Quite often analysis will require the simultaneous fits of spectra from multiple regions, such as cluster radial annuli (e.g., Figure 19). In the Abell 1795 radial profile there are 10 regions for a total of 30 EPIC spectra plus the one RASS spectrum. With 58 parameters per spectrum and 31 spectra, that makes a total of 1798 parameters. With so many parameters, the aim again is to link or freeze as many as possible in order to reduce the number of free parameters. As with the full-field spectra, while the normalizations for the MOS Al K and Si K lines and the pn Al K and Cu lines will vary with annulus, the energies and widths will not and therefore can be linked. The cosmic background parameters can also be linked for all spectra providing the region solid angle scale factors have all been set correctly. The cluster emission component and absorption should be the same for the same annuli from the three EPIC instruments, and so can be linked. While the residual SPF contamination may not have a constant spectrum over the field of view, as a first approximation it can be assumed to be so. Thus the spectral parameters (the spectral indices and break energy for the broken power laws) can be linked (the MOS detectors together). However, the distribution over the CCDs is not uniform and the tool proton_scale can be used to generate appropriate scale factors. These scale factors should all be fixed to their appropriate values. When running proton_scale, three numbers are printed out. The last of these (“Scaled Ave Flux”) is the normalization for the soft proton events for that region and instrument. Pick one of your regions and then scale all the others to it. The scaling is just the ratio of that region's normalization to the normalization of the chosen region. The scale factors for the SPF contribution should be used to link all of the regions to one active parameter (in Xspec, newpar B = A * F where the B is the parameter number to be linked, the A is the parameter number of the active parameter, and F is the scaling between the two parameters). Normalizations for the SPF contamination, Gaussian lines, and cluster emission should be frozen at 0 for the RASS data and the solid angle normalization should be frozen to 1 (the units of the RASS spectrum provided by the X-ray background tool is in units of arcmin. proton_scale can be run in two modes, either on individual spectra with command line input or on groups where the spectra are identified in an ascii file.
With these linkages the number of parameters that are fit are already reduced from to about 80, but we are not done yet. The Galactic absorption in the cosmic background component can be frozen to the Galactic column (also provided by the HEASARC X-ray Background Tool) and the extragalactic power law index can be frozen to the canonical value ( ). If the solid angle scale factor has been set in units of square arc minutes, the extragalactic power law normalization can be set to its canonical value as well ( in Xspec units, equivalent to a normalization of 10.5 photons keV cm s sr). However, when point sources have been removed the normalization should be adjusted. The XMM-ESAS package provides the task point-source which will calculate the scale factor using several different models and user-selected parameters. The redshift for the cluster in the various annuli should also be the same so it can be linked. The number of free parameters for the initial fit attempt is still too large for an easy convergence so a few more parameters can be temporarily frozen. The temperatures of the cosmic components, the energies and widths of the Gaussian instrumental components, and the redshift of the cluster if it is known. This can reduce the number of free parameters to about 55.
After an initial fit some of the frozen parameters should be thawed. These include the energies and widths of the instrumental Gaussians (note, however, that if the source emission is bright relative to the Gaussians the width should remain frozen). It may also become clear that other parameters can be linked, but this is where scientific intuition comes into play. In this case, there are insufficient statistics to significantly constrain the fitted abundances in the outer annuli of the cluster so they can be linked to those of inner annuli successively until significance is achieved. The upper panel of Figure 20 shows the best fit for the ten extracted annuli from the Abell 1795 observation. Even after Xspec has converged in fitting the data, it may have only found a local minimum. This will often become apparent when the parameter confidence interval is being determined (e.g., using the Xspec tool steppar). So the fitting process will need to be redone to reflect the better fit.