Spectral Formation in X-Ray Pulsars and Associated Identities Involving the Laguerre Polynomials

TL;DR

This paper analytically solves the PDE governing radiation spectrum formation in X-ray pulsars, revealing new mathematical identities involving Laguerre polynomials and providing physical insights into observed spectra.

Contribution

It introduces an analytical Green's function solution for photon scattering in X-ray pulsars and establishes a novel summation formula with Laguerre polynomials.

Findings

Analytical Green's function for radiation spectrum formation

New summation formula involving Laguerre polynomials

Physical insights into X-ray pulsar spectra formation

Abstract

The partial differential equation governing the formation of the radiation spectrum in the shocked plasma accreting onto an x-ray pulsar is solved to obtain the associated Green's function describing the scattering of monochromatic radiation injected into the plasma at a fixed altitude above the star. Collisions between the photons and the infalling electrons cause both the ordered and random components of the plasma energy to be transferred to the radiation, which escapes by diffusing through the walls of the accretion column. The analytical solution for the Green's function provides important physical insight into the formation of the observed spectra in x-ray pulsars. Interesting mathematical aspects of the problem include the establishment of a new summation formula involving the Laguerre polynomials, based on the calculation of the photon number density via integration of the Green's function.