描述X射线脉冲星光谱形成的超几何格林函数的精确解
Exact Solution for the Hypergeometric Green's Function Describing Spectral Formation in X-Ray Pulsars
摘要：
本文采用一种包含超几何函数的本征函数展开方法，求解了含辐射激波的X射线脉冲星吸积柱中辐射输运的偏微分方程。该过程产生的格林函数，它描述了单色辐射的散射注入到列从位于附近的恒星表面的源的精确解。注入的光子和下落的电子之间的碰撞使辐射在扩散通过气体时获得能量，并通过柱壁逐渐逃逸。冲击波的存在增强了辐射的能量，并在高能量下产生了幂律谱，这是费米过程的典型特征。格林函数的解析解为X射线谱的形成过程提供了重要的物理见解，它也直接关系到这些源的光谱数据的解释。其他有趣的数学方面的问题，包括建立一个封闭形式的表达的正交特征函数的二次归一化积分，并推导出一个新的求和公式，涉及产品的超几何函数。通过取一般表达式的各种极限，我们还发展了雅可比多项式的新的线性和双线性生成函数。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
