Linear Polarization in a Black Hole, 4U~1957+115 and its X-ray Spectral Analysis

We present a new spectro-polarimetric method for estimation of the inclination of X-ray binaries using the linear polarization (LP) value in black hole (BH) sources, based on the early paper by Sunyaev \& Titarchuk (1985, ST85). The X-ray LP arises from multiple scattering of initially soft photons in a hot, optically thick Compton cloud (CC) using a flat geometry. ST85 showed that the LP degree $P$ depends on the binary inclination $i$ and remains independent of the photon energy. Moreover, the $P$ follows a characteristic angular distribution based on the CC optical depth $\tau_0$, in particular for $\tau_0 >$ 10 the LP follows the Chandrasekhar (1950) distribution. In light of the vast number of recent IXPE spectro-polarimetric observations of BHs, testing this approach in combination with spectral analysis of these sources by other missions is desirable. In particular, the BH source, 4U 1957+115 was observed with IXPE in the high/soft state for which $P\sim 2\%$ was found (Marra et al. 2024). To apply the new method, in addition to knowing $P$ for a given source, it is necessary to estimate the CC optical depth $\tau_0$ and then to calculate of the best-fit photon index $\Gamma$ and the plasma temperature $kT_e$. We present the X-ray spectral analysis of 4U~1957+11 using data from IXPE, NuSTAR, NICER, RXTE}, Swift, Suzaku and ASCA. We show that the X-ray source spectra are well described by the Comptonization model with $\Gamma$ varying from 1.5 to 3. We find a monotonic increase in $\Gamma$ with the mass accretion rate, $\dot M$, and a final saturation of the index at $\Gamma\sim3$ for high $\dot M$. We determine a BH mass by the scaling method: $M_{1957}=4.8 \pm 1.8~M_{\odot}$, assuming a source distance of 22 kpc, using H 1743-322, 4U 1630-47, and GRS 1915+105 as reference sources.