In several cases, analyzing simulated data is a powerful tool to demonstrate feasibility. For example:

*To support an observing proposal*. That is, to demonstrate what constraints a proposed observation would yield.*To support a hardware proposal*. If a response matrix is generated, it can be used to demonstrate what kind of science could be done with a new instrument.*To support a theoretical paper*. A theorist could write a paper describing a model, and then show how these model spectra would appear when observed. This, of course, is very like the first case.

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^{-2} s^{-1}, 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^{-2}. 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) Model: phabs[1]( raymond[2] ) Input parameter value, delta, min, bot, top, and max values for ... Current: 1 0.001 0 0 1E+05 1E+06 phabs:nH>0.21 Current: 1 0.01 0.008 0.008 64 64 raymond:kT>0.86 Current: 1 -0.001 0 0 5 5 raymond:Abundanc>0.27 Current: 0 -0.001 0 0 2 2 raymond:Redshift>/* --------------------------------------------------------------------------- --------------------------------------------------------------------------- Model: phabs[1]( raymond[2] ) Model Fit Model Component Parameter Unit Value par par comp 1 1 1 phabs nH 10^22 0.2100 +/- 0. 2 2 2 raymond kT keV 0.8600 +/- 0. 3 3 2 raymond Abundanc 0.2700 frozen 4 4 2 raymond Redshift 0. frozen 5 5 2 raymond norm 1.000 +/- 0. --------------------------------------------------------------------------- ---------------------------------------------------------------------------

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 0.2802 photons ( 4.9626E-10 ergs)cm**-2 s**-1 ( 0.500- 4.500) XSPEC> newpar 5 0.014 3 variable fit parameters XSPEC>flux Model flux 3.9235E-03 photons ( 6.9476E-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^{-2} s^{-1} 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 ... 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 - `cornorm` can be left at its
default value).

Use counting statistics in creating fake data? (y) / Input optional fake file prefix (max 4 chars): / Fake data filename (s0c1g0234p40e1_512_1av0_8i.fak) [/ to use default]: ngc4472_sis.fak T, cornorm (1, 0): 40000 Net count rate (cts/s) for file 1 0.3563 +/- 3.0221E-03 using response (RMF) file... s0c1g0234p40e1_512_1av0_8i.rsp Chi-Squared = 188.6545 using 512 PHA bins. Reduced chi-squared = 0.3706375 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>ignore 1-15 76-** Chi-Squared = 63.30437 using 60 PHA bins. Reduced chi-squared = 1.110603 for 57 degrees of freedom Null hypothesis probability = 0.264

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 55.3176 -3 0.2309 0.8569 0.2772 0. 1.4322E-02 55.2946 -4 0.2320 0.8565 0.2784 0. 1.4322E-02 55.2945 -5 0.2321 0.8565 0.2784 0. 1.4322E-02 --------------------------------------------------------------------------- Variances and Principal axes : 1 2 3 5 1.51E-08 | -0.03 -0.01 0.03 1.00 1.02E-05 | 0.39 0.91 -0.11 0.03 9.98E-05 | -0.91 0.40 0.11 -0.03 2.32E-04 | -0.15 -0.06 -0.99 0.03 --------------------------------------------------------------------------- --------------------------------------------------------------------------- Model: phabs[1]( raymond[2] ) Model Fit Model Component Parameter Unit Value par par comp 1 1 1 phabs nH 10^22 0.2321 +/- 0.9426E-02 2 2 2 raymond kT keV 0.8565 +/- 0.5048E-02 3 3 2 raymond Abundanc 0.2784 +/- 0.1510E-01 4 4 2 raymond Redshift 0. frozen 5 5 2 raymond norm 1.4322E-02 +/- 0.5423E-03 --------------------------------------------------------------------------- --------------------------------------------------------------------------- Chi-Squared = 55.29454 using 60 PHA bins. Reduced chi-squared = 0.9874024 for 56 degrees of freedom Null hypothesis probability = 0.502 XSPEC>err 1 2 3 Parameter Confidence Range ( 2.706) 1 0.217009 0.248102 2 0.847909 0.864666 3 0.254807 0.304992

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.