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Solving the Astrophysical N-body Problem More Efficiently

Illustration of four planets orbiting a star and the general equations of motion for any gravitational n-body system.
Illustration of four planets orbiting a star and the general equations of motion for any gravitational n-body system. Illustration by Jochen Marschall

A new algorithm developed by a grantee of the Foundation’s Science program has the potential to trace the evolution of large astrophysical systems, such as galaxies, planetary systems, or clusters of stars.

The finding is shared in a paper published in the July 2023 issue of Monthly Notices of the Royal Astronomical Society. One of its two authors is researcher David Hernandez, whose contributions were supported by the Heising-Simons Foundation as part of the interdisciplinary CycloAstro collaboration. CycloAstro integrates science from paleoclimatology, geology, and planetary astronomy to constrain Solar System evolution, Earth-Moon dynamics, paleoclimate change, and geological time.

Astrophysical systems are in constant motion, with some of them exhibiting a large range of time scales. For example, a stellar cluster might have two stars orbiting each other every several days, while the time to cross the entire cluster is millions of years. This enormous range of time scales makes capturing both rapid and slow changes a huge challenge. Yet, capturing these time scales is vital in order to understand these system’s formation and evolution.

Enter the “N-body problem”—a large set of equations which describes the relative motion of all the massive gravitationally-interacting objects (numbering N) in the system. These problems are usually solved by numerical integration—approximations that calculate how each object changes position over a short time interval, then repeating this process to reveal how the system configuration advances in time. Using integrators for different time scales can introduce large numerical errors, which is what this new algorithm solves.

Hernandez, alongside fellow researcher Walter Dehnen, have created an algorithm that combines integrators while respecting a time-symmetry condition, a criterion that does not change the path of the solution if time is reversed. The technique was applied to planetary systems with highly eccentric orbits and close encounters between objects, demonstrating that errors could be reduced by several orders of magnitude with no appreciable extra computational work.

The publication can be found here.

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