/10/93 MS1019+5139

701544 157.83 50.89 161.44 54.44 10777 7628 22/10/93.-23/10/93 P,LXJ 10313+5053

600057 158.13 54 40 156.20 52.80 3979 2109 20/04/91-21/04/91 HARO2

900149 158.78 54.24 13135 9671 01/11/91-02/11/91 UMA CLOUD 1

4361 3978 16/04/92 UMA CLOUD 1

600156 159.69 53.50 156.61 54.06 9114 6267 17/11/91-18/11/91 NGC 3310

180
TABLE B.5

171 +50 OBSERVATIONS

ROR R.A. Dec. l b Exp. Exp. Date Target

(32000) (J2000) (Total) (Sel.)

800102 145 73 46.99 171.63 48.28 13941 8826 17/11/91-20/11/91 A851

700167 146.25 47.50 170.77 48.52 546 213 21/04/91 HS 47.5/22 FIELD

700 647 20/11/92 HS 47.5/22 FIELD

1451 1317 20/10/93 HS 47.5/22 FIELD

700168 14625 47,00 171.51 48.63 716 566 19/11/92 HS 47.5/22 FIELD

1408 1222 25/04/93 HS 47.5/22 FIELD

700169 146.25 46.50 172.25 48.73 2394 1808 24/04/93-25/04/93 HS 47.5/22 FIELD

700451 146.25 4800 170.04 48.41 3383 2523 10/05/92-19/05/92 HS 47.5/22 FIELD If.

700158 146.63 46.25 172.55 49.04 2505 1777 14/04/91-18/04/91 HS 47.5/22 FIELD

70017] 14663 46.75 171.80 48.94 2272 1792 14/04/91-18/04/92 HS 47.5/22 FIELD

700172 14663 47.25 171.06 48.83 1077 776 21/04/91-23/04/91 HS 47.5/22 FIELD

700454 146.63 47.75 170.32 48.72 3183 2990 09/05/92-19/05/92 HS 47.5/22 FIELD If.

300329 146,65 44,78 174.75 49,32 5422 3739 09/ii/93-I0/Ii/93 DV UMA

700166 147 00 47.50 170,60 49.02 2157 1682 14/04/91-19/04/91 HS 47.5/22 FIELD

700170 147.00 46.50 172,09 49.24 1358 779 20/04/91 HS 47.5/22 FIELD

1032 819 21/Ii/92 HS 47.5/22 FIELD

700182 147.00 47.00 171.35 49.13 2816 2358 I0/05/91 HS 47.5/22 FIELD

700457 147.00 48.00 169.86 48.90 3236 3085 09/05/92-19/05/92 HS 47.5/22 FIELD II_

700157 14738 46.25 172.38 49.54 2751 2477 15/11/91-16/11/91 HS 47,5/22 FIELD

700181 147.38 46.75 171.63 49.44 2900 2579 15/11/91-20/11/91 HS 47,5/22 FIELD

700453 147,38 47.75 170.14 49.21 4617 3238 10/05/92--19/05/92 HS 47,5]22 FIELD If.

700173 147.38 47.25 170.88 49.33 1504 II00 23/04/91 HS 47,5/22 FIELD

575 363 II/ii/92 HS 47.5/22 FIELD

700159 147.75 46.50 171.92 49.74 2604 1858 17/04/91-21/04/91 HS 47.5/22 FIELD

700165 147.75 47.50 170.42 49.51 1953 1873 15/ii/91-17/i 1/91 HS 47,5/22 FIELD

700180 14775 47.00 171.17 49.63 1041 613 21/04/91 HS 47.5/22 FIELD

1354 1298 19/11/92-20/11/92 HS 47.5/22 FIELD

700456 147.75 48.00 169,67 49.39 3420 3041 I0/05/92-19/05/92 HS 47.5]22 FIELD II.

700156 148,13 46.25 172.21 50.05 1757 1442 18/04/91-20/04/91 HR 47.5/22 FIELD

700174 148.13 47.25 170.70 49.82 2910 1985 I0/i1/91 HS 47.5/22 FIELD

700179 148.13 46.75 171.45 49.94 2063 1288 17/04/91-22/04/91 HS 47.5]22 FIELD

2103 1973 11/11/91-12/I 1/91 HS 47.5/22 FIELD

700452 148.13 47.75 169.95 49.70 2911 2761 11/05/92-19/05/92 HS 47.5/22 FIELD II.

700160 148.50 46.50 171.74 50.24 2553 2422 16/11/91 HS 47.5/22 FIELD

700176 148.50 47.50 170.23 50.00 2337 1592 17/04/91 HS 47.5/22 FIELD

700178 148.50 47.00 170.98 50.13 2944 1719 15/ii/91 HS 47.5/22 FIELD

700455 148.50 48.00 169.48 49.87 3487 3336 11/05/92-18/05/92 HS 47.5/22 FIELD II.

700046 14854 49.26 167.59 49.56 3503 3029 13/04/91-14/04/91 MSS0950+49

700155 148.88 46.25 172.03 50.56 2690 1620 15/11/91 HS 47.5/22 FIELD

700175 148 88 47.25 170.51 50.31 2082 1742 10/11/91 HS 47.5/22 FIELD

7¢)0177 14888 4675 171.27 50.44 1890 1715 25/04/91-06/05/91 HS 47.5/22 FIELD

700450 148 88 47.75 16975 50,19 3679 2899 12/05/92-18/05/92 HS 47.5/22 FIELD If.

700150 14925 48.50 168.52 50.22 545 528 07/05/91 HS 47.5/22 FIELD

1418 1243 23/04/93 HS 47.5/22 FIELD

1922 1638 20/10/93 HS 47.5/22 FIELD

700153 149 25 48.00 169.27 50.36 2594 1900 10/11/91 HS 47.5/22 FIELD

700161 14925 46.50 171,55 50.75 2586 1807 20/11/92-21/11/92 HS 47.5/22 FIELD

700184 149 25 47.50 170.03 50.49 3240 3063 15/I 1/91-21/I 1/9] HS 47.5/22 FIELD

700187 149 25 47.00 170.79 50.62 2524 1889 16/11/91 HS 47.5/22 FIELD

701402 14958 47.42 170.05 50.73 3583 2830 13/11/93 0955+476

700151 149 63 48.25 168.78 50.53 1326 1204 17/11/91 HS 47.5/22 FIELD

1298 1266 24/04/91-29/04/91 HS 47.5/22 FIELD

7/J0154 14963 47.75 169.54 50.67 3094 2922 07/05/91-09/05/91 HS 47.5/22 FIELD

700185 149 63 47.25 170.30 50.81 2124 1803 16/11/91 HS 47.5/22 FIELD

7¢_0188 149 63 46.75 171.06 50.94 2593 2324 ii/05/91-12/05/91 HS 47.5/22 FIELD

2885 2536 16/II/91-18/] I/_1 HS 47.5/22 FIELD

7_)0149 15000 48.50 168.29 50.70 1805 1665 16/11/91 HS 47.5/22 FIELD

7r_0152 150 00 48.00 169.05 50.84 1576 1427 15/04/91-16/04/9! HS 47.5/22 FIELD

569 491 23/04/91 HS 47.5/22 FIELD

7o0162 150 00 46.50 17135 51.25 3042 2505 08/05/92-09/u5/9'.' HS 47.5/22 FIELD

7¢_0183 150 00 47.50 169.81 50.98 2588 1745 15/II/91-16/I i/91 HS 47,5/22 FIELD

7_0186 150 00 47.00 17058 51.12 1594 1261 06]05/91 HS 47.5/22 FIELD

716 664 21/11/92 HS 47,5/22 FIELD

7_0163 15038 46.75 17085 5143 2145 1807 I0/05/91 HS 47.5/22 FIELD

7_0164 150 38 47.25 |7008 51.30 890 638 17/11/91 HS 47.5/22 FIELD

1748 1706 23/04/91 HS 47.5/22 FIELD

7_|546 152 05 47.09 169 80 52.43 2145 1640 24/i0/93 B.XJ 10081+4705

181
TABLE B.6

220+84 OBSERVATIONS

ROR R.A. Dec, l b Exp. Exp. Date Target

(J2000) (J2000) (TotaJ) (Sel.)

700223 185.38 28.23 201+76 83.29 15223 12087 14/06/91-15/06191 ON 231

701056 185.34 30.18 186.34 82.73 10969 10307 18/06/92-22/06/92 2A 1218+304

701203 184.62 29.81 190.68 82,28 6608 6266 16/06/92-21/06/92 MS1215.9+3005

2916 2572 09/12/92 MS1215+9+3005

701353 184.61 29.81 19070 82.27 3146 2717 17/12/93-18/12/93 MKN766

700221 184.46 30.12 188.87 82.05 21999 20253 15/06/91-16/06/91 B2 1215+30

700970 184.61 30.04 189,09 82,20 19870 14489 21/12/92-23/12/93 MKN766FIELD

701413 185.03 29.28 193.78 82.77 3413 2937 16/12/93 NGC 4278

700232 181,18 27.90 205.98 79.62 27859 10580 04/06/91-05/06/91 1202+281

600444 181.66 28.17 20435 80.03 15075 13551 24/05/93-26/05/93 NGC 4104

201170 183.00 27.50 208,24 81,23 2166 2064 16/06/92-17/06/92 CB8

201169 184.00 25,00 225.92 81.76 1814 1743 15/06/92-16/06/92 CB7

200529 184,15 26,49 215.65 82.20 3194 2020 28/12/91-31/12/91 PG 1214+268

201168 184.67 25.50 22360 82.48 1813 1741 18/06/92 CB6

201175 184.67 23.00 240,24 81.51 4031 3112 29/06/92 CB13

201167 185.33 2600 220,74 83.17 1678 1610 18/06/92 CB5

201176 185.50 27.00 212.38 83.45 4374 3993 19/06/92 CB14

200307 185,75 25.95 221.82 83,53 16526 11616 13/06/91-14/06/91 COM15

201166 186.00 26.50 217.24 83.85 1964 1872 20/06/92 CB4 "

201174 186.00 22.50 24808 " 82,24 2354 2206 14/12/92-17/12/92 CB12

700308 186.10 24.61 23392 83.42 5624 2945 19/12/91-29/05/92 MSS 1221+24

700388 186.41 24,98 231,84 8382 8347 6453 01/06/92-05/06/92 1223+252

201165 186+67 27.00 212,85 84,49 1999 1938 21/06/92 CB3

201173 186.67 23.00 247.95 83.04 2262 2144 08/12/92-09/12/92 CBll

201164 187.33 27.50 207.34 85.07 2040 1701 20/06/92 CB2

201172 187.33 23.50 24770 83.82 2152 1669 29/06/92 CB1O

201163 188.00 2800 200,37 85,61 1810 1740 21/06/92 CB]

201171 188,00 2400 247,38 84.62 2200 2085 16/12/92 CB9

600162 188.48 26.27 224.31 85.99 12015 8649 15/12/91-04/01/92 NGC 4565

600415 189.09 25.99 230.74 86.44 9202 7940 30/06/92-04/07/92 NGC 4585

10519 9496 08/12/92-15/12/92 NGC 4565

600595 188.99 27.96 198.41 86.47 17353 16100 01/07/92-02/07/92 NGC 4559

701097 189.42 26.71 219.92 86.91 5424 5085 30/06/92-01/07/92 IC 3599

701098 189.42 26.71 219+92 86.91 1772 1688 16/06/92 IC 3599

701099 189.42 26.71 219.92 86.91 1875 1807 17/06/92 IC 3599

701100 189,42 26,71 219,92 86,91 2011 1937 18/06/92 IC 3599

701528 189.42 26.71 219.92 86.91 3434 2001 17/06/92-18/06/92 IC 3599

800475 190.32 26.23 233.96 8756 6797 4167 15/06/93-03/07/93 NGC 4615

TABLE B.7

295-89 OBSERVATIONS

ROR R.A. Dec. i b Exp. Exp. Date TsJrget

(J2000) (J2000) (TotDJ) (Sel.)

800069 9.88 -28.81 359.66 -86.88 7465 6733 23/12/91-04/01/92 SCULPTOR C,D

800070 11,38 -29.64 329 97 -87.17 6779 6273 23/12/91-02/01/92 SCULPTOR B

700275 13,02 -29.09 29884 -88,03 24461 16725 01/06/92-12/06/92 SGP2

700133 13.75 -28.33 26987 -8856 1569 1530 04/01192 005459-281943

19466 16675 03/06/92-15/06/92 005459-281943

900496 13,75 -2833 26987 -8856 9415 8616 19/12/93 SGP3

22693 17315 30/06/94-01/07/94 SGP3

701223 14.37 -27.64 234 16 -88.56 48943 42843 03/06/92-08/06/92 GSGP4

700081 14,49 -26.22 181.36 -8828 3817 3664 28/12/91 0055-264

700528 14,53 -27.64 232,35 -8843 12516 10805 08/12/92-14/12/92 SGP4

18185 16917 04/06/92-15/06/92 SGP4

700121 15,57 -27,33 218.34 -87.58 6157 6016 05/01/92-06/01/92 Q0059-2735

2191 1926 05/06/92-15/06/92 Q0059-2735

6724 5412 08/12/92-09/12/92 Q0059-2735

182
TABLE B.8

045+00 OBSERVATIONS

ROR R.A. Dec. l b Exp, Exp. Date Tin'get

(J2000) (J2000) (Tot,.l) (Sel.)

600281 282,96 13.46 45.00 6.01 3696 2745 12/10/92-13/I0/92 PLANE 45-i

600282 283.75 13.07 45.00 5.14 2421 1712 13/10/92 PLANE 45-2

1123 789 23/03/93 PLANE 45-2

12.69 2916 2609 16/10/92-17/10/92 PLANE 45-3

12,30 3629 3543 28/10/92-29/10/92 PLANE 45-4

11.91 4304 3760 28/10/92-29/10/92 PLANE 45-5

11.52 3728 2930 18/I0/92-19/I0/92 PLANE 45-6

11.12 4922 3594 19/10/92 PLANE 45-7

10.72 1813 1764 19/I0/92 PLANE 45-8

714 664 27/09/93 PLANE 45-8

822 799 30/03/94 PLANE 45-8

10.32 4340 3911 19/10/92-20/10/92 PLANE 45-9

9.92 628 522 31/10/92 PLANE 45-10

3092 1862 26/03/93-27/03/93 PLANE 45-10

9.52 2628 2497 28/10/92-03/II/92 PLANE 45-11

1004 887 30/09/93 PLANE 45-11

9.12 926 870 28/10/92 PLANE 45-12

2518 1683 28/03/93 PLANE 45-12

8.71 4431 3550 30/10/92-02/I I/92 PLANE 45-13

8.31 3638 3083 27/10/92-28/10/92 PLANE 45-14

7.90 2580 2351 30/I0/92 PLANE 45-15

1526 1236 30/03/93 PLANE 45-15

600283 284,54

600284 285.32

600285 286.10

600286 285.88

600287 287.65

600288 288,43

600289 289.20

600290 289.9?

600291 290.74

600292 291.50

600293 292,27

600294 293,03

600295 293.79

45.00 4.28

45.00 3.43

45.00 2.57

45.01 1.71

45.00 0.86

45.00 -D.00

45.00 -0.86

45.00 -1.72

45.00 -2.58

45.00 -3.43

45.00 -4.29

45.00 -5,14

45.00 -6.00

183
TABLE B.9

178-22 OBSERVATIONS

ROR R.A. Dec. Exp. Exp. Date Target

(32000) (J2000) (Total) (Sel.)

200912 63.87 16.67 1658 1479.5 I0/09/92 HYADES C

23740 15615.6 26/02/93-03/03/93 HYADES C

200981 64.18 16.82 4495 3678.6 23/02/93-04/03/93 VLM HYADES 3

201316 64.37 20.58 4158 2356.3 14/02/93-16/02/93 LEER I

201604 64.86 21.14 7671 7098.3 31/08/93-02/09/93 HD27295

200775 64.95 15.63 4096 3312.3 22/02/93-23/02/93 HYADES FIELD KI

200776 65.22 13.86 22995 21977.0 30/08/91-31/08/91 HYADES FIELD FI

200441 65.30 14.70 10987 6684.5 24/02/91 HYADES, POSITION 1

200442 65.30 17.53 19948 10074.5 24/02/91-27/02/91 HYADES, POSITION 2

201370 65.38 18.42 4977 3263.1 11/09/92 HYADES FIELD F4

13662 11343.4 23/02/93.24/02/93 HYADES FIELD F4

201315 65.50 19.53 645 590.6 23/02/93 T TAU

1545 1442.9 14/08/93-15/08/93 T TAU

700044 65.58 19.85 4704 2888.5 26/02/91-27/02/91 MSS0419-I-I 9

200020 66.21 15.10 40412 23226.9 27/02/91-01/03/91 HYADESFIELD

200777 66.40 15.94 16296 13433.3 30/08/91 Hyades field F2

201369 66.60 16.85 15551 13045.8 24/02/93-27/02/93 HYADES FIELD G I

201314 66.78 18.78 2693 2084.0 12/09/92 UX TAU

1397 1323.3 24/02/93.26/02/93 UX TAU

200982 67.09 13.82 7727 5754.0 04/03/93.13/03/93 VLM HYADES 4

200083 67.15 16.97 2929 1226.4 25/02/91-27/02/91 VAN RHIJN 16

200576 67.15 19.18 1541 1245.6 12/03/92 EPSILON TAU

200942 67.21 13.05 7374 4868.8 09/09/92-I0/09/92 79 TAU

200778 67.53 15.64 1920 1842.6 12/03/92 HYADES FIELD F3

201368 67.53 15.64 16606 11140.1 10/09/92-12/09/92 HYADES FIELD F3

200945 67,65 13.72 4101 2767.0 09/09/92 83 TAU

900353 67.85 17.96 7724 6367,3 21/02/93.24/02/93 L1551

200553 67.85 12.39 10961 9849,1 27/02/92-12/03/92 HYADES B

200911 67.85 12.39 17472 10987.0 09/09/92-11/09/92 HYADES B

13766 9807.5 15/02/93-03/03/93 HYADES B

201313 68,07 18,03 4033 3852.6 10/09/92 V 826

200443 68.13 17.53 20074 12696.2 07/03/91-10/03/91 HYADES, POSITION 3

200444 68.13 14.70 14593 9900.1 06/03/91-07/03/91 HYADES, POSITION 4

201747 68.64 1550 19618 18354.8 31/08/93-04/09/93 HYADES FIELD G2

200073 68.98 16.51 6833 3559.2 24/02/91 ALPHATAU

184
Appendix C: Reduction Scripts

Abstract

This appendix contains the description of a number of UNIX shell scripts that

are useful for quasi-automating the ESAS data-reduction process. These shell scripts

should not be used until the use of the FORTRAN programs described tn the previous

section is well understood and a test reduction has been successfully completed.

C.1. UNIX Scripts for Data Reduction

When reducing ROSATdata for diffuse background studies, it is often neccessary to execute

the same program for each ROSA T band, typing

in nearly the same long parameter lists multiple times. These scripts are intended to alleviate some of that tedium. However useful these

scripts may be, they are not programmed with

any semblance of elegance, and they can be difficult to read. Much of the difficulty arises from

the need to feed to programs input parameters

as if they were typed from the keyboard, with

all the requisite carriage returns. The method

for solving this problem is not elegant:

programname <<E

inputlinel [CR]

inputline2 [CR]

inputline3 [CR]

E

Here the "<<E" means to take all the text down

to the matching "E" including carriage returns

("[CR]") as input. The "E" was an arbitrary

choice, it could have been any character string

that is unlikely to occur as a program parameter.

C.2. Prescription for Use

0.) Setup. I suggest creating a file called

"'reduce log". The name is important; if it is

not named properly the "cleanup" script will

not work properly. The "reduce.log" file should

contain the information as shown for a sample

observation in Figure C.1. These lines can be

copied from the "public_contents" file.

1.) The initial processing. The "initial"

script executes every program necessary to create the observed light curve and the light curve

of the scattered solar X-ray contamination.

2.) Fitting the LTE and SSX. The easiest way of fitting the LTE and SSX is to run the

FORTRAN program RATE_FIT_AUTO. However, if you want to have a bit more involvement

with the fitting process (which is laudable, and

even recommended the first few times through)

run RATE_FIT on band R4 with various orders of LTE polynomial and various orders of

SSX scaling polynomial. Rather than running

through all of these combinations by hand, let

the computer do all the possible combinations

and save plots of the results. This is the function of the "rate.fit.4" script. Once it has produced QDP format plot files of all of the possible

combinations of SSX and LTE polynomial orders, I use IDL routines to review the plot files.

You can then compare these results with that of

RATE_FIT_AUTO.

Once you have decided which orders of polynomials to use for fitting the LTE and SSX, use

the FORTRAN routine RATE_FIT to fit the

other bands. Then check the output QDP plot

files to check each band.

3.) The middle processing. The "middle"

script casts the count images, the exposure images, and the scattered solar X-ray images.

4.) After-pulse fitting. As with the reductions unaided by scripts, run CAST.PART (setting afterpulse counts to zero) until it outputs

the total number particle background counts;

the total number of scattered solar X-ray counts

can be found in the output from TILT, which

was executed by the "middle" script. Then use

the FORTRAN routine FIT_AP to determine

the number of afterpulse counts. After fitting, I

suggest saving the last several lines of output,

those shown in Figure C.2, into a file named

"fit_ap.log", so that it can be incorporated into

the final reduction log by the "cleanup" script.

185
OBSERVATION_REQUESTID = "600057"

FOLLOWUP_I NDEX = ....

PRINCIPALINVESTIGATOR = "LOOSE, DR, HANS-HERMANN"

INSTRUMENT_NAME = "PSPC B"

RIGHT.ASCENSION = " 10h 32m 31.20s"

DECLINATION = " 54d 24m O.OOs"

GALACTIC.LONGITUDE = "156d 12m 2.44s"

GALACTIC_LATITUDE = " 52d 48m 4.74s"

UT_START_TIME = "20-APR-1991 00:01: 18"

UT_END_TINE = "21-APR-1991 00:26: 12"

TOTAL_ACCEPTED_SECONDS = "3979"

Fig. C.1.--The header text to be saved in thelog file.

6.0 22 23 6.000000 6.050000

ICHISQ = 18.3, DOF = tg, EEDCHISQ = 0.965

[PB = 416.t55 AP = 26.858

Fig. C.2.-- Output from fit_ap to be saved in log file.

5.) The final processing. The '_final"

script casts the background images (i.e., runs

CAST_PART) and calculates the count rate images.

6.) Cleanup. After finishing the reduction,

there remain a large number of very small files

that were used to transfer fit coefficients from

one program to another, such as "rate_fit_lte.8.dat',

"rate.:fit_ssx_8.dat'. and "T0.dat'. The "cleanup"

script tidies all of these data files away into the

single file TRANSFER.BND, from which they

can be re-extracted if neccessary. All of the

".log" files listed in Table C.2 are tidied away

into the single file LOGFILE.

C.3. Scripts

The scripts and their usage are listed below

in the order in which they are to be used.

C.3.2. rate.fit.4

Purpose: to fit the LTE and SSX contributions to band R4. This script fits band

R4 using increasing orders of polynomial

to fit the LTE. It executes the fits using

no SSX component (IS0 and IS1 set to

zero), using only a scaling factor for the

SSX component (IS0 variable, IS1 set to

zero), and allowing a time variable SSX

component (IS0 and IS1 both variable).

Plots of the fits are placed in QDP format

files named plot_4.n.0.qdp, plot_4.n.l.qdp,

and plot_4.n.2.qdp for a fit to the LTE

using up to nth order polynomials, and

SSX fits as above. This script is superceded by the FORTRAN routine

RATE_FIT_AUTO, and is included here

for pedagogical purposes only.

Usage: rate.fit.4

C.3.1. initial

Purpose: to do all of the processing before

editing the data and fitting the LTE and

SSX contributions. This script executes

the FORTRAN programs VALID_TIMES,

RATE, and AO.

Usage: init ial ha,he gain

C.3.3. middle

Purpose:to do all the processing after the LTE

and SSX fitting, but before fitting the AP

contribution. This script executes the FORTRAN programs LTE, CAST_DATA, CAST_EXP,

TILT, and CAST_SSX.

Usage: middle name gain tilt

186
TABLE C.1

REDUCTION SHELL PARAMETERS

Name Definition and Units

ap

gain

nasne

tilt

the number of _t'terpulsecounts in the exposure

the gain state of the observation

1 high gain data (before 11 October 1991)

2 low gain data (after11 October 1991)

3 high gain data, but using the R1L (RS) band

the root name of the data files

a flag denoting the need to run tilt

0 there were no memsured solar photons

(i.e, the ISO and IS1 from RATE.FIT were both 0

for all bands) and thus tilt should not be run

1 tilt should be run

TABLE C,2

LOG FILES AND THEIR ORIGINS

Name Source

reduce3og created by hand

r_te_flt.log output of RATE.FIT

Ire.log output of LTE

tilt.log output of TILT

fit_ap.}og created by band

from the output of F1T_AP

xxx_diag out created by CAST_DATA

and augmented by CAST.PART

finad.Jrnage.log output of FINAL_IMAGE

187
C.3.4. final

Purpose: to execute all processing after fitting

the AP contribution. This script executes

the FORTRAN programs CAST_PART and

FINAL-IMAGE.

Usage: final name gain ap

C.3.5. cleanup

Purpose: to consolidate into one archive file

all of the transfer data files produced in

the reduction process (such as ssx_x.dat,

lte_x.dat). The final output file is TRANSFER.BND which, if executed as a shell

script, upacks itself into all of the original files. (This is a neat trick learned from

Kernighan _ Pike (1984).) This script also

takes all of the log files listed in Table C.2

and consolidates them into one file named

LOGFILE.

Usage: cleanup

C.3.6. clearout

Purpose: to remove all files not neccessary for

forming mosaics.

Usage: clearout

REFERENCES

Kernighan, B. W., & Pike, R. 1984, The UNIX Programming Environment (Prentice Hall)

188
Appendix D: Mosaicking

Abstract

Thechapterdescribestheprocessof creatinga mosaicfroma numberof overlappingROSAT PSPC or HRI pointings. Whereas a previous chapter provided a

theoretical basis for the method of mosaicking, this chapter provides a step-by-step

prescription for using the FORTRAN progams developed by Kuntz and Snowden to

make a mosaic. The programs described in this chapter, as well as this documentation,

are avialable through the HEASARC.

D.1. Introduction

This chapter describes the FORTRAN programs in detail, as well as some UNIX scripts

that make the mosaicking process less onerous.

D.2. Accessing and Compiling

D.2.1. Retrieving Files

Most of the neccessary files will have been

acquired with the data reduction software (as

described in Appendix B of Part I). There are a

few additional data files that are needed which

are to obtained through other sites.

The ROSAT All-Sky Survey li: The ROSAT

All-Sky Survey II (Snowden et al, 1997) is now

available. The maps can be retreived from

http://www.rosat.mpegarching.mpg.de /survey /sxrb /12 / fits.html

The ROSAT Bright Source Catalogue: The

ROSA T Bright Source Catalogue can be retrieved via Internet browser from

http://www.rosat.mpegarching, mpg.de/survey/rass-bsc/rass-bsc1.2rxs.cat.gz

or may be obtained by anonymous ftp from

ftp.rosat.mpe-garching.mpg.de, where it can be

found in

archive/survey/rass-bsc/rass-bsc-1.2rxs.cat.gz.

D.2.2. Compiling

Before compiling the mosaicking software, a

few directory names will need to be changed. In

addition to those listed in Part I, the RASSTR

variable in get_rass_mosaic.f, get_rass_point.f and

offset_dc.fwill need to be changed to point to the

directory containing the ROSA TAll-Sky Survey

images. Further, in the program get_rass_bsc.f

the CATFIL variable will need to be changed

to point to the file containing the Bright Source

Catalogue.

UNIX: This release of the mosaicking software includes the file make.mosaic which is a

standard UNIX makefile. The architecture dependent FITSIO filename to be replaced is fitsdec.for. The EXE variable should be changed

to contain the name of the directory into which

you wish the executables to be placed. It will

also be neccessary to edit the FFLAGS variable

for your system.

VMS: The package can be compiled using

the VAX script mosmc.lnk.

D.3. Prescription

This is a brief description of the process of

making a mosaic, listing the FORTRAN routines in the Extended Source Analysis Software

(Kuntz & Snowden 1998) to be used and the order in which to apply them. For detailed description of what inputs and files each FORTRAN routine requires, and suggested settings

for some of the input parameters, see the program descriptions in Appendix D.5.

When forming mosaics at 1⁄4 keV it is important to remember to reduce all of the component

pointings using the same band, that is, do not

attempt a mosaic where some of the pointings

have band R1L and others band R1. Unless all

of the component pointings were taken in the

189
highgainmode,reducethehighgainobservationswiththelowgainbanddefinitions.

Themosaickingsoftwareusesthecountfiles

(PSPC:b_n.fits, HRI: counts.fts), exposure files

(PSPC: expose_n.fits, HRI: expose.fits), and background files (PSPC: back_n.fits, HRI: back.fits),

rather than the final products of the individual exposure reduction (PSPC: ratc_n.fits, HI:U:

rate.fits). The object of the mosaicking software is to produce four mosaics. The first three

are analogous to the count, exposure, and background images used in the individual exposure

reduction. The fourth mosaic is the LTE offset

mosaic. These four mosaics can then be combined analogously to the individual exposure images,

counts - background - LTEof f set

Y'Qte ""

exposure

(D.1)

Assuming that the uncertainty in the model

background counts is much less than _,

uncertainty : (D.2)

exposure

The mosaicking software will produce a mosaic in a zenith equal-area (ZEA) projection 1,

one of the standard projections of the FITS format (Greisen & Calabretta 1993) _. This projection has the advantage of preserving the angular

sizes of features, and thus allowing simple computation of physically meaningful quantities directly from maps (Each pixel has the same solid

angle, though not the same shape on the sky.)

The projection can be in one of three coordinate

frames: equatorial, ecliptic, or galactic.

Listed at the end of the description of each

step, if appropriate, is the name of a useful

UNIX shell script for executing that step in a

quasi-automated way

t If, in a spherical coordinate system, a point has a zenith

angle, azimuthal angle coordinate (0, _), then the map

coordinate of the point, in the polar coordinates (radius, angle) will be (scaling factor _/2(1 - sin0),¢). See

(Greisen & Calabre t ta 1993) fi_r a more complete description. The advantages of this projection for studies of

the diffuse background are relatively low distortion over

moderate angular scales and pixels having equal solid

angles.

2Although this software follows the standards, do not expect other software to t,_ so careful. For example, ZEA

projection images including a pole will not have the correct coordinates when displayed in SAOIMAGE.

0.) Reduce the Individual Pointings.

Once the individual pointings are reduced, place

each in its own directory, and make sure that

the count, exposure, and background files have

their standard names: b_n.fits, e_ose_n.fits, and

back_n.fits for the PSPC, counts.fits, expose.fits,

and back.fits for the HRI. Figure D.1 depicts

the necesary directory structure. Create a file

named dir.dat containing the path name of all

the directories, one per line. For UNIX systems,

make sure that the name of each directory is

terminated with a "/".

1.) Fix Broken Exposures. A "broken exposure" is a set of (sub)exposures with exactly

the same pointing direction. This is often the result of an observation being scheduled over two

or more half-year observing periods, but can be

multiple observations proposed for the same target (e.g. SN 1987a) and processed with the same

nominal pointing direction. In order to detect

point sources to the faintest possible limit, the

various sub-exposures should be summed, that

is, the count files (b_n.fits or counts.fits), the exposure files (expose_n.fits or e_t_ose.fits), and the

background files (back_n.fits or back.fits) all need

to be summed over the subexposures for each

band of interest. The summation can be accomplished with the program FIX._BREAK, which

is described in Part I. Two exposures that are

not part of a single observation should also be

"fixed" so long as they have exactly the same

pointing direction. The best directory structure

to adopt for broken exposures is shown in Figure D.1. Remember to update the dzr.dat file

accordingly.

The resultant "summed exposure" should be

used for source detection, and can be used to

form the mosaic if the mask for the summed exposure is formed from the sum of the exposure

files.

Useful UNIX shell script: add.obs

2.) Detect Point Sources. As has been

noted above, the detection of point sources in

the mosaic is actually a multiple step process.

The first step is to detect the point sources in

each exposure. In the case of broken exposures,

detect the point sources in the summed exposure, not in the subexposures. The detection of

point sources is done by the FORTRAN routine

DETECT by a sliding box method At every

point, the counts within 1.5 PSF radii are compared to the counts within an annulus from 1.5

190
data rp900001

rp900002

-b_ll.fits

-b_2.fits

-back_ll.fits

-back.2.fits

-diag_82.dat

--expose_ll.fits

--expose_2.fits

-mask_82.fits

_- rp900002n00

rp900002n01

-b_li.fits

-b_2.fits

-back_ll.fits

-back.2.fits

-diag_82.dat

-expose_ll.fits

-expose_2.fits

-mask_82.fits

-b_ll.fits

-b.2.fits

-back_ll.fits

-back_2.fits

-diag_82.dat

_-expose_l l,fits

_--expose_2.fits

-rnask_82.fits

-b_ll.fits

-b_2.fits

-back_ll.fits

-back_2.fits

-diag_82.dat

-expose_l l.fits

---expose_2.fits

-mask_82.fits

Fig. D. 1.-- A sample directory tree for a mosaic of the Rli and R2 bands made from two exposures.

Exposure rp900002 contains two subexposures, rp900002n00 and rp900002n01. The image files in

the directory rpg00002 contain the sums of the images in the subdirectories, and will be used to

create the mosaic.

191
to 2.5 PSF radii to determine the statistical significance. If the statistical significance is greater

than that set by the user (typically 4a, with no

minimum number of counts needed to form a

source), then a point source is "detected".

The DETECT routine automatically forms a

image mask eliminating the point sources. The

mask files created at this point will be called

the "local" mask files as they are based on local

detections using the individual images.

The suggested parameter settings are given

in Table D.1. Point source detection is usually

executed on wide-band images in order to have

the highest count rates, and thus the lowest detection limit. The two PSPC bands usually used

are R1L2 and R47.

Useful UNIX shell script: multi.detect.

3.) Determine the Point Source Detection Limits. Produce a map of the pointsource detection limit over the mosaic. This

task is accomplished with the FORTRAN routine PS_LIMIT. Nora bene: To determine the

point-source limit at a given position the routine uses the mean count rate for the pointing. If

there are a large number of small point sources,

the mean count rate may be artificially raised,

thus making the calculated point source detection limit somewhat worse than the true point

source detection limit. Unless there are large

bright features in the mosaic, we recommend not

using a mask for this step.

4.) Merge the Point Source Lists. Reject false detections and low quality duplicate

detections, where the quality is based upon

the point source detection limit at the position

of the point source. The FORTRAN routine

MERGE_DETECT reads the point source detection lists for all of the exposures and rejects

detections on the basis of the point source detection map produced in step 3. Accepted detections are kept in a "master list",

sources_accept_rim.fits.

Nota bene: Check the images of the "rejected"

point sources for possible problems (see §2.5).

5.) Mask Out the Point Sources. Create

masks for all of the exposures using the "master

list" of detections, using the FORTRAN routine MASK. Specify zero as the minimum count

rate (for reasons that will soon become apparent). The masks formed in this step will be

called "global" mask files since they use information from the entire mosaic. If the region

includes bright areas of extended emission, they

should be masked out for the determination of

the relative LTE offsets. The FORTRAN routine MAKEMASK allows the user to produce

modified masks.

6.) Cast Images into the Mosaic Projection. Map the counts, exposures, and background counts from the projection of the individual exposures into the mosaic projection

using the FORTRAN routine MERGE_RAW 3

The units of the output images will be counts,

seconds, and counts for the count, exposure, and

model background mosaics, respectively.

If you do not want point sources to appear

in your final mosaic, you should use the global

mask files when you recreate the count, exposure, and background mosaic. If you are in pursuit of a pretty picture, and point sources are

not important one way or the other, then it is

a good idea to remove them. Due to the position dependent PSF, point sources in the regions

of overlap can develop a complicated core-halo

structure which, since is does not occur everywhere in the mosaic, can be distracting and/or

confusing.

Useful UNIX shell script: merge.comp.

7.) Determine the LTE Offsets. Although much of this document is devoted to the

measurement of the LTE offsets, the same percentage of time need not be devoted to the actual process of calculation. How the offsets are

determined depends upon the ROSAT detector

used.

PSPC (if the RASS is available): For

programming reasons, the process of determining the offsets for the PSPC is divided into several steps.

7a.) Extract Sources from the BSC. As

described above, when one compares a pointing to the RASS, it is essential to have removed

point sources from the pointing to the same level

as have been removed from the RASS. Create a

list of the sources removed from the RASS by extracting those sources from the ROSAT Bright

Source Catalogue using the FORTRAN routine

GET_RASS_BSC.

3This is a change in name since the previous version due

to UNIX having a merge command.

192
TABLED.1

SUGGESTED DETECT/MASK PARAMETERS

Detector Band Significance Count Exclusion Minimum

Rate Radius Radius

(Sigma) (cnts/s) (PSF Rad.) (Pixels)

PSPC RIL2 4 0 1.5 8

PSPC R.47 4 0 i .5 1

HRI 4 0 2 3

BSC RI2 0.025 20 14

BSC R47 0.04 20 14

7b.) Mask the BSC Sources. To form

image masks removing the BSC sources, use

the FORTRAN routine MASK. The routine will

query for the minimum count rate for sources

to be removed, these values are listed in Table 3.1 for various bands. By specifying a nonzero rate, the routine knows to put the masks

in files named bsc_mask_n.fits rather than overwriting the standard mask files.

7e.) Calculate the Direct-Comparison

(DC) LTE Offsets. For each energy band

compare each pointing to the RASS and calculate the mean difference, as described in § 2.1.

This task is accomplished by the FORTRAN

routine OFFSET_DC. This routine produces the

file offset_dc.n.out, which is used in the next

step. The routine queries for a mask file, use

the masks produced in the previous step. Nora

bene: This routine produces many small FITS

files in the data subdirectories (pntd_rate_n.fits,

pntd_stg_n.fits, pntd_mask_n.fits, rass_rate..n.]its,

and rass._ig_n.fits for each band for each pointing) which can consume vast amounts of space.

They can be deleted after being used for diagnostic purposes.

7d.) Calculate the Weighted Comparison (WC) LTE Offsets. For each energy

band, determine the count rates in the exposure

overlap regions, and solve for the LTE offsets

as described in §2.3. This is accomplished using the FORTRAN routine OFFSET. The routine will query for image masks, use the standard masks (mask_n.fits), not the masks using

only the BSC point sources. The results will be

stored in ASCII files.

Useful UNIX shell script: solve.offset.

PSPC (if the RASS is unavailable): The

process of determining the LTE offsets for the

PSPC when the RASS is not available is much

more simple, though one should be cautious of

the results (as per § 2.2 above).

7.) Calculate the Self-Consistent (SC)

LTE Offsets. This task is accomplished with

the FORTRAN routine OFFSET_SC. Once the

the program has finished, rename the file off"

set_sc_n.out to match the output of the FORTRAN routine OFFSET, offset_n.out.

HILI: The process of determining the LTE

offsets for the HILl is much simpler as there is no

good way of direct comparison with the RASS,

which was created with the PSPC.

7.) Calculate the Self-Consistent (SC)

LTE Offsets. This task is accomplished with

the FORTRAN routine OFFSET_SC. Once the

the program has finished, rename the offset..sc_n.out

file to match the output of the FORTRAN routine OFFSET, o ffset_n.out.

Useful UNIX shell script: solve.hri.off.

8.) Create Mosaic of LTE Offsets Create

a mosaic image of the LTE offset counts using

the FORTRAN routine MERGE_OFFSET. The

units of this mosaic will be counts.

Useful UNIX shell script: merge.offset.

9.) Calculate Rate Calculate the final rate

and significance images (units will be counts

s-1 arcmin -2) using the FORTRAN routine

MERGE_/MAGE. As with the FINAL_IMAGE

routine, images can be made from a range of energy bands as well as from a single energy band.

Useful UNIX shell script: merge.image.

D.3.1. Further Refinements

Those who would apply the mosaicking software to fields at high galactic latitude or fields

193
withlowabsorbingcolumnswillfindafewmore

complications.

Extended Objects such as galaxy clusters may

or may not have been detected by the point

source detection algorithm, but in any case, will

not be sufficiently masked. Many of these regions will only be recognizable in the final rate

images. Therefore, one must make the mosaic interatively. Once the rate image has been

created, determine the position of the centers

of these objects and the radius of the region

which they effect, both measures typically being in mosaic pixels. Convert the positions to

decimal degrees in whatever coordinate system

you are using, and create a detection file of the

same format as that produced by DETECT or

MERGE.DETECT. Then use the MASK routine to create a new set of extended source masks

that include only these objects. Remember, the

MASK routine expects a radius in pixels, not the

mosaic pixels, but pixels of the original pointings (14:947). The extended source masks can

then be multiplied by the regular masks, using

ROARITH, to get the mask to be used when calculating offsets. It may also be a good idea to

use the extend source mask files to improve the

bsc_mask_nn.fits masks. Then one must remerge

the components and recalculate the offsets.

Given that you will have created a rate mosaic

to detect the extended sources, one may as well

improved the point source removal before remerging the components. Use BETTER_LIMIT

to calculate a better value for the point source

detection limit, and use its output to re-run

MERGE.DETECT. (You will have to do some

renaming of files so that M ERG E_DETECT will

be able to use the output from BETTER_LIMIT.)

D.5. Program Descriptions

The programs in this section are used to mosaic individual observations (processed by the

software in § 3) into larger area maps. They

require that the names for the fits files produced by the reduction software to remain unchanged and that the files are found in separate directories. Specifically, for the PSPC, they

use the b_n.fits, expose_n.fits, back_n.fits, and

mask_re.fits files where n is the band number

and m is a band or band range (e.g., b_5.fits,

expose_5.fits, back_5.fits, and mask__ 7.fits). For

the HRI, they use the counts.fits, expose.fits,

back.fits, and mask.fits files.

The output files for the mosaic programs are

in the zenith equal-area projection (ZEA). For

small regions, there is little difference between

the ZEA projection and the tangential projection (-_ 1% at a radius of 9*). If your favorite

image display package doesn't support ZEA, the

header information can be changed from ZEA to

TAN.

The programs are listed in alphabetical order.

D.4. A Final Word

The software described in this document was

developed over several years by S. L. Snowden

in support of the ROSAT project. Much of the

mosaicking software has been modified by K D.

Kuntz in the process of his dissertation work.

Although a great deal of effort has been expended to make sure that the software behaves

as described, the software changed a great deal

in the process of writing this document. Therefore, do not be surprised if it does not act exactly

as advertised.

194
TABLE D.2

MOSAICKING PROGRAMS

Program Description

PS_LIMIT

MERGE_DETECT

MASK

MERGE_COMP

OFFSET_DC

OFFSET_SC

OFFSET

MERGE_OFF

MERGE_IMAGE

ADAPT_MERGE

determines the point source detection limit over the mosaic

merges point source detection files for all pointings

masks out the point sources from individual exposures

merges event, exposure, or background maps into a mosaic

determines the mean difference between RASS

and pointing count rates

determines the relative offsets in the zero levels

by comparing the overlap regions between pointings (Obsolete)

combines the output of OFFSET_DC and the method of

OFFSET.SC to determine absolute LTE offsets

uses OFFSET output to create an offset count mosaic

same as FINAL-IMAGE but runs on mosaic images

same as ADAPT but runs on mosaic images

I.q5
TABLE D.3

FILES PRODUCED BY MOSAICKING PROGRAMS

Name Producing Can Be

Program Deleted

Main Directory

back_n.fits, back.fits MERGE.COMP

counts_n.fits, counts.fits MERGE_COMP

covariance.n.fits OFFSET x/

covarianceac_n.fits OFFSET.SC _/

dir.dat

expose_n.fits, expose.fits MERGE_COMP

mask.n.fits, m,_sk.fits MERGE.IMAGE

otf-n.fits MERGE.OFFSET

off'set .dc.n.out OFFSET.DC x/

offsct.dc.n.dat OFFSET_DC _/

offset .n .out OFFSET

offset.n.dat OFFSET x/

off'set .sc.n .out OFFSET.SC

offset .sc.n .dat OFFSET.SC _/

psexp.nm.fits PS_LIMIT

psBd.nm.fits PS.LIMIT

pslim.nm.fits PS_LIMIT

psrad.nm.fits PS.LIMIT _/

rate.n.fits MERGE_IMAGE

rej.xxx.acc.fits MERGE.DETECT _/

rej .xxx.rej.fits MERGE_DETECT _/

sig.n.fits MERGE.IMAGE

sources.accept .nm.dst MERGE.DETECT

sources.duplic.nm.dst MERGE.DETECT _/

sources.rejectmm.dat MERGE.DETECT

Individual Pointing Directories

diag.nm.dst DETECT

muk.rLm.fits DETECT

pntd.mssk.n.fits OFFSET_DC x/

pntd.rate_n.fits OFFSET.DC _/

pntd.sig.n.fits OFFSET.DC _/

rass.ra, t e.n.fits OFFSET_DC _/

raas.sig_n.fits OFFSET.DC x/

• A "_' indicates a file that is output for diagnostic purposes that need not be retained for continued analysis.

196
ADAPT_MERGE HRI, PSPC

To smooth the full 512 x 512 mosaicked images with an adaptive filtering algorithm. For each

unmasked pixel, the program will include neighboring pixels within a circle of increasing radius

until a selected number of counts is reached. The original pixe] is then given the average surface

brightness for the pixels within the circle.

Interactive Input

1) Coordinate control - n

The coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, or 3 for

galactic coordinates.

2) Number of counts for smoothing - n

This is the number of counts that the adaptive filter will use as its smoothing kernel.

3) Fractional limit for masking - f

ADAPT_MERGE will take the mask (either input file or exposure map), determine the average

non-zero value of the mask, and mask out any pixels where the mask value is less than the

average mask value multiplied by the fraction. Values of 0.02 for PSPC data and 0.2 for HRI

data are reasonable when using exposure maps.

4) Band range - nl, n_

The range of bands to be smoothed for the image. For the PSPC, e.g., enter 2 2 for the R2

band, 4 5 for the R45 (4a- keV) band, 8 2 for the R1L2 band, or 4 7 for the R47 band covering

bands R4, RS, R6, and RT. For the HRI enter 0 0.

5) Mask name - file name

Enter the file name of an image to use as a mask. Enter a carriage return to use the exposure

map.

File Input

1) Observation data set FITS files - none

2) Images - PSPC: counts_n.fits, back_n.fits, expose_n.fits, HRI: counts.fits, back.fits, expose.fits

These are the count, model background count, and exposure images. The PSPC files must

refer to individual bands. Therefore, MERGE, OFFSET, and MERGE_OFF must be run

separately for all desired bands before ADAPT_MERGE is run.

3) ASCII files -- none

ASCII File Output

None

Image Output

1) Smoothed image - PSPC: smooth.n.fits, HRI: smooth.fits

A FITS file of the smoothed image where for the PSPC output n is the band or band range.

The file contains the count rate in units of counts s-1 arcmin -2.

Screen Output

Program diagnostic information.

197
BETTER__LIMIT HRI, PSPC

To determine the point source detection limit at every point in a mosaic. For every pointing included in the mosaic, BETTER.LIMIT casts the exposure map into the mosaic coordinate system,

determines the size of the PSF at each pixel, and calculates the point source detection limit at that

pixel for that pointing. This is then compared to the point source detection limit at that pixel for

all other pointings, and the best (i.e., lowest) limit is chosen.

This routine is better than PSLIMIT, but requires a measurement of the true cosmic background count rate over the mosaic. Since PS_LIMIT uses the mean count rate (including point

sources) it will produce point source detection limits that are somewhat higher than the actual

limits, as can be seen by comparing Figure D.2 with Figure D.7. The output of MERGE_DETECT

is normally very little changed by use of BETTER_LIMIT rather than PSLIMIT.

0.015

!

' °°°°f

o.ooo + , R47

• I , , , , J , ,

0.000 0.005 0.010 0.015 0.000 0.005 0.010 0.015

Detection Limits Detection Limits

Fig. D.2.-- A comparison of measured count rates and the calculated point source detection limit

for BETTER_LIMIT. The few points that lie below the line (point sources that lie below the local

point source detection limit) are due to the local cosmic background rate being below the mean

cosmic background rate. See Figure C22 for a similar plot for PS_LIMIT.

Interactive Input

1) Coordinate control - n

Set the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, or 3 for

galactic coordinates. Should be identical to that used in MERGE_COMP.

2) Central coordinates - fl, f2

The central coordinates for the mosaic in decimal degrees in the coordinate system selected

in 1). Should be identical to those used in MERGE_COMP.

3) Scale factor for mosaic - f

The pixel size for the mosaic in arc minutes. Should be identical to that used in MERGE_COMP.

4) Radius limit - f

The radius of the region in the individual pointings, in arc minutes, to be considered (applied

to the sky image rather than in detector coordinates). This value should include the entire

field as the extreme edge of one field may still be better than the center of another. For the

PSPC a limit of 6ff will include the entire field; for the HRI xx _ Note that this is not the

198
samevalue used for casting the mosaics.

5) Band range - nl, n2

The range of bands to be combined for the final image (i.e., "2 2" will produce band R2, "4

5" will produce band R45 as a 43-keV band, and "8 2" will produce band RIL2, "4 7" will

produce band R47 combining the bands R4, R5, R6, and R7). Enter "0 0" for Hill data.

6) Cosmic background count rate - f

The cosmic background rate in counts s-1 arcmin-_.

7) Statistical significance and count-rate threshold - fl, f2

The statistical significance can be entered either as the Poisson (i.e., 3_r = 0.9973002) probability or the value in sigmas (allowed values: 1., 2., 2.5, 3., 3.5, 4., 4.5, 5., 5.5). The count-rate

threshold in units of counts s-1 may be set to zero.

8) Mask file name for masking of the data - filename

The data from each pointing will be masked using files with the input file name in the pointing

directory. If no additional masking is desired, enter a carriage return.

File Input

1) Images - PSPC: expose_n.fits and a mask if desired from each directory listed in dir.dat, HRI:

expose.fits and a mask if desired from each directory listed in dir.dat

2) ASCII files - dir.dat or dir.add.dat

The file dir.dat contains a list of directory path names for the various observations to be

mosaicked. This allows BETTER_LIMIT to find the observations to be mosaicked from the

directory where it is being run.

ASCII File Output

None

Image File Output

1) The FITS file pslim_n.flts, the best point source detection limit at each pixel in the mosaic. The

units are counts s-1.

2) The FITS file psfld_n.fits, the number (line number in the dir.dat file) of the pointing containing

the best point source detection limit at each pixel in the mosaic.

3) The FITS file psexp_n.fits, the length of the exposure corresponding to the best point source

detection limit.

4) The FITS file psrad_n.fits, the PSF radius corresponding to the best point souce detection limit.

The units are arcminutes.

Screen Output

Program diagnostic output.

199
GET_RASS_BSC PSPC

To extract from the ROSAT Bright Source Catalogue (BSC) a list of all of the sources in a mosaic.

Interactive Input

1) Band range- nl, n2

The range of bands from which the point source count rate is to be calculated

2) Coordinate control- n

Set the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, 3 for

galactic coordinates.

3) Mosaic image name - file name

The name of a mosaic file image (can be counts, exposure, background, etc., so long as the

header has the correct information for the mosaic.)

File Input

1) Images- as selected above.

ASCII File Output

1) bsc_eztract_n.dat- A list of point sources in nearly the same format as produced by the DETECT

routine.

Image File Output

None.

Screen Output

Program diagnostic output.

"' ' "u ..... _ ......... ' ' '_ ...... 'u' 'O' 'D" ' ' .........

400 _-- O O O D

. D [3 rl

- O_ I-I

350 --

300 _ [] rl

_' eso

200 - "k- D o _. o

150 lD [][]

IOOL ........ ,.,'.,......... I ......... i ......... i ......... ).-

0 100 200 300 400 500

Pixels

IFig. D.3.-- Detected point sources in a mosaic compared to point sources in the Bright Source

Catalogue. The detections were made in the RIL2 band. D: BSC detections over 0.025 counts

s-1, x: detections from individual pointings over 0.025 counts s -1, +: detections from individual pointings under 0.025 counts s -l. Quasi-circular and irregular regions are previous removed

sou rces.

200
GET__RASS_MOSAIC PSPC

To determine the offset between a mosaic and the RASS. This program determines which RASS

panel has its center closest to the mosaic, and recasts the mosaic into the RASS coordinate system.

For all pixels occupied by the mosaic, the program determines the mean and standard deviation

of the count rate of the RASS. All RASS pixels exceding a user specified number of standard

deviations from the mean are discarded. The offset,

offset = mosaic- RASS (D.3)

is calculated from the remaining pixels.

Interactive Input

1) Band to be compared - n

la) R1L/R1 band ratio - f

If the band to be compared is R1L, then the R1L/R1 band ratio is needed to convert the

RASS flux to the same band as the mosaic.

2) Sigma limit - f

The value in standard deviations over which pixels are discarded.

3) Coordinate control- n

Set the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, 3 for

galactic coordinates.

File Input

l) Images - counts_n.fits, back_n.fits, expose_n.fits for the moasic, 9z_zz_wrn.fits and g_sn.fits

for the RASS.

ASCII File Output

None

Image File Output

1) mosc_rate_n.fits

A 512x512 FITS image in the RASS coordinate system containing the mosaic count rate.

2) mosc_sig_n.fits

A 512x512 FITS image in the RASS coordinate system containing the mosaic count rate

uncertainty.

3) mosc.mask_n.fits

A 512x512 FITS image in the RASS coordinate system containing the mosaic mask.

Screen Output

Diagnostic output:

X Limizs : 234 287

Y Limits : 129 161

The extent in RASS pixels of the mosaic.

Number of pixels in field: 1062.000

Number of pixels discarded: 2.000000

The number of RASS pixels in the mosaic, and the number of those discarded for being too far

from the mean.

Offse¢ : 53.01571

201
+/- : 2.079403

RASS : 189.9938

+/- " 1.793882

Pointed: 213.7634

+/- : 0.4400081

RASS rate has been corrected from

band R1 to band RIL using RIL/RI= 0.8227000

The mosaic - RASS, RASS, and mosaic respectively.

202
GET_RASS_POINT PSPC

To determine the offset between a single pointing and the RASS. This program determines which

RASSpanel has its center closest to the pointing, and recasts the pointing into the RASS coordinate

system. For all pixels occupied by the pointing, the program determines the mean and standard

deviation of the count rate of the RASS. All RASS pixels exceding a user specified number of

standard deviations from the mean are discarded. The offset,

offset = pointing - RASS (D.4)

is calculated from the remaining pixels.

Interactive Input

1) Band to be compared - n

la) R1L/R1 band ratio - f

If the band to be compared is R1L, then the R1L/R1 band ratio is needed to convert the

RASS fiux to the same band as the pointing.

2) Sigma limit - f

The value in standard deviations over which pixels are discarded.

3) Mask file name - file name

The data will be masked using files with the input file name in the individual directories. The

mask files must be in the same directories as the data files. If no additional masking is desired,

enter a carriage return.

File Input

1) Images - b_n.fits, back_n.fits, expose_n.fits for the moasic, gxxz'xxz'_.fits and gz'xxz'z'xsn.fits for

the RASS.

ASCII File Output

None

Image File Output

1) pntd_rate_n.lits

A 41 × 41 FITS image in the RASS coordinate system containing the mosaic count rate.

2) pntd_sig_nfits

A 41 xztl FITS image in the RASS coordinate system containing the mosaic count rate uncertainty.

3) pntd_mask_n.fits

A 41×41 FITS image in the RASS coordinate system containing the mosaic mask.

4) rass_rate_n fits

A 41x41 FITS image in the RASS coordinate system containing the mosaic mask.

5) rass._sig_n.fits

A 41×41 FITS image in the RASS coordinate system containing the mosaic mask.

It should be noted that for pointing band RIL, the RASS images will be band R1, will not be

scaled to band R1 L, and will be named rass..xxxx_l.fits.

Screen Output

Diagnostic output:

X Limits: 234 287

Y Limits: 129 161

The extent in RASS pixels of the mosaic.

203
Number of pixels in field:

Number of pixels discarded:

96.00000

O.O000000E+O0

The number of RASS pixels in the mosaic, and the number of those discarded for being too far

from the mean.

Offset : 21.96925

+/- : 6.681984

RASS : 198.4002

+/- : 5.841153

Pointed: 172.1212

+/- : 1.135728

RASS rate has been corrected from

band El to band RIL using RIL/RI= 0.8227000

The mosaic - RASS, RASS, and mosaic respectively.

204
MASK PSPC

To use the list of "accepted" detections to mask point sources from pointings.

Interactive Input

1) Band range - nl, n2

The range of bands over which the point sources were detected.

2) Point source list file - file name

The name of the file containing the point sources to be masked, usually the file sources_accept_n.dat.

3) Masking radius - f

The radius (in PSF radii) to be removed around point sources.

4) Minimum mask size - n

The minimum radius, in pixels, to be removed around point sources.

5) Minimum count rate - f

The minimum count rate to be accepted as a source. This value is of particular value when

creating masks from source lists extracted from the ROSA T Bright Source Catalogue. If the

minimum count rate is not zero, the output mask will be in bsc_mask_n.fits rather than in

mask_n.fits.

File Input

I) Fixed input - the sources_accept_n.dat file produced by MERGE_DETECT.

ASCII Output

None

Image Output

A mask files, mask_n.fits or, if the minimum count rate is not set to zero, bsc_mask_n.fits.

Screen Output

For each source masked from the pointing, a line such as

1 65.67660 16.05800 381 247 14

will appear. It contains the number of the field in which the point source had its best detection,

the R.A. and dec. of the point sources, the X and Y pixel position of the source in this pointing,

and the mask radius in pixels.

205
MERGE_COMP HRI, PSPC

To merge the various component (count, background, and exposure) images from separate observations of different pointings into a large-area mosaic.

Interactive Input

1) Coordinate control - n

Set'the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, or 3 for

galactic coordinates.

2) Central coordinates - fl, f2

The central coordinates for the mosaic in decimal degrees in the coordinate system selected

in 1).

3) Scale factor for mosaic - jr

The pixel size for the mosaic in arc minutes.

4) Radius limit - st

The radius of the region in the individual pointings, in arc minutes, to be cast into the mosaic

(applied to the sky image rather than in detector coordinates). For the PSPC a limit of

53 t serves reasonably well to exclude the worst problems caused by the large point response

function at large off-axis angles. For the HRI typically the entire field can be used.

5) Band to be cast - n

The program will cast only one band at a time. Enter 0 for HRI data.

6) Component to be cast - n

The program will cast only one component at a time. Enter 1 for counts, 2 for exposure, or

3 for background counts. For the PSPC, the program FINAL_IMAGE must be run on the

individual bands to create the back_milts files for the background counts. All files to be cast

must have the same names as they were labelled when generated, i.e., b_n.fits, expose_milts,

and back_milts. Counts.fits, expose.fits, and back.fits for HRI data.

7) Mask file name - file name

The data will be masked using files with the input file name in the individual directories. The

mask files must be in the same directories as the data files. If no additional masking is desired,

enter a carriage return.

8) Add flag - n

Enter 1 to add to an existing image, 0 to cast a new image. If data are to be added to an

existing image then MERGE_COMP will query for the file name. If data are to be added to

an existing image the new directories must be listed in the file dir_add.dat.

File Input

1) Images - PSPC: b_n.fits or back_n.fits or expose_n.fits and a mask if desired from each directory

listed in dir.dat. HPd: counts.fits or back.fits or expose.fits and a mask if desired from each

directory listed in dir.dat

The program will query for the component to be mosaicked and a mask name.

2) ASCII files - dmdat or dzr_add.dat

The file dir.dat contains a list of directory path names for the various observations to be

mosaicked. This allows MERGE_COMP to find the observations to be mosaicked from the

directory where it is being run.

ASCII File Output

None

Image Output

1) Mosaicked image- PSPC: counts_n.fits or ezpose_n.fits or back_n.fits, HRI: counts.fits or ezpose fits or back fits

206
Theoutputfiledependsonthechosencomponent.NotethatforPSPCdataexpose_n.fits and

back_n.fitshavethesamenamesasthefilesfortheindividualobservations.FortheHRIall

fileshavethesamenames.

ScreenOutput

Programdiagnosticinformation.

207
MERGE__DETECT HRI, PSPC

To merge the lists of point source detections made in the separate pointings, discarding duplicate

detections.

Method

This process is accomplished in three steps. First, every point source detection is checked to

see if it was made in the pointing which has the best point source detection limit at the position

of the detection. If it is, the detection is "accepted" and is output to the file sources_accept_n.dat,

if not, the detection is rejected. Second, the list of rejects is compared to the list of accepted

detections. If the reject falls within the "match radius" (a user defined multiple of the PSF radius)

of an accepted detection (or an accepted detection falls with that same multiple of the PSF radius

of the rejected detection) the detection is considered a "duplicate" detection and is output to the

file sources_duplic_n.dat. Third, rejected detections that do not "duplicate" accepted detections

are output to the file sources_re3ect_n.dat. Figure D.4 shows the distribution of the three types

of detections over a mosaic. One can see that a "match radius" of 1.5 PSF provides reasonable

matches.

400

350

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0

:_ 200

150

100

0

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go

I O0 200 300 400 500

Mosaic X (pixels)

Fig. D.4.-- Detected point sources in a mosaic. Solid: accepted detections, Light: duplicate

detections, Dashed: rejected detections. In all cases, the 1.5 PSF radius is drawn. The mosaic

scale is 1.% per pixel. Instances of two overlapping accepted detections are due to both detections

being made in the same pointing.

For each rejected detection a 201 x 201 pixel region surrounding the rejected detection position

is extracted and placed in the file re3.xxz_rej.fits, where "xxx" is the line number of the detection in

the sources_re3ect_n.dat file. Similarly, a 201 x 201 pixel region surrounding the rejected detection

position is extracted from the pointing with the best point source detection limit (at that position)

and placed in the file rej.xxz_acc.fits. The user should examine the rejected detections carefully.

As described in the text, some "rejected" sources may be unmasked hot-spots which should be

removed before the mosaic is created. Also described in the text is a problem due to a point source

208
fallingontheboundaryof twopointings.FigureD.5showssomeextractedregions,for boththe

rejecteddetections(left)andthesameregioninthe "bestdetection"pointing(right),similarto

Figure3.13.Notethat thetop rowisthereverseof thebottomrow. In thiscase,thedetection

ineachpointingfell intheregionforwhichtheotherpointingwasto beaccepted.Thisproblem

doesnotoccuroften,andiseasilynoticedina briefexaminationof the"rejected"sources.

©

©

©

,%

h, _ .

©

O

Fig. D.5.-- Comparison of "rejected" sources to better exposures of the same regions. Left two

panels: four "rejected detections". The leftmost panel is overplotted with the local PSF. Right two

panels: the same positions (R.A., dec.) in the pointings with the best point source detection limit

at those positions. The right-most panel is overplotted with the focal PSF. Each panel has been

scaled by the extremes within that panel. Note that the bottom row is the reverse of the top row.

Interactive Input

1) Band range - nl, n2

The range of bands over which point sources were detected.

2) Coordinate control - n

Set the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, or 3 for

galactic coordinates.

3) PSF scale - f

The multiple of PSF radii within which two sources must reside in order to be deemed the

same object.

4) Postage stamp flag - n

Enter 1 if the extracted image files re).x__rej.fits (containing the region around the rejected

detection xxx extracted from the pointing in which the detect was made) raend re)_z_z_acc.fits

(containing the region around the rejected detection xxx _xtracted from the region with the

best detection limit) are to be formed, otherwise, 0.

File Input

1) Images - psfld_n.fits and psrad_n.fits,

output by PS_LIMIT.

2) ASCII files - dzr.dat

The file dir.dat contains a list of directory path names for the various observations to be

mosaicked. This allows MERGE_DETECT to find the observations to be compared from the

209
directorywhereit isbeingrun.

3) ASCII files - dzag_n.dat, output by DETECT in each pointing directory.

ASCII File Output

1) sources_accept_n.dat- a list of "accepted" detections.

2) sources_duplic_n.dat- a list of "duplicate" detections.

3) sources_re3ect_n.dat - a list of rejected detections that were not duplicates.

Image File Output

1) FITS files - rej.z__rej.fits, a 201 × 201 pixel image surounding the rejected detection extracted from the pointing in which the rejected detection was made. The number zzz is the

corresponding line number in sources_reject_n.fits.

2) FITS files - rej_x__acc.fits, a 201 x 201 pixel image surounding the rejected detection extracted

from the pointing which has the best point source detection at the position of the rejected

detection. The number x_ is the corresponding line number in sources_reject.n.fits.

Screen Output

Program diagnostic information.

TABLE D.4

COLUMNS IN DETECT, MERGE_DETECT, AND GET_RASS_BSC OUTPUT

Column DETECT MERGE.DETECT GET.P, ASS laSC Format

1 detection number field number index number 16

2 image X (pixel) mosaic X (pixel) mosaic X (pixel) 16

3 image Y (pixel) mosaic Y (pixel) mosaic Y (pixel) 16

4 R.A (decimai degrees) B.A (decimal degrees) R.A. (decimal degrees) F9.4

5 Dec (decimal degrees) Dec (decimal Degrees) Dec. (decimal Degrees) F9.4

6 Count Rate (cnts/s) Count Rate (cnts/s) Count Rate (cnts/s) F12.7

7 Significance Significance BSC Probability/10 4 F13.8

210
MERGE_IMAGE HRI, PSPC

To combine the count, modeled background count, modeled offset count, exposure map, and mask

(usually just the exposure map) into final surface-brightness (count rate and count-rate uncertainty) images. Nearly the same as FINAL_IMAGE but runs on mosaic images.

Interactive Input

1) Coordinate control - n

Set the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, or 3 for

galactic coordinates.

2) Band range - hi, n2

The range of bands to be combined for the final image (i.e., "2 2" will produce band R2, "4

5" will produce band R45 as a _ keV band, and "8 2" will produce band RIL2, "4 7" will

produce band R47 combining the bands R4, R5, R6, and R7). Enter "0 0" for HRI data.

3) Fractional limit for masking - f

ADAPT will take the mask (either input file or exposure map), determine the average nonzero value of the mask, and mask out any pixels where the mask value is less than the average

mask value multiplied by the fraction. A value of 0.02 for is reasonable in most cases.

4) Mask name - file name

Enter the file name of an image to use as a mask. Enter a carriage return to use the exposure

map.

File Input

1) Images - PSPC: counts_n.fits, back_n.fits, ezposc_n.fits, off_n.fits and a mask if desired, HRI:

counts.fits, back.fits, expose.fits, off.fits and a mask if desired

The program will query for the band range to be mosaicked and a mask name.

2) ASCII files - none

ASCII File Output

None

Image Output

1) Final mosaicked images - PSPC: rate_n.fits, sig_n.fits, mask_fits, HRI: rate.fits, sig.fits,

mask.fits

Fits files of the mosaicked count-rate, count-rate uncertainty, and an output mask. The PSPC

files rate_milts and sig.n.fits (where n is the band number or band range) and HRI files rate.fits

and sig.fits contain count rate and count-rate uncertainty (in units of counts s-1 arcmin-_)

for the full 512 x 512 image.

Screen Output

Program diagnostic information.

211
MERGE__IMAGE_BIN HRI, PSPC

To combine the count, modeled background count, modeled offset count, exposure map, and mask

(usually just the exposure map) into final surface-brightness (count rate and count-rate uncertainty) images. Nearly the same as MERGEAMAGE, but bins by a factor of 2, 4, 8, or 16.

Interactive Input

1) Coordinate control - n

Set the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, or 3 for

galactic coordinates.

2) Band range - nl, n2

The range of bands to be combined for the final image (i.e., "2 2" will produce band R2, "4

5" will produce band R45 as a _ keV band, and "8 2" will produce band R1L2, "4 7" will

produce band R47 combining the bands R4, R5, R6, and R7). Enter "0 0" for HRI data.

3) Binning parameter - n

The multiple by which to bin the output image.

4) Fractional limit for masking - f

ADAPT will take the mask (either input file or exposure map), determine the average nonzero value of the mask, and mask out any pixels where the mask value is less than the average

mask value multiplied by the fraction. A value of 0.02 for is reasonable in most cases.

5) Mask name - file name

Enter the file name of an image to use as a mask. Enter a carriage return to use the exposure

map.

File Input

1) Images - PSPC: counts_n.fits, back_n.fits, expose_n.fits, off_n.fits and a mask if desired, HRI:

counts.fits, back.fits, expose.fits, off.fits and a mask if desired

The program will query for the band range to be mosaicked and a mask name.

2) ASCII files - none

ASCII File Output

None

Image Output

1) Final mosaicked images - PSPC: bin_x_rate.nfits, bin.x_sig_n.fits, bin.x_mask_.fits, HRI:

bin..v_rate.fits, bin_x_sig.fits, bin_x_mask.fits,

FITS files of the mosaicked count-rate, count-rate uncertainty, and an output mask. The x is

the binning factor. The PSPC files btn.x.rate.n.fits and bin.x_sig_n.fits (where n is the band

number or band range)and HRI files bm_x_rate.fits and bin.x.sig.fits contain count rate and

count-rate uncertainty (in units of counts s-1 arcmin -2) for the full 512 × 512 image.

Screen Output

Program diagnostic information.

212
MERGE_OFF HRI, PSPC

Uses the output of OFFSET (offset_n.out for PSPC data for band n; offset.out for HRI data) and

exposure arrays to create offset count maps for the individual observations. When cast with the

count, background, and exposure maps, the offset count maps will produce a relatively uniform

zero level across the field.

Interactive Input

1) Coordinate control - n

Set the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, or 3 for

galactic coordinates. Should be identical to that used in MERGE.

2) Central coordinates - fl, .t"2

The central coordinates for the mosaic in decimal degrees in the coordinate system selected

in 1). Should be identical to those used in MERGE.

3) Scale factor for mosaic - f

The pixel size for the mosaic in arc minutes. Should be identical to that used in MERGE.

4) Radius limit - f

The radius of the region in the individual pointings, in arc minutes, to be cast into the mosaic

(applied to the sky image rather than in detector coordinates). For the PSPC a limit of

53' serves reasonably well to exclude the worst problems caused by the large point response

function at large off-axis angles. For the HRI typically the entire field can be used. Should

be identical to that used in MERGE.

5) Band to be cast - n

The program will cast only one band at a time. Enter 0 for HRI data.

6) Mask file name - file name

The data will be masked using files with the input file name in the individual directories. The

mask flies must be in the same directories as the data files. If no additional masking is desired,

enter a carriage return.

7) Add flag - n

Enter 1 to add to an existing image, 0 to cast a new image. If data are to be added to an

existing image then MERGE will query for the file name. If data are to be added to an existing

image the new directories must be listed in the file dir_add.dat.

File Input

1) Images - PSPC: expose_n.fits and a mask if desired from each directory listed in offset_n.dat,

HRI: expose.fits and a mask if desired from each directory listed in overlap,dat

The program will query for the mask name.

2) ASCII files - PSPC: offset_n.dat, HRI: offset.dat

The files offset_n.dat and overlap.dat are output from OFFSET and contain a list of directory

path names for the various observations to be mosaicked and offset count rates. This allows

MERGE_OFF to find the observations to be mosaicked from the directory where it is being

run and to scale the exposure maps for mosaicking.

ASCII File Output

None

Image Output

1) Mosaicked image- PSPC: off_n.fits, HRI: of].fits

Mosaic of the model offset counts

Screen Output

Program diagnostic information.

213
OFFSET HRI, PSPC

To calculate the LTE offsets for all of the pointings within a mosaic using both the self-consistent

and direct-comparison equations. The relative offsets between all adjacent pointings in their regions

of overlap are determined, the LTE offsets determined by OFFSET_DC are read in, and a singularvalue decomposition finds the optimal solution. The relative weight of the direct-comparison offsets

can be adjusted. This routine determines the absolute LTE offsets, unlike OFFSET.SC, which only

determines relative offsets. See § 2.3 for a full explanation.

The use of point-source masks is highly recommended as bright sources can contribute substantially to the mean count rate in a region of overlap. Bright extended regions may also need to be

masked.

Interactive Input

1) Coordinate control - n

Set the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, or 3 for

galactic coordinates.

2) Central coordinates for analysis - fl, f$

The central coordinates for the mosaic in decimal degrees.

3) Scale factor for mosaic - f

The pixel size for the mosaic in arc minutes. This can be the same size as for the mosaic.

4) Radius limit - f"

The radius of the region in the individual pointings, in arc minutes, to be cast into the mosaic

(applied to the sky image rather than in detector coordinates). If a point-source mask is used,

generally the entire field can be used (default to 100'). If no point-source mask is used, a

limit of 53' for PSPC data will again serve reasonably well (hopefully) to exclude the worst

problems caused by the large point response function at large off-axis angles. The entire field

can be used for the HPd.

5) Weightings - jr

The relative weightings to be given to the direct comparison equations.

6) Band to be analyzed - n

The program will analyze only one band at a time. Enter 0 for HPd data.

7) Mask file name - file name

The data will be masked using files with the input file name in the individual directories. The

mask files must be in the same directories as the data files. If no additional masking is desired,

enter a carriage return.

File Input

1) Images - PSPC: b_n.fits, back_n.fits, expose_n.fits and a mask if desired from each directory

listed in dir.dat, HPd: counts.fits, back.fits, expose.fits and a mask if desired from each directory

listed in dir.dat

The program will query for the band to be analyzed and a mask name.

2) ASCII files - dir.dat or dir_add.dat

The file dir.dat contains a list of directory path names for the various observations to be

mosaicked. This allows OFFSET to find the observations to be compared from the directory

where it is being run.

3) ASCII file - offset_tic.out This file contains the out put of OFFSET_DC, the difference in count

rate between each pointing and the corresponding region of the RASS. (This program does do

a consistency check between the directories listed in d_r.dat and those listed in ogset_dc.out.)

ASCII File Output

1) File offset_n.dat (o_set.dat for HPJ data) contains the record of the individual overlap region

214
offsetsplusthefinaloffsetvalues.

2) Fileoffset_n.out (offset.out for HRI data) contains the directory name of each pointing, the

central coordinates of the individual observations, the LTE offset solution for each pointing,

and the LTE offset uncertainties. This file is required as input to the program MERGE_OFF.

Image Output

l) FITS file covariance_n.fits which contains the covariance matrix for the solution.

Screen Output

Program diagnostic information.

TABLE D.5

COLUMNS IN OFFSET(_XX) OUTPUT

Column Quantity Units Format

1 Directory name a50

2 R.A. decimal degrees f10.2

3 dec. decimal degrees f10.2

4 LTE Offset counts s -1 arcmin -2 f10.2

5 LTE Offset uncertainty counts s -1 arcmm -2 f10.2

215
OFFSET_DC HRI, PSPC

To determine the LTE offset for all of the pointings within a mosaic by calculating the mean

difference between each pointing and the same region in the RASS. The point source mask is used

to remove the point sources from the pointed images. This removal is important; point sources

can contribute significantly to the mean pointed count rate. As discussed in § 2.1, this method

of measuring LTE offset can produce problems. We recommend that the LTE offsets from this

routine not be used by themselves, but be combined with the self-consistent solution by using the

OFFSET routine.

Interactive Input

1) Band to be compared - n

la) R1L/R1 band ratio - f

If the band to be compared is R1L, then the R1L/R1 band ratio is needed to convert the

RASS flux to the same band as the mosaic.

2) Sigma limit - f

The value in standard deviations over which pixels are discarded.

3) Coordinate control - n

Set the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, or 3 for

galactic coordinates.

4) Central coordinates for analysis - fl, f2

The central coordinates for the mosaic in decimal degrees.

5) Mask file name - file name

The data will be masked using files with the input file name in the individual directories. The

mask files must be in the same directories as the data files. If no additional masking is desired,

enter a carriage return.

ASCII File Output

1) File offset_dc_n.dat (offset_dc.dat for HILl data) contains the record of the individual overlap

region offsets plus the final offset values.

2) File offset_dc_n, out (offset_dc. out for HRI data) contains the directory name of each pointing, the

central coordinates of the individual observations, the LTE offset solution for each pointing,

and the LTE offset uncertainties. This file is required as input to the program MERGE_OFF.

Image File Output

1) pntd_rate.n.fits

A 41×41 FITS image in the RASS coordinate system containing the mosaic count rate.

2) pntd_sig_n.fits

A 41x41 FITS image in the RASS coordinate system containing the mosaic count rate uncertainty.

3) pntd_mask_n.fits

A 41x41 FITS image in the RASS coordinate system containing the mosaic mask.

4) rass_rat e_n. fits

A 41x41 FITS image in the RASS coordinate system containing the mosaic mask.

5) rass..sig_n.fits

A 41x41 FITS image in the RASS coordinate system containing the mosaic mask.

It should be noted that for pointing band R1L, the RASS images will be band R1, will not be

scaled to band R1L, and will be named rass_xxxx_l.fits.

Screen Output

216
Programdiagnosticoutput.

217
OFFSET_SC HRI, PSPC

To determine the relative offsets in the zero levels of the observations to be merged. While the

resulting zero level after the observations have been merged may still include an additive offset

from true zero, it will be considerably more uniform across the field. OFFSET_SC determines the

relative offsets between all adjacent pointings in their region of overlap, and then uses a single-value

deconvolution algorithm to solve for the best-fit group of offsets. The use of point-source masks

is highly recommended as bright sources can mess up the works. Bright extended regions may

also need to be masked. Unless there is some good reason to use this program (such as using HRI

images), the OFFSET_DC/OFFSET combination should be used instead.

Interactive Input

1) Coordinate control - n

Set the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, or 3 for

galactic coordinates.

2) Central coordinates for analysis - fI, f_

The central coordinates for the mosaic in decimal degrees in the equatorial coordinate system.

3) Scale factor for mosaic - f

The pixel size for the mosaic in arc minutes. This can be the same size as for the mosaic.

4) Radius limit - f

The radius of the region in the individual pointings, in arc minutes, to be cast into the mosaic

(applied to the sky image rather than in detector coordinates). If a point-source mask is used,

generally the entire field can be used (default to 100'). If no point-source mask is used, a

limit of 53 _ for PSPC data will again serve reasonably well (hopefully) to exclude the worst

problems caused by the large point response function at large off-axis angles. The entire field

can be used for the HRI.

5) Band to be analyzed - n

The program will analyze only one band at a time. Enter 0 for HRI data.

6) Mask file name - file name

The data will be masked using files with the input file name in the individual directories. The

mask files must be in the same directories as the data files. If no additional masking is desired,

enter a carriage return.

File Input

1) Images - PSPC: b_n.fits, back_n.fits, ezpose_n.flts and a mask if desired from each directory

listed in dir.dat, HRI: counts.fits, back.fits, expose.fits and a mask if desired from each directory

listed in dir.dat

The program will query for the band to be analyzed and a mask name.

2) ASCII files - dir.dat or dir_add.dat

The file dir.dat contains a list of directory path names for the various observations to be

mosaicked. This allows OFFSET to find the observations to be compared from the directory

where it is being run.

ASCII File Output

l) Overlap diagnostic data - PSPC: offset_sc_n.dat, HRI: offset_sc.dat

Contains the record of the individual overlap region offsets plus the final offset values.

2) Overlap parameters - PSPC: offset_sc_n.dat, HRI: offset_sc.dat

Contains in addition to the final results, the directories and central coordinates of the individual observations to be used as input to the program MERGE_OFF.

Image Output

Nolle

218
ScreenOutput

Programdiagnosticinformation.

219
PS_LIMIT HRI, PSPC

To determine the point source detection limit at every point in a mosaic. For every pointing

included in the mosaic, PS_LIMIT casts the exposure map into the mosaic coordinate system,

determines the size of the PSF at each pixel, and calculates the point source detection limit at

that pixel for that pointing. This is then compared to the point source detection limit at that

pixel for all other pointings, and the best (i.e. lowest) limit is chosen.

The detection limit is calculated from the mean count rate, which includes the point sources.

Thus the mean background rate will be somewhat higher than the true background rate, thus

increasing the point source detection limit. For the purposes of MERGE_DETECT, this inaccuracy

in the true point source detection limit is inconsequential due MERGE_DETECT looking at relative

point source detection limits. One can imagine situations in which the inaccuracy could cause

difficulties for MERGE_DETECT, but in practice these difficulties have not arisen.

0.015

o 0.010

in

0.005

0.000

0.000

+ +++++++ ++ / i +++ _+ + /

+..+++ / +++ ++++++++>/+

+ ,, / t + _ + +/

++ +/ +

, +++/' J.-'_4 ._ _+ +++

+ _ _++.+++_ 4++++++,+ +_

+ R47

0.005 0.010 0.015 0.000 0.005 0.010 0.015

Detection Limits Detection Limits

Fig. D.6.-- A comparison of measured count rates and the calculated point source detection limit

for PS_LIMIT. The few points that lie below the line (point sources that lie below the local point

source detection limit) are due to the local cosmic background rate being below the mean cosmic

background rate. See Figure C18 for a similar plot from BETTER_LIMIT.

Interactive Input

l) Coordinate control - n

Set the coordinate system for the mosaic image. Enter 1 for ecliptic, 2 for equatorial, or 3 for

galactic coordinates. Should be identical to that used in MERGE.

2) Central coordinates - fl, f2

The central coordinates for the mosaic in decimal degrees in the coordinate system selected

in 1). Should be identical to those used in MERGE.

3) ScaLe factor for mosaic - f

The pixel size for the mosaic in arc minutes. Should be identical to that used in MERGE.

4) Radius limit - f

The radius of the region in the individual pointings, in arc minutes, to be considered (applied

to the sky image rather than in detector coordinates). This value should include the entire

field as the extreme edge of one field may still be better than the center of another. For the

220
PSPCa limit of 60'will includetheentirefield;fortheHRIxx' Notethat this isnot the

samevalueusedforcastingthemosaics.

5)Bandrange- nl, n2

The range of bands to be combined for the final image (i.e., "2 2" will produce band R2, "4

5" will produce band R45 as a 43-keV band, and "8 2" will produce band RIL2, "4 7" will

produce band R47 combining the bands R4, RS, R6, and R7). Enter "0 0" for HRI data.

6) Statistical significance and count-rate threshold - fl, f2

The statistical significance can be entered either as the Poisson (i.e., 3_r = 0.9973002) probability or the value in sigmas (allowed values: 1., 2., 2.5, 3., 3.5, 4., 4.5, 5., 5.5). The count-rate

threshold in units of counts s -x may be set to zero.

8) Mask file name for masking of the data - filename

The data from each pointing will be masked using files with the input file name in the pointing

directory. If no additional masking is desired, enter a carriage return.

File Input

1) Images - PSPC: expose_n.fits, counts_n.fits and a mask if desired from each directory listed in

dir.dat, HRI: expose.fits, counts.fits and a mask if desired from each directory listed in dir.dat

2) ASCII files - dir.dat or dir_add.dat

The file dir.dat contains a list of directory path names for the various observations to be

mosaicked. This allows PS_LIMIT to find the observations to be mosaicked from the directory

where it is being run.

ASCII File Output

None

Image File Output

1) The FITS file pslim_n.fits, the best point source detection limit at each pixel i_ the mosaic. The

units are counts s -x.

2) The FITS file psfld_n.fits, the number (line number in the dir.dat file) of the pointing containing

the best point source detection limit at each pixel in the mosaic.

3) The FITS file psezp_n.fits, the length of the exposure corresponding to the best point source

detection limit.

4) The FITS file psrad.n.fits, the PSF radius corresponding to the best point souce detection limit.

The units are arcminutes.

Screen Output

Program diagnostic output.

221
Fig.D.7.--Twoof theimagesoutputby PS_LIMIT: the field number image and the point source

detection limit image. Top. The image containing the number of the pointing with the best point

source detection limit. Bottom: The log10 of the point source detection limit. The grey scale runs

from-3.1 to-1.0.

222
ROARITH HRI, PSPC

To perform simple mathematical operations on images; adding two images together, subtracting

one from another, multiplying two images, or dividing one by another.

Interactive Input

1) The name of the first input FITS file.

2) The code of the operation to perform, '%" for addition, "-" for subtraction, "*" for multiplication, and "/" for division. 3) The name of the second input FITS file.

4) The name of the output FITS file.

File Input

1) Two 512x512 FITS images.

Image Output

The output FITS image.

223
D.6. UNIX Scripts

When making a mosaic, one does the same

thing to many different exposures (such as DETECT and MASK), or one does the same thing

mask

multi.detect

To execute DETECT on each individual exposure. This routine assumes that every directory named dir.dat contains an individual expofor each different band (such as MERGE_RAW sure. It removes any files created by DETECT

and OFFSET). This section describes some UNIX in the past and runs DETECT for the R1L2 and

script files that make these processes easier. In R47 bands.

order to understand the order in which they are

used, see § D.3.

Caveat: These scripts were created as they

were needed. They are not models of good shell

programming, nor are they anything that anyone with the slightest modicum of shell programming experience could not do, and better.

As a matter of organization, it is a good idea

to put subexposures into subdirectories of the

directory containing the summed exposure. A

model directory tree is shown in Figure D.1

Table D.6 contains definitions of parameters

required by these shell scripts. The parameters

required for any script can be found by typing

the name of the script without any arguments.

The scripts are listed in alphabetical order.

Their method of employ should be clear from

the prescription in § D.3.

add.obs

To add together two subexposures of the same

field. It is assumed that the names of the subdirectories are in the file dir.dat. The result is

put into the current directory.

Usage: add.obs

merge.comp

To create a mosaicked image of a given component (i.e., count, exposure, or background) for

all bands of interest (R1L, R2, R4, R5, R6, and

R7).

Usage" merge.comp coor long iati scal excl

mask comp

merge.image

To create a mosaicked image of the count rate

and the associated "uncertainty for each band of

interest (R1L, R2, R4, R5, R6, and R7), as well

as the combinations R1L2, R45, R67, and R47.

Usage: merge.image coor frac

merge.off

To create a mosaicked image of the LTE offsets far all bands of interest (R1L, R2, R4, R5,

R6, and RT).

Usage: merge.offset coor long lati scal excl

Usage: multi.detect sgma 1L2cr 47cr exrd

1L2rd 47rd 1L2cr 47cr

multi.mask

To execute MASK on each individual exposure. This routine assumes that every directory

named dmdat contains an individual exposure.

It removes any mask files (mask.xx.fits) created

by DETECT or MASK in the past and runs

MASK for the R1L2 and R47 bands. Note that

the name of the files containing the point source

lists should include whatever part of the path

is needed to point to these files from within the

directory of an individual pointing.

Usage: multi.mask 1L2sf 47sf exrd 1L2rd

47rd

multi.move

To move a file of one name, "oldname', to

a new name, "newname", in every directory

named ?p*. This is useful to move all of the

mask files produced by DETECT before creating new ones with MASK.

Usage: multi.move oldname newname

multi.remove

To remove a file named "name" from every

directory named in dtr.dat. This is useful for

correcting many mistakes, but has the potential

for making many more. To use wild-cards, enclose the file name in quotation marks.

Usage: multi.remove name

solve.off

To execute OFFSET and OFFSET_DC for

every band of interest (R1L, R2, R4, R5, R6,

RT).

Usage: solve.off long lati scal excl

224
TABLE D.6

MOSAICKING SHELL PARAMETERS

Name Definition and Units

47cr

I L2cr

47rd

IL2rd

47sf

IL2sf

coor

comp

excl

exrd

fr_c

lati

long

mask

scal

sgma

minimum valid countrate for a point source

in band R47 in units of counts/second

minimum valid countrate for a point source

in band R1L2 in units of counts/second

minimum radius around a point source to remove

for band R47 in units of pixels

minimum radius around a point source to remove

for band RIL2 in units of pixels

name of file containing the "master list"

of sources detected in band Rd7

name of file containing the "master list"

of sources detected in band R1L2

coordinate system to use for mosaic

1 ecliptic coordinates

2 equatorial coordinates

3 galactic coordinates

component to be c_t into a mosaic

1 counts

2 exposure

3 background

radius beyond which exposure is to be discarded

radius around point source to remove

in umts of PSF radii

the minimum fraction of the mean exposure

to be accepted as good

latitude of center of mosaic

longitude of center of mosa, ic

mask flag

0 If no mask is to be used

I if mask Is to be used

scale of mosaic in arcmin/pixel

minimum significance for a point source detection

225
REFERENCES

Greisen, E. W., & Calabretta, M. 1993, Representations of celestial coordinates in fits

Kuntz, K. D., & Snowden, S. L. 1998, Cookbook

for analysis procedures for rosat xrt observatiom of extended objects and the diffuse

background, part ii: Mosaics, Technical report, NASA/GSFC

226
Appendix E: The TAE in the Southern Galactic Hemisphere

Abstract

WehavemadeapreliminaryinvestigationofthespectrumoftheTAEinthesouthernGalactichemisphere. Although the emission in the southern Galactic hemisphere

has poorer statistics than the emission in the northern Galactic hemisphere, we find

the component temperatures to be consistent with those of the north.

We calculated the SED of the TAE in the

southern Galactic hemisphere using the method

described in Chapter 6. In order to exclude contamination by Loop I, we excluded regions with

< 60 o and £ > 2700 . This is a somewhat

larger region than used in the northern Galactic

hemisphere, but is justified by examination of

the RAS$ R45 band image. We have restricted

analysis to the region b > 45 °. We also excluded

regions with NH > 3.0 × 102° cm -2.

Assuming the same soft TAE component temperature found for the northern Galactic hemisphere, we attempted the standard local/distant

separation. For the local component we found a

peak R2/R1 ratio of 1.06, a FWHM of 0.89-1.27,

and corresponding model temperatures log T =

6.11 and 6.02-6.25, assuming a foreground absorbing column of N! = 4.0 x 10 is cm -2. The

temperature of the local component is consistent with the results from the northern Galactic hemisphere, logT g 1,_+0 is For the local

component we found a peak R2/R1 of 0.69, a

FWHM of 0.41-1.22, and corresponding model

temperatures log T = 5.90 and 5.72-6.25. Note

that the upper bounds of neither of the distributions correspond to a Raymond & Smith model

temperature.

The reason for the large difference between

these results and those for the northern Galactic

hemisphere is poor statistics. Figure E.1, shows

a histogram of the R2/R1 band ratio for both

the local and distant components. Since the

number of pixels available for analysis is only

0.75 that available in the north, both the local and the distant distributions are somewhat

broader than those for the northern hemisphere.

The flux of the distant component in the south

is _ 0.67 that of the north, but with similar uncertainties. Thus, the distribution for the distant component is much broader, and it is difficult to determine the mode of the distribution.

The peak value of the R2/R1 ratio of the distant

component could be placed anywhere from 0.7

to 0.9 (5.9 < logT < 6.03). The distribution

does suggest that the temperature in the southern Galactic hemisphere is somewhat lower than

in the north. These results are consistent with

those of Snowden et al. (1998), and a lower temperature for the sourthern Galactic hemisphere

can be inferred from Figure 11 of Snowden et al.

(2000).

REFERENCES

Snowden, S. L., Egger, R., Finkbeiner, D., Freyberg,

M. J., & Plucinsky, P. P. 1998, ApJ, 493, 715

Snowden, S. L., Freyberg, M. J., Kuntz, K. D., &

Sanders, W. T. 2000, ApJS, in press

227
7O

6O

5O

40

r_

E

7= 30

2o

to

3o

¢..

20

z

i i .... i

,6"_.6 L

: ;= .=n:, ......... n=.m;n :

,6"_. 6

0.0 0.5 1.0 1.5 2.0

X2/Xl

Fig. E.I.--The distributionof the R2/RI band

ratiosforthe localand distantcomponents. The

squiggle above the histogram shows the conversion to Raymond & Smith model temperature. For the distant component, both the

R2/RI and Raymond & Smith conversion are

afterdeabsorption. For the localcomponent,

the histogram is of the R2/RI derived from /_

values; the temperature conversion assumes a

4 x I0Is cm -: absorption. Note that only the

lower branch of the temperature conversionisof

interestas the upper branch would over-predict

the R47 flux.

228
Appendix F: A Note on the Lx to Count Rate Conversions

Abstract

This appendix describes the method by which we calculated the relation between

the intrinsic stellar X-ray luminosity and the hardness ratio.

This section describes the intrinsic X-ray luminosity, Lx, to count rate conversion used in

Chapter 7 for the Schmitt (1997) and Schmitt,

Fleming, & Giampapa (1995) data. Unlike other

data sets where the original count rate to Lx

conversion used a single factor, Schmitt (1997)

and Schmitt, Fleming, & Giampapa (1995) use

the function

(5.3HR+ 8.31) x 10-1_ergs cm-2counts -1

(F.I)

where

R47 - R1L_

HR = R4r + R1L2" (F.2)

and Rxx is the count rate in band XX. Thus,

the conversion factor depends upon the hardness

ratio which, in turn, depends upon the intrinsic

X-ray luminosity of the star.

Schmitt (1997) and Schmitt, Fleming, & Giampapa (1995) have shown that the hardness ratio is correlated with the X-ray luminosity Lx.

These correlations are shown in F.1. The function

H R = a + blog Lx (F.3)

was fitted to the correlation for each spectral

type using the procedure of Akritas & Bershady

(1996), OLS(Y--X), for data with errors in both

dependent and independent variables 1. The fit

values are listed in Table F. 1.

1 Only the uncertainties in R_r + Rt L_ are published; one

can calculate the the uncertainties in Lx and HR in a

straight forward way. Since

iogLx = Iog(5.3HR + 8.31 • 10-t2)(Pq7 + RzL_)(4_rd 2)

(F.4)

where d is in centimeters, the uncertainty is

a|ogLx = (In10)(R47 +RzL2)"

I have estimated uncertainties in HR in the following

manner; since /Ll7 and RLL_ are independent measuremerits

aa_,+a,L== _a,,-R,L_ (F.6)

-0.2

-0.4

-0.8

-0.8

- 1.0

@J

0.0

-0.2

-0.4

-0.0

0.2 ! ..... - * - -

-0.2

-o. i

-O.8

-0.8

-I.0

0.2 ............. " • - ' 1 .

: ( "

i

-1.0 !

s-':i M(, ly) .......

-0.1: : : "E "

_ -0.4

-0.8

-I.8

Im 2e lift m 21D 30

L

Fig. F.1.--Fit of HR vs. logLx for each spectral type. Dotted Line: Fit excluding URBs,

Solid Line: Fit including URBs. Data points of

URBs are shown as squares.

229
TABLE F.I

FITS TO THE HR - Lx RELATION

Spectral Excluding URB a Including URB

Type a b a b

F - 18.094-1.99 0.615 -9.714-1.99 0.3264"0.069

G - 10.654-2.22 0.359"4-0.079 -8.92+1.28 0.2984-0.046

K -9.4104-1.85 0.3144-0.067 -8.164-1.29 0.272-I'0.046

M (early) -5.104-0.62 0.1694-0.022 -5.804-0.66 0.1944-0.024

M (]ate) -7.664-1.54 0.2704-0.05? -7.224-1.49 0.2534-0.055

Unversal Slope Fits

F -9.544-0.99 0.3174-0.031 -8.414-0.55 0.2804-0.021

G -9.454-0.99 0.3174-0.031 -8.414-0.55 0.2804-0.021

K -9.484-0.96 0.317"I"0.031 -8.394"0.54 0.2804-0.021

M (early) -9,17"1"0.98 0.3174-0.031 -8.15"1-0.54 0.2804-0.021

M (late) -8.924-0.94 0.3174-0.031 -7.944-0.53 0.2804-0.021

aUn-Resolved Binaries

From the catalogue of Schmitt (1997) and

Schmitt, Fleming, & Giampapa (1995) all stars

for which only upper limits existed for /Lt7 +

Rlr_, or for which HR was listed as -1 (due to

R47 being only an upper limit) were discarded.

Measurements for unresolved binaries (URBs)

are also suspect, since the flux must be divided

between the members in some more or less arbitrary fashion. Thus fits were made both including and excluding the UBBs. As can be seen

from Figure F.1, there is a substantial intrinsic

scatter around any linear relation. Thus, given

the small number of stars in any single spectral

class, fits are quite uncertain (despite the small

formal uncertainties), and the situation is made

much worse by excluding the unresolved binaries.

The slopes from fits including the unresolved

binaries have overlapping 1_ confidence intervals, leading one to suspect that 1.) the log Lx -

and thus

aHR = [HRIcrR47+R:L:t P,,47 -- RII,2 P..47 + RIL2 "

(F.7)

It is interesting to note that _rLx ',* 0.4V_ whereas

OHR ", 1.4V_ for very soft or very hard sources.

The published values of O/¢4_+RIL _ often fall below

_/P_7 + RI L2, the lower limit imposed by Poisson counting statistics. In these cases I have set c,/_s,+j%L _ =

"_P'-4": + R1L2

HR relation has a single "universal" slope for

all spectral classes (a suspicion first voiced by

Schmitt (1997)), perhaps with an offset for each

spectral class, and that 2.) the large differences

in the slopes from the fits excluding the unresolved binaries is due to the large uncertainties

inherent in fitting a small number of intrinsically

scattered points. Acting upon these suspicions,

we employed an iterative scheme for determining

a single "universal" slope all spectral classes 2.

The "universal slope" fit can be seen in Figures F.3 and F.2, where the data for each spectral type has been given the appropriate offset.

A "universal slope" was determined both including and excluding unresolved binaries. Although

the two slopes are different, their 1_ confidence

regions overlap. The offsets for individual spectral classes span an interval comparable to the

uncertainties in HR. The "universal slope" fits

are compared to the data from individual spec21n the first step, I calculated the means, • and _" from

the data for each spectral class. I then calculated the HR

offset needed to place all of the centroids, the (_,_)i, for

all of the spectral classes onto the fitted relation for one

arbitrarily chosen class. These offsets were added to the

HR, and all the data from all the spectral classes were

merged and fit by a single line. In the second step, I determined the offsets needed to place the centroids on the

new fit. Those offsets were added to the (original) HR,

and step two was repeated until the value of the slope

did not change between iterations. Only a few iterations

were needed.

230
i.O -_-

URB j

×

0,0

-l. '' '" ' ' 30 -0.4

25 _'6 27 z8 20

L,

-o.8

-o.8

Fig. F.2.-- Fit of HR vs. log Lx for all spectral -1.o

classes together using the iterative fitting tech- o.2

nique described in the text. Data from URBs o.0

are included. -o.z

_ -0.4

-0.0

-0.0

-1.0

0.2

0.0

tral classes in Figure F.4.

REFERENCES

Akritas, M. G., & Bershady, M. A. 1996, ApJ, 470,

706

Schmitt, 3. H. M. M. 1997, A&A, 318, 215

Schmitt, J. H. M. M., Fleming, T. A., & Giampapa,

M. S. 1995, ApJ, 450, 392

l.O

0.5

0.0

-0.5

-1.0

25 26 27 28 29 30

L.

Fig. F.3.-- Fit of HR vs. log Lx for all spectral

classes together using the iterative fitting technique described in the text. Data from URBs

are not included.

Fig. F.4.-- Comparison of the "Universal SLope"

fit to the data for each spectral type. Dotted

Line: Fit excluding URBs, Solid Line: Fit including URBs. Data points of URBs are shown

as squares.

231

Appendix G: The Uncertainty in IV(0)

Abstract

Thisappendixdiscussesvariousmethodsforfindingtheuncertaintyin W(g). We

present a more efficient form of the "shuffle" method, and show that in noise dominated

images that it is equivalent to other methods used for determining the uncertainty in

W(0). However, we also show that in the presence of structures with amplitudes

larger than the noise the shuffle method (or, indeed, any method that relies on the

independence of neighboring pixels) can produce uncertainties that are a factor of a

few too large at small O and a factor of _ 10 too small at large 0.

While writing a proposal to extend this work,

we attempted to find a method that would allow us to determine the uncertainty in the ACF

without reducing each pointing and calculating

the ACF. Clearly, the uncertainty will be proportional to _, where n is the number of pairs

of pixels used to calculate the ACF on a given

scale. The uncertainty in R can be easily estimated from the length of the exposure and the

expected cosmic background rate. However, the

calculation of the uncertainty in RR' involves a

substantial correlation term, aRn, 2 , which has a

magnitude similar to o'_. Thus, an ab imtw estimation of the uncertainty in the ACF is not

practicable.

There are two common methods for determining the uncertainty once the autocorre]ation

function has been calculated. Both of these

methods are discussed by Chen et al. (1994).

The first method is to take the variance at

each scale,

-we,)= [ -

calculation 1.

The second method requires the image to be

randomly "shuffled"; for all of the pixels to be

reordered. The ACF is then measured upon this

new image. The process is repeated many times

in order to measure the variance in the distribution of W(0) produced by all of the trial shuffles. Chen et al. (1994) found this latter method

consistent with the former and preferred it because it "takes into account the correlation of

the data". What is not clear is how many shufties need to be executed to determine the variance.

We have developed a more efficient form of

this latter method; a "maximal shu_e'. If we

consider an image to be a list of n pixels, then

there are (n- 1)n/2 possible pixel pairs. For

each pixel pair, we may calculate whatever form

of the ACF is desired, and then calculate the

variance of that quantity, _;:_. The uncertainty

of W(8) is then given by ¢/v/'_, where ne is

the number of pairs of pixels used to calculate

W(9). This method has the advantage over previous shuffle methods; it provides the maximum

possible number of shuffles with a minimum of

(Z[(R - - ])

Then, by the standard propagation of uncertainties and the application of the central limit

theorem, the uncertainty in the mean will be

aw(e)/_f_ where ne is the number of pairs in

the sum. This method has the advantage that

it is easy to calculate, but it slows the calculation of the autocorrelation, an already slow

2 computing.

lG'-Uods, "" Consideration given the appropriate will show that conditions, these threemustmeth- be

consistent. All three methods rely upon the central limit theorem by setting the uncertainty of

the quantity RR' to be the variance of that

quantity divided by the root of the number

I On a DECa with a minimal amount of memory (64M,

the standard amount) an autocorrelation of a 512 x 512

pixel image containing a mosaic takes up to 24 hours,

using a FORTRAN algorithm. The same mosaic on a

PC, using IDL, takes about 2 hours.

233
of samplesmade.Thethreemethodsvary in

the number of samples from which the variance

of RR' is determined. The "maximal shuffle"

method uses the maximum number of independent samples of RR' to determine the variance.

The shuffling method takes a large number of

samples in a duplicitive manner. The variance

method uses a small number of samples, just

the values of RR _ used to calculate the ACF on

a particular scale. Chen et al. (1994) suggested

that if the pixels are strongly correlated, then

the variance method will significantly overestimate the true variance of RR'.

Chen et al. (1994) state a preference for a

shuffling method for measuring the uncertainty

in W(O) "because it takes into account the correlation of the data". They find that the shuffling

method produced uncertainties consistent with

those produced by the variance method, where

the variance method is presumably the application of Equation G.1.

We have compared the shutting method with

the results of the variance method used in Chapter 10, which is not the same as the variance

method used by Chen et al. (1994) 2. In practice,

we find that the shuffling method can produce

an uncertainty that is a factor a few larger than

the variance method at small 8, and that the

shuffling method can produce and uncertainty

that is a factor ,,- 10 smaller than the variance

method at large 0. (See Figure G.1.)

This difference can be understood in the following terms. In the presence of correlations,

the samples are no longer independent and so

the uncertainty of the mean of a subregion is

not the square root of variance of all of the pixels divided by the root of the number of pixels in the subregion. If we look at the variance of IF where I and I' are separated by a

distance that is much shorter than the correlation length, that variance will be much smaller

than one calculated from an equivalent number

of pixel pairs drawn at random from over a larger

region. Thus, shuffling will overestimate the uncertainty at small 0. If we look at the variance of

ll' where I and I' are separated by a distance

that is much on the order of than the correlation length, that variance will be much larger

than one calculated from an equivalent number

of pixel pairs drawn at random from over a larger

region. Thus, shuffling will underestimate the

uncertainty at 0 comparable to the correlation

length.

Therefore, for images with structure, the variance method is superior to the shuffling method.

REFERENCES

Chen, L.-W., Fabian, A. C., Warwick, R. S.,

Branduardi-Raymont, G., & Barber, C. R.

1994, MNRAS, 266, 846

2Since the method given by Equation G.1 is computationally intensive, it is to be avoided even in cases where

the noise dominates. It is computationaily little more

costly to calculate WIS=. By) than it is to calculate W(8).

It is then trivial to compute the variance from all the W

for which 83+8 2 =8 _.

234
gr.

(.9

{.9

{.9

rL

[.9

r_

086+74

0.10

o.o51

0.00

-0.05

-0.10

0.I0

0.05

0.00

-0.05

-0.10

0.10

0.05

0.00

-0.05

-0.I0

0.2

 R82 ""--k__

::-10'

-

k__

]

75'

-2' !

-- | !

0.1

0.0

-0.1

-o.2 L10'

o.2__' I_

0.1 F

o.o_

i

-0.1 "-

-0.2 [5'

0.2 k

0.1_

0.0 _

-o.1 L2' -0.2 --

o.o

7

, | g I

i

i

|

0.5 1.0 1.5 2.0 2.5

Log,0(O (arcminutes))

Fig. G.1.-- Comparison of the uncertainty in X(8) determined by the variance and shuffle methods. The histogram line is the ACF, the shaded region is the smoothed ACF plus-or-minus the

uncertainty determined by the suffle method• The irregular solid lines are the smoothed ACF

plus-or-minus the uncertainty determined by the variance method. Note that band R82 is the

same as band R1L2.

235

Appendix H: Fluctuation Figures

Abstract

Because there are seven data sets, two bands, three exclusion radii (i.e., maximal

PSF sizes), and two "modes" (raw and high-pass filtered), the number of different

data sets to display is quite large. Chapter 10 extracts the most interesting data for

display and discussion, but the critic justifiably wants to examine the data themselves

to confirm the analysis. In order to make Chapter 10 manageable, the bulk of the data

display has been exiled to this appendix. The ACFs are displayed first in four figures.

They are followed by four plot of _1/I in terms of ab(¢)/1.

237
A

A

0.2

0.1

0.0

-0.1

-0.2

02

0.1

0.0

-0.1

-0.2

0.2

0,1

0.0

-0,1

-0.2

A

Band R82 lO' PSF

I

335+76

086+74

138+52

02

0,0 " '

-0,1

-0.2 161+51

0.2

O.l

0.0

-0.1

-0.2

0.2

0.1

0.0

-0.1

-0.2

0.2

0,!

0.0

-0.1

-0,2

0.0

171+50

.>20+84

295-89

0.2

0.1

0.0

-0,1

-0.2

0.2

0.5 i.0 1.5 2.0 2.5

_lJe(a_minut,=e))

Band R82 2' PSF

3.0

0.!

0.0

-o.I

-0.2

0.2

O.l

o.0

-0.1

-0.2

0.2

0.1,

_ 0.0

-0. l

-0.2

0.2

0.!

o,o

-0.1

-0.2

0.2

0.!

ib

D

-0!

-0.2

0.2

0.|

0.0

-0.1

-0.2

0.0

_ J,I

035+76 ---'

I

086+74

161+51 I r ....

.171+50

.... _ I

220+84 .......

A

A

Band R82 5' PSF

0.2

0,1

O,I

-0.',

-0.;

0.2

0.!

0.0

-0.1

-0.2

0.2

0.!

0.0

-0.1

-0.2

935+78

:)86+74 _ 1

.138+52

L61+51

0.2

0.1

0.0

-0.!

-0.2

0.2

0.1

0.0

-0.!

-0.2 171 + 50

0.2!

0.0

-0.!

-o.2 220+64

0.2 0.! _

0.0

-0.!

-o.z .>95-89

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Log_e(areminut_i))

Band R82

0.2

0.1

0.0

-0.1

-0.2

0.2

0.1

-0.1

-0.2

0.11

0.1

_ 0.0

-0.1

-0.2

0.2

0.l

_ 0.0

-0.1

-0.2

0.2

0.!

_ o.o -o.li

-0.2

0.2

0.!

_ 0.0

-0.1

-0.2

0.2

_ 0.0

-0. i

-0.2

035+78

086+74

......... -__ . • . .

138+52

161+51

171+50

220+64

295-69

05 1.0 1.5 2.0 2.5 3.0 0,0 0,2 0.4 0.0 0.8 1,0 t.2 1.4

_l_¢(au'cmiau_o) ) _l_S(m_miau_m))

Fig. H.I.-- ACFs for band RIL2(=RB2). Top Left: for 10' maximum PSF. Top R=ght: for 5'

maximum PSF. Bottom Left." for 2' maximum PSF. Shaded regions in the first three panels show

the uncertainty• Bottom Right: ACFs for all three PSF sizes. Solid: 10', Dashed: 5', and

Dotted: 2'.

238
A

Band R45 10' PSF

o,!

-o.1

-0.2 335+78

0.1

o.o -

-0.!

-0.2 )86+74

0.2

O.t

0.[

-0.1

-o._ 138+52

0.2

0.1

-0.1

-o._ L61+51

0._

O.l

-O.t

-o.2 171+50

0,2 0.! _

A

_ 0.0

-0.1

-Ola 220+84

0.2 0.1 ..... .MA

0.0

-0.1 _-_

-o.z 295-89

0.0 0.5 1.0 1.5 2.0 2.5

_g_e(Ir©miau_=))

Band R45 2' PSF

q-- _wvTrmw _

3.0

Band R45 5' PSF

0.2 _

_ 0.1

_ 0,0

-o.1 1035+78

-0.2

0.2

-o.! 1086+74

-0.2

0,2 :

-0 !

-o'2 138+52 vq L

o.o _ "-'----"_

l= -O.I -

°.2-

D: -0.1 - +

_ o.o

":L'

_ 0.1

_ 0.0

_= -0" l,o=_-- RO .... I 1

0.0 0.5 I .O 1.5 2.0 2.5 3,0

Lol[de(&rcminute=))

Band R45

086+74

1220+84

295-89

0.2 0.4 0.6 0.8 1,0 1,2 1.4

Lo_,(8(srcmtnutes))

Fig. H.2.-- ACFs for band R45. Legend as for previous figure.

239
A

A

A

A

A

A

Band R82 10' PSF High-Pass Filtered

0.020

O,OLO

0.000

-0.010

-o.ozo 035+78

0.020

0.010

0.000

-0.010

-0.0Z0 )86+74

0.02*0

O.OlO

0.000

-O.OlO

-0.020

0.020

0.010

0.000

-0.010

-0.020

0.020

O.OlO

0.000

-0.010

-O.OZO

0.020

0.010

0.000

-0.010

-o.ozo .J20+84

0.020

0.010

0.000

-0.010

-o,om .)95-89

0.0 0.5

Y

138+52

: .......

181+51

t71+50

....... d

"'""' "Vql

1.0 1.5 2.0 2.5

I_.(l(arcmmuum))

3.0

Bend R82 2' PSF Hilh-Pass Filtered

o.1°'" :,+,L_.._

_"-o2-°'°'° .......... - ........

0.00"2 >9

-0+1

-0.2

0.2 °, :oodLeS___

0.0 0.6 1.0 l.S 2.0 2+5

LoI,.( II ( _r'cmm u teu ) )

0.020

0,010

0.000

-O.OlO

-0.020

0.020

0.010

0.000

-0.010

-0.020

0+020

0,010

0.000

-0.010

A

0.2

0.I

. 0.0

-O.l

-0.1!

0.2

0.!

_ 0.0

-0.1

-0.2

0.2

_ 0.!

_ 0,0

-0.!

-0.2

0.2

0+1

_ 0.0

-0.!

-0.2

0.2

0.1

_ 0.0

-0.1

-0.2

0.2

0+1

_ 0.0

-O.l

- -0.2

0.2

0.!

_ 0.0

-0.1

-0.2

035+78

086+74

........ ""....... "-. ---i ..... " "

:138+52

161+51

171+50

220+84

:295-89

3.0 0.0 0.2 0.4 0.6 0.8 ! .0 ! .2 1.4

Lol_(@ ( Ircmio tllt_• ) )

Fig. H,3.-- ACFs for high-pass filtered images of band RIL2(=R82). Legend as for previous

figure. Note the changes of scales.

240
Band R45 lO' PSF High-Pass Filtered

0.020

0.010

-0.020

0.020

O.OtO

0,000

-0,010

-0.020

0.020

0.010

0.000

-0.010

-0.020

0.020

0.010

0.000

-0.010

-0o20 161+51

0.020

0.010

0.000

-0.010

-0.020

0.020

A 0.010

0,000

-0.010

-0.020 .>20+84

0.020

0.010

0°000

-0.010

-o.ozo ?95-89

0.0 0.5 1.0 1.0 2.0 2.5

Lo¢,_e(arcmJau_m))

Band R45 2' PSF High-Pass Filtered

3.0

A o.! • !

i 0.0

-0.1

-0.2

0.2

0. l

0.0

-0.1

-0.2

0.2

0.1

0.0

-0.12 -0.:

0.2

0A

0.0

-0.!

-0.2

0.2

_ 0,1

_ 0.0

-0.1

-0.2

0.2

_ 0.1

_ 0.0

-O.I

-0.2

0.2

_ 0.1

-O,I

-0.2

0.0 0.5 1.0 1.5 2.0 2.0 3.0

Lot._e(arcminute=}}

0.020

0.010

0.000

-0.010

-0.020

0.020

0.010

0.000

-0.010

-0.020i

0.020 _

0.000

-0.010

Band R45 5' PSF High-Pass Filtered

0.000

-o.o,o 1161+51

-0.020

0.020

0.010

O,OlO

-0.02[

0.010

0.000

-0.010

-o.o2o >-95- 89

0.0 0.5 1,0 1.5 2.0 2.5

Log,_'_'( arem|l= ULeB)}

Band R45 High-Pass Filtered

3.0

0.2

0.I

0.0

-0.1

-0.2

0.2

0.1 A

0.0

-O.l

-0.2

0.2

0.!

-0.1

-0.2

0.2

0.!

0.0

-0.1

-0.2

0.2

0,1

0.0

-0.1

035+78

086+74

138+52

161+51

i -j :.)'_ _

'-...... , _---.:-I ."_'_-- :':¢1 '

-0.2 171+50

0.2

0. i

0.0

-0.!

-o2 220+84

0.2

0.1

0.0

-0A-

-o.= 295-89

0.0 0.2 O.4

!

.

0.6 0.8 1.0 1.2 1.4

Lol.(@(arcmmute=)}

Fig. H.4.-- ACFs for high-pass filtered images of band R45. Legend as for previous figure. Note

the changes of scales.

241
IOO

80

- 60

) 4o

_ 20

Io8

80

40

20

,o8

=.. 8o

-- 60

) 4o

_ 20

) 4o

_ 20

-- 60

) 4o

_ 20

) ,o

o

v 20

_ 4o

_ 20

-- 60

_ 4o

v 20

_ 4O

w _

_ 20

-- 60

20

o

o

Band R82 lO' PSF

035+78

086+74

138+52

" _ • , 161+51

...._1 171+_o

1_O 40 80 80 1oo 120

_(arcmta)

Band R82 2' PSF

035+78

08,6+74

138+52

161+51

-- -. 171+50

_._ 220+64

295-89

1oo

BO

_ 8o

d 4o

1o8

80

4O

_ 20

1o8

80

4O

_ 20

1o8

ao

_ '°4O

_ 20

1o8

8O

40

_ 20

_o8

'it'°40

20

o

Band R82 5' PSF

035+78

006+74

138+52

161+51

171+50

220+84

-_., J_. 295-89

20 40 60 BO I O0 120

Band RB2

_._" 035+78"

__,...-,-----'_ - 138+_Z

_t

_ _..,.-.-, .... :

_. 171 -PO{]

___

20 40 60 50 i00 120 0 tO 20 30 40 50

_(*_mia) _(*r=mm)

6o

Fig. H.5.-- Measures of ab(¢)/l for band R1L2(=R82). Top Left: for 10' maximum PSF. Top

Right: for 5' maximum PSF. Bottom Left: for 2' maximum PSF. Shaded regions in the first three

panels show the uncertainty. Bottom Right." Measures of t_b(¢)/I for all three PSF sizes. Solid:

10', Dashed: 5', and Dotted: 2'.

242
%

20

15

I0

5

-g

20

15

10

5

0

"-_ I0

20

___

20

• 15

Band R45 I0' PSF

20

035+78

086+74

,, , 138+52

_ 161+51

' 84

295-89'

-Oi , ,

0 20 40 _ 8'0 100 120

Band R45 2' PSF

IO

15F o

035+,8:

to

_o

s.. 15

5

20

3-_ 15

_ _0

_g

20

_ s

20+

_,. 15

*__

*__

086+?4

138+52

161+51

.: ' _! 171+50

_ . ,

220+84

295-89

15

iI2O

15

0--

-5

20

_5

0

O;

_0

_5

-g

10

_5

0

-5

2O

0

15

_ _°5

_ 0

t_

_.. lO

t_

_ 10

+ oi

"i++

s_, LO

%o

% s

% s

Is

Band R45 5' PSF

035+78

086+74

138+52

t

171+50

_----.--.-..,,,_ _k_ 220+841

........ 295-891

2'o ,o ;o ;o

#,(,rcmi_)

Band R45

100 120

....... ;.++ + . ". +741

I

......

• -_ m _,pl_;, _' ' :b_'

....._.___,L-_ '_ _'61+51:=

_1¢1 - _

... . ** r- -

20 40 60 80 100 120 10 :_0 30 40 50 #tO

Fig. H.6.-- Measures of a_(¢)/l for band R45. Legend as for the previous figure.

243
v

Band R82 10" PSF

035+78

J_ 086+74

_+52

__' 161+51

171+50 _

_: o _,

I 84: i

,, e ,o _, ;o ,;o ,,o

Band R82 2' PSF

035+78

" _ 086+74

I _,k., 138+52

-1

-i _J_ 161+51

,g

8

-2

-2

s... 8

-2

s.., 8

-2

Band R82 5' PSF

035+78

086+74

138+52

161+51

171+50

220+84

295-89

80 I O0 ! 20

5

171+50

.... 220+84

___L 295-89

20 40 80 I$0 1 O0 120 0 20 40 80

_(srcmin) _(srcmm)

295-89

80 I00 120

Fig. H.7.-- Measures of ab(¢)/l for high-pass filtered images of band R1L2(=R82). Legend as for

the previous figure.

244
Band R45 lO' PSF

•'iF

035+78

086 + 74

161+51 1

2 .... 171+50-_

|

-4 _ . , .

0 20 4,0 60 BO |00 ]20

@{srcmin)

Band R45 2' PSF

2 035+78

086+74

138+52

161+51

2

i

¥

20

_ 171+50

220+84

295-89

40 60 80 l O0 120

_(ercmm)

-I

t

-2

s. 8

0

-2

:s_ 6!

-2

-41

Band R45 5' PSF

,i

035+78

086+74

138+52

'i

181+51

220+84 _

20 40

295-89

8O

1_(sPemm)

Band R45

BO ! O0 ! 20

!

4

(

il

.,%-_. _.

035+78

086+74

138+52-

2

161+51

_I_ 171 +50

-_P 220+84

m[-

-,=i 2_5-89:

20 40 60 80 I O0 120

@(ar©min)

Fig. H.8.-- Measures of o'_(¢_)1I for high-pass filtered images of band R45. Legend as for the

previous figure.

245

Appendix I: Calculating the Suppression Factor

Abstract

This appendix provides a rough derivation of the factor by which fluctuations of

scale ef are suppressed by high-pass filtering with a threshold of es. We also show,

non-rigorously, that the ca}culated value of the excess fluctuations at scale ¢ is the

linear combination of the fluctuations at all other scales.

Assume that an image I contains only fluctuations of scale eF and amplitude _rl.r (i.e., the

variance of pixels of size eF in image I). Assume also that the image is pixellated at scale

_bv, where eF/¢P _" a few. At the pixels size,

the variance will be

= (I.1)

(N.B., with respect to the text in Chapter 10

this is ¢_, not a_). The smoothed image, S,

is the original image smoothed by es, where

es/$f _ a few. The variance of the smoothed

image on the scale of ep will be

_ (¢F_ 2

aS, p = (I.2)

Fig. 1.1.-- Cartoon demonstrating values used

in calculating the suppression factor. The meanings of the variables is given in the text.

The high-pass filtered image. H, is constructed

by subtracting S from I, so the variance of the

high-pass filtered image will be

_2H,p _ ff _l,p + ITS,p 2 _2_s, p (I.3)

where als is the correlation of the image and the

smoothed image. It is shown in the footnote 1that

the correlation term is given by

2 2 (I.8)

_Is, p ~ _l,p \ es / '

thus,

o"2 ~ [1- (¢F'_2I (I.9)

.,,, J

This formulation, of course, makes the assumption that there are no fluctuations at any other

scale, which is clearly not the case.

Since the fluctuations at different scales are

statistically independent, one can pursue a similar course assuming that there are fluctuations

IThe correlation term may be calculated as follows. Assume that a pixel in the initial image is the central pixel

of a smoothing region of area n pixeis and has the value

P. The sum of all of the other pixels in the smoothing

region is Q. The mean value of a pixel is M The value

of the same pixel in the low-pass filter will be (S + Q)/n.

The correlation between the value of the pixel in the initial image, and the same pixel in the low-pass filtered

image is given by

This expression can be expanded as

1 -- + (1-5)

n _ n

and if we make the approximation that

then

1 2

a_s _ _ae. (I.7)

247
at both CF and Cp. Then the variance of the

original image on the scale of Cp is

,,p _-p _r_. (I.10)

The variance in the smoothed image is then

[¢pp._2 2 (OF)2 (_F,_2 (I.11)

The variance in the high-pass-filtered image is

then

(I.12)

Thus, the variance at a given scale is the linear sum of the variances at all scales. The variance at a given scale of a high-pass-filtered image

is the linear sum of the variances, each multiplied by [l - (CRIes) _] where Cf is the scale of

the fluctuation.

248
Glossary

ACF- AutoCorrelationFunction,see Equation 10.7.

AGN - Active Galactic Nuclei, one of the major sources of the cosmic X-ray background. These

objects are typically a nuisance for those studying the truly diffuse background.

ASCA - Advanced Satellite for Cosmology and Astrophysics (Tanaka, Inoue, &Ho[t 1994).

BCES - Bivariate Correlated Errors and Scatter, a linear regression model developed by Akritas

_: Bershady (1996) for data with errors in both variables.

CFG -Chen, Fabian, & Gendreau (1997)

CLF - The Cumulative Luminosity Function, the number of sources brighter than a given luminosity (or brightness) as a function of luminosity (or brightness).

COBE - The COsmic Background Explorer satellite (Boggess et al. 1992).

DII_BE - The Diffuse Infra-Red Background Experiment, an instrument on COBE (Silverberg

et al. 1993).

EPL - The Extragalactic Power Law, the emission due to unresolved extragalactic sources, typically AGN and QSOs. The spectrum of the emission in the 0.5-2.0 keV energy range is well

modeled as a power-law with a photon index of --- 1.4.

HEASARC - The High Energy Astrophysics Science Archive Research Center, a service of the

GSFC LHEA.

HR - Hardness Ratio, usually defined for ROSAT PSPC bands as (P_ - Rs2)/(P_ + Rs2), but

is may be defined differently by other authors.

HRI - The High Resolution Imager, the secondary instrument on ROSAT.

HVC - High Velocity Cloud, a cloud of neutral hydrogen with a line-of-sight velocity not produced

by galactic rotation and [v[ > 100 km s-1.

IDL - Interactive Data Language, a relatively expensive high level programming language developed and marketed by Research Systems Inc. that emulates FORTRAN, but has extensive

visualization support and similarly extensive function libraries.

IGM - The InterGalactic Medium.

IMF - The Initial Mass Function, the number of stars of a given mass created per unit time per

unit mass.

]HAS - The Infra-Red Astronomical Satellite, which created an all-sky survey at 12 pro, 24 pm,

60 pm and 100pro.

ISM - The InterStellar Medium, all that which occupies the space between the stars.

IVC - Intermediate Velocity Cloud, a cloud of neutral hydrogen with a line-of-sight velocity not

produced by galactic rotation and 40 km s-1 < [v[ < 100 km s -1.

LDS - The Leiden-Dwingeloo Survey of galactic neutral hydrogen (Hartmann & Burton 1997).

See Chapter 2 for a description.

249
LHB- TheLocalHotBubble,theirregularregionofmilliondegreegasthatextendsfor50to 100

pcin everydirectionfromthesun.TheLHBiscontainedwithinbutdoesnotfill theLocal

Cavity.Thesourceofthegasin theLHBisunknownbutgenerallyattributedto asequence

ofsupernovae, eachonereheatingthecavityleftbythepreviousone(Smith1997).Themost

recentobservational studyof theLHBisSnowdenet al. (1998).(Audiencesnot familiar

withX-rayastronomyhavebeenknowntoexpressabsolutedisbeliefin theexistenceof this

phenomenonasit violatestheprinciplethattheearthisnotinapreferredlocation.It should

benotedthat this principle is cosmological, and has no application to galactic astronomy of

the ISM.)

LIC - The Local Interstellar Cloud, a tenuous asymmetric cloud of neutral hydrogen surrounding

the sun and extending a few parsecs in every direction. The cloud is detected in absorption

in lines of sight to nearby stars (Lallement et al. 1995) There are other similar features

within the Local Cavity, see the work on the Local ISM by Lallement, Vidal-Madjar, &

Ferlat (1986).

LF - The Luminosity Function, the number of objects with a given luminosity (or brightness) as

a function of luminosity (or brightness).

LMB - The Lanapton, Margon, and Bowyer criteria for determining a probability interval (Lampton, Margon, & Bowyer 1976).

Local Cavity - An irregular region surrounding the sun with a lower than average density of

neutral gas, and having a clear boundary at which the column density increases abruptly.

The Local Cavity was first discussed by Knapp (1975), was studied by Paresee (1984) and

Fri_zla & York (1983), and most recently mapped by Sfeir et al. (1999).

Local Fluff- The Local Interstellar Cloud.

LTE - Long-Term Enhancements. A temporally variable component of the SXRB of unknown

source. The time scales of variability range from hours to months. See Chapter 2.

PSDL - The Point Source Detection Limit, the flux of the faintest point source detectable.

PSF - The Point Spread Function, the function describing the fraction of flux due to a point

source detected at a given radius and position angle from the expected detector location of

the point source.

PSPC - The Position Sensitive Proportional Counter, the primary instrument on ROSAT.

See Chapter 2 for a more detailed description.

RASS - The ROSAT All-Sky Survey. See Chapter 2.

ROSAT - The ROntgen SATellite, a satellite sensitive to soft X-ray emission (0.1-2 keV) which

produced an All-Sky Survey (the RASS ) as well as a substantial archive of pointed observations. See Chapter 2 for a more detailed description.

ILlS - Raymond and Smith model thermal spectra (Raymond & Smith 1977).

SED - The Spectral Energy Distribution

SFD - Schlegel, Finkbeiner, & Davis (1998), a set of destriped and recalibrated IRAS 100 pm

maps.

SFR- The Stellar Formation Rate, the mass of stars (in solar masses) created per unit time.

SXRB - The Soft X-Ray Background. The exact energy range of the SXRB depends upon the

author, but generally is taken to mean 0.1-2 keV.

250
TAE - The Trans-Absorptive Emission, the X-ray emission originating beyond the absorption of

the Galactic ISM. The TAE includes emission from the Galactic halo as well as emission due

to extragalactic point sources and any diffuse extragalactic emission.

URB - UnResolved Binary.

WHIM - The Warm-Hot Intergalactic Medium, intergalactic gas with temperatures from 10 s K

to 107 K.

XSPEC - A common program for spectral analysis in the X-ray energy band (Arnaud 1996).

REFERENCES

Akritas, M. G., & Bershady, M. A. 1996, ApJ, 470, 706

Arnaud, K. A. 1996, in Astronomical Data Analysis Software and Systems V, ed. G. Jacoby & J. Barnes,

17

Boggess, N., et al. 1992, ApJ, 397, 420

Chen, L.-W., Fabian, A. C., & Gendreau, K. C. 1997, MNRAS, 285, 449

Frisch, P. C., & York, D. E. 1983, ApJ, 271, L59

Hartmann, D., & Burton, W. B. 1997, Atlas of Galactic Neutral Hydrogen (Cambridge University)

Knapp, G. R. 1975, AJ, 80, 111

LaUement, R., Ferlet, R., Lagrange, A. M., Lemoine, M., & Vidal-Madjar, A. 1995, A&A, 304,461

Lallement, R., Vidal-Madjar, A., & Ferlat, R. 1986, A&A, 168, 225

Lampton, M., Margon, B., & Bowyer, S. 1976, ApJ, 208, 177

Paresce, F. 1984, A J, 89, 1022

Raymond, J. C., & Smith, B. W. 1977, ApJS, 35

Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525

Sfeir, D., Lallement, R., Crifo, F., & Welsh, B. Y. 1999, A&A, submitted

Silverberg, R. F., Hanser, M. G., Boggess, N. W., Kelsal], T. J., Moseley, S. H., & Murdock, T. L. 1993,

Proc. SPIE, 2019, 180

Smith, R. K. 1997, Ph.D. thesis, University of Wisconsin

Snowden, S. L., Egger, R., Finkbeiner, D., Freyberg, M. J., & Plucinsky, P. P. 1998, ApJ, 493, 715

Tanaka, Y., Inoue, H., & Holt, S. S. 1994, PASJ, 46, L37

251

REFERENCES

Abell, G. O., Corwin, H. G., & Oiowin, R. P. 1989, ApJS, 70, 1

Akritas, M. G., & Bershady, M. A. 1996, ApJ, 470, 706

Arabadjis, J. S., & Bregman, J. N. 1999, ApJ, 510, 806, submitted

Arendt, R. G., et al. 1998, ApJ, 508, 74

Arnaud, K. A. 1996, in Astronomical Data Analysis Software and Systems V, ed. G. Jacoby & J. Barnes,

17

Babu, G. J., & Feigelson, E. D. 1996, Astrostatistics (Chapman & Hall)

Balucinska-Church, M., & McCammon, D. 1992, ApJ, 400, 699

Barber, C., Warwick, R. S., & Snowden, S. L. 1995, in RSntgenstrahlung from the Universe, ed. H. Zimmermann, J. Triimper, & H. Yorke, MPE Report 263, 319

Barber, C. R., Roberts, T. P., & Warwick, R. S. 1996, MNRAS, 282, 157

Barber, C. R., & Warwick, R. S. 1994, MNRAS, 267, 270

Benjamin, R. A., Venn, K. A., Hiltgen, D. D., & Sneden, C. 1996, ApJ, 464, 836

Be_eU, M. S., & StringfeUow, G. S. 1993, ARA&A, 31, 433

Bienaym6, O., Robin, A. C., & Crdzd, M. 1987, A&A, 180, 94

Bloch, J. J., Jahoda, K., Juda, M., McCammon, D., Sanders, W. T., & Snowden, S. L. 1986, ApJ, 308,

L59

Bogged, N., et al. 1992, ApJ, 397, 420

Boulanger, F., Abergel, A., Bernard, J.-P., Burton, W. B., D_sert, F.-X., Hartmarm, D., Lagache, G., &

Puget, J.-L. 1996, A&A, 312, 256

Bower, R. G., et al. 1996, MNRAS, 281, 59

Bowyer, C. S., Field, G. B., & Mack, J. E. 1968, Nature, 217, 32

Boyle, B. J., Wilkes, B. J., & Eivis, M. 1997, MNRAS, 285, 511

Breg-man, J. N., & Houk, J. C. 1997, ApJ, 485, 159

Briel, U. G., & Henry, J. P. 1995, A&A, 302, L9

Bryan, G. L., & Norman, M. L. 1998, ApJ, 495, 80

Bunner, A. N., Coleman, P. L., Hraushaar, W. L., McCammon, D., Palmieri, T. M., Shilepsky, A., &

Ulmer, M. 1969, Nature, 233, 1222

Buries, S., & Tytler, D. 1998, ApJ, 499, 699

Burrows, D. N. 1989, ApJ, 340, 775

Burrows, D. N., McCammon, D., Sanders, W. T., & Kraushaar, W. L. 1984, ApJ, 287, 208

Burrows, D. N., & Mendenhall, J. A. 1991, Nature, 351, 629

Carballo, R., Warwick, R. S., Barcons, X., Gonzalez-Serrano, J. I., Barber, C. R., Martinez-Gonzalez, E.,

Peres-Fournon, I., & Burgos, J. 1995, MNRAS, 277, 1312

Cen, R., Kang, H., Ostriker, J. P., & Ryu, D. 1995, ApJ, 451, 436

Cen, R., & Ostriker, J. P. 1999, ApJ, 514, 1

Chert, L.-W., Fabian, A. C., & Gendreau, K. C. 1997, MNRAS, 285, 449

Chea, L.-W., Fabian, A. C., Warwick, R. S., Brandwrdi-Raymont, G., & Barber, C. R. 1994, MNRAS,

266, 846

Cox, D. P. 1998, in The Local Bubble, ed. D. Breitschwerdt & M. F_eyber$, 121

Cox, D. P., & Reynolds, R. J. 1987, ARA&A, 25, 303

Cox, D. P., & Snowden, S. L. 1986, Advances in Space Research, 6(2), 97

Cravens, T. E. 1996, Geophysical Research Letters, 24, 105

Cui, W., Sanders, W. T., McCammon, D., Snowden, S. L., & Womble, D. S. 1996, ApJ, 468, 117

Daldem, M., Weaver, K. A., & Heckman, T. M. 1998, ApJS, 118, 401

Danly, L., Albert, C. E., & Kuntz, K. D. 1993, ApJ, 416, L29

Dav_, R., et al. 2000, ApJ, submitted

Davidsen, A., Shulman, S., Fritz, G., Meekins, J. F., Henry, R. C., & Friedman, H. 1972, ApJ, 177, 629

Deiker, S., Kelley, R. L., Lesser, A., McCammon, D., Porter, F. S., Stable, W. T. S. C. K., & Szymkowiak,

A. E. 1997, in Proceedings of the Seventh [nternational Workshop on Low Temperature Detectors,

113

Dempsey, R. C., Linsky, J. L., Fleming, T. A., & Schmitt, J. H. M. M. 1993, ApJS, 86, 599

DomgSrgen, H., & Mathis, J. S. 1994, ApJ, 428, 647

Donahue, M., Aidering, G., & Stocke, J. T. 1995, ApJ, 450, L45

Fiegelson, E. D., & Nelson, P. [. 1985, ApJ, 293, 192

253
Field, G. B. 1965, ApJ, 142, 531

Fixsen, D. J., Dwek, E., Mather, J. C., Bennett, C. L., & ShMer, R. A. 1998, ApJ, 508, 123

Fleming, T. A., Molendi, S., Maccacaro, T., & Wolter, A. 1995, ApJS, 99, 701

Fleming, T. A., Snowden, S. L., Pfeflermarm, E., Briel, U., & Greiner, J. 1996, A&A, 316, 147

Freyberg, M. J. 1998, in The Local Bubble, ed. D. Breitschwerdt & M. Freyberg, 113

Frisch, P. C., & York, D. E. 1983, ApJ, 271, L59

Fukugita, M., Hogan, C. J., & Peebles, P. J. E. 1998, ApJ, 503, 518

Gehrels, N., & Williams, E. D. 1993, ApJ, 418, L25

Greisen, E. W., & Calabretta, M. 1993, Representations of celestial coordinates in fits

Gruber, R. 1992, in Data Analysis in Astronomy IV, ed. V. D. Gesfl, 153

Guillout, P., Haywood, M., Motch, C., & Robin, A. C. 1996, A&A, 316, 89

Hartmann, D., & Burton, W. B. 1997, Atlas of Galactic Neutral Hydrogen (Cambridge University)

Hartmann, D., Kalberla, P. M. W., Burton, W. B., & Mebold, U. 1996, A&A, 119, 115

Hasinger, G., Burg, R., Giacconi, R., Hartner, G., Schmidt, M., Triimper, J., & Zamorani, G. 1993, A&A,

275, 1

Hasinger, G., Burg, R., Giacconi, R., Schmidt, M., Triimper, J., & Zamorani, G. 1998, A&A, 329, 482

Hanser, M. G., et al. 1998, ApJ, 508, 25

Haywood, M., Robin, A. C., & Cr6z_, M. 1997, A&A, 320, 428

Henry, T. J., & McCarthy, D. W. 1993, A J, 106, 773

Herbstmeier, U., Heithauaen, A., & Mebold, U. 1993, A&A, 272, 514

Herbstmeier, U., Mebold, U., Snowden, S. L., Hartmann, D., Burton, W. B., Moritz, P., Kalberla, P.

M. W., & Egger, R. 1995, A&A, 298, 606

Heraquist, L., Katz, N., Weinberg, D. H., & Miralda-Escud_, J. 1996, ApJ, 457, L51

Hiinsch, M., Schmitt, J. H. M. M., Schr6der, K.-P., & Zickgraf, F.-J. 1998, A&A, 330, 225

Hiinsch, M., Schmitt, J. H. M. M., Sterzik, M. F., & Voges, W. 1999, A&AS, 135, 319

Hutchinson, I. B., Warwick, R. S., & Willingale, R. 1998, in The Local Bubble, ed. D. Breitschwerdt &

M. Ft'eyberg, 283

Jakolamn, P., & Kalm, S. M. 1986, ApJ, 309, 682

Juda, M., Bloch, J. J., Edwards, B. C., McCammon, D., Sanders, W. T., Snowden, S. L., & Zhang, J.

1991, ApJ, 367, 182

Kalberla, P. M. W. 1978, Ph.D. thesis, University of Bonn

Kalberla, P. M. W., Westphalen, G., Mebold, U., Hartmann, D., & Burton, W. B. 1997, in The Local

Bubble, ed. D. Breitschwerdt & M. Freyberg, 479

KelsaU, T., et al. 1998, Ap.], 508, 44

Kernighan, B. W., & Pike, R. 1984, The UNIX Programming Environment (Prentice Hall)

Kelp, J. 1994, A&A, 289, 597

Kerp, J., Burton, W. B., Egger, R., Fi'eyberg, M. J., Hartmarm, D., Kalberla, P. M. W., Mebold, U., &

Pietz, J. 1999, A&A, 342, 213

Knapp, G. R. 1975, A J, 80, 111

Krautter, J., et al. 1999, A&A, 350, 743

Kroupa, P., Tout, C. A., & Gilmore, G. 1993, MNRAS, 262, 545

Kull, A., & B6hringer, H. 1999, At, A, 341, 23

Kuntz, K. D. 2000, Ph.D. thesis, University of Maryland

Kuntz, K. D., & Danly, L. 1992, PASP, 104, 1256

Kuntz, K. D., & Denly, L. 1996, ApJ, 457, 703

Kuntz, K. D., & Snowden, S. L. 1998, Cookbook for analysis procedures for rosat XRT Observations of

Extended Objects and the Diffuse Background, Part ii: Mosaics, Technical report, NASA/GSFC

Kuntz, K. D., & Snowden, S. L. 2000, ApJ

Kuntz, K. D., Snowden, S. L., & Verter, F. 1997, ApJ, 484, 245

Kuntz, K. D., Snowden, S. L., & Warwick, R. S. 1998, in The Local Bubble, ed. D. Breitschwerdt &

M. Freyberg, 329

LaUement, R., Ferlet, R., Lagrange, A. M., Lemoine, M., & Vidal-Madjar, A. 1995, A&A, 304, 461

Lallement, R., Vidal-Madjar, A., & Ferlat, R. 1986, A&A, 168, 225

Lampton, M., Margon, B., & Bowyer, S. 1976, ApJ, 208, 177

Landini, M., & Monsignori-Fossi, B. C. 1990, A&AS, 82, 229

Laor, A., Fiore, F., Elvis, M., Wilkes, B. J., & McDoweU, J. C. 1997, ApJ, 477, 93

Lilienthal, D., Wermmacher, A., Herhstmeier, U., & Meboid, U. 1991, A&A, 250, 150

Lisse, C. M., et al. 1996, Science, 274, 205

Lockman, F. J. 1984, ApJ, 283, 90

Lockraan, F. J., & Gehman, C. S. 1991, ApJ, 382, 182

Lockman, F. J., Jahoda, K., & McCammon, D. 1986, ApJ, 302, 432

Long, K. S., Agrawal, P. C., & Garmire, G. P. 1976, ApJ, 206, 411

Mather, J. C., et al. 1994, ApJ, 420, 439

254
McCammon, D.,Meyer,S.S.,Sanders, W.T.,& Williamson, F.O.1976,ApJ,209,46

McHardy,I.M.,etal.1998,MNRAS,295,641

McKee.C.F.,& Ostricker, J.P.1977,ApJ,218,148

Mewe,R.,Gronenschild, E.,&vandenOord,G.H.J.1985,A&AS,62,197

Miyaji,T.,Ishisaki,Y.,Oga.sal_, Y.,Ueda,Y.,Freyberg, M.J.,Hasinger, G.,&Tanaka,Y.1998,A&A,

334,L13

Morrison, R.,& McCammon, D.1983,ApJ,270,119

Mortiz,P.,Wennmacher, A.,Herbstmeier, U.Mebold,U.,Egger,R.,&Snowden, S.1998,A&A,336,682

Mushotzky, R.F.,Cowie,L.L.,Barger,A.J.,&Arnaud,K.A.2000,Nature,inpress

Ostriker,J.P.,& Cen,R.1996,ApJ,464,27

Ottmarm,R.,Fleming,T. A.,&Pasqnini, L. 1997,A&A,322,785

Ottmann, R., & Schmitt, J. H. M. M. 1992, A&A, 256, 421

Paresce, F. 1984, A J, 89, 1022

Parmar, A. N., Guainazzi, M., Oosterbroek, T., Orr, A., Favata, F., Lumb, D., & Malizia, A. 1999, A&A,

submitted

Peebles, P. J. E. 1993, Principles of Physical Cosmology (Princeton University Press)

Pierre, M., Bryan, G., & Gastaud, R. 2000, A&A, submitted

Pietz, J., Kerp, J., Kalberla, P., Burton, W. B., Hartmann, D., & Mebold, U. 1998, A&A, 332, 55

Plucinsky, P. P., Snowden, S. L., Briel, U. G., Hasinger, G., & Pfeffermarm, E. 1993, ApJ, 418, 519

Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. 1986, Numerical Recipes (Cambridge

University Press)

Prieto, M. A., Ha.singer, G., & Snowden, S. L. 1996, A&A, 120, 187

Rand, R. J. 1998, Proc. Astron. Soc. Aust., 15, 106

Rand, R. J., Kulkarni, S. R., & Hester, J. J. 1992, ApJ, 396, 97

Rand, R. J., & Stone, J. M. 1996, A J, 111, 190

Randich, S., & Schmitt, J. H. M. M. 1995, A&A, 298, 115

Randich, S., Schmitt, J. H. M. M., & Prosser, C. 1996, A&A, 313, 815

Raymond, g. C. 1992, ApJ, 384, 502

Raymond, J. C., & Smith, B. W. 1977, ApJS, 35

Reach, W. T., Koo, B.-C., & Heiles, C. 1994, ApJ, 429, 672

Reach, W. T., Wall, W. F., & Odegard, N. 1998, ApJ, 507, 507

Read, A. M., Ponman, T. J., & Strickland, D. K. 1997, MNRAS, 286, 626,668

Reynolds, R. J. 1985, ApJ, 333, 341

Reynolds, R. J. 1991, in The Interstellar Disk-Halo Connection in Galaxies, ed. H. Bloemen, 67

Reynolds, R. J. 1999, private communication

Reynolds, R. J., Tufte, S. L., Haffner, L. M., Jaenig, K., & Percival, J. W. 1998, Proc. Astron. Soc. Aust.,

15, 14

Reynolds, R. J., Tufte, S. L., Kung, D. T., McCullough, P. R., & Heiles, C. 1995, ApJ, 448, 715

Sanders, W. T., Burrows, D. N., McCammon, D., & Kraushaar, W. L. 1983, in Supernova Remnants and

their X-ray Emission, ed. J. Danziger & P. Gorenstien, 361

Sanders, W. T., Edgar, R. J., Liedahl, D. A., & Morgenthaler, J. P. 1998, in The Local Bubble, ed.

D. Breitschwerdt & M. Freyberg, 83

Sanders, W. T., Kraushaar, W. L., Nousek, J. A., & Fried, P. M. 1977, ApJ, 217, 1,87

Savage, B. D., Bohlin, R. C., Drake, J. F., & Budich, W. 1977, ApJ, 216, 291

Scalo, J. M. 1986, F£mdamentals of Co6mic Physics, 11, 1

Scharf, C., Donahue, M., Voit, G. M., Rosati, P., & Postman, M. 2000, ApJ, 528, L73

Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525

Schmidt, M., et al. 1998, A&A, 329, 495

Schmitt, J. H. M. M. 1985, ApJ, 293, 178

Schmitt, J. H. M. M. 1997, A&A, 318, 215

Schmitt, J. H. M. M., Fleming, T. A., & Giampapa, M. S. 1995, ApJ, 450, 392

Schmitt, J. H. M. M., Snowden, S. L., Aschenbach, B., Hasinger, G., Pfeffermann, E., Predehl, P., &

Trfimper, J. 1991, Nature, 349, 583

Schmutzler, T., & Tscharnuter, W. M. 1993, A&A, 273, 318

SchrSder, K.-P., Hiinsch, M., & Schmitt, J. H. M. M. 1998, A&A. 335, 591

Sfeir, D., Lallement, R., Crifo, F., & Welsh, B. Y. 1999, A&A, submitted

Shelton, R. L. 1998, ApJ, 504, 785

Shelton, R. L. 1999a, ApJ, 521,217

Shelton, R. L. 1999b, private communication

Sidher, S. D., Summer, T. J., Quenby, J. J., & Gambhir, M. 1996, A&A, 305, 308

Silverberg, R. F., Hauser, M. G., Boggess, N. W., Kelsall, T. J., Moseley, S. H., & Murdock, T. L. 1993,

Proc. SPIE, 2019, 180

Smith, R. K. 1997, Ph.D. thesis, University of Wisconsin

255
Snowden, S. L.

Snowden, S. L.,

Snowden, S. L.,

Snowden, S. L.,

Snowden, S. L.,

Snowden, S. L.,

Snowden, S. L.,

Snowden, S. L.,

430, 601

1998, ApJS, 117, 223

Cox, D. P., McCammon, D., & Sanders, W. T. 1990, ApJ, 354, 211

Egger, R., Finkbeiner, D., Freyberg, M. J., & Plucinsky, P. P. 1998, ApJ, 493, 715

et al. 1997, ApJ, 485, 125

& Freyberg, M. J. 1993, ApJ, 404,403

Freyberg, M. J., Kuntz, K. D., & Sanders, W. T. 2000, ApJS, in press

et al. 1995, ApJ, 454, 643

Hasinger, G., Jahoda, K., Lockman, F. J., McCammon, D., & Sanders, W. T. 1994a, ApJ,

Snowden, S. L., & Kuntz, K. D. 1998, Cookbook for analysis procdures for rosat XRT Observations of

Extended Objects and the Diffuse Background, Part |: Individual Observations, Technical report,

NASA/GSFC

Snowden, S. L., McCammon, D., Burrows, D. N., & Mendenhall, J. A. 1994b, ApJ, 424, "714

Snowden, S. L., McCammon, D., & Verter, F. 1993, ApJ, 409, L21

Snowden, S. L., Mebold, U., Hirth, W., Herbstmeier, U., & Schmitt, J. H. M. M. 1991, Science, 252, 1529

Snowden, S. L., & Pietsch, W. 1995, ApJ, 452, 627

Snowden, S. L., Plucitmky, P. P., Briel, U., Hasinger, G., & Pfeffermann, E. 1992, ApJ, 393, 819

Snowden, S. L., Schmitt, J. H. M. M., & Edwards, B. C. 1990, ApJ, 364, 118

Snowden, S. L., Turner, T. J., & Freyberg, M. J. 1999, A&A

Sodroski, T. J., Odegard, N., Arendt, R. G., Dwek, E., Weiland, J. L., Hauser, M. G., & Kelsall, T. 1997,

ApJ, 480, 1"73

Soltan, A., & Ha_inger, G. 1994, A&A, 288, 77

Soltan, A. M., Hasinger, G., Egger, R., Snowden, S.,& Triimper, J. 1996, A&A, 305, 17

Spitzer, L. 1956, ApJ, 124, 20

Stark, A. A., Gammie, C. F., Wilson, R. W., Bally, J., Linke, R. A., Heiles, C., & Hurwitz, M. 1992, ApJS,

79, 77

Stauffer, J. R., Caillault, J.-P., Gagnd, M., Prosser, C. F., & Hartman, L. W. 1994, ApJS, 91, 625

Stem, R. A., Schmitt, J. H. M. M., & Kahabka, P. T. 1995, ApJ, 448, 683

Strang, G. 1976, Linear Algebra and its Applications (Academic Press)

Tanaka, Y., lnoue, H., & Holt, S. S. 1994, PASJ, 46, L37

Vallerga, J., & Slavin, J. 1998, in The Local Bubble, ed. D. Breitschwerdt & M. Freyberg, 79

Vikhlinin, A., & Forman, W. 1995, ApJ, 455, L109

Voges, W., et al. 1992, in Proceedings of Satellite Symposium 3, Space Science with Particular Emphasis

on High-Energy Astrophysics, ed. T. D. Guyenne & J. J. Hunt, 223

Voges, W., et al. 1996, http:/]www.rosat.mpe-garching.mpg.de/survey/rass-bsc/

Voges, W., et al. 1999, A&A, 349, 389

Vogler, A., Pietsch, W., & Kahablm, P. 1996, A&A, 305, 74

Wakker, B. P., & Boulanger, F. 1986, A&A, 170, 84

Wang, Q. D. 1997, in The Local Bubble, ed. D. Breitschwerdt & M. Freyberg, 503

Wang, Q. D., Walterbos, R. A. M., Steakley, M. F., Norman, C. A., & Braun, R. 1995, ApJ, 439, 176

Warwick, R., Hutchinson, I., Willingale, R., Kuntz, K., & Snowden, S. 1997, in The Local Bubble, ed.

D. Breitschwerdt & M. Freyber 8, 321

Warwick, R. S., & Roberts, T. P. 1998, Astronomische Nachrichten, 319, 59

Weliachew, L., Sancisi, R., & Gudlin, M. 1978, A&A, 65, 37

Wu, K. K. S., Fabian, A. C., & Nulsen, P. E. J. 1999, MNRAS, in press

Yah, M., Sadeghpour, H. R., & Dalgarno, A. 1998, ApJ, 496, 1044

Zamorani, G., et al. 1999, A&A, 346, 731

Zhang, Y., Anninos, P., Norman, M. L., & Meiksin, A. 1997, ApJ, 485, 496

256

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4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Small-Scale Spatial Fluctuations in the Soft X-Ray Background 662

6. AUTHOR(S)

K.D. Kuntz

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS (ES)

Goddard Space Flight Center

Greenbelt, Maryland 20771

9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS (ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

8. PEFORMING ORGANIZATION

REPORT NUMBER

2001-01596-0

10. SPONSORINGIMONITORING

AGENCYREPORTNUMBER

TM--2001-209981

11.SUPPLEMENTARY NOTES

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13. ABSTRACT (Maximum 200 words)

In order to isolate the diffuse extragalactic component of the soft X-ray background, we have used a combination of ROSAT

All-Sky Survey and IRAS 100 pm data to separate the soft X-ray background into five components. We find a Local Hot

Bubble similar to that described by Snowden et al (1998). We make a first calculation of the contribution by unresolved

Galactic stars to the diffuse background. We constrain the normalization of the Extragalactic Power Law (the contribution

of the unresolved extragalactic point sources such as AGN, QSO's, and normal galaxies) to 9.5_-/-0.9 keV cm 2s-_ sr_keV -',

assuming a power-law index of 1.46. We show that the remaining emission, which is some combination of Galactic halo

emission and the putative diffuse extragalactic emission, must be composed of at least two components which we have

characterized by thermal spectra. The softer component has log T ~ 6.08 and a patchy distribution; thus it is most probably

part of the Galactic halo. The harder component has log T - 6.46 and is nearly isotropic; some portion may be due to the

Galactic halo and some portion may be due to the diffuse extragalactic emission. The maximum upper limit to the strength

of the emission by the diffuse extragalactic component is the total of the hard component, ~ 7.4+1.0 keV cm--" s-_ sr -_ keV -_ in

the _ keV band. We have made the first direct measure of the fluctuations due to the diffuse extragalactic emission in the _4

keV band. Physical arguments suggest that small angular scale (- 10") fluctuations in the Local Hot Bubble or the Galactic

halo will have very short dissipation times (- 105 years). Therefore, the fluctuation spectrum of the soft X-ray background

should measure the distribution of the diffuse extragalactic emission. Using mosaics of deep, overlapping PSPC pointings,

we find an autocorrelation function value of - 0.0025 for 10" < 0 < 20", and a value consistent with zero on larger scales.

Measurement of the fluctuations with a (5 1/I method produces consistent results.

14. SUBJECT TERMS

Soft x-ray background; ROSAT; IRAS.

17. oFSECURITY CLASSIFICATIONunclassifiedREPORT 118" oFSECURITY CLASSIFICATIONunclassifiedTHIS PAGE I

NSN 7540-01-280-5500

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OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

256

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UL

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