The formal underpinnings of the response functions used in X-ray spectral analysis

This work provides an in-depth mathematical description of the response functions that are used for spatial and spectral analysis of X-ray data. The use of such functions is well known to anyone familiar with the analysis of X-ray data where they may be identified with the quantities contained in the ancillary response file (ARF), the redistribution matrix file (RMF), and the exposure map. Starting from first principles, explicit mathematical expressions for these functions, for both imaging and dispersive modes, are arrived at in terms of the underlying instrumental characteristics of the telescope including the effects of pointing motion. The response functions are presented in the context of integral equations relating the expected detector count rate to the source spectrum incident upon the telescope. Their application to the analysis of several source distributions is considered. These include multiple, possibly overlapping, spectrally distinct point sources, as well as extended sources. Assumptions and limitations behind the usage of these functions, as well as their practical computation, are addressed.