# Hybrid Symplectic Integrators for Planetary Dynamics

## TL;DR

This paper reevaluates the switching function criteria in hybrid symplectic integrators like MERCURY, proposing simpler alternatives that maintain accuracy in planetary dynamics simulations.

## Contribution

It clarifies the criteria for choosing switching functions and introduces a simpler scheme using a Heaviside function without sacrificing accuracy.

## Key findings

- Heaviside switching function performs as well as polynomial functions in short-term accuracy.
- Original polynomial switching function choice in MERCURY was based on incorrect motivation.
- Simpler switching schemes can be used without loss of performance.

## Abstract

Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high order integration scheme for the duration of the encounter while otherwise using a standard 2nd order Wisdom-Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching function that determines which parts of the Hamiltonian are integrated with the high order integrator. We show that the original motivation for choosing a polynomial switching function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching functions including an infinitely differentiable function and a Heaviside function. We find that using a Heaviside function leads to a significantly simpler scheme compared to MERCURY, while maintaining the same accuracy in short term simulations.