The use of Kepler solver in numerical integrations of quasi-Keplerian orbits

TL;DR

This paper introduces a modified Kepler solver for numerical integration of quasi-Keplerian orbits, improving accuracy and conservation of orbital elements over long timescales, especially in perturbed two-body problems.

Contribution

A new correction method for Kepler solvers that enhances accuracy and conserves orbital elements in numerical integrations of perturbed two-body systems.

Findings

Reduces errors in post-Newtonian two-body simulations

Eliminates secular drifts over 10^8-year integrations

Outperforms uncorrected integrators in accuracy

Abstract

A Kepler solver is an analytical method used to solve a two-body problem. In this paper, we propose a new correction method by slightly modifying the Kepler solver. The only change to the analytical solutions is that the obtainment of the eccentric anomaly relies on the true anomaly that is associated to a unit radial vector calculated by an integrator. This scheme rigorously conserves all integrals and orbital elements except the mean longitude. However, the Kepler energy, angular momentum vector and Laplace-Runge-Lenz vector for perturbed Kepler problems are slowly-varying quantities. However, their integral invariant relations give the quantities high-precision values that directly govern five slowly-varying orbital elements. These elements combined with the eccentric anomaly determine the desired numerical solutions. The newly proposed method can considerably reduce various errors for a post-Newtonian two-body problem compared with an uncorrected integrator, making it suitable for a dissipative two-body problem. Spurious secular changes of some elements or quasi-integrals in the outer solar system may be caused by short integration times of the fourth-order Runge-Kutta algorithm. However, they can be eliminated in a long integration time of 10^8 years by the proposed method, similar to Wisdom-Holman second-order symplectic integrator. The proposed method has an advantage over the symplectic algorithm in the accuracy but gives a larger slope to the phase error growth.