## Fitting Models to Data: An Old Example from EXOSAT
Our first example uses very old data which is much simpler than more
modern observations and so can be used to better illustrate the basics
of XSPEC analysis. The sample files are found at
%xspec as in this example:
%xspec XSPEC version: 12.12.1 Build Date/Time: Mon Feb 7 15:09:57 2022 XSPEC12>data s54405.pha 1 spectrum in use Spectral Data File: s54405.pha Spectrum 1 Net count rate (cts/s) for Spectrum:1 3.783e+00 +/- 1.367e-01 Assigned to Data Group 1 and Plot Group 1 Noticed Channels: 1-125 Telescope: EXOSAT Instrument: ME Channel Type: PHA Exposure Time: 2.358e+04 sec Using fit statistic: chi Using Response (RMF) File s54405.rsp for Source 1 The data command tells the program to read the data as well as the response file that is named in the header of the data file. In general, XSPEC commands can be truncated, provided they remain unambiguous. Since the default extension of a data file is .pha, and the abbreviation of data to the first two letters is unambiguous, the above command can be abbreviated to da s54405, if desired. To obtain help on the data command, or on any other command, type help command followed by the name of the command. One of the first things most users will want to do at this stage - even before fitting models - is to look at their data. The plotting device should be set first, with the command cpd (change plotting device). Here, we use the pgplot X-Window server, /xs.
XSPEC12> cpd /xs There are more than 50 different things that can be plotted, all related in some way to the data, the model, the fit and the instrument. To see them, type:
XSPEC12> plot ? plot data/models/fits etc Syntax: plot commands: background chain chisq contour counts integprob data delchi dem emodel eemodel efficiency eqw eufspec eeufspec foldmodel goodness icounts insensitivity lcounts ldata margin model ratio residuals sensitivity sum ufspec Multi-panel plots are created by entering multiple options e.g. data chisq The most fundamental is the data plotted against instrument channel (data); next most fundamental, and more informative, is the data plotted against channel energy. To do this plot, use the XSPEC command setplot energy. Figure 4.1 shows the result of the commands:
XSPEC12> setplot energy XSPEC12> plot data Note the label on the y-axis has no cm so the plot is not corrected for the effective area of the detector.
We are now ready to fit the data with a model. Models in XSPEC are
specified using the model command, followed by an algebraic
expression of a combination of model components. There are two basic
kinds of model components:
More generally, XSPEC allows three types of modifying components: convolutions and mixing models in addition to the multiplicative type. Since there must be a source, there must be least one additive component in a model, but there is no restriction on the number of modifying components. To see what components are available, just type model :
XSPEC12>model Additive Models: agauss c6vmekl eqpair nei rnei vraymond agnsed carbatm eqtherm nlapec sedov vrnei agnslim cemekl equil npshock sirf vsedov apec cevmkl expdec nsa slimbh vtapec bapec cflow ezdiskbb nsagrav smaug vvapec bbody compLS gadem nsatmos snapec vvgnei bbodyrad compPS gaussian nsmax srcut vvnei bexrav compST gnei nsmaxg sresc vvnpshock bexriv compTT grad nsx ssa vvpshock bkn2pow compbb grbcomp nteea step vvrnei bknpower compmag grbjet nthComp tapec vvsedov bmc comptb grbm optxagn vapec vvtapec bremss compth hatm optxagnf vbremss vvwdem brnei cph jet pegpwrlw vcph vwdem btapec cplinear kerrbb pexmon vequil wdem bvapec cutoffpl kerrd pexrav vgadem zagauss bvrnei disk kerrdisk pexriv vgnei zbbody bvtapec diskbb kyrline plcabs vmcflow zbknpower bvvapec diskir laor posm vmeka zbremss bvvrnei diskline laor2 powerlaw vmekal zcutoffpl bvvtapec diskm logpar pshock vnei zgauss bwcycl disko lorentz qsosed vnpshock zkerrbb c6mekl diskpbb meka raymond voigt zlogpar c6pmekl diskpn mekal redge vpshock zpowerlw c6pvmkl eplogpar mkcflow refsch Multiplicative Models: SSS_ice constant ismdust polpow wndabs zphabs TBabs cyclabs log10con pwab xion zredden TBfeo dust logconst redden xscat zsmdust TBgas edge lyman smedge zTBabs zvarabs TBgrain expabs notch spexpcut zbabs zvfeabs TBpcf expfac olivineabs spline zdust zvphabs TBrel gabs pcfabs swind1 zedge zwabs TBvarabs heilin phabs uvred zhighect zwndabs absori highecut plabs varabs zigm zxipab acisabs hrefl polconst vphabs zpcfabs zxipcf cabs ismabs pollin wabs Convolution Models: cflux ireflect kyconv reflect thcomp xilconv clumin kdblur lsmooth rfxconv vashift zashift cpflux kdblur2 partcov rgsxsrc vmshift zmshift gsmooth kerrconv rdblur simpl Mixing Models: ascac monomass polrot recorn suzpsf xmmpsf clmass nfwmass projct Pile-up Models: pileup Table models may be used with the commands atable/mtable/etable atable{</path/to/tablemodel.mod>} and are described at: heasarc.gsfc.nasa.gov/docs/heasarc/ofwg/docs/general/ogip_92_009/ogip_92_009.html Additional models are available at: heasarc.gsfc.nasa.gov/docs/xanadu/xspec/newmodels.html For information about a specific component, type help model followed by the name of the component:
XSPEC12>help model apec Given the quality of our data, as shown by the plot, we'll choose an absorbed power law, specified as follows :
XSPEC12> model phabs(powerlaw) Or, abbreviating unambiguously:
XSPEC12> mo pha(po) The user is then prompted for the initial values of the parameters. Entering return or / in response to a prompt uses the default values. We could also have set all parameters to their default values by entering /* at the first prompt. As well as the parameter values themselves, users also may specify step sizes and ranges (value, delta, min, bot, top, and max values), but here, we'll enter the defaults:
XSPEC12>mo pha(po) Input parameter value, delta, min, bot, top, and max values for ... 1 0.001( 0.01) 0 0 100000 1E+06 1:phabs:nH>/* ======================================================================== Model: phabs<1>*powerlaw<2> Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 1.00000 +/- 0.0 2 2 powerlaw PhoIndex 1.00000 +/- 0.0 3 2 powerlaw norm 1.00000 +/- 0.0 ________________________________________________________________________ Fit statistic : Chi-Squared 4.878354e+08 using 125 bins. Test statistic : Chi-Squared 4.878354e+08 using 125 bins. Null hypothesis probability of 0.000000e+00 with 122 degrees of freedom Current data and model not fit yet. The current statistic is and is huge for the initial, default values - mostly because the power law normalization is two orders of magnitude too large. This is easily fixed using the renorm command.
XSPEC12> renorm Fit statistic : Chi-Squared 845.91 using 125 bins. Test statistic : Chi-Squared 845.91 using 125 bins. Null hypothesis probability of 1.09e-108 with 122 degrees of freedom Current data and model not fit yet.
We are not quite ready to fit the data (and obtain a better ), because
not all of the 125 PHA bins should be included in the fitting: some
are below the lower discriminator of the instrument and therefore do
not contain valid data; some have imperfect background subtraction at
the margins of the pass band; and some may not contain enough counts
for to be strictly meaningful. To find out which channels to discard
(ignore in XSPEC terminology), consult mission-specific documentation
that will include information about discriminator settings, background
subtraction problems and other issues. For the mature missions in the
HEASARC archives, this information already has been encoded in the
spectral files using a “QUALITY” column according to the standard described in
OGIP Memo OGIP/92-007
XSPEC12> ignore bad ignore: 40 channels ignored from source number 1 Fit statistic : Chi-Squared 793.46 using 85 bins. Test statistic : Chi-Squared 793.46 using 85 bins. Null hypothesis probability of 5.97e-117 with 82 degrees of freedom Current data and model not fit yet. Then plot with:
XSPEC12> plot ldata chi We get a warning that the fit is not current because no fit has been performed yet.
Giving two options for the plot command generates a plot with vertically stacked windows. Up to six options can be given to the plot command at a time. Forty channels were rejected because they were flagged as bad - but do we need to ignore any more? Figure 4.2 shows the result of plotting the data and the model (in the upper window) and the contributions to (in the lower window). We see that above about 15 keV the S/N becomes small. We also see, comparing Figure 4.2 with Figure 4.1, which bad channels were ignored. Although visual inspection is not the most rigorous method for deciding which channels to ignore (more on this subject later), it's good enough for now, and will at least prevent us from getting grossly misleading results from the fitting. To ignore energies above 15 keV:
XSPEC12> ignore 15.0-** 78 channels (48-125) ignored in spectrum # 1 Fit statistic : Chi-Squared 715.30 using 45 bins. Test statistic : Chi-Squared 715.30 using 45 bins. Null hypothesis probability of 2.42e-123 with 42 degrees of freedom Current data and model not fit yet. If the ignore command is handed a real number it assumes energy in keV while if it is handed an integer it will assume channel number. The “**” just means the highest energy. Starting a range with “**” means the lowest energy. The inverse of ignore is notice, which has the same syntax. We are now ready to fit the data. Fitting is initiated by the command fit. As the fit proceeds, the screen displays the status of the fit for each iteration until either the fit converges to the minimum , or we are asked whether the fit is to go through another set of iterations to find the minimum. The default number of iterations before prompting is ten.
XSPEC12>fit Parameters Chi-Squared |beta|/N Lvl 1:nH 2:PhoIndex 3:norm 450.421 150.593 -3 0.0916817 1.61266 0.00388600 412.275 63000.6 -3 0.283518 2.30662 0.00911916 53.9571 27976.3 -4 0.529631 2.14207 0.0121535 43.8301 4648.87 -5 0.565375 2.23873 0.0130851 43.8179 125.675 -6 0.552335 2.23611 0.0130313 43.8179 0.622244 -7 0.551351 2.23578 0.0130239 ======================================== Variances and Principal Axes 1 2 3 4.7830E-08| -0.0025 -0.0151 0.9999 2.3114E-03| 0.3929 -0.9195 -0.0129 9.0741E-02| 0.9196 0.3928 0.0082 ---------------------------------------- ==================================== Covariance Matrix 1 2 3 7.709e-02 3.194e-02 6.736e-04 3.194e-02 1.595e-02 3.201e-04 6.736e-04 3.201e-04 6.553e-06 ------------------------------------ ======================================================================== Model phabs<1>*powerlaw<2> Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 0.551351 +/- 0.277654 2 2 powerlaw PhoIndex 2.23578 +/- 0.126308 3 2 powerlaw norm 1.30239E-02 +/- 2.55995E-03 ________________________________________________________________________ Fit statistic : Chi-Squared 43.82 using 45 bins. Test statistic : Chi-Squared 43.82 using 45 bins. Null hypothesis probability of 3.94e-01 with 42 degrees of freedom There is a fair amount of information here so we will unpack it a bit at a time. One line is written out after each fit iteration. The columns labeled Chi-Squared and Parameters are obvious. The other two provide additional information on fit convergence. At each step in the fit a numerical derivative of the statistic with respect to the parameters is calculated. We call the vector of these derivatives beta. At the best-fit the norm of beta should be zero so we write out beta divided by the number of parameters as a check. The actual default convergence criterion is when the fit statistic does not change significantly between iterations so it is possible for the fit to end while beta is still significantly different from zero. The beta/N column helps us spot this case. The Lvl column also indicates how the fit is converging and should generally decrease. While not necessary for this simple model, for more complicated models only varying the norms on the first iteration helps the fit proper get started in a reasonable region of parameter space. At the end of the fit XSPEC writes out the Variances and Principal Axes and Covariance Matrix sections. These are both based on the second derivatives of the statistic with respect to the parameters. Generally, the larger these second derivatives, the better determined the parameter (think of the case of a parabola in 1-D). The Covariance Matrix is the inverse of the matrix of second derivatives. The Variances and Principal Axes section is based on an eigenvector decomposition of the matrix of second derivatives and indicates which parameters are correlated. We can see in this case that the first eigenvector depends almost entirely on the powerlaw norm while the other two are combinations of the nH and powerlaw PhoIndex. This tells us that the norm is independent but the other two parameters are correlated. The next section shows the best-fit parameters and error estimates. The latter are just the square roots of the diagonal elements of the covariance matrix so implicitly assume that the parameter space is multidimensional Gaussian with all parameters independent. We already know in this case that the parameters are not independent so these error estimates should only be considered guidelines to help us determine the true errors later. The final section shows the statistic values at the end of the fit. XSPEC defines a fit statistic, used to determine the best-fit parameters and errors, and test statistic, used to decide whether this model and parameters provide a good fit to the data. By default, both statistics are . When the test statistic is we can also calculate the the null hypothesis probability. This latter is the probability of getting a value of as large or larger than observed if the model is correct. If this probability is small then the model is not a good fit. The null hypothesis probability can be calculated analytically for but not for some other test statistics so XSPEC provides another way of determining the meaning of the statistic value. The goodness command performs simulations of the data based on the current model and parameters and compares the statistic values calculated with that for the real data. If the observed statistic is larger than the values for the simulated data this implies that the real data do not come from the model. Occasionally the fit of one of the simulated spectra will generate an error and cause the goodness command to fail; repeating the command will usually solve this problem. To see how goodness works, we will use the command for this case (where it is not necessary)
XSPEC12>goodness 1000 Parameter distribution is derived from fit covariance matrix. 61.20% of realizations are < best test statistic 43.82 (sim) (fit) XSPEC12>plot goodness A little more than half of the simulations give a statistic value less than that observed, consistent with this being a good fit. Figure 4.3 shows a histogram of the values from the simulations with the observed value shown by the vertical dotted line. So the statistic implies the fit is good but it is still always a good idea to look at the data and residuals to check for any systematic differences that may not be caught by the test. To see the fit and the residuals, we produce figure 4.4 using the command
XSPEC12>plot data resid
Now that we think we have the correct model we need to determine how well the parameters are determined. The screen output at the end of the fit shows the best-fitting parameter values, as well as approximations to their errors. These errors should be regarded as indications of the uncertainties in the parameters and should not be quoted in publications. The true errors, i.e. the confidence ranges, are obtained using the error command. We want to run error on all three parameters which is an intrinsically parallel operation so we can use XSPEC's support for multiple cores and run the error estimations in parallel:
XSPEC12>parallel error 3 XSPEC12>error 1 2 3 Parameter Confidence Range (2.706) 1 0.109522 1.03334 (-0.441776,0.482041) 2 2.03727 2.44823 (-0.198497,0.212462) 3 0.00953569 0.0181505 (-0.00348792,0.00512687) Here, the numbers 1, 2, 3 refer to the parameter numbers in the Model par column of the output at the end of the fit. For the first parameter, the column of absorbing hydrogen atoms, the 90% confidence range is to , corresponding to an increase in of 2.706. The reason these “better” errors are not given automatically as part of the fit output is that they entail further fitting. When the model is simple, this does not require much CPU, but for complicated models the extra time can be considerable. The error for each parameter is determined allowing the other two parameters to vary freely. If the parameters are uncorrelated this is all the information we need to know. However, we have an indication from the covariance matrix at the end of the fit that the column and photon index are correlated. To investigate this further we can use the command steppar to run a grid over these two parameters:
XSPEC12>steppar 1 0.0 1.5 25 2 1.5 3.0 25 Chi-Squared Delta nH PhoIndex Chi-Squared 1 2 162.65 118.83 0 0 0 1.5 171.34 127.53 1 0.06 0 1.5 180.35 136.53 2 0.12 0 1.5 189.64 145.82 3 0.18 0 1.5 199.2 155.38 4 0.24 0 1.5 . . . . . . . 318.01 274.2 4 0.24 25 3 336.24 292.42 3 0.18 25 3 355.29 311.47 2 0.12 25 3 375.18 331.37 1 0.06 25 3 395.94 352.12 0 0 25 3 and make the contour plot shown in figure 4.5 using:
XSPEC12>plot contour
What else can we do with the fit? One thing is to derive the flux of the model. The data by themselves only give the instrument-dependent count rate. The model, on the other hand, is an estimate of the true spectrum emitted. In XSPEC, the model is defined in physical units independent of the instrument. The command flux integrates the current model over the range specified by the user:
XSPEC12> flux 2 10 Model Flux 0.0035392 photons (2.2323e-11 ergs/cm^2/s) range (2.0000 - 10.000 keV) Here we have chosen the standard X-ray range of 2-10 keV and find that the energy flux is ergs/cm/s. Note that flux will integrate only within the energy range of the current response matrix. If the model flux outside this range is desired - in effect, an extrapolation beyond the data - then the energies command should be used. This command defines a set of energies on which the model will be calculated. The resulting model is then remapped onto the response energies for convolution with the response matrix. For example, if we want to know the flux of our model in the ROSAT PSPC band of 0.2-2 keV, we enter:
XSPEC12>energies extend low 0.2 100 Models will use response energies extended to: Low: 0.2 in 100 log bins Fit statistic : Chi-Squared 43.82 using 45 bins. Test statistic : Chi-Squared 43.82 using 45 bins. Null hypothesis probability of 3.94e-01 with 42 degrees of freedom Current data and model not fit yet. XSPEC12>flux 0.2 2. Model Flux 0.0042889 photons (8.7877e-12 ergs/cm^2/s) range (0.20000 - 2.0000 keV) The energy flux, at ergs/cm/s is lower in this band but the photon flux is higher. The model energies can be reset to the response energies using energies reset. Calculating the flux is not usually enough, we want its uncertainty as well. The best way to do this is to use the cflux model. Suppose further that what we really want is the flux without the absorption, then we include the new cflux model by:
XSPEC12>editmod pha*cflux(pow) Input parameter value, delta, min, bot, top, and max values for ... 0.5 -0.1( 0.005) 0 0 1e+06 1e+06 2:cflux:Emin>0.2 10 -0.1( 0.1) 0 0 1e+06 1e+06 3:cflux:Emax>2.0 -12 0.01( 0.12) -100 -100 100 100 4:cflux:lg10Flux>-10.3 Fit statistic : Chi-Squared 51.71 using 45 bins. Test statistic : Chi-Squared 51.71 using 45 bins. Null hypothesis probability of 1.22e-01 with 41 degrees of freedom Current data and model not fit yet. ======================================================================== Model phabs<1>*cflux<2>*powerlaw<3> Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 0.551483 +/- 0.277863 2 2 cflux Emin keV 0.200000 frozen 3 2 cflux Emax keV 2.00000 frozen 4 2 cflux lg10Flux cgs -10.3000 +/- 0.0 5 3 powerlaw PhoIndex 2.23579 +/- 0.126392 6 3 powerlaw norm 1.30242E-02 +/- 2.57317E-03 ________________________________________________________________________ The Emin and Emax parameters are set to the energy range over which we want the flux to be calculated. We also have to fix the norm of the powerlaw because the normalization of the model will now be determined by the lg10Flux parameter. This is done using the freeze command:
XSPEC12>freeze 6 We now run fit to get the best-fit value of lg10Flux as -10.2792:
XSPEC12>fit Warning: renorm - no variable model to allow renormalization Parameters Chi-Squared |beta|/N Lvl 1:nH 4:lg10Flux 5:PhoIndex 43.821 153.715 -3 0.542840 -10.2824 2.23172 43.8179 2.65056 -4 0.550962 -10.2792 2.23570 ======================================== Variances and Principal Axes 1 4 5 3.3544E-05| 0.0629 -0.8010 0.5954 3.1521E-03| 0.5045 -0.4892 -0.7114 1.0233E-01| 0.8611 0.3451 0.3733 ---------------------------------------- ==================================== Covariance Matrix 1 2 3 7.669e-02 2.963e-02 3.177e-02 2.963e-02 1.296e-02 1.427e-02 3.177e-02 1.427e-02 1.587e-02 ------------------------------------ ======================================================================== Model phabs<1>*cflux<2>*powerlaw<3> Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 0.550962 +/- 0.276921 2 2 cflux Emin keV 0.200000 frozen 3 2 cflux Emax keV 2.00000 frozen 4 2 cflux lg10Flux cgs -10.2792 +/- 0.113860 5 3 powerlaw PhoIndex 2.23570 +/- 0.125973 6 3 powerlaw norm 1.30242E-02 frozen ________________________________________________________________________ Fit statistic : Chi-Squared 43.82 using 45 bins. Test statistic : Chi-Squared 43.82 using 45 bins. Null hypothesis probability of 3.94e-01 with 42 degrees of freedom Then find the error on that parameter:
XSPEC12>error 4 Parameter Confidence Range (2.706) 4 -10.4578 -10.0801 (-0.178718,0.198982) for a 90% confidence range on the 0.2-2 keV unabsorbed flux of - ergs/cm/s. The fit, as we've remarked, is good, and the parameters are constrained. But unless the purpose of our investigation is merely to measure a photon index, it's a good idea to check whether alternative models can fit the data just as well. We also should derive upper limits to components such as iron emission lines and additional continua, which, although not evident in the data nor required for a good fit, are nevertheless important to constrain. First, let's try an absorbed black body:
XSPEC12>mo pha(bb) Input parameter value, delta, min, bot, top, and max values for ... 1 0.001( 0.01) 0 0 100000 1e+06 1:phabs:nH>/* ======================================================================== Model phabs<1>*bbody<2> Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 1.00000 +/- 0.0 2 2 bbody kT keV 3.00000 +/- 0.0 3 2 bbody norm 1.00000 +/- 0.0 ________________________________________________________________________ Fit statistic : Chi-Squared = 3.377094e+09 using 45 PHA bins. Test statistic : Chi-Squared = 3.377094e+09 using 45 PHA bins. Reduced chi-squared = 8.040700e+07 for 42 degrees of freedom Null hypothesis probability = 0.000000e+00 Current data and model not fit yet. XSPEC12>fit Parameters Chi-Squared |beta|/N Lvl 1:nH 2:kT 3:norm 1532.94 63.1918 0 0.332069 3.01550 0.000673517 1521.42 111571 0 0.153909 2.96509 0.000613317 1490.16 170627 0 0.0624033 2.87563 0.000569789 1443.32 204857 0 0.0178739 2.76602 0.000534628 1388.24 227421 0 0.00734630 2.64698 0.000503822 1324.93 244501 0 0.00243756 2.52364 0.000475538 1254.69 259287 0 0.000178082 2.39808 0.000449095 1178.52 272841 0 5.59277e-05 2.27059 0.000423584 1085.63 287146 0 2.59575e-05 2.13543 0.000401354 985.199 290402 0 1.33741e-06 1.99739 0.000379210 Number of trials exceeded: continue fitting? Y ... ... 123.773 23.7694 -3 6.35802e-09 0.890290 0.000278598 Number of trials exceeded: continue fitting? ***Warning: Zero alpha-matrix diagonal element for parameter 1 Parameter 1 is pegged at 6.35802e-09 due to zero or negative pivot element, likely caused by the fit being insensitive to the parameter. 123.773 0.374813 -3 6.35802e-09 0.890204 0.000278596 ============================== Variances and Principal Axes 2 3 2.2370E-11| -0.0000 1.0000 2.8677E-04| 1.0000 0.0000 ------------------------------ ======================== Covariance Matrix 1 2 2.868e-04 9.336e-09 9.336e-09 2.267e-11 ------------------------ ======================================================================== Model phabs<1>*bbody<2> Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 6.35802E-09 +/- -1.00000 2 2 bbody kT keV 0.890204 +/- 1.69342E-02 3 2 bbody norm 2.78596E-04 +/- 4.76175E-06 ________________________________________________________________________ Fit statistic : Chi-Squared 123.77 using 45 bins. Test statistic : Chi-Squared 123.77 using 45 bins. Null hypothesis probability of 5.42e-10 with 42 degrees of freedom
Note that after each set of 10 iterations you are asked whether you
want to continue. Replying Note that the fit has written out a warning about the first parameter and its estimated error is written as -1. This indicates that the fit is unable to constrain the parameter and it should be considered indeterminate. This usually indicates that the model is not appropriate. One thing to check in this case is that the model component has any contribution within the energy range being calculated. Plotting the data and residuals again we obtain Figure 4.6. The black body fit is obviously not a good one. Not only is large, but the best-fitting N is indeterminate. Inspection of the residuals confirms this: the pronounced wave-like shape is indicative of a bad choice of overall continuum.
Let's try thermal bremsstrahlung next:
XSPEC12>mo pha(br) Input parameter value, delta, min, bot, top, and max values for ... 1 0.001( 0.01) 0 0 100000 1e+06 1:phabs:nH>/* ======================================================================== Model phabs<1>*bremss<2> Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 1.00000 +/- 0.0 2 2 bremss kT keV 7.00000 +/- 0.0 3 2 bremss norm 1.00000 +/- 0.0 ________________________________________________________________________ Fit statistic : Chi-Squared 4.561670e+07 using 45 bins. Test statistic : Chi-Squared 4.561670e+07 using 45 bins. Null hypothesis probability of 0.000000e+00 with 42 degrees of freedom Current data and model not fit yet. XSPEC12>fit Parameters Chi-Squared |beta|/N Lvl 1:nH 2:kT 3:norm 102.905 23.3691 -3 0.274774 6.15759 0.00726465 46.547 16309.7 -4 0.0366323 5.60534 0.00785786 ... ... 42.3441 3135.61 -8 1.65741e-06 5.64294 0.00792237 ======================================== Variances and Principal Axes 1 2 3 1.8456E-08| -0.0015 0.0006 1.0000 1.3122E-02| 0.9776 0.2103 0.0013 6.6860E-01| -0.2103 0.9776 -0.0009 ---------------------------------------- ==================================== Covariance Matrix 1 2 3 4.211e-02 -1.348e-01 1.388e-04 -1.348e-01 6.396e-01 -5.632e-04 1.388e-04 -5.632e-04 5.441e-07 ------------------------------------ ======================================================================== Model phabs<1>*bremss<2> Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 1.65741E-06 +/- 0.205196 2 2 bremss kT keV 5.64294 +/- 0.799757 3 2 bremss norm 7.92237E-03 +/- 7.37654E-04 ________________________________________________________________________ Fit statistic : Chi-Squared 42.34 using 45 bins. Test statistic : Chi-Squared 42.34 using 45 bins. Null hypothesis probability of 4.56e-01 with 42 degrees of freedom Bremsstrahlung is a better fit than the black body - and is as good as the power law - although it shares the low N. With two good fits, the power law and the bremsstrahlung, it's time to scrutinize their parameters in more detail. First, we reset our fit to the powerlaw (output omitted):
XSPEC12>mo pha(po) From the EXOSAT database on HEASARC, we know that the target in question, 1E1048.1-5937, has a Galactic latitude of 24 arcmin, i.e., almost on the plane of the Galaxy. In fact, the database also provides the value of the Galactic N based on 21-cm radio observations. At cm, it is higher than the 90 percent-confidence upper limit from the power-law fit. Perhaps, then, the power-law fit is not so good after all. What we can do is fix (freeze in XSPEC terminology) the value of N at the Galactic value and refit the power law. Although we won't get a good fit, the shape of the residuals might give us a clue to what is missing. To freeze a parameter in XSPEC, use the command freeze followed by the parameter number, like this:
XSPEC12> freeze 1 The inverse of freeze is thaw:
XSPEC12> thaw 1
Alternatively, parameters can be frozen using the newpar command, which allows all the quantities associated with a parameter to be changed. We can flip between frozen and thawed states by entering 0 after the new parameter value. In our case, we want N frozen at cm, so we go back to the power law best fit and do the following :
XSPEC12>newpar 1 Current value, delta, min, bot, top, and max values 0.551351 0.001(0.00551351) 0 0 100000 1e+06 1:phabs[1]:nH:1>4,0 Fit statistic : Chi-Squared 800.74 using 45 bins. Test statistic : Chi-Squared 800.74 using 45 bins. Null hypothesis probability of 2.82e-140 with 43 degrees of freedom Current data and model not fit yet. The same result can be obtained by putting everything onto the command line, i.e., newpar 1 4, 0, or by issuing the two commands, newpar 1 4 followed by freeze 1. Now, if we fit and plot again, we get the following model (Fig. 4.7).
XSPEC12>fit ... ======================================================================== Model phabs<1>*powerlaw<2> Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 4.00000 frozen 2 2 powerlaw PhoIndex 3.55530 +/- 6.72007E-02 3 2 powerlaw norm 0.109349 +/- 8.78816E-03 ________________________________________________________________________ Fit statistic : Chi-Squared 131.82 using 45 bins. The fit is not good. In Figure 4.7 we can see why: there appears to be a surplus of softer photons, perhaps indicating a second continuum component. To investigate this possibility we can add a component to our model. The editmod command lets us do this without having to respecify the model from scratch. Here, we'll add a black body component.
XSPEC12>editmod pha(po+bb) Input parameter value, delta, min, bot, top, and max values for ... 3 0.01( 0.03) 0.0001 0.01 100 200 4:bbody:kT>2,0 1 0.01( 0.01) 0 0 1e+20 1e+24 5:bbody:norm>1e-5 Fit statistic : Chi-Squared 128.71 using 45 bins. Test statistic : Chi-Squared 128.71 using 45 bins. Null hypothesis probability of 9.87e-11 with 42 degrees of freedom Current data and model not fit yet. ======================================================================== Model phabs<1>(powerlaw<2> + bbody<3>) Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 4.00000 frozen 2 2 powerlaw PhoIndex 3.55530 +/- 6.72007E-02 3 2 powerlaw norm 0.109349 +/- 8.78816E-03 4 3 bbody kT keV 2.00000 frozen 5 3 bbody norm 1.00000E-05 +/- 0.0 ________________________________________________________________________
Notice that in specifying the initial values of the black body, we have frozen kT at 2 keV (the canonical temperature for nuclear burning on the surface of a neutron star in a low-mass X-ray binary) and started the normalization small. Without these measures, the fit might “lose its way”. Now, if we fit, we get (not showing all the iterations this time):
======================================================================== Model phabs<1>(powerlaw<2> + bbody<3>) Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 4.00000 frozen 2 2 powerlaw PhoIndex 4.82939 +/- 0.159361 3 2 powerlaw norm 0.333962 +/- 4.81488E-02 4 3 bbody kT keV 2.00000 frozen 5 3 bbody norm 2.27796E-04 +/- 2.06724E-05 ________________________________________________________________________ Fit statistic : Chi-Squared 68.45 using 45 PHA bins. The fit is better than the one with just a power law and the fixed Galactic column, but it is still not good. Thawing the black body temperature and fitting does of course improve the fit but the powerl law index becomes even steeper. Looking at this odd model with the command
XSPEC12> plot model We see, in Figure 4.8, that the black body and the power law have changed places, in that the power law provides the soft photons required by the high absorption, while the black body provides the harder photons. We could continue to search for a plausible, well-fitting model, but the data, with their limited signal-to-noise and energy resolution, probably don't warrant it (the original investigators published only the power law fit). There is, however, one final, useful thing to do with the data: derive an upper limit to the presence of a fluorescent iron emission line. First we delete the black body component using delcomp then thaw N and refit to recover our original best fit. Now, we add a gaussian emission line of fixed energy and width then fit to get:
======================================================================== Model phabs<1>(powerlaw<2> + gaussian<3>) Source No.: 1 Active/On Model Model Component Parameter Unit Value par comp 1 1 phabs nH 10^22 0.772654 +/- 0.328763 2 2 powerlaw PhoIndex 2.38047 +/- 0.166698 3 2 powerlaw norm 1.58948E-02 +/- 3.94215E-03 4 3 gaussian LineE keV 6.40000 frozen 5 3 gaussian Sigma keV 0.100000 frozen 6 3 gaussian norm 7.46264E-05 +/- 4.74341E-05 ________________________________________________________________________ The energy and width have to be frozen because, in the absence of an obvious line in the data, the fit would be completely unable to converge on meaningful values. Besides, our aim is to see how bright a line at 6.4 keV can be and still not ruin the fit. To do this, we fit first and then use the error command to derive the maximum allowable iron line normalization. We then set the normalization at this maximum value with newpar and, finally, derive the equivalent width using the eqwidth command. That is:
XSPEC12>err 6 Parameter Confidence Range (2.706) ***Warning: Parameter pegged at hard limit: 0 6 0 0.000151075 (-7.46482e-05,7.64265e-05) XSPEC12>new 6 0.000151075 Fit statistic : Chi-Squared 46.06 using 45 bins. Test statistic : Chi-Squared 46.06 using 45 bins. Null hypothesis probability of 2.71e-01 with 41 degrees of freedom Current data and model not fit yet. XSPEC12>eqwidth 3 Data group number: 1 Additive group equiv width for Component 3: 0.782901 keV Things to note: - The true minimum value of the gaussian normalization is less than zero, but the error command stopped searching for a of 2.706 when the minimum value hit zero, the “hard” lower limit of the parameter. Hard limits can be adjusted with the newpar command, and they correspond to the quantities min and max associated with the parameter values.
- The command eqwidth takes the component number as its argument.
- The upper limit on the equivalent width of a 6.4 keV emission line is high (783 eV)!
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