Exact Solution for the Hypergeometric Green's Function Describing Spectral Formation in X-Ray Pulsars

TL;DR

This paper derives an exact analytical Green's function solution for radiation transport in X-ray pulsar accretion columns, revealing how shock-enhanced energization leads to characteristic high-energy spectra.

Contribution

It provides the first exact solution using hypergeometric functions for the Green's function in this context, offering new mathematical formulas and physical insights.

Findings

Exact Green's function solution for radiation transport

Power-law high-energy spectrum due to shock effects

New mathematical formulas for hypergeometric and Jacobi polynomials

Abstract

An eigenfunction expansion method involving hypergeometric functions is used to solve the partial differential equation governing the transport of radiation in an X-ray pulsar accretion column containing a radiative shock. The procedure yields the exact solution for the Green's function, which describes the scattering of monochromatic radiation injected into the column from a source located near the surface of the star. Collisions between the injected photons and the infalling electrons cause the radiation to gain energy as it diffuses through the gas and gradually escapes by passing through the walls of the column. The presence of the shock enhances the energization of the radiation and creates a power-law spectrum at high energies, which is typical for a Fermi process. The analytical solution for the Green's function provides important physical insight into the spectral formation process in X-ray pulsars, and it also has direct relevance for the interpretation of spectral data for these sources. Additional interesting mathematical aspects of the problem include the establishment of a closed-form expression for the quadratic normalization integrals of the orthogonal eigenfunctions, and the derivation of a new summation formula involving products of hypergeometric functions. By taking various limits of the general expressions, we also develop new linear and bilinear generating functions for the Jacobi polynomials.