FINAL REPORT 

0 3 z f//5- 

SOLAR FLARE PHYSICS 

*4::.jA $ ;+: June 8, 1997 ;.+ 

NAG 5-2001 (UMD 01-5-26120) 

A Grant from the Goddard Space Flight Center, 

National Aeronautics and Space Administration 

January 1, 1997 to June 30, 1997 

Institution: Astronomy Department, University of Maryland 

College Park, MD 20742 

Edward J. Schmahl 

Principal Investigator / 

Sot. Sec. No. 

(301) 405-1506 

Mukul R. Kundu 

Co-Principal Investigator 

Professor of Astronomy 
Solar Flare Physics

We have continued our previous efforts in studies of fourier imaging methods

applied to hard X-ray flares. We have performed physical and theoretical

analysis of rotating collimator grids submitted to GSFC for future space

or suborbital missions involving the imaging of solar flares in hard X-rays.

In particular, we have simulated the performance of the High Energy Solar

Spectroscopic Imager (HESSI), using pseudo-flare images provided by Dr.

David Alexander at Lockheed-Martin Palo Alto Research Lab. We have

computed count rates that HESSI would record for these simulated flares,

and reconstructed images from the count rates. These results were presented at the "HXT Image Reconstruction Workshop, January 13-16, 1997,

at Lockheed-Martin Palo Alto Research Lab.

The following pages show the parameters of the HESSI simulation (Section

I), the mathematics of image synthesis (Section II), and the results of reconstructing the images provided by Dr. Alexander. In general, from this

study we conclude that the HESSI telescope will provide better imaging of

solar flares than previous hard X-ray imagers.
SECTION I

THE PARAMETERS OF THE HESSI SIMULATION
HESSI IMAGING PARAMETERS

• PITCHES: 2, 3.5, 6, 10.4, 18, 31.2, 54,93.5, 162 arc sec

• ROTATION RATE: 15 RPM => All Fourier components in 2 sec

• FIELD OF VIEW: 1/50 radian = 1.1 degree

• PHOTON TAGGING: - 2 microsec

SIMULATION OF MODEL HESSI COUNT

RATES

• Select Ncoll collimators from the 7 (or possibly 9) grid pairs.

• Select number of count bins (Nbins) appropriate to pitch, time interval and photon flux:

O Mean count rate - 1/4 photon flux on subcollimator

O Number of bins < 1/10 count rate x time interval

O Number of pitch cycles=radial distance/pitch

O Number of bins > 2 x number of pitch cycles

• Transparancy Matrix: 64 x 64 x Nbins X Ncoll

• Randomize photons by one of the following equivalent methods:

0 Introduce exponentially distributed time delays

0 Shift the photons' trajectories

0 Poisson transform the model source

0 Poisson transform the count rate (preferred method)

• Multiply the model source by transparancy matrix for each collimator and time bin.

• Total the products for each time bin to obtain binned count rate.
RECONSTRUCTION OF IMAGES

Back Projection and Spatially-Oriented Methods

• Multiply each Transmission Matrix by count rate and sum, producing a "Dirty Map"

• Construct a Point-Spread Function (PSF) from the Transmission Matrix.

• Deconvolve the Dirty Map using one of the following:

O Cornwell MEM

O Hogbom CLEAN

O Richardson-Lucy Maximum Likelyhood

O Fourier Inversion

• Disadvantages:

O Not straightforward to include Chi-squared statistics, which are based on the count rates in the

time domain.

O Requires high count rates to define a useful PSF.

O PSF is not exactly spatially invariant.

Reconstruction Constrained by Count Rate Chi-Squared

• Sakao's Method:

O Maximize Entropy for a model of the source;

O Iteratively modify the source model, minimizing the Chi squared of the count

O Stop when Chi squared equals 0.5.

• Advantages:

O Convergence is relatively fast;

O Can watch the convergence process;

O The Chi-squared value can be used for goodness of fit.

• Disdvantages:

0 Convergence depends on many parameters;

0 Convergence is not unique: one chisquared, many entropies;

0 Some parts of the count-rate profiles are fit better than others.
1. HESI collimators

The High Energy Spectroscopic Imager (HESI) is a Rotation Modulation Collimator (RMC) telescope

proposed to be launched by NASA to image hard X-ray and gamma ray flares during solar maximum

around the turn of the century. One of its possible configurations is 9 parallel collimators with grids that

subtend angles ranging from 2 to 162 arc seconds. The image below shows small segments of the grids.

Fi rids with 77 deg relative rotations

The above image is shown with poor resolution, and the finer grids are highly aliased. Click it to see the

larger and more detailed version.

The grid pitches in the above figure are (reading from the bottom up):2.00, 3.46, 6.00, 10.39, 18.00,

31.18, 54.00, 93.53, 162.00 arc seconds. The field of view of each square is 128 x 128 arcseconds.

The pitches are derived from a powerlaw with base b=sqrt(3). That is, pitch(N)=2*b^N, where

N=0,1,2,3,4,5,6,7,8. This means that every other grid in the sequence have pitches in a 1:3 ratio.

HESSI Imaging Parameters

• PITCHES: 2, 3.5, 6, 10.4, 18, 31.2, 54,93.5, 162 arc sec

• ROTATION RATE: 15 RPM => All Fourier components in 2 sec

• FIELD OF VIEW: 1/50 radian = 1.1 degree
• PHOTONTAGGING: - 2 microscc

2. UV plane and comparison with HXT

The spatial frequencies that an RMC is sensitive to are inversely proportional to the pitches of its

collimator grids. For a given instantaneous orientation of a collimator, the sensitive frequency is

represented by a point in the wave number plane (called the U,V plane). As the collimator rotates, the

locus of sensitive spatial frequencies traces out a circle in the U,V plane. For HESI, there are 9 such

circles, as shown below.

Fig. 2a: HESI: Circles at radii= 1/pitch

The spatial frequencies that the Hard Xray Telescope (HXT) on Yohkoh is sensitive to are shown below.

Fig. 2b: HXT: U,V Points
" ..i;

Since the HXT telescope does not rotate, the spatial frequencies it is sensitive to are represented by

discrete points in the U.V plane. There are 64 such points, compared with the potential thousands of

points in HESI's U,V plane.

3. How modulation occurs (high photon rates)

For a point source with high flux, the detectors behind each collimator record a "chopped" signal which

provides the modulation profile for that collimator. If the source is constant in time and space, the

modulation is regular and almost sinusoidal. To compute this, we take the projected image of a

collimator against the sky, and select a single source point of that projected image. As the image rotates,

the brightness of that source point changes with time, going from zero to 100% of the peak brightness.

The projected field seen through an RMC is shown for 100 rotation angles in the figure below.

Fig. 3: The projected field of a single collimator at I00 rotation angles between 0 and 180 degrees.

4. Modulation for a point source

To illustrate the modulation of an extended source by the HESI collimators, we use a point source

model. The modulation is computed by multiplying the instantaneous point-source response by the

model map and totaling the product for that time. The RMCs are assumed to rotate at 15 RPM, so the

collimators rotate 180 degrees in 2.0 s. The total is obtained at 50 intervals during the 180 degrees of

rotation.

COUNT RATE FOR POINT SOURCE

o.o,'_ _ FI I'7

o.o,o IhllrJL FLFL

0 50 100 1 _0

Rotation Angle (d_j¢'ll_)

Fig. 4: Plot of modulation vs time for a single collimator.

In general, for extended sources, modulation profiles show the following characteristics:

• The coarsest collimator shows the highest modulation efficiency;

• The mean value for each 0.5 s interval is close to the true flux;

• The modulation amplitude goes to zero for the finest grids because an extended source is over

resolved.

5. HESI time-tagge¢l photons

Because the HESI instrument records the time of arrival and the energy of photons, sections 3 and 4

above are really a "swindle". To pass from a model image to time-tagged photon lists, one must go

through several steps. We assume the image is for a 0.5 s period.

1. For each pixel of the image, take the photons and spread them out at random times in the 0.5 s

interval. (This process introduces Poisson statistics into the data).

2. Spread the time-randomized photons, while maintaining their vectorial directions, shift their paths

to random set of points of incidence into the collimators.

3. Trace the path of the photons, and list the times of arrival of those photons that successfully pass

through both grids.

The figure below illustrates step 2.

NON--RANDOM FLUX FROM POINT SOURCE RANDOMIZED FLUX FRONI POINT SOURCE
E

C1 0O

c- 0o00

Q._

c- oO

cQ_

c0 x0

L__ I f I

r_0 I0oZ

<_

n-"

C_

_Jw

0 13_

(D

if-)

c,i
0.5-S MODULATION PROFILES- PITCH(8)=160.845 (Alexander Model 1)

Ld

__1

b_

o

Pr"

Q_

Z

o

i--

5

123

0

2_

(.)

:3-

{,0

"10

('-)

q_

O

O..

C

o

8

o

O

J

TIME (1 block=0.5 s)

bO

G4

°.

O

£o

:&

03
0.5-S MODULATION PROFILES: PITCH(7)=92.9461 (Alexonder Model 1)

w

LI_

o

rY

Q_

Z

O

"t

rm

O

(D

tY

TIME (1 block=0.5 s)
i,i

._1

i,

o

o/

I:1_

Z

O

t--

"t

tm

0

Od

0.5-S MODULATION PROFILES: PITCH(6)=53.7105 (Alexonder Model 1)

m./x/vv'

"-_/x/k/V

/X/X/"

TIME

/X/X/"

R3

:3-

d)

CD

3

o

tm

c

--°

O

-b

8

o

o

,.

0

°.

to

to
0.5-S MODULATION PROFILES: PITCH(5)=31.0376 (Alexander Model 1)

i,i

__1

LI_

o

n_

n

z

0

)--

a

o

TIME (1 block=0.5 s)
LU

--J

I,

0

n.-

13..

Z

0

t--

S

C_

0

(.)

c_

0.5-S MODULATION PROFILES: PITCH(4 =17.9356 (Alexonder Model 1

TIME (1 block=0.5 s)

O9

-ID

:G"

db

ED

_>

3

o

Q.

c

Q

o

-6

8

o

0

,°

O_
SECTION II

THE MATHEMATICS OF IMAGE SYSTHESIS

|ll i

SECTION I

RECONSTRUCTING PSEUDO IMAGES
MODEL 1

FRAME 49

HESSI 1- AND 2-S SIMULATIONS
-j

,I
MODEL 2

FRAME 49

HESSI 1- AND 2-S SIMULATIONS
%

C..

_J
%i
MODEL 3

FRAME 49

HESSI 1- AND 2-S SIMULATIONS
r

_r
J_0
_t,k

_J

t

0

crf

"t
MODEL 4

FRAME 49

HESSI 1- AND 2-S SIMULATIONS
.jo

.-)J

)°
41,,,

.b

4..-.

n.

q
MODEL 5

FRAME 49

HESSI 1-S SIMULATIONS
J

o_

_4

_°
MODEL 5

FRAME 49

HESSI 2-S SIMULATIONS
ii
0
C_J