## Simultaneous FittingXSPEC 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: *The same target is observed at several epochs but, although the source stays constant, the response matrix has changed.*When this happens, the data files cannot be added together; they have to be fitted separately. Fitting the data files simultaneously yields tighter constraints.*The same target is observed with different instruments.*All the instruments on Suzaku, for example, observe in the same direction simultaneously. As far as XSPEC is concerned, this is just like the previous case: two data files with two responses fitted simultaneously with the same model.*Different targets are observed, but the user wants to fit the same model to each data file with some parameters shared and some allowed to vary separately.*For example, if we have a series of spectra from a variable AGN, we might want to fit them simultaneously with a model that has the same, common photon index but separately vary the normalization and absorption.
As an example we will look at a case of fitting the same model to two
different data files but where not all the parameters are
identical. Again, this is an older dataset that provides a simpler
illustration than more modern data. 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
XSPEC12>data 1:1 losum 2:2 hisum 2 spectra in use Spectral Data File: losum.pha Spectrum 1 Net count rate (cts/s) for Spectrum:1 1.401e+02 +/- 3.549e-01 Assigned to Data Group 1 and Plot Group 1 Noticed Channels: 1-48 Telescope: GINGA Instrument: LAC Channel Type: PHA Exposure Time: 1 sec Using fit statistic: chi Using test statistic: chi Using Response (RMF) File ginga_lac.rsp for Source 1 Spectral Data File: hisum.pha Spectrum 2 Net count rate (cts/s) for Spectrum:2 1.371e+03 +/- 3.123e+00 Assigned to Data Group 2 and Plot Group 2 Noticed Channels: 1-48 Telescope: GINGA Instrument: LAC Channel Type: PHA Exposure Time: 1 sec Using fit statistic: chi Using test statistic: chi Using Response (RMF) File ginga_lac.rsp for Source 1 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, and hi3.pha, say), rather than one, the six files would be specified as da 1:1 lo1 1:2 lo2 1:3 lo3 2:4 hi1 2:5 hi2 2:6 hi3. The ignore command works on file number, and does not take group number into account. So, to ignore channels 1–3 and 37–48 of both files:
XSPEC12> 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:
XSPEC12> mo phabs * highecut (po) After defining the model, we will be 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:
XSPEC12>mo phabs*highecut(po) Input parameter value, delta, min, bot, top, and max values for ... Current: 1 0.001 0 0 1E+05 1E+06 DataGroup 1:phabs:nH>100 Current: 10 0.01 0.0001 0.01 1E+06 1E+06 DataGroup 1:highecut:cutoffE> Current: 15 0.01 0.0001 0.01 1E+06 1E+06 DataGroup 1:highecut:foldE> Current: 1 0.01 -3 -2 9 10 DataGroup 1:powerlaw:PhoIndex> Current: 1 0.01 0 0 1E+24 1E+24 DataGroup 1:powerlaw:norm> Current: 100 0.001 0 0 1E+05 1E+06 DataGroup 2:phabs:nH>1 Current: 10 0.01 0.0001 0.01 1E+06 1E+06 DataGroup 2:highecut:cutoffE>/* ======================================================================== Model phabs<1>*highecut<2>*powerlaw<3> Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp Data group: 1 1 1 phabs nH 10^22 100.000 +/- 0.0 2 2 highecut cutoffE keV 10.0000 +/- 0.0 3 2 highecut foldE keV 15.0000 +/- 0.0 4 3 powerlaw PhoIndex 1.00000 +/- 0.0 5 3 powerlaw norm 1.00000 +/- 0.0 Data group: 2 6 1 phabs nH 10^22 1.00000 +/- 0.0 7 2 highecut cutoffE keV 10.0000 = p2 8 2 highecut foldE keV 15.0000 = p3 9 3 powerlaw PhoIndex 1.00000 = p4 10 3 powerlaw norm 1.00000 = p5 ________________________________________________________________________ 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:
XSPEC12>newpar 10 1 Fit statistic : Chi-Squared 1.590267e+07 using 33 bins. Chi-Squared 4.084839e+06 using 33 bins. Total fit statistic 1.998751e+07 with 59 d.o.f. Test statistic : Chi-Squared 1.998751e+07 using 66 bins. Null hypothesis probability of 0.000000e+00 with 59 degrees of freedom Current data and model not fit yet. XSPEC12>fit ... ======================================================================== Model phabs<1>*highecut<2>*powerlaw<3> Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp Data group: 1 1 1 phabs nH 10^22 20.6316 +/- 0.185277 2 2 highecut cutoffE keV 14.6866 +/- 5.59794E-02 3 2 highecut foldE keV 7.42555 +/- 9.02094E-02 4 3 powerlaw PhoIndex 1.18928 +/- 6.33109E-03 5 3 powerlaw norm 5.95641E-02 +/- 9.44254E-04 Data group: 2 6 1 phabs nH 10^22 1.31016 +/- 3.86477E-02 7 2 highecut cutoffE keV 14.6866 = p2 8 2 highecut foldE keV 7.42555 = p3 9 3 powerlaw PhoIndex 1.18928 = p4 10 3 powerlaw norm 0.313970 +/- 4.52066E-03 ________________________________________________________________________ Fit statistic : Chi-Squared 13992.93 using 33 bins. Chi-Squared 1316.02 using 33 bins. Total fit statistic 15308.95 with 59 d.o.f. After fitting, this decoupling reduces by a factor of six to 15,308, 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 tried.
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