Cornell University
FORCAST
Memo:
TM-FOR99-11
Subject:
FORCAST Sensitivity vs. SOFIA PSF –
Gaussian Analysis
Distribution:
Sean Casey, FORCAST Team
Author:
Terry Herter
Date:
19-Aug-1999
Version:
1.4
Orig. Date:
09-Aug-1999
Posted:
19-Aug-1999
This memo estimates the impact of degraded SOFIA pointing stability on FORCAST sensitivity for different wavelengths.
We make the following assumptions:
1.
We have explicitly assumed here that the PSF is well sampled and that the observations are background limited.
2.
All contributions are treated as gaussian beams, that is,
 x2 y2 
P ( x, y ) 
exp  2  2 
 x y
  x  y 
F
Where F is the total flux obtained by integrating over the entire beam. The radial rms is
rms2 D
 x2   y2 


2


1/ 2
and the 1-D rms is
 y2 
 
 2 
rms1D
1/ 2
The signal-to-noise ratio (SNR) vs. the extraction distance, a, is given by:
SNR(a) 
F
(B x y )1 / 2
1  exp( a 2 )
a
Where a = ax/x = ay/y defines the elliptical extraction region and B is the background intensity. Note
that the maximum SNR occurs for a = 1.121, or an encircled energy of 0.715 (see Figure 1). This gives
SNRmax 
0.638F
(B x y )1 / 2
3.
The widths () are computed by adding in quadrature all the contributions to the seeing and pointing.
4.
The optical quality term is given by opt = 0.591” (= d80/2.54), where d80 is the 80% encircled energy
diameter. This give 80% encircled energy within the diameter d80 = 1.5” as per the SOFIA optical
specifications (see Figure 2)
5.
The nominal rms pointing is taken to be nom = 0.2” radial (xp = yp = nom = rms2D. Here the subscript p indicates the pointing ).
6.
For diffraction we take diff = 0.0595” (m) obtained by setting the gaussian FWHM (1.6651) equal
to the diffraction FWHM (1.2/D). A gaussian with  =diff gives about the same 80% encircled energy radius as pure diffraction.
TM-FOR99-11
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Cornell University
FORCAST
7.
The degraded pointing has an axial ratio of 2:1 so that yp = 2xp. Defining rms1D as the 1-D rms in
the y direction, yp = sqrt(2)rms1D, and rms2D = sqrt(5/4) rms1D = 1.12 rms1D.
8.
Boundary layer and other atmospheric seeing related contributions to the PSF have been ignored.
The merit function, MF, which is the factor by which the SNR decreases from the nominal value, is given
by:
2
2
2
 opt
  nom
  diff
MF 
4
2
2
2
2
( opt
  x2   diff
)( opt
  y2   diff
)
or
2
2
2
 opt
  nom
  diff
MF (rms1D ) 
4
(
2
opt
rms12D
2
2
2

  diff
)( opt
 2  rms12D   diff
)
2
The results of this calculation are given in Table 1 below. Integration times with scale as the square of this
function. See Figure 3 for a plot of 1/MF(rms1D).
Table 1: Sensitivity Loss vs. Image Quality
Merit Function (square for time factor)
1-D RMS
5 (m)
10 (m)
20 (m)
30 (m)
0.2
1.01
1.01
1.00
1.00
0.3
1.07
1.05
1.02
1.01
0.4
1.15
1.10
1.04
1.02
0.5
1.23
1.16
1.07
1.04
0.6
1.33
1.23
1.10
1.05
0.7
1.44
1.30
1.14
1.08
0.8
1.55
1.38
1.18
1.10
0.9
1.66
1.47
1.23
1.12
1.0
1.78
1.56
1.27
1.15
1.1
1.90
1.65
1.32
1.18
1.2
2.03
1.74
1.37
1.21
A future calculation will include pixelation and true diffraction in the analysis. However, these estimates
should be pretty good for now, especially for the longer wavelengths.
TM-FOR99-11
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Cornell University
a
FORCAST
0.001  0.1  5
1
0.8
0.6
SNR ( a )
ENC ( a )
0.4
0.2
0
0
1
2
3
4
5
a
Figure 1: SNR vs. extraction size (solid line) in units of a = ax/x =
ay/y, that is, the extraction regions follow contours of constant amDo a plot working in units of /D for x axis.
plitude. The encircled energy (dashed line) is also shown.
1
0.8
0.6
G( x )
A( x )
0.4
0.2
0
0
1
2
3
x
4
5
6
Figure 2: Comparison of encircled energy for gaussian (solid line)
and diffraction pattern (dashed line). The FWHM of the gaussian has
been chosen to match the FWHM of the diffraction telescope. x is in
units of /D.
TM-FOR99-11
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Cornell University
FOM (   rms )
FORCAST
1
SNorm(   rms )
rms k
k
10
1
SNorm 5  rms
k
SNorm 10  rms
SNorm 20  rms
SNorm 30  rms
k
0.8
k
k
0.6
0.4
0
0.2
0.4
0.6
rms
k
0.8
1
Figure 2: Decrease in SNR vs. rms (of long axis) guiding stability.
Starting from the bottom, the curves are for 5, 10, 20 and 30 m respectively. The guiding blur has y = 2x. rms2D = 1.12rms1D. Table 1 is the inverse of these numbers. Note that the calculation is
normalized to rms1D = 0.2, roughly the nominal pointing spec.
Supporting Files
File
Description
Optimal_SNR.mcd
Computes SNR vs. extraction size and sensitivity change for pointing degradation
assuming gaussian beams. (MathCad 7.0 file)
Revision History
Version
Date
1.0
10-Aug-99
First released
1.3
14-Aug-99
Corrected some typos. Changed rms1D equation to make things clearer.
1.4
19-Aug-99
Added support files and revision history table.
TM-FOR99-11
Comments/Changes
Page 4