NASA/TMm2000-209973

for

of

a Methodology

Measurements

and

the Solar Corona

11August 1999 in Elazig,Turkey

and

Center

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NASA/TMm2000-209973

MACS, An Instrument and a Methodology

for Simultaneous and Global Measurements

of the Coronal Electron Temperature and

the Solar Wind Velocity on the Solar Corona

Nelson Leslie Reginald

University of Delaware, Newark

National Aeronautics and

Space Administration

Goddard Space Flight Center

Greenbelt, Maryland 20771

December 2000
NASA Center for AeroSpace Information

7121 Standard Drive

Hanover, MD 21076-1320

Price Code: A17

Available from:

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22161

Price Code: A10
MACS, AN INSTRUMENT, AND A METHODOLOGY FOR

SIMULTANEOUS AND GLOBAL MEASUREMENTS OF THE CORONAL

ELECTRON TEMPERATURE AND THE SOLAR WIND VELOCITY

ON THE SOLAR CORONA

by

Nelson Leslie Reginald

A dissertation submitted to the Faculty of the University of Delaware in partial

fulfillment of the requirements for the degree of Doctor of Philosophy with a major in

Physics

Fall 2000

© 2000 Nelson Leslie Reginald

All Rights Reserved
- MACS, AN INSTRUMENT, AND A METHODOLOGY FOR

SIMULTANEOUS AND GLOBAL MEASUREMENTS OF THE CORONAL

ELECTRON TEMPERATURE AND THE SOLAR WIND VELOCITY

ON THE SOLAR CORONA

Approved _ %

Henry R. Glyde, Ph.D.

Chairman, Department of Physics and Astronomy

Vice Provost for Academic Programs and Planning
tce, r_ify that I have .read this _ssertadon _md thai in my opinion it meets the academic

a_d p,rofession_ standard required by the University as a dissertation for the degree of

Doctor of Philosophy,

Dem_.ot_ J, Mullah, Ph.D. "

Professor in charge of dissertation

[ certify" ttmt I have read this dissertation and that in my opinion it meets the academic

and professional st_mdard required by the University as a dissertation for the degree of

Doctor of Philosophy.

ce_ify that Ii have read this dissertation and tt_at in my opinion i_ meets _he academic

and proik, ssional standard required by the Universit}, as a dissertation for the degree of

Doctor of Philosophy,

Si

Sta_fley P. Owocki, Ph,D.

Member of dissertation committee

t ceru_ that I have _ead tt_:_sdisserta_m_ and that m m} opmmn _t meets tt_e academic

_d professional st_mdard required by the University as a disse_ation tbr the degree of

Doctor of Philosophy,

cerfii\'_' that _[have mad thisdissertation and that in my opinion it meets the academic

George H. Watson,

Mere[mr of dissertation committee
Acknowledgements

I am greatly proud for the opportunity afforded to conduct my Ph.D. dissertation

research at the Solar Physics Division of the NASA's Goddard Space Flight Center in

Greenbelt, Maryland, USA under the able guidance of Dr. Joseph M. Davila. I am also

thankful for the opportunity provided to participate in the assembly of the SERTS (Solar

Extreme Ultraviolet Rocket Telescope) payload and its launch on the 26 July 2000 from

the White Sands Missile Range, New Mexico.

In this regard I wish to convey my hearty thanks to Dr. Dermott J. Mullan for the

trust and faith in recommending me for the graduate research position offered by Dr.

Davila, for providing the funds to attend the International Summer School on " The Sun

As Seen From Space" in L'Aquila, Italy (1997) and for serving as the chairman of the

Ph.D. thesis committee.

This research that spanned over the years 1998 to 2000 involved in the theoretical

modeling, the design and construction of an instrument, an eclipse expedition to Turkey

(1999), the analyses and travel to the AAS meetings in Chicago (1999) & Nevada (2000),

the TRACE workshop in California (1999), the AGU conference in Washington D.C

iv
(2000) and the IAP "Eclipses and the Solar Corona" conference in France (2000), thus

molding me for a successful scientific career.

My sincere thanks are also extended to Mr. Chuck Condor for helping with all the

encouragement and logistics for the eclipse expedition to Turkey, and to Mr. Larry White

for the long hours spent on the engineering drawings and fabrication of MACS.

I also wish to convey my regards to the staff of my high school at S. Thomas'

College, Mount Lavinia, Sri Lanka and my undergraduate institute at the University of

Peradeniya in Sri Lanka, my MS thesis advisor and chairman Prof. James MacDonald at

the University of Delaware and for the full tuition scholarship and teaching assistantship

awarded by the University of Delaware, which enabled me to travel to the USA in 1994

in pursuance with my graduate studies.

My sincere thanks is also extended to Miss. Elizabeth (Sugar Bug) Chapman for

her help in preparing the text.

V
Dedicated to my loving parents

Gnaniah J. Reginald & Beryl A. Reginald

vi
TABLE OF CONTENTS

LIST OF TABLES ............................................................................... xii

LIST OF FIGURES ................................................................................ xv

ABSTRACT ................................................................................... xxxviii

Chapter

1 INTRODUCTION ......................................................................... 1

1.1 Introduction to the solar corona .................................................. 1

1.2 Other methods to measure the solar wind velocity and the coronal

temperature ....................................................................... 12

2 FORMATION OF THE K-CORONA AND THE FOCUS OF THIS

RESEARCH .............................................................................. 18

2.1

2.2

2.3

2.4

2.5

2.6

K-corona .......................................................................... 18

The photospheric spectrum ...................................................... 18

The scattering source ............................................................. 20

The effects on the incident radiation by the scattering source .............. 21

The K-coronal spectrum ......................................................... 22

Further explanation for the formation of the temperature sensitive

anti- nodes ........................................................................ 29

vii
3

4

2.9

2.10

Theinfluenceof thesolarwind ontheK-coronalspectrum................ 38

Simultaneousdeterminationof thecoronaltemperatureandthe

solarwind velocity............................................................... 44

Globaldeterminationof thecoronaltemperatureandthe solarwind

velocity............................................................................ 47

Theessenceof this dissertation................................................. 48

THE INSTRUMENT .................................................................... 49

3.1 Overview of the instrument ...................................................... 49

3.2 Description of the optical elements in detail .................................. 58

3.2.1 The telescope ............................................................ 58

3.2.2 The fibers .................................................................. 61

3.2.3 The lenses .................................................................. 65

3.2.4 The grating ................................................................. 67

3.2.5 The camera ................................................................. 68

3.3 Analytical expressions for the integration time, the spectral

resolution and the spatial resolution .......................................... 69

3.4 Cost of the instrument .......................................................... 87

OBSERVATIONAL RESULTS AND ANALYSES .............................. 88

4.1

4.2

4.3

Observational site ................................................................ 88

Spectra observed by MACS ..................................................... 90

Identifying the individual fiber location on the corona ...................... 93

viii
5

6

7

4.4

4.5

4.6

Thetelescope-spectrometer sensitivitycurve............................. 108

Application of the intensity correction factor ............................. 115

Determination of the Thermal electron temperature and the Solar

wind velocity for each fiber location ....................................... 116

Sources of error and error bars .................................................. 132

Summary ........................................................................ 135

MODIFIED CRAM'S THEORY AND ITS DEPENDENCE ON

VARIOUS PARAMETERS .......................................................... 136

5.1 Modified Cram' s theory ....................................................... 136

5.2 Dependence on the electron density distribution function ................ 140

5.3 Dependence on the temperature profile ...................................... 155

5.4 Dependence on the wind profile .............................................. 158

5.5 Dependence on the numerical method ........................................ 162

COMPARISON BETWEEN AND OBSERVATION AND THEORY ...... 167

6.1 An unexpected phenomenon noticed in the observations ................. 167

6.2 Comparison between theory and observation .............................. 169

6.3 Remedial measures ............................................................. 179

CONCLUSION ......................................................................... 185

7.1 Summary of this thesis ......................................................... 185

7.2 Using a space platform .......................................................... 188

ix
Appendix

A THE THEORY OF FORMATION OF THE K-CORONA ...................

B

197

A. 1 The general theory of scattering of radiation by free electrons .......... 197

A.2 The total power radiated per unit solid angle for the cases where

the electric vector is aligned parallel and perpendicular to the

scattering plane ................................................................. 206

A.3 Contribution due to Compton scattering .................................... 216

A.4 Expression for the total K-coronal scattered intensity ..................... 218

A.5 Expressions for the dependent variables in equation (A.78) in terms

of the independent variables .................................................. 226

A.6 Expression for the incident solar intensity on the coronal electrons .... 230

A.7 Final expression for the observed intensity ................................. 233

CODE .................................................................................... 235

B. 1 The basic equation to be solved .............................................. 235

B.2 Approximations to the _d_' integral ....................................... 237

B.3 b, Q_/a and Q_a in terms of the independent variables co, q) and x ..... 239

B.4 Rewriting the basic equation with all parameters expressed in terms

of the independent variables (co, % x, y) .................................... 241

B.5 Changing the independent variable cos co to cos e ........................ 243

B.6 Expression for the extraterrestrial solar irradiance f(_) .................. 247

B.7 Expressions for the limb-darkening coefficient u 1(_) ..................... 248

X
B.8 Expression for the coronal electron density model

N e (_-5 +p 2 R solar / ............................................................ 250

B.9 Expressions for the coronal temperature model T (x/-x2 + p2 R solar)

and the solar wind model W(qFx-7 + p2Rsolar) .............................. 251

B.10 Evaluation of the integrals ..................................................... 252

B.11 The symbols used in the code ................................................ 264

B.12 The IDLcode ................................................................... 267

C A STREAMER MODEL ............................................................. 276

C. 1 The effects due to streamers .................................................. 276

REFERENCES .................................................................................. 285

xi
LIST OF TABLES

3.1

3.2

3.3

3.4

A table showing the values for the angle of diffraction, the linear dispersion,

the line width and the resolvable theoretical wavelength range for three

different wavelengths using the physical parameters of the optical components

used in MACS. In column 3 the reciprocal linear dispersion is calculated per

pixel, which is the dimension of a pixel in the CCD camera used in MACS ...... 79

The transmission efficiencies of the fibers, the lenses and the grating for three

different wavelengths. This information is from figure (3.9), figure (3.11) and

figure (3.12) ............................................................................... 83

The list of lenses and the associated number of surfaces through which the

incident light had to pass before entering the detector .............................. 83

The effective transmission effciency ('c_)err of MACS for the wavelengths

3600.0, 4000.0 and 4500.0 angstroms. The values are based on the

information from tables (3.2) and (3.3) ................................................ 84

xii
3.5

3.6

4.2

6.1

TheK-coronalbrightnessat 1.1and 1.6solarradii for threedifferent

wavelengthsduringthesolarmaximum.Thevalueswereobtainedfrom Allen

(1973)from pages172and176.Theunitsarein ergs/sec/cmZ/angs/steradians.

(1 Joule=107ergs)......................................................................... 84

Lowerlimit of theintegrationtime for theK-coronal spectrumfor MACS

basedonequation(3.36)................................................................. 85

Costassociatedwith theconstructionof MACS..................................... 87

List of theemissionlinesreportedby Athay andOrrall (1957)of aprominence

from thetotal solareclipseof 1952.In thispapertheSrII line at 4077.7has

beenerroneouslyrecordedas3077.7angstrom.In thefollowing tableonly the

emissionlinesthatcouldlie within thewavelengthrangeof MACS is

enumerated................................................................................. 96

List of theemissionlinesreportedby Jefferies,SmithandSmith(1959)from

theflareof 18September1957.Thistableonly includesthoseemissionlines

categorizedasvery strongandstrong.Also in this following tableonlythe

emissionlinesthatcouldlie within thewavelengthrangeof MACS is

enumerated................................................................................ 100

Comparisonof thewind andthetemperaturesensitiveintensityratiosbetween

theobservationalandthetheoreticalK-coronalspectrum,for fiber #04.These

valuesarebasedonfigure (6.5)........................................................ 177

xiii
B.1

B.4

Thedependenceof theextraterrestrialsolarirradianceandthelimb-darkening

functionswith thewavelength..Thecolumns1and4 arefrom section(B.6)

andcolumns2 and3 arefrom section(B.7)......................................... 259

Cumulativevaluesof thecolumns2,3and4 of table(B.1)........................ 260

Multiplying column4 of table(B.2) with column2 and3of table(B.2).This

is theformatof thetableusedby thecode........................................... 260

Theformat of theinput informationonthewind velocity in km/sec,

isothermaltemperaturein MK andthedistanceto theline of sightin SR....... 275

xiv
LIST OF FIGURES

1.1

1.2

1.3

2.1

2.2

The solar corona as seen during the solar eclipse of February 26, 1998.

Picture: NASA eclipse library ........................................................... 1

This plot shows the variation of the E, K and F components of the solar

coronal brightness with height above the solar limb .................................. 7

A cross section of the sun to highlight the prominent parts of

it and the temperature scales between the core, photosphere and the

corona. Picture: SOHO pictures library ................................................. 9

This is a plot of the extraterrestrial solar irradiance spectrum. This

was obtained by starting with the ground-based Fourier Transform

Spectrometer at the McMath/Pierce Solar Telescope at Kitt Peak,

Arizona, and then correcting for wavelength-dependent absorption

in the Earth's atmosphere ................................................................ 19

The photospheric radiation is emitted from a point S on the surface of

the sun. The electrons at point P in the corona scatter some of the

incident radiation towards a terrestrial observer at E, which is known as

K-coronal radiation ....................................................................... 23

xv
2.3

2.4

2.5

2.6

2.7

2.8

This plot shows the modeled K-coronal intensity spectra against

wavelength. The composite plots are for different isothermal

temperatures assumed for the solar corona for a given line of sight at

1.1 solar radii .............................................................................. 24

Modeled K-coronal intensity spectrum against wavelength for different

isothermal coronal temperatures for a given line of sight at 1.3 solar radii ....... 25

Modeled K-coronal intensity spectrum against wavelength for different

isothermal coronal temperatures for a given line of sight at 1.5 solar radii ....... 26

This is a plot of the K-coronal intensity ratios at 3850.0 to 4100.0

angstroms against the assumed isothermal coronal temperatures. The

ratios were calculated from the modeled K-coronal spectra for

different temperatures shown in figure (2.3). These values pertain to

a line of sight at 1.1 solar radii .......................................................... 27

An input intensity spectrum containing an absorption line centered

at 4000.0 angstrom with a FWHM of 40.0 angstrom in an otherwise

uniform continuum ........................................................................ 29

The theoretical scattered intensity spectrum by the coronal electrons for

an input spectrum given by figure (2.7). The theory assumes an

isothermal corona. The plots show that the smearing of the absorption

line in the scattered increases with increasing temperatures ........................ 30

xvi
2.9

2.10

2.11

2.12

An inputintensityspectrumcontainingtwo absorptionlinescentered

at 4000.0and4500.0angstromswith aFWHM of 40.0angstromin an

otherwiseuniform continuum........................................................... 31

Thetheoreticalscatteredintensityspectrumby thecoronalelectronsfor

aninput spectrumgivenby figure (2.9).Thetheoryassumesan

isothermalcorona.Thedifferencein thedepthsof theabsorptionlines

in thescatteredspectrumis dueto thewavelengthdependent

limb-darkeningcoefficient............................................................... 32

An inputintensityspectrumconsistingof two absorptionlinescenteredat

4000.0and4500.0angstromswhileanemissionline centeredat 4250.0

angstromandwith a FWHM of 40.0angstromfor theabsorptionand

emissionlinesin anotherwiseuniform continuum................................... 33

Thetheoreticalscatteredintensityspectrumby thecoronalelectronsfor an

input spectrumgivenby figure (2.11).Thetheoryassumesanisothermal

corona.Thedifferencein thedepthsof theabsorptionlinesin the

scatteredspectrumisdueto thewavelengthdependentlimb-darkening

coefficient................................................................................. 34

xvii
2.13

2.14

2.15

2.16

Thisfigure showsthetheoreticalK-coronalintensityspectrafor four

isothermalcoronaltemperatureswhoseelectrondensityis givenby

equation(5.8).Theabsoluteintensityof thedisk centerphotospheric

spectrumis alsoshown.Theverticalbarsatthebottomof thefigure

indicatetherelativestrengthsof thethreecoronallines"c", and

nineflashspectrumlines"f'. Reproducedfrom Cram(1976)...................... 37

Thisis aschematicdrawingto illustratetheredshiftingof the

photosphericradiationscatteredoff thecoronalelectrons,which are

radiallyflowing awayfrom the sunat auniformvelocity (w).

Electron2 is locatedat theclosestpoint ontheline of sightto thecenter

of thesun.Electrons1and3 arelocatedsymmetricallyoneithersides

of electron2............................................................................... 39

Thisfigure showsthescaleof thethermalDopplerbroadeningandthe

redshiftassociatedwith thesolarwind velocity in thewavelengthscale

in comparisonto thefeaturesof thephotosphericspectrum.Thescaleof

theredshiftis shownby areddot, whichis highlightedby acircle.

Thiscomparisonis madefor anincidentwavelengthof 4000.0angstromon

thecoronalelectrons,whichareradially outflowingat avelocity of

300.0km/secin amillion degreeKelvin corona...................................... 40

ModeledK-coronalintensityagainstwavelengthfor differentisothermal

coronaltemperatures,for line of sightat 1.1solarradii andawind

velocity of 400.0km/sec................................................................. 41

ooo

XVlll
2.17

2.18

2.19

2.20

3.1

3.2

Modeled K-coronal intensity against wavelength for different isothermal

coronal temperatures, for line of sight at 1.1 solar radii and a wind

velocity of 800.0 km/sec ................................................................ 42

Modeled K-coronal intensity against wavelength for solar wind velocities

of 0.0 km/sec to 900.0 km/sec in intervals of 100.0 km/sec, for line of sight

at 1.1 solar radii and an isothermal coronal temperature of 1.0 MK .............. 43

The intensity ratio at 3850.0 to 4100.0 angstroms against isothermal

coronal temperatures for a range of solar wind velocities with the line of

sight at 1.1 solar radii. The horizontal spread in the low end of the

temperature scale is N0.02 MK over the velocity range of 0.0 km/sec

to 900.0 km/sec ........................................................................... 45

The intensity ratio at 3987.0 to 4233.0 angstroms against solar wind velocities

for a range of isothermal coronal temperatures with the line of sight at 1.1

solar radii. To determine the wind velocity, first the temperature has to

be determined from figure (2.19). Then the K-coronal models need to be

created for a range of wind velocities at this temperature and plotted in

figure (2.20) ............................................................................... 46

Meade 12-inch, F/10.0 Schmidt-Cassegrain telescope with auto

tracking capabilities ...................................................................... 51

1=/6.3 focal reducer reduces the focal length by a factor of 0.63. The device

shown in figure (3.4) was placed at the focal plane behind the 1=/6.3 focal

reducer ..................................................................................... 52

xix
3.3

3.4

3.5

3.6

3.7

3.8

3.9

A drawingdepictingthelocationof thefibertips ontheimageplane

of thetelescope.Theinnerandouterrings areat 1.1and 1.5solar

radii,respectively.Duringtheeclipsethelunarshadowreplacesthesun......... 53

Theglassplateimbeddedwith twenty-onefibers.Theinnerandtheouter

circlesarelocatedat 1.1and1.5solarradii,respectively,ontheimageplane

of thesun.Thefiber atthecenterwasto recordthebackgroundsignalduring

theeclipse.................................................................................. 54

Thisis apictureof thespectrograph.Thecoronallight wasfedinto

thespectrographby twenty-onefibers thatwerevertically arrangedata

regularspacing.Itslocationin the spectrographwasatthefocalpoint of

thecollimatinglens....................................................................... 55

Thisis apictureof theinsideof the spectrographbetweenthegratingand

thecameralens.Baffling wereusedto preventall ordersbut thefirst order

from enteringintothecameralens...................................................... 56

Thisis apictureof thethermoelectricallycooledCCD camerausedin

theexperiment.Thefront endhousedthecameralensthatfocusedthe

dispersedlight from thegratingonto thecamera.................................... 57

Schematicdrawingontheimageformationby theprimarymirror of

thetelescope............................................................................... 59

Transmissionefficiencyof thefibersusedin MACS andmarkedasUV/VIS.

Theregionof wavelengthinterestin MACS wasfrom

3500.0-4500.0angstroms............................................................... 62

XX
3.10

3.11

3.12

3.13

3.14

3.15

3.16

3.17

3.18

Schematic diagram depicting the light ray path at the focal plane of the

telescope into a fiber optic cable ........................................................ 63

Reflection properties for the anti UV-reflective coating on the collimating

and the camera lenses used in the spectrograph ....................................... 66

Transmission properties at normal incidence for the anti UV-reflection coating

on the transmission grating. Here the back surface reflection is not

considered. The substrate material is fused silica. The black squares pertain

to the grating used in MACS ............................................................ 67

Quantum efficiency (QE) curve for the back thinned SITE

512.0>(512.0, 24.0gin square chip used in MACS. This chip also featured a

full well depth in excess of 300,000.0 electrons, a dynamic range of

N 90.0 dB and a readout time of N 5.0 seconds with an ISA card .................. 68

Schematic diagrams of the optical components that make up MACS and

the parameters that determine the spatial and the spectral resolutions ............ 69

Schematic diagram showing the projected width and height of a single fiber

on the camera plane ...................................................................... 71

Schematic diagram of the dispersing element showing the angular and the

linear dispersions ......................................................................... 72

Schematic layout of the slit spectrometer. In this diagram collimated light

is incident on the grating ................................................................. 73

Schematic diagrams showing the diffraction limited spectral resolution and

the resolving power ....................................................................... 76

xxi
4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

A photographof theAugust1999eclipsefrom theBlackSeaarea.

Thephotographalsorevealsadetachedprominenceeruptionabovethe

southwestlimb............................................................................ 89

A mapshowingthepathof theeclipseoverTurkey.

Courtesy:NASA ReferencePublication1398........................................ 89

A pictureshowingthelayoutof MACS andthetelescope.Thetelescopeis

onapolarmountandMACS is laid out onthemetalbox........................... 90

Thespectrafrom twenty-onefibersexposedto theskywhile four other

fibersexposedto themercurycalibrationlampin Elazig,Turkey................. 91

Thecoronalspectrarecordedby MACS for theexposuretimes4.0,8.0and

83.0seconds,respectively.Eachverticalline correspondsto aspectrum

from a singlefiber. It is alsoclearthatfive of thefibers showemissionlines... 93

Thispictureshowstheenvisagedlocationof thetwenty-onefibers in the

focalplaneduringtheeclipse.Theinnerandtheouterringscorrespondto 1.1

and1.5solarradii,respectively.Thepictureisnot theactualeclipseon 11

August1999.Thefiber in thecenterwasexpectedto recordthe

backgroundsignal........................................................................ 94

Thespectrumrecordedby thefiber #19in the4.0secondsexposure.The

emissionlinesandtheir correspondingwavelengthsandpixel positions

areidentified.............................................................................. 97

Theregressionfit betweentheemissionline peaksandthepeakpositions

of thefiber #19in the4.0 secondsexposure.Thetableincorporatedshows

thewavelengthscalebetweenanytwo-emissionlines.............................. 98

xxii
4.9

4.10

4.11

4.12

4.13

4.14

Thespectrumrecordedby thefiber #20in the4.0secondsexposure.This

toomatcheswith all theemissionlinesrecordedby thefiber#19as

shownin figure (4.7)..................................................................... 99

Thespectrumrecordedby thefiber #22in the4.0secondsexposure.The

emissionlinesandtheir correspondingwavelengthsandpixel positionsare

identified.................................................................................. 101

Theregressionfit betweentheemissionline peaksandthepeakpositions

of thefiber#19in the4.0 secondsexposure.Thetableincorporated

showsthewavelengthscalebetweenanytwo-emissionlines..................... 102

Thisis acomparisonbetweensolarimageby SOHOin 304.0angstromat

11:14UT andaneclipsephotographtakenfrom theBlackSeaat N11"16UT.

Theorientationwasmadeby matchingtheprominenceeruptionN68o

southwestin theSOHOimage.Theorientationof thepolaraxisof thesun

with respectto thepolaraxisof theearthwasobtainedfrom thep-angle

datagivenin, 'The AstronomicalAlmanac1999'for 11August1999........... 104

Imagesof the suntakenby theLASCOinstrumentonSOHOonthe11

August1999.Thenumbersdenotethetimein UT. C2andC3arecoronal

imagesof thesunfrom 2.0- 6.0solarradii and3.7- 32.0solar

radii,respectively,usinganexternallyocculteddisk............................... 105

Imagesof the suntakenby theEIT instrumentonSOHOonthe 11August

1999.Thenumbersdenotethetimein UT........................................... 106

ooo

XXlll
4.15

4.16

4.17

4.18

4.19

4.20

4.21

Images of the sun taken by the radiograph in Nancy, France and the

x-ray telescope on Yohkoh on the 11 August 1999 ................................. 107

Plots of the sky spectrum recorded by fiber #06 in Elazig, Turkey before and

after correcting for Rayleigh scattering. The plots have been normalized at

3968.5 Angstrom ......................................................................... 109

Plots of the extraterrestrial photospheric intensity spectrum and the

terrestrial photospheric intensity spectrum recorded by fiber #06. The

extraterrestrial photospheric spectrum is a smoothed version of the

extraterrestrial photospheric spectrum shown in figure (2.1) ...................... 110

Plots of the smoothed extraterrestrial photospheric intensity spectrum and

the terrestrial photospheric intensity spectrum recorded by fiber #06 ............ 111

Plots of the smoothed extraterrestrial photospheric intensity spectrum,

terrestrial photospheric intensity spectrum recorded by fiber #06 and the

ratio between these two against wavelength .......................................... 112

Polynomial fit made to the ratio between the extraterrestrial

photospheric spectrum and the terrestrial photospheric intensity spectrum.

The numbers on the plot are the coefficients of the polynomial fit ............... 113

Application of the polynomial fit shown in figure (4.20) to the

terrestrial photospheric intensity spectrum to obtain the

extraterrestrial photospheric intensity. Also plotted for comparison purpose is

the smoothed extraterrestrial photospheric intensity spectrum as shown in

figure (4.18) .............................................................................. 114

xxiv
4.22

4.23

4.24

4.25

4.26

4.27

4.28

4.29

4.30

Theseplots showthecoronalspectrarecordedby thetwenty-onefibers

locatedatvariouslatitudesandheightsonthecoronaasdepictedin figure

(4.12).Fiber#23wascenteredonthemoon.Theabovespectrawere

recordedin the 18.0secondsexposure.Thex-coordinateis in pixelunits...... 117

Thecoronalspectrumrecordedby fiber#06......................................... 119

TheterrestrialK-coronal spectrumreducedfrom thecoronalspectrum

recordedby fiber#06.................................................................... 120

Thisplot showstheextraterrestrialK-coronal spectrumderivedfor fiber#06.

Thewind andthetemperaturesensitiveintensityratiosare1.245and

1.488,respectively....................................................................... 121

Thetemperaturesensitiveplot. This assignsathermalelectrontemperature

of N1.73MK for fiber#06.............................................................. 122

Thewind sensitiveplot. This assignsasolarwind velocity of N467.0km/sec

for fiber#06.............................................................................. 122

Thetemperaturesensitiveplot. This assignsathermalelectrontemperature

of _1.29MK for fiber#04.............................................................. 123

Thewind sensitiveplot. This assignsasolarwind velocity of _300.0km/sec

for fiber#04.............................................................................. 123

Thisplot showstheextraterrestrialK-coronal spectrumfor fiber #10.The

wind andthetemperaturesensitiveintensityratiosare1.248and

1.524,respectively....................................................................... 124

xxv
4.31

4.32

4.33

4.34

4.35

4.36

4.37

The temperature sensitive plot. This assigns a thermal electron temperature

of N1.34 MK for fiber #10 .............................................................. 125

The wind sensitive plot. This assigns a solar wind velocity of N571.0 km/sec

for fiber #10 .............................................................................. 125

This plot shows the terrestrial coronal spectrum recorded by fiber #14.

The continuum is a measure of the terrestrial F+K corona. The E-corona is

removed by making Gaussian fits of the form given in equation (4.1) to the

individual emission lines and fitting the background given by equation (4.2)

to the continuum ......................................................................... 126

This plot shows the extraterrestrial K-coronal spectrum for fiber #14. This

is following the elimination of the E-corona from figure (4.33) .................. 127

The temperature sensitive plot. This assigns a thermal electron temperature

of _0.73 MK for fiber #14 .............................................................. 128

The wind sensitive plot. This assigns a solar wind velocity of _819.0 km/sec

for fiber #14 .............................................................................. 128

Plot showing the variation in the temperature-sensitive intensity ratio, from

the theoretical models without streamers, for a streamer of geometry described

in Appendix-C and density enhanced by a factor of 50.0 together with

various streamer inclinations in front and behind the solar limb .................. 129

xxvi
4.38

4.39

5.1

5.2

Thethermalelectrontemperatureandthesolarwindvelocity determinedfor

solarcoronallocationsobservedby fibers#04,06,08, 10,12,14& 22that

arelocatedat 1.1solarradii andfiber#15& 17locatedat 1.5solarradii.

Thesevaluesarebasedonthetheoreticalmodels.Thetemperatureandthe

wind velocity aremeasuredin MK andkm/sec,respectively.Howeverthe

wind measurementsseemtoofar high.In thisregardit is acknowledgedthat

theexperimentwasnot successfulin measuringthewindvelocitiesand

theabsolutevaluesshouldbedisregarded............................................ 131

Thespectrumrecordedby fiber #23,which wascenteredonthelunar

shadowduringtheeclipse.Its purposewasto measurethe

backgroundcounts....................................................................... 132

Thisis aschematicdiagramshowingthemathematicaldescriptionof

thescatteringphenomenonwherethephotosphericradiationfrom thesun

is scatteredby thefreeelectronslying alongtheline of sightof anobserver.

Theline of sightis at adistance(9 ) from the center of the sun ................... 137

A plot showing the solar radial log-intensity variation as a function of solar

radii (r) from the center of the sun. The solid curve represents the

measurements by November and Koutchmy (1996) while the dashed

curve represents the measurements by Baumbach (1937) based on several

eclipses observations .................................................................... 142

xxvii
5.3

5.4

5.5

5.6

5.7

A plot showingthelog of theelectrondensityvariationasafunctionof the

solarradii (r) from thecenterof thesun.Thesolidcurverepresents

thecalculationsby NovemberandKoutchmy(1996)whilethedashed

curverepresentsthecalculationsby Banmbach(1937)basedon several

eclipsesobservations.................................................................... 143

A photographof thesolarcoronaduringthetotal solareclipseof 11August

1999wherethesunwasapproachingthemaximumphase.Herethecorona

is uniformlybright andcircularin shape............................................. 144

A photographof thesolarcoronaduringthetotal solareclipseof 4

November1994wherethesunwasapproachingtheminimumphase.Here

thecoronais elongatedin theequatorialregionsandbrighterthanthe

polarregions.............................................................................. 144

A seriesof plots showingtheradialdependenceof themeanelectrondensity

in coronalholescomputedfrom varioussetsof polarizationbrightnessdata.

Thisplot wasobtainedfrom Cranmeret al. (1999)................................. 148

A plot showingtheelectrondensitiesfor thenorth(solidline) andsouth

polarcoronalholes(dashedline) andthenorth(dottedline) andsouth

(dot-dashedline)polarcoronalraysasa functionof height.Reproduced

from FisherandGuhathakurta(1995)................................................. 151

xxviii
5.8

5.9

5.10

5.11

A comparisonof theelectronnumberdensityfunctionsbasedonthemodels

by Banmbach(1937,equation(5.7)),Ichimotoet al. (1996,

equation(5.8)),Newkirk (1961,equation(5.9)),Cranmeret al. (1999,

equation(5.10))andGuhathakurtaandHolzer(1994,equation(5.12))......... 152

Plotsof thetheoreticalK-coronalintensityspectrabasedontheelectron

numberdensityfunctionsgivenby Banmbach(1937,equation(5.7)),

Ichimotoet al. (1996,equation(5.8)),Newkirk (1961,equation(5.9)),

Cranmeret al. (1999,equation(5.10))andGuhathakurtaandHolzer

(1994,equation(5.12))andtheline of sightat2.0 solarradii from thecenter

of thesun.Thecurveshavebeennormalizedat4000.0angstrom................ 153

Plotsof thetheoreticalK-coronalintensityspectrabasedontheelectron

numberdensityfunctionsgivenby Banmbach(1937,equation(5.7)),

Ichimotoet al. (1996,equation(5.8)),Newkirk (1961,equation(5.9)),

Cranmeret al. (1999,equation(5.10))andGuhathakurtaand

Holzer(1994,equation(5.12))andtheline of sightat 1.1solarradii from

thecenterof thesun.Thecurveshavebeennormalizedat4000.0angstrom.... 154

Thisis aplot of thetemperatureprofile alongtheline of sightat 1.3solar

radiibasedontheassumptionof aconductivecoronaChapman(1957).

Thetemperatureat thepoint of intersectionof theline of sightandtheplane

of thesolarlimb is 1.0MK ............................................................. 156

xxix
5.12

5.13

5.14

5.15

Thisis aplot of themodeledK-coronalintensityspectrafor anisothermal

coronaof 1.0MK anda modelwith thetemperatureprofilesgiven

by T(r) = 1.0×(1.3/r) z/7and T(r) = 1.0+ (1.0+ 9.0* (r - 1,75)2) -1 . Here

(r) is the distance between the center and points along the line of sight 1.3

solar radii. The temperature at the point of intersection of the line of sight and

the plane of the solar limb is 1.0 MK .................................................. 157

Coronal wind profile based on the Parker model for an isothermal corona. In

the ideal solution the velocity is equal to the isothermal sound speed at the

critical radius and starts with subsonic velocities at the lower boundary

and reaches supersonic velocities at large distances ................................ 159

Comparison of the theoretical K-coronal intensity spectrum for an

isothermal corona at 1.0 MK, zero wind velocity and line of sight at 1.5 solar

radii with the intensity spectrum due to the bulk flow velocity that naturally

arises with the Parker wind model for an isothermal corona as given in

equation (5.14) ........................................................................... 160

Comparison between two theoretical K-coronal intensity spectra due to

isothermal coronal temperatures of 1.0 and 1.5 MK and with the inclusion of

the Parker wind model for an isothermal corona .................................... 161

XXX
5.16

6.1

6.2

Plot of the polarization from theoretical modeling for the formation of the

K-corona, at a given isothermal coronal temperature of 1.0 MK, for lines of

sight at 1.1, 1.3 and 1.5 solar radii. The behavior of the polarization conforms

to the observations. That is, the polarization is almost wavelength

independent and increases with coronal heights ..................................... 163

The spectrum recorded by the fiber located in the center of the lunar

shadow during the eclipse. The x-coordinate corresponds to wavelength

scale. The two prominent peaks are associated with the Calcium K & H

lines corresponding to 3933.7 and 3968.5 angstroms, respectively ............... 168

Theoretical isothermal K-coronal models fitted to the observed

K-coronal intensity spectrum to determine the temperature by Ichimoto

et al. (1996). The lower curves show the differences between the

observational spectrum and the theoretical models. The model that showed

the least difference determined the temperature ..................................... 169

xxxi
6.3

6.4

6.5

Thisplot showsthecomparisonbetweenexperimentallyobtained

K-coronalspectrumforfiber # 04with theoreticalmodelsfor various

isothermaltemperatures.Fromtheintensityratio methodthewind velocity

andthetemperaturededucedfor fiber # 04are,300.0km/secand

1.29MK, respectively.ThetheoreticalK-coronalmodelsplottedin figure

(6.3)correspondto awind velocity of 300.0km/secandisothermal

coronaltemperaturesof 0.5MK, 1.0MK, 1.5MK and2.0MK. In sucha

scenariotheexperimentallyobtainedK-coronalspectrummaybeexpectedto

takeapositionbetweenthetheoreticalmodelsfor 1.0MK and 1.5MK.

(Expt:= Experimental,Theo:Theoretical)........................................... 172

A plot of thesuperpositionof figure (6.1)andfigure (6.3).Theshapeof

thespectrumrecordedby thefiber locatedin thecenterof thelunar

shadowcorrespondswith theshapeof theobservedK-coronalspectrumby

fiber#04.Theexpectedlocationof theobservedspectrumis between

thetheoreticalisothermalK-coronalspectrafor 1.0MK and1.5MK ........... 174

A plot showingtheobservedK-coronalspectrumfor fiber#04alongwith

thetheoreticalK-coronalspectrumfor thewind velocity andtemperature

derivedfrom fiber #04. The plot also includes the difference between

the observational and the theoretical plots along with the scattered light

spectrum observed by the fiber in the center of the lunar shadow ................. 175

xxxii
6.6

6.7

6.8

Plotsshowingtheeffectonthe shapeof theobservedK-coronal spectrumof

fiber# 04,asshownin figure (6.5),in comparisonwith theshapeof

thetheoreticalK-coronalspectrumby subtractingvariousfractionsof

thescatteredspectrumfrom theobservedK-coronalspectrum.A, B, C, D, E

andF correspondto a subtractionof 0.0%,15.0%,10.0%,8.0%, 5.0%and

3.0%,respectively,of thescatteredlight spectrumrecordedby thefiberthat

waslocatedin thecenterof thelunarshadowfrom thespectrumrecordedby

fiber#04.Theseplots showthattheobservedspectrumcloselyfollows the

shapeof thetheoreticalspectrumwith variousfractionalsubtractionof

thescatteredspectrum.................................................................. 178

Theextrafibersplacedin theshadowof themoonandoutsidethefield of

view in orderto recordtheinstrumentalstraylight distribution.................. 181

A schematicdiagramshowingtheintroductionof averythin high

transmissionglasssurface(AB) inclinedat45oto theopticalaxiswith a

semi-reflectingcoatingonthesidefacingthefiberopticplatealongwith

thefibersembeddedonanopticalflat. This alterationwill enable

thesimultaneousimagingof theimageformedonthefiber opticplate.......... 184

xxxiii
7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

Thepathenvisagedfor theunmannedSolarProbeSpacecraft.At the

perihelion(C) andthepolesthespacecraftwill befour andeightsolar

radii,respectively,from thecenterof thesun.Thepositionof thespacecraft

(P) is givenby (r, 0 ) with respectto thecenterof thesun.In thispicture

theparabolicpath,thedetectorplaneandthecenterof the sunarein the

sameplane................................................................................ 188

ThetheoreticalK-coronalintensityvariationfor variousdetectorelevations

for 0.0km/secwind velocity andthespacecraftpositionedat thepole(P)...... 192

ThetheoreticalK-coronalintensityvariationfor variousdetectorelevations

for 400.0km/secwind velocity andthespacecraftpositionedat thepole(P)... 192

Thevariationof thewind-sensitiveintensityratioagainst'y for various

wind velocities at T=1.0 MK and the spacecraft positioned at the pole .......... 193

The variation of the temperature-sensitive intensity ratio against 7 for

various temperatures at W=400.0 km/sec and the spacecraft positioned at the

pole ........................................................................................ 193

Temperature-sensitive intensity ratioVS Temperature for various wind

velocities at 7 = 0.0 degrees and the spacecraft positioned at the pole .......... 194

Wind-sensitive intensity ratioVS Wind velocity for various temperatures at

7 = 0.0 degrees and the spacecraft positioned at the pole ......................... 194

Temperature-sensitive intensity ratioVS Temperature for various wind

velocities with intensities measured through filters of fifty angstrom

bandpass centered at those wavelengths for line of sight at 1.1 solar radii ...... 195

xxxiv
7.9

A.1

A.2

A.5

A.6

A.9

Wind-sensitiveintensityratioVSWind velocitiesfor varioustemperatures

with intensitiesmeasuredthroughfilters of fifty angstrombandpasscentered

at thosewavelengthsfor line of sightat 1.1solarradii............................ 196

p

The general coordinates where r is a field point and r__. is a source point ..... 197

The general coordinates of the incident and scattered radiation and

the orientations of the parallel and the perpendicular components of the

electric vectors with respect to the scattering plane ................................. 206

Propagation of a monochromatic plane EM waves ................................. 210

Coordinates for the rotational transformation of the scattering plane through

an angle o_ about the line of sight ..................................................... 212

Geometrical configuration for the scattering of photospheric radiation by

coronal electrons ........................................................................ 214

Geometrical configuration for the formation of the K- Corona.

Photospheric radiation emitted from a point S on the Sun is scattered from

an electron E towards an observer O. The solar wind on the electron is radial

and blows in the direction CE .......................................................... 215

Scattering of radiation off an electron also known as Compton scattering ...... 216

Construction to obtain an expression for the scattered intensity in the rest

frame of the observer. Consider a coronal electron with velocity u subjected

to radial solar wind velocity w in a coordinate system where the x-axis

bisects the supplement of the scattering angle ....................................... 219

Highlighted map of the radial and the scattering planes of figure (A.6) ......... 226

XXXV
A. 10

A. 11

B.1

B.2

B.3

C.1

C.2

C.3

C.4

Highlight of the spherical triangle formed by EDJI in figure (A.6) ............... 228

Highlight of the triangle ESC of figure (A.6) ........................................ 229

The plot of extraterrestrial solar irradiance VS wavelength obtained with a

Fourier Transform Spectrometer at the McMath/Pierce Solar Telescope on

Kitt Peak, Arizona ....................................................................... 247

The plot of limb-darkening coefficient versus wavelength with

linear approximation super imposed .................................................. 248

The plot of electron number density versus the radial distance from the solar

surface for the two models given by equation (B.31) and equation (B.32) ..... 251

Illustrating the integration procedure of the first integral in equation (B.44)... 261

Plot of the cumulative y-axis values of figure (B.4) against the

wavelength 1"1'........................................................................... 262

Model streamer of thickness d in front of the solar limb with an electron

density enhancement by a factor fs and thermal electron temperature Ts ...... 277

Plot to show the effect of streamers on the K-coronal spectrum with d=0.5

SR, f_ =4.0, T_ =T=I.0 MK and at angles E =-45.0 and +45.0 degrees ......... 280

Model streamers of similar dimensions at -45.0 and -45.0 degrees above and

below the plane of the solar limb, respectively ...................................... 281

Plot to show the effect of streamers on the K-coronal spectrum with d=0.5 SR,

f_ =4.0, T_ =T=I.0 MK and spread over various angles e ......................... 282

xxxvi
C.5

C.6

Effectsby streamersontheratio I(4233)/I(3987)for W=400km/secand

rho=l. 1SR.Thehorizontallinesshowtheintensityratio for agivensolar

wind velocity of 400.0km/secandline of sightat 1.1solarradii, andfor

differentisothermalcoronaltemperaturein theabsenceof streamers........... 283

Effectsby streamersontheratio I(4100)/I(3850)for W=400km/secand

rho=l. 1SR.Thehorizontallinesshowtheintensityratio for agivensolar

wind velocity of 400.0km/secandline of sightat 1.1solarradii, andfor

differentisothermalcoronaltemperaturein theabsenceof streamers........... 284

xxxvii
Abstract

The determination of the radial and latitudinal temperature and wind profiles of

the solar corona is of great importance in understanding the coronal heating mechanism

and the dynamics of coronal expansion. Cram (1976) presented the theory for the

formation of the K-coronal spectrum and identified two important observations. He

observed the existence of temperature sensitive anti-nodes at certain wavelengths in the

theoretical K-coronal spectra. The anti-nodes are separated by temperature-insensitive

nodes. Remarkably, Cram showed that the wavelengths of the nodes and anti-nodes are

almost independent of altitude above the solar limb. Because of these features, Cram

suggested that the intensity ratios at two anti-nodes could be used as a diagnostic of the

electron temperature in the K-corona. Based on this temperature diagnostic technique

prescribed by Cram a slit-based spectroscopic study was performed by Ichimoto et.al

(1996) on the solar corona in conjunction with the total solar eclipse of 3 November 1994

in Putre, Chile to determine the temperature profile of the solar corona. In this thesis

Cram's theory has been extended to incorporate the role of the solar wind in the

formation of the K-corona, and we have identified both temperature and wind sensitive

intensity ratios. The instrument, MACS, for Multi Aperture Coronal Spectrometer, a fiber

optic based spectrograph, was designed for global and simultaneous measurement of the

ooo

XXXV111
thermal electron temperature and the solar wind velocity in the solar corona. The first

ever experiment of this nature was conducted in conjunction with the total solar eclipse of

11 August 1999 in Elazig, Turkey. In this instrument one end of each of twenty fiber

optic tips were positioned in the focal plane of the telescope in such a way that we could

observe conditions simultaneously at many different latitudes and two different radial

distances in the solar corona. The other ends of the fibers were vertically aligned and

placed at the primary focus of the collimating lens of the spectrograph to obtain

simultaneous and global spectra on the solar corona. By isolating the K-coronal spectrum

from the spectrum recorded by each fiber the temperature and the wind sensitive intensity

ratios were calculated to obtain simultaneous and global measurements of the thermal

electron temperature and the solar wind velocity. We were successful in obtaining

reliable estimates of the coronal temperature at many positions in the corona. This is the

first time that simultaneous measurements of coronal temperatures have been obtained at

so many points. However due to instrumental scattering encountered during observations,

reliable estimates of the wind velocity turned out to be impossible to obtain. Although

remedial measures were taken prior to observation, this task proved to be difficult owing

to the inability to replicate the conditions expected during an eclipse in the laboratory.

The full extent of the instrumental scattering was apparent only when we analyzed the

observational sequence. Nevertheless the experience obtained from this very first attempt

to simultaneously and globally measure both the wind velocity and the temperature on

the solar corona have provided valuable information to conduct any future observations

successfully.

xxxix
Chapter 1

INTRODUCTION

1.1 Introduction to the solar corona

The solar corona is the region which begins at an altitude of N2000 km above the

solar photosphere, and then extends out to many solar radii. This region becomes visible

to the naked eye during a solar eclipse as seen in figure (1.1).

Figure (1.1). The solar corona as seen during the

solar eclipse of February 26, 1998. Picture: NASA

eclipse library
Accounts of solar eclipses date back to at least the fourteenth century BC.

However it is not clear that the solar corona was ever mentioned. The first clear mention

of the corona was from Kepler in his Astronomiae pars Optica, although he seemed to

have attributed the corona to be a property of the moon. With the advent of photographic

methods and spectroscopy in the nineteenth century experimental studies were initiated

on the nature of the sun during eclipses. By then eclipse path predictions were also

possible. Edmond Halley was the first to predict the path of an eclipse in 1715 using the

Newton's law of gravity.

At the 1860 solar eclipse, comparisons of photographs by De la Rue and Secchi

from different locations confirmed that prominences were solar in nature. Prominences

are cloud-like features with a reddish color, often seen during solar eclipses off the sun's

limb. This confirmation also added weight to the fact the corona too was solar in nature.

Further discoveries followed with each subsequent solar eclipse. In the solar

eclipse of 1871, Janssen discovered and identified absorption features in the solar

spectrum. In 1814, Fraunhofer had been the first to label certain absorption features in the

spectrum of the sun, and Janssen was able to identify some of Frannhofer's lines in the

coronal spectrum. Most of the Fraunhofer features in the solar spectrum are atomic lines

that are formed in the solar photosphere, although some are caused by molecules in the

earth's atmosphere. Janssen's discovery of the solar absorption lines in the coronal

spectrum confirmed that the corona was also solar in nature.

2
At the sameeclipseof 1871,throughspectroscopicobservationsLockyer showed

that the coronal gasesextendedN500,000km abovethe solarlimb. This observation

providedthefirst cluethat thecoronawasalsoveryhot. Laterobservationshaveshowed

that the coronaextendedto overthirty solarradii abovethe solarlimb. Becauseof this

greatspatialextent,it canbeinferredthat thecoronalgasis veryhot. As to how hot the

gasactuallyis, welook to thecoronalspectrumfor ananswer:this is a major aim of the

presentwork.Howeverit wasalreadyknownin 1957(from work of E.N. Parker)thatthe

coronaltemperatureis sohigh that it has importantdynamicaleffects.To seethis, we

notethat the extendednature of the solar coronais a result of a competitionbetween

gravityandgaspressure:theradialgradientof gaspressuretendsto forcematerialaway

from the sun,whereasgravity tendsto hold thematerialback.In theinnercorona,within

oneortwo solarradii of thesurface,theseopposingtendenciesbalanceeachotheralmost

exactly.However,beyonda few solarradii, Parker(1965)showedthat gravity losesthe

competition,and the corona acceleratesaway from the sun to form the solar wind.

Spacecraftmeasurementsof thewind flux showthat if the coronawerenot continually

re-suppliedwith material,thewind wouldemptythecoronain atime-scaleof afew days.

With the dawn of the spectroscopicage it was now possible to identify the

elementspresentin hot gasesthrough spectrumanalyses.Hot gasesemit photonsat

wavelengths,which are precisely characteristicof the elementspresentin the gases.

Howeverthecoronal spectrumposeda greatproblembecausemost of the wavelengths

couldnot be identifiedwith known elementsonEarth.Edlen(1937)identified someof
theemissionlines as originatingin forbidden transitions of the element iron in highly

stripped stages of ionization, FeX and FeXIV. Forbidden transitions are those which are

not allowed by electric dipole selection rules: the transitions must be induced by

magnetic dipole or electric quadrupole interactions. For iron to be in the ionization stages

FeX or FeXIV requires that the electron temperature in the coronal gas be at least 1.0

MK. For the forbidden transitions to occur, the ambient gas must have densities that are

lower than a certain value. The upper limit of the density depends on the details of the

transition. Golub and Pasachoff (1997) give a very detailed account of the history that led

to the discovery of the solar corona and its properties.

Figure (1.1) shows the solar corona during an eclipse. It is also evident from this

picture that the coronal intensity peaks around the solar equator and gradually diminishes

as it approaches the solar poles. Due to the reduced coronal intensity around the solar

poles, these regions are called eoronal holes. Although the coronal light intensity is small

in these holes compared to the equatorial region, the holes are also the region for the fast

solar wind. Figure (1.1) also shows solar plumes that radiate out from the north pole and

south pole of the sun tracing its magnetic field.

In the solar wind, material consisting of protons, electrons and a mixture of heavy

elements flow out from the sun into interplanetary space. This ionized flow is so highly

conducting that it drags along the magnetic field from the sun. The spatial extent of the

solar wind is unknown: eventually, the wind runs into the interstellar medium and is

4
terminatedat a shock.Spacecrafthasnot yet locatedthe terminationshockof the solar

wind, butestimatesputit at adistanceof order 100AU from thesun.

In essencethe solarcoronais anextremelyhot (> million Kelvin), highly ionizedgas

surroundingthe sun. This is visible only during a total solar eclipseas a white light

region extending to severalsolar radii, displaying streamers,plumes and loops. Its

appearancechangesduring the solar cycle. At solar maximum it consistsof many

structuresaroundthe disk, but at the solarminimum it is dominatedby large coronal

holesat eachpoleandsheet-likestructureneartheequator.

The brightnessof the solarcoronasurroundingthe solar disk is composedof three

maincomponents,namely;

ao K- (Kontinuierlich) corona: The K-corona is due to the scattering of the

photospheric light incident on the rapidly moving free electrons in the solar corona.

The free electrons are a result of the electrons stripped off the coronal gas elements

due to extremely hot temperature in the corona. Due to the high thermal velocity of

these electrons they tend to broaden the photospheric spectrum, which consists of

narrow lines, giving rise to a continuous spectrum. This scattering process is

commonly known as Thomson scattering.
b. E- (Emission) corona: The E-corona is due to the emission from coronal ions,

especially in highly ionized states. These emission lines are in many cases forbidden

transitions resulting from atomic transitions from the highly ionized ions in the

corona. Although these transitions are difficult to replicate in laboratories, however,

they are not over ruled by selection rules for atomic transitions.

C. F- (Fraunhofer) corona: The F-corona is due to photospheric light being scattered

by dust particles in interplanetary space. Dust particles which are refractory enough to

be able to survive at radial distances of a few solar radii also emit radiation in the near

and mid-infrared wavelength regimes. The F-corona dominates the visible coronal

brightness from about three solar radii distance from the center of the sun and has an

increasing contribution to the total coronal brightness at longer wavelength.

Figure (1.2) shows the radial brightness distribution of the F-corona, the K-corona

and the E-corona. From figure (1.2) it is evident that the K-corona dominates in the

coronal brightness to a height of N 2.0 solar radii above the limb and the F-corona begins

to dominate beyond that height.

6
iiii i

Figure (1.2). This plot shows the variation of the E,

K and F components of the solar coronal brightness

with height above the solar limb.

As for the reason for the existence of the solar corona there does not exist a firm

answer. Observations of other stars have revealed that many stars have coronae and many

others do not. Empirically, the solar corona is densest and hottest in regions of closed

magnetic fields ("loops"), and is coolest and fastest moving in regions of open fields

("coronal holes"). These results indicate that magnetic fields play an important role in the

coronal heating process.

7
Figure(1.3) showsa crosssectionof the sunwith someof its prominentfeatures

andthetemperaturescalesbetweenthecore,photosphereandthecorona.Thecoreis the

innermostpart of the sun where energy is generatedby nuclear reactions and its

temperatureis N15,000,000K. Thephotosphereis thevisible surfaceof thesunandalso

the layer which emits the light the humaneye sees.Its temperatureis N 6,000K. The

coronais theoutermostlayerin thesolaratmosphereandconsistsof highly rarefiedgas.

This layer begins at altitudes of about _ 2000 km abovethe photosphereand its

temperatureis _1,000,000K. Thecoronais visible to thenakedeyeonly duringa solar

eclipse.The regionbetweenthe photosphereand the coronais the chromosphere.The

region demarcatingthe chromosphereand the corona is the transition region that

featuresa sharptemperaturerisefrom _104K to _10 6 K within a range of N 500 kin. The

coronal streamers are large-scale magnetic structures in the corona.

From the point of view of physics, the most interesting aspect of the corona is the

huge increase in temperature that occurs as we move from the photosphere (_ 6000 K) up

into the corona (_1.0 MK). Now, coronal material must be replenished every few days

(due to the solar wind): the only source of this material is the photospheric gas.

Therefore, the creation of the corona is a continual process that involves heating the 6000

K gas to several million K. It is impossible for a thermal process, i.e. one based on solely

on exchange of internal energy, to do this. Therefore, a variety of non-thermal processes

have been considered in the literature (Narain and Ulmschneider, 1996). The energy
budgetof thecoronais not very demanding.Of the total energyflow throughthe solar

surface,onepartin 105sufficesto heatthecorona(Tsuneta,1996).

Figure (1.3). A cross section of the sun to highlight the prominent

parts of it and the temperature scales between the core,

photosphere and the corona.

Picture: SOHO pictures library

The most likely source of non-thermal energy input to the solar corona is

associated with disturbances generated in the solar convective layers, as shown in figure

(1.3). These disturbances in the solar interior are manifested as supergranulations,

granulations, magnetic flux tubes, and wave motions on the solar photosphere and

acoustic energy above the photosphere. Now the question is to determine, which of these

9
energyformscanpropagateupwardsandthendissipatein the solarcorona.Howeverthis

processis furthercompoundedby the needfor thewidely differing powerrequirements

for the different coronal structures.The following are someof the popular methodof

non-thermalenergydepositionin the solarcorona.

The most favored candidatesare the Ohmic dissipationof the coronal electric

currents and the viscous dissipation of magnetohydrodynamic(MHD) waves and

turbulence(NarainandUlmschneider1990). The coronalelectriccurrentsarebelieved

to be generatedby the twisting of the magneticflux tubes and, dependingon the

resistivity of the coronalplasma,dissipatethroughJouleheatingin the coronalplasma.

The MHD disturbancesarebelievedto propagatealongthe magneticfields protruding

from thesolarsurfaceinto thesolarcoronaanddissipatetheirenergy.TheseMHD waves

canbevisualizedasa combinationof the longitudinal soundwavesand the transverse

Alfven waves.Alfven waves arewavesgeneratedby the disturbanceof the magnetic

field lines andthe soundwavescould beproducedby pressureperturbation.SeeZirker

(1993)for a surveyof coronalheatingtheories,andconditionsa successfultheorymust

satisfy.

Thesunis alsooccasionallythe siteof transientreleasesof energyin theform of

flares.Theseeventsemit high-energyparticlesandshockwavesthat maycreateaurorae

on the earth.Althoughthe reasonfor the solarwind is well understoodto be driven by

gaspressuredueto thehigh temperatureof thegas,however,thedriving mechanismthat

10
causesthe solar wind flow velocity to changefrom subsonicvelocities to supersonic

velocitiesduringits flow in the solarcoronais yet to beestablished.The first analytical

treatmentof the solarwind modelis due to Parker(1965).It wasbasedon a thermally

driven wind model. Howeverthe large velocitiespredictedat 1.0AU do not seemto

matchthe physicalparametersat the coronalbase.Hereagainthe treatmentof the solar

wind asapurely thermallydrivenwind cannotproducethe experimentallyobservedhigh

velocities for the solar wind and someform of other non-thermalenergy deposition

mechanismsareneededto describeits transformationinto supersonicvelocities.Solar

Wind Nine(1998)andthereferencesthereingivea summaryof thepresentsituation.

Sincethecoronaitself doesnot containanyheat sourcesthat couldpossiblyheat

thecoronato abovemillion degreesor drive the solarwind flow to supersonicvelocities

thepossibilitiesneedto belinked to variousphenomenaon thesolar surfaceitself.And,

asa naturalconsequenceto link varioussolarsurfacephenomenawith the propertiesof

the solar corona it is important to measurethe coronal propertiesas accuratelyas

possible.In thisregardtwo of themostimportantphysicalparametersonthesolarcorona

arethetemperatureandthe solarwind velocity at differentlatitudesandradii in thesolar

corona.Thereforemeasurementof the coronaltemperatureand the solarwind velocity,

simultaneouslyand globally on the solarcorona,would help in gaining further insight

into the stateof thematterandits dynamicsin the solarcoronaandits associationwith

thesolarsurfacephenomena.

11
Thefocusof this researchis on themeasurementof thecoronaltemperatureand

the solar wind velocity, simultaneouslyand globally, on the solar corona from the

measurementof a singleobservablequantity.In section(1.2) someof themethodsused

by otherresearchersfor the determinationof thecoronaltemperatureandthe solarwind

velocity arediscussed.

1.2 Other methods to measure the solar

temperature

The following are some of the techniques

coronal temperature and the solar wind velocity.

wind velocity and the coronal

adopted in the measurement of the

(a) Measurements of the radial intensity distribution of the white light corona (Saito

1965, Koutchmy 1971, Guhathakurta et al. 1992) allow one to derive the scale-height

of electron density. This can be converted to an equivalent electron temperature if one

assumes hydrostatic equilibrium.

(b) Emission line intensities of various lines have been used to determine the

temperature of coronal electrons (Gabriel 1971, Nakada et al. 1975, Tsubaki 1975,

Guhathakurta et al. 1992). These emission line theories rely on the ionization

balances, various excitation mechanisms and atomic constants. Some of these

quantities are subject to major changes owing to new discoveries in atomic physics.

Ionization balance occurs when the rate at which an ion loses electrons per second (as

12
aresultof collisional ionizationby freeelectrons)is balancedby therateat whichthe

ion gains electronsper second (as a result of recombination).The excitation

mechanismthat is assumedto bepredominantis collisionalexcitation.This modelis

a simpletwo-levelatomconsistingof thegroundlevel m and an excited state n. The

population of the upper state is set by a balance between collisional excitation from m

to n due to electron-atom collisions, and by spontaneous radiative decay from n to m

via an allowed atomic transition. For the coronal conditions the induced emission

from n to m, the photoexcitation from m to n and the collisional de-excitation from

impacts with electrons are considered negligible. The atomic constants that are

inherent in this model are the Einstein coefficients for spontaneous emission,

statistical weights of the various atomic levels, collisional excitation coefficient and

the collision strength factor. The latter is a slowly varying function of the incident

electron energy and involves difficult quantum mechanical calculations. In this regard

we have preliminary evidence to suggest that Thomson scattering could be partly

responsible for the emission lines in the EUV region using the Thomson scattering

code developed as part of this dissertation work.

(c) Hara et al. (1994) used the ratio of the soft X-ray intensities in different energy

bands to determine the coronal 'color temperature'. They used the X-ray analysis

filters onboard Yohkoh to formulate a filter ratio method to determine the temperature

of the quiet corona and have reported a value of 2.7 MK (Hara et al. 1992). Here

photon energy ratios are determined for different filters. These are again functions of

13
the wavelength and temperature dependent emissivities and the response

characteristicsof thefilters. Haraet al. (1994)andWithbroe (1988)containsreviews

of coronaltemperaturemeasurementsby variousothergroups.

Withbroe et al. (1985) measuredthe temperatureof coronal ions using the

Dopplerwidths of emissionlines. The Dopplerwidth measuresthewavelengthshift

dueto motion of the gasresponsiblefor the observedemissionline, which couldbe

determinedby comparisonwith a calibrationlampfor the sameemissionline. Here

the motion is attributedto the kinetic temperature,which includesboth thermaland

non-thermalmotions.Howeverto isolatethe thermaleffect on the motion the nonthermaleffectsonthemotion, suchasmotionsdueto turbulenceandwaves,haveto

beeliminated.

(e) Radio observational techniques have also been used to infer the temperature of

coronal electrons (Zirin 1966). This method involves the radio observation of the

quiet sun in determining the coronal temperature. From each level of the solar

atmosphere only emission at frequencies higher than the plasma frequency may

escape. Therefore, by tuning on to different frequencies the radio emission down to

the region where the plasma frequency matches to the tuned frequency could be

measured. This will include levels on the solar atmosphere with different

temperatures too. If these temperatures are irregular the hotter regions are heavily

weighted. For low optical depth the radio emission intensity is a product of the source

14
function and the optical depth where optical depth is a measureof how far the

radiationwill travelbeforebeingabsorbedor scattered,whichis alsoafunction of the

frequency.The radiation in the radio region is assumedto be due to free-free

emissionneglectingothernon-thermalprocesseswherethe free-freeemissionis due

to decelerationof anunboundelectronduringits passagecloseto anionizedatom.In

additionthePlanckfunction is assumedto representhe sourcefunction.Hereagain

thesearebesetwith uncertaintiesassociatedwith the opticaldeptheffects,refraction

and the emissionmechanisms.Furthermore,the presenceof magneticfields could

produceother cut-off-frequenciesassociatedwith gyroresonance,which complicates

the propagation.Tiffs reversalof the aboveprocedureis alsopossiblewhereradar

signalsaredirectedat the coronaand arebouncedoff the layer wherethe plasma

frequencyis greaterthanthefrequencyof theincomingsignal.

(r) Doppler dimming technique has been used to measure solar wind velocity. In this

technique (Strachan et al. 1993) the aim is to measure the intensity of a

chromospheric line which has been resonantly scattered off coronal material, ff the

coronal material were to be at rest relative to the chromosphere, then the resonant

scattering would be optimized, and the intensity of the scattered line would be

maximized. However in the event that the coronal material has a non-zero velocity

relative to the chromosphere, there is more or less significant reduction in the overlap

of the line profiles in the chromosphere and corona. As a result of this mis-match

between chromospheric and coronal line profiles, the efficiency of the resonant

15
scatteringis reduced.The scatteredline becomesdimmer, the larger the velocity

difference between corona and chromosphere.Quantitative interpretation of an

observeddimmingin termsof wind velocity is subjectto anumberof uncertainties:

amongtheseare the emissionmechanismof the line, and the associatedatomic

constants.

(g) Solarwind velocitieshavealsobeenmeasuredusingremotesensingtechniques.

Hereaknownradiosignalis madeto propagatethroughthe solarcoronaandthrough

the detectionof fluctuationsin variouspropertiesof the radio signal is utilized to

determinethe motions of density inhomogeneitiesin the solar corona. For these

remotesensingtechniques,a varietyof methodsareavailable,dependingonwhether

the distant sourceis a natural source(broadband)or artificial (narrow-band).The

fluctuationswhich thesolarcoronacreatesin intensityor in phaseor in frequencyor

in line-width contain information on coronal density inhomogeneitieson a broad

varietyof length scales,from a few kilometersup to tensof thousandsof kilometers

(Yakovlev and Mullan 1996).Observationscan also be madefrom widely spaced

ground stationsor from singlestations(Coleset al. 1986,Watanabe& Schwenn

1989,Efimov 1994).Heretheresultsareacknowledgedto be substantiallylowerthan

theDopplerdimmingtechniques(Strachanet al. 1993).Thesetoo couldbebesetwith

problems owing to solar radio interference,carrier

sensitivityto variationsin thedensityof theionosphere.

signal broadening and the

In additiontheyalsodetect

themotionsof densityinhomogeneitiesthatincludenot only the bulk outflow of the

16
solar wind, but may also include wave motions if the latter have a compressive

component.A reviewonthis subjectis foundin Bird (1982).

In all of thetechniquessummarizedabove,the aim is to measurethe temperatureof

thecoronalelectronsor ions,or thesolarwind speed.Thetechniquesdependin different

waysona numberof parameters,eachof which is subjectto uncertainty. Howeverthe

theoreticalideautilized in this dissertationwork is basedon anwell-understoodsimple

physicalprinciple,whichis basedonThomsonscatteringof thephotosphericradiationby

thecoronalfree electrons,which is responsiblefor the formation of the K-corona.Here

thequantitymeasured,which is theintensity,is dispersedovera wavelengthregion.The

scatteredintensity over this wavelength region is dependenton both the coronal

temperatureand the solarwind velocity with certain regionsin the wavelengthspread

being sensitiveto eitherthe temperatureor the wind velocity. This featureof selective

sensitivityis exploitedto formulatetemperatureandvelocity diagnostics.In chapter-2the

physical overview for the formation of the K-corona is explained,together with a

methodologyto derivethethermalelectrontemperatureandthe solarwind velocity from

theK-coronalspectrum.

17
Chapter 2

FORMATION OF THE K-CORONA AND THE FOCUS OF THIS RESERACH

2.1 The K-corona

The K-coronal spectrum, as described in chapter-l, is the coronal brightness

attributed to the photospheric spectrum scattered off the free electrons in the solar corona.

The existence of the free electrons is due to the high temperatures prevalent in the solar

corona. This scattering phenomenon is commonly known as Thomson scattering. This

brightness is only revealed during a total solar eclipse.

2.2 The photospheric spectrum

Figure (2.1) shows the solar spectrum in the visible region, as it would appear

above the earth's atmosphere. The shape of the photospheric spectrum is similar in

nature, however, with a much greater flux. As evident from the solar spectrum the visible

region is replete with absorption lines. Among the deep and conspicuous absorption

lines, which were labeled by Fraunhofer, are the calcium H and K lines at 3968.5

angstrom and 3933.7 angstrom, respectively. The dark Frannhofer lines observed in the

18
solar spectrumare causedby selectiveprocessof line scatteringor line absorption,

accompaniedby continuumabsorption.

25O

_00

150

i

-_ i00

©

50

0

extra LerrestrialJrradianee VS observed wave]engLh

Figure (2.1). This is a plot of the extraterrestrial solar

irradiance spectrum. This was obtained by starting with

the ground-based Fourier Transform Spectrometer at

the McMath/Pierce Solar Telescope at Kitt Peak,

Arizona, and then correcting for wavelength-dependent

absorption in the Earth's atmosphere

19
2.3 The scattering source

The scattering source for the formation of the K-corona is the free electrons in the

solar corona. For a corona of one million Kelvin these free electrons have a mean thermal

velocity of N 5500 km/sec and given by equation (2.1).

v = _ where (2.1)

k = Boltzmann constant = 1.38 x 10" 23 J/K

m = Electron mass = 9.11x10" 31 kg

T = Coronal temperature ~ 106K

In an equilibrium state the electron velocity distribution can be described as Maxwelian

in the solar corona where the electron number density in the vicinity of the solar limb is

N10 s cm 3. This means that due to frequent collisions the velocity distribution is the same

in all coordinates. However cataclysmic phenomena such as flares or coronal mass

ejection can cause a deviation from a Maxwellian distribution locally. Furthermore, in the

case of dilute plasma where collisions are infrequent the velocities may differ in different

directions. This may be more pronounced in the interplanetary space where the particles

stream from the sun in one direction.

20
2.4 The effects on the incident radiation by the scattering source

In section (2.3) it was shown that a free coronal electrons move with an average

thermal velocity of N5500.0 km/sec in a corona of million degree Kelvin. Such motions

are called Thermal Motions. This also means that a photon of wavelength (_) incident

in an electron will be Doppler broadened by equation (2.2).

A_ v ~ :5500_ x vkm/sec where I

for T = 1.0 MK

c = 3 x l0 s km/sec

(2.2)

From equation (2.2) for an incident wavelength at 4000.0 angstrom the

broadening is N70.0 angstroms. This phenomenon is called Thermal Doppler

Broadening. This broadening increases with increasing thermal velocities for the

electrons, which in turn depend on the coronal temperature. This phenomenon could be

expected to cause all the narrow absorption lines, as evident in figure (2.1), to smooth

into a continuum upon incidence with the coronal electrons. However weak depressions

are still expected in the vicinity of the deep Frannhofer lines thus smoothening the smallscale depressions while preserving the large-scale depressions. This near-total

obliteration of the deep Frannhofer lines by the thermal Doppler broadening served as

one of the first clues for a hot corona. From equation (2.1) and equation (2.2) it is

evident that the extent of smoothening (A_) is related to the coronal temperature (T) by

equation (2.3).

Ax ~ 4T (2.3)

21
Once the radiation is incident on the electrons the scattering process is

independentof wavelengthand equalnumbersof photonsare scatteredforwards and

backwards.Thatis the scatteringis coherentin therestframeof the electrons.Sincethe

scattering sourceitself is in thermal motion the radiation reaching the observeris

Doppler shifted. In addition to the thermal velocity the coronal electronsmay also

acquirevelocity dueto turbulence.

2.5 The K-coronal spectrum

The photospheric radiation scattered off the free coronal electrons is known as the

K-coronal radiation. Cram (1976) presented a physical model for the formation of the Kcorona based on the Thomson scattering of the photospheric radiation by the free

electrons in the solar corona. In figure (2.2) a schematic drawing of Cram's model is

shown. Here a known electron density distribution, determined by other independent

means, is assumed for all positions along the line of sight. The normal to the line of sight

from the center of the sun is at a distance ( P ).

Each point on the line of sight has certain number of electrons and is subject to

collisions with photons that make up the photospheric spectrum. The electrons at a

particular point on the line of sight are exposed to the same photospheric spectrum from

all the points on the solar surface that they see and are subjected to Thomson scattering.

However the photospheric radiation incident on these electrons are also subjected to the

limb-darkening effect. That is the brightness of the incident radiation is decreased as the

22
radiationfrom theedgestravelthroughthickerandthickerlevelsonthesolaratmosphere

to reachtheelectrons.Thelimb darkeningis wavelengthdependent.Thelimb-darkening

effectis mostnoticeablein thevisiblewavelengthregion.

The thermalDoppler broadenedscatteredradiationobservedby an observeron

earth,dueto Thomsonscattering,is the accumulationof all theradiationthat is scattered

in thedirectionof theline of sightof theobserverby all theelectronslying alongtheline

of sight.

Figure (2.2). The photospheric radiation is emitted from a point S

on the surface of the sun. The electrons at point P in the corona

scatter some of the incident radiation towards a terrestrial

observer at E, which is known as K-coronal radiation.

23
Figure (2.3) shows a modeled spectrumfor the scatteredwavelengthfor an

independentlymeasuredelectronnumberdensity,givenby equation(5.7), alongtheline

of sight at 1.1 solarradii from the centerof the sun. Here an isothermalcoronawas

assumedandtheplot showsmodelsfor differentisothermalcoronaltemperatures.

5.0

4.5

4,0

m

2,0

3800

i i i I i

w 1ml/_ o.ooooc_

T _ o_ooooo

rb_ _ 1.1DO00

Modeled K-coro:al inten_iLy VS WavelengLh

''1'''1''' '''1''

:>" I

" 398"?

_ ./":_411

x(_)/x(2)= :.o41es44

x(2)/x(1)= :.o_19:1_

) , l , J ) I i ) , l , i ) li i , , l , i ,

_800 4000 4200 4400 4800

angska-oms

Figure (2.3). This plot shows the modeled K-coronal

intensity spectra against wavelength. The composite plots

are for different isothermal temperatures assumed for the

solar corona for a given line of sight at 1.1 solar radii.

4800

As predicted in section (2.3) the modeled K-coronal intensity spectrum, as

depicted in figure (2.3) for a line of sight at 1.1 solar radii, is very much smoother than

the shape of the radiation incident on the coronal electrons. From the shape of the

incident radiation spectrum, as is shown in figure (2.1), the vertical spread in the intensity

24
reflectstheexistenceof millions of narrowabsorptionlinestogetherwith wideabsorption

lines. Thesemillions of narrow lines in the incident radiation have been completely

washedaway,asrevealedin figure (2.3),giving riseto a continuousspectrum.However

thelarge-scaledepressionsarepreserved.

Figure (2.4) and figure (2.5) showthe modeledK-coronal intensity spectrafor

linesof sightat 1.3and1.5solarradii,respectively.

1.20

1.00

0,90

0,80

m

_0,70

0.60

0,50

3800

Modeled K-coronal intensity VS WavelengLh

' ' ' [ ' ' ' I ' ' ' [ ' ' ' I ' ' ' I ' ' '

rla_ _ l_OO_

............................................................................................................................................... 7-i

.}.::..........

l)_s98v

;!i}/

-. .. i'iTi;'

::i>_a. ::-• 2"f_/

\:Z./

8800 4000

X(3}/X(2} = 1.0420506

X(2)/XO} = 1.oolvoo_

7(s)/Y(2} = 1.o#le3ee

7(2)/Y(1} = 1.232908a

4£00 4400 4600

angoLa-ores

4800

Figure (2.4). Modeled K-coronal intensity spectrum against

wavelength for different isothermal coronal temperatures for

a given line of sight at 1.3 solar radii.

25
040

0,8,5

080

ts

g

0,20

Modeled K-coronal inkensity VS WavelengLh

:...%..._ ........ !.._._ ?.._................................................................................................... :.._i./i418

- . ..(: / X(aI/X(Z) = 1.0422B6a

.-'_../ x(z)/xO) = :.oalvoo_

............ _ Y(S)/Y(2} = 1.0917754

7(2)/Y(1} = 1.2SlS46e

o,15

3600 3800 4000 42,00 4400 4600 4800

angsLToms

Figure (2.5). Modeled K-coronal intensity spectrum against

wavelength for different isothermal coronal temperatures

for a given line of sight at 1.5 solar radii.

From these modeled K-coronal intensity spectra for lines of sight at different

heights off the solar limb, as depicted in figure (2.3), figure (2.4) and figure (2.5), Cram

(1976) made two very important observations.

(a) The existence of temperature insensitive nodes in the K-coronal intensity spectra

for various isothermal coronal temperatures.

(b) The remarkable independence of the positions of these nodes on the wavelength

scale for lines of sight at various heights above the solar limb.

26
From the abovetwo propertiesfor the K-coronal spectrumit is now possibleto

implement a temperaturediagnostic scheme for the measurementof the coronal

temperatureby meansof measuringthe K-coronal intensity spectrum.For this purpose

suitableintensityratioshaveto be determinedat wavelengthpositionswheresufficient

temperaturedependentparameterizationof the shapecouldbeconsidered.In this regard,

from figure (2.3)-(2.5),it is evidentthatthesuitablewavelengthpositionsoccurat3850.0

angstromand4100.0angstroms,respectively,which arealsoanti-nodes.Figure (2.6)is a

plot of theK-coronalintensityratiosat 3850.0angstromto 4100.0angstrom,from theKcoronalmodelsfor differentisothermalcoronaltemperatures,againsthetemperatures.

1 70

1.65

,_. 1.80

>..

= 1,55

1,5CI

1,45

0,5

\.

,.,

1(4-100}/'1(385D) VS Temperature

Mnd km/sec

-,.

\

\..,

'\1

-...

1 i p i [ i i i i

1.0

Temperature I,.414

I

I

[ i p I I

1.5 2.0

rho in solar radius 1.10_0000

Figure (2.6). This is a plot of the K-coronal intensity

ratios at 3850 .0 to 4100.0 angstroms against the

assumed isothermal coronal temperatures. The ratios

were calculated from the modeled K-coronal spectra for

different temperatures shown in figure (2.3). These

values pertain to a line of sight at 1.1 solar radii.

27
It is apparentfrom figure (2.6) that thecoronaltemperaturecouldbe determinedby

measuringtheK-coronalintensityratio at 4100.0to 3850.0angstromsfrom anobserved

K-coronal spectrum.Also from figure (2.6)it canbe shownthat if the aboveK-coronal

intensity ratio could be determinedwithin uncertaintyof _ 1.0% then the coronal

temperaturecould be inferred within an uncertainty of _+0.2 MK. However it is

unfortunatethattheintensitymeasurementat 3850.0angstromlies in theblueendof the

spectrum,which is subjectto atmosphericabsorption.This requires special chemical

coatedopticsfor signalenhancement.Below 3700.0angstromdifficulties arisefrom the

wavelengthdependentatmosphericextinctionwhile above4700.0angstromthespectrum

becomesrelativelytemperatureinsensitive.

Thefeasibilityof this methodfor thedeterminationof the electrontemperaturein the

solar corona,as first suggestedby Cram (1976),was demonstratedby Ichimoto et al.

(1996). Ichimotoet al. (1996)useda slit spectrographfor spectroscopicobservationon

the solar coronaduringthe total solareclipseof 3 November1994in Putre,Chile. By

isolating the K-coronal spectraform the observedtotal coronal spectratemperaturesof

1.07MK and 1.71MK weredeterminedfor thebaseof a coronalhole and a streamer,

respectively.Theyalsoreportedcirrus coveringof thesky duringthe totality, whichhad

causedsomedifficulties in theanalyses.

28
2.6 Further explanation for the formation of the temperature sensitive antinodes

To adequately answer the reason for the formation of the temperature sensitive

anti-nodes three simple intensity profiles are considered as input intensity spectra in order

to understand the scattered intensity spectra produced by the theory. Figure (2.7) shows

an absorption line centered at 4000.0 angstrom with a FWHM (Full width at half

maximum) of 40.0 angstrom on an otherwise uniform continuum. In this example the

intensity spectrum shown by figure (2.7) is assumed to be the sun spectrum as observed

from earth.

m

e_'

rj

D1

I

E

aJ

Input spectrum to the scattering code

I i I I

, , , , I , , , , , , , , I , , , , I

5500 400(_ 450(_ 50(JO

Wowlength in ongatroma

i i i i

55(_

Figure (2.7). An input intensity spectrum containing an

absorption line centered at 4000.0 angstrom with a

FWHM of 40.0 angstrom in an otherwise uniform

continuum.

29
Figure (2.8) showsthe theoreticalscatteredK-coronal intensity spectrumfor an

input spectrumgivenby figure (2.7).Thisintensityspectrumwasobtainedby allowingto

be scatteredoff the coronal electrons.The theory assumesan isothermal corona of

temperatures0.5MK. 1.0MK, 1.5MK and2.0 MK. Theessentialfeatureobservedin

the scatteredintensity spectrumis that the smearingof photonsby coronalelectronsfills

in the deepestabsorptionthus makingthe absorptionat the line centershallower.This

smearingprocessincreaseswith increasingtemperatures,which in essenceincreasesthe

ThermalDopplerBroadeningasgiven by equation(2.2).The essentialpoint is that the

intensityofradiationatline centerincreasesasthetemperatureof thescatteringelectrons

increases.

&

E

_4

0.04_

3.04Q i

0,0,55 i

_.030

0.025

a,02ci

360C,

Me, deled scaLLered intensity VS wuvelengLh

";: ,,,

\;,,

!".

,/ !

;,('

t :'

',,.¢,'

T uk r-'.sr-s'--o--, : ::,::,::

,he _R 1.1Bfl03

I I I i I I I I I I I I I I I I I I I I I I I

58 O0 40013 4200 44,90 4600 4800

angsLroms

Figure (2.8). The theoretical scattered intensity spectrum

by the coronal electrons for an input spectrum given by

figure (2.7). The theory assumes an isothermal corona.

The plots show that the smearing of the absorption line in

the scattered increases with increasing temperatures.

30
In figure (2.8), thegradientd(Intensity)/d(Temperature)is positiveat line center.

In orderto ascertainthe behaviorfor an input intensity spectrumwith two identical

absorptionfeaturesconsiderfigure (2.9)wheretwo strongabsorptionlinesarecenteredat

400.0and4500.0angstromsin anotherwiseuniform continuum.

g.

g

el'

_J

Input spectrum to the scattering code

I I I I

550O

Figure (2.9). An input intensity spectrum containing two

absorption lines centered at 4000.0 and 4500.0 angstroms

with a FWHM of 40.0 angstrom in an otherwise uniform

continuum.

Figure (2.10) shows the theoretical scattered K-coronal intensity spectrum for an

input spectrum given by figure (2.9). Here we see that whereas the centers of absorption

lines are filled in more and more as the temperature increases, the continuum between the

line centers become progressively fainter as temperature increases. Therefore, at

wavelength between the two lines (4100 - 4400 angstroms), the intensity of the

continuum has a gradient d(Intensity)/d(Temperature) which is negative. This is in

31
striking contrastto the positivegradientd(Intensity)/d(Temperature)at line centers.It is

this differentbehaviorbetweenline andnon-lineintensitiesthat givesrise to anti-nodes,

andprovidesthebasisfor Cram'smethod.Thereasonfor thedifferencein the depthsof

the absorptionlines is due to the effect of the temperaturedependentlimb-darkening

coefficient.

O. 050

0.045

_- 0,040

g

c

0,0,35

E

_, 0.030

O. O 2.5

i

i

i

0.020 i I

3600

i i [ i

'_" kmJ_ B.OBBg_,

:: LT- °::_--..

\

rh_ _R l.J BG_

Modeled scattered intensity VS wavelength

' ' ' ' ' I ' ' ' I '

_,, =,

i i [ i i i

7

, J

, I , _ , I , , , I , , , I , _ , I , _ ,

,3800 4000 4200 4400 4600 4600

angstroms

Figure (2.10). The theoretical scattered intensity spectrum by the

coronal electrons for an input spectrum given by figure (2.9). The

theory assumes an isothermal corona. The difference in the depths of

the absorption lines in the scattered spectrum is due to the wavelength

dependent limb-darkening coefficient.

32
Now considerthe implications of introducing an emission line between the two

absorption lines in an otherwise uniform continuum as shown in figure (2.11). Here the

absorption lines are centered at 4000.0 and 4500.0 angstroms while the emission line is

centered at 4250.0 angstrom with both the absorption and the emission lines with a

FWHM of 40.0 angstrom. Figure (2.12) shows the theoretical scattered K-coronal

spectrum for an input spectrum given by figure (2.11).

m

el'

_d

g

I

E

_.c

Input spectrum to the sc.atter[n 9 code

5000 550(] 400r2 458(} 5000

Wavelength in angatrema

5500

Figure (2.11). An input intensity spectrum consisting of two

absorption lines centered at 4000.0 and 4500.0 angstroms while

an emission line centered at 4250.0 angstrom and with a

FWHM of 40.0 angstrom for the absorption and emission lines

in an otherwise uniform continuum.

33
0.080

0.070

0,060

,__

c

0,050

E

,_ 0,040

0.050

i i i I i

v_ km/_ 0.00000o

T _K o._oo_c_o

the SR 1.10000

Modeled scattered intensityVS wavelength

''1'''1'''1' I i

J

'.j

5600 3600 4000 4200 4400 4600

angstroms

4600

Figure (2.12). The theoretical scattered intensity spectrum by the

coronal electrons for an input spectrum given by figure (2.11). The

theory assumes an isothermal corona. The difference in the depths

of the absorption lines in the scattered spectrum is due to the

wavelength dependent limb-darkening coefficient.

With the introduction of the emission line it is evident that in the scattered Kcoronal intensity spectrum, as shown in figure (2.12), the absorption lines become

shallower while the emission line peak becomes reduced in strength. This again gives rise

to the intensity on one side of the node to decrease while the intensity on the other side

increase. In analogy, when the input spectra of the examples shown in figure (2.7), figure

(2.9) and figure (2.11) are substituted by the photospheric spectrum the dips in the

photospheric spectrum are progressively filled in while the higher points are

progressively reduced, giving rise to anti-nodes. This is true of an integral over

34
wavelengthof the product of the photosphericspectrumand a Gaussiankernel whose

half-widthliesin therangeof 50- 200 angstroms.

In the solar spectrum,the anti-nodesin the wavelengthregion 3900 - 4500

angstromsowetheir existencemainly to two wide and strongabsorptionfeatures:(1) a

pair of lineslabeledH andK by Fraunhoferat 3968.5and3933.7angstrom,respectively;

(2) the combinedeffectsof the FraunhoferG bandat 4300 - 4320 angstromsand the

FraunhoferfeatureHv at 4340.5angstrom.The G-bandis a collectionof closelyspaced

absorptionlinespredominantlydueto thediatomicradicalCH andiron.

In essence,the uniquenessof the spectralregion around4000 angstromfor the

formationof theanti-nodesarisesfrom the following coincidences.On the onehand,the

H andK lineshappento provideessentiallya singleabsorptionfeaturewith a width of

tens of angstroms.On the other hand, the only molecular band that contributes

significantlyto thevisible solarspectrum(theG-band:almost100detectablelineswithin

arangeof 60 angstroms)happento lie within a few tensof angstromsof a strongatomic

line Hr. Both of thesestrongabsorptionfeaturesarereadily identifiable in the lowresolutionsolarspectrumshownin figure (2.13): the figure is takenfrom Cram (1976)

with his theoreticalK-coronal spectrasuperposed.Notethat in low-resolution,the solar

photosphericspectrumbetween4000and4300angstromshascorrugationsthat give the

impressionof "quasi-emissionlines". For this reason,when we were modeling the

35
scatteringof syntheticsolar spectraabove,as shownin figure (2.11), we included an

emissionfeaturebetweenthetwo absorptionlines.

Are thereanyotherpartsof the solarspectrumthatmightbeusefulto testCram's

idea?It seemsunlikely. If wego shortwardin wavelengthfrom 3800angstrom,the solar

flux becomesrapidly so low that it is difficult to obtainadequatesignalto noiseratio.

And if we go to longerwavelengthswe do not find pairs of strongabsorptionfeatures

that are wide enoughto remain detectablewhen we smooththem by many tens of

angstroms.

The differencein the intensity of the photosphericspectragiven by figure (2.1)

and figure (2.13)is due to the units of measurement.In figure (2.1) the photospheric

intensityin givenin units of ergs/sec.cm2.angstromwhile in figure (2.13)the intensityis

given in units of ergs/sec.cm2.angstrom.steradians. To reconcile the photospheric

intensitiesbetweenfigure (2.13)and figure (2.1) the latter needsto be divided by the

solid angleof the sun'sdisk, whichis 6.8x 10-s. The differencesin the intensitiesof the

K-coronal spectra,givenby figure (2.3)and(2.13),oweto thedifferentelectronnumber

densityprofilesandtheline of sightchosenfor theanalyses.

36
Figure (2.13). This figure shows the theoretical K-coronal

intensity spectra for four isothermal coronal

temperatures whose electron density is given by equation

(5.8). The absolute intensity of the disk center

photospheric spectrum is also shown. The vertical bars at

the bottom of the figure indicate the relative strengths of

the three coronal lines "c", and nine flash spectrum lines

Reproduced from Cram (1976).

37
2.7 The influenceof the solar wind on the K-coronal spectrum

The solar wind is an inherent property of the hot solar corona, a constant outflow

of gas into interplanetary space. There the coronal electrons in the vicinity of the solar

limb can be assumed to have a bulk outflow velocity equivalent to the solar wind

velocity. In the vicinity of the sun this flow is radial and begins to take a spiral pattern at

large distances, which is attributed to the rotation of the sun. However these flow

velocities are latitude dependent with the fastest outflows occurring in the coronal holes.

Figure (2.14) is a schematic diagram to explain the redshift induced on the Kcoronal spectrum by the radially outflowing coronal electrons due to the solar wind. For

the illustration of the above consider three electrons located at positions 1,2 and 3 on the

line of sight. All three electrons are embedded in a radially directed out flow: the

magnitude of the outflow velocity is the same at three electrons, but the velocity vectors

point in different directions. Electrons 1 and 3 are symmetrically located on either sides

of electron 2 while electron 2 is located at the closest point on the line of sight to the

center of the sun. Since they are all moving away from the sun the photospheric radiation

incident on these electrons are redshifted in the frame of reference of the electrons.

Furthermore, for Thomson scattering, the photospheric radiation scattered off these

electrons is wavelength independent in the reference frame of the electrons and they

scatter a fraction of this incident radiation along the line of sight of the observer. In

figure (2.14), the electrons 1 and 3 have common line of sight velocity magnitudes, away

and towards the observer, respectively. This allows for the scattered radiation off

38
electrons1and3 to beredandblue shifted,respectively,in theframeofreferenceof the

observer,thus renderinga cancelingeffect on the observedradiation from thesetwo

electrons.Howeverthe line of sightvelocity of electron2 is zero,which simply scatters

theredshiftedradiationincidenton it without causinganymorewavelengthshifts.Thus

theDopplershifting of theradiationscatteredby thesethreeelectronsis dominatedby the

effectdueto electron2, whichis anoverallredshiftingof thescatteredradiationalongthe

line of sight.

D

Figure (2.14). This is a schematic drawing to illustrate the redshifting

of the photospheric radiation scattered off the coronal electrons,

which are radially flowing away from the sun at a uniform velocity

(w). Electron 2 is located at the closest point on the line of sight to the

center of the sun. Electrons 1 and 3 are located symmetrically on

either sides of electron 2.

39
The magnitude of this redshift on the K-coronal spectrum for an incident radiation

of 4000.0 angstrom on the coronal electrons with an outflow velocity of 300.0 km/sec in

a million degree Kelvin Corona is, from equation (2.2), N4.0 angstrom. Figure (2.15) is a

reproduction of figure (2.1) with horizontal bars to emphasize the magnitude of the

thermal Doppler broadening and the redshift associated with the solar wind velocity. For

this illustration these quantities are calculated for an incident wavelength of 4000.0

angstrom on the coronal electrons, which are radially outflowing at a velocity of 300.0

km/sec in a million degree Kelvin corona.

50

Redshift due to solar wind velocity

as indicated by the red dot within the circle

Figure (2.15). This figure shows the scale of the thermal

Doppler broadening and the redshift associated with the

solar wind velocity in the wavelength scale in comparison to

the features of the photospheric spectrum. The scale of the

redshift is shown by a red dot, which is highlighted by a

circle. This comparison is made for an incident wavelength

of 4000.0 angstrom on the coronal electrons, which are

radially outflowing at a velocity of 300.0 km/sec in a million

degree Kelvin corona.

4O
The effect of the solar wind velocity in the model for the formation of the Kcorona was considered to be negligible by Cram (1976). However an interesting

additional property was revealed by the inclusion of a constant radial solar wind velocity

to Cram's model. That is the temperature insensitive nodes, which were observed by

Cram (1976), were now observed to shift in wavelength positions with increasing solar

wind velocity while maintaining the remarkable independence of the nodal positions with

height above the solar limb. Figure (2.16) and figure (2.17) are K-coronal intensity

models for different isothermal coronal temperatures at a line of sight at 1.1 solar radii

with assumed radial solar wind velocities of 400.0 km/sec and 800.0 km/sec,

respectively.

4.0

g

_0@ 3800

ii i

4000 4200 44_0 4_0_ 480_

nngsta'om_

Figure (2.16). Modeled K-coronal intensity

against wavelength for different isothermal

coronal temperatures, for line of sight at 1.1

solar radii and a wind velocity of 400.0 km/sec.

41
4.0

_ (1,5

g

_800

_deled K-coroznl intensity VS Wa-¢elength

I I I I

://4;_1

/i

_ • .i['i _ i

5994

Y(S)/Y(t) - J.s_ta2as

i i I , I i li _ i i [ I i

4£00 44t70 41_0_)

angsLrom_

,i/

i I i ] I i I I

3(1oo 4000

i

4_]0(1

Figure (2.17). Modeled K-coronal intensity

against wavelength for different isothermal

coronal temperatures, for line of sight at 1.1

solar radii and a wind velocity of 800.0 km/sec.

Comparing figure (2.16) and figure (2.17) with figure (2.3), which is a K-coronal

model for zero wind velocity, it is evident that the temperature insensitive nodes shift in

wavelength positions with increasing solar

property pertaining to the shifting in the

wind velocities. This newfound physical

wavelength positions of the temperature

insensitive nodes with increasing solar wind velocities can be utilized to construct wind

sensitive diagnostics.

42
Figure (2.18) is a plot of the K-coronal intensity model for different wind

velocities ranging from 0.0 km/sec to 900.0 km/sec in intervals of 100.0 km/sec for an

isothermal coronal temperature of 1.0 MK with the line of sight at 1.1 solar radii. It is

evident from figure (2.18) on the existence of a wind insensitive node at 4233.0 angstrom

with the maximum spread in intensity (anti-node) at 3987.0 angstrom. This property

holds for other isothermal coronal temperatures and heights above the solar limb too.

.5.0

4.5

4,0

._w

t_

g 3,5

¢-.1"

E

t_

ro 3,0

2.5

Modded K-coronal intensity VS Wavelength

T &IK 1.0D(]_O

2,0 i i I 1 I l

5600 5600 4000 4200 4400

angstroms

l l l l l l

4600

l

4800

Figure (2.18). Modeled K-coronal intensity against

wavelength for solar wind velocities of 0.0 km/sec to 900.0

km/sec in intervals of 100.0 km/sec, for line of sight at 1.1

solar radii and an isothermal coronal temperature of 1.0 MK.

43
Thusthe summaryof the theoreticalobservationsonthemodifiedversionof the Cram's

model,with theinclusionof thesolarwind velocity is,

(a) Theexistenceof temperaturesensitiveanti-nodesatcertainwavelengthsin the Kcoronalintensityspectrumprovidesameaningfultemperaturediagnostics.

(b) The wavelengthsof the anti-nodesareremarkablyindependentof heightsabove

thesolarlimb.

(c) The simultaneous existence of nodes in the spectrum, where temperature effects

are negligible, enables us to develop a diagnostic of wind speed that is more or less

independent of temperature.

2.8 Simultaneous determination of the coronal temperature and the solar wind

velocity

From the theoretical properties observed from the modified Cram's model it is

evident on the existence of both temperature and wind diagnostics. Figure (2.19) is a plot

of the intensity ratios at the anti-nodes 3850.0 to 4100.0 angstroms against isothermal

coronal temperatures for a range of solar wind velocities from 0.0 km/sec to 900.0 km/sec

and with the line of sight at 1.1 solar radii. It too reveals the feasibility of the coronal

temperature measurement within an uncertainty of + 0.2 MK if the above intensity ratio

at the anti-nodes 3850.0 to 4100.0 angstroms could be experimentally determined within

44
anuncertaintyof + 1.0 %. The width at the low end of the temperature range is N0.02

MK over a wind velocity range of 0.0 km/sec to 900.0 km/sec.

1.70

1,65

s

1.60

13D

03

0

0

,,--4

.e 1.55

I--4

I(4100)/I(3850) VS Temperature

i i i [

wind km/s_e

I

I

0,00000000

IO0.O000Q

_00.0000¢

300.0000C

400,0000C

500.0000C

@O0.O000C

700.0000C

800,00000

900.00000

1.50

1,45

0.5

line of sight in sdlar radii

I

I

i i i I i

1.0

Figure (2.19). The intensity ratio at 3850.0 to 4100.0

angstroms against isothermal coronal temperatures for a

range of solar wind velocities with the line of sight at 1.1

solar radii. The horizontal spread in the low end of the

temperature scale is ~0.02 MK over the velocity range of

0.0 km/sec to 900.0 km/sec.

Figure (2.20) is a plot of the intensity ratios of the wind sensitive anti-node at

3987.0 to the wind insensitive node at 4233.0 angstroms against solar wind velocities for

a range of isothermal coronal temperatures with the line of sight at 1.1 solar radii. The

solar wind velocity can be determined by first determining the temperature from figure

(2.19) and then by plotting in figure (2.20) the intensity ratios of the wind sensitive anti45
node at 3987.0 to the wind insensitivenode at 4233.0 angstromsagainstthe wind

velocitiesat this temperature. Here the solar wind velocity measurementsarehighly

sensitiveto the wind-sensitiveintensityratio measurements.An uncertaintyof + 1.0 %

in the wind-sensitive intensity ratio measurements can cause an uncertainty of + 200.0

km/sec at 0.50 MK. This uncertainty increases with increasing coronal temperatures.

I ,_BO

] .270

1.260

---.

g"

[%2

-_ 1.250

1 ._40

i i i I l i l

i i

i I

i i

l[Ite of sight inlsolar

/ .t "_ . - I ...-"

.x t

._<_:/7- 1- " ) I

I I I

I I I

I(_zss)/'I(sfs?) vs wired

]]emperafiure ._dlf

I / I

I / I

i _:_OOO000O

I / l

t 1" I.. o(::,ooo©q?

r a di i 1,1 OO05!,_0 :, ':::: :':::?: : _'#:

/" I

._.- ;i:"()i'):i.iK_) 75

J

.q -" I

/ I j _ I

/" I I

• . . .-

.-f .."r_ .-'" .I

/ i ' " "

J .230 , I , I l I l I '

0 _00 400 600 _00 lO00

Wind kin/see

Figure (2.20). The intensity ratio at 3987.0 to 4233.0

angstroms against solar wind velocities for a range of

isothermal coronal temperatures with the line of sight at

1.1 solar radii. To determine the wind velocity, first the

temperature has to be determined from figure (2.19).

Then the K-coronal models need to be created for a

range of wind velocities at this temperature and plotted

in figure (2.20).

46
2.9 Global determination of the coronal temperature and the solar wind velocity

In section (2.7) a methodology was developed to simultaneously determine both

the coronal temperature and the solar wind velocity from the same wavelength dispersed

K-coronal intensity spectrum. That is, by measuring the intensity ratios at 3850.0

angstrom to 4100.0 angstrom and 3987.0 angstrom to 4233.0 angstrom, from the

observed K-coronal spectrum. The former is termed the, 'temperature-sensitive'

intensity ratio, which allows for the temperature measurements from figure (2.19). The

latter is termed the, 'wind-sensitive' intensity ratio, which allows for the wind

measurements from figure (2.20). However for the wind measurement, further theoretical

K-coronal models need to be created for various wind velocities using the measured

temperature as the isothermal coronal temperature. From these models the wind-sensitive

intensity ratios are plotted against the wind velocities assumed for those models in figure

(2.20). Now by calculating the wind-sensitive intensity ratio from the observed Kcoronal spectrum its corresponding wind velocity can be obtained.

In order to extend this measurement globally on the solar corona the wavelength

dispersed K-coronal intensities need to be measured simultaneously and globally on the

solar corona. Chapter-3 explains an instrument designed for simultaneous and global

measurement of the wavelength dispersed coronal intensity spectra from multiple

locations on the solar corona. Chapter-4 discusses the results from the very first attempt

to measure both the coronal temperature and the solar wind velocity, simultaneously and

47
globally, on the solar corona.This experimentwas performedduring the total solar

eclipseof 11August1999in Elazig,Turkey.

2.10 The essenceof this dissertation

In this dissertation the influence of the solar wind velocity was added to the

Cram's model to search for both coronal temperature and solar wind velocity diagnostics.

Based on the feasibility for the measurement of the coronal temperature and solar wind

velocity diagnostics, form a single spectroscopic observation of the solar corona, an

instrument was designed that would globally and simultaneously measures both these

quantities at twenty different locations on the solar corona (Reginald & Davila, Paper I,

2000). The very first experimental effort was carried out in conjunction with the total

solar eclipse of 11 August 1999 in Elazig, Turkey and the results were analyzed to obtain

both the coronal temperatures and the solar wind velocities at multiple locations

(Reginald & Davila, Paper II, 2000).

48
Chapter 3

THE INSTRUMENT

3.1 Overview of the instrument

Basically, the instrument designed was a spectrograph to obtain coronal spectra of

the solar corona during the total solar eclipse of August 11, 1999.

However this design incorporated a new feature in the way the light was collected

and fed to the spectrograph in obtaining spectra. The exception from the usual slit-based

spectrograph was that light was collected at the focal plane of the telescope via fiber

optics. The fiber optic tips at the focal plane of the telescope were positioned at various

latitudes and radii on the image of the sun formed on the focal plane of the telescope

during the solar eclipse. This method served two very important purposes.

1. This way the instrument was able to obtain simultaneous coronal spectra from

different latitudes and radii on the solar corona.

49
o Also as a complimentary the telescope was spared the task of carrying the weight of

the spectrograph. This was because they were detached systems connected via fiber

optic cables.

In essence the instrument was composed of three vital components.

1. The telescope imaged the sun during the eclipse on the focal plane of the telescope.

2. The fiber optic tips, positioned at various latitudes and radii on the sun's image on the

focal plane of the telescope, carried the coronal light to the spectrograph.

3. The spectrograph produced simultaneous spectra from the light received by the

individual fibers.

The telescope used in this experiment was a Schmidt-Cassegrain telescope with a

primary mirror diameter of 12-inches and a focal ratio of 10.0. A focal reducer was

attached to reduce the focal ratio to 6.3. Schmidt-Cassegrain telescopes by design are free

from coma, spherical aberration and astigmatism. The telescope also contained autotracking capability. Figure (3.1) is a picture of the telescope used in this experiment and

produced by the Meade Corporation. Figure (3.2) is a picture of the focal reducer

attached to the back end of the telescope that increased the field of view by 56.0 % at the

expense of lowering the magnifying power by 36.0 %.

50
At the focalplaneof thetelescopewasa glassplateembeddedwith twenty-onefiber

optic tips with teneachspreadin equalanglesin acircularloop correspondingto 1.1and

1.5solarradii onthesun'simageandoneat thecenterof theframe.Theonein thecenter

of theframewasusedto recordthebackgroundsignalusingthelunar shadowduringthe

eclipse.Anotherfour fiberswereattachedto amercurycalibrationlamp.

Figure (3.1). Meade 12-inch, F/10.0 SchmidtCassegrain telescope with auto tracking

capabilities.

51
Figure (3.2). F/6.3 focal reducer reduces the focal length by a

factor of 0.63. The device shown in figure (3.4) was placed at

the focal plane behind the F/6.3 focal reducer.

Figure (3.3) shows a schematic of the glass plate, as it would appear when the sun is

in focus. Figure (3.4) shows the picture of the fiber optic tips embedded on the glass

plate. The front end of the glass plate, where the fiber ends were exposed to the coronal

light, was roughened with the back end smoothened. This allowed for the focusing of the

sun prior to the eclipse on the front end of the glass while observing the image going in

and out of focus from the back end. The fiber in the center was to record the background

signal during the eclipse.

52
Figure (3.3) A drawing depicting the location of the fiber tips on

the image plane of the telescope. The inner and outer rings are at

1.1 and 1.5 solar radii, respectively. During the eclipse the lunar

shadow replaces the sun.

In figure (3.4) is the glass plate imbedded with twenty-one fibers that was placed

at the focal plane of the telescope. The coronal light entering the fibers, which were

located at the focal plane of the telescope and shown in figure (3.4), were all vertically

aligned at the other end. The other end substitutes the position of the slit in a slit-based

spectrograph. This slit, made up of multiple fibers at regular spacing between neighbors,

was placed at the prime focus of the collimating lens of the spectrograph. In figures (3.5)

and (3.6) are images of the spectrograph.

53
Figure (3.4). The glass plate imbedded with twenty-one fibers. The inner

and the outer circles are located at 1.1 and 1.5 solar radii, respectively, on

the image plane of the sun. The fiber at the center was to record the

background signal during the eclipse.

From figure (3.5) it is apparent that, apart from the capability of this setup to

simultaneously produce spectra, this methodology also allowed for the telescope and the

spectrograph to be two independent units connected via light fiber optic cables. This is

not possible in a slit-based spectrograph. In such a case the spectrograph needs to be

attached to the back end of the telescope with the slit at the focal points of both the

telescope and the collimating lens of the spectrograph. The purpose of the collimating

lens is to transmit collimated light to the diffraction grating for dispersion.

54
_ Fiberopticlink to thespectrograph

Figure (3.5). This is a picture of the spectrograph. The coronal light was fed

into the spectrograph by twenty-one fibers that were vertically arranged at a

regular spacing. Its location in the spectrograph was at the focal point of the

collimating lens.

Figure (3.6) shows the inside of the spectrograph between the grating and the

camera lens that focuses the dispersed light to the camera in the backend. The

spectrograph body contained many features that made it light tight from stray light that

could have found its way into the spectrograph. Also the setscrews ensured that the many

parts of the spectrograph fitted in place during assembly as a single unit.

55
Figure (3.6). This is a picture of the inside of the spectrograph between the

grating and the camera lens. Baffling were used to prevent all orders but the

first order from entering into the camera lens.

In figure (3.7) is a picture of the CCD camera that was used in this experiment to

record the light dispersed in the first order in the wavelength region of 3500.0 to 4500.0

angstroms. A thermoelectric cooler cooled the CCD device. Baffling were placed to

restrict only the dispersed light in the first order from entering the camera. All the other

orders were subjected to multiple reflections within the saw-tooth shaped grooves in the

baffles for absorption by the black-coated baffling material.

56
Figure (3.7). This is a picture of the thermoelectrically cooled CCD

camera used in the experiment. The front end housed the camera lens

that focused the dispersed light from the grating on to the camera.

As a general overview of the instrument the following could be said. The

telescope imaged the eclipsed sun on to a glass plate containing twenty-one fibers, which

was located at the focal plane of the telescope. The fibers picked up the light from

various locations on the sun's corona and the moon's center. This light was fed to a

spectrograph via the fiber optic cables. Then the grating simultaneously dispersed lights

from all the fibers. The camera located at the backend of the spectrograph recorded the

dispersed light in the first order in the wavelength region of 3500.0 to 4500.0 angstroms.

In view of this instruments ability to record simultaneous spectra it was deemed

appropriate to name this system the Multi Aperture Coronal Spectrometer with the

acronym (MACS).

57
3.2 Description of the optical elements in detail

In the following subsections all the optical elements of MACS will be discussed

in details.

3.2.1 The telescope

The telescope used in MACS was a 12-inch F/10.0 Schmidt-Cassegrain

manufactured by the Meade Corporation. The f-number was reduced to F/6.3 using a

focal reducer in order to increase the plate scale. The LX200 model used in MACS also

featured auto tracking and a super wedge for polar alignment. By design the SchmidtCassegrain type telescopes eliminate image distortion due to coma, spherical aberration

and astigmatism, which are explained below. These optical systems are also called

anastigmats and consist of at least three optical elements. A Schmidt-Cassegrain

telescope consists of a concave and convex mirrors as the primary and the secondary,

respectively, and a thin aspheric corrector plate in the front.

Spherical aberration: Parallel rays are not brought to a focus at a point, but

along a line. Therefore off-axis rays are brought to a focus closer to the lens than are onaxis rays.

Coma: Oft-axis rays do not quite converge at the focal plane, thus rendering oft'-

axis points with tails, reminiscent of comets, hence the name.

58
Astigmatism: Rays of different orientations having different focal lengths

resultingin thedistortionof theimage.Raysof light from horizontalandverticallinesin

a planeon the objectarenot focusedto the sameplaneon the edgesof the image.Off'-

axispoints areblurredin theirradialor tangentialdirection,andfocusingcanreduceone

at theexpenseof theother.

Theplate scale(P) of thetelescopeis definedasthe spatialareaseenin the sky,

which is measuredin arcseconds,correspondingto the spatialscalein the focalplaneof

thetelescope,which is measuredin micrometersandshownin figure (3.8).

I

Figure (3.8). Schematic drawing on the image formation by the I

primary mirror of the telescope. I

59
Fromfigure (3.8)theexpressionfor angleA is givenby equation(3.1).

D (3.1) Tan(A) = --

F

Since (A) is a small angle, expressing (A) in radians, where (Tan (A) = A), and then

converting radians into arcseconds, where (1 radian = 206265) arcseconds, gives the

following expression for (A), as shown in equation (3.2).

A - 206265x D (3.2)

F

From the definition of the plate scale (P) it could be written as shown in equation (3.3).

A

p=m

D

206265 x D / F

D

206265

F

Dx m

D

206265

DxF/#

arcseconds/t_m

(3.3)

60
Substituting for the diameter of the primary D in tam (12-inch = 12x2.54 cm =

12x2.54x 10 4 g m) and the f-number of the telescope (F/# = 6.3) in equation (3.3) gives

the following value for the plate scale (P) as shown in equation (3.4).

[P = 0.107 arcseconds/lam] (3.4)

3.2.2 The fibers

The fibers linking the focal plane of the telescope and the spectrograph

were twenty-one fibers with four more connected to a mercury calibration lamp. Each of

these fibers was 1.0 m in length with a core diameter of 200.0 g in. Figure (3.9) shows the

transmission efficiency of the fibers, marked as UV/VIS, for the region of wavelength

interest which was from 3500.0 - 4500.0 angstroms. These fibers were purchased from

Oriel Corporation.

61
iiiiiiiiiiiiii

iiiiiiiiiiiiii

:i:i;i:}:i:i:!

iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!_iiiiiiiiiiiiii_i!iiiiiiiiiii!i_iiiiiiiiiiiiiiii!s_i_iiiiiiiiiiiiiii_i_!iiiiiiiiiii_ii_@!iiiiiiiiiii_i_

Figure (3.9) Transmission efficiency of the fibers used in

MACS and marked as UV/VIS. The region of wavelength

interest in MACS was from 3500.0 -4500.0 angstroms.

All optical fibers transmit light signals by total internal reflection. That is, if a ray

of light in a medium of refractive index n_ strikes the interface with another medium of

refractive index n2 (n2 < nl) at an angle 0, and if0 is greater than 0 c , the ray is totally

reflected back in to the first medium where 0 c is given by equation (3.5).

{)e = sin- l(nl /nz ) (3.5)

62
D/2

iiiiiiiii iiiiii

iiiiiiiiiiiiiiiiiiiiiiiii iii i pi i i iiii i ig iiii iii h iiii ° ieiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

Figure (3.10). Schematic diagram depicting the light ray path at

the focal plane of the telescope into a fiber optic cable.

Figure (3.10) is a schematic drawing depicting the light ray path at the focal plane

of the telescope into a fiber optic cable. At point (A) applying the Shell's law gives the

relationship between angles (a and (_) and refractive indices (no and nl) by equation

(3.6).

sin (a) × n o = sin(m)× n 1 (3.6)

Now from triangle ABC the expression for angle 0 is given by equation (3.7).

0.= 90-_

sin(0) : cos(c_) (3.7)

: _l-sin2 (c_)

63
At thecritical angle(0= 0c) usingequations(3.5),(3.6)and(3.7)theexpressionfor the

acceptanceangle(a) in termsof therefractiveindicesis givenby equation(3.8).

sin(a)= 4n12 -nee andn o ~ 1 (3.8)

2

n o

The sine of the acceptance angle (a) is defined as the numerical aperture (NA). Also from

figure (3.10), the expression for sin(a) is given by equation (3.9).

sin(a) - D / 2

h

1

2h_

D

1

m

2(F/#)

(3.9)

The fibers used in MACS have a high purity UV grade silica core and silica glass

cladding with a numerical aperture of 0.27. This corresponds to an acceptance cone angle

(2a) of 31.0 degrees and an f-number (F/#) of 1.9.

All the light rays entering the fiber within the acceptance cone is totally reflected

at the core-cladding boundary inside the fiber and the signal is transmitted to the other

end. That is, all light rays within the acceptance cone entering the fiber at an angle less

than the acceptance angle (a) strike the core-cladding interface at an angle greater than

the critical angle 0 c . This fulfils the requirement for total internal reflection at the corecladding boundary.

64
Thef-numberof thetelescopeusedin MACS wasF/6.3,whichcorrespondsto an

acceptanceconeof 9.1 degrees.Howeverthis valuewasmuchlessthanthe acceptance

coneof 31.0degreesfor the fibers.Thereforeall thelight that wasfocusedontoeachof

thefibersat thefocalplaneof thetelescopewasexpectedto betransmitted.

The emergentconeform the otherend of the fiber is dependentupon the input

illumination, thefiber propertiesand the layout of thefiber bundle.For long fibers,the

fiber propertiesdominate,while for short fibers (< 1.0 to 2.0 m) the input conditions

dominate.Since the fibers in MACS were 1.0 m in length the emergentcone was

expectedto closelyresembletheinputconditions.

3.2.3 The lenses

The spectrographcontainedtwo lensesof diameter8.0 cm eachwith fnumbersF/3.9 and F/2.0. The former functionedas a collimator to collimatethe light

givenoutby thefibers.Thisenabledthegratingto receivecollimatedlight for dispersion.

Thelatteractedasa focusinglensto focusthedispersedlight onto thecamera.Spindler

& Hoyer manufacturedtheseachromaticlenses.An anti UV-reflective coating was

appliedonthelenssurfacesin orderto enhancethetransmissionbelow4000.0angstrom.

Figure(3.11)showsthereflectionpropertiesfor theantiUV-reflectivecoating.Very low

reflection ensuresgreatertransmission.This was achievedwith ARB2 UV chemical

coating.

65
Figure (3.11). Reflection properties for the anti UV-reflective

coating on the collimating and the camera lenses used in the

spectrograph.

66
3.2.4 The grating

Thedispersingelementusedin MACS wasa squaretransmissiongrating

of dimensions6.0× 6.0 cm 2 with 6000.0 fines per cm and anti UV-reflective coating for

enhanced transmission. Figure (3.12) shows the transmission properties for the anti UVreflective coating. This grating was manufactured by American Holographic.

Efficiency

Wavelength ×10 4 Angstrom

Figure (3.12). Transmission properties at normal incidence for the

anti UV-reflection coating on the transmission grating. Here the

back surface reflection is not considered. The substrate material is

fused silica. The black squares pertain to the grating used in

MACS.

67
3.2.5 The camera

The camera used in MACS was a CCD camera with a back thinned SITE

512.0X512.0, 24.0_tm square chip and controlled with PMIS camera-control software.

PMIS contains a powerful command interpreter designed for automating repetitive or

complex actions. The camera can also be thermoelectrically cooled from 50.0 o - 60.0 o C

below ambient and was manufactured by Apogee Instruments, Inc. Figure (3.13) is a plot

of the quantum efficiency (QE) curve for the for the CCD chip in the camera.

Figure (3.13). Quantum efficiency (QE) curve for the back thinned SITE

512.0X512.0, 24.0 _tm square chip used in MACS. This chip also featured

a full well depth in excess of 300,000.0 electrons, a dynamic range of ~ 90.0

dB and a readout time of ~ 5.0 seconds with an ISA card.

68
3.3 Analytical expressions for the integration time, the spectral resolution and

the spatial resolution

Figure (3.14). Schematic diagrams of the optical components that make

up MACS and the parameters that determine the spatial and the

spectral resolutions.

Figure (3.14) is a schematic diagram of the optical components that contributed to

the operation of MACS and the physical parameters of the different components that

determined the spectral and the spatial resolutions. In MACS the fibers substituted for a

slit in a slit based spectrograph.

69
Considera singlefiber of diameter(dr) and approximatethe circular end of the

fiber to a squareof side(dr).Theprojectedfiber width (w) andheight(h) on thecamera

focalplaneare,respectively,givenby equation(3.10)andequation(3.11).

(3.10)

In equation (3.10) ([3) is the angle of incidence on the grating and is zero for normal

incidence, which is the case for MACS. The direction of the dispersion is along (w).

For normal incidence the grating equation is given by equation (3.12).

d sin(m) = mE (3.12)

In equation (3.12) (d) is the ruling density, ((_) the diffraction angle, (m) the diffraction

order and (_,) the wavelength.

70
Figure (3.15). Schematic diagram showing the projected width and

height of a single fiber on the camera plane.

Figure (3.15) is a schematic diagram showing the projected width and height of a single

fiber on the camera plane.

For collimated light incident on the grating the angular dispersion is given by

(do_/d_), with the dispersion parallel to the rulings on the grating. Then the linear

dispersion is given by (f2 x (do_ / d_) ) where (t'2) is the focal length of the camera lens.

71
Usingequation(3.12)theangulardispersion(AD) andthelineardispersion(LD)

aregiven,respectively,by equation(3.13)andequation(3.14).

AD d_

m

dcos(a)

(3.13)

dl

LD -

dE

d(E

= f2

f2 m

dcos(_)

(3.14)

Figure (3.16). Schematic diagram of the dispersing element showing

the angular and the linear dispersions.

72
Figure (3.17). Schematic layout of the slit spectrometer. In this diagram

collimated light is incident on the grating.

From figure (3.17) the height and width of the entrance slit (single fiber) subtend angles

co and qo on the sky and the collimating lens, respectively, and are given by equation

(3.15) and equation (3.16).

Dp

(,0= m

F (3.15)

df

fl

(3.16)

73
If for the CCD detector the pixel width is (p) then it needs to be decided on how

many pixels are to be allowed to match the image width (w). The usual convention is to

consider a proper match to be one where two pixels cover the width (w). However in

situations constrained by, time, amount of light expected from the object to be studied,

transmission efficiency of the optics and resources available in obtaining the desired

optical components, the decision rests on the compromise between resolution and

allowing for sufficient transmission of light to the detector.

In the absence of such constraints, say (n) number of pixels are to be matched to

the image width (w), then from equations (3.10), (3.11) and (3.15) the expression for the

pixel width (w) is given by equation (3.17).

np =w

• cos@)

_(f= ]cos(l)

= Dp _11/_-1 ) _

= (ODD f2 COS([_)

d 1 cos(o 0

(3.17)

In equation 3.17) the relationship (fl/dl = F/Dp) is used together with the physical

parameter for the beamwidth (dl) at the collimating lens, as shown in figure (3.17). Here

it is assumed that the grating diameter is at least (dl).

74
Considerthespectrometerslit of width (dr) illuminatedwith light of wavelengths

()_) and()_+ d)_). Theslit imagein eachof thesewavelengthswill have a width (w) on

the detector. From equation (3.14) the separation between the centers of these two images

is given by equation (3.18).

f2 m

(3.18)

Now defining spectral purity (8)_) as the wavelength difference for which

(Al = w ), ensuring the images are just resolved. Substituting this condition in equation

(3.18) and also using equation (3.10) the spectral purity is given by equation (3.19).

dcos(_)

6_. = w

f2 m

dfdcos([_)

mf 1

(3.19)

where ( [_ = 0 ) for collimated light incident on the grating. However it needs to be noted

that the expression for the spectral purity given by equation (3.19) cannot be smaller than

the spectral purity limit (_)_o) set by diffraction. The resolving power (R) of a grating is

the ratio between the smallest change of wavelength that the grating can resolve and the

wavelength at which it is operating. The resolving power is given by equation (3.20).

R - (3.20)

75
The spectralpurity (8_o) derivedin equation(5.19)doesnot takeinto account

the limit on the imagesizesetby diffraction. What follows here is the expressionfor

spectralresolution(8_o) andresolvingpower (Ro)limited by diffraction. The spectral

purity (8_) cannot be smallerthan (_o) while the resolving power (R) cannotbe

greaterthanthediffractionlimited resolvingpower(Ro).

Figure (3.18). Schematic diagrams showing the diffraction limited spectral

resolution and the resolving power.

76
From figure (3.18)thepathdifference(DO) betweencontributionsfrom the centerand

theedgeof thegratingis givenby equation(3.21).

t

DO = --sin(6c0 (3.21)

2

Then the associated phase difference ({) is given by equation (3.22).

2K

= --×DO

=

2

\ /

2x L'Z o

: --X--

2f 2

(3.22)

The intensity distribution (Iz) at the focal plane, due to diffraction, is given by equation

(3.23).

I Z = I 0

sin2(_) (3.23)

2

The first minimum of equation (3.23) is for ( { ) given by equation (3.24).

2x × L'Z o

=-2-

(3.24)

From equation (3.14) two wavelengths that are differing by (_)_), with each

0

producing a diffraction pattern at the focal plane, as shown in figure (3.18), will be

displaced by ( 8l, ) and given by equation (3.25).

f2 m

_l° - dc-o-_s(cz)_o

(3.25)

77
Theclosestthetwo wavelengthpatternscouldbewithout merginginto eachother

is given by the Rayleigh's criterion. As depictedin figure (3.18) the two wavelength

patternsdiffering by (8_) could just be resolvedif the maximum of one pattern

O

coincides with the first minimum of the other. In this case (81 = Z ). Substituting (Zo)

O O

form equation (3.24) in equation (3.25) and from figure (3.18) using (L'= Leos(o0)

gives the following expression for ( 8_ ° ) given by equation (3.26).

&o

Lm

Nm

Ro

(3.26)

where (N) is the total number of grooves in the grating.

The theoretical resolving power R is then given by equation (3.27).

R o =Nm (3.27)

78
Thevaluesfor thephysicalparametersof the opticalcomponentsin MACS aregivenin

equation(3.28)andtable(3.1).

m = 1.0

3 =0 °

N = 3.6 x 10 4

fl = 31.0 cm

f2 = 16.0 cm

d = 1.67 x 10 -3 mm

de = 200.0 _tm

pixel = 24.0 _tm

R 0 = 3.6x104

(3.28)

Table (3.1). A table showing the values for the angle of diffraction, the linear

dispersion, the line width and the resolvable theoretical wavelength range for

three different wavelengths using the physical parameters of the optical

components used in MACS. In column 3 the reciprocal linear dispersion is

calculated per pixel, which is the dimension of a pixel in the CCD camera

used in MACS.

wavelength

o

A

diffraction

angle

degree

reciprocal

linear

dispersion

spectral

purity

o

-- A/pixel

LD

6_ A

resolvable

theoretical

wavelength

range

o

6_ o A

3600.0 12.47 2.44 10.77 10.00

4000.0 13.86 2.43 10.77 9.00

4500.0 15.63 2.41 10.77 8.00

79
The exposure time allowed to record the spectra is dependent on the source

brightness, the light collecting area, the transmission efficiency of the telescopespectrograph system, the slit dimension and the sensitivity of the detector. Consider a

source of brightness given by equation (3.29).

f_ Joules/sec/cm'2/angs/steradians (3.29)

The flux received by the Schmidt-Cassegrain telescope is then given by equation (3.30).

D s 2 / Joules/sec/angs/steradians

(3.30)

where (Dp) and (Ds) are, respectively, the diameters of the primary and the secondary

mirrors of the telescope.

The flux through a single fiber at the focal plane of the telescope is then given by

equation (3.31).

fk X 7C[ Dp2 -- os2 ] dr dr

k )

Joules/sec/angs (3.31)

In equation (3.31), (dr/F ) is the angle subtended by the height and width of a single

fiber on the sky and F the effective focal length of the telescope.

80
Let ('C_) be the wavelength

spectrometercombination.

dependenttransmissionefficiency of the telescopeThespectralflux receivedby a singlepixel (E) is thengivenby equation(3.32).

Dp 2_Ds 2]drdrS)_ p2

4 )F F wh

× "c)_Joules/see (3.32)

In equation (3.32) (w) and (h) are the projected fiber width and height on the detector and

given by equations (3.10) and (3.11), respectively, (_)_) the spectral purity and (p) the

dimensions of a square pixel. From equation (3.32) the number photons of wavelength

(E) and energy (h e/E ) incident on a pixel per unit time (Nphotons) is then given by

P

equation (3.33) where (hp) and (e) are the Planck constant and the speed of light,

respectively.

E

see'l

Nph°t°ns -- h e/E

P

(3.33)

The basic detection mechanism of a CCD is related to the photoelectric effect

where the light incident on the semiconductor produces electron-hole pairs. These

electrons are then trapped in potential wells produced by numerous small electrodes. The

maximum number of electrons that could be contained in each a pixel is called the well

depth (WD). The quantum efficiency (QE) is the ratio of the actual number of photons

detected to the number of photons incident at a given wavelength ()_). If the time taken

81
for a pixel to reachthewell depth(WD) is (T) thenthewavelengthdependentintegration

time (T) is givenby equation(3.34).

h c/X

T = P x(WD) seconds

E x x (QE)x

(3.34)

The physical parameters that are used in determining the integration time (T) are given in

equation (3.35).

Dp = 30.5 cm,

dr = 200.0 tam,

p = 24.0 tam,

f2 = 16.0 cm,

d = 1.67 x 10"3 mm,

D s = 10.2 cm

F = 192.0 cm

fl = 31.0 cm

m = 1.0

o_ = sin-l( m_L_

t

0

_:0 0 ,

w: <1,t77-) ,

c = 3.0 x 108 msl

h = 6.63 x 10 TM Joules/sec

P

WD = 300,000

0

QE(_L : 3600 A) : 0.32

0

QE(_L : 4000 A) : 0.60

0

QE(_L : 4500 A) : 0.70

6_L = 10.77 A

(3.35)

The transmission effciency ('C_) is a wavelength dependent parameter that

depends on the transmission efficiency of the collimating lens, the fibers, the camera lens

and the grating used in MACS and its wavelength dependent values are listed in table

82
(3.2). The numberof optical surfacesthat contributeto the transmissionefficiency are

listed in table (3.3). Table (3.4) gives effective transmissionefficiency at different

wavelengthby consideringthe wavelengthdependentransmissionefficienciesgivenin

table (3.2) andthe numberof optical surfacesthroughwhich the the light has to pass

before enteringthe detectorand given in table (3.3). Table (3.5) gives the K-coronal

brightnessat 1.1and1.6solarradii atdifferentwavelengths.

Table (3.2), The transmission efficiencies of the fibers, the lenses and the

grating for three different wavelengths. This information is from figure (3.9),

figure (3.11) and figure (3.12).

Wavelength Fibers Lenses Grating

Angstrom

3600.0 0.70 1.0 0.33

4000.0 0.90 0.99 0.32

4500.0 0.92 0.98 0.29

Table (3.3). The list of lenses and the associated number of surfaces through

which the incident light had to pass before entering the detector.

Optical element Number of surfaces

Corrector plate 2

Focal reducer 4

Collimating lens 4

Camera lens 4

CCD glass protector 2

83
eft

Table (3.4) The effective transmission effciency (z_) of MACS for the

wavelengths 3600.0, 4000.0 and 4500.0 angstroms. The values are based on the

information from tables (3.2) and (3.3).

Wavelength in angstrom

3600.0

4000.0

Effective transmission efficiency

elf X _#NS (NS=number of surfaces) "_ : "_fiber _lenses X "_grating

0.23

0.25

4500.0 0.19

Table (3.5) The K-coronal brightness at 1.1 and 1.6 solar radii for three

different wavelengths during the solar maximum. The values were obtained

from Allen (1973) from pages 172 and 176. The units are in

ergs/sec/cm2/angs/steradians. (1 Joule=107 ergs)

Wavelength

Angstrom

3600.0

4000.0

4500.0

f_ x 104

at 1.1 solar radii

6.21

8.71

12.33

f_ x 104

at 1.6 solar radii

4.26

5.99

8.47

84
Substitutingequation(3.32)in equation(3.34)theexpressionfor theintegrationtime (T)

is givenby equation(3.36).

(h c/_) × (WD)

T = P seconds (3.36)

f_x_ p -Ds dr dr _E p- x'cerr x (QE)_

F F wh

Using the values for the physical parameters from equation (3.35) and tables (3.4) and

(3.5) gives the following chart, as shown in table (3.6), for the lower limit of the

integration time (T) for the K-corona at 1.1 and 1.6 solar radii.

I MACS Table (3.6). basedLower on limit of the integration time for the K-coronal spectrum for equation (3.36).

Wavelength

angstrom

T in seconds

at 1.1 solar radii

T in seconds

at 1.6 solar radii

3600.0 122.0 177.0

4000.0 38.0 55.0

4500.0 27.0 39.0

85
Howeverthe actualintegrationtime will dependonthe amountof light absorbed

by the variousopticalelements,theamountof light escapingthe light collectingareaof

theopticalelementsandfurthercontributionsfrom theF-corona.

As for thespatialresolutionof MACS thewidth of a singlefiber of 200.0 gm and

aplatescaleof 0.107arcseconds/gmcorrespondsto a spatialscaleof 21.4" (0.022solar

radii). Taking the height of a singlefiber as200.0 gm and from equation(3.11)the

projectedheightof thefiber on the spatialdirectionof the detectoris givenby equation

(3.37).

h = 200lam × 16.0 em

31.0 em

= 103.23lam

(3.37)

This corresponds to (103.23tam/24.0tam) 4.30 pixels in the spatial direction. Therefore

form the spatial scale per fiber and the projected height of the fiber given in equation

(3.37) the spatial resolution is given by equation (3.38).

21.4"

Spatial resolution -

4.30 pixels

= 4.98" / pixel

~ 5.19 × 10 -3 solar radii / pixel

(3.38)

86
3.4 Cost of the instrument

I Table (3.7). Cost associated with the construction of MACS.

Item $ 000 Manufacturer

Telescope 6.79 Meade Corporation

Solar filters 1.02 Thousand Oak Optical

Fibers 4.05 Oriel Instruments

Collimating lens 1.39 Spindler & Hoyer

Camera lens 1.58 Spindler & Hoyer

Grating 2.35 American Holographic

CCD camera 10.5 Apogee Instruments

Camera software 0.60 PMIS

Computer 2.90 DPS 9000

GPS 0.41 Eagle Map Guide Pro

Glass base 0.45 United Glass Company

Spectrograph body 7.25 Lorr Company

Generator 0.77 Honda

Total 40.31

87
Chapter 4

OBSERVATIONAL RESULTS AND ANALYSES

4.1 Observational site

The coronal intensity measurements on the total solar eclipse of 11 August 1999

were conducted by Lake Hazar in Elazig, Turkey. The totality at this location

(38°41 ' N, 039°14'E ) lasted for 124 seconds. The second contact was at N 11.36 GMT.

Also no clouds could be reported for the entire duration of the totality.

Figure (4.1) is a photograph of the total solar eclipse of 11 August 1999 taken

from the Black Sea area. From figure (4.1) it is evident that the coronal brightness was

almost uniformly distributed all around the sun. This was expected of the sun that was

approaching maximum phase in 2000. This photograph also reveals a detached

prominence in the southwest limb during the totality. Figure (4.2) depicts the path of the

eclipse where the maximum duration of the totality occurred in Romania. Figure (4.3)

shows the layout of MACS. The spectrograph lies on the metal box. The telescope,

CCD-camera, and the computer were powered from a portable generator.

88
Figure (4.1). A photograph of the

August 1999 eclipse from the Black

Sea area. The photograph also reveals

a detached prominence eruption above

the southwest limb.

Total Solar gelipse of 1999 August I I

[;R,LRE |_ "['UE [CLIPS]_ |'AII! lliROL:f;I] "|'UI_I<E_', S_RI _. _.Xl.} [l_kQ

38" N

40_E 42_E n4"E

Longitude

Figure (4.2). A map showing the path of

eclipse over Turkey.

Courtesy: NASA Reference Publication 1398

the

89
Figure (4.3). A picture showing the layout of MACS and the telescope.

The telescope is on a polar mount and MACS is laid out on the metal box.

4.2 Spectra observed by MACS

Figure (4.4) shows the spectra due to the twenty-one fibers being exposed to the

sky and the remaining four fibers being exposed to the mercury calibration lamp in

Elazig, Turkey. The horizontal lines indicate the prominent absorption lines in the shy

spectrum and the four bright dots on the two ends correspond to the emission line at

4046.7 angstrom in the mercury calibration lamp. This information was used to determine

the wavelength scale and was determined to have a linear dispersion of 2.39

angstrom/pixel. The positions numbered from 1 to 4 are the locations corresponding to

9O
the temperatureand the wind sensitivewavelengthpositions, which is explainedin

chapter-2.

q_

}i

:'2

['2;

2 _i:_

SO,::_

Figure (4.4). The spectra from twenty-one fibers exposed to the sky

while four other fibers exposed to the mercury calibration lamp in

Elazig, Turkey.

MACS was exposed to the coronal light during totality for duration of 4.0, 18.0

and 83.0-seconds. Figures (4.5 a), (4.5 b) and (4.5 e) show the spectra recorded by

MACS for the above duration, respectively. Each vertical line corresponds to the

spectrum from a single fiber with a total of twenty-one vertical lines corresponding to the

twenty-one fibers located in the focal plane of the telescope.

91
(b)

._.: :_.f:_",i_

)!

92
(c)

Figure (4.5 a, b, and c). The coronal spectra recorded by

MACS for the exposure times 4.0, 8.0 and 83.0 seconds,

respectively. Each vertical line corresponds to a spectrum

from a single fiber. It is also clear that five of the fibers

show emission lines.

4.3 Identifying the individual fiber location on the corona

Prior to the eclipse the sun was focused to the position identified by the moon

during eclipse, as shown in figure (4.6). A solar filter attached at the front end of the

telescope facilitated this process. The telescope was then allowed to track. Watching the

direction of movement of the sun' s image upon slewing the telescope due north and south

with the auto controls the fiber #19 was oriented at the position perceived to be the

earth's South Pole. Its relation to the heliographic coordinates was determined from the

value listed for the P-angle for that day in, 'The Astronomical Almanac 1999'.

93
Figure (4.6) showsthe envisagedlocation of the twenty-onefibers in the focal

planeof the telescopeduring the eclipse.Howeverthe envisagedlocation could have

drifted dueto imperfecttrackingor minute movementsof the wholetelescopestructure

for not being anchoredon hard ground. Time constraintsand the remotenessof the

locationcompoundedtheseproblems.The term "envisaged"is emphasizedbecauseof

the incapabilityof MACS at the presentstateto simultaneouslyrecord the image.This

capabilitywouldhaveverifiedtheexactlocationof thefibertingsduringtheeclipse.

Figure (4.6). This picture shows the envisaged location of the twenty-one

fibers in the focal plane during the eclipse. The inner and the outer rings

correspond to 1.1 and 1.5 solar radii, respectively. The picture is not the

actual eclipse on 11 August 1999. The fiber in the center was expected to

record the background signal.

94
In orderto identify the fiber orientation to a reasonableaccuracy,someother

informationshownin thecoronalspectrawasutilized. Thistaskwasfurtheraidedby the

verydistinctiveemissionlinesrecordedin the spectraof five of thefibers asrevealedin

figure (4.5a), figure (4.5b), andfigure (4.5e).Theseemissionlinesaremostpertinentto

be from fibers exposedto prominenceeruptions.Using imaging data at totality from

other spaceand groundbasedsolarinstrumentsthat are dedicatedto observespecific

solarfeatures,it wasnow possibleto infer the solarfeaturesthat wouldhavecontributed

to theemissionlinesin thosefibersandaccordinglyorientthefibers.

The fibers that show emissionlines, going from left to right in figure (4.5 a),

figure (4.5b), andfigure (4.5c), arethe fibersnumbered22, 20, 19, 15and 14in figure

(4.6).All the emissionlines of fiber #19 correspondto the signaturesof a prominence.

Table(4.1) gives alist of prominencelinesreportedby Athay andOrrall (1957)from a

prominencein the westlimb of the total solar eclipseof 1952.Figure (4.7) showsthe

spectrumrecordedby fiber #19. In this paperan emissionline of Sr II listed at 3077.7

angstromneedsto be correctedas4077.7angstrom.Comparisonbetweentable (4.1) and

figure (4.7) revealsthat the fiber #19 had recordedall the prominenceemissionlines

reportedby Athay and Orrall (1957), that falls within the wavelengthrange of the

detectorin MACS.

95
Table (4.1) List of the emission lines reported by Athay and Orrall

(1957) of a prominence from the total solar eclipse of 1952. In this

paper the Sr II line at 4077.7 has been erroneously recorded as

3077.7 angstrom. In the following table only the emission lines that

could lie within the wavelength range of MACS is enumerated.

Emission line Wavelength (angstrom)

Sr II 4215.5

H6 (2-6) 4101.7

Sr II 4077.7

He I 4026.3

Ca II H 3968.5

Ca II K 3933.7

H8 (2-8) 3889.1

H9 (2-9) 3835.4

Figure (4.7) shows the spectrum recorded by the fiber #19 in the 4.0 seconds

exposure. This is after averaging over three pixels about its center and subtracting the

dark signal. However the spectrum is not yet corrected for the wavelength sensitivity of

the telescope-spectrometer system.

96
1.2 x ] 04

1,0x104

8,0_r103

6,0xi03

4,0z103

2,0xi03

0--

_ 275,419

-2,Ox103 I I

250

i i

_ 4-a i .J,.._rll

Coronal spectrum for the 4 seconds exposure in fiber number 19

' ' ' .... _i ' li ' ' '' ' ' ' ' -

4.077i? :_.h!ii 4-O.L!:.: :: H:Si

41ol .:' H_AI is_;s._ CoEE-_i

_ SgSqTiCoN

300 550

Pl×el number - spectral dlrect[on

_:89 ! H.° 3835,4 H_

t

!

4 @_'.91 _: 4,52.4-1. _ _

i ] I p

400 460

Figure (4.7). The spectrum recorded by the fiber #19 in the 4.0

seconds exposure. The emission lines and their corresponding

wavelengths and pixel positions are identified.

From figure (4.7) and table (4.1) it is apparent that the fiber #19 had recorded all

the prominence lines that fall within the wavelength range of the detector. The yellow

curves superimposed on the spectrum are the ganssian curve fits. The wavelengths

correspond to the peaks of the ganssian curves. However it needs to be yet ascertained

whether the emission lines were correctly identified. This was verified by constructing a

regression fit between the emission peaks and the peak positions. Figure (4.8) shows the

result of the regression fit and the wavelength scale between different emission lines.

These values give an average linear dispersion per pixel of 2.39 angstrom and matches

with the linear dispersion per pixel calculated from the optical parameters of MACS.

97
"3

_2

......ill

i:::

i=

EnJ_slon-Iin_s

i'"

:<,r,i....

i:i

i'_:._',!.y'_'_"__._'_i__'.r_ _'_',._ _:__'_"i:h_:._,.,_'_._"_i,_'.:,_'.:i_:=, ," li_,:_._,_'<_,.,_'_',ii:i__':<_:_-': _: _i_._'.,'" :_":i h"_ i:h,:._'.4 _::,-_'__':__':.,'_i,_" _._'..__'.:_'.'__,'_'_:_

_,_ correlation = 0,999995 "==

............... _"---... .... linear dlapera[_n _,n,.3_/plxel = 2,N}459_7 ..... _i

"%-..._. y = 21334_73 4- -0_4-17760 x

Srn 140 Srll l-lel CelI-H Cell--t< @ Hi! I Hi}

Srll 13_0000_000 2.575252_7 2._715651 2_5875895 2.58875,_4 2_Z_,B2577 2_391225S 2.Z_O6;_TD

H6 2,375232?0..30000000 2.553194G 2,@64595 2,4004-Z,64 2,Z_371620 2,599_}752 2,Z,i73&?_5

,,==

Srll 2_5713651 2.5551945 O_(]ODO0000 2_4-321146 2.411074-[:, 2_4,D46500 2_4059486 24Q18166

Hel 2,5875895 2.4064-5<35 @,4-52114-6 0,0000000_3 2,5926008 2,Z496263 2,5962646 2,5{i57678

CelI--H 2.3t_75B4- 2.4004_64 2.4-11074-9 2.392600i_ t'J.OOOOOrJ'DB 2.J_4710_, 2._9_,9541 2.Z;_4-274-D

.... i

Call--I_ 2,58_,2577 2.597162_0 2,4046500 2,_896263 2,;_84.7106 O.OC{_O0000 2,4-101508 2,;_t767_,1

H_, Z,5912258 Z,399875Z 2.40594-88, 2,5982.646 2.,39891_1 2,4.101505 O.O00C_DBO0 2,5_74-16B

..==

• H9 2,5906870 2.5975525 2.470181_6 2,,5937678 Z,5942740 2,Z_976781 2,5874-168 0.00 00000

.,;)_; C_ :; >'_X':' 4,:?'? b 4- i '5(-'. _-::}_5r;_ 45_3_:)

Figure (4.8). The regression fit between the emission line peaks

and the peak positions of the fiber #19 in the 4.0 seconds

exposure. The table incorporated shows the wavelength scale

between any two-emission lines.

Furthermore, the spectrum recorded by the fiber #20 also shows all the emission

lines from a prominence as listed in table (4.1). Figure (4.9) is the spectrum recorded by

the fiber #20.

98
C_ronol apBotrurn for th{_ 4 aecond_ expc-sure in fiber number 20

6.0_104 , , _ r T ' ' ' r _ , , , _ _ _ , , ,

5,0_104

4.0×104

_ 3,o×1c0

E

Z2.0x104

1,0x104

0

25D 3OD

Figure (4.9). The spectrum recorded by the fiber #20 in the 4.0

seconds exposure. This too matches with all the emission lines

recorded by the fiber #19 as shown in figure (4.7).

In table (4.2) is a list of emission lines, categorized as very strong and strong,

from the flare of 18 September 1957 and reported by Jefferies, Smith and Smith (1959).

Figure (4.10) shows the spectrum recorded by the fiber #22 in the 4.0 seconds exposure.

This is after averaging over three pixels about its center and subtracting the dark signal.

However the spectrum is not yet corrected for the wavelength sensitivity of the telescopespectrometer system.

99
Table (4.2).List of the emissionlines reported by Jefferies, Smith

and Smith (1959)from the flare of 18 September1957.This table

only includes thoseemissionlines categorizedasvery strong and

strong. Also in this following table only the emissionlines that

could lie within thewavelengthrange of MACS is enumerated.

Emission line Wavelength Category

angstrom

H 8 3889.1 strong

Ca II- K 3933.7 very strong

Ca II-H 3968.5 very strong

H e 3970.1 strong

H _ 4101.7 very strong

H ,,/ 4340.5 very strong

HeI 4471.7 strong

Figure (4.10) showing the spectrum recorded by the fiber #20 does indicate that

all but the H e line at 3970.1 angstrom listed in table (4.1) to be present. Since the linear

dispersion is N2.40 angstrom/pixel and, with the H e line at 3970.1 angstrom being only

2.0 angstrom away from the very strong Ca II-H line at 3968.5 angstrom, this may have

caused this line to blend with the stronger line.

100
5.0::104

4.D_104

,5,D×I04

8

"5

2.,D_;I04

1 D_I 04

Coronal spectrum for the 4 seconds e_po_ure in fiber number 22

220_7449c

396_

i

i !

E

i t

i 5935.7

i I

_ £._I-_1 ,:' i-t(I 4_+:i _-:_i-i,_i 4.1 _3I,:L ! ti_j

I

F i E

i '

i ,2_I':

_.....--_ ............. _ '__ "S_ \_,_., ......

.;::._:,:_ 7"_Y::5: _.,X.i.5-t'_::'_::!3_,i I _9 O r,;:72 2

50 1 DO 150 2.0D

Pi_el number - apectr_l dlrecli_n

#.i 1-18i

_..-J

._i'26

250

JToGgi

0 _ h i

Figure (4.10). The spectrum recorded by the fiber #22 in the 4.0

seconds exposure. The emission lines and their corresponding

wavelengths and pixel positions are identified.

Figure (4.11) shows the result of the regression fit and the wavelength scale

between different emission lines of figure (4.10). These values give an average linear

dispersion per pixel of 2.40 and matches with the linear dispersion per pixel calculated

from the optical parameters for MACS, as listed in table (3.1).

101
Figure (4.11). The regression fit between the emission line peaks

and the peak positions of the fiber #19 in the 4.0 seconds

exposure. The table incorporated shows the wavelength scale

between any two-emission lines.

Based on the comparison on the spectra recorded by the fiber#19, fiber#20 and

fiber #22 and the prominent emission lines expected of a prominence or a flare as listed

in tables (4.1) and (4.2), respectively, it is reasonable to assume that these fibers were

exposed to a prominence or a flare or to both of these features.

102
In order to identify the individual fiber location on the corona as accurately as

possible the following analyses were performed.

. Identifying the fibers exposed to prominence spectrum. The fibers #19, 20 and 22

were positively identified to have recorded the typical emission lines due to a

prominence.

. Orienting an eclipse photograph showing prominent eruptions with EIT images from

SOHO on the day of the eclipse and matching these features. This matching was

performed using the conspicuous prominence eruption identified at N 68 o southwest

in the EIT picture in 304 angstrom by SOHO at 11:26 UT with an eclipse photograph

taken at Nll:16 UT from the Black Sea.

. Drawing the polar axis of the earth, which is p-angle from the polar axis of the sun.

The p-angle, which is the orientation of the polar axis of the sun with respect to the

polar axis of the earth, was obtained from, 'The Astronomical Almanac 1999'.

. Placing fiber #19 at the intersection point of the South Pole with the orbital circle at

1.5 solar radii. Prior to the eclipse the sun was focused to the position identified by

the moon during eclipse, as shown in figure (4.6). A solar filter attached at the front

end of the telescope facilitated this process. The telescope was then allowed to track.

Watching the direction of movement of the sun's image upon slewing the telescope

103
duenorthand southwith theautocontrolsthe fiber #19 wasorientedat the position

perceivedto betheearth'sSouthPole. Thelocationsof theotherfibers couldbethen

determinedbasedontheirpositionswith respectto fiber #19.

Figure (4.12) shows the envisaged positions of the fibers on the corona based upon the

above procedure.

Figure (4.12). This is a comparison between solar image by SOHO in

304.0 angstrom at 11:14 UT and an eclipse photograph taken from the

Black Sea at ~ ll:16UT. The orientation was made by matching the

prominence eruption ~ 680 southwest in the SOHO image. The orientation

of the polar axis of the sun with respect to the polar axis of the earth was

obtained from the p-angle data given in, 'The Astronomical Almanac

1999' for 11 August 1999.

104
Figure (4.13), figure (4.14) and figure (4.15) are picturesof the sun takenby

LASCO instrument on SOHO, EIT instrument on SOHO and radio image from Nancy

and x-ray image by Yohkoh, respectively, on 11 August 1999.

Figure (4.13). Images of the sun taken by the LASCO instrument

on SOHO on the 11 August 1999. The numbers denote the time in

UT. C2 and C3 are coronal images of the sun from 2.0 - 6.0 solar

radii and 3.7 - 32.0 solar radii, respectively, using an externally

occulted disk.

105
10:55 ................................ 11:48 12:27

11:08 11:42 12:21

11:01 11:54 12:33

11:14 12:02 19:19

Figure (4.14). Images of the sun taken by the EIT instrument

on SOHO on the 11 August 1999. The numbers denote the

time in UT.

106
YOHKOH 14:42 UT 164 MHz 10:45 UT 327 MHz 10:45 UT

Figure (4.15). Images of the sun taken by the radiograph in Nancy, France

and the x-ray telescope on Yohkoh on the 11 August 1999.

In figure (4.13) the coronal images of the sun were obtained from the C2 and C3

coronagraphs, of the LASCO (Large Angle and Spectrometric Coronagraph) instrument

on SOHO, providing coronal images of the sun from 2.0 - 6.0 solar radii and 3.7 - 32.0

solar radii, respectively. These pictures reveal large-scale activities in the form of

streamers in the southeast limb as compared to the rest of the limb. From figure (4.12) the

fibers #19, 20, 21 and 22 were located in this region of high streamer activity.

In figure (4.14) are EIT (Extreme ultraviolet Imaging Telescope) on SOHO are

images of the sun through various filters centered at different wavelengths as indicated on

the images. These are images, through selected bandpasses in Fe IX/X at 171.0 A, Fe

XII at 195.0 A, Fe XV at 284.0 A and He II at 304.0 A, reflect temperatures of N1.0

MK, N2.0 MK, _3.0 MK and _4.0 MK, respectively.

107
In figure (4.15) areradio imagesfrom Nancy, Franceand a x-ray imagefrom

Yohkoh. The radio imagein the centerrevealsthat the eastlimb was either hotter or

lesserin electronnumberdensitythanthewestlimb. Thebright spotontheimageonthe

right showsthesubflare#8662reportedat southl4east04on11August1999.

4.4 The telescope-spectrometer sensitivity curve

In section 4.4 the wavelength correction factor to convert terrestrial intensity

observation to extraterrestrial intensity is discussed. The intensity recorded by MACS

differs from the extraterrestrial intensity primarily due to the following two reasons.

lo The wavelength dependency of the physical parameters such as transmission

efficiency, focal length, dispersion and quantum efficiency, etc of the optical

components of MACS.

2. The wavelength dependency of the measured coronal intensity on the transmission

efficiency and the scattering in the earth's atmosphere.

In order to find a correction term to convert terrestrial intensity observation to

extraterrestrial intensity for each of the fibers the following steps were followed. This

follows the procedure adopted by Ichimoto et al. (1996).

108
. The dark-current subtracted terrestrial sky spectrum observed by each fiber was

corrected for Rayleigh scattering (X -4 ). The sky represents the F-component of the

solar spectrum and also closely resembles the solar spectrum itself. However the

Rayleigh scattering affects the overall distribution of the sky spectrum. Thus the

Rayleigh corrected sky spectrum was assumed to closely represent the terrestrial solar

spectrum at the observation site. Figure (4.16) shows the sky spectrum recorded by

fiber #06 before and after correcting for Rayleigh scattering.

E

2

E

o 5

*6

_4

o

o

o

E 5

o

o

,_ 2

E

c

0

3700

Sky spectrum recorded by {]ber #06 in Elaz[ 9 corrected {or Rayleigh scattering

'''''''''1 '' '' ''''' I' ''i'' 'i'''l '' '' '''''1'''''''''1'''''''''1 '' '' '' ''

I

i

Elozlg £I,;yll

i

I

I

i

I

i

I

I

i

I

I

I

I

I

i

I

I

I

I

i

i

I

i

.... . i

I

i

('Rayte[gh corrected}

/f: "...'

¢

/

I

/tji

! tli

.5800 3900 4000 41 O0 4200 4300

Waveleng-[h in angstrom

i

4400

Figure (4.16). Plots of the sky spectrum recorded by fiber #06 in Elazig,

Turkey before and after correcting for Rayleigh scattering. The plots

have been normalized at 3968.5 Angstrom.

109
. The terrestrial photospheric spectrum (Rayleigh corrected sky spectrum) recorded by

MACS was calibrated with a wavelength scale using prominent absorption lines Ca II

K & H at 3933.7 & 3968.5 angstrom, respectively and H_ at 4101.7 angstrom. The

positions of these absorption lines were matched with the corresponding positions of

the same absorption lines in the extraterrestrial photospheric spectrum. These

intensities were then divided by their respective intensities at 3968.5 angstrom. Figure

(4.17) shows the terrestrial photospheric spectrum (Rayleigh corrected sky spectrum)

recorded by fiber #06 and the extraterrestrial photospheric spectrum.

0

S700

/{ii i

8800 3900 4400

Figure (4.17). Plots of the extraterrestrial photospheric

intensity spectrum and the terrestrial photospheric

intensity spectrum recorded by fiber #06. The

extraterrestrial photospheric spectrum is a smoothed

version of the extraterrestrial photospheric spectrum

shown in figure (2.1).

110
. The extraterrestrial photospheric spectrum was smoothed until it matched the

resolution of the terrestrial photospheric spectrum recorded by MACS. Figure (4.18)

shows the terrestrial photospheric spectrum recorded by fiber #06 and the

extraterrestrial photospheric spectrum smoothed in resolution to match the resolution

of the terrestrial photospheric spectrum.

"6

E

o

m

a

E

E

¢,t

o

E

E

/

/ //\

_900 4-000 4100 4200

Wavelenqth ]n ang._Lrom

Figure (4.18). Plots of the smoothed extraterrestrial

photospheric intensity spectrum and the terrestrial

photospheric intensity spectrum recorded by fiber #06.

111
. The ratios between the extraterrestrial and the terrestrial photospheric spectra

were calculated against wavelength. Figure (4.19) shows plots of the smoothed

extraterrestrial photospheric intensity spectrum, terrestrial photospheric intensity

spectrum recorded by fiber #06 and the ratio between them.

i

5900 400f_' 4100 4200

W'aveleng_hin angstrom

Figure (4.19). Plots of the smoothed extraterrestrial

photospheric intensity spectrum, terrestrial photospheric

intensity spectrum recorded by fiber #06 and the ratio between

these two against wavelength.

112
. A polynomial fit was made to the ratios between the extraterrestrial and the

terrestrial photospheric spectra that were calculated against wavelength. Figure

(4.20) shows the polynomial fit made to the ratio between the smoothed

extraterrestrial photospheric intensity spectrum, and the terrestrial photospheric

intensity spectrum recorded by fiber #06.

2.5

2,0

S

"6

o

o=

r_

"u

0,5

0,0

W_velength dependent intensity' correction factor

.... I ......... I ......... I ......... I ....

A 0 = 25045.208

A 1 = -26.1847B9

A 2 = 0.0083976463

A 3 = 7.2216525e--008

A 4 = --3.7537644e--010

A 5 = --S.061945Be--014

A 6 = 6.363391Be--017

A 7 = --1.1336005e--020

A 6 = 6.7819400e--025

//,/

p

/,

/,

/-

f,

./-

f

j.,

f = _um(_3, [} for i = _h, 8.

t_._-estri_i _ntensi_y,./f = e_trcJt_restri_i intreat:it

3900 4000 4100 4200

Wsvelength ( ,_ ) in angstrom

Figure (4.20). Polynomial fit made to the ratio between the

extraterrestrial photospheric spectrum and the terrestrial

photospheric intensity spectrum. The numbers on the plot are

the coefficients of the polynomial fit.

113
Figure (4.21) shows the application of the polynomial fit to the terrestrial

photospheric intensity spectrum to obtain the extraterrestrial photospheric intensity. This

indicates a reasonably good fit. Also plotted in figure (4.12), for comparison purpose, is

the smoothed extraterrestrial photospheric intensity spectrum as shown in figure (4.18).

o

4

o

.3

E

z

v 2

_6

E 1

Fitting Elozig sky spectrum to the extroterrestrlul photospheric spectrum

_ !i_/::.._.:r.,. , ,:,, ,',.,/ i_,

.... Rayleigh corrected Eloz[g _ky _pectrurn ol fiber _06

_ _ _ _ ............ I Rd";"lcJ_v_ll _ori'a,:..t_,J Eioz[_ _k,,..,",_itti___f_ct,:,_" I _ _ _ _ _ _ _ _ _ I _ _ _ _ _ I _ _ _ _ '_ _ _ _ -_ I _ _ _ _

,_900 4000 4100 4200

Wo,_el_ngJh in ongsf.rorn

Figure (4.21). Application of the polynomial fit shown in figure

(4.20) to the terrestrial photospheric intensity spectrum to

obtain the extraterrestrial photospheric intensity. Also plotted

for comparison purpose is the smoothed extraterrestrial

photospheric intensity spectrum as shown in figure (4.18).

114
4.5 Application of the intensity correction factor

The extraterrestrial photospheric intensity spectrum differs from the terrestrial

photospheric intensity spectrum measured by MACS primarily due to the atmosphere and

the optical elements in MACS. Therefore the ratio between the extraterrestrial and the

terrestrial photospheric intensity at a given wavelength could be written as shown in

equation (4.1).

IpT (_)/IpT (_0)

IEX (E)/IEx(E0) = f(A,z) -f(E)

Z(E) = transmission efficiency of MACS

A(E) = atmospheric attenuation

(4.1)

where I p, andI Ex are the photospheric intensity measured by MACS from ground and

the extraterrestrial photospheric intensity, respectively. Here both the extraterrestrial and

the terrestrial intensities are divided by their respective intensities at a reference

wavelength _,o. In producing figure (4.18) the reference wavelength _,o was selected to

be the Ca-II H line at 3968.5 angstrom. The same is true for the measurement of any

other extraterrestrial intensity spectrum including the measurement of the solar coronal

intensity measurement during an eclipse.

115
Using equation (4.1) the terrestrialcoronal intensity at a given wavelengthS,

measuredby MACS duringthe eclipsewould correspondto the extraterrestrialcoronal

intensitygivenby equation(4.2).

IECX(_)= _ X IEcx(_0) X'C (_ IC (_0) _GVW

I c (_) : terrestrial coronal intensity

IECX(_)= extraterrestrial coronal intensity

(4.2)

From equation (4.2) the extraterrestrial coronal intensity ratio at wavelengths _1 and _,2

is given by equation (4.3).

IICx(_l)_ f(_21 IC(_l) .--C-- x

IEx _--zzi _ Ic ()_z)

(4.3)

4.6 Determination of the Thermal electron temperature and the Solar wind

velocity for each fiber location

The coronal spectrum recorded by each fiber contains the K, F and E coronal

spectra. In order to determine the thermal electron temperature and the solar wind

velocity, as per the methodology discussed in chapter-2, the K-coronal spectrum has to be

isolated. Figure (4.22) shows the coronal spectrum recorded by the twenty-one fibers in

the 18.0 seconds exposure.

116
......... \I0 

i!iS_i .........

_v_ .... _::::::.::== _ " _

_°

................... _ ........... i _: i.......................... ii::.................. i _

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

__i_ _

...........

......... I '_

_ ........... ,__-._

. ....-.: _

i . \_ I _:

Figure (4.22). These plots show the coronal spectra recorded

by the twenty-one fibers located at various latitudes and

heights on the corona as depicted in figure (4.12). Fiber #23

was centered on the moon. The above spectra were recorded in

the 18.0 seconds exposure. The x-coordinate is in pixel units.

117
Comparisonbetweenfigure (4.22)and(4.12)it is evidentthatthefibersbelowthe

equatorprominently display emissionlines. From fiber #23, which was locatedat the

centerof the moon,was expectedto record the backgroundcounts in addition to the

coronal light reflected off the clouds and re-reflectedoff the moon. However the

observationalsitein Elazig, Turkeyreportedlybeing free of cloudsthe spectrumfrom

fiber #23 only revealsthe extent of the instrumental stray light. This stray light is

believed,largely,to haveoriginatedin thetelescopeandcouldposea significanterrorin

the wind measurement.This is becausethe scatteringis significantlylargein the bright

end of the spectrumin MACS. This in turn could give a largercountnumberfor the

intensity at 4233.0angstrom,which is usedin determiningthe wind sensitiveintensity

ratio. Thisis evidentin thespectrumof fiber #23.In this regardit aprudentto eliminate

at least1-%of thecountsin thebright endof the spectrum.Thefibersrecordingemission

linescanbeattributedto exposureto prominencespectrumasdiscussedin section(4.3).

Figure(4.23)is the coronalspectrumrecordedby fiber #06.As perfigure (4.12)

this fiber was locatedmidwaybetweenthe solarequatorandthe solarNorth Poleat 1.1

solarradii. Also LASCO imagesin figure (4.13) confirm this region free of streamer

activities.Thefollowing stepswerecarriedoutin orderto isolatetheK-coronalspectrum

in fiber#06 andcloselyresemblethemethodologydescribedby Ichimotoet.al(1996).

118
. It was assumed that the Rayleigh corrected sky spectrum to closely resemble the

F-component of the solar spectrum. On this basis a fraction of the sky component

corresponding to about 1-% of the counts at 4250.0 angstrom in the coronal

spectrum was subtracted.

5000

E

_,4ooo

u2

%

_=3000

o

o

E

= 2000 .1

=

oo

'S

E

z 1000

E

Coronal spectrum recorded by fiber #06

_ _ _ _ _ i _ _ _ _ _ _ , _ , i , , , _ _ , _ _ _ i _ _ _ _ _ _ , _ , i , , ,

fib@r#136

/

i

//

/

-I I I I I I I I I I F F I F I I I I I I I I I I I I I I I I F F I F I I I I I I I

3900 4000 4100 4200

Wavelength ]n angstrom

I Figure (4.23). The coronal spectrum recorded by fiber #06.

. A third order polynomial fit was made to the difference between the coronal spectrum

substracted by the F-coronal spectrum. This was possible in the above case since the

coronal spectrum of fiber #06 is devoid of any significant emission lines. In the cases

with strong emission lines ganssian fits would be made and subtracted with the

background added to the continuum. In order to account for the stray light a constant

value was subtracted from the coronal spectrum in order to make the overall variation

119
with wavelength(the ratio of the averageintensity in 3700-3800angstromandthe

intensity around4250 angstrom)to be identical with that of the brightestspectrum

recordedby a fiber. Forfiber #06this correctionamountedto 55digital units in a 16-

bit dynamicrange.Figure (4.24) showsthe reducedK-coronal spectrumfrom fiber

#06.

5000

4O00

= 5000

o

u

E

m

2000

2

1000

Terrestrial K-coronal spectrum recorded by fiber #06

i i i i i i i i i i i i i i i i j i i i i i i i i i I i i i i i i i i i j i i i i

TetTe _dol E+F+K S __

j,_f'/

3900 4000 4100 4200

Wavelength ]n angstrom

Figure (4.24). The terrestrial K-coronal spectrum reduced from

the coronal spectrum recorded by fiber #06.

. The terrestrial K-coronal spectrum was adjusted by the wavelength sensitivity

curve in figure (4.20). This would transform the terrestrial K-coronal spectrum to

the extraterrestrial K-coronal spectrum. This curve was used to determine the

wind and the temperature sensitive ratios using equation (4.2). Figure (4.25)

120
.

shows the extraterrestrial K-coronal intensity derived for fiber #06 together with

the wind and the temperature sensitive intensity ratios.

2400

2200

ExtraLerrestria[ k,_-coroncsl spectrum, calculated for fiber #06

2000

1400

1200

1000

3900 4000 4100 4200

Wavelengih in angstrom

Figure (4.25). This plot shows the extraterrestrial Kcoronal spectrum derived for fiber #06. The wind and the

temperature sensitive intensity ratios are 1.245 and

1.488, respectively.

The theoretical temperature and the wind sensitive plots for the desired line of sight

were then used to predict the thermal electron temperature and the solar wind

velocity. The figure (4.26) and figure (4.27) show the locations of the temperature

and the wind sensitive intensity ratios, derived from figure (4.25), on the theoretical

thermal electron temperature and the solar wind velocity plots at 1.1 solar radii,

respectively. These plots associate a thermal electron temperature of _1.73 MK and a

solar wind velocity of _467.0 km/sec for fiber #06's location on the solar corona. The

error bars will be discussed in section (4.7).

121
g

1,45

0.5

..........1.488

i i i

1(41oo}/'1(38so) vs T_mp_,-_t_r_

%__._'

.........

1,0 1,5

Temperature MK

rio _n solar rodlus 1.1@0D000

2.0

Figure (4.26). The temperature sensitive plot. This

assigns a thermal electron temperature of ~1.73

MK for fiber #06.

I 260

g

s

.d 1.250

(42_.3)/(.39571 VS Wlnd

I I

i i i

mperature MK /. i

I ."

I

I ."

I O_'_Joo00000

I -"

._" _ 00¢8C,0 S

x' I

iz'I I

I .-i

i I -

245 " "_I .-F " '"I . .4' " ..... 'i

____ ........... L........ :_:;; ....... _",_.;._;._:::::_;;; ................ L; .............................. _...............

- /'A" :-' ' I I I i

_4':;: " I I I I 1

" ;, I, I,467 •0 I, 1,

I I , I :

t • • ] ± L L ] J t L • J J t L • ± J

200 400 600 800 1000

_]nd km/_ec

d_o i;',_:.I_r radiu_ 1.100_000

Figure (4.27). The wind sensitive plot. This assigns

a solar wind velocity of ~467.0 km/sec for fiber

#06.

122
1['4-100)/1(3850) VS Tempe.rat ....

v

1.531 ]

Figure (4.28). The temperature sensitive plot.

This assigns a thermal electron temperature of

~1.29 MK for fiber #04.

1.260

_n

,o

1.250

1.240

1.23(]

g

(4233)/ (3987) VS Wirad

I [

i I I

14Fr,perotur e MK ._" _ I 4

0.66o00000 _

I - I

M" I._.,oeooo o i

-1 i

i _ :::::<> < i

t I ,

I • . •

i A -'f" " i

•" 242 " .... " ' ..... ' Z

' 4" ' " " ...... '

I _' I-." .- I

- _ .. .......

- .i - .- i i <

:< ! i3ooo. !

i

L t J- • ] _ [ L J _ i L • J J t L •

200 #Og 500 _oo 1000

Wlnd kin/gee

Figure (4.29). The wind sensitive plot. This

assigns a solar wind velocity of ~300.0 km/sec for

fiber #04.

123
Similarly figure (4.28) and figure (4.29) show the locations of the temperature and

the wind sensitive intensity ratios on the theoretical thermal electron temperature and the

solar wind velocity plots at 1.1 solar radii, respectively. These plots associate a thermal

electron temperature of N1.29 MK and a solar wind velocity of N300.0 km/sec for fiber

#06's location on the solar corona. Figure (4.30) shows the extraterrestrial K-coronal

spectrum derived for fiber #10. Figure (4.31) and figure (4.32) give a thermal electron

temperature of _1.34 MK and a solar wind velocity of _571.0 km/sec for fiber #10's

location on the solar corona.

4500

4000

o

u

_; _500

E

=

z

3000

2500

2000

Extraterrestrial K-coronal spectrum calculated for fiber #10

i i , , i i I ' i , I I , i I I I i I I , i , I I , I I I , , I I , , , J I i ,

iZa : E:.z:[i. l,'z$1z.rxi: .. ":2;:? ! (t2i_:_

i '[-<2F15 ,"1( "4S8"?' = ! £,C_£,8 ";!5

7

J Y

9900

==:_!ifi:" I i p i I i i p i :iii::p i i i p i i i J p 4v93i

4000 41 O0 4200

Wo_elengIh in ongstrom

Figure (4.30). This plot shows the extraterrestrial Kcoronal spectrum for fiber #10. The wind and the

temperature sensitive intensity ratios are 1.248 and

1.524, respectively.

124
1(4100},i1(3_50) VS Te'naera_ure

_s

Figure (4.31). The temperature sensitive plot. This

assigns a thermal electron temperature of ~1.34

MK for fiber #10.

! 280 I I

! 260

Figure (4.32). The wind sensitive plot. This assigns

a solar wind velocity of ~571.0 km/sec for fiber

#10.

125
Figure (4.33) shows the coronal spectrum recorded by fiber #14 that includes

prominent emission lines in calsium and hydrogen. The continuum was obtained by

fitting a ganssian function of the form given by equation (4.4) to the emission lines and

fitting equation (4.5) to the continuum.

z2

f(X)=A0e 2 +A3+A4 X+A5X2 where Z=

x-A 1

A 2

(4.4)

A 3 +A4X+A5X2

sooo

4000

200g

Caronal specLrum recorded by Fibe_ #141n the 4-seconds exposure

i

i

i

i/i

ii

t /

-'"J 3933,7 AO! .3O6,S 5 A0

.............._...'._.:

•,J '"

/,

/4

/.J

j/

........ conti ",uurn (K :f)

.... E+K+F coro_o

: i ': 42'_'(L7 >0

Ca II K ! !C_ iiH

l

3900 4000 4-1O0 42_0

Wavelength in ong_trorn

Figure (4.33). This plot shows the terrestrial coronal

spectrum recorded by fiber #14. The continuum is a

measure of the terrestrial F+K corona. The E-corona is

removed by making Gaussian fits of the form given in

equation (4.1) to the individual emission lines and fitting

the background given by equation (4.2) to the continuum.

(4.5)

126
Following the stepsdescribedabovefigure (4.34) showsthe extraterrestrialKcoronalspectrumreducedfor fiber#14.

Extm[errestrlaJ K-corsnul spectrum calculated For fiber #14

4000

3000 --

"8_ 2000

z

/ i i

1000

3900 4000 41 O0 4200

Wovelenq_h in cnq_+-rom

Figure (4.34). This plot shows the extraterrestrial Kcoronal spectrum for fiber #14. This is following the

elimination of the E-corona from figure (4.33).

From figure (4.35) and figure (4.36) the thermal electron temperature and the

solar wind velocity derived for fiber #14 are N 0.73 MK and N 819.0 km/sec, respectively.

However from figure (4.12) and figure (4.13) it is evident of a streamer and some bright

structures in the field of view of fiber #14. This fairly low temperature could result from

one of two possibilities. Firstly the fiber may have seen the corona very close to the limb

based on the possibilities of errors arising from improper alignment of the fiber optic

plate in the focal plane of the telescope or improper tracking.

127
1.65

0.73

0.00000000

100,00000

o0000050

500,00000

400 00000

500.00000

_0( (00,'0

700,00000

800,06000

900.00000

Figure (4.35) The temperature sensitive plot. This

assigns a thermal electron temperature of ~0.73

MK for fiber #14.

1,270

_" 1,250

1(4233]/1(5987) VS Wind

I

Tempe atur_ MK i

J

I / I

1.268f l /" " / '

i

/ ,.

/

// • ! _"

,.2 " . - i

.S . " i

% i _ i

*- i

i i [8 9.o I

i i

± J _ L L • • ] ± t • ] ± i L _ J J t

2OO 400 600 600 1000

Wind k _/'sec

_,) rr, ;-,siermdlL_ 1.108,0000

Figure (4.36). The wind sensitive plot. This assigns

a solar wind velocity of ~819.0 km/sec for fiber

#14.

128
Secondly, it is apparent that local density enhancement may have resulted at

points of intersection of the streamer with the line of sight of fiber #14. Although the

actual geometry nor the magnitude of the density enhancement are known, nevertheless,

using a trial geometry and an assumed magnitude for density enhancement it can be

shown that the temperature determined for fiber #14 may differ widely from the true

value. This phenomemon is described in Appendix-C. In the following example the

geometry described in Appendix-C was used along with an assumed density

enhancement by a factor of 50.0.

wind

1.70

819.0

_I 9.0

_ i.'i i}

6E _,o

1,40

1.30 i i

-100

_z_ 1.60

u_

%

v

1,50

1(4100)/1{5850) '<S Streamer angle

I I

_m/sec J-_- ...... temp MK

_19_..,_':2 ........... Z:..__..... ]l'_ ............................. _.ac_

r_ '-_ i _- .i ,

i 4 " ....... F_ t1

I

I' • '" ''¢- '", .. i '.. i

'. I ,.. .

...................................................... ,=y:;.......... =:; ............................. :;_:. -- _-.LI

I fronL of I,he imb beh]r_'dLh_.ra_b_ I

der ity enhancement facf-_ r 50.O/strearner _idth 0.5 SR/rPo 1.1000000

I

I

-50 0 50

r_ngle in decjree_

0.5G

0,7_

i 0

0

I

Figure (4.37). Plot showing the variation in the

temperature-sensitive intensity ratio, from the

theoretical models without streamers, for a streamer of

geometry described in Appendix-C and density

enhanced by a factor of 50.0 together with various

streamer inclinations in front and behind the solar limb.

129
In figure (4.37)thehorozontallinesshowthetemperature-sensitiveintensityratio

for modelswithout streamers.Thecurveswith the matchingcolorsshowthevariationof

this ratio with streamersof the abovegeometry and density enhancementlocated at

variousanglesin front andbehindthelimb. The two arrowsindicatethat a coronawith

temperature1.0MK andwind velocity 819.0km/secalongwith a streamerenhancedin

densityby a factor 50.0and inclined at N 18.0oor N58.0° degreesin front of the limb

couldhavegiven the sametemperature-sensitiveintensityratio asthetheoreticalmodel

without streamersfor an isothermal coronaof temperature0.73 MK and solar wind

velocity 819.0km/sec.This exampleshowsthe sensitivityof the streamerscrossingthe

linesof sightof thefibersin determiningthetemepratureandthewind velocity.However

this cannotbeconsidereda detrimentalfactorin the broaderscienceof determiningthe

densityenhancementassociatedwith streamers,theirtemperaturesandvelocities.

My belief for the low temperaturerecordedby fiber #14 has the following

interpretation.From figure (4.1) and (4.14)it is apparentthat the fiber #14 was closely

placedto theextendedprominence.It is very likely in this this scenariothat the line of

sightintegrationfor fiber #14 containedcool materialassociatedwith a prominenceand

is a signatureof its likely temperature.

From similar analysesthe schematicdiagramin figure (4.38) showsthe values

determinedfor the thermalelectrontemperatureand the solarwind velocity for fibers

#04,06, 08, 10,12, 14& 22 locatedat 1.1solarradii andfiber #15 & 17locatedat 1.5

130
solarradii. Unfortunatelyit hasnot beensuccessfulin determiningthewind velocitiesfor

fibers # 15 and 22. The spectrumrecordedby thesefibers show a gradual declinein

countsabove4200.0angstromfor reasons,which arenot obvious.All the temperature

and wind valuesdeterminedfor the abovefibers arebasedon the theoreticalmodels

without considerationfor streamers.

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: .............. ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

3 7.4° ................

Figure (4.38). The thermal electron temperature and the solar wind

velocity determined for solar coronal locations observed by fibers #04, 06,

08, 10, 12, 14 & 22 that are located at 1.1 solar radii and fiber #15 & 17

located at 1.5 solar radii. These values are based on the theoretical

models. The temperature and the wind velocity are measured in MK and

km/sec, respectively. However the wind measurements seem too far high.

In this regard it is acknowledged that the experiment was not successful in

measuring the wind velocities and the absolute values should be

disregarded.

131
4.7 Sources of error and error bars

The sources of systematic errors could have resulted from improper alignment of

the fiber optic plate in the focal plane of the telescope and tracking inadequecies. These

practical reasons for these problems in the present status of MACS are highlighted in

chapter-7 with suggested remedial measures.

Another disappointment in this experiment reflects from the spectrum recorded by

finer #23, which was centered in the center of the lunar shadow to record the background

counts. The spectrum recorded by fiber #23, which was intended to measure the

background counts is shown in figure (4.39).

/,/

/ii

Figure (4.39). The spectrum recorded

by fiber #23, which was centered on

the lunar shadow during the eclipse.

Its purpose was to measure the

background counts.

132
Figure(4.39)highlightsthe scaleof the instrumentalscatteringspeciallyin thehigh end

of the wavelengthregionthat is sensitiveto the wind-sensitiveintensity ratio giving a

higher countat 4233.0angstrom.An attemptto quantify its influenceon the spectra

recordedby the otherfibers is given in chapter-6.This may bethe reasonfor obtaining

very large measurementsfor the wind velocities.The wind velocitiesexpectedat 1.1

solar radii are of the order of N100.0 km/sec. In this regard the absolutevalues

determinedfor thewind velocitiesshouldbedisregarded.Howevertheratiosof thewind

velocitiesin the equatorialregionsto thecoronalhole regionsdo makesensewherethe

wind velocities at the poles are of the order N3.0 times the wind velocities in the

equatorialregions.Instrumental scatteringcould also be responsiblefor causingthe

coronallight recordedby anumberof fibersunusablefor analyses.

constraints associated

constraint associated

However scattering is an inevitable consequencearising from the financial

with obtaining superior optics. Also in respect to the time

with the eclipse duration it was not possible to do repeat

measurementsfor randomstatistics.Howeverthis effort will be feasiblewith MACS

attachedto a coronagraph.

As for the temperature-sensitiveintensity ratio one of the concern was the

associtionof thewavelengthpositionsto bein closeproximity to strongemissionlines.

Forexampletheproximity of the4100angstromwavelengthpositionto theemissionline

of H6 at 4101.7angstrom.

133
In additionthereis also an error associated with the derivation of the instrumental

wavelength sensitivity curve. Figure (4.21) shows the application of the wavelength

sensitivity curve to the Rayleigh corrected sky spectrum to obtain the extraterrestrial

photospheric spectrum (red curve). Matching this extraterrestrial photospheric spectrum

(red curve) with what is considered as the true extraterrestrail photospheric spectrum

(green curve), as depicted in figure (4.21), the overall deficiency is 0.44%. This is the

difference between the area under these two curves. However this does not mean that

point to point differences are las low as 0.44%. At the temperature-sensitive wavelength

locations at 3850.0 and 4100.0 angstroms, these errors are - 4.1% and + 1.9 %,

respectively. This can cause the temperature to be under estimated by 5.7 %, which

translates to 1.14 MK. The ideal remedy to this problem is an absolute wavelength

calibration of MACS with sources of known strengths. However this does not still

account for the instrumental scattering.

Although no clouds were reported during observation, yet the haze due to

convection that is associated with the very dry terrain in Elazig, Turkey is another hidden

consequence on the accurate coronal brightness measurements.

The influence of the numerical procedure and other physical parameters, such as

the electron number density profile, used in creating the shape of the K-coronal intensity

spectrum is discussed in chapter-5.

134
4.8 Summary

It has been demonstratedthat the concept of MACS could be used as an

instrument for global and simultaneousmeasurementsof the thermal electron

temperatureand the solar wind velocity in the solar corona.The inauguralmissionof

MACS was beset with problemsassociatedwith instrumental scattering,lacking an

imaging device to image the eclipseto ascertaintracking reliability and the possible

atmosphericdust pollution due to the experimentallocationin a dry terrain devoid of

vegetation.Howevertheremedialmeasuresarevery simplein natureandtiedto adequete

financialresources.Using MACS with a coronagraphwill also eliminatethe time and

terrainconstraintsassociatedwith eclipses.Using MACS in a spaceplatform that could

takeit very closeto thesolarsurfaceandlook into thecoronawill alsoeliminatetheneed

for a coronagraphand the problemsassociatedwith the F-corona.This would be very

promisingin light of the scatteringandthe F-coronaconcernsassociatedwith terrestrial

observations.The remedial measuresand the usesof other platforms for MACS are

detailedin chapter-7.

135
Chapter 5

MODIFIED CRAM'S THEORY AND ITS DEPENDENCE ON VARIOUS

PARAMETERS

5.1 Modified Cram's theory

Lawrence Cram (1976) formulated the theory for the formation of the K-corona.

The modified Cram's theory includes the influence of the solar wind velocity in the

formation of the K-corona, which is a continuous spectrum. The reasons for the

formation of this continuous spectrum and the influence of the solar wind velocity are

discussed in chapter-2. Figure (5.1) is a schematic diagram showing the mathematical

description of the scattering phenomenon where the photospheric radiation from the sun

is scattered by the free electrons lying along the line of sight of an observer. This

scattering phenomenon is popularly known as Thomson scattering while the scattered

radiation contributes to the formation of the K-corona. A detailed treatment of the

scattered intensity, which includes the influence of the solar wind velocity, is provided in

Appendix-A. Equation (5.1) gives the intensity of the radiation scattered at an observed

wavelength (_) by the free electron density distribution along the line of sight at height

( 9 ) above the solar limb.

136
Figure (5.1). This is a schematic diagram showing the

mathematical description of the scattering phenomenon where

the photospheric radiation from the sun is scattered by the free

electrons lying along the line of sight of an observer. The line of

sight is at a distance ( p ) from the center of the sun.

+_ 2_

IRa0_'PRsolar) = f f

--_ 0 COS OJ* 0

Ne(rRsolar) X QoRa (_,O)x

p

2_f_Ab I(_, c0,x)exp

1

f id_'dq)dcoso) d(XRsolar)×

2b 2 cos cow (rR solar )radial

)_'1+

C

2Ab

where O - (//,1)

//parallel to the radial plane

I perpendicular to the radial plane

(5.1)

137
Q/_ : 1-_n _T (sin2 _ +cos 2 _cos 2 O)

Q ,R_: 1-_n _T (COS2_ + sin2 _cos20 )

/

i(_,,co, x) = l/TC_ Rs°lar )AU/2/.1- u `l__U-[-031u, ,cos

O = n - cos-m (sin cosin k0sin Z + cos cocos Z

= sin-1 k[_____)sin cosin qo

/

= io-l/1 /

0 = sin-m (r sin co)

r 2 = x 2 + p2

(5.2)

138
Thedistancesr, 9 and x are expressed in solar radii as follows.

r --> rRsola r

p --> PRsola r

X --> xRsola r

(5.3)

The physical parameters that are inherent in equation (5.1) are given in equation (5.4).

u 1 (_') = limb darkening coefficient

f(U) = extraterrestrial solar irradiance

N e (rRsola r ) = electron density model

T(rRsola r ) = coronal temperature model

W(rRsola r ) = solar wind model

(5.4)

The detailed derivation of equation (5.1) (=Appendix-A.98) is shown in

Appendix-A. The computer code written in IDL to solve equation (5.1) is given in

Appendix-B. From the physical parameters that are inherent in equation (5.1), as shown

in equation (5.4), it is evident that the shape of the theoretical K-coronal intensity

spectrum is dependent upon the models used for the coronal temperature, the solar wind

velocity and the electron density distribution structures. In addition the numerical

outcome of equation (5.1) can also be affected by the numerical methods employed in

solving the various integrals. However the extraterrestrial solar irradiance spectrum and

the limb darkening coefficients could be considered as well-established quantities. As

such, it is prudent to analyze the dependence of the shape of the theoretical K-coronal

intensity spectrum on the coronal temperature, the solar wind velocity and the electron

density distribution structures and the numerical methods used in solving the integrals.

139
5.2 Dependence on the electron density distribution function

The bright corona seen during a solar eclipse is due to the scattering of the

photospheric light by the free coronal electrons (F-corona), by dust (F-corona) and the

emission lines (E-corona). The F-corona can be easily eliminated with the use of

polarizes while the E-corona could be identified with the prominent peaks in an otherwise

continuous spectrum. These peaks could be easily removed, as demonstrated in figure

(4.33), to isolate the K-corona. The point that needs to be made here is that the most

straightforward way of measuring the coronal electron density distribution is by

measuring the K-coronal intensity distribution during a solar eclipse that is filtered

through polarizes orientated at three different angles to eliminate the K-corona. This

information can then be used to determine the electron density distribution that would

have yielded the measured intensity distribution due the scattering of the photospheric

spectrum by the free coronal electrons. This method is most suitable for electron density

distribution during the maximum phase of the sun, which allows for the corona to assume

azimuth symmetry.

One of the pioneering efforts in this regard was due to Banmbach (1937).

Baumbach, from an analysis of the photometric material of ten eclipses from 1905 to

1929, deduced the expression given in equation (5.5) for the distribution of brightness in

the solar corona. This calculation was based upon averaging for all the observations and

for all values of the position angle with the brightness of the center of the solar disk taken

as 10 6 .

140
0.0532 1.425 2.565

I(r)- r2.S + r7.----_ -_ 1717.o (5,5)

From equation (5.5) it is apparent from the last term that the brightness falls off

most rapidly in the innermost region of the corona while at large distances the brightness

is proportional to the first term, which decreases fairly slowly. This formula gives only a

smoothed, average value for the distribution of the brightness. On a brighter note

November and Koutchmy (1996) report of a best ever achieved white-light coronal

observations. This unique opportunity was afforded by the path of total solar eclipse of

11 July 1991 passing above the 3.6 m Canada-France-Hawaii Telescope (CFHT) on

Manna Kea, Hawaii, which allowed for high spatial resolution white-light coronal

observations with the most modem detectors that was denied to Baumbach (1937). The

sun was at its maximum phase during this eclipse. Like Baumbach (1937), November

and Koutchmy (1996) use the sum of three power -law terms to best approximate three

regions of the curve, namely, r < 1.1 SR, 1.1 SR < r < 1.5 SR, and r > 1.5 SR (SR =

solar radii), to obtain the brightness distribution given by equation (5.6). Here again I(r )

is in units of 106 of the solar disk center and (r) in units of solar radii.

0.0551 1.939 3.670

I(r)- r2.S _- r7.---U-+ rlS. o (5.6)

141
Figure(5.2) showsthe comparison between the brightness distributions given by

equation (5,5) and equation (5,6). November and Koutchmy (1996) conclude that the

small difference in the slope to be smaller than the uncertainties in the measurements.

Log (I)

1.

0._

O.

-11.5

Log (Intensity) VS Solar radii

I I r" ' l I I

-

- -

V " -

/I I . I I, I_ [ _-

1. 1.1 1,2 1,.3 1/- 1,R 1.6 1.7

r (Solar radii)

Figure (5.2). A plot showing the solar radial log-intensity

variation as a function of solar radii (r) from the center of the

sun. The solid curve represents the measurements by

November and Koutchmy (1996) while the dashed curve

represents the measurements by Baumbach (1937) based on

several eclipses observations.

The coronal medium is optically thin. For Thomson scattering in optically thin

plasma, the measured K-coronal intensity is proportional to the integral through the line

of sight of the electron density times a local dilution factor, which is the solar intensity

illuminating the scattering region of the corona. One consequence of this property is that

the average radial intensity variation determines the mean radial variation of the electron

density. Figure (5.3) shows the electron density distribution derived by Banmbach

(1937) and November and Koutchmy (1996). Here an isotropic scattering function is

assumed. This is not a bad choice for the inner corona since the illumination dilution

142
factorbecomesequivalentto thenearlyisotropic scatteringform becauseit representsa

superpositionof intensitycontributionsfrom awiderangeof solardiskangles.

Log(N e ) em3 Log (Electron number density) VS Solar radii

( 1 I I I I I ---- ]

J

'_N I I I I I L I "-_

_1 12 1.3 1,,4 "1,.,_ 1,1_. "F,,,7

r (Solar radii)

Figure (5.3). A plot showing the log of the electron density variation

as a function of the solar radii (r) from the center of the sun. The

solid curve represents the calculations by November and Koutchmy

(1996) while the dashed curve represents the calculations by

Baumbach (1937) based on several eclipses observations.

The electron number density derived by Banmbach (1937) is given by equation (5.7). A

detailed derivation of equation (5.7) is also shown in Shklovskii (1965).

.0.036 1.55 2.99.

Ne(r) = 108 X (7+ r----6-_.o+ r--_-._.o) cm 3

(5.7)

It is true that the total brightness of the corona changes from one eclipse to another.

Generally speaking the coefficients of equation (5.7) too should change from eclipse to

eclipse. At sunspot maximum, that is when the sun is at its maximum phase, the corona

has approximately a circular form and uniformly bright as shown in figure (5.4). At

sunspot minimum, that is when the sun is at its minimum phase, the corona is elongated

in the equatorial regions and brighter than the polar regions as shown in figure (5.5).

143
Figure (5.4). A photograph of the solar

corona during the total solar eclipse of

11 August 1999 where the sun was

approaching the maximum phase.

Here the corona is uniformly bright

and circular in shape.

Figure (5.5). A photograph of the solar

corona during the total solar eclipse of

4 November 1994 where the sun was

approaching the minimum phase. Here

the corona is elongated in the

equatorial regions and brighter than

the polar regions.

144
For severalreasonsaslisted below, the electronnumberdensity distribution given in

equation(5.7) was selectedasinput parameterin equation(5.1)for thecomputationof

thetheoreticalK-coronalintensity.

(a) This derivationof theelectronnumberdensitygiven in equation(5.7)wasbased

uponten eclipseobservations.Althoughtheobservationswereperformedalmost60-

80yearsagotheshapeof thedensitydistributionwasreconfirmedby Novemberand

Koutchmy (1996)using themost modemtelescopefor the samepurpose.Sincethe

reconfirmationwasin conjunctionwith the sunin maximumphasein July 1991,this

densitydistributioncouldbe consideredsuitablefor the total solareclipseof August

1999,which againcoincidedwith the maximumphaseof the sun.The comparison

between the electron number densities determined by Banmbach (1937) and

NovemberandKoutchmy(1996)is shownin figure (5.3).

(b) During the maximumphaseof the sunthe coronalbrightnessis approximately

uniform all aroundthesun,which is not truefor the sunat theminimumphasewhere

the coronalbrightnessis concentratedin the equatorialregions. For a sunat the

minimumphasea singledensitydistributionfunction may not hold for all latitudes.

Most of the otherdensitymodelsarespecificregionsand in particularhold for the

minimumphaseof thesun,which aredetailedbelow. Thereforein conjunctionwith

themaximumphaseof the sunwherethecoronalbrightnessis uniform all aroundthe

sunthedensitymodelgivenby equation(5.7)seemto bemostappropriate.

145
(c) The method employed in determining the electron density given by equation (5.7)

is an average over so many other coronal features. In determining the theoretical Kcoronal intensity from equation (5.1) it is difficult to integrate over the different

coronal features that may cross the line of sight. For such a calculation to be possible

individual coronal features have to be identified along with their electron density

contrast. In light of the difficulties in such a process it is again most appropriate to

use an electron density model averaged over all such coronal features that may occur

in conjunction with the maximum phase of the sun.

(d) The very definition of the white-light corona is the brightness produced by the

scattering of the photospheric light by the free coronal electrons, which is identified

as the K-corona. Although this may be contaminated by the scattering by dust

particles, which is identified as F-corona, this contamination can easily quantified by

the use of polarizes and eliminated. Also in the region of interest of the experiment

described in this dissertation, which is from the limb to 1.5 solar radii, the

contribution by F-corona is negligible. Therefore it is prudent to use an electron

number density calculation determined by the measurement of the brightness of the

white-light corona, which is the case for the electron number density determined by

Banmbach (1937) and November and Koutchmy (1996).

146
TheelectrondensitymodelusedCram(1976),in formulatingthepropertiesof theKcoronalintensity spectrumfor the determinationof the thermalelectrontemperaturein

thecorona,is givenin equation(5.8).

N e (r) = 1.67 × 10 (4"0+4"04/r) cm3 (5.8)

Ichimoto et al. (1996) used the same electron density model given in equation (5.8) in

determining the coronal temperatures. Their eclipse observation of February 1994

coincided with the sun at its minimum phase where the coronal brightness is prominently

displayed in an elongated shape in the equatorial regions of the sun.

According to Cram (1976) the model given in equation (5.8) agrees with the Van de

Hulst (1950) minimum equator model within 2% for line of sight between 1.5 and 2.0

solar radii. This model is appropriate for the temperature determination by Ichimoto et al.

(1996) because their observations were confined to regions between 1.5 and 2.0 solar

radii and also their observations on the total solar eclipse of February 1994 coincided

with the minimum phase of the sun.

Another model for the electron number density was due to Newkirk (1961). This

model was based on radio and optical observations. Here the effects of refraction were

considered while neglecting the effects of magnetic fields. The optical depth effects

inherent in any radio observations are discussed in section (1.2). Separate values for the

electron number densities were obtained for the average corona, the polar caps and the

147
activeregions. The approximateformula for the electronnumberdensitypresentedby

Newkirk (1961)is shownin equation(5.9).

N e (r) = 4.2 x 10 (4"0+4"32/r) cm3 (5.9)

Figure (5.6). A series of plots showing the radial dependence

of the mean electron density in coronal holes computed

from various sets of polarization brightness data. This plot

was obtained from Cranmer et al. (1999).

148
Variousotherresearchgroupshavecomputedradialdependenceof themeanelectron

density specific to coronal holes. Thesecalculationswere basedon various setsKcoronalpolarizationbrightness.Figure(5.6)is aplot from Cranmeret al. (1999)showing

theelectronnumberdensityprofilescalculatedby variousresearchgroups.

In figure (5.6) the heavy solid line is due to Cranmeret al. (1999) using the

UVCS/WLC aboard the SOHO satellite. And the electron number density profile

obtainedby Cranmeret al. (1999)is givenby equation(5.10).The observeddatain this

caseisrestrictedto regionsgreaterthan1.5solarradii where(r) is in solarradii.

N e (r) = 108 × (3.890 × r -10.5 .__0.00869 × r-2.57) cm3 (5.1o)

The filled circles in figure (5.6) is due to Strachen et al. (1993) using rocket based

coronograph data. Here they considered the corona to be unstructured and spherically

symmetric. However this assumption is reasonable for considerations restricted to the

coronal hole. Here again electron density profile is restricted to the coronal hole region at

heights above 1.5 solar radii. The electron number density profile obtained by Strachen et

al. (1993) is given by equation (5.11) where (r) is in solar radii.

Ne (r) = 108 × (0.152×r -6"71+ 0.0193×r 2"4°) cm 3 (5.11)

149
Using data obtained by Spartan 201-01, Guhathakurtaand Holzer (1994) have

presentedan electrondensity profile for the polar coronalhole region specific to the

minimumphaseof the solarcycle.This againis valid for heightsabove2.0 solarradii

from the centerof the sun. The electrondensitymodel by Guhathakurtaand Holzer

(1994)is givenby equation(5.12)where(r) is in solarradii.

S e (r) = 108 × (0.81 × r -16.87 + 0.80 X r-8.45 + 0.014 × r2"8 ) cm3 (5.12)

The electron density measurements by Fisher and Guhathakurta (1995) were based on

the use of both the Mk-HI K-coronameter from Manna Loa and White Light Coronagraph

of the space-borne Spartan 201. This study coincided with the maximum phase of the sun

in 1990. However the observations were restricted to the northern and southern coronal

holes. Their measurements extended from 1.16 solar radii to 5.5 solar radii. The

uncertainties reported for the electron density measurements at 1.16 - 1.3 and 1.4 - 1.8

solar radii are N15% and (30 % - 35 %), respectively. This latter region coincided with

data set where Mk-HI and Spartan 201 data were cross calibrated. Figure (5.7) is a plot of

the electron density measurements by Fisher and Guhathakurta (1995).

150
1,i 107 i.... 1 I

I

" _,_.

1 2 3 4 5 6

Radial dlstance r in units of Re

Figure (5.7). A plot showing the electron

densities for the north (solid line) and south

polar coronal holes (dashed line) and the north

(dotted line) and south (dot-dashed line) polar

coronal rays as a function of height. Reproduced

from Fisher and Guhathakurta (1995).

Figure (5.8) is a comparison of the electron number density functions based on the

models by Baumbach (1937, equation (5.7)), Ichimoto et al. (1996, equation (5.8)),

Newkirk (1961, equation (5.9)), Cranmer et al. (1999, equation (5.10)) and Guhathakurta

and Holzer (1994, equation (5.12)). Figure (5.9) is a comparison of the shapes of the Kcoronal intensity spectra based on the Banmbach (1937, equation (5.7)), Ichimoto et al.

(1996, equation (5.8)), Newkirk (1961, equation (5.9)), Cranmer et al. (1999, equation

(5.10)) and Guhathakurta and Holzer (1994, equation (5.12)) models for the electron

number density. These plots are normalized to intensity at 4000.0 angstrom and modeled

151
for line of sightat 2.0 solarradii abovethe centerof the sun.Thisplot is simply for the

comparisonof the shapesof thetheoreticalK-coronalintensityspectrabasedondifferent

electrondensitymodels.Thetheoryby Cram (1976)for thedeterminationof thethermal

electron temperatureof the solar corona is basedupon the shapeof the K-coronal

intensity spectrumand not its absolutevalues.It is evident from figure (5.9) that no

discernibledifferencescould be seenin the shapesof the theoreticalK-coronal spectra

for five differentelectrondensitymodels.

'!

.9

i

o

Y

10

Electron number 4e_si±y VS radius

i , i i i i i i ] i i i i i i i i i ] i i i i i i i i i ] i i i i i i i i i ] i i i i

i\

 ̧,

\'\,.  ̧ ̧ ̧ ̧% ̧i ̧ ̧ ̧

1_0¢[_'((0.038/t_1.5) + (1 55/r_8) + (2 99/r-1t))- Baumbach (19_7}

Q i! > i (i (!- {.:. ]-D ¢ i))i]/':' i: ---L;::.:,,,'.:.:!.::':.:: i L (]11] ii

i [) >ii.-ii 1) i [.(i / :11#i11- ?(] 1]{]1 !)-i i - ] ::'> 11---: :; .- :;:<; (;':) i i;;:i!,; ]

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

1 t 4 5

radlus in solar uniLs

Figure (5.8). A comparison of the electron number density

functions based on the models by Baumbach (1937, equation

(5.7)), Ichimoto et al. (1996, equation (5.8)), Newkirk (1961,

equation (5.9)), Cranmer et al. (1999, equation (5.10)) and

Guhathakurta and Holzer (1994, equation (5.12)).

152
E

£

a

q

¥

N

o

c

1.5[

1,0

0,5

5600

Modeled obsep_ed intens[t/ for different electron number density profiles VS wavelength

' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' '

T _K 1.oaaoo _'

//

the SR 2.oaaao /_

/

/

/

/

/

1! equatlon {5.7), Baumbach (1937)

2! equatlon {5.8), Ichimoto {1996)

3! equatlon {5.9), NewkJrk {1961)

4! equatlon {5.10), Cranmer eL al. /,1999)

5! equatlon {5.12), guhathaku_a & Holzer {1994)

3_100 4000 4200 #400 4600

angstroms

i i

#800

Figure (5.9). Plots of the theoretical K-coronal intensity spectra

based on the electron number density functions given by

Baumbach (1937, equation (5.7)), Ichimoto et al. (1996, equation

(5.8)), Newkirk (1961, equation (5.9)), Cranmer et al. (1999,

equation (5.10)) and Guhathakurta and Holzer (1994, equation

(5.12)) and the line of sight at 2.0 solar radii from the center of

the sun. The curves have been normalized at 4000.0 angstrom.

Figure (5.10) is another plot the theoretical K-coronal intensity based on different

density models for the line of sight at 1.1 solar radii. Here again no discerning differences

could be observed in the shapes of the K-coronal intensity distribution based on the

different electron density models. However the absolute values would differ for the

different electron density models although it is irrelevant to this experiment.

153
1.5 i i

E

&

c

o

o,

rq

o

c

Modeled observed intensity for different ±emperoture profiles VS wovelength

/

rh_ SR 1.1afloo

/.,.,_ _ .....

/'

//

/

1: equatlon {5.7), Baumboch (1937)

2: equotion {5.8), Ichimoto {1996)

3: equotion {5.9), Hewkirk {1961)

4: equotion {5.10), Cranmer et ol. (.1999)

5: equotion {5.1 2), Cuhothokurta a, Holzer (1994)

5800 5800 4000 4200 4400 4600

angsLroms

i

4800

Figure (5.10). Plots of the theoretical K-coronal intensity spectra

based on the electron number density functions given by

Baumbach (1937, equation (5.7)), Ichimoto et al. (1996, equation

(5.8)), Newkirk (1961, equation (5.9)), Cranmer et al. (1999,

equation (5.10)) and Guhathakurta and Holzer (1994, equation

(5.12)) and the line of sight at 1.1 solar radii from the center of

the sun. The curves have been normalized at 4000.0 angstrom.

In summary the electron density measurements given by equations (5.8) to (5.12) are

specific to minimum phase of the sun, or specific to the coronal holes or specific to

heights above the region of interest of MACS. The region of interest of MACS extends

from 1.1 solar radii to 1.5 solar radii and the experiment coincided with the maximum

phase of the sun. In this regard it was decided to select the electron number density given

by Baumbach (1937), as given by equation (5.7), as input parameter for the theoretical

calculation of the K-coronal intensity spectrum. Here the calculations were based on ten

154
eclipse observations, which avoids the scattering inherent with coronagraphs, and the

region of interest extended from 1.0 solar radius to N3.0 solar radii. Although these

measurements were conducted about 60 - 80 years ago, this distribution function was

reconfirmed by November and Koutchmy (1996). They had a unique opportunity of

performing the same experiment using the most modern 3.6-m CFHT telescope in Hawaii

in conjunction with the total solar eclipse of July 1991, where the sun was at its

maximum phase.

5.3 Dependence on the temperature profile

Here what need to be investigated are the differences in the theoretical K-coronal

intensity profiles based on the assumption of an isothermal corona and temperature

profiles with a radial dependence. Usually the radial dependence of the coronal

temperature profiles is based on other experimental results. Since the purpose of this

dissertation is to determine the temperature and the wind profiles at several coronal

heights and latitudes, it will not be helpful in substituting radial temperature profiles

determined experimentally. However an important theoretical consideration for the

temperature profile of the solar corona is due to Chapman (1957). He assumed that above

a certain height in the corona losses by emission to be insignificant compared to

conductive losses and derived the following radial dependence for the temperature profile

as given by equation (5.13) where (r) is in solar radii.

T(r) = T o xr -_/7

(5.13)

155
For comparisonbetweenanisothermalcoronaanda temperatureprofile givenby

equation(5.13)assumetheline of sightat 1.3solarradii with theelectronnumberdensity

profile givenby equation(5.7).Also assumethat for the caseof the temperatureprofile

givenby equation(5.13)that thetemperatureat theintersectionbetweenthe line of sight

at 1.3solarradii andtheplaneof thesolarlimb to be 1.0MK andthento fall off asr 2/7.

For sucha casethetemperatureprofile is givenby equation(5.14)andits variationalong

theline of sightis givenby figure (5.11).Here(r) is thedistanceto points alongtheline

of sightfrom thecenterof the sun.The isothermalcoronaassumesa temperatureof 1.0

MK all alongtheline of sight.

1.0

0,0

_3.6

T(r)=l_o_O.s/r)z/7

...... I ......... I ......... l ......... l .......

3 4

radius in solar unihs

Figure (5.11). This is a plot of the temperature profile along the

line of sight at 1.3 solar radii based on the assumption of a

conductive corona Chapman (1957). The temperature at the

point of intersection of the line of sight and the plane of the solar

limb is 1.0 MK.

156
Figure(5.12)is a plot of the theoreticalK-coronalintensity spectradueto anisothermal

corona of temperature 1.0 MK and due to temperature profiles given by

T(r)=l.0×(1.3/r) z/7 and T(r)=l.0+(1.0+9.0*(r-l.75)z) -_. The latter is a

hypothetical temperatureprofile assumedby Cram (1976). From figure (5.12) no

discernibledifferencescouldbeseenbetweenthespectradueto anisothermalcoronaand

themodelwith atemperatureprofile givenby T(r) = 1.0×(1.3/r) z/7.

1.5

E

_' 1.0

o

o

o

3

¥

N

o

E

0.s

O3

Modeled observed intensity for different ternperat,Jre profiles VS wavelength

km}_ o.oooooo 1 3 ._--'_- ..........

z rJ

rho 5R 1.300_0 _-if

S. S

&/

/

/

/,<"

//

../'

1: isothermol corono

2: T(r)= 1.0.{1.3,/r) 2/7

3: T(r)= 1.0+ (1.0+9.0_(r-- 1.75) 2) -1

0,0 _ _ _ I _ _ _ I _ _ _ I _ _ _ I p _ p I p

5600 5800 4000 4200 4400 4600

angstroms

F

4800

Figure (5.12). This is a plot of the modeled K-coronal intensity

spectra for an isothermal corona of 1.0 MK and a model with

the temperature profiles given by T(r)=l.0×(1.3/r) 2/7 and

T(r) = 1.0 + (1.0+ 9.0" (r -1.75) 2)-1. Here (r) is the distance

between the center and points along the line of sight 1.3 solar

radii. The temperature at the point of intersection of the line of

sight and the plane of the solar limb is 1.0 MK.

157
5.4 Dependence on the wind profile

Here again what need to be investigated are the differences in the theoretical Kcoronal intensity profiles based on the assumption of an isothermal corona and wind

profiles with a radial dependence. Usually the radial dependence of the coronal wind

profiles is based on other experimental results. However one important wind profile

based purely on theoretical consideration is due to Parker (1958). The wind structures

due to isothermal and non-isothermal coronal conditions, driven by sound waves, driven

by dust, driven by lines, driven by the combined effect of the magnetic fields and rotation

and driven by Alfven waves are detailed in Lamers & Cassinelli (1999).

Consider an isothermal wind in which the gas is subject to two forces: the inward

directed gravity and the outward directed gradient of the gas pressure. It has been shown

that the momentum equation has many solutions, depending on the boundary conditions

but only one of them, the critical solution starts subsonic at the lower boundary of the

wind and reaches supersonic velocities at large distances. The solution to the wind

structure in such a case is given by equation (5.14).

V X

exp( ) r _ r z )

GMs°lar _(_---_/

r e - , a =

2a 2

where

(5.14)

158
This phenomenonrevealsthat a wind is generatedas a natural consequenceof an

isothermal corona. In equation (5.14), G, T, to, R, a and rc are the universal

gravitational constant,isothermaltemperature,mean atomic weight of the particles

expressedin units of the massof the proton,universalgas constant,isothermalsound

speedand the critical radius,respectively. For solarcomposition ta is 0.602.For an

isothermaltemperatureof 1.0MK theisothermalsoundspeed(a) andthecritical radius

(re) are117.5km/secand6.9solarradii,respectively.Figure(5.13) showstheprofile of

thewind fortheabovecase.

5 .CI

2.5

2,_

_- 1.5

1,0

0.5

0,0

Solar whld velocity (based an Perker model) VS solar radlus

I i i i I i i i I

/

/

j_

J

J

/

/

/

/?

/

,/

/

._ i i I i i i I i i i I i i i

2 4 6

r/r e

i i i

Figure (5.13). Coronal wind profile based on the Parker

model for an isothermal corona. In the ideal solution

the velocity is equal to the isothermal sound speed at

the critical radius and starts with subsonic velocities at

the lower boundary and reaches supersonic velocities at

large distances.

159
The Parkerwind model shownin figure (5.13)is a naturalconsequenceof an

isothermalcorona.Therefore,evenif the solarwind is neglectedin thedeterminationfor

thetheoreticalK-coronal intensityspectrumfor anisothermalcorona,it is worthwhileto

investigatetheeffectonthis spectrumby thebulk flow velocity introducedby theParker

modelnaturally.In figure (5.14)thecomparisonis madebetweena theoreticalK-coronal

spectrumfor anisothermalcoronaof 1.0MK, zerowind velocity andline of sightat 1.5

solarradii with the spectrumdueto thebulk flow velocity that naturallyariseswith the

introductionof theParkerwind modelgivenin equation(5.14).

1.5

E

£

o

s

o

E

-_ 0.5

Modeled observed lntensi'_y fDr different vdnd profiles VS wovelengkh

jjf

/

/'

//

/

//

1_ hcludin 9 Perl_er wind rnodol for on i_othermd corone of 1.0 MK t Iin_ of £ight et 1 8 SR

2: E_thelmol ccr_mc of LO M_': wlth z_r_, _[nd ,,_;,_clL,! & iin_ o_ ̧ sigh_ ,_t _.5 SI_

0,0 , i , I , , , I , , , I , , i I , , , I i

3600 8600 4000 4200 4400 4800

ongstrorns

1 i

4800

Figure (5.14). Comparison of the theoretical K-coronal

intensity spectrum for an isothermal corona at 1.0 MK,

zero wind velocity and line of sight at 1.5 solar radii with

the intensity spectrum due to the bulk flow velocity that

naturally arises with the Parker wind model for an

isothermal corona as given in equation (5.14).

160
In figure (5.14)no discernibledifferencescouldbe observedbetweenthetwo Kcoronalintensity spectra.Thereasonsfor this aretwofold. First, from the Parkerwind

profile shownin figure (5.13)it is evidentthat thesolarwind velocity at 1.5solarradii is

very small.Second,althoughthe wind profile shownin figure (5.13)indicatesthat the

wind valuesaresignificantat largedistances,nevertheless,this effectis negligibledueto

the drop in the electronnumber density at large distances.Figure (5.15) showsthe

theoreticalK-coronalintensity spectradueto isothermalcoronaltemperaturesof 1.0and

1.5MK with theinclusion of the Parkerwind modelfor an isothermalcoronagivenin

equation(5.14). As expectedthe nodesbegin to appearmainly due to the temperature

differences.

E

=_ 1,0

q

g

.?-

¥

E

o

=

Modeled obser-_ed in__enslky for dlfferenL wlnd prefiles VB _avelength

' I _ ' _ I ' ' _ i _ ' ' I _ ' _ I ' '

_, Dj- sg_y

/

/

.j?/

..2"

1: Isotherre,cl coron_ of 1.0 MK Mth Parker Mnd model & llne of siejht ot 1.5 SR

2: _<:th_i'm_l core:Re ct i.5 HFI ,_iti_ P_rk_r ,<rid mod_-I & Ir_ ,=,f _i_'t _t !.2 SP,

3600 380t0 4000 4200 440D 4600

ar, gstrorn_

i

460_

Figure (5.15). Comparison between two theoretical Kcoronal intensity spectra due to isothermal coronal

temperatures of 1.0 and 1.5 MK and with the inclusion of

the Parker wind model for an isothermal corona.

161
5.5 Dependence on the numerical method

The theoretical model for the K-corona, as shown in equation (5.1), consists of

four coupled integrations. In the absence of an analytical solution this equation could be

solved only through a four-dimensional numerical quadrature. While acknowledging the

existence of multitude of numerical methods, the numerical methods used in solving

equation (5.1) were based on procedures presented in the Handbook of Mathematical

Functions (Ed. M. Abromowitz and L.A. Stegun). Two of the integrals were calculated

using the Trapezoidal Composite Rule while the other two were solved using a Ganssian

Quadrature. There is reason to believe in the results since they can reproduce the

theoretical results obtained by Cram (1976) and Ichimoto et al. (1996). This numerical

procedure was also sufficient to determine the polarization components to confirm that

the degree of polarization is almost wavelength independent and that it increases with

heights above the solar limb. The degree of polarization (p) is defined by equation (5.15).

Ra Ra ijj

p(Z,p) = iR/a(Z,p)+iRa(Z,p) where

I/_0_,p) -- Intensity parallel to the radial plane

I_ _ (Z, p) -- Intensity perpendicular to the radial plane

(5.15)

Figure (5.16) shows a plot of the degree of polarization, for a given isothermal coronal

temperature of 1.0 MK, for lines of sight at 1.1, 1.3 and 1.5 solar radii. It is evident from

figure (5.16) that the polarization components were calculated with sufficient accuracy to

confirm the behavior expected of the polarization. That is, they are wavelength

independent and increase with heights above the solar limb.

162
s

%

polarizawon VS observed wavelength

I ' ' i ' I I

0,60 _ km/sec O.O0'O000

[ MK 1,00{3'00

rhc SR 1. I ODO'3 I .:'!.;::l:)_l:::. i i

, =

0.50

OAO r

0,3© I

3500

(}(3 il _}-i] i i

..i---i

Ca..:,i('ik;i " "................................ i ..............................

._---_ ..... . ...... i i "

3800 4000 4200 4400 4600

angstroms

Figure (5.16). Plot of the polarization from theoretical

modeling for the formation of the K-corona, at a given

isothermal coronal temperature of 1.0 MK, for lines of

sight at 1.1, 1.3 and 1.5 solar radii. The behavior of the

polarization conforms to the observations. That is, the

polarization is almost wavelength independent and

increases with coronal heights.

The numerical code to solve equation (5.1) was written in IDL (Interactive Data

Language) and took N100 minutes to calculate the intensity spectrum from 3700 - 4700

angstroms in a Pentium-II 233 MHz PC. Of the four integrals two were solved using the

Trapezoidal Composite Rule while the other two were solved using Ganssian Quadrature.

In the situation where the Trapezoidal Composite Rule was used, the quadrature points

were divided into higher and higher number of points to check for differences in the

results. The final number of quadrature points was selected based on the point where the

overall results for the intensity converges to within five decimal places. As for the other

163
two integrals,thesewere solved using Ganssianquadratures.Their integration limits

allowed for the use of Hermite polynomialsand its associatedweights. That is, these

integralshadtheform givenby equation(5.16).

f(x)e -x dx= f hi i where

__ i=0

h i = zeros of Hermite polynomials

w i = weights

(5.16)

However the line of sight integration was of the form given by equation (5.17), which

was converted to the form given by equation (5.16), which follows the procedure given in

the Handbook of Mathematical Functions (Ed. M. Abromowitz and L.A. Stegun, pg.

924).

+_ O

.... i=O

hi = zeros of Hermite polynomials

W i = weights

where

(5.17)

In equation (5.17) the function f(x) is of the following form given by equation (5.18),

which is a convergent function without any singularities in the region of interest.

f(x)

0.036

1

I-(4x2+p2)6,.- +(_/-_+p2 l ]x

_/-_ + p2 4x2 +02 ____[2 -1

(5.18)

164
For exampleassumeta=1.0 and 9=1.1 in equation (5.18). Then the equation

(5.18) reduces to equation (5.19).

0.036 1.55 2.99 ×_- 92 (5.19)

Solving equation (5.19) by using the trapezoidal composite role, where the integration

limit is restricted to x = +5000 and dividing this region into 100000 intervals of width

0.1, the result yields a value of 1.1954237. The restriction in the x-range is perfectly

valid because the integrand rapidly falls to zero about x=0. Performing this operation

over a range of values for ta is very time consuming. However by using the Ganssian

quadrature over a Hermite polynomial roots of just twenty points, as specified in equation

(5.17), the result yields a value of 1.1900812. This value is accurate to within 1.0 % of

the result yielded by the trapezoidal composite rule and is also computed very much

faster. This is also acceptable in light of the limits imposed by the instrument. An

analytical solution also exists for equation (5.19) as shown in equation (5.20) (Standard

Mathematical Tables and Formulae).

a+l-bc

i X a m b (m+xb)cdxa+l

where a > -1,b > 0,m > 0,c > --, 17- Gamma function

b

(5.20)

165
Writing equation(5.19),asshownin equation(5.21),andusingequation(5.20) yields a

value of 1.1954237,which matcheswith the value obtainedby using the trapezoidal

compositerule.

if[ (__(_)_ 0.036 1.55 (_l2.99 _)× x_+_ 1 --_"x

= 2X /0,036X i°x o (02 -{.X 2) 7/4 {. 1,55X o (02 -{.X 2 (02 -{.X 2

(5.21)

The detailed derivation of the theory and the code written in IDL is provided in

Appendix - A and B, respectively. Throughout the code the accuracy of the calculations

were checked using the built in CHECK_MATH function, which returns the accumulated

math error status, and manually. There were no errors reported.

166
Chapter 6

COMPARISON BETWEEN OBSERVATION AND THEORY

6.1 An unexpected phenomenon noticed in the observations

As shown in figure (3.4) the fiber optic plate in the focal plane of the telescope

contained a single fiber in the center of the frame. During the observations this fiber was

positioned at the center of the lunar shadow. The purpose of this fiber was to measure the

background signal and was to be subtracted from the signals measured by the other fibers

that were exposed to the coronal light.

However the shape of the signal measured by the fiber located in the center of the

lunar shadow only provided a clue that the observations were affected by scattering.

Figure (6.1) shows the signal recorded by the fiber at the center of the lunar shadow.

This prominently shows the peaks of the Calcium H and K lines. The counts recorded by

this fiber are not negligible compared to the counts recorded by fibers exposed to the

coronal light. This indicates that we have a problem with scattering light. The extent of

scattering could not be envisaged in advance for the simple reason that the true

observational condition could not be replicated in the laboratory. However Cram (1976)

167
pointedout that the bright sourcesof scatteredlight at the time of a solar eclipseare

prominencesandthechromosphere,andthesehavea spectrumquiteunlike thatof either

the F or K corona. The only remedialmeasurethat was takenin orderto minimize

scatteringwas to maintaincleanoptical surfacesof the optical componentsin MACS.

Althoughthe extentof scatteringin the different wavelengthregionsis very difficult to

quantify,nevertheless,it is a hardlessonto belearntfor anyfuture observations.In this

regardsomeof the remedialmeasuresfor any future observationsare highlighted in

section(6.3).

g

u

2000

E

J000

Fiber ]n the center of the lunar shadow

b

171/I ;i!'t,

/

/

/

// Calciu_ II _ _ H

3700 3800 3900 4000 4100 4200

_&uelerNth in angstrom

i

4500

Figure (6.1). The spectrum recorded by the fiber located in

the center of the lunar shadow during the eclipse. The xcoordinate corresponds to wavelength scale. The two

prominent peaks are associated with the Calcium K & H

lines corresponding to 3933.7 and 3968.5 angstroms,

respectively.

168
6.2 Comparison between theory and observation

In chapter-2 temperature and wind sensitive intensity ratios were identified at

particular wavelengths. These intensity ratios were at wavelength positions

4100.0/3850.0 and 4233.0/3987.0, respectively, for temperature and wind measurements.

In chapter-2, we also showed how to determine the temperature and the wind velocity

using the above intensity ratios calculated from the K-coronal intensity spectrum. In

contrast to our ratio technique, Ichimoto et al. (1996) chose to determine the coronal

temperature by fitting theoretical isothermal K-coronal models to the observed K-coronal

spectrum, as shown in figure (6.2).

2000

Figure (6.2). Theoretical isothermal K-coronal models fitted

to the observed K-coronal intensity spectrum to determine

the temperature by Ichimoto et al. (1996). The lower curves

show the differences between the observational spectrum and

the theoretical models. The model that showed the least

difference determined the temperature.

169
The sametemperaturemeasurement,as shownin figure (6.2), could also be

determinedfrom themeasurementof thetemperaturesensitiveintensityratio. Suchratio

calculations were performed using filters centered at the temperature sensitive

wavelength positions by J.M. Pasachoff and his team from Williams College,

Massachusettsduring the total solar eclipse of 11 August 1999 in Romania.N.L.

Reginaldand J.M. Davila performedthe theoreticalcalculationsfor predictionsof the

temperaturefor theintensity ratiosfor J.M. Pasachoffandhis team.No resultshaveyet

beenpublishedby Pasachoffet al.

In chapter-4 the intensity ratio technique was employed to determine the

temperatureand the wind velocity for severalfiber locations.The wind measurements

wereconsidereda total failure in this very first attemptto simultaneouslyand globally

determinebothtemperatureandwind velocity at twentydifferent locationson the solar

corona.Thereasonfor thisis thatthewind measurementis very sensitiveto theintensity

at 4233.0angstrom,which is at thehigh endof the wavelengthmeasurementby MACS.

As revealedin figure (6.1) the instrumentalscatteringwashigh at the high end of the

wavelengthregionthuspredictingvery high valuesfor the wind velocity measurements.

As for thetemperatureit too sufferedfrom theinstrumentalscattering.Thereasonfor this

is that thetemperatureis sensitiveto theintensitymeasurementat 3850.0angstromand

4100.0angstrom.As revealedin figure (6.1) the strongCalcium Hand K lines in the

region of 3900-4000 angstrom seem to have contributed strongly to instrumental

scattering.Predictingthecontributiondueto scatteringonthecoronalintensitymeasured

170
by individual fibers is further compoundedby its dependencyon wavelength.At this

juncture the knowledge of instrumental scatteringonly provides awarenessof the

problemof instrumentalscatteringand remedialmeasuresthat need to be considered

beforeany futureobservations.Thesolehandicapin suchobservationsis theinability to

replicatetheexperimentin thelaboratoryin theabsenceof a coronagraph.

Figure (6.3) showsthe comparisonbetweenexperimentallyobtainedK-coronal

spectrumfor fiber # 04 with theoreticalmodelsfor various isothermaltemperatures.

Fromtheintensityratio methodthewindvelocity andthetemperaturededucedfor fiber #

04are300.0km/secand1.29MK, respectively.ThetheoreticalK-coronalmodelsplotted

in figure (6.3) correspondto a wind velocity of 300.0 km/sec and isothermalcoronal

temperaturesof 0.5MK, 1.0 MK, 1.5 MK and 2.0 MK. In such a scenariothe

experimentallyobtained K-coronal spectrummay be expectedto lie between the

theoreticalmodelsfor 1.0MK and 1.5MK. Howeverit is obviousthat theexperimental

K-coronalspectrumdeviatesfrom thepredictions.This deviationcanbedividedinto four

distinctregionsonthewavelengthscale.

1. 3800- below:In this regiontheexperimentalcurvefallsbelowthepredictions.

2. 3800- 4000:In thisregiontheexperimentalcurverisesabovethepredictions.

3. 4000- 4200:In thisregiontheexperimentalcurvefallsbelowthepredictions.

4. 4200- above:In thisregiontheexperimentalcurverisesabovethepredictions.

171
1.Gi

£

E

O

1.2

E

O

a

7-

I

- g,8

3900

Cornparlson of inCenslty recorded by fiber #04 wiCh f_he theoretlcal models

i i i i i i [ i i i i i i i i i [ i i i i i i i i i I i i i i i i i i i I i i i i

_1 _. Th_. 11_. ]1_ The_.

• :

_ : .* __.,.--"

4000 4100 4200

angstroms

Figure (6.3). This plot shows the comparison between

experimentally obtained K-coronal spectrum for fiber # 04 with

theoretical models for various isothermal temperatures. From the

intensity ratio method the wind velocity and the temperature

deduced for fiber # 04 are, 300.0 km/sec and 1.29 MK,

respectively. The theoretical K-coronal models plotted in figure

(6.3) correspond to a wind velocity of 300.0 km/sec and

isothermal coronal temperatures of 0.5MK, 1.0 MK, 1.5 MK and

2.0 MK. In such a scenario the experimentally obtained Kcoronal spectrum may be expected to take a position between the

theoretical models for 1.0 MK and 1.5 MK. (Expt: =

Experimental, Theo: Theoretical)

The comparison between observation and theory for the K-coronal spectrum recorded

by fiber # 04, as shown in figure (6.3), needs to be considered in conjunction with the

extent of instrumental scattering that is depicted in figure (6.1). Figure (6.1) too can be

divided into four distinct regions. Assuming that the signal expected of this fiber located

in the center of the lunar shadow to produce a straight line with a positive gradient

172
signalingahigherbackgroundcountat higherwavelengths.Basedonthis assertion,and

assumingthe signalrecordedin the 4000- 4200angstromto representthe correcttrue

backgroundsignal,figure (6.2)canbedividedintofour distinctregions.

1. 3800 - below: Someoptical componentin MACS seemsto possessa very low

sensitivityto theblueregionof the spectrum.

2. 3800- 4000:TheCalciumII K & H linescontributealargesignalin thisregion.

3. 4000- 4200:Assumedto representhetruebackgroundcount.

4. 4200- above:Scatteringis againsignificantin thehigh endof thespectrum.

Theabovepropertiespertainingto thefour distinctregionsin the scatteredspectrum

is similar to the patternof deviation of the observedK-coronal spectrumof fiber # 04

from the theoreticalmodels, as shown in figure (6.3). That is, the very low blue

sensitivity in the wavelengthregion 3800 - below and the higher counts from the

scatteringof the CalciumII K & H lines in the region3800- 4000angstrommayhave

causedthe observedK-coronal spectrumto dip andrise, respectively,in theseregions.

The scatteringassociatedwith the high end of the wavelengthregion,i.e. 4200- above,

may have causedthe observedK-coronal spectrumto againrise abovethe predicted

level. Figure (6.4) is a plot of the superpositionof figure (6.1) and figure (6.3). This

showsthat thereis a certainamountof similaritybetween,on theonehand,theshapeof

the scatteredspectrum,and on the otherhand,the deviationbetweenthe predictedand

observedK-coronal spectra.In figure (6.4), all spectraarenormalizedto unity at 4000.0

173
angstrom.It is reasonableto infer from figure (6.4)that someproportionof the scattered

light spectrumrecordedby the fiber located in the center of the lunar shadowhas

contaminatedspectrumrecordedby fiber #04.This couldexplainthedeviationsbetween

theobservedspectrumandthepredictions.

E

,o

_zn

E

o

I

¢o

-'d

3900 4000 4100 4200

angstroms

Figure (6.4). A plot of the superposition of figure (6.1) and figure

(6.3). The shape of the spectrum recorded by the fiber located in

the center of the lunar shadow corresponds with the shape of the

observed K-coronal spectrum by fiber #04. The expected location

of the observed spectrum is between the theoretical isothermal Kcoronal spectra for 1.0 MK and 1.5 MK.

Figure (6.5) shows the superposition of the observed K-coronal spectrum for fiber

#04 and the theoretical K-coronal spectrum for a wind velocity and isothermal

temperature of 300.0 km/sec and 1.29 MK, respectively, and the difference between the

two. The wind velocity and isothermal temperature of 300.0 km/sec and 1.29 MK,

174
respectively,for fiber #04 weredeterminedfrom theintensityratio method.Fromfigure

(6.5) it is apparentthat the deviationsbetweenthe observationaland the theoretical

spectraarelargerat wavelengthsof 3900- 3950 angstroms(wherethe scatteredlight

spectrumpeaks),andaresmalleratwavelengthsaround4100angstromIn thisregard,the

deviationsbetweenobservedandpredictedspectraexhibitcertaingrosssimilaritiesto the

shapeof the scatteredlight spectrum.Even a small percentageof the scatteredlight

spectrumbeingcontaminatedwith fiber #04 could havecausedits spectrumto deviate

from thetheoreticalprediction.

2

2

_w

o

I

2.5

2.0

1.5

1.0

0,5

o.o

-0.5

Comparison between theory and obser_'ation for fiber #04-

...... I ......... I ......... I ......... I ....

'R km/a_ 3 Q_C,.£]rJ_

T _k I _.9Qo2,

SR 1.1 D_]eo

__ ob_ erva_ian_dl

__ thoor_t_cal

............ .:? :i ?"., ;E_::)K:2 :E: _::2-:_ ::-I--?E

__ scat_ererl iight specburn

/ J"L

.:' \ /"-_

/

/ ',

/ _'_. ...................... ,f22222222._:;:.'_-_'"_==_-:'::_

.... ". ...............:Z&-...................... / -

/, .......,:. '_,=,,x=- ............... _: _

"j......... ;:_-_-_ ..........

, , _ , _ _ I _ _ _ _ , _ p p _ I , _ , , _ , , _ _ I _ _ _ _ _ _ , _ , I , , , _

3900 4000 4100 4200

angstroms

Figure (6.5). A plot showing the observed K-coronal spectrum

for fiber #04 along with the theoretical K-coronal spectrum for

the wind velocity and temperature derived from fiber #04. The

plot also includes the difference between the observational and

the theoretical plots along with the scattered light spectrum

observed by the fiber in the center of the lunar shadow.

175
In figure (6.5) the areasunder the observationaland the theoretical spectraare

0.22168834and 0.22163162,respectively.That is, the differencebetweenthe areasis

almost negligible. Nevertheless, since the brightest sources of scattered light

(chromosphericor prominencematerial) have spectrawhich are quite different from

those of F- or K-corona, we can now see that even a slight contamination of our coronal

data by such scattered light can have a serious effect on the quantitative interpretation of

our results. This was pointed out by Cram (1976) who suggested that any observations

should avoid such bright sources. However in MACS twenty fibers were exposed to all

around the sun and at different heights above the solar limb. As a result, it is almost

inevitable that some of our fibers were unfortunately exposed to bright prominence

material. The effects due to exposure to these bright sources could be avoided only

through a scatter free instrument. In this regard every effort was made to keep

instrumental scattering to a minimum. The present experiment, the very first of its kind,

reveals the extent and effects of instrumental scattering and the need for more remedial

measures in this regard. The same is true of other fibers that were exposed to the coronal

light: each was contaminated by scattered light to a greater or lesser extent.

Table (6.1) gives the wind and the temperature sensitive intensity ratios for fiber #04

from figure (6.5). From table (6.1) it is apparent that the wind and temperature sensitive

intensity ratios are well matched in spite of the fact that the observational spectrum

deviates from the prediction. This may be somewhat fortuitous, since if the amount of

scattered light had been different, the observed spectrum might have matched a totally

176
differenttheoreticalspectrum.Thismatteris furthercompoundedby thenon-existenceof

any mechanismsto quantify the degreeof contaminationof instrumentalstraylight on

theindividual fiber. In this regardit appearsthatin spiteof theproblemscausedby the

instrumental scatteringthe temperaturesratios were somehowmaintained.Thus the

accuracyof thetemperatureis to within the errorestimatesmadein chapter-(4.7). That

is to within N_+6% (_+1.2MK). As for thewind measurements,asdiscussedin chapter-

(4.7), its measurementwasdeemedto havebeen a failure in the currentexperimental

effort. Howeverthe high wind valuesarein line with theuncertaintyof _+200km/sec

for ameasurementuncertaintyof _+1%.

Table (6.1). Comparison of the wind and the temperature sensitive

intensity ratios between the observational and the theoretical K-coronal

spectrum, for fiber #04. These values are based on figure (6.5).

Fiber #04 Temp: I(4100)/I(3850) Wind: I(4233)/I(3987)

Observational 1.531 1.242

Theoretical 1.529 1.242

Figure (6.6) shows plots of the observed K-coronal spectrum of fiber #04, as

shown in figure (6.5), corrected for the scattered light in a crude manner. Here various

fractions of the scattered light recorded by the fiber located in the center of the lunar

shadow were subtracted from the spectrum recorded by fiber #04. It is evident from

figure (6.6) that the observed K-coronal spectrum then begins to follow closely the shape

of the theoretical K-coronal spectrum implying the influence of the scattered light.

177
CCrrpor_soMbs_lq Lheor_ <:ridoLservc]tion fc:r fiber _04

I I I ....

2o ........

!s

/

i

oa ...... st+oct' ' ' ' ' ' '4oloo...... 411_o......... 421rio....

CQm7qri_on bet_,een Lhecry mnd obaewa*Jen for fiber ,5404

2s I I I ....

',, ................ ._:: ::::: :::;777Z2_[[[7][77 _ _.

7

41./7:'-':::: .............................. ".................. 7

oo

o<5 ' ' ' ' ' '_[o'_' ' i i , , , ,4oIo,, , , , , i i i411o,, , , , , , , ,421o,, , ,

O@mparf_an between theory and ob_e_aLion far fiber #04

t

_o .......... / ?_//' I_ _t_o-oti_0= _ l

/ _'\ ......... =: :=:::::72722_7

/

k;:::::::::;; :_" -........... .........................

o k

_o _ooo 41oo 4zoo

rii

I_i_1

o,

Figure (6.6). Plots showing the effect on the shape of the observed Kcoronal spectrum of fiber # 04, as shown in figure (6.5), in comparison

with the shape of the theoretical K-coronal spectrum by subtracting

various fractions of the scattered spectrum from the observed K-coronal

spectrum. A, B, C, D, E and F correspond to a subtraction of 0.0%,

15.0%, 10.0%, 8.0 %, 5.0% and 3.0%, respectively, of the scattered light

spectrum recorded by the fiber that was located in the center of the lunar

shadow from the spectrum recorded by fiber #04. These plots show that

the observed spectrum closely follows the shape of the theoretical

spectrum with various fractional subtraction of the scattered spectrum.

178
A similar pattern as above follows for all the other fibers for which temperatures

were determined in chapter- 4. Instrumental scattering can be minimized only by paying

careful attention to the optical quality of the components of the instrument. The stricter

the demands on optical quality, the higher the cost will be. The very first operation of

MACS in conjunction with the total solar eclipse of 11 August 1999 has shed much light

and understanding on the effects of instrumental scattering associated with such

observations. Apart from better optics, other remedial measures envisaged for future

operation of MACS are listed in section (6.3).

6.3 Remedial measures

The principle and the method described in this thesis is unique in its character that

allows for the simultaneous and global measurements of the thermal electron

temperatures and the solar wind velocities on the solar corona using the shape of the Kcoronal spectrum. However the steps described in section (4.6) in isolating the K-coronal

spectrum from a terrestrial observation in conjunction with an eclipse are beset with

problems. The main problems that arise from terrestrial observation during an eclipse are

time constraints and state of the sky. Apart from these, there are other difficulties that

arise in connection with the data analysis. We summarize these here.

. In order to determine the wavelength sensitivity of our instrument, we have assumed

that the sky spectrum recorded by MACS (corrected for Rayleigh scattering) should

match the photospheric spectrum depicted in figure (2.1). The comparison between

the two spectra shown in figure (4.21) demonstrates clearly the nature of the

179
o

difficulties we face in this regard, especially close to the Ca line at 3933.7 angstrom

and regions at 4000.0 angstrom. Since this method depends on the shape of the Kcoronal spectrum these inadequacies, although seem to be irrelevant, do affect the

shape of the K-coronal spectrum in a significant manner. However a near perfect

wavelength sensitivity curve could be obtained with the use of absolute wavelength

calibration using sources of known strengths as commonly done for all advanced

instruments with optical elements. This suggests that in this particular experiment the

derivation of the wavelength sensitivity curve for MACS was limited in its accuracy

by choice.

Another problem arises from our assumption that the F-corona must be allowed for in

our Rayleigh-corrected MACS sky spectrum. The intensity of the F-corona (which

depends on the properties of interplanetary dust between the sun and the earth) has

some wavelength dependence, perhaps proportional to x_-. However on the

assumption that the F-corona is unpolarized while the K-corona is polarized (although

only orthogonal scattering gives complete polarization as per equations (A.48) and

(A.49) there exists an easy practical means to determine the contribution from Fcorona. This is done by observing the corona through a polarized filter with the

polarization axis at minimum of three known angular positions. A detailed

explanation could be found in Golub and Pasachoff (1997). The only concern that

hinders this easy maneuver in conjunction with an eclipse is the time constraints. This

suggests that the ideal platform to use MACS be in tandem with a coronagraph.

180
. The instrumental stray light could be reduced to a great extent only through superior

optics. A fair estimate of the distribution of the instrumental stray light could also be

estimated by placing many more fibers in the shadow of the moon and outside the

field of view in the focal plane of the telescope as depicted in figure (6.7).

Figure (6.7). The extra fibers placed in the shadow of

the moon and outside the field of view in order to

record the instrumental stray light distribution.

. Another important deficiency in MACS in its present state is in its inability to

simultaneously visually image the focal plane during the observations. This prevented

us from knowing whether the fibers stayed in focus at their intended locations.

Deviations from the intended positions can arise from slight movements of the

telescope due to wind, and also due to deficiencies in the tracking mechanism. This

understanding could be easily satisfied by attaching a CCD-based video camera to the

viewfinder of the telescope that in turn is co-aligned with the telescope. The live

181
videoimageof thesunthroughasunfilter projectedonaTV screen,atleastfrom one

hour prior to the first contact,canbe placedon a suitable2_D grid and its course

followed for tracking purposes.The bright pixels prior to the first contactshould

appearasdarkbetweenthe secondandthethird contactswith the sunfilter removed.

In orderto achievethis goal, the viewfinder andthe telescopeshouldbe perfectly

aligned.Althoughthis enablesusto monitorthetrackingefficiencyit doesnot prove

thecenteringof the sunon the imageplaneof the telescope.To accomplishthis the

following maneuvercouldbefollowed.

(a). First centerthesun'simagethroughthesunfilter onareticleeyepiece.Thereticle

patternin theMeadebrandis a doublecrossline with two concentriccircles.At this

stageif the viewfinderis perfectly alignedwith the telescopethenthe sun'simage

seenthrough a video recorderattachedto the viewfinder shoulddisplay the sun's

image at the centerof the detector.Centeringon the reticle eyepieceand making

adjustmentsto the telescope'sorientationscanbe carriedout alternativelyuntil the

imageremainscenteredontheviewfinderandshouldsatisfythetrackingefficiency.

(b). Thenattacha camerawith the lensesdemountedfor prime focusphotographyof

thesunthroughasunfilter at thefocalplaneof thetelescope.For theMeadebrandof

12-inchSchmidt_Cassegraintelescopesoneendof the#62-T adapterconnectsto the

F/6.3focalreducerandtheotherendthreadsinto auniversalT-adapterwith bayonetmount unique to the brand of 35ram cameraand focus the image. This should

facilitatethefocusingfor theexperimentsincethefiber opticplatetoo will belocated

182
at theprime focusof thetelescopeduring observations.If step(a) was successfully

performedthentheimageshouldbecenteredtoo.

Howeverthe abovefeaturesdo not yet allow for the simultaneousimaging of the

coronal image formed on the front surface of the fiber optic plate. The following optical

feature as shown in figure (6.8) may be contemplated for MACS in order to obtain

simultaneous imaging of the image formed on the focal plane. In figure (6.8) the fibers

are embedded on an optical flat (well polished reflecting surface facing the telescope)

along with a very thin high transmission glass surface (AB) inclined at 45 o to the optical

axis and with a semi-reflecting coating on the side facing the fiber optic plate. The

reflected image from (AB) is a copy of the image formed on the fiber optic plate with

dark dots marking the light transmitted through the fibers. Such an image will identify the

actual location observed by each of the individual fibers on the corona. This is ideal to be

used in attachment to a coronagarph. This is because any additional photons lost in the

transmission through the additional optical component could be compensated by a longer

integration time.

183
Reflect iOn Of theimag

form_d6n_h b_ pfi_plan

Figure (6.8). A schematic diagram showing the introduction of a very

thin high transmission glass surface (AB) inclined at 450 to the optical

axis with a semi-reflecting coating on the side facing the fiber optic plate

along with the fibers embedded on an optical flat. This alteration will

enable the simultaneous imaging of the image formed on the fiber optic

plate.

The proposal (NRA:99-OSS-01) to build a coronagraph to accommodate MACS

at a cost of N$ 500,000.0 has been approved by the NASA's Office of Space Science.

NASA's Goddard Space Flight Center and The Catholic University of America will

jointly build this instrument over the next three years. Possibilities for observing sites

include Sacramento Peak, Hawaii and the Canaries.

184
Chapter 7

CONCLUSION

7.1 Summary of this thesis

The methodology presented in this thesis is unique in its character: it allows for

the simultaneous and global measurement of the temperature and the solar wind velocity

in the solar corona. This line of thinking originated from Cram's (1976) theoretical

formulation for the formation of the K-corona and its properties. The physical properties

that were taken into consideration in the formulation of the theory included the effects

due to thermal Doppler broadening, limb darkening effects, angular dependence of the

Thomson scattering and the cumulative contribution from different positions along the

line of sight. The properties that were derived were the existence of temperature sensitive

anti-nodes in the K-coronal spectrum and the remarkable independence of the

temperature insensitive nodes with height above the solar limb. It was suggested that

these properties could be exploited to measure the coronal temperature from the Kcoronal intensity profile. Ichimoto et al. (1996) demonstrated the feasibility of Cram's

method by analyzing the K-coronal spectrum obtained using a slit spectrograph during

the total solar eclipse of 3 November 1994 in Putre, Chile.

185
In this thesisCram's theorywas extendedby including the effects of coronal

expansionin the formationof theK-corona.While confirming thepropertiesderivedby

Cram,anadditionalpropertywasdiscoveredwith theintroductionof thesolarwind. This

allows for the simultaneous determinationof both the temperature and the wind

velocity from the sameK-coronalspectralmeasurement.An overviewof the theoryand

of themethodcanbefound in chapters-1and2. Thedependenceof thetheoreticalmodel

on the selectionof the electronnumberdensitymodel, different temperatureand wind

profilesandthenumericalproceduresarediscussedin chapter-5.

In this thesis, insteadof a slit spectrographa fiber optic spectrograpghwas

designedwhosefeaturesallowed for global determinationof the K-coronal spectraall

aroundthe solarlimb andat differentheights.Theinstrumentis discussedin chapter-3.

Combiningthetheoreticalpredictiononthe feasibility to determineboththe temperature

and the wind velocity from a singleK-coronal spectrumandthe instrumentsability to

globally measurethe K-coronal spectra,the instrumentwas namedMACS for Multi

ApertureCoronalSpectrometer.Thevery first effort of its kind wastestedin conjunction

with thetotal solareclipseof 11August1999in Elazig,Turkey.Theobservationalresults

arediscussedin chapter-4.Weweresuccessfulin determiningtheelectrontemperatureat

variouspointsin theinnercorona.

However,animportant(andunwelcome)aspectof our observationsconcernsthe

degradationof the data due to instrumentalscattering.Becauseof unexpectadlyhigh

186
levels of scattering,we were not successfulin obtaining reliable estimatesof wind

velocity.Althoughremedialmeasuresweretakenin expectationof possibleinstrumental

scattering,the actual magnitudeof the scatteredlight wasnot realized.Our efforts to

reducescatteringwerecompoundedby theinability to replicatetheobservingconditions

in a laboratoryandthe lack of fundsto obtainhigh quality opticalcomponents.Oneof

the vital ingredientrequiredto minimize instrumentalscatteringis careful selectionof

high quality optical components.Another important featurefor the successof MACS

wouldbe to performanabsolutewavelengthcalibrationin the laboratory.Theextentof

theinstrumentalscatteringandsomeof theremedialmeasuresarediscussedin chapter-6.

Despitethe lack of successin determiningthe wind velocity, our successin

determiningthe coronal temperatureusing MACS has alreadybeen recognizedas a

successfulproof-of-conceptby a funding agency.A proposalto mateMACS concept

with a coronagraphwas submittedto NASA and it receivedfavourablereview. As a

result,fundshavebeenawardedfor anupgradedversionof our experimentin which the

scatteredlight problemshouldbemuchreduced.It is hopedthat theupgradedversionof

MACS will be ready for the solar eclipseof 21 June2001 in Africa. At this eclipse

MACS is to becomplimentedwith afilter basedsystem,wherefilters wouldbecentered

at thetemperatureandwind sensitivewavelengthpositions.The conceptof MACS has

alsobeenrecommededasaninstrumentin theproposalfor theinstrumentalpackagethat

would fly aboardthe SolarProbe.A brief review of the SolarProbeis givenin section

(7.2).

187
7.2 Using a spacecraft platform

A space platform envisaged for implementing the theoretical concept of MACS is

the unmanned Solar Probe Spacecraft that will approach the sun at four solar radii at its

perihelion. In this proposal a series of filters centered on the temperature and the wind

sensitive wavelength positions is proposed. This would allow for both temperature and

wind velocity measurements.Figure (7.1) shows the parabolic path of Solar Probe.

gentsiiiit_iii_h_iiisuniiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

iiiiiii_iiiii

A

iiiiiiiiiiiiiiiiiiiiiiii

iiiiiiNiiiii

::::::::::::::::::

iiiiii iiiii"

iiiiiiDiiiii

Figure (7.1). The path envisaged for the unmanned Solar Probe

Spacecraft. At the perihelion (C) and the poles the spacecraft will be four

and eight solar radii, respectively, from the center of the sun. The

position of the spacecraft (P) is given by (r, 0 ) with respect to the center

of the sun. In this picture the parabolic path, the detector plane and the

center of the sun are in the same plane.

188
Theequationof theparabolicorbit of the SolarProbeasdepictedin figure (7.1)in polar

coordinatesis givenby equation(7.1).

2.0x FC

r(0) -

1- cos(0)

8.0R

1- cos(0)

where R is the solar radius & FC = 4R

(7.1)

The angle (Z made by the tangent at P with the x-axis is given by equation (7.2).

t_ = tan-I ( 1 -- COS(0) ]

k )

(7.2)

The derivation of equation (7.2) is as follows.

y = r sin(0) --> dy = r cos(0)d0 + sin(0)dr

x = r cos(0) --->dx = -r sin(0)d0 + cos(0)dr

d_y_y= r cos(0)d0 + sin(0)dr = r cos(0) + sin(0)(dr / dO)

dx -rsin(0)d0 + cos(0)dr - rsin(0) + cos(0)(dr / dO)

dr -2xFCxsin(0) &usingrfrom equation (1) d--O: (1 - cos(0)) 2

tan(t_)- dy _ 1-cos(0)

dx sin(0)

2xFC

1- cos(0)

From the triangle APE the relationship between o_, 13and 0 is given by equation (7.3).

0:_+_-_:0-_ (7.3)

189
Supposingthedetectordirectionis y degrees from the tangent to the orbit at position (P)

then the perpendicular distance 9 from the center of the sun to the line of sight of the

detector is given by equation (7.4).

p = r sin(J3 + T) (7.4)

At this instance the distance between the spacecraft and the intersection between the line

of sight and the normal from the center of the sun is given by equation (7.5).

PG = r cos(J3 + T) (7.5)

In order to prevent the detector being pointed directly at the sun the angle _/ will have to

satisfy the following condition given by equation (7.6).

T- > -([_ - [_min) measured in the (-) direction

7+ < 2To- ([_ + [_min) measured in the (+) direction

wh r mn/

(7.6)

In determining the theoretical K-coronal spectrum observed by the Solar Probe the only

change occurs in the x-integration (along the line of sight) as shown in equation (7.7).

dx ---> dx (7.7)

--_ -PG

190
The aboveintegrationmaybeperformedthe following way asshownin equation(7.8).

Thefirst andthe secondintegrationsareperformedusingtrapezoidalcompositerule and

Laguerreintegration,respectively.

+_ 0 +_

I (x).x+I (x).x

-PG -PG 0

0 +_

: ff(x)dx + fe-X(f(x)e+X)dx

-PG 0

i=O i=O

0- (-PG)

where h - & z i = -PG + i x h

2O

(7.8)

Figure (7.2) and figure (7.3) show for solar wind velocities 0.0 and 400.0 km/sec,

respectively, the theoretical K-coronal intensity variation for various detector elevations

with respect to the tangent when the spacecraft is positioned at the pole (P). Figure (7.4)

shows the variation of the wind-sensitive intensity ratio against the detector tilt 7 for

various wind velocities at a constant temperature of 1.0 MK and the spacecraft positioned

at the pole. Figure (7.5) shows the variation of the temperature-sensitive intensity ratio

against the detector tilt 7 for various temepratures at a constant wind velocity of 400.0

km/sec and the spacecraft positioned at the pole. Figures (7.6) and (7.7) show the

variation of the temperature-sensitive intensity ratio against temperature for various wind

velocities and the variation of the wind-sensitive intensity ratio against wind velocitiy for

various temperatures, respectively, for a given detector tilt angle',/ and the spacecraft

positioned at the pole.

191
[

[ o _eou

0.006 _:,_.

i 0.0:i ;i

0.004 _,x_

-2

<

observed absolute ]ntens]ty VS observed wavelength

I I I I I

1 ao _0u guoD Moo

1,00 _,oo _o,oo 7o,00

!.am _°.,_u __.o,--, zooz,

• :]:;/ . ;_ _..........

s /

i"

c,ooo[ , , _ I _ , , I _ _ _ I , _ , I _ , , I , , ,

3600 3800 4000 4200 4400 4600

an_s_rorn_

i

i

i

i

i

i

i

4800

Figure (7.2) The theoretical K-coronal intensity variation

for various detector elevations for 0.0 km/sec wind velocity

and the spacecraft positioned at the pole (P).

< 0,004

<

g

0.002

400.o i.oo

-_.o 1.oo

O,O00

3600 4800

observed absolute ]n±ensffy VS observed wavelengkh

I I I I I

B.Da _o.og 11u.o

e.o_ _0.oo 7o,eo

a.go _o.og 4_oo

_.,_-,u o,_-s,_ _:, ,:,_ j_.J

./// _J--

................... // / f .........................................

3800 4000 4200 4400 4600

angsLrom_

Figure (7.3). The theoretical K-coronal intensity variation

for various detector elevations for 400.0 km/sec wind

velocity and the spacecraft positioned at the pole (P).

192
1,260

_ 125oi

-- 1,240

I(42_5)/I(3987) VS T for 50 F_ngstrom bandpass

I I ' ' ' I ' ' ' I ' ' ' I ' ' i

_/.d

!!!1!! ....... i

..... ..- . .......... .............. .......... ..... .......

... -.-- _.---"

.....

1.260 .............................................................................................

1.220 i p I p _ , I _ _ _ I p _ , I _ , _ I , _ , l

0 20 40 60 _0 100

_,' in deg_es

Figure (7.4). The variation of the wind-sensitive intensity

ratio against ? for various wind velocities at T=I.0 MK

and the spacecraft positioned at the pole.

1.60

1,55

-&

1,30

1,'t5

1.40

1.55

1(4100)/i(385@ VS 7 for 50 Angstrom b<_ndposs

...... '- ' I I I I I

- ' - - -. _ _ - _ km/_= • )I_ r sR e

.............. ::: [! it:: ,i:!_::

L . .

-- . , .

. . .. .

-- --. -... ...

"-,..

•.. _ . . ....... _... _.....

- _ , , , I , , , I p _ , I , _ I , , ', " I-. _ .-

0 20 40 60 80 100

7 degrees

Figure (7.5). The variation of the temperature-sensitive

intensity ratio against ? for various temperatures at

W=400.0 km/sec and the spacecraft positioned at the pole.

193
1.60

<.

v

1,55

1,50

1,45

0,5

J(4100}/1(3850) vS T_mperatur_ far SO angstrom bandpass

i i i i i i i i i i i

0.00000000

IO0.OOC)O0

_:%_, _Fhc%_°

'%,sh ,,_,%%%_I

1,0 1,5

Temperature MK

rho in solar radlu£ 1 1000000

2.0

Figure (7.6). Temperature-sensitive intensity ratioVS

Temperature for various wind velocities at _/ = 0.0

degrees and the spacecraft positioned at the pole.

1.280

1,270

1,260

_o

<.

-=.

1.240

1,250

1{4255)/1(5987) VS Wind for 50 angsLrom bandpass

m I I I 7 _

_p , s_ 6 a, 9 7'e'9

]o.5_m &ae £o.a o._%_.,a /

_..... :::':: s::!i :::::: : :2!::!:: -'

./

.,-

/

....... ,, i iiii iiiiii i -::/", "

.--"_

200 400 600 800

Wlad in kin/see

Figure (7.7). Wind-sensitive intensity ratioVS Wind

velocity for various temperatures at _ = 0.0 degrees and

the spacecraft positioned at the pole.

194
Theseplots show the promisebehind the theoreticalconceptof MACS being

employedfrom a spaceplatformdeepinsidethe solar coronafor themeasurementof the

coronaltemperatureandthesolarwind velocitythusavoidingtheF-coronaandtheuseof

a coronagraph.Eventhe spectrographcould be avoidedwith the use of optical filters

centeredat the temperatureand wind sensitivewavelengths.Thetemperatureand wind

sensitiveintensityratiosremainalmostthesameevenatfifty angstrombandpass.Figures

(7.8) and(7.9)showthe temperatureandwind sensitiveintensityratios,respectively,for

theuseof fiters centeredat thosewavelengthswith a fifty angstrombandpassfor line of

sightintegrationfrom _+ooat 1.1solarradii. This couldbeverified by thecomparisonof

figures(7.8)and(7.9)with figures(2.3)and(2.20),respectively.

l(4100}/1(3850) VS Temperature for 50 angstrom bandpass

1 ._{J • 1 1 F 1 r l • T • • 1

wlnd km/s_c

O.O0000OO0

1,65 - I O0.OOO,O0

?_:_ 20(%00w39 !.{i_ 500 OOOO0

"%1_'_. ,m,s.c,c, ooo

_. _ soo ooooo

•%_,_ ISO(,Oi)O0C;

":_'a_. 700.08@30 "_ _I: &13'.'.,,31;,O(10 *' _,_"._, soe aoc, oo

__ 1._o- "@.}),_:.

a %,_%,

1,55

1.45 _ , , , I , , , p I _ p _

0,5 1,0 1,5 2.0

Temperature MP,

rho in _olar radiu£ 1.1000000

Figure (7.8). Temperature-sensitive intensity ratioVS

Temperature for various wind velocities with intensities

measured through filters of fifty angstrom bandpass

centered at those wavelengths for line of sight at 1.1 solar

radii.

195
1,2_70

1•250

1 •2_40

' j

q

q

q

q

q

q

q

•./T

y'"- -" " I

. --: > { i i ..... i

"= i I

.- i i

; i

I I

i i i J p i p J p i i

2(10 400

Wind km/_ec

1(425.3}/1(.3.987)vs Wlr,d far 50 ar,gstro,n bandpass

I ' ' I ' ' I ' [ ' I ' i

I /

I

o.5_6ooeo

I -

I .- 1 _}_;,x'.,{},3_? i

._,'i _i:i£):}:i:_i::i.

/

z

. < -1 ..... 1

1

1

i

, i

I 1

I

, 1

I i

I l

I 1

i

, 1

I , p , I ,

600 800

1.230

0 1000

r,ht_ _n _rairJr ta<Ji_J_: i i0,300G0

Figure (7.9). Wind-sensitive intensity ratioVS Wind

velocities for various temperatures with intensities

measured through filters of fifty angstrom bandpass

centered at those wavelengths for line of sight at 1.1 solar

radii.

In conclusion, with the approval of funding for a coronagraph for MACS the

future is very optimistic as a tool for simultaneous and global coronal temperature and

wind measurements. There is also good reason to hope that the concept can be applied as

an instrument on a space-based platform.

196
Appendix A

THE THEORY OF FORMATION OF THE K-CORONA

A.1 The general theory on scattering of radiation by free electrons

Figure (A.1). The general coordinates where

P

point and r is a source point.

is a field

197
Consider a system whose charges and currents are varying in time. There is no

loss in generality by restricting our considerations to potentials, field and radiation from

localized systems that vary sinusoidally in time.

The time dependent charge and current densities could be written, respectively, as given

by equation (A.1) and equation (A.2), based on coordinates shown in figure (A.1).

p(r_,,t ) = p(r) e- icot (A.1)

, icot

J(r ,t) = J(r)e- (A.2)

And the continuity equation is given by equation (A.3).

V.J + 6)9 = V.J + (-kop) = 0 (A.3)

at

The scalar and vector potentials for the charge distributions are given, respectively, by

equation (A.4) and equation (A.5).

1 p(_)eikr-_

V(r) - .[ d3_ (A.4)

4_e o r - r'

A(r) = B° j _(r-")eik r - r_'

4n r-_

d3r_. ' (A.5)

198
In the far radiation zone,wherer >> [] and r >> r/, the denominatorIr- r/I can be

consideredto beindependentof r_/ although the argument of the complex exponential is

not. Thus, equation (A.5) could be written as equation (A.6).

_to e ikr

A(r) -

4re r

t

_j(r,)e-ik(r.r /r) d3r ,

(A.6)

Also if the source dimension r/ is small compared to the wavelength, then k r/ << 1 which

justifies writing equation (A.6) as equation (A.7).

A(r) - --t_° eikr- _; (-ik)l- '(r'r' "_l

4x r l l! _J_(r_TJd3r-'

(A.7)

The above is true since the electron diameter of 5x 10 -15 m is very small compared to the

wavelength of visible light of 5x 10.o7 m. Therefore the expansion of the exponential

term in the integrand of equation (A.6) is justified by the summation given equation

(A.8).

' ) (-ik) l

e-ik r.r /r =_ _ (r._/r)l (A.8)

l l!

The scattering of electromagnetic radiation by systems whose physical

dimensions are small compared with the wavelength of the wave being scattered, it is

reasonable to assume that the incident radiation as inducing electric and magnetic

multipoles and these to oscillate in definite phase relationship with the incident wave and

199
alsoradiateenergyin directionsdifferentfrom thedirectionof theincidentwave. For the

casewherethewavelengthof the incidentradiationis very long comparedto the sizeof

the scatterer,only the lowestmultipoles,usually the electric andmagneticdipoles(1=0)

are important.Thereforefor electric dipole radiation

equation(A.9).

A(r) go eikr - fJ_.(£)d3£

4g r

Consider the following mathematical operation

(1=0) equation (A.7) reduces to

(A.9)

v'.(xLJ)=(V'x').j+x'(V'4)

=Jx+x'(V'4)

•.Jx=v'.(x2J)-x'(V'4)

fJxd'_' = fg "".(x_J)d" 3"r -fx "(g "..J)d" 3"r

P P P 3 P

=_x J.ds-_x (V ._J)d" r

Using equation(1.3)

, ,Op _ ,

= _x J.ds+_x --d-_r

Ot

Usingequation(1.1)andintegrating over a large enough volume

-i¢of x'p d3r '

and for all components of J

P P P P

f J(r ) d3r =- i6o_r'p(r ) d3r = -imp

p P P

where dipole moment p =_r p(r )d3r

(A.10)

200
Equation(A.11)canbewrittenfrom equation(A.9) andequation(A.10).

- io>t_° eikr

A(r) -- p

4_ r

(A.11)

Then, using equation (A.11) the magnetic field is given by equation (A.12).

B= VxA(r) i_OlJo V_(eikr p }- 47z --_-- x

and using polar coordinates

V(r, 0,0) = _ 0_0_+ _ 0_0_+ __ c3

Or r 00 r sin(0) 0_)

B- o)klJ o ( 1 _e ikr ^ _ _--_r)-T_×p (A.12)

From Maxwell's equations for outside the source (l=0) the magnetic and the electric

fields are related by equation (A.13).

0E ico

V×B = Iao% -" +laJ =--_-E -

Ot e

where c2 = -- 1 and k = --co

_t e c

0 0

(A.13)

From equation (A.13) the electric field is given by equation (A.14).

.2

IC E(r):--VxB(r)

fO

(A.14)

201
Andusingthefollowing vectoridentities

a ×(B×c)= (8.c)a-(8.a)c

v ×(8×B)=8(v.B)-B(v×8)+ (B.v)8- (8.v)B

and the relations

2

V._ =-,

r

v< =3andp, p(r)

on equation (A.14) the expression for the electric field is given by equation (A.15).

E(r) : ckmbt° eikr

_ _ 3

4x r 4xe r

0

(A.15)

In the radiation zone kr>>l where k = co/e the equations (A.15) and (A.12) reduce to

equation (A.16).

ikr

ckmg o e

E(r)_--

4_ r

ikr

kmbt o e

_(_)_

4g r

__.(r)=cB(_)×_

_xp

(A.16)

202
ThePoyntingvector(averagepowerradiatedperunit area)is givenby equation(A.17).

E×B

(s) - -_ -" Watts/m 2 (A.17)

2go

Using equation (A.16) and the vector identity

(A×B)×C=(A.C)B-_.C)AandD.(D×E)=0 and k=co/c the Poynting vector

given by equation (A.17) reduces to equation (A.18).

(s)

32nZrZe

Watts/m 2 (A. 18)

Then the total power radiated through an area da is given by equation (A.19).

dL = (S).da = (S).?r2df_ Watts (A.19)

Substituting equation (A.18) in (A.19) gives equation (A.20).

dL4 4

co _t° _/(_×P/2_'_r2df2_ co _t___O_O _/(_×p/2df 2 Watts

32n2r2c 32n2c

(A.20)

Now for an oscillating electric dipole the induced moment p is given by equation (A.21).

- icot

P=Po e (A.21)

203
Differentiatingequation(A.21)twicewith respectto timegivesequation(A.22).

]i =o)2p (A.22)

Substituting equation (A.22) in (A.20) gives equation (A.23).

dL - _t° (_ × _)2d_ Watts 32n2c (A.23)

For a single electron the induced moment p is given by

p(t) = er'(t) (A.24)

P

where r is a source point with respect to its origin.

From the equation of motion in the non-relativistic case

mr'=e(E+v×B)_= eE (A.25)

where m is the mass of the electron.

Substituting equation (A.25) in (A.24) gives equation (A.26).

m

(A.26)

204
Again substituting equation (2.26) in (2.23) gives equation (A.27).

dL _me 2 ) 32x

(A.27)

Diving and multiplying equation (A.27) by 6xe o gives equation (A.28).

dL : _ _Teoc(_ x E) 2 d_ (A.28) 16x

where (_w is the Thomson scattering cross section given by equation (A.29).

_T- -

6xe o (mc 2 )

(A.29)

Therefore from equation (A.28) the power radiated per unit solid angle is given by

equation (A.30).

dL 3

d_2 16x _T (_ z E) 2 Watts/solid angle] (A.30)

205
A.2 Total power radiated per solid angle for the cases where the electric vector is

aligned parallel and perpendicular to the scattering plane

Consider the case where the electric vector is parallel to the scattering plane. In

figure (A.2) nin and no. t are unit vectors in the directions of the incident and the

scattered radiation and they both lie on the scattering plane and ® is the scattering angle.

^ Sc

E// is the unit vector of the component of the electric vector parallel to the scattering

^ Sc

plane and Ez is the unit vector of the component of the electric vector perpendicular to

the scattering plane.

Figure (A.2). The general coordinates of the incident and scattered

radiation and the orientations of the parallel and the perpendicular

components of the electric vectors with respect to the scattering plane.

206
A

The direction cosines of _in and non t are given by equation (A.31) and equation (A.32),

respectively.

 -in=/sin/O¢)c°s/O sinO0' (A.31)

fi-.out : (1,0,0) (A.32)

A A

From figure (A.2) the dot product ofnin and nou t gives equation (A.33)

fi-.in'fiout = fi_in nil.out cos(O) (A.33)

and the square of its cross product gives equation (A.34).

(fi-.in X_out )2 = -qin 2 _out 2sin2(O) (A.34)

^ Sc ^ Sc

The direction cosines for the electric vectors E// and E± are, respectively, given by

equation (A.35) and equation (A.36).

^ Sc

E// = (-sin(n-O),cos(n-O),O)= (- sin O,- cos O,O) (A.35)

^ Sc

E. = (o,o,1) (A.36)

207
The sum of the crossproducts of equation (A.32)

equation(A.37).

with (A.34) and (A.35) is gives

^ Sc

_out X E//

j l_ i j k

^ ^ Sc "_ "_

+ n out )< E1 = 0 + 0

O - cos O 0

=-cos O_ + ]

(A.37)

From figure (A.2) using the relation _ = _o.t and equation (A.37) the following

relations are obtained as given by equation (A.38) and equation (A.39).

^ Sc Sc )2 )2 _×E//E// =(E/_ c sin2 (90- O) = (E_/C)2 cos20 (A.38)

^ Sc Sc 12 _xE±E± = (Esc) 2

(A.39)

Now substituting the equation (A.38) and equation

equation (A.40) and equation (A.41), respectively.

(A.39) in equation (A.30) give

dL] sc 3 (eoc(ES/C)2 )cos2 ®

d_ )// - 16-re (ST

(A.40)

dL) sc 3 (eoc(ESC)2)

)± - 16-re (ST (A.41)

208
But for isotropicradiationequation(A.42)is true.

(A.42)

Substituting equation (A.42) in (A.40) and (A.41) give equation (A.43) and (A.44),

respectively.

(A.43)

a-d). la-,_ )

(A.44)

209
Consideraplanemonochromaticwaveasshownin figure (A.3).

Figure (A.3). Propagation of a monochromatic plane EM waves.

Using the Poynting vector S = 1E × B the energy flux density transported by the plane

monochromatic electromagnetic wave depicted in figure (A.3) is given by equation

(A.45).

S : ceoEo z cosZ (kx- cot + 6)1_ (A.45)

210
andaveragingequation(A.45) overacyclegivesequation(A.46).

(A.46)

Fromequation(A.46)the averagepowerperunit areatransportedby anelectromagnetic

waveis calledtheintensityandis givenby equation(A.47).

Io--(s)--1⁄2C_oEo _ (A.47)

Substituting equation (A.47) in equation (A.43) and equation (A.44) give equation (A.48)

and equation (A.49), respectively.

,_ /.L/_ 3 o

d_ )// 167z _TI° c°s2 ® Joules/sec.steradian. A.electron

(A.48)

s_ _(dL/s_ 3 o

I± _)± - 16n C_TI° Jonles/sec.steradian.A.electron

(A.49)

where I o is the incident radiation.

211
Consideraradialplaneinclinedat ananglet_ to the scattering plane, as shown in figure

(A.4). The variation of intensity through an angle [3 is given by equation (A.50).

I = I M cos 2 [3

where I M is the maximum intensity.

(A.50)

Figure (A.4). Coordinates for the rotational transformation of the

scattering plane through an angle _ about the line of sight.

212
From figure (A.4) andusing equation(A.50) equation(A.51) and equation(A.52) are

obtainedfor theintensityparallelandperpendicularto theradialplane,respectively.

=,:cos cos ( /

_-I/_c cos 2 _ + IS,c sin 2

(A.51)

,.:,cos cos/ /

= I/_¢ sin 2 t_ + IS,¢ cos 2 t_

(A.52)

Substituting equation (A.48) and equation (A.49) in equation (A.51) and equation (A.52)

give equation (A.53) and equation (A.54), respectively.

R. _ ( dL _R. 3 _Xi0 (sin 2 t_ + cos 2 _)COS 2 0_)

: I sc sin 2 t_ + I/_c cos 2 t_

- Q/_" (t_, ®)I o Joules/sec.steradian.A o .electron

(A.53)

R. _(dL _R. 3 _TIo(COS 2t_+cos2®sin 2 t_)

= I sc cos 2 c_ + I/__ sin 2 c_

- Q _" (t_, ®)I o Joules/sec.steradian.A o .electron

(A.54)

213
Using the scatteringtheoriesderivedabovethe samecould be appliedfor the

scatteringof photosphericradiation by the coronalelectrons.Figure (A.5) showsthe

correspondinggeometricalconfigurationsdepictedby figures(A.2) and(A.4)for thecase

of photosphericradiation being scatteredby the coronalelectronsto an observeron

earth.

Figure (A.5). Geometrical configuration for the scattering of photospheric

radiation by coronal electrons.

214
Thefollowing figure (A.6) showsthegeometricalconfigurationfor theformation

of theK-Corona,which alsoincorporates the effect due to the radial solar wind.

Figure (A.6). Geometrical configuration for the formation of the K- Corona.

Photospheric radiation emitted from a point S on the Sun is scattered from an

electron E towards an observer O. The solar wind on the electron is radial and

blows in the direction CE.

215
A.3 Contribution due to Compton scattering

In the rest frame of the electron the scattering of the photospheric light incident on

the electron is coherent. The red shift due to the Compton effect is very negligible. To

quantify the contribution due to Compton effect, consider radiation of wavelengthk

incident on a stationary electron, as shown in figure (A.7). Let the radiation scattered off

the electron be of wavelength k' and the velocity of the electron now be 2. The

scattering angles for the radiation and the electron are, O and q0, respectively.

Sca_teredradia{f6

Figure (A.7). Scattering of radiation off an electron also known as Compton

scattering.

216
Fromconservationof energyin figure (A.7) equation(A.55)is obtained.

hc hc

+ mc 2

(A.55)

From figure (A.7) using conservation of momentum in the X the Y directions,

respectively, equation (A.56) and equation (A.57) are obtained.

h h mv

Ic°' ,cV //

h mv

where

h = Planck cons tan t = 6.63 x 10 TM J.s

m = electron mass = 9.11x 10 31 Kg

c = speed of light = 3 x l0 s msl

Eliminating ¢p and v from equation (A.55), (A.56) and (A.57), the wavelength shift is

given by equation (A.58).

t,z : z'- z : h(1- cos0)

mc

= 0.0243A ° (1 - cos 0) < 0.0486A °

(A.58)

217
A.4 Expression for the total K-Coronal scattered intensity

From figure (A.6) for the calculation of the total K-Coronal scattered intensity in the rest

frame of the electron, first, it is necessary to consider an electron velocity distribution over a

volume element at P in order to determine the number density. Consider a Maxwellian velocity

distribution for the coronal electrons. Then the number density at the point P in the velocity

interval (u_u,u + du) is given by equation (A.59).

[-£/

fe (_) : Ne(P) x_n (_ _ e q2 where

[2kTe __5508_e kms-1

q(P) = mean electron thermal velocity = _ me

k = Boltzmann constant = 1.38 × 10 -23 JK-I

m 3 = electron mass = 9.11× 10 TMkg

T e in 106 Kelvins

(A.59)

218
However, in the rest frame of the observer, the scattered radiation off a moving

electron will be altered in wavelength from the monochromatic radiation incident on that

electron. Consider a coronal electron with velocity u subjected to radial solar wind

velocity w in a coordinate system where the x-axis bisects the supplement of the

scattering angle, as shown in figure (A.8).

Y O(U

Figure (A.8). Construction to obtain an expression for the scattered

intensity in the rest frame of the observer. Consider a coronal electron with

velocity u subjected to radial solar wind velocity w in a coordinate system

where the x-axis bisects the supplement of the scattering angle.

219
Fromfigure (A.8)thedirectioncosinesof theunit vectorsnin and nout

equation (A.60) and equation (A.61).

are then given by

nan = (- cosy, siny,0) (A.60)

n out : (cos y, sin y,0) (A.61)

And the net velocity vector V is given by equation (A.62).

V:(Ux --]-Wx,Uy -]-Wy,Uz -]-Wz) (A.62)

From Doppler effect, in the rest frame

wavelength given by equation (A.63).

of the observer, the observed radiation has

1

1 n°"t

)b observed = )_scattered C

1 + n o.t .V

C

(A.63)

The relationship between )bobserved and )bphotosphere is given by equation (A.64).

)b scattered : )b photosphere

n. oV

1 -F -.in

C

C

(A.64)

220
Fromequation(A.63)andequation(A.64)eliminating_scattered gives equation (A.65).

observed : _ photosphere

1 1

+cCv+nnnincCvV/ (A.65)

Since the speed of light c>> V equation (A.65) can be reduced to equation (A.66).

/la (_in 5 _°ut )'V---" ] 1

_ °bserved _-__ ph°t°sphere (nin-n )V

1 c°ut "_

(A.66)

By Taylor expansion of equation (A.66) it could be reduced to equation (A.67).

#

_observed _ _photosphere [ 114

\

(_in -n°ut)'Vc ]1 (_in- nout )'V

C

 phoosphere/l n°/

(A.67)

221
Fromequation(A.60),(A.61) and(A.62)equation(A.68)couldbeobtained.

-noo,lv=[ "c cos Ux+wx/l (A.6S)

From equation (A.67) and equation (A.68) and using the notations used in figure (A.8)

equation (A.69) could be obtained, which relates between )_obsevedand )_photosphere•

= X'/1-2cos_

_= _ observed

_P = _photosphere

U x -t- W x

where

(A.69)

Equation (A.79) is obtained by rearranging equation (A.69).

/ 2c°_7X' _x - IX'/1 2c°/7-Wx )- XI = 0 (A.70)

Equation (A.70) satisfies the condition that scattering is coherent in the rest frame of the

electron.

222
Now, from figure (A.8) and equation (A.53), (A.54), (A.59),

scatteredfrom anelectronat pointPis givenby equation(A.71).

(A.70) the intensity

I/_" (X, nout )+ I ,R"(X,nout)= I(X', nout )x (Q/_" (o¢, 0)+ Q ,R"(0¢, ®))X ND

where

ND =_je o,-. - ,,x-

(t-_-q)-_ L( c 5 ,---Wxc -_' ,J-x,J-y,J-z

(A.71)

The expression for ND in equation (A.71) reduces to equation (A.72).

2 2

+_

ND Ne(P) +f -u_q2 f _Uzq2 3 e dUy e du z x

(_q)....

2

Ne(P)X(_-q)x(_-q)xexp

(aq)'

=/_P>i_xpL--X'/1

(2Ab)

2bWxc }

wh r , =q 'anOc .=cos =cos( )

2 N

_k/1 1

/

2-

(A.72)

223
Fromequation(A.71)and(A.72)thescatteredintensityfrom pointPis givenby equation

(A.73).

IR"(_,nout) = I(_',nin)×QR"(_,O)× Ne(P)

2 x]-x-Ab exp

where O - (//, A_)

/

-_' 1 2bWx

£

2Ab

(A.73)

To obtain an expression for the total observed scattered intensity, from figure (A.6), the

equation (A.73) needs to be integrated over the following parameters.

1. All wavelengths from each point on the photosphere

ida' (A.74)

o

2. From all points on the photosphere

2_ co* 2_ 1

Idq_Isincodco- Idq_ Ideosco

0 0 0 cos co*

(A.75)

3. From all points along the line of sight

+_

fax (A.76)

224
From figure (A.6) andfigure (A.8) the radial solarwind velocity hascomponentsgiven

by equation(A.77).

w = (-cos cocosV,coscosinV,sinco)- (Wx,wy, w z ) (A.77)

Using equation (A.74), (A.76), (A.76) and (A.77) in (A.73) gives the expression for the

total observed intensity for a given observed wavelength X at a given line of sight

distance P from the center of the Sun (see figure (A.6)), as given by equation (A.78).

+_2_ I I I1 i

--_ 0 COSO)* 0

Ne() r X QoRa (_,O)x

1

2x/-_-Ah I (_" _in ) exp

_ -- _'/1 + 2b2 c°s cow(r)r"di"l C /

2Ab

2-

where O - (//, _L)

//parallel to the radial plane

_Lperpendicular to the radial plane

(A.78)

225
A.5 Expressions for the dependent variables in equation (A.78) in terms of the

independent variables

Figure (A.9) is a highlight from figure (A.6) of the triangles formed by SEW

(scattering plane), CEF (radial plane) and CSE.

o

E

X

Figure (A.9). Highlighted map of the radial and the scattering planes of figure

(A.6).

From figure (A.9) equation (A.79) and equation (A.80) can be obtained.

ES = _ (sin cocos % sin cosin % cos co)

EF = EF (0, sin Z, cos Z)

(A.79)

(A.80)

226
And thedotproductof equation(A.79)andequation(A.80)givesequation(A.81).

_.E_FF= E S _ (sin o)sin (psin X + cos o)cosx)

: ESEFcos( - O)

(A.81)

From equation (A.81) the expression for ® is given by equation (A.82).

0 = 7z- cos -1 (sin (osin (psin Z ÷ cos (ocos Z) (A.82)

Also from figure (A.9) the expression for angle Z is given by equation (A.83).

X

cos Z = - (A.83)

r

From figure (A.6) and measuring distances in solar radius the expression for angle co* is

given by equation (A.84).

sin co* -

HC 1

EC r (A.84)

227
From figure (A.6) considerthe sphericaltriangleformedby EDJI, as shownin figure

(A.10).

J

O iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

Figure (A.IO). Highlight of the spherical triangle formed by EDJI in figure

(A.6).

From figure (A.10) the relationship between the angles is given by equation (A.85).

sin t_ sin q_

sin _o sin(x - ®) (A.85)

228
Fromequation(A.85)theexpressionfor angle_zis givenby equation(A.86).

t_ = sin -1 ( si--_(_ _)-)sincosin- q) )

Figure (A.11) shows the triangle ECS from figure (A.6).

(A.86)

S iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

C iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

Figure (A.11). Highlight of the triangle ESC of figure (A.6).

From figure (A.11) the relationship between the angles 0 and co is given by equation

(A.87).

1 r

sinco sin(n-0) (A.87)

.'. sin 0 = r sin co

229
A.6 Expression for the incident solar intensity on the coronal electrons

In order to evaluate equation (A.78) the incident solar intensity I(_',nin ) need to

be known. The following expressions are from Astrophysical Quantities by Allen. Let

I(_', 0) be the intensity of the solar continuum at an angle 0 from the center of the disk

and I(U,0) the continuum intensity at the center of the disk. The results may be fitted by

the expression given by equation (A.88).

I(E', 0) = 1 - u 2 - v 2 + u 2 cos 0 + v 2 cos 2 0

I(U,0)

where u 2 and v 2 are limb darkening constants

(A.88)

Or less accurately equation (A.88) can be written as equation (A.89).

I(U,0)

I(_L',0) : 1 -u 1 + u 1 cos0

where cos 0 is the heliocentric angle shown in figure (A.6).

(A.89)

230
And for determiningthevalue of u_ it is preferableto makea fit at cos0= 0.5, which

givestheexpressiongivenby equation(A.90).

1

cos 2 0 = i (3 cos 0 - 1) at cosO= 0.5 and

I0¢,0) _12 iO¢,o)-l-u2-v 2+u2cosO+v 2 (3cos0-1)

=1-- /U2+-_V 2 + U 2+ v /cos0

-1-Ul +ulcosO where

(A.90)

The ratio of mean to central intensity is given by equation (A.91).

1 i0¢,0) -_v2

1

=l---_U 1

(A.91)

From equation (A.90) and (A.91) the ratio of I0¢,0)/F0¢ ) is given by equation (A.92).

I0¢,0 )_ 1-Ul +UlCOS0

F()¢) 1- 1 (A.92)

231
Theemittanceof solarsurfaceperunit wavelengthrangeis givenby equation(A.93).

E(U) : rtF()_') (A.93)

And the solarflux outsidethe Earth atmosphereperunit areaand wavelengthrangeis

givenby equation(A.94).

f(_')=E(_I"s°r/ -2-6--

=_ v(_')

(A.94)

From equation (A.90) and equation (A.92) the incident intensity on the coronal electrons

is given by equation (A.95).

Ul + Ul cos0 f(_,_

1 Ksol. r )

1 - --u 1

3

(A.95)

In equation (A.94) and equation (A.95) AU is the Sun-Earth distance. The wavelength

dependent limb-darkening coefficient U_ can be obtained from Astrophysical Quantities

by A llen

232
A.7 Final expression for the observed intensity

From figure (A.6) and equation (A.78) all the following expressions are given in

terms of solar radii, as shown in equation (A.96).

--_ rR solar

--->OR solar

xR solar

(A.96)

And the observed intensity is given by equation (A.97).

+_2_ 1

IRa(_'PRsolar) = ff f

-_ 0 coso)* 0

Ne (rRsolar) X QRa (_, Q) X

ida' dq) d cos co d(XRsola r ) X

1 I(_', co, x)exp -

2_qt-_Ab

- U/1 + 2b2 COS cow(rRs°larc )radial

2Ab

where 0 - (//, L)

//parallel to the radial plane

I perpendicular to the radial plane

(A.97)

The expressions for physical parameters in equation (A.97) in terms of independent

variables are given in equation (A.98). Equation (A.99) gives the parameters in equation

(A.97) for which suitable models or actual measurements need to be used.

233
Q//Ra = __(_3 T (Sin2 + (Z COS 2 O) (g COS 2

16n

QRa = --(_3 T(COS 2 t_ + sin 2 0_COS 2 O)

16n

b:cos :cos/ )

c

i(_,, c0, x)= 1(T__ R)/AU/2 s°lar l1 -- Ul0/fl --_U+ llUl COS

O = n - cos -1 (sin _sin tosin Z + cos cocos Z)

CZ= sin-I / sin _sin to /

\ /

 :cos-l/ )

:s,o-l/1 /

0 = sin-1 (r sin CO)

r 2 =X 2 ___p2

(A.98)

u I (_') = limb darkening coefficient

f(_') = extraterrestrial solar irradiance

N e (rRsolar) = electron density model

T(rRsola r ) = coronal temperature model

W(rRsola r ) = solar wind model

(A.99)

234
Appendix B

CODE

B.1 The basic equation to be solved

The basic equation to be solved is equation (A.97) as developed in Appendix A

and reproduced below. The observed K-Coronal intensity is given by equation (B.1). The

expressions for physical parameters in equation (B.1) in terms of independent variables

are given in equation (B.2). Equation (B.3) gives the parameters in equation (B.1) for

which suitable models or actual measurements need to be used.

+_2r_

IoRa(_,PRsolar) = f f

--_ 0 COSO)* 0

Ne (rRsolar) × QoRa (_, Q) ×

1

f i d_' d c°s (° dq) d(XRsolar ) )<

m

_'/1 +

2b 2 COS60w (rR solar )radial

C

2Ab

where 0 - (//, l)

//parallel to the radial plane

_Lperpendicular to the radial plane

(B.1)

235
Q// 3 (sin 2 -_- 0_ COS 2 O) Ra =--(_T 0_ COS 2

16re

QRa = --(_T3 (c0s2 (z + sin2 (_ COS2 O)

16re

/

C

I(;L,,CO,x) = l(T_ t Rs°lar )AU/2/'1-- If/ Ul1----UI31 + Ul COS0

O = x - cos -1 (sin cosin tpsin Z + cos cocos Z)

sin o)sin _

• Ill co = sin-1

0 = sin-I (r sin co)

r 2 :X 2 .__p2

(B.2)

Ul (_t) ---- limb darkening coefficient

f(_') = extraterrestrial solar irradiance

N e (rRsolar) : electron density model

T(rR solar ) : coronal temperature model

W(rRsola r ) = solar wind model

(B.3)

236
B.2 Approximations to the f d_' integral

(B.4).

Consider the exponent in equation (B.1) be represented by y as shown in equation

y - where A - q_' (B.4)

2Ab e

The first approximation to be made on equation (B.4) is given in equation (B.5).

A = q_' __>q_

c c (B.5)

This approximation is justified on the grounds that the speed of light c (3 x l0 s )

km/sec is much larger than the thermal electron velocity q (5500 km/sec) so that a

difference of about 10 Angstroms between _ and _' would hardly have any effect on the

value of A.

Then differentiating equation (B.4) with respect to _' together with the approximation

given by equation (B.5) gives equation (B.6).

W

1 + 2b 2 cos co--

dy c

d_' 2Ab

(B.6)

237
Thesecondapproximationto bemadeontheintegrationlimits in equation(B.4)is given

by equation(B.7).

When )_' --> 0 then y -->

where c = 3 × lO s km / sec

q _ 5500 km/sec

b__l

... y ---> oo

)_ )_ c

2Ab 2q)_b 2qb

e

(B.7)

When )_' --> _o / wW/

1 + 2b 2 COS

then y -_ ._o 2Ab c -_ -_o

Using the limits derived in equation (B.7), the exponent in equation (B.1) together with

all other )_' dependent parameters reduces the integral over I d)_' to equation (B.8).

i/ Jf e J f (_'') d)_'= 1 f )_ + y(2Ab)

2Ab l+2b 2 cos cow l+2b 2 cos(0--w

C C

)_+y(EAb) and )_' =

W

1 + 2b 2 cos co--

C

)_(l+2qbY)

= C

W

1 + 2b 2 cos (1) --

C

(B.8)

238
B.3 b, Q/RR and Q RR in terms of the independent variables co, _o and x

From equation (B.2) the expression for b is given by equation (B.9).

b = cos 7 and 2',/= x - ®

(B.9)

Using the trigonometric identity cos(Zy) = 2cosZ(y) - 1 on equation (B.9) together with

expressions for eos(x- ®), sinz, cos Z and r from equation (B.2) gives the following

expression for b, as shown in equation (B.10).

1 1

b: _(1 +cos(Zv))_

: _(1 +cos(_-0))_

___(ll 1

= + sin cosin Z sin q_+ cos cocos Z)_

1

=-_-11 + sinco 29_5_ sin q_+ cos co x 9 2

4X 2 + 9 2 4X 2 +

1

= 1 1 + 41_cos_4x_9+p_sin q_+ cos co 4x_+9 2

(B.10)

239
Fromequation(B.2)theexpressionfor Q/Rais givenby equation(B.11).

Ra --

3

16re

3

16re

3

16re

3

16re

3 C_T (sin2 tX + cos 20_ cos 2 O)

16re

---c_ T (sin 2 o_+ (1- sin 2 ®0cos 2 ®)

- --c_ T (cos 2 ® + sin 2 o_(1 - cos 2 ®))

- --c_ T (1 - sin 2 ® + sin 2 o_(1 - cos 2 ®))

- --c_ T (1 + sin 2 o_sin 2 ® - sin 2 ®)

(B.11)

Now, using the relationships sin(27) = sin ® and sin _x -

sin cosin q)

sin ®

from equation (B.2)

and the trigonometric identities sin(2y)= 2sinycosy and sin27+ cos 2 y= 1 on equation

(B.11) reduces equation (B.11) to equation (B.12).

QRa__ 16_'_3(_T( 1 +sin 2 cosin 2 q__ 4(1_ cos 2 y)cos 2 Y)

(B.12)

The expression for cos 7 as a function of (co, % x) is given by equation (B.IO).

240
Again,from equation(B.2) theexpressionfor n R"is givenby equation(B.13).

QR,_ 3 c_T(cos 2_+sin 2_cOs 20)

16n

3

=--Or(I-sin 2 _+ sin 2 _cos 2 O)

16n

3

:--C_T (1-- sin 2 _(1- cos 2 ®))

16n

= --3 (YT _ -- sin _ sin o)

16n

3 / sin2 6°sin 2 )

- 0 v 1 - qOsin2 0

16n sin 2 0

3

= 16---_or (1- sin2 60sin 2 qo)

(B.13)

And using the relation sin t_ -

equation (B.14).

sincosinq) from equation (B.2) in equation (B.13) gives

sin ®

Q""- 30T(1-sin2cosin2q) )

(B.14)

B.4 Rewriting the basic equation with all parameters expressed in terms of the

independent variables (co, % x, y)

With change of variables introduced in section B.2 and the relationships

developed in section B.3 the expression for the observed K-Coronal intensity is given by

equation (B.15). The expressions for the physical parameters in equation (B.15) in terms

of the independent variables are given in equation (B.16). Equation (B.17) gives the

241
parametersin equation(B.15) for which suitablemodelsor actualmeasurementsneedto

beusedin termsof thegivendistanceP and observed wavelength )_.

+_2_ 1 +®

Ra ['_ R Io [/_,P solar): f f f fdydq) dcos0)d(XRsolar) X

--_ 0 COS 8" --_

Se(4X 2 -[- p2Rsolar)XQRa(o),(_,X)X

___ I(y, c0, % x) exp[- Yz] w

1 + 2b z cos co--

C

where O - (//, _1_)

//parallel to the radial plane

_1_perpendicular to the radial plane

(B.15)

b :

Q/Ra__ 3

16X

QRa__ 3

16x

A = q_____

C

(

1 [ 1+41-COS 2C04 x2+pP 2 sin qo+ cos co

\

-- -- (5T (l + sin 2 cosin2_-4(1-b2_o 2)

-- -- (YT (1 -- sin 2 0) sin 2 _0)

cos _0" =cos(sin -1 4X 2 1+92 t

COS0 : cos(sin-I (4x 2 +9 2 sin co_

P

1/2

(B.16)

242
u i (_) = limb darkening coefficient

f (_) = extraterrestrial solar irradiance

W

1 + 2b 2 cos co --

c

N e (4 x 2 -t- p 2 R solar ) = electron density model

2

T(_fx 7 + p Rsolar ) : coronal temperature model

W(4x2 + p2 Rsolar )= solar wind model

(B.17)

B.5 Changing the independent variable cos co to cos 0

Consider the integral in equation (B.15) as shown in equation (B.18).

1

dcosco

COS tD*

(B.18)

From equation (B.2) the expression for cosco*and 0 are given by equation (B.19) and

equation (B.20), respectively.

cos co* : X/1 - sin 2 co*

= I 1 X2+p1 2

(B.19)

0 = sin-1 (4x 2 + 0 2 sin co)

sin 0

• sin co =

X --]- 0 2

_ ]1- COS 2 0

• cosco:cos(sin-l(Jl_2c_°.s2__20

x +o

(B.20)

243
Substitutingcos0 = ta equation(B.20)canbewrittenasshownin equation(B.21).

cosco=cos/sin-1 ([1--t't 2

(B.21)

Differentiating equation (B.21) with respect to g gives equation (B.22).

d cos co

d_t sin/sin

x2+92 I 1 x2+92 1-_t 2

+pz

_t

_/x_+ o_/x _+o 5+_-l

X

(B.22)

And the limits on the integration

1

f d cos co will change to the following form.

COS (D

From equation (B.16) the expression for g can be written as equation (B.23).

_t =COSO

= _/1 - sin 2 0

= x/1 - r 2 sin 2 co

(B.23)

when cosco = 1 then _t = 1 and

whencosco=cosco = 1 X 2 +[32

then _t = 0

244
Incorporatingthe changeof variablefrom coto 0 thebasicequation(B.15) changesto

equation(B.24).). Theexpressionsfor physicalparametersin equation(B.24)in termsof

independentvariablesaregiven in equation(B.25).Equation(B.26)givestheparameters

in equation(B.24)for which suitablemodelsor actualmeasurementsneedto beusedin

termsof distance9 and observed wavelength )_.

+_2_ 1 +®

IRa(_',DRsolar)= f ff IdY dq)d_t d(XRsolar )×

-_0 0-_

N e (4x 2 -[- D2 Rsolar )× Q_a (g,q),x)×

g

×

_/-_ + O__/x_+0 5+g_-I

1

l+ Zb_ w

C

where 0 - (//,_1_)

//parallel to the radial plane

_1_perpendicular to the radial plane

(B.24)

245
/1/2

1 _2 P sin q) + [3 P

QH

16x

[3 = cos sin-I

2

A = q____Z

C

--i--u-u f

I(y,_t,q),x) K_ us°lar ) / 1-_ ,

(B.25)

u i (_) = limb darkening coefficient

f (_) = extraterrestrial solar irradiance

1 + 2b2_ w

C

N e (4X 2 + p 2 Rsola r )= electron density model

2

T(_/x 2 + p Rsola r) = coronal temperature model

W(ffx2 + p2 Rsolar )= solar wind model

(B.26)

246
B.6 Expressionfor the extraterrestrial solar irradiance f(_)

250

extra_terrestrialirradiance VS observed wavelength

_oo

m

__ _oo

h

50

o

3000 3500 4000 4500 5000 5500

Figure (B.1). The plot of extraterrestrial solar irradiance VS

wavelength obtained with a Fourier Transform Spectrometer at the

McMath/Pierce Solar Telescope on Kitt Peak, Arizona.

The spectrum of extraterrestrial solar irradiance shown in figure (B.1) was

downloaded from ftp.noao.edu/fts/fluxatL The spectrum was then recreated in intervals

of 0.0025 Angstroms through linear interpolation. The reasons for this change will be

apparent in section (B.10). Here f(_) is the wavelength dependent extraterrestrial solar

irradiance with wavelength { measured in Angstroms.

247
B.7 Expressions for the limb-darkening coefficient u_ (_)

1.5

_J

1_0

co

Kto

U

<7",

cco.s

F/

E

I

't

limb darken[n I coefficient VS wavelength

:, _.._,._._:,,' _-U <_;:. U _ _I._../_ ,:__..!'_ _ _L_ .................. *_._ _.:_U/j _,_

_...... iii

,:?:..9_:!2]---(S:----:!I220:O:_ZO,i20,'d-,!:JO.<:_ ':_f.::i b200 L:: 3700 .,_.

o.s62_(,_-sToo.o),(oml2,,'-1_oc,_o/fors700 _oseooA

\<L i

: --.: j

0,0 I E I i I I i [ I I I I

2000 4000 60O0 8000

wavelength in angstroms

Figure (B.2). The plot of limb-darkening coefficient versus wavelength

with linear approximation super imposed.

248
Using the wavelength dependent limb-darkening coefficient from Astrophysical

Quantities by Allen the following linear fits were made for the different wavelength

regions.

(1) 3000 - 3200 Angstroms

538.0 0.11× (B.27) Ul 200.0 200.0

(2) 3200 - 3700 Angstroms

UI(_-_) : 0.922 (_- 3220.0)×0.120 (B.28)

480.0

(3) 3700 - 5000 Angstroms

[I 1 (_) ---- 0.862 - (_ - 3700.0)× 0.212

1300.0

(B.29)

(4) 5000 - 5500 Angstroms

[ll(_,_=k_._] 625.0 0.06×_ (B.30)

500.0 500.0

In the above equations _ is the wavelength measured in Angstroms.

249
/, \

B.8 Expression for the coronal electron density model N e/4X 2 + p 2R solar )

If the eclipse takes place at sunspot maximum, the corona has an approximate

circular form. This means that the coronal rays are most probably equally distributed

between the polar and the equatorial regions of the Sun. Since the Sun is expected to

reach sunspot maximum in the year 2000 the solar corona at the August 11, 1999 total

solar eclipse will assumed to be of circular form. This assumption justifies using a radial

dependent expression for the electron number density.

From the analysis of the photometric data on ten eclipses from 1905 to 1929

Banmbach (1937) deduced the following expression for the coronal electron density.

Ne (r)= 108(0.036r -Ls + 1.55r -6 + 2.99r-'6)cm 3

where r = ff-_ + P 2

(B.31)

Cram (1976) used the expression given in equation (B.32).

N e (r) : 1.67 × 10 (4+4"04/r)cm 3

where r=4x z+p z

(B.32)

Equation (B.32) agrees within 2% in the region 1.5 < r < 2.0 to the minimum equator

model given by Van de Hulst (1950). In this code the electron density model given by

equation (B.31) was used. Figure (B.3) shows the electron density plots given by

equation (B.31) and equation (B.32).

250
I

s

0

6

.3

4

=

(3

0

©

Electron number density VS radius

1,OeS*((O,O36/r_l,54(l'.55/r_.6) (2,g,5/r_16}) Baumbach (19a7)

l

...

J

0 2 4 _ _ 10 12

radius in solar units

Figure (B.3). The plot of electron number density

versus the radial distance from the solar surface for

the two models given by equation (B.31) and

equation (B.32).

B.9 Expressions for the coronal temperature model T X2-_ - p R solar )and the

solar wind model W(X/-_ + p2Rsola r )

In this code the corona is considered to be isothermal which implies that that the

temperature T is a constant. As regards the solar wind W, it is considered to be radial,

isotropic and constant. The idea is to construct multiple models for different

combinations of temperature T and solar wind velocity W. Nevertheless, the code does

not restrict from applying a coordinate dependent temperature and wind profiles that are

measured independently.

251
B.10 Evaluation of the integrals

Rewriting equation (B.24) incorporating all of the parameters introduced

sections B.6 to B.9 give equation (B.33) and (B.24) for I/_a and I,R" , respectively.

in

+_2n 1 +_

I/_a(_,,p) = constant x f_ _dy dq_d_t dxx

-_ 0 0 -_

(o + +, + + +

g

×

X ___ 192 4X 2 ___ 192 + _[2 _ 1

(_+(_-_)_i._-.(_-b_)b_)×

1-.,(_) +_.,(_)f(_)×

w

1 + 2b 213

c

constant =

where

3X (_T(AU) 2

167CS/2Rsolar

b = 1 l+p X2 +p2 sinq_+

_ = c°s/sin-l/I12+_22 //

= _'(l+2qby) and q = _T

1+ 2b2[_ w

c

1/2

(B.33)

252
+_2_ 1 +_

I ,aa(_,9) : constantx f ff fdy dq3d_tdxx

-_ 0 0 -_

g

×

_/x_+o_,jx_+o_+_-I

_--_.,(_)

3

exp[- Y2 ] where

1+2b2[3 w

C

constant =

+9

2 )-16 /X

3XC_T (AU) 2

16x s /2R Solar

b:_ l[ l+p_.__[1-_+92_ sin(p-_

[3 :c°s[sin-l[i 1_ i2 /]

)_(l+2q bYJ

_= c

1+2b2[3 w

(B.34)

253
(YT = 6.677 X 10 -2s cm 2

X 10

R solar =6.96 10 cm

AU = 1.496 x 1013 cm

(B.35)

From equation

equation (B.36).

(B.33) and equation (B.34) the total observed intensity is given by

ITot,l (_, 19): I/_a (_, 9) + I_a (_, 9) (B.36)

1. Evaluation of the f A(x)dx integral

The above integral is evaluated using a 20-point Hermite polynomial expansion as

shown in equation (B.37).

fA(x)dx : f (A(x)e x_)_-X_dx (B.37)

20

_ Z A(hi )_(hi)2 Wi

i=l

where h i and w i are the Hermite polynomials and the associated weights, respectively.

254
2. Evaluationof the

2_

f B ((p)d(p integral

o

This integral

equation (B.38).

is evaluated using 20-point trapezoidal composite rule as shown in

f B((p_l(p --- 2 x Z B((pj )- B((po )- B((P2o ) where

0 j=0

2x -0

h B -

2O

q)j =0+h B zj

(B.38)

o

1

Evaluation of the f C(B)dB integral

o

This integral

equation (B.39).

is evaluated using 10-point trapezoidal composite rule as shown in

1 he/ 1o )

C(_._)d_._ _ T 2 X Z C(_._j )- C(_._ o )- C(_._lO ) where

0 k=0

1-0

h c -

10

_[k =0+hc xk

(B.39)

4. Evaluation of the

+_

f D(y)e-/dy integral

This integral is evaluated using a 12-point Hermite polynomial expansion as shown in

equation (B.40).

+_ 12

f D(y)e-/dY _ Z D(hl )Wl (B,40)

__ l=l

where h I and w I are the Hermite polynomials and the associated weights, respectively.

255
+_

Consider the integral I D(y)e-Y2 dy its y dependent parameters and the expansion outlines

in equation (B.40) in equation (B.41).

+_

f D(y)e_/dy - +f 1 - u, (_) + _u, (_)

3

where _ : _(y)

12 ) f(_l)Wl

____ Z 1 -- I11 (_ l ) "_ _'[[ll (_,

1 u

l=l 1--_- 1(_1)

12 12

----Z 1--Ul(_l) f(_l)_Vl "_'[Z Ul(_l)

;--_-UZ -i---- f(_l )Wl

1=1 )_._ 1:1 l___Ul(_l )

(B.41)

_(1+ 2qbhl )

where _l : C

1 + 2b213 w

c

Due to the rapid intensity variations associated with the Frannhofer lines, as

evident in figure (B.1), this effect is included by taking the mean value for f(_l ) over

each interval in the Hermite polynomial expansion coefficients h I . The mean is then

given by equation (B.42).

256
(B.42)

In order to evaluate the integral in equation (B.42) the following procedure was

followed in order to expedite the calculation.

1. The extraterrestrial solar irradiance spectrum was recreated in intervals of 0.0025

Angstroms ('q') '

257
2. For a given setof valuesfor (x,%B) from the integrals (I dx, I d(p, I dB) the values

(11upper, 1"1lower'1"1R) were calculated.

3. (rlupper, rl.ower) were assigned new values (rl'pper, rl_ower) which are divisible by 0.0025

and of(n') (nu r'n,ow r)

4. The values (f(_upper)_f(_lower )) nOW corresponded to (_pper'_ower)"

5. This procedure allowed the integral to be expanded in intervals of 0.0025 Angstroms.

o Within the range of 0.0025 Angstroms the limb-darkening coefficient H 1

extraterrestrial solar irradiance f could be assumed to be a constant. This allows for

writing equation (B.42) as shown in equation (B.43).

_._ TIupp'r Ul(_ ) f (TI_ITI

1 'qup(er l__pl!q 2 f(q)dq + _ I -i ---

]_R l]l_wer l_3][l, (]_) I"11..... 1-_][l, (]_)

f 1]_ower +0.0025A ° 1]_ower +0.0050A °

I 1-u'(TI) f(Tl)dTl+ I 1---uz!TI2 f(TI):ITI+

Tl' .... ' -- 3_[11 (_) Tl_ower =FO.O02' A 0 ' -- 3_[11 (_) I

_m

TIR Tlupper

........ + I I--_lu I-(TI) f (TI_ITI

/rl{ow_r +0.0025A ° rl;owe _ +0.0050A °

I u' (TI)f(Tl)dTl + I U' (TI)f (TI)::ITI+

Tl_.... 1 -- 1 U] (q) Tl_ower +0"002SA01 - -u,(n) 1

3 3

]]upper

........ + I u,(q) f (TI)::Iq

_upper-0.002'A0 1 -- 1 . , (n)

3

(B.43)

258
Now to elaborateon the method,by which theintegrationis expedited,considerthe

following table.Columns1 and 4 arefrom sectionB.6 and columns2 and 3 arefrom

sectionB.7. HereT is divisibleby 0.00025.

Table (B.1). The dependence of the extraterrestrial solar irradiance and the

limb-darkening functions with the wavelength.. The columns 1 and 4 are from

section (B.6) and columns 2 and 3 are from section (B.7).

Wavelength 1"1 1-u,(n) u,(n)

in Nanometers

(nm)

1 u

1--_ 1(]])

1 u

1--_ 1(]])

Extraterrestrial

Solar

irradiance f (q)

T A1 B1 C1

T+0.00025 A2 B2 C2

T+0.00050 A3 B3 C3

T+0.00075 A4 B4 C4

T+0.00100 A5 B5 C5

T+0.00125 A6 B6 C6

T+0.00150 A7 B7 C7

T+0.00175 A8 B8 C8

T+0.00200 A9 B9 C9

continued continued continued continued

In the code T is 300.00025 nm and then increases in intervals of 0.00025 nm to

550.00000 nm.

259
Now createanewtablewith eachrowreflectingthecumulativesumof table(B.1)for the

columns2, 3 and4.

I Table (B.2). Cumulative values of the columns 2,3 and 4 of table (B.1).

T AI*C1 BI*C1 C1

T+0.00025 A2"C2 B2"C2 C2

T+0.00050 A3"C3 B3"C3 C3

continued continued continued continued

In the code the following format is used, where columns 2 and 3 are multiplied by

column 4 of table (B.2).

Table (B.3). Multiplying column 4 of table (B.2) with column 2 and 3 of table (B.2).

This is the format of the table used by the code.

T

T+0.00025

T+0.00050

T+0.00075

continued

AI*C1

(AI*C1)+(A2*C2)

(AI*C1)+(A2*C2)+

(A3"C3)

(AI*C1)+(A2*C2)+

(A3"C3)+ (A4"C4)

continued

BI*C1

(B 1"C1)+(B2"C2)

(B 1*C 1)+(B2"C2)+

(B3"C3)

(B 1*C 1)+(B2"C2)+

(B3"C3)+ (B4"C4)

continued

260
Goingback to equation(B.43) theintegralscouldbe written asa summation,as

shownin equation(B.44). Figure (B.4) illustratesthe evaluationof the first integral of

equation(B.44) wherethe integrationundereachinterval is approximatedasthe area

underthesquare.Thisprocedureisjustified ongroundof thevery smallinterval.

1 TluPper [._ Tlupper __ll(T 1

_-R I 1-u I(I]) f(l])th]+_ I 1! f(l])th]

I I

,11.... ----Ul(Tl)3 ,11.... l--_Ul(11)

ATI Tl_dr 1 !]I)f(Tl ) _-_ATI Tl_dr UI(TI )

[n_:t_,°_.l,1-_u,(l]) _ _:_;t°g,_a,,1--1U,(I])

of 0.002SA ° of 0.002SA °

where M] = 0.0025A o

-13' -13' ]1 R -- upper lower

(B.44)

iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii_ 3ooooo25n_ I_iii

Figure (B.4). Illustrating the integration procedure of the first

integral in equation (B.44).

261
Taking the cumulativevalue of the parameterreflectedon the y-axis of figure (B.4)

wouldgivethefollowing, asshownin figure (B.5).

Row R

Row P

300.(1(1025 nm

550.(1(1(1(10 ira1

Figure (B.5). Plot of the cumulative y-axis values of figure (B.4) against the

wavelength q'.

The same procedure as above could be performed for the second integral in equation

(B.44) too.

262
In termsof expeditingthecalculationof theintegralpresentedin equation(B.42),

equation(B.44) allowsthe codedo thecalculationin two easystepsin conjunctionwith

table(B.3)andillustratedby figure (B.5)for thefirst integral,asfollows.

. Find the corresponding row numbers, say, rows (R,P), respectively, which equal the

values tq'upper' q.'wer ) in column 1 of table (B.3) and as depicted in figure (B.5).

. Find the values in columns 2 and 3 corresponding to the rows (R,P) in table (B.3) and

as shown by the arrows cutting across the y-axis in figure (B.5).

. Now calculate [(2(R)- 2(P))+ _t(3(R)- 3(P))]× Aq where the numbers correspond

qR

to the column numbers. The first part 2(R) - 2(P) corresponds to the first integral in

equation (B.44) and the second part 3(R)- 3(P) corresponds to the second integral

in equation (B.44).

equation (B.44).

In figure (B.5), (2(R)- 2,P/I(_] evaluates the first integral in

qR

263
B.11 The symbols used in the code

The following symbols have been used in the code.

x ---) x

q_--_ y

_t ---) z

1"1 --_ cetaU

upper

1"1 --->cetaL

lower

q' --_ a(0,u)

upper

P

q lower _ a(0, t)

h i ---> hermite_12(l)

w I --->weights _12(1)

h i ---> hermite_20(i)

w i --->weights _20(i)

b --> bb

p --->rho

)_ --_ lambda

(B.45)

In the code table (B.3) is given by 3000_5500_cum_irradiance.dat

264
Fromequation(B.2)thefollowing relationshipscouldbederived.

angle _ 1 - sin (co)

sin(O)

+p2

_ sin(cos-I (COS 0))

+O2

, n(co,l(./)

m

X + 0 2

(B.46)

z-cos(w) _l)) : cos(sin -1 (angle angle _

(B.47)

angle _ 3 -_ c°s(Z)_x2+p 2x I

(B.48)

4- sin(z) _3)) = sin(cos -1 (angle angle _

(B.49)

angle _5 - cos(n- O)

= cos(co ) cos (Z) + sin (co)sin _ )sin (q))

= angle _ 2 × angle _ 3 + angle _ 1 × angle

_4×sin(y)

(B.50)

angle _ 6 -= (ncos-- ?)(angle _ 5)

(B.51)

265
bb -b

= cos(T)

:cos/ /

= COS angle_-_ 6

From equation (B.16) the following relationships could be derived.

cc_t- 16____x__ Q_.

3G T

= 1.0 - sin 2 (_o)sin 2((0)

= 1.0 - sin 2(y )(angle _1)2

(B.52)

(B.53)

: 1.0 + sin2 ((p)sin 2((0)- 4(1 - b 2 )b 2

: 1.0+ sin2 (y)(angle_ 1) 2 -4(1- (bb) 2 _bb) 2

(B.54)

266
B.12 The IDL code

The following is the code written in IDL where (;) is a comment symbol and ($) a

command continuation symbol.

;README

;DECIDE ON THE RADIAL WIND VELOCITY IN KM\SEC

;DECIDE ON THE ISOTHERMAL CORONAL TEMPERATURE IN MK

;DECIDE ON THE VALUES FOR RHO IN SOLAR RADIUS

;000.0 0.50 1.10

;400.0 2.00 1.30

;800.0 1.50 1.40

;ENTER THE ABOVE INFORMATION IN BATCH FILE.PRO

;COMPILE ECLIPSE SIMULATION.PRO

;RUN ECLIPSE SP_IULATION.PRO

;THE DATA WILL BE STORED AS

;'ES_w000.0_T0.50_rl.10.dat'

;'ES_w400.0_T2.00_rl.30.dat'

;'ES_w800.0_T1.50_rl.40.dat'

;CALULATING AVERAGE INTENSITY IN THE REGION [cetaU,cetaL]

function KURL CZ_ I ABLE,a,z,cetaU,cetaL

u=long(rouT_d((double(cel-aU)-300.00025d0)/0.00025dO )) ;a(0,0)=300.00025

t=long(round( (double(cetaL )-300°OOO25dO )/O.OOO25dO )) ;a(0,0)=300.00025

interval=a(0,u)-a(0J)

irradiance=((a(l ,u)-a(1,0)+z* (a(2_u)-a(2,t)))*0.00025d0/[ n_erva

;nficrowatts/cm squared/nm

return,irradiance ;nficrowatts/cm squared/nm

end

;CALCULATE FOURTH INTEGRAL PARAMETRS AND THE VALUE

function calculate_fourth____integral=value ,a,lambda,rho,hernfite____l 2,weights_l 2,x,y,z,w,T

q ....5508.0*sqrt(T) ;thermal electron velocity in km/sec

c =-3.0d5 ;velocity of light in km/sec

final sum__/l=dbla _:r(2, I )

angle____1=sin (acos(z))/sqrt(xa2+filoa2 ;sin(omega)

ang Ie____2=cos(asin (angle.___1)) ;cos(omega)

267
angIe.___3 =x/sqr_-(xA2+rho,*'2)

angle____4= sin(acos(a_:_gle____3))

;cos(chi)

;sin(chi)

a:_gle5=a_gle 2*angle 3 + a_gle 1*_mg]e.__/-_*sin(y)

angle 6=acos(angle5)

r=sqrt(xa2+ ¢ho" 2 ) ;radius

aa=l o0dt*(0,036/(rA1.5) + 1 o55/(r%) + 2,55/(:_-16)) ;Baumbach model

bb=.,cos(angle_6/2 O) ;cos(gamma)

c c.__J=i1,0-(sin(y)A2)*(angle. 1A2) ;tangential

ccr=l ,0+(sin(y)A2)*(ang]e 1A2)-4,0"(1,0-bb,"2)*(bbA2);radial

dd=( 1,0*z)/(sqrt(x A2+rhoA2)*sqrl-(xA2+d]oA2+zA2-1.0))

;changing d(cos(omega))to d(mu)

ee=l ,()+2,0*bb,*'2*ar_gle____2*(w/c) ;wind component

N=I 1 ; Nth order hennite polynomial expansion is used

sm_i___._,angentiaI=0.0

sum .i radial =0.0

for i=0,N do begh_; first_loop

if (i eq 0 ) l-}_enbegin

cetaL=(I ambda-lambda* (q/c) *bb*hem_i 1-e.___ i12(i))/ee

;nanometers

end else if (i ne ()) then begin

c e mL= (1ambd a -1am b da'" (q/c )" b b * (hermi re.___12 (i}+iherm ite ____ 12 (i- 1)_)/ee

manometers

endif

if (i eq 11) then begin

cel-aU=tlambda-Ian_bda*(q/c)*bb*hermi{,e___.12(i))/ee

;nanometers

end else if (i ne 11 ) fhen begin

cetaU =i.lambda.-Jambda _:vq/c) "bb*(hermite_ 12(i)+hermi_e_ 12(i+ 1 _))/ee

manometers

endif

if (((cemU or cetaL) It 300o0(i1()25d0) or ((cemU or cetaL) gt 550,0000()d0)) then begin

fmal__m__4_,O_O_.---99_ _" ...... _.... 9cccc ..)..)9_)99_9ccc ..)9._)9_99

final_sum_4( 1,0)=9999999999995)999

gol-o,termin al-e.__4

endif

268
;CALCULATING THE AVERAGE INTENSITY CORRESPONDING TO

;[cetaU,cetaL]

irradia_ce=KURUCZ__TABLE(a,z,ce_.aU,cel-aL} ;microwatts/cm squared/nm

FC=ir.radia_ce* l.Od-7 ;W/cm squared/Angstroms

sum____[__.mngem-ia l=sum_ i____tangential+weighls _.12 (i)* (FC)

sum i radial ....-sum i radial +weights_12(i)e(FC)

el:_d%r ;loop_i

fi rml____sum.___4 (0,0) =su m___i__ta_:_ ge_:__ia1*a a*cc_j*dd* ( 1.0/ee)

final_sum_4(l,0) .... sum i radiaI*aa*cc_r*dd*(1,0/ee)

1-erminatefi:

remm,fina[___sum___4

end

;CALCULATE THIRD INTEGRAL PARAMETRS AND THE VALUE

function calculate_third__megral_value ,a,lambda,dlo,hermite_ 12,weights_ 12,x,y,w,T

:THI_<.D INTEGRAL BOUNDS

a3=0.0

b3=1.0

n3= 10,()

h3=(b3.-a3)/n3

f3=-fix(n3)

sum .z___t=dbl arr(G + 1)

sum____z____r=dblarr(G+ 1)

total ____stm_____z___.t=0o0

t oral.___sum___z____r=0o0

final sum 3 =dblarr (2,1)

{br r:0,f3 do begin

z=a3+r*h3 ;mu=cos(theta)

.... .... / ..... "1 .... <(3 ..... r

final sum 4=calculate fourth h_te_-ral "_alue $

(a,lambda,rho,hemfite____l 2,weights.___ 12,x,y,z,w,T)

if ( final sum_4(0,0) eq 999999999999.9999) then begin

final_sum_3,=firml sum 4

goto, _.emfinate 3

endif

su m____z____t(r)=final ____stm_____4(0,0)

tota [___.s an____.z___.t=to ta [___.s ank_z__t+fhm l____sum____4 (0,())

269
sm_k__.z___:r(r)=fin al,___.._m_k__.4( 1,0)

tota[___sum____z____r=tota[___s_mk__z.___r+final.___sum___4(1,0)

endfor

final____sm:a____3 (0,0)=(h3/2,0)*(2.0*total.___sm:¢___z t-sm:¢___z____t(0)-sum____z____t(n3))

fina[__.stem___.3 (1,0)=(h3/2,0)*(2.0*total____st_m____z___£r-su m____z____r(0)- sum.___z____r(n3))

terrainate.___3:

return,fi:nal____sm_k___3

end

;CALCULATE SECOND INTEGRAL PARAMETERS AND THE VALUE

fun c tion c alcu ! ate____sec ond____i ntegral ____v a! ue, a, 1arab da,:d_ o ,hermi te____12, we i gh ts____ 12, x, w, T

;SECOND _NTEGRAL BOUN©S

,% -. a_:0,0

b2=2,0*!pi

n2=20.0

h2=(b2-a2)/n2

f2=fix(n2)

sum_yj::::dblarr(f2+ 1)

sum .y___r=dbl art( _2+ 1)

1-oral.__s_.N:n.___yt=0.0

total____stmk___y____r=0o0

fin a [__.strek__.2=dblarr(2,1 )

for v=-0,f2 do begin

y,:::.a2+v*h2 ; in radians

t:inal stmk__3:calculate fl:_ird__imegraI 5,alue $

(a,I ambda,dl o,hemUte.___12,wei ghts___.12, x ,y,w,T)

if (fhml____sm:_-k___3(0,0)eq 999999999999,9999) _hen begi:n

finaI.___stm_.___2=fina l___.._m_:k__.3

goro, tem_ina_e_2

endif

s u m .y____t (v)=fin a[___stun 3 (0,0)

tota[___sum____y.___t=to mI_ sum___y.___t+fi na[___sum____3(0,0)

sam___y____r(v)=finaI___s_m__ 3 (1,0)

tota[___sum___y.___r=_otal,___.._m____y____r+fhlal____sum____3 (1,0)

endfor

fin a l___.sam___.2 (0,0)=(h2/2,0)* (2,0*to_-aI._st_m._y___.t-sum____y____t(())-suo-k___y____t(n2))

final____sum___2(1,0) =(h2/2.())* (2 .()* total____sum____y____r- s mk__.y____r(0)- su m___y.___ r(n2))

term inae____2:

270
returu,fi:_:ml____sunL__2

end

;CALCULATE FIRST INTEGRAL PARAMETERS AND THE VALUE

functioncalculate_firstintegral_value $

,a,lambda,rho,hennite_ 12,weights_l 2,hermite_20,weights_20,w,T

;};[RST }INTE.;GRAL USI/NG HERIb,,1H"[-);POLYNOMI[ALS

n 1,:::19

sum .x t=dblarr(n ] + 1)

total.__sum.__x .t=0.0

s um___.x____r=dbIarr(n1+ 1)

total.___sum___x___;r=(}.0

inten sity=d blan'(2,1 )

for h= 0,nl do begin

x=hermite_20(h}

fi nal____sum___2=calcula_e____second___in_egral____value$

(a, 1ambda,:_:![_o, hem:_ ire____12, w d gh ts___. 12, x,w, T)

if (final ...............sum 2(0,0") eq 99 99.-_-_ 999 999.-_-.. _ _ 99 9 99.-) __ 9' then beoin_,

intensity=final_sum2

goto, terminate 1

endif

sum___x__t(l:_)=final___.sum___.2(0,0)*weigihts____20(h)*exp(x*x)

tota l.___.sum___.x____t=total____su m____x.___t+ stm_.___.x____t(h)

su m___x_r(ih) =final____sum___2(1,0)*weigl:_ts_20(h)*ex p (x'x)

totaI_sum x r=-total_sum x r+sum_x_r(h)

end%r

sigma=6.677E-25 ;Thomson Scattering Cross Section cm squared

AU=1.49597870E+13 ;cm-symbol D in CRAM

solar_ radius=6o96E+l() ;cm-symbol R in CRAM

con ._tant= (3.0" sigma* (AU)A2.0)t( 16.0" solar radiu s* ( !ply' (2.5))

h:_tensi_y(0,0)=coT_stanl-*_otal.__,_um.__ x .t ;Watts/cm squared/Angstroms/steradians

inten,dty( 1,0)=constant _total____sum____x.___r ;Watts/cm squared/Angstroms/steradians

terminate.___1 :

retnrn,inten sity

end

pri:_:_t,'smrting time ', systi me(0)

271
;FILES TO BE READ

;READING THE KURUCZ EXTRATERRESTRIAL IRRADIANCE AT 0.00025A

;INTERVALS FROM 3000A TO 5500A

a=dN a rr(3,1000000_)

opera;2, B000___.55()()___cum____irradiance.daff

readf,2,a

close,2

;READING THE ROOTS OF 12TH ORDER HERMITE POLYNOMIALS

;HANDBOOK OF TABLES OF MATHEMATICS:SELBY 3RD EDITION PAGE

;843

heml i_e____12=d blare 1,12)

opem;3,_ermite 12 rootsAat'

madf,3,hermite____l 2

close,3

;READING THE WEIGHTS OF 12TH ORDER HERMITE POLYNOMIALS

;HANDBOOK OF TABLES OF MATHEMATICS:SELBY 3RD EDITION PAGE

;843

weigh_s___.l2 =dbIarr(l, 12)

opera;4, _ermite.___ 12____weights oda_"

read f,4,weights____ 12

dose,4

;READING THE ROOTS OF 20TH ORDER HERMITE POLYNOMIALS

;HANDBOOK OF MATHEMATIC FUNCTIONS EDITED BY M.ABRAMOWITZ

;AND L.A. STEGUN PAGE 924

hermite_20=:dblarr(1,20)

openr,5,'hermite.___20___roots.daff

readf,5,hermite___20

close,5

272
;READING THE WEIGHTS OF 20TH ORDER HERMITE POLYNOMIALS

;HANDBOOK OF MATHEMATIC FUNCTIONS EDITED BY M.ABRAMOWITZ

;AND L.A. STEGUN PAGE 924

wei ghts .20 =d bl art(i1,20)

opera',6, _ermite.___ 2{i)____weights°dal"

readf,6,weights____20

close,6

;READING THE BATCH FILE.PRO

;THIS FILE CONTAINS THE WIND,TEMPERATURE AND RHO VALUES

dam:read ascii( l)atch___.file.pro ',$

commenCsymbol ::::_xxx .x ',data star_=: 1)

data=__elemems=n elements(dam.field 1)

counte:r=(data____eleme:nts/3)- 1

for j=0,counter do begin

wind= datadie]dl(3*j+0);wind in km\sec

temp= dal-a.fieldl (3*j+ 1);Temperature in MK

rhoo= da-a.fieId l(3*j+2);rho in solar radius

slr____w=st:ri:_:_g(wind)

-parrs=str sep(srr_w,',)

wl =:parts[O]

w2=strmid(pa rl-s[! ],0,1)

s_r____t=string(_emp)

parts=str:__sep (str_____, '°)

_1=parts[O]

t2=-srrmid (par, s[ 11,0,2)

s _r____r=slrin g (rh o o )

parts=stc_.sep (str____r,'.)

rl =par_s[0]

r2,:::.s_rmid(parts[ l ],0,2)

w=double(wh:_d)

T=double(temp)

rho=double(rhoo)

print,'wind = ',w,' knf,,_ec temp = ,temp,'MK $

273
rho= ;rhoo,' solarradius_

;RECORDING THE OUTPUT

openw,9,'ES .w %trim(w 1,'(i 3,3 )')+', '+_rim (w2,'(i 1,1 ))+' .T5

+ tri m(t 1, '(i 1.1_ ).---..-i-.trm_tt_, 0 _o2 ) ).-i-.___.:_: +tr m (r 1, '(i 1o1) ).-i-.N

+trim (r2, '(i2°2) )+ '.dat'

for r=-0,1000 do begin ;370.0 nm to 470.0 nm

Iambda=370.0d0 + r*0, ld0 ;nano meters

intensity=calctdate.___firsL__integraI__va hte $

(a,lambda,d_o,hemfite.___l 2,weights____12,hernfite____20,weights.___20,w,T)

if (intensity(0,0)eq 999999999999.9999) [hen begin

prinh?ntensity(0,0) ....999999999999,9999 for $

]ambda = ;]ambda

gol-o, imen sity___oraL_of range

endif

totaI=_intensi_y=intensity(0,0)+imer_sity(1 ,());Watts/cm squared/Angstroms/steradia_s

pfiT_t,'lambda= 'J ambda* 10.0 ,' ./\_ gsl-roms',' $

intensi_y = ;_otal.__jntensity* 1.0E+7 ,$

' ergs/sec/cm squared/Angs/steradians'

;print,'time = ,systime(0)

prinff, 9, torm *° a [= '(,4L::15°__'}",Iambd a °._+10.0, to ta I=_in ten sity, in ten sity (0,0), $

intensity(1,0)

flush, 9

inten si ty.___ou.L__ofrange:

end%r ; the r loop for intensity

close, 9

end%r ; the j loop for bach file

print,'ending 1-ime ',sysfime(0)

end

274
An exampleof theinput datainformationgivenin BATCHFILE.PRO

Table (B.4).The format of the input information on the wind velocity in

km/sec, isothermal temperature in MK and the distance to the line of

sight in SR.

Wind(km/sec)

400.0

300.0

continued

XXX,X

Temperamre(MK)

1.50

0.50

continued

XX,X

RHO(..SR)

1.10

1.50

continued

XX,X

275
Appendix C

A STREAMER MODEL

C.1 The effects due to streamers

Due to the structural nature of the solar corona it is important to consider the

effects on the K-corona due to coronal inhomogeneities. In this regard consider a

streamer imbedded in a spherically symmetric isothermal corona. The model streamer

considered has the following structure as depicted in figure (C.1). The streamer is of

thickness d and the extremes of the streamer, namely, RT and UV are equidistant from

the solar center C and are inclined at an angle e to the plane of the solar limb. Within the

streamer the electron number density is enhanced by a factor f and the thermal electron

s

temperature is T s . From figure (C.1) the expressions for X1 (=FV) and X2 (=FT) are

given by equation (C.1).

[12= P× tan(e) d / 2 and

cos(e)

p × tan(e)-¢ d/2 cos(e)

(c.1)

276
F iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

Y

iiiiii iiiii__..`iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

iiiiiiiii

Figure (C.1). Model streamer of thickness d in front of the solar

limb with an electron density enhancement by a factor fs and

thermal electron temperature Ts.

And in the streamer model considered the electron density is enhanced by a factor f_

in the region between x 2 and x_ along the line of sight. The incorporation of the effect

due to the streamer in the code, as described in section (B.12), is as follows.

1. For specified values for d, 9 and e calculate x_ and x 2 .

2. In the x-integration (integration along the line of sight) if x lies between and including

x_ and x 2 then multiply the electron number density by a factor f_ and substitute T_

for T where f_ and Ts are specified.

277
3. Perform the x-integration by trapezoidalcompositerule instead of the Hermite

polynomialexpansion.This was foundto be necessaryin orderto pick up sufficient

x-values in the region [xlandx2]. Since the Hermite polynomial expansion

considered in the code is of twentieth order there may not be sometimes any x-values

in the region [x_ and x 21.

4. The x-integration is substituted by the following commands

;CALCULATE FIRST INTEGRAL PARAMETERS AND THE VALUE

flmction calculate first_integralvalue $ ,a,lambda,rho,hermite_12,weights_12,w,T

;NRST }XNTE.;GRAL BOU N DS

al ....... 6.0 ;lower bound at - 6.0 solar radii

bl =+6.0 ;upper bound at +6.0 solar radii

nl =120.0 ;region divided into 120 equal intervals

h l=(bl --al)/n 1

f] :fix(n ] )

su m___x___.t=dbIarr(fl+ 1)

sum x r=dblarr(f] + 1)

tom] ___sum__x 1-=O.0

total____sunL__x____r=0o0

intensity=dblarr(2,1)

278
%r h=0_fl do begin

x=a 1+h*h 1

finaI_st_m .2=calculal-e.__ second___inl-egral vahJe $

(a,lambda,rho,hermite____l 2,weights.___ 12,x,w,T)

if (final.__ s_.lm__2(0,0) eq 999999999999.9999) 1-hen begin

inte_sitv=fi_ml sum 2 • j ----' ....

goto, terminate___.1

endif

su m____x___!(l:l) =fin aI___s_m____2(0,0)

total____su m___x___t=tota L__s_m____x____ H--su m____x.___t (h)

sun____.x____r(h)=fSn aI___.sum.2( 1,0)

tota l,___.s tm:____.x____r=tota I____ stml___.x____r+ stml___.x____r(l:x)

endfor

Although x-integration need to be considered in the interval [-_,+_ ] it is

sufficient to restrict the integration to [-6.0, +6.0] solar radii due to the rapid falloff in the

electron number density.

279
Considera streamermodelasdepictedby figure (C.1) with a streamerwidth (d)

of 0.5 solar radii, density enhancementfactor (fs) of 4.0 within the streamerand

assumingthe thermal electron temperature(Ts) within the streamerto be equal to

isothermalcoronaltemperature(T) for the rest of the corona.Figure (C.2) showsthe

effect of streamers,at angles(e) - 45.0degrees(front of the limb) and+ 45.0 degrees

(behindthelimb) to thelimb of thesolarcorona,ontheK-coronalspectrum.

5.5

5.0

4,5

,__

_a

%

g

=

4,0

E

u

m

g

; 3,5

5.0

i i I i

angle dog -._5.O(_D

kmJ_ _)O.fl00

r LIK l.OBflO0

the NR l.lDfl_

_:'r: II(H}

C_ II!K)

observed total intensity VS observed _avelength

///

/

I I I I I

_800

J

/

/

/

2,5 , , I _ R , I , , :., I , , , I ,

5600 40(30 4200 4400 4600

angsi:roms

I

4800

Figure (C.2). Plot to show the effect of streamers on the Kcoronal spectrum with d=0.5 SR, fs=4.0, Ts =T=I.0 MK and at

angles g =-45.0 and +45.0 degrees.

280
Although the two streamersconsideredin figure (C.2) aresymmetricaboutthe

planeof thesolarlimb theasymmetryin thetwo theoreticalspectrais dueto thefact that

the streamersscatterthrough different angles in reaching the observer.Figure (C.3)

illustrates this point.

®

a F

Figure (C.3). Model streamers of similar dimensions at -45.0 and -

45.0 degrees above and below the plane of the solar limb,

respectively.

Figure (C.4) shows the effect of streamers, at series of angles (e) in front and

behind the plane of the solar limb, on the K-coronal spectrum. As per the streamer model

considered the intensity of the K-coronal spectrum decreases as the location of the

streamer moves away from the plane of the solar limb.

281
I

£

8.

cq"

E

r_

co

&

12

2

3600

observed total intensity VS observed wavelength

' ' I ' ' i' I ' ' ' I ' ' I ' ' ' I ' '

i

'_" km/_ 4J_O._](}O ./.,

.,."

..-'

T _K 1.OOO_O //.

the SR 1.1 Bfl(]O ......................... .., ..-;;//"

.1

Ca i_y) ,'

[,"/' ,Ay" ]/'"

,{" 4" i

: z ..:l i ,,--t 's

I _ z/ : j

, i ,-.+ ij"

................... ,_" i y .<<<......... i .,,-_>__ "

................................. i k

,5600 4000 4200 4400 4600

_ng_trorns

I

4800

Figure (C.4). Plot to show the effect of streamers on the K-coronal

spectrum with d=0.5 SR, fs =4.0, Ts =T=I.0 MK and spread over

various angles e.

The following figures (C.5) to (C.6) show the effect on the wind and temperature

sensitive intensity ratios I(4233)/I(3987) and I(4100)/I(3850), respectively, by a streamer

of the shape depicted by figure (C.1) and with a streamer width (d) of 2.0 solar radii,

density enhancement factor (f,) of 4.0 within the streamer and assuming the thermal

electron temperature (T=) within the streamer to be equal to isothermal coronal

temperature (T). The horizontal lines show the intensity ratio for a given solar wind

speed, isothermal coronal temperature and line of sight in the absence of streamers. The

curves are the corresponding ratios with a streamer of the shape depicted in figure (C.1)

282
but at variousanglesin front andbehind the solarlimb. This is only oneexampleof a

streamermodel.

1.260

1.255

1.250

O_

t_3

C-.l

_- 1.245

1,240

1(4233)/1(5987) VS Streamer- angle

rho = 1.1 SR I

I

,, wind km/sec- I tern p

MF

I ',

............. =._:2:z................... o-d

! ..... I

I I

I I

I I

i i

I I

i .. I

I I

L__4_o__o............. = :- :-n:-: 1-272:-2:-u_-z:=:._____......!

i.................... I

L - _4_GO42..... =._ __.= =:Z :2 : 2 21 F2 :LI :_ :=.:.= ==_ ......... 2_

I I

_ __4_c_42._ _-=-_::_L................................ 2_

I front of the limb beh-ind _he. I-'_mbl

4-0@. 0

@::.)0,0

400,0

_i !i ̧:

"i

2,

m

o r.

1,235 , , _ I _ , , I , , _ I , , , I _ _ , I , _

-60 -40 -20 0 20 40 60

angle in degrees

Figure (C.5). Effects by streamers on the ratio I(4233)/I(3987) for

W=400 km/sec and rho=l.1 SR. The horizontal lines show the

intensity ratio for a given solar wind velocity of 400.0 km/sec and

line of sight at 1.1 solar radii, and for different isothermal coronal

temperature in the absence of streamers.

From figure (C.5) it is apparent that even for a simple streamer structure shown in

figure (C.1) the wind sensitive intensity ratio can differ from the ratios predicted by

models that do not consider streamer structures. The horizontal lines in figure (C.5) are

from models that do not consider any streamer structures. A similar occurrence could also

be seen for the temperature sensitive intensity ratio as shown in figure (C.6).

283
1.07

1.65

1.60

o

Lr3

03

"-. 1.55

o

(=3

__ 4-00.@

1.50

•i-OO, 0

1.45 40 0 O

1.40 , ,

-60

rho = 1.1

1(4100}/'1(5850) VS Streamer clngle

' ' ' I ' ' ' I ' ' '

: _, w!nd km/sec

L_ _4_022,_Q.............. : _:.

I

I

I

I

I

i

' ' ' I ' ' ' I '

temp HI4" I

i

..................... Q_5_

r

i-- __4_OG.O.............. :_ ........................

I

L__4_022._

............. _:Z'-_,_ ....................

I

I £CZO,O............. 2:L-:_l ...................... i

I--- ,......... _,_ -_ .,.., --[-.,

L- _¢_OZL_ ............. 0:±'_ _ ....... "_L_L-: ::: _L_-:-:_'__

i .......

I I '--. ..-4

I front of the limb I ...... ----beh_d.btle-l[m i

I _ _ , I , _ , I _ p , I , _ , I ,

-¢0 -20 0 20 40

tingle in degrees

SR

:X:I:_

1:

'3 :_-

I I

6O

Figure (C.6). Effects by streamers on the ratio I(4100)/I(3850) for

W=400 km/sec and rho=l.1 SR. The horizontal lines show the

intensity ratio for a given solar wind velocity of 400.0 km/sec and

line of sight at 1.1 solar radii, and for different isothermal coronal

temperature in the absence of streamers.

Unfortunately it is not an easy task to incorporate realistic streamer models simply

by observing an eclipse. Nevertheless figure (C.5) and figure (C.6) are examples that

show the effect on the wind and the temperature sensitive intensity ratios by streamers.

284
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288
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

December 2000 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

MACS, An Instrument and a Methodology for Simultaneous and Global

Measurements of the Coronal Electron Temperature and the Solar Wind 682

Velocity on the Solar Corona

6. AUTHOR(S)

Nelson L. Reginald

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

2001-00786-0

10. SPONSORING / MONITORING

AGENCY REPORT NUMBER

TM--2000-209973

11. SUPPLEMENTARY NOTES

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Report available from the NASA Center for AeroSpace Information,

7121 Standard Drive, Hanover, MD 21076-1320. (301) 621-0390.

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13. ABSTRACT (Maximum 200 words)

In Cram's (1976) theory for the formation of the K-coronal spectrum he observed the existence of temperature sensitive anti-nodes, which were separated by temperature insensitive nodes, at certain wavelengths in the K-coronal spectrum. Cram also showed these properties were remarkably independent of

altitude above the solar limb. In this thesis Cram's theory has been extended to incorporate the role of the

solar wind in the formation of the K-corona, and we have identified both temperature and wind sensitive

intensity ratios. The instrument, MACS, for Multi Aperture Coronal Spectrometer, a fiber optic based

spectrograph, was designed for global and simultaneous measurements of the thermal electron temperature and the solar wind velocity in the solar corona. The first ever experiment of this nature was conducted in conjunction with the total solar eclipse of 11 August 1999 in Elazig, Turkey. Here twenty fiber

optic tips were positioned in the focal plane of the telescope to observe simultaneously at many different

latitudes and two different radial distances in the solar corona. The other ends were vertically stacked

and placed at the primary focus of the spectrograph. By isolating the K-coronal spectrum from each fiber

the temperature and the wind sensitive intensity ratios were calculated.

14.SUBJECTTERMS

Solar physics, solar corona, solar wind, MACS.

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