HESSI Variables and Equations used for Image Reconstruction

t_i= time = Binary microseconds from reference start time
N_i= counts = Number of events per time bin
x^A_i, y^A_i= x_asp, y_asp = Yaw, pitch offset of sun-center from imager axis in spacecraft coordinates for the ith time bin
X_m, Y_m= X_pixoff, Y_pixoff = Pixel m offset from map center in inertial coordinates
X^M, Y^M= X_map, Y_map = Map center offset from Sun center in inertial coordinates (user input)
x^M, y^M= x_map_sc, y_map_sc = Map center offset from imager axis in S/C coords

x_map_sc = x_asp + X_map*cos(roll_angle) - Y_map*sin(roll_angle)
y_map_sc = y_asp + Y_map*cos(roll_angle) + X_map*sin(roll_angle)

$x^M = x^A + X^M cos(\alpha) - Y^M sin(\alpha)$
$y^M = y^A + Y^M cos(\alpha) + X^M sin(\alpha)$
$\alpha$ = roll_angle = Roll angle (radians), of spacecraft X-axis (measured clockwise from celestial N)
$\beta$ = grid_angle = Grid slit orientation in spacecraft coordinates (m radians) (from parameter file)
$\tau$ = livetime = Binary fraction giving duty cycle of detector
T= gridtran = Grid-response-matrix transmission
A_h= modamp = Grid-response-matrix amplitude
Q_h= peak_resp_offset = Peak grid-response-matrix offset (from hsi_grm-Step 5)
p= ang_pitch = Angular pitch of subcollimator grids (from parameter file)
$\Theta_i$ = phase_map_ctr = Phase of map center relative to line of maximum response (Step 6)

phase_map_ctr = 2*!pi*h*(x_map_sc*cos(grid_angle) + y_map_sc*sin(grid_angle) + peak_resp_offset)/ang_pitch

$\Theta_i = 2 \pi h (x^M cos(\beta)+ y^M *sin(\beta) + Q_h)/p$
$\Lambda$ = phase_pixel = Phase of pixel m relative to phase of map center

= 2*!pi*h*(X_pixoff[m]*cos(roll_angle-grid_angle) - Y_pixoff[m]*sin(roll_angle-grid_angle)) / ang_pitch

$\Lambda= 2 \pi h (X_m cos(\alpha-\beta)-Y_m sin(\alpha-\beta))/p$
E_mi= counts_exp = Expected counts in time bin i from pixel m:

= F[m,i]*livetime * gridtran*( 1 + modamp*cos(phase_map_ctr + phase_pixel) )

$E_{mi}= F_{mi} \tau T (1+A_h cos(\Theta_i+\Lambda_m)$
F= flux = Postulated flux

Previous: HESSI Variables and Equations

Edward J. Schmahl
1999-06-10