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Highlight 3: TRACE (a Time-Reversible Code for Astrophysical Close Encounters) (Project 5)

TRACE (a Time-Reversible Code for Astrophysical Close Encounters) – a novel open-source time-reversible hybrid integrator for the planetary N-body problem, a strict improvement in both speed and accuracy over previous state-of-the-art methods (Lu, Hernandez & Rein, 2024). It allows simulating a wide diversity of planetary systems, including our own, during strong chaotic evolution (e.g., possible ejection of Mercury).

The gravitational N-body problem does not admit a convenient exact analytic solution, so to study the problem of planetary motion we must turn to methods of numerical integration (e.g. Wisdom & Holman 1991). We have developed a novel hybrid integrator, TRACE (Lu et al., 2024), which offers an excellent blend of speed and accuracy for chaotic systems in the planetary N-body regime – a dominant central mass orbited by many smaller bodies, as is appropriate for our solar system and many other planetary systems.

TRACE is based on the time-reversible algorithm of Hernandez & Dehnen (2023), and offers two material advantages over previous state-of-the-art hybrid integrators such as MERCURY (Chambers 1999) and MERCURIUS (Rein et. al, 2019). The first is much faster resolution of close encounters between pairs of planets. The second is accurate resolution of close encounters between planets and their host stars, which previous hybrid integrators cannot handle at all. TRACE has been released and fully integrated into the popular open-source REBOUND (Rein & Liu 2012) environment.

Previous hybrid methods have seen widespread use in astronomy, and have been presented in highly cited papers. TRACE can potentially replace these methods. This project was led by graduate student Tiger Lu (Project 5), with input from David M. Hernandez (Project 5) and Hanno Rein (University of Toronto). This tool can, moreover, be used by CycloAstro team members in investigations of chaotic solar system evolution.

Fig. 1: Jacobi constant error for a test of the restricted three-body problem, comparing the integrators TRACE (Lu, Hernandez & Rein 2024), MERCURIUS (Rein et. al 2019) and WHFAST (Rein & Tamayo 2015). TRACE is as accurate as the previous best hybrid integrator MERCURIUS, but runs over an order of magnitude faster. Adapted from Figure 2 in Lu, Hernandez & Rein (2024).

Fig. 2: Statistics from an chaotic ensemble of 500 scattering simulations, comparing the performance of TRACE, MERCURIUS, Bulirsch-Stoer and the present gold standard for accuracy in N-body simulations IAS15 (Rein & Spiegel 2015). We compare the number of surviving planets at the end of the simulation (top subplot), energy error (middle subplot) and runtime (bottom subplot). TRACE matches the population-level results of IAS15, at a fraction of the runtime. Adapted from Lu, Hernandez & Rein (2024).

References:

Chambers, J. (1999). “A hybrid symplectic integrator that permits close encounters between massive bodies.” Monthly Notices of the Royal Astronomical Society, Volume 304, Issue 4. https://ui.adsabs.harvard.edu/abs/1999MNRAS.304..793C/abstract

Hernandez, D. and Dehnen, W. (2023). “Switching integrators reversibly in the astrophysical N-body problem.” Monthly Notices of the Royal Astronomical Society, Volume 522, Issue 3. https://ui.adsabs.harvard.edu/abs/2023MNRAS.522.4639H/abstract

Lu, T., Hernandez, D. and Rein, H. (2024). “TRACE: a code for time-reversible astrophysical close encounters.” Monthly Notices of the Royal Astronomical Society, Volume 533, Issue 3. https://ui.adsabs.harvard.edu/abs/2024MNRAS.533.3708L/abstract

Rein, H. and Liu, S. “REBOUND: an open-source multi-purpose N-body code for collisional dynamics.” Astronomy & Astrophysics, Volume 537, id.A128. https://ui.adsabs.harvard.edu/abs/2012A%26A...537A.128R/abstract

Rein, H. and Spiegel, D. (2015). “IAS15: a fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits.” Monthly Notices of the Royal Astronomical Society, Volume 446, Issue 2. https://ui.adsabs.harvard.edu/abs/2015MNRAS.446.1424R/abstract

Rein, H. and Tamayo, D. (2015). “WHFAST: a fast and unbiased implementation of a symplectic Wisdom-Holman integrator for long-term gravitational simulations.” Monthly Notices of the Royal Astronomical Society, Volume 452, Issue 1. https://ui.adsabs.harvard.edu/abs/2015MNRAS.452..376R/abstract

Rein, H., Hernandez, D., Tamayo, D., Brown, G., Eckels, E., Holmes, E., Lau, M., Leblanc, R. and Silburt, A. (2019). “Hybrid symplectic integrators for planetary dynamics.” Monthly Notices of the Royal Astronomical Society, Volume 485, Issue 4. https://ui.adsabs.harvard.edu/abs/2019MNRAS.485.5490R/abstract

Wisdom, J. and Holman, M. (1991). “Symplectic maps for the N-body problem.” Astronomical Journal, Vol. 102. https://ui.adsabs.harvard.edu/abs/1991AJ....102.1528W/abstract

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