Simultaneous solution of Kompaneets equation and Radiative Transfer equation in the photon energy range 1 - 125 KeV

TL;DR

This paper develops a method to solve the coupled radiative transfer and Kompaneets equations to model the spectrum of 1-125 keV photons in a hot electron gas, considering both isotropic and anisotropic incident radiation.

Contribution

It presents a simultaneous solution of the radiative transfer and Kompaneets equations in a plane parallel medium, incorporating free-free processes and electron gas emission for the first time in this energy range.

Findings

Spectrum becomes angle independent after multiple Compton scatterings.

The model accurately predicts emergent and reflected radiation spectra.

Inclusion of free-free emission improves spectral modeling.

Abstract

Radiative transfer equation in plane parallel geometry and Kompaneets equation is solved simultaneously to obtain theoretical spectrum of 1-125 KeV photon energy range. Diffuse radiation field is calculated using time-independent radiative transfer equation in plane parallel geometry, which is developed using discrete space theory (DST) of radiative transfer in a homogeneous medium for different optical depths. We assumed free-free emission and absorption and emission due to electron gas to be operating in the medium. The three terms $n, n^2$ and $\displaystyle \bigg({\frac {\partial n}{\partial x_k}}\bigg)$ where $n$ is photon phase density and $\displaystyle x_k= \bigg({\frac {h \nu} {k T_e}} \bigg) $, in Kompaneets equation and those due to free-free emission are utilized to calculate the change in the photon phase density in a hot electron gas. Two types of incident radiation are considered: (1) isotropic radiation with the modified black body radiation $I^{MB}$ [1] and (2) anisotropic radiation which is angle dependent. The emergent radiation at $\tau=0$ and reflected radiation $\tau=\tau_{max}$ are calculated by using the diffuse radiation from the medium. The emergent and reflected radiation contain the free-free emission and emission from the hot electron gas. Kompaneets equation gives the changes in photon phase densities in different types of media. Although the initial spectrum is angle dependent, the Kompaneets equation gives a spectrum which is angle independent after several Compton scattering times.