General form for Pseudo-Newtonian Potentials, imitating Schwarzschild geodesics

Abstract

We propose a new, general form for a pseudo-Newtonian gravitational potential (PNP), expressed as a series of Paczy\'nski-Wiita-like functions with the addition of increasing negative powers of $r$ with arbitrary coefficients. We present a procedure for determining these coefficients to construct a custom PNP that replicates key features of Schwarzschild geodesics for a test particle near a black hole. As an example, we construct potentials set to reproduce (I) the presence of an innermost stable circular orbit at the $r=6$ (geometric units), with the correct infall velocity for small deviations (on the geodesic universal infall), (II) the periapsis advance at large distances, and (III) the presence of a marginally bound circular orbit with specific angular momentum $L=4$, and the periapsis advance of parabolic orbits close to it. We compare the performance of our examples against the Paczy\'nski-Wiita potential and other existing potentials. Finally, we discuss the limitations and advantages of our formulation.