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Accretion disks around stellar-mass black holes (BHs) are observed to emit radiation peaking in the X-rays (at about 1 keV) when the source is in the thermal state. This emission is expected to be polarized, because of electron scattering in the disk, with a linear polarization fraction up to 12 percents, depending on the inclination angle (Chandrasekhar, 1960). Photons will experience a rotation of the polarization plane due to General Relativity effects (Connors & Stark, 1977, Stark & Connors, 1977, Ishihara et al., 1988). X-ray polarization measurements can be then crucial in assessing strong gravity effects around accreting stellar-mass BHs, providing independent constraints on their spin. Such kind of measurements should be within reach of the new-generation X-ray polarimeters like IXPE (recently approved by NASA and scheduled to be launched in 2021) and eXTP (approved by Chinese Academy of Sciences). In this work, we present a model for simulating the spectral and polarization properties of X-ray radiation emitted by the accretion disk. Starting from the model developed by Dovciak et al. (2008), we incorporate the contribution of returning radiation, i.e. of photons that are deflected back to the disk before they eventually reach the observer. We develop a specific code to calculate the change in polarization angle for photons traveling between two different locations in the disk, accounting for absorption by the disk material.

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Anomalous X-ray pulsars (AXPs) and Soft gamma repeaters (SGRs) form together a single class of astrophysical sources characterized by the emission of strong X-ray bursts and persistent emission with luminosity 10^31 - 10^36 erg/s in the 0.2-10 keV energy range. These objects are commonly associate to magnetars, i.e. neutron stars endowed with ultra-strong magnetic fields. New-generation X-ray polarimeters like IXPE (NASA SMEX program) and eXTP (Chinese Academy of Science) to be launched in the next decade, will play a key role in assessing the nature of these sources by directly probing the star magnetic field. In fact, in the highly magnetized environment radiation is expected to be strongly polarized and such a measure will be easily within reach of IXPE and eXTP. Polarization measurements will eventually confirm the presence of ultra-strong magnetic fields, probing the magnetar scenario. In this talk I will discuss theoretical expectations, within the magnetar scenario, for the polarization signature of AXPs and SGRs and present numerical simulations for the detector response of the polarimeters currently under construction. I will also show how these sources can be used to test vacuum birefringence, a QED effect predicted by Heisemberg and Euler in the Thirties and not experimentally verified as yet.

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