2025A&A...704A..87T

Context. We present a theoretical framework for the formation of X-ray linear polarization in a 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. We demonstrated that the degree of linear polarization is independent of the photon energy and follows a characteristic angular distribution determined by the optical depth, τ₀. For τ₀ > 5, the linear polarization follows the Chandrasekhar classical distribution. The IXPE observations of several BHs' X-ray binaries and of one Seyfert-1 galaxy confirm our theoretical prediction regarding the values of the linear polarization, P.
Aims. 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. These polarization measurements demonstrate that the polarization degree, P, is almost independent of energy.
Methods. We estimated the CC optical depth, τ₀, for all BHs observed by IXPE, using the plot of P versus μ = 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 obtained the photon index, Γ, and, analytically determined the CC plasma temperature, k_BT_e.
Results. Different BHs - in particular, Cyg X-1, 4U 1630-47, LMC X-1, 4U 1957+115, Swift J1727.8-1613, GX 339-4, and the Seyfert-1 BH, NGC 4151 - exhibit polarization at the 1%-8% level nearly independently of energy. k_BT_e is in the range of 5-90 keV, with a smaller value in the high-soft state with respect to the low-hard state. Remarkably, a polarization vector parallel to the CC plane can be excluded solely based on spectral constraints, in agreement with the IXPE observations.
Conclusions. Using IXPE results on polarization and the inclination of the system, we estimated τ₀. Using the photon index, Γ, and τ₀, we derived the plasma temperature, k_BT_e, without any free parameters. We find a similarity between the physical parameter and the IXPE findings and we provide evidence suggesting that the CC exhibits a flat geometry.