A Symplectic Integrator for Hill's Equations

TL;DR

This paper introduces a symplectic integrator based on a generalized leapfrog method for Hill's equations, enhancing long-term orbital simulations in astrophysics with improved accuracy and efficiency.

Contribution

A novel symplectic integrator tailored for Hill's equations, implemented in PKDGRAV, enabling stable long-term integrations and efficient collision detection.

Findings

Demonstrates minimal secular drift in orbital elements over many dynamical times

Efficient collision searching via linear extrapolation of particle positions

Validated on simple orbital scenarios

Abstract

Hill's equations are an approximation that is useful in a number of areas of astrophysics including planetary rings and planetesimal disks. We derive a symplectic method for integrating Hill's equations based on a generalized leapfrog. This method is implemented in the parallel N-body code, PKDGRAV and tested on some simple orbits. The method demonstrates a lack of secular changes in orbital elements, making it a very useful technique for integrating Hill's equations over many dynamical times. Furthermore, the method allows for efficient collision searching using linear extrapolation of particle positions.