Analytical techniques for polarimetric imaging of accretion flows in the Schwarzschild metric

Emission from an accretion disk around compact objects, such as neutron stars and black holes, is expected to be significantly polarized. The polarization can be used to put constraints on the geometrical and physical parameters of the compact sources - their radii, masses, and spins - as well as to determine the orbital parameters. The radiation escaping from the innermost parts of the disk is strongly affected by the gravitational field of the compact object and the relativistic velocities of the matter. The straightforward calculation of the observed polarization signatures involves a computationally expensive ray-tracing technique. At the same time, having fast computational routines for direct data fitting is becoming increasingly important in light of the currently observed images of the accretion flow around the supermassive black hole in M 87 by the Event Horizon Telescope and infrared polarization signatures coming from Sgr A*, as well as the upcoming X-ray polarization measurements by the Imaging X-ray Polarimetry Explorer and enhanced X-ray Timing and Polarimetry mission. In this work, we obtain an exact analytical expression for the rotation angle of the polarization plane in the Schwarzschild metric accounting for the effects of light bending and relativistic aberration. We show that the calculation of the observed flux, polarization degree, and polarization angle as a function of energy can be performed analytically with a high level of accuracy using an approximate light-bending formula, eliminating the need for the precomputed tabular models in fitting routines.