---
abstract: |
    We present [[WHFast]{}]{}, a fast and accurate implementation of a Wisdom-Holman symplectic integrator for long-term orbit integrations of planetary systems. [[WHFast]{}]{}is significantly faster and conserves energy better than all other Wisdom-Holman integrators tested.

    We achieve this by significantly improving the Kepler-solver and ensuring numerical stability of coordinate transformations to and from Jacobi coordinates. These refinements allow us to remove the linear secular trend in the energy error that is present in other implementations. For small enough timesteps we achieve Brouwer’s law, i.e. the energy error is dominated by an unbiased random walk due to floating-point round-off errors.

    We implement symplectic correctors up to order eleven that significantly reduce the energy error. We also implement a symplectic tangent map for the variational equations. This allows us to efficiently calculate two widely used chaos indicators the Lyapunov characteristic number (LCN) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO).

    [[WHFast]{}]{}is freely available as a flexible C package, as a shared library, and as an easy-to-use python module.
author:
- |
    Hanno Rein$^{1,2}$ and Daniel Tamayo$^{1,3,4}$\
    $^1$ Department of Physical and Environmental Sciences, University of Toronto at Scarborough, Toronto, Ontario M1C 1A4, Canada\
    $^2$ Department of Astronomy and Astrophysics, 