X-ray Spectro-polarimetry with Photoelectric Polarimeters

We derive a generalization of forward fitting for X-ray spectroscopy to include linear polarization of X-ray sources, appropriate for the anticipated next generation of space-based photoelectric polarimeters. We show that the inclusion of polarization sensitivity requires joint fitting to three observed spectra, one for each of the Stoke's parameters, I(E), U(E), and Q(E). The equations for Stoke's I(E) (the total intensity spectrum) are identical to the familiar case with no polarization sensitivity, and for which the model-predicted spectrum is obtained by a convolution of the source spectrum, F(E'), with the familiar energy response function, e(E')*R(E', E), where e(E') and R(E', E) are the effective area and energy redistribution matrix, respectively. In addition to the energy spectrum, the two new relations for U(E) and Q(E) include the source polarization fraction and position angle versus energy, a(E'), and psi'_0(E'), respectively, and the model-predicted spectra for these relations are obtained by a convolution with the "modulated" energy response function, m(E')*e(E')R(E, E'), where m(E') is the energy-dependent modulation fraction that quantifies a polarimeter's angular response to 100% polarized radiation. We present results of simulations with response parameters appropriate for the proposed PRAXyS Small Explorer observatory to illustrate the procedures and methods, and we discuss some aspects of photoelectric polarimeters with relevance to understanding their calibration and operation.