Plane Polarization in Comptonization process : a Monte Carlo study

High energies emissions observed in X-ray binaries (XRBs), active galactic nuclei (AGNs) are linearly polarized. The prominent mechanism for X-ray is the Comptonization process. We revisit the theory for polarization in Compton scattering with unpolarized electrons, and note that the ($k \times k'$)-coordinate (in which, ($k \times k'$) acts as a $z$-axis, here $k$ and $k'$ are incident and scattered photon momentum respectively) is more convenient to describe it. Interestingly, for a fixed scattering plane the degree of polarization PD after single scattering for random oriented low-energy unpolarized incident photons is $\sim$0.33. At the scattering angle $\theta$ = 0 or $\theta \equiv$ [0,25$^o$], the modulation curve of $k'$ exhibits the same PD and PA (angle of polarization) of $k$, and even the distribution of projection of electric vector of $k'$ ($k'_e$) on perpendicular plane to the $k$ indicates same (so, an essential criteria for detector designing). We compute the polarization state in Comptonization process using Monte Carlo methods with considering a simple spherical corona. We obtain the PD of emergent photons as a function of $\theta$-angle (or alternatively, the disk inclination angle $i$) on a meridian plane (i.e., the laws of darkening, formulated by \citealp{Chandrasekhar1946}) after single scattering with unpolarized incident photons. To explore the energy dependency we consider a general spectral parameter set corresponding to hard and soft states of XRBs, we find that for average scattering no. $\langle N_{sc}\rangle$ $\sim$1.1 the PD is independent of energy and PA $\sim 90^o$ ($k'_e$ is parallel to the disk plane), and for $\langle N_{sc}\rangle$ $\sim$5 the PD value is maximum for $i=45^o$. We also compare the results qualitatively with observation of IXPE for five sources.