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 Paczyński–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 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 Paczyński–Wiita potential and other existing potentials. Finally, we discuss the limitations and advantages of our formulation.