Orbital precession due to central-force perturbations

TL;DR

This paper derives a versatile formula for calculating orbital precession caused by various central-force perturbations, enabling more precise constraints on hypothetical new forces affecting planetary orbits.

Contribution

It provides a general integral formula for orbital precession under arbitrary central-force perturbations, including analytic solutions for specific potentials, improving upon previous methods.

Findings

The formula is straightforward to evaluate numerically.

Analytic solutions are obtained for specific potentials involving hypergeometric functions.

Results reproduce known precession formulas for general relativity, constant forces, and cosmological constant.

Abstract

We calculate the precession of Keplerian orbits under the influence of arbitrary central-force perturbations. Our result is in the form of a one-dimensional integral that is straightforward to evaluate numerically. We demonstrate the effectiveness of our formula for the case of the Yukawa potential. We obtain analytic results for potentials of the form V(r) = \alpha r^n and V(r) = \alpha \ln(r/\lambda) in terms of the hypergeometric function {_2F_1} (1/2-n/2,1-n/2; 2; e^2), where e is the eccentricity. Our results reproduce the known general relativistic (n=-3), constant force (n=1), and cosmological constant (n=2) precession formulas. Planetary precessions are often used to constrain the sizes of hypothetical new weak forces--our results allow for more precise, and often stronger, constraints on such proposed new forces.