The spectral shape of the SPF emission is highly variable. (See Kuntz & Snowden (2008) for a more complete discussion.) There is some evidence that, on average, the stronger flares have harder spectra, but such information is of little use when determining the spectral shape for a single observation. The mean spectrum at a given flare strength can be reasonably well fit with a broken power law. A typical flared observation will contain flares at various strengths with, presumably, different spectral shapes. Thus, rather than attempting to create a precise characterization of the spectral shape, we attempt to reduce the strength of the soft proton flare sufficiently so that error in the spectral shape will be insignificant. This is the reason why one filters with espfilt to remove the bulk of the soft proton component rather than attempting to fit that component.
Since protons are not photons, the effective area curve should not be used. It would, for instance, produce edges where there are none in the soft proton flare spectrum. The case for or against using the redistribution function is more complex. Presumably, the deposition of energy with depth for soft protons will be different from photons. Conversely, there will still be losses both near the surface and at depth. Thus, one does expect the response to a monoenergetic beam of soft protons to have a finite energy width. Although I am not aware of a measured redistribution function for soft protons for the XMM CCDs, it is not strictly necessary. Since the energy spectrum for soft protons is not expected to have sharp features, the difference in shape between the proper redistributed spectrum and initial spectrum should not be strong. (We note that there has been work recently published on the redistribution function for soft proton flares. It has not yet been incorporated into ESAS.)
Thus, one can model the soft proton component with a broken power law which has not been folded through the instrument response, that is, without using the effective area file (the .arf) and using a special redistribution file (the .rmf) for an infinitely narrow line spread function. For Xspec V12 and higher this fitting is accomplished by adding a separate model with a diagonal unitary matrix. This is demonstrated in the XMM-ESAS examples in the Xspec *.xcm files. The diagonal matrices were previously supplied with the ESAS CalDB files. This is no longer the case. The diagonal response files are now accessible from the ESAS web site as mos1-diag.rsp.gz, mos2-diag.rsp.gz, and pn-diag.rsp.gz.
For moderately SPF affected data, the SPF spectral parameters are usually robustly fit. For a simple power law the fitted index is typically in the range 0.5-1.0 but can vary between 0.1 and 1.4. However, in certain circumstances the index can blow up to relatively large or small (even negative) numbers. In these circumstances it is probably reasonable to freeze the index at a reasonable value (e.g., 0.1 for small or negative indices and 1.4 for large indices). The SPF index for the PN is likely to be different from that of the MOS detectors so in general they should not be linked, nor should any of the normalizations between different instruments be linked. This is a bit of an art form and it is not possible at this time to make any definitive rules. In addition, use of a broken power law spectrum with a break at 3.0 keV can be warranted. For broken power law fits the indices share the lower reasonable limit but may range to 2.5 or higher for the upper range. When the fitted value for the SP normalization drops below a few times arcmin the values are effectively zero. There is no practical difference between normalizations of and .