Pseudo-Newtonian Potentials for Nearly Parabolic Orbits

We describe a pseudo-Newtonian potential which, to within 1% error at all angular momenta, reproduces the precession due to general relativity of particles whose specific orbital energy is small compared to c^2 in the Schwarzschild metric. For bound orbits, the constraint of low energy is equivalent to requiring the apoapsis of a particle to be large compared to the Schwarzschild radius. Such low-energy orbits are ubiquitous close to supermassive black holes in galactic nuclei, but the potential is relevant in any context containing particles on low-energy orbits. Like the more complex post-Newtonian expressions, the potential correctly reproduces the precession in the far field, but also correctly reproduces the position and magnitude of the logarithmic divergence in precession for low angular momentum orbits. An additional advantage lies in its simplicity, both in computation and implementation. We also provide two simpler, but less accurate potentials, for cases where orbits always remain at large angular momenta, or when the extra accuracy is not needed. In all of the presented cases, the accuracy in precession in low-energy orbits exceeds that of the well-known potential of Paczyński & Wiita, which has ~30% error in the precession at all angular momenta.