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 6s X-ray pulsar 1E1048.1-5937 was observed by EXOSAT in June 1985 for 20 ks. In this example, we'll conduct a general investigation of the spectrum from the Medium Energy (ME) instrument, i.e. follow the same sort of steps as the original investigators (Seward, Charles & Smale, 1986). The ME spectrum and corresponding response matrix were obtained from the HEASARC On-line service. Once installed, XSPEC is invoked by typing

%xspec

as in this example:

%xspec XSPEC version: 12.9.0 Build Date/Time: Sun Jun 28 17:58:18 2015 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 test 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 data delchi dem emodel eemodel efficiency eufspec eeufspec foldmodel goodness icounts insensitivity lcounts ldata margin model ratio residuals sensitivity sum ufspec Multi-panel plots are created by entering multiple commands e.g. "plot 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. The word ``normalized'' indicates that this plot has been divided by the value of the EFFAREA keyword in the response file. Usually this is unity so can be ignored. The label also has no cm-2 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: *additive*, which represent X-Ray sources of different
kinds. After being convolved with the instrument response, the
components prescribe the number of counts per energy bin (e.g., a
bremsstrahlung continuum); and *multiplicative* models components, which
represent phenomena that modify the observed X-Radiation
(e.g. reddening or an absorption edge). They apply an energy-dependent
multiplicative factor to the source radiation before the result is
convolved with the instrumental response.

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 apec bapec bbody bbodyrad bexrav bexriv bkn2pow bknpower bmc bremss bvapec bvvapec c6mekl c6pmekl c6pvmkl c6vmekl cemekl cevmkl cflow compLS compPS compST compTT compbb compmag comptb compth cplinear cutoffpl disk diskbb diskir diskline diskm disko diskpbb diskpn eplogpar eqpair eqtherm equil expdec ezdiskbb gadem gaussian gnei grad grbm kerrbb kerrd kerrdisk laor laor2 logpar lorentz meka mekal mkcflow nei nlapec npshock nsa nsagrav nsatmos nsmax nsmaxg nsx nteea nthComp optxagn optxagnf pegpwrlw pexmon pexrav pexriv plcabs posm powerlaw pshock raymond redge refsch rnei sedov sirf smaug srcut sresc step vapec vbremss vequil vgadem vgnei vmcflow vmeka vmekal vnei vnpshock vpshock vraymond vrnei vsedov vvapec vvgnei vvnei vvnpshock vvpshock vvrnei vvsedov zagauss zbbody zbremss zgauss zpowerlw Multiplicative Models: SSS_ice TBabs TBgrain TBvarabs absori acisabs cabs constant cyclabs dust edge expabs expfac gabs heilin highecut hrefl lyman notch pcfabs phabs plabs pwab recorn redden smedge spexpcut spline swind1 uvred varabs vphabs wabs wndabs xion zTBabs zbabs zdust zedge zhighect zigm zpcfabs zphabs zredden zsmdust zvarabs zvfeabs zvphabs zwabs zwndabs zxipcf Convolution Models: cflux cpflux gsmooth ireflect kdblur kdblur2 kerrconv lsmooth partcov rdblur reflect rgsxsrc simpl zashift zmshift Mixing Models: ascac projct suzpsf xmmpsf Pile-up Models: pileup Additional models are available at : legacy.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 = 4.864244e+08 using 125 PHA bins. Test statistic : Chi-Squared = 4.864244e+08 using 125 PHA bins. Reduced chi-squared = 3.987085e+06 for 122 degrees of freedom Null hypothesis probability = 0.000000e+00 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 = 852.19 using 125 PHA bins. Test statistic : Chi-Squared = 852.19 using 125 PHA bins. Reduced chi-squared = 6.9852 for 122 degrees of freedom Null hypothesis probability = 7.320765e-110 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 headers of the spectral files as a list of ``bad'' channels. Simply issue the command:

XSPEC12> ignore bad ignore: 40 channels ignored from source number 1 Fit statistic : Chi-Squared = 799.74 using 85 PHA bins. Test statistic : Chi-Squared = 799.74 using 85 PHA bins. Reduced chi-squared = 9.7529 for 82 degrees of freedom Null hypothesis probability = 3.545709e-118 Current data and model not fit yet. XSPEC12> plot ldata chi

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 = 721.56 using 45 PHA bins. Test statistic : Chi-Squared = 721.56 using 45 PHA bins. Reduced chi-squared = 17.180 for 42 degrees of freedom Null hypothesis probability = 1.253044e-124 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 451.814 150.854 -3 0.0961968 1.60719 0.00386478 413.575 63220.2 -3 0.264111 2.30580 0.00908043 53.9398 28104.3 -4 0.517263 2.14118 0.0121356 43.816 4617.17 -5 0.551755 2.23947 0.0130926 43.802 139.682 -6 0.538816 2.23680 0.0130394 43.802 0.58082 -7 0.537846 2.23646 0.0130320 ======================================== Variances and Principal Axes 1 2 3 4.7889E-08| -0.0025 -0.0151 0.9999 8.6827E-02| -0.9153 -0.4026 -0.0084 2.2916E-03| -0.4027 0.9153 0.0128 ---------------------------------------- ==================================== Covariance Matrix 1 2 3 7.312e-02 3.115e-02 6.565e-04 3.115e-02 1.599e-02 3.208e-04 6.565e-04 3.208e-04 6.562e-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 0.537846 +/- 0.270409 2 2 powerlaw PhoIndex 2.23646 +/- 0.126458 3 2 powerlaw norm 1.30320E-02 +/- 2.56170E-03 ________________________________________________________________________ Fit statistic : Chi-Squared = 43.80 using 45 PHA bins. Test statistic : Chi-Squared = 43.80 using 45 PHA bins. Reduced chi-squared = 1.043 for 42 degrees of freedom Null hypothesis probability = 3.949507e-01

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. Note that on the first iteration only the powerlaw norm is varied. While not necessary 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 reduced and 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. To see how this works we will use the command for this case (where it is not necessary)

XSPEC12>goodness 1000 50.40% of realizations are < best fit statistic 43.80 (nosim) XSPEC12>plot goodness

Approximately 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 0.107599 1.00722 (-0.430231,0.469393) 2 2.03775 2.44916 (-0.198718,0.212699) 3 0.00954178 0.0181617 (-0.00349016,0.00512979)

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 . This corresponds to an excursion 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 2 162.65 118.84 0 0 0 1.5 171.59 127.79 1 0.06 0 1.5 180.87 137.06 2 0.12 0 1.5 190.44 146.64 3 0.18 0 1.5 200.29 156.49 4 0.24 0 1.5 . . . . . . . 316.02 272.22 4 0.24 25 3 334.68 290.88 3 0.18 25 3 354.2 310.4 2 0.12 25 3 374.62 330.82 1 0.06 25 3 395.94 352.14 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 0.003539 photons (2.2321e-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 command energies 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 = 43.80 using 45 PHA bins. Test statistic : Chi-Squared = 43.80 using 45 PHA bins. Reduced chi-squared = 1.043 for 42 degrees of freedom Null hypothesis probability = 3.949507e-01 Current data and model not fit yet. XSPEC12>flux 0.2 2. Model Flux 0.0043484 photons (8.8419e-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 = 52.01 using 45 PHA bins. Test statistic : Chi-Squared = 52.01 using 45 PHA bins. Reduced chi-squared = 1.268 for 41 degrees of freedom Null hypothesis probability = 1.163983e-01 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 0.537843 +/- 0.270399 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 2.23646 +/- 0.126455 6 3 powerlaw norm 1.30320E-02 +/- 2.56146E-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 -10.2903 then:

XSPEC12>error 4 Parameter Confidence Range (2.706) 4 -10.458 -10.0789 (-0.167672,0.211462)

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 1535.61 63.3168 0 0.334306 3.01647 0.000673086 1523.48 112166 0 0.157480 2.96616 0.000613284 1491.73 170831 0 0.0668724 2.87680 0.000570109 1444.74 204638 0 0.0228530 2.76753 0.000535209 1387.84 226856 0 0.00206016 2.64901 0.000504576 1325.39 243767 0 0.000851460 2.52617 0.000476469 1256.03 258162 0 0.000299777 2.40098 0.000449911 1180.44 271941 0 4.82004e-05 2.27529 0.000425076 1093.77 284819 0 1.99048e-05 2.14630 0.000402842 976.95 291176 0 9.75848e-06 1.99181 0.000380678 Number of trials exceeded: continue fitting? Y ... ... 123.773 25.397 -8 1.87147e-08 0.890295 0.000278599 Number of trials exceeded: continue fitting? ***Warning: Zero alpha-matrix diagonal element for parameter 1 Parameter 1 is pegged at 1.87147e-08 due to zero or negative pivot element, likely caused by the fit being insensitive to the parameter. 123.773 1.92501 -3 1.87147e-08 0.890205 0.000278596 ============================== Variances and Principal Axes 2 3 2.8677E-04| -1.0000 -0.0000 2.2370E-11| 0.0000 -1.0000 ------------------------------ ======================== Covariance Matrix 1 2 2.867e-04 9.315e-09 9.315e-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 2.24621E-09 +/- -1.00000 2 2 bbody kT keV 0.890178 +/- 1.69312E-02 3 2 bbody norm 2.78595E-04 +/- 4.76156E-06 ________________________________________________________________________ Fit statistic : Chi-Squared = 123.77 using 45 PHA bins. Test statistic : Chi-Squared = 123.77 using 45 PHA bins. Reduced chi-squared = 2.9470 for 42 degrees of freedom Null hypothesis probability = 5.417154e-10

Note that after each set of 10 iterations you are asked whether you
want to continue. Replying *no* at these prompts is a good idea if
the fit is not converging quickly. Conversely, to avoid having to keep
answering the question, i.e., to increase the number of iterations
before the prompting question appears, begin the fit with, say
fit **100**. This command will put the fit through 100
iterations before pausing. To automatically answer yes to all such
questions use the command query yes.

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 = 4.534834e+07 using 45 PHA bins. Test statistic : Chi-Squared = 4.534834e+07 using 45 PHA bins. Reduced chi-squared = 1.079722e+06 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 105.28 24.2507 -3 0.273734 6.18714 0.00724161 46.8022 16593.4 -4 0.0371200 5.59937 0.00785588 ... ... 40.0373 270.662 0 7.62629e-05 5.28989 0.00830799 ======================================== Variances and Principal Axes 1 2 3 1.9514E-08| -0.0016 0.0007 1.0000 1.1574E-02| 0.9736 0.2281 0.0014 5.3111E-01| 0.2281 -0.9736 0.0011 ---------------------------------------- ==================================== Covariance Matrix 1 2 3 3.862e-02 -1.148e-01 1.431e-04 -1.148e-01 5.015e-01 -5.379e-04 1.431e-04 -5.379e-04 6.290e-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 7.62629E-05 +/- 0.196509 2 2 bremss kT keV 5.28989 +/- 0.708184 3 2 bremss norm 8.30799E-03 +/- 7.93082E-04 ________________________________________________________________________ Fit statistic : Chi-Squared = 40.04 using 45 PHA bins. Test statistic : Chi-Squared = 40.04 using 45 PHA bins. Reduced chi-squared = 0.9533 for 42 degrees of freedom Null hypothesis probability = 5.574343e-01

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 , 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 0.537843 0.001(0.00537843) 0 0 100000 1e+06 1:phabs[1]:nH:1>4,0 Fit statistic : Chi-Squared = 823.34 using 45 PHA bins. Test statistic : Chi-Squared = 823.34 using 45 PHA bins. Reduced chi-squared = 19.148 for 43 degrees of freedom Null hypothesis probability = 6.152922e-145 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 3.59784 +/- 6.76670E-02 3 2 powerlaw norm 0.116579 +/- 9.43208E-03 ________________________________________________________________________ Fit statistic : Chi-Squared = 136.04 using 45 PHA 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+24 1e+24 5:bbody:norm>1e-5 Fit statistic : Chi-Squared = 132.76 using 45 PHA bins. Test statistic : Chi-Squared = 132.76 using 45 PHA bins. Reduced chi-squared = 3.1610 for 42 degrees of freedom Null hypothesis probability = 2.387580e-11 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 3.59784 +/- 6.76670E-02 3 2 powerlaw norm 0.116579 +/- 9.43208E-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 4.89633 +/- 0.158893 3 2 powerlaw norm 0.365391 +/- 5.26695E-02 4 3 bbody kT keV 2.00000 frozen 5 3 bbody norm 2.29745E-04 +/- 2.03915E-05 ________________________________________________________________________ Fit statistic : Chi-Squared = 69.53 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 0.753994 +/- 0.320347 2 2 powerlaw PhoIndex 2.38165 +/- 0.166974 3 2 powerlaw norm 1.59131E-02 +/- 3.94933E-03 4 3 gaussian LineE keV 6.40000 frozen 5 3 gaussian Sigma keV 0.100000 frozen 6 3 gaussian norm 7.47374E-05 +/- 4.74253E-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 0.000151164 (-7.476e-05,7.64036e-05) XSPEC12>new 6 0.000151164 Fit statistic : Chi-Squared = 46.03 using 45 PHA bins. Test statistic : Chi-Squared = 46.03 using 45 PHA bins. Reduced chi-squared = 1.123 for 41 degrees of freedom Null hypothesis probability = 2.717072e-01 Current data and model not fit yet. XSPEC12>eqwidth 3 Data group number: 1 Additive group equiv width for Component 3: 0.784169 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 (784 eV)!