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

This work provides an in-depth mathematical description of the response fun ctions that are used for spatial and spectral analysis of X-ray data. The u se 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 th e ancillary response file (ARF), the redistribution matrix file (RMF), and the exposure map. Starting from first principles, explicit mathematical exp ressions for these functions, for both imaging and dispersive modes, are ar rived at in terms of the underlying instrumental characteristics of the tel escope including the effects of pointing motion. The response functions are presented in the context of integral equations relating the expected detec tor count rate to the source spectrum incident upon the telescope. Their ap plication to the analysis of several source distributions is considered. Th ese include multiple, possibly overlapping, spectrally distinct point sourc es, as well as extended sources. Assumptions and limitations behind the usa ge of these functions, as well as their practical computation, are addresse d.