In several cases, analyzing simulated data is a powerful tool to demonstrate feasibility. For example:
Here, we'll use XSPEC to see how an ASCA observation of the elliptical galaxy NGC 4472 can constrain the condition of the hot gas. The first step is to define a model on which to base the simulation. The way XSPEC creates simulated data is to take the current model, convolve it with the current response matrix, while adding noise appropriate to the integration time specified. Once created, the simulated data can be analyzed in the same way as real data to derive confidence limits.
We begin by looking in the literature for the best estimate of the NGC
4472 spectrum. BBXRT observed the galaxy in 1990 and the results were
published in Serlemitsos et al., (1993). They found a flux in the
0.5-4.5 keV range of 
erg cm 
s 
, a
temperature range of 0.74 < kT < 0.98, an abundance range (as a
fraction of solar) of 0.09 < A < 0.46 and a column range of 
cm . A Raymond-Smith spectral
model was found to give a good fit. We specify this model at first with
the median parameter values, except for the normalization of the
Raymond-Smith, which we leave at its default value of unity at first (but
adjust later):
XSPEC> mo pha(ray)
mo = phabs[1]( raymond[2] )
Input parameter value, delta, min, bot, top, and max values for ...
Mod parameter 1 of component 1 phabs nH 10^22
1.000 1.0000E-03 0.0000E+00 0.0000E+00 1.0000E+05 1.0000E+06
0.21
Mod parameter 2 of component 2 raymond kT keV
1.000 1.0000E-02 8.0000E-03 8.0000E-03 64.00 64.00
0.86
Mod parameter 3 of component 2 raymond Abundanc
1.000 -1.0000E-03 0.0000E+00 0.0000E+00 5.000 5.000
0.27
Mod parameter 4 of component 2 raymond Redshift
0.0000E+00 -1.0000E-03 0.0000E+00 0.0000E+00 2.000 2.000
Mod parameter 5 of component 2 raymond norm
1.000 1.0000E-02 0.0000E+00 0.0000E+00 1.0000E+24 1.0000E+24
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mo = phabs[1]( raymond[2] )
Model Fit Model Component Parameter Unit Value
par par comp
1 1 1 phabs nH 10^22 0.2100 +/- 0.0000E+00
2 2 2 raymond kT keV 0.8600 +/- 0.0000E+00
3 3 2 raymond Abundanc 0.2700 frozen
4 4 2 raymond Redshift 0.0000E+00 frozen
5 5 2 raymond norm 1.000 +/- 0.0000E+00
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3 variable fit parameters
We now can derive the correct normalization by using the commands dummyrsp, flux and newpar. That is, we'll determine the flux of the model with the normalization of unity (this requires a response matrix to cover the BBXRT band--we use a dummy response here). We then work out the new normalization and reset it:
XSPEC> dummy 0.5 4.5
XSPEC> flux 0.5 4.5
Model flux 0.2673 photons ( 4.7803E-10 ergs)cm**-2 s**-1 ( 0.500- 4.500)
XSPEC> newpar 5 0.014
3 variable fit parameters
XSPEC> flux
Model flux 3.7424E-03 photons ( 6.6924E-12 ergs)cm**-2 s**-1 ( 0.500- 4.500)
Here, we have changed the value of the normalization (the fifth
parameter) from 1 to 
to give the flux observed by BBXRT (
erg cm s in the energy range 0.5-4.5).
The simulation is initiated with the command fakeit. If the argument none is given, the user will be prompted for the name of the response matrix. If no argument is given, the current response will be used:
XSPEC> fakeit none
For fake data, file # 1 needs response file: s0c1g0234p40e1_512_1av0_8i.rsp
... and ancillary response file : none
There then follows a series of prompts asking the user to specify whether he or she wants counting statistics (yes!), the name of the fake data file (ngc4472_sis.fak in our example), and the integration time T (40,000 seconds--the other quantities A, Bkg, cornorm can be left at their default values).
Use counting statistics in creating fake data? (y) Input optional fake file prefix (max 12 chars): Override default values for file # 1 Fake data filename (sis_small.fak): ngc4472_sis.fak T, A, Bkg, cornorm ( 1.0000 , 1.0000 , 1.0000 , 0. ): 40000 Net count rate (cts/s) for file 1 0.3503 +/- 2.9973E-03 using response (RMF) file... s0c1g0234p40e1_512_1av0_8i.rsp Chi-Squared = 168.1253 using 512 PHA bins. Reduced chi-squared = 0.3303052 for 509 degrees of freedom Null hypothesis probability = 1.00
We now have created a file containing a simulated spectrum of NGC 4472. As is usual before fitting, we need to check which channels to ignore. This time, we'll examine the actual numbers of counts in each channel and reject those that have fewer than 20 per channel. We use iplot counts and see that our criterion requires us to ignore channels 1-15 and 76-512:
XSPEC> ign 1-15 76-** Chi-Squared = 53.13371 using 60 PHA bins. Reduced chi-squared = 0.9321703 for 57 degrees of freedom Null hypothesis probability = 0.621
As expected, 
is reasonable even before fitting because the
model and the data have the same shape. But the point of this simulation
is to determine confidence ranges. First, we thaw the value of the
abundance (fixed by default), fit and then use the error command:
XSPEC> thaw 3
Number of variable fit parameters = 4
XSPEC> fit
Chi-Squared Lvl Fit param # 1 2 3 4
5
47.3996 -3 0.2011 0.8696 0.2705 0.0000E+00
1.3923E-02
47.3954 -4 0.2013 0.8693 0.2707 0.0000E+00
1.3928E-02
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Variances and Principal axes :
1 2 3 5
1.45E-08 | -0.03 -0.01 0.03 1.00
9.81E-06 | 0.41 0.91 -0.10 0.03
8.11E-05 | -0.91 0.41 0.07 -0.03
2.14E-04 | -0.10 -0.06 -0.99 0.03
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mo = phabs[1]( raymond[2] )
Model Fit Model Component Parameter Unit Value
par par comp
1 1 1 phabs nH 10^22 0.2013 +/- 0.8399E-02
2 2 2 raymond kT keV 0.8693 +/- 0.4775E-02
3 3 2 raymond Abundanc 0.2707 +/- 0.1453E-01
4 4 2 raymond Redshift 0.0000E+00 frozen
5 5 2 raymond norm 1.3928E-02 +/- 0.5187E-03
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Chi-Squared = 47.39543 using 60 PHA bins.
Reduced chi-squared = 0.8463470 for 56 degrees of freedom
Null hypothesis probability = 0.787
XSPEC> err 1 2 3
Parameter Confidence Range ( 2.706)
1 0.187992 0.215481
2 0.861300 0.876959
3 0.248081 0.296034
These confidence ranges show that an ASCA observation would definitely constrain the parameters, especially the column and abundance, more tightly than the original BBXRT observation. Of course, whether these constraints are sufficient depends on the theories being tested. When producing and analyzing simulated data, it is crucial to keep in mind the purpose of the proposed observation, for the potential parameter space that can be covered with simulations is almost limitless.