Lev Titarchuk, Paolo Soffitta, Elena Seifina, Fabio Muleri, Romana Mikusincova

We present a theoretical framework for the formation of X-ray linear polarization in black hole (BH) source. The X-ray linear polarization originates from up-scatterings of initially soft photons within a hot, optically thick Compton cloud (CC) characterized by a flat geometry. The IXPE observations of several BHs X-ray binaries and of one Seyfert-1 galaxy confirm our theoretical prediction regarding values of the linear polarization P. The goal of this paper is to demonstrate that the main physical parameters of these Galactic and extragalactic sources can be derived without any free parameter using the polarization and X-ray spectral measurements. We estimate the CC optical depth, {\tau}0 for all BHs observed by IXPE, using the plot of P vs {\mu} = cos i, where i is an observer inclination with respect to the normal, and considering the values P for a given source. Using X ray spectral analysis, we obtain the photon index {\Gamma} and, analytically determined the CC plasma temperature kBTe. Different BHs (Cyg X 1, 4U 1630 47, LMC X 1, 4U 1957+115, Swift J1727.8 1613, GX 339 4) and the Seyfert 1 NGC 4151 exhibited polarization between 1 and 8 percent level nearly independently of energy. kBTe is in the range between 5 and 90 keV with a smaller value in High/Soft State with respect to Low/Hard State. We find a similarity between the physical parameter and the IXPE findings and we provide evidence suggesting that the CC exhibits a flat geometry.