whfast: a fast and unbiased implementation of a symplectic Wisdom–Holman integrator for long-term gravitational simulations
Open accessAbstract
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 11 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 and the Mean Exponential Growth factor of Nearby Orbits. whfast is freely available as a flexible C package, as a shared library, and as an easy-to-use python module.
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In the absence of P9, objects shown in Figure 1 have a dynamical lifetime on the order of ∼100 Myr.
“in the absence of P9, objects shown in Figure 1 have a dynamical lifetime on the order of ∼100 Myr; Rein & Tamayo 2015”
From Generation of Low-inclination, Neptune-crossing Trans-Neptunian Objects by Planet Nine (2024) · cites this paper as Rein & Tamayo 2015
The paper’s Figure 1 caption describes tests of an orbital integrator, not the objects in the citing document. Searches for P9, Planet Nine, and a roughly 100 Myr dynamical lifetime found no passage in our copy addressing that claim; Section 4.2 instead discusses integrator tests of the outer Solar system over 12,000 years. (LLM, conf=0.78)
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Research works printed but not linked: 7
- ISO. 2011, ISO/IEC/IEEE 60559:2011 Information Technology -Micro- processor Systems -Floating-Point arithmetic, 58
- Murray C. D., Dermott S. F., 2000, Solar System Dynamics. Cambridge Univ. Press, Cambridge
- Plummer H. C. K., 1918, An Introductory Treatise on Dynamical Astron- omy. Cambridge Univ. Press, Cambridge
- Stiefel E. L., Scheifele G., 1971, Linear and Regular Celestial Mechanics (Die Grundlehren d. math. Wissenschaften, Band 174). Springer, Berlin
- Sussman G. J., Wisdom J., 2001, Structure and Interpretation of Classical Mechanics. MIT Press, Cambridge, MA
- Wisdom J. L., 1981, PhD thesis, California Institute of Technology, Pasadena Wisdom J., 2006, AJ, 131, 2294
- Wisdom J., Holman M., Touma J., 1996, Fields Institute Communications, Vol. 10, p. 217
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Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1
Gillon, M., Triaud, A. H. M. J., Demory, B.-O., Jehin, E., Agol, E., Deck, K. M., Lederer, S. M., de Wit, J., Burdanov, A., Ingalls, J. G., et al. (2017) · Nature · doi:10.1038/nature21360 · Cited by 1,555 · see citing works
Constraints on the size and dynamics of the J1407b ring system
Rieder, S., & Kenworthy, M. A. (2016) · Astronomy & Astrophysics · doi:10.1051/0004-6361/201629567 · Cited by 23 · see citing works
“We run simulations of this model using AMUSE1 (Pelupessy et al. 2013; Portegies Zwart et al. 2013) with the Rebound/WHFast2 (Rein & Liu 2012; Rein & Tamayo 2015) N-body integrator.”
MethodGeneration of Low-inclination, Neptune-crossing Trans-Neptunian Objects by Planet Nine
Batygin, K., Morbidelli, A., Brown, M. E., & Nesvorný, D. (2024) · The Astrophysical Journal Letters · doi:10.3847/2041-8213/ad3cd2 · Cited by 11 · see citing works
“These footprints were recorded at 1 Myr intervals, far exceeding the typical Lyapunov time of the particles 2 .”
BackgroundInfluential
- Authors
- Rein, H., & Tamayo, D.
- Affiliations
- University of Toronto (CA)
- Corresponding author
- Rein, Hanno
- Venue
- Monthly Notices of the Royal Astronomical Society
- Publisher
- Oxford University Press (OUP)
- Published
- 2015-07-04
- Volume
- 452
- Issue
- 1
- Pages
- 376-388
- ISSN
- 0035-8711 (ISSN-L)
- 1365-2966
- OpenAlex
- W1864108573
- Cited by
- 467
- References
- 0
- Citations by year
- 2026: 33 · 2025: 61 · 2024: 55 · 2023: 51 · 2022: 52 · 2021: 47 · 2020: 47 · 2019: 43 · 2018: 45 · 2017: 21 · 2016: 9 · 2015: 1
- FWCI
- 14.42
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Monthly Notices of the Royal Astronomical Society (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HighWire Press Open Archive (submittedVersion)
- arXiv (Cornell University)
- Funders
- Natural Sciences and Engineering Research Council of Canada (RGPIN-2014-04553, RGPIN-2014)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Stellar, planetary, and galactic studies
- Physical Sciences › Mathematics › Numerical Analysis › Numerical methods for differential equations
- Physical Sciences › Engineering › Aerospace Engineering › Spacecraft Dynamics and Control
- Keywords
- symplectic integrators, planetary systems, energy conservation (physics), Jacobi coordinates, coordinate transformation, variational equations, chaos indicators, Lyapunov characteristic numbers
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