University of Toronto, Toronto, Ontario, M5S 3H4, Canada\
    $^3$ Canadian Institute for Theoretical Astrophysics, 60 St. George St, University of Toronto, Toronto, Ontario M5S 3H8, Canada\
    $^4$ Centre for Planetary Sciences Fellow
bibliography:
- 'full.bib'
date: 'Submitted: 8th May 2015. Accepted: 2nd June 2015.'
title: '[[WHFast]{}]{}: A fast and unbiased implementation of a symplectic Wisdom-Holman integrator for long term gravitational simulations'
---

methods: numerical — gravitation — planets and satellites: dynamical evolution and stability

Introduction {#sec:intro}
============

Celestial mechanics, the field that deals with the motion of celestial objects, has been an active field of research since the days of Newton and Kepler. Analytic solutions only exist for a few special cases. Historically, the main driver for the development of perturbation theory has been the problem of planets orbiting the Sun. Because the central body is so much more massive than the planets, it is profitable to ask how the small mutual tugs between the planets modify the Keplerian orbits they would each individually follow around the Sun in the absence of the other bodies. This analytical approach has been, and continues to be, successful in explaining many important features of planetary orbits. However, the Solar System is chaotic, and 