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General form for Pseudo-Newtonian Potentials, imitating Schwarzschild geodesics

Itamar Ben Arosh Arad, Reem Sari

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

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 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 (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 , 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.

Paper Structure

This paper contains 5 sections, 35 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The General PNP Form
  3. The ISCO Geodesic
  4. precession rates
  5. Examples and Comparison

Figures (2)

  • Figure 1: Comparison of two pseudo‑Newtonian potentials with $N_1=7$ (purple line) and $N_1=1$ (turquoise line), constructed using the constraints presented in sections §\ref{['sec:ISCO']} and §\ref{['sec:prec']}, alongside the PW potential (Equation \ref{['PWp']}, green dotted line), Wegg's potential (Equation \ref{['WeggPot']}, red dashed line) and the exact GR results (solid black line). Panel (I): Effective potential versus $r$ for $L=4$, corresponding to the angular momentum of the marginally bound circular orbit. Panel (II): Precession per orbit (in units of $\pi$) for a parabolic trajectory, plotted against $L-4$ in logarithmic scale, illustrating the logarithmic divergence as $L \rightarrow 4^+$. Panel (III): Precession per orbit (in units of $\pi$) for a parabolic trajectory, versus $L$ in the far field regime ($L\gg4$, log‑log scale), demonstrating the $\frac{1}{L^2}$ scaling of the leading correction to the precession.
  • Figure 2: Comparison of two pseudo‑Newtonian potentials with $N_1=7$ (purple line) and $N_1=1$ (turquoise line), constructed using the constraints presented in sections §\ref{['sec:ISCO']} and §\ref{['sec:prec']}, alongside the PW potential (Equation \ref{['PWp']}, green dotted line), Wegg's potential (Equation \ref{['WeggPot']}, red dashed line) and the exact GR results (solid black line). (a) Deviation in radial velocity $\Delta\dot{r}=\dot{r}_{pnp}-\dot{r}_{gr}$ plotted against $r$, Between $r=4$ to the ISCO radius ($r=6$) on the GUI. Wegg's potential result is shifted such that its ISCO aligns at $r=6$ for comparison. (b) The Effective potential for $L_{ISCO}$ versus $r$, highlighting the stationary inflection at $r=6$ for all curves besides Wegg's potential, and the monotonic decline toward $-\infty$ as $r\rightarrow 0$, indicative of the plunge.