XSPEC has the very useful facility of allowing models to be fitted simultaneously to more than one data file. It is even possible to group files together and to fit different models simultaneously. Reasons for fitting in this manner include:
Other scenarios are possible--the important thing is to recognize the flexibility of XSPEC in this regard.
As an example of the first case, we'll fit two spectra derived from two separate Einstein Solid State Spectrometer (SSS) observations of the cooling-flow cluster Abell 496. Although the two observations were carried out on consecutive days (in August 1979), the response is different, due to the variable build-up of ice on the detector. This problem bedeviled analysis during the mission; however, it has now been calibrated successfully and is incorporated into the response matrices associated with the spectral files in the HEASARC archive. The SSS also provides an example of how background correction files are used in XSPEC.
To fit the same model with the same set of parameters to more than one data file, simply enter the names of the data files after the data command:
XSPEC> data sa496b.pha sa496c.pha Net count rate (cts/s) for file 1 0.7806 +/- 9.3808E+05( 86.9% total) using response (RMF) file... sa496b.rsp using background file... sa496b.bck using correction file... sa496b.cor Net count rate (cts/s) for file 2 0.8002 +/- 9.3808E+05( 86.7% total) using response (RMF) file... sa496c.rsp using background file... sa496c.bck using correction file... sa496c.cor Net correction flux for file 1= 8.4469E-04 Net correction flux for file 2= 8.7577E-04 2 data sets are in use
As the messages indicate, XSPEC also has read in the associated:
These files are all listed in the headers of the data files (sa496b.pha & sa496c.pha).
To ignore channels, the file number (1 & 2 in this example) precedes the range of channels to be ignored. Here, we wish to ignore, for both files, channels 1-15 and channels 100-128. This can be done by specifying the files one after the other with the range of ignored channels:
XSPEC> ignore 1:1-15 1:100-128 2:1-15 2:100-128
or by specifying the range of file number with the channel range:
XSPEC> ignore 1-2:1-15 100-128
In this example, we'll fit a cooling-flow model under the reasonable assumption that the small SSS field of view sees mostly just the cool gas in the middle of the cluster. We'll freeze the values of the maximum temperature (the temperature from which the gas cools) and of the abundance to the values found by instruments such as the Ginga LAC and the EXOSAT ME, which observed the entire cluster. The minimum gas temperature is frozen at 0.1 keV; the ``slope'' is frozen at zero (isobaric cooling) and the normalization is given an initial value of 100 solar masses per year:
XSPEC> mo pha(cflow)
mo = phabs[1]( cflow[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
0.045
Mod parameter 2 of component 2 cflow slope
0.0000E+00 1.0000E-02 -5.000 -5.000 5.000 5.000
0,0
Mod parameter 3 of component 2 cflow lowT keV
0.1000 1.0000E-03 8.0800E-02 8.0800E-02 79.90 79.90
0.1,0
Mod parameter 4 of component 2 cflow highT keV
4.000 1.0000E-03 8.0800E-02 8.0800E-02 79.90 79.90
4,0
Mod parameter 5 of component 2 cflow Abundanc
1.000 1.0000E-02 0.0000E+00 0.0000E+00 5.000 5.000
0.5,0
Mod parameter 6 of component 2 cflow Redshift
0.0000E+00 -0.1000 0.0000E+00 0.0000E+00 100.0 100.0
0.032
Mod parameter 7 of component 2 cflow norm
1.000 1.0000E-02 0.0000E+00 0.0000E+00 1.0000E+24 1.0000E+24
100
---------------------------------------------------------------------------
---------------------------------------------------------------------------
mo = phabs[1]( cflow[2] )
Model Fit Model Component Parameter Unit Value
par par comp
1 1 1 phabs nH 10^22 4.5000E-02 +/- 0.0000E+00
2 2 2 cflow slope 0.0000E+00 frozen
3 3 2 cflow lowT keV 0.1000 frozen
4 4 2 cflow highT keV 4.000 frozen
5 5 2 cflow Abundanc 0.5000 frozen
6 6 2 cflow Redshift 3.2000E-02 frozen
7 7 2 cflow norm 100.0 +/- 0.0000E+00
---------------------------------------------------------------------------
---------------------------------------------------------------------------
2 variable fit parameters
Chi-Squared = 2705.736 using 168 PHA bins.
Reduced chi-squared = 16.29961 for 166 degrees of freedom
Null hypothesis probability = 0.000E+00
XSPEC> fit
Chi-Squared Lvl Fit param # 1 2 3 4
5 6 7
401.110 -3 0.1889 0.0000E+00 0.1000 4.000
0.5000 3.2000E-02 285.1
364.046 -4 0.2298 0.0000E+00 0.1000 4.000
0.5000 3.2000E-02 316.1
363.536 -5 0.2351 0.0000E+00 0.1000 4.000
0.5000 3.2000E-02 319.8
363.530 -6 0.2358 0.0000E+00 0.1000 4.000
0.5000 3.2000E-02 320.2
---------------------------------------------------------------------------
Variances and Principal axes :
1 7
3.04E-05 | -1.00 0.00
3.38E+01 | 0.00 -1.00
---------------------------------------------------------------------------
---------------------------------------------------------------------------
mo = phabs[1]( cflow[2] )
Model Fit Model Component Parameter Unit Value
par par comp
1 1 1 phabs nH 10^22 0.2358 +/- 0.8515E-02
2 2 2 cflow slope 0.0000E+00 frozen
3 3 2 cflow lowT keV 0.1000 frozen
4 4 2 cflow highT keV 4.000 frozen
5 5 2 cflow Abundanc 0.5000 frozen
6 6 2 cflow Redshift 3.2000E-02 frozen
7 7 2 cflow norm 320.2 +/- 5.816
---------------------------------------------------------------------------
---------------------------------------------------------------------------
Chi-Squared = 363.5299 using 168 PHA bins.
Reduced chi-squared = 2.189939 for 166 degrees of freedom
Null hypothesis probability = 8.325E-17
As we can see, 
is not good, but the high statistic could
be because we have
yet to adjust the correction file. Correction files in XSPEC take
into account detector features that cannot be completely prescribed
ab initio and which must be fitted at the same time as the model.
Einstein SSS spectra, for example, have a background feature the
level of which varies unpredictably. Its spectral form is contained in
the correction file, but its normalization is determined by fitting.
This fitting is set in motion using the command recornrm (
reset the correction-file normalization):
XSPEC> reco 1
File # Correction
1 0.4022 +/- 0.0673
After correction norm adjustment 0.402 +/- 0.067
Chi-Squared = 327.7646 using 168 PHA bins.
Reduced chi-squared = 1.974485 for 166 degrees of freedom
Null hypothesis probability = 1.091E-12
XSPEC> reco 2
File # Correction
2 0.4775 +/- 0.0597
After correction norm adjustment 0.477 +/- 0.060
Chi-Squared = 263.8333 using 168 PHA bins.
Reduced chi-squared = 1.589336 for 166 degrees of freedom
Null hypothesis probability = 2.025E-06
This process is iterative, and, in order to work, must be used in tandem
with fitting the model. Successive fits and recorrections are applied
until the fit is stable, i.e., until further improvement in no
longer results. Of course, this procedure is only worthwhile when the
model gives a reasonably good account of the data. Eventually, we end
up at:
XSPEC> fit
Chi-Squared Lvl Fit param # 1 2 3 4
5 6 7
189.931 -3 0.2791 0.0000E+00 0.1000 4.000
0.5000 3.2000E-02 300.4
---------------------------------------------------------------------------
Variances and Principal axes :
1 7
4.85E-05 | -1.00 0.00
3.84E+01 | 0.00 -1.00
---------------------------------------------------------------------------
---------------------------------------------------------------------------
mo = phabs[1]( cflow[2] )
Model Fit Model Component Parameter Unit Value
par par comp
1 1 1 phabs nH 10^22 0.2791 +/- 0.1072E-01
2 2 2 cflow slope 0.0000E+00 frozen
3 3 2 cflow lowT keV 0.1000 frozen
4 4 2 cflow highT keV 4.000 frozen
5 5 2 cflow Abundanc 0.5000 frozen
6 6 2 cflow Redshift 3.2000E-02 frozen
7 7 2 cflow norm 300.4 +/- 6.198
---------------------------------------------------------------------------
---------------------------------------------------------------------------
Chi-Squared = 189.9309 using 168 PHA bins.
Reduced chi-squared = 1.144162 for 166 degrees of freedom
Null hypothesis probability = 9.831E-02
The final value of is much better than the original, but is
not quite acceptable. However, the current model has only two free
parameters: further explorations of parameter space would undoubtedly
improve the fit.
We'll leave this example and move on to look at another kind of simultaneous fitting: one where the same model is fitted to two different data files. This time, not all the parameters will be identical. The massive X-ray binary Centaurus X-3 was observed with the LAC on Ginga in 1989. Its flux level before eclipse was much lower than the level after eclipse. Here, we'll use XSPEC to see whether spectra from these two phases can be fitted with the same model, which differs only in the amount of absorption. This kind of fitting relies on introducing an extra dimension, the group, to the indexing of the data files. The files in each group share the same model but not necessarily the same parameter values, which may be shared as common to all the groups or varied separately from group to group. Although each group may contain more than one file, there is only one file in each of the two groups in this example. Groups are specified with the data command, with the group number preceding the file number, like this:
XSPEC> da 1:1 losum 2:2 hisum Net count rate (cts/s) for file 1 140.1 +/- 0.3549 using response (RMF) file... ginga_lac.rsp Net count rate (cts/s) for file 2 1371. +/- 3.123 using response (RMF) file... ginga_lac.rsp 2 data sets are in use
Here, the first group makes up the file losum.pha, which contains the spectrum of all the low, pre-eclipse emission. The second group makes up the second file, hisum.pha, which contains all the high, post-eclipse emission. Note that file number is ``absolute'' in the sense that it is independent of group number. Thus, if there were three files in each of the two groups (lo1.pha, lo2.pha, lo3.pha, hi1.pha, hi2.pha & hi3.pha, say), rather than one, the six files would be specified as follows:
XSPEC> da 1:1 lo1 1:2 lo2 1:3 lo3 2:4 hi1 2:5 hi2 2:6 hi3
The ignore command works, as usual, on file number, and does not take group number into account. So, to ignore channels 1-3 and 37-48 of both files:
XSPEC> ignore 1-2:1-3 37-48
The model we'll use at first to fit the two files is an absorbed power law with a high-energy cut-off:
XSPEC> mo phabs * highecut (po)
After defining the model, the user is prompted for two sets of parameter
values, one for the first group of data files (losum.pha), the
other for the second group (hisum.pha). Here, we'll enter the
absorption column of the first group as 
cm 
and enter
the default values for all the other parameters in the first group. Now,
when it comes to the second group of parameters, we enter a column of
cm and then enter defaults for the other
parameters. The rule being applied here is as follows: to tie parameters in the
second group to their equivalents in the first group, take the default
when entering the second-group parameters; to allow parameters in the second
group to vary independently of their equivalents in the first group,
enter different values explicitly. Like this:
XSPEC> mo phabs * highecut (po)
mo = phabs[1]*highecut[2]( powerlaw[3] )
Input parameter value, delta, min, bot, top, and max values for ...
Mod parameter 1 of component 1 phabs nH 10^22 for data group 1
1.000 1.0000E-03 0.0000E+00 0.0000E+00 1.0000E+05 1.0000E+06
100
Mod parameter 2 of component 2 highecut cutoffE keV for data group 1
10.00 1.0000E-02 1.0000E-04 1.0000E-02 100.0 200.0
Mod parameter 3 of component 2 highecut foldE keV for data group 1
15.00 1.0000E-02 1.0000E-04 1.0000E-02 100.0 200.0
Mod parameter 4 of component 3 powerlaw PhoIndex for data group 1
1.000 1.0000E-02 -3.000 -2.000 9.000 10.00
Mod parameter 5 of component 3 powerlaw norm for data group 1
1.000 1.0000E-02 0.0000E+00 0.0000E+00 1.0000E+24 1.0000E+24
Mod parameter 6 of component 4 phabs nH 10^22 for data group 2
100.0 1.0000E-03 0.0000E+00 0.0000E+00 1.0000E+05 1.0000E+06
1
Mod parameter 7 of component 5 highecut cutoffE keV for data group 2
10.00 1.0000E-02 1.0000E-04 1.0000E-02 100.0 200.0
Mod parameter 8 of component 5 highecut foldE keV for data group 2
15.00 1.0000E-02 1.0000E-04 1.0000E-02 100.0 200.0
Mod parameter 9 of component 6 powerlaw PhoIndex for data group 2
1.000 1.0000E-02 -3.000 -2.000 9.000 10.00
Mod parameter 10 of component 6 powerlaw norm for data group 2
1.000 1.0000E-02 0.0000E+00 0.0000E+00 1.0000E+24 1.0000E+24
---------------------------------------------------------------------------
---------------------------------------------------------------------------
mo = phabs[1]*highecut[2]( powerlaw[3] )
Model Fit Model Component Parameter Unit Value Data
par par comp group
1 1 1 phabs nH 10^22 100.0 +/- 0.0000E+00 1
2 2 2 highecut cutoffE keV 10.00 +/- 0.0000E+00 1
3 3 2 highecut foldE keV 15.00 +/- 0.0000E+00 1
4 4 3 powerlaw PhoIndex 1.000 +/- 0.0000E+00 1
5 5 3 powerlaw norm 1.000 +/- 0.0000E+00 1
6 6 4 phabs nH 10^22 1.000 +/- 0.0000E+00 2
7 2 5 highecut cutoffE keV 10.00 = par 2 2
8 3 5 highecut foldE keV 15.00 = par 3 2
9 4 6 powerlaw PhoIndex 1.000 = par 4 2
10 5 6 powerlaw norm 1.000 = par 5 2
---------------------------------------------------------------------------
---------------------------------------------------------------------------
6 variable fit parameters
Chi-Squared = 1.9666626E+07 using 66 PHA bins.
Reduced chi-squared = 327777.1 for 60 degrees of freedom
Null hypothesis probability = 0.000E+00
Notice how the summary of the model, displayed immediately above, is
different now that we have two groups, as opposed to one (as in all the
previous examples). We can see that of the 10 model parameters, 6 are
free (i.e., 4 of the second group parameters are tied to their
equivalents in the first group). Fitting this model results in a huge
(not shown here), because our assumption that only a change in
absorption can account for the spectral variation before and after
eclipse is clearly wrong. Perhaps scattering also plays a role in
reducing the flux before eclipse. This could be modeled (simply at
first) by allowing the normalization of the power law to be smaller
before eclipse than after eclipse. To decouple tied parameters, we
change the parameter value in the second group to a value--any
value--different from that in the first group (changing the value in
the first group has the effect of changing both without decoupling). As
usual, the newpar command is used:
XSPEC> newpar 10 1
7 variable fit parameters
Chi-Squared = 1.9666626E+07 using 66 PHA bins.
Reduced chi-squared = 333332.7 for 59 degrees of freedom
Null hypothesis probability = 0.000E+00
After fitting (not shown), this decoupling reduces by a
factor of six to 15,478, but this is still too high. Indeed, this simple
attempt to account for the spectral variability in terms of ``blanket''
cold absorption and scattering does not work. More sophisticated models,
involving additional components and partial absorption, should be
investigated.