Post-Newtonian spherically symmetrical accretion

Abstract

The objective of this work is to investigate the influence of the corrections to the spherical symmetrical accretion of an infinity gas cloud characterized by a polytropic equation into a massive object due to the post-Newtonian approximation. Starting with the steady state post-Newtonian hydrodynamic equations for the mass, mass-energy, and momentum densities, the post-Newtonian Bernoulli equation is derived. The post-Newtonian corrections to the critical values of the flow velocity, sound velocity and radial distance are obtained from the system of hydrodynamics equations in spherical coordinates. It was considered that the ratio of the sound velocity far the massive body and the speed of light was of order ${a}_{\ensuremath{\infty}}/c={10}^{\ensuremath{-}2}$. The analysis of the solution led to following results: the Mach number for the Newtonian and post-Newtonian accretion have practically the same values for radial distances of order of the critical radial distance; by decreasing the radial distance the Mach number for the Newtonian accretion is bigger than the one for the post-Newtonian accretion; the difference between the Newtonian and post-Newtonian Mach numbers when the ratio ${a}_{\ensuremath{\infty}}/c\ensuremath{\ll}{10}^{\ensuremath{-}2}$ is insignificant; the effect of the correction terms in post-Newtonian Bernoulli equation is more perceptive for the lowest values of the radial distance; the solutions for ${a}_{\ensuremath{\infty}}/c>{10}^{\ensuremath{-}2}$ does not lead to a continuous inflow and outflow velocity at the critical point; the comparison of the solutions with those that follow from a relativistic Bernoulli equation shows that the dependence of the Mach number with the radial distance of the former is bigger than the Newtonian and post-Newtonian Mach numbers.