Abstract
We present a new mixed variable symplectic (MVS) integrator for planetary systems that fully resolves close encounters. The method is based on a time regularisation that allows keeping the stability properties of the symplectic integrators while also reducing the effective step size when two planets encounter. We used a high-order MVS scheme so that it was possible to integrate with large time-steps far away from close encounters. We show that this algorithm is able to resolve almost exact collisions (i.e. with a mutual separation of a fraction of the physical radius) while using the same time-step as in a weakly perturbed problem such as the solar system. We demonstrate the long-term behaviour in systems of six super-Earths that experience strong scattering for 50 kyr. We compare our algorithm to hybrid methods such as MERCURY and show that for an equivalent cost, we obtain better energy conservation.
Highlights
Precise long-term integration of planetary systems is still a challenge today
Because of the chaotic nature of planetary dynamics, statistical studies are often necessary, which require running multiple simulations with close initial conditions (Laskar & Gastineau 2009). This remark is true for unstable systems that can experience strong planet scattering caused by close encounters
Symplectic schemes incorporate the symmetries of Hamiltonian systems, and as a result, usually conserve the energy and angular momentum better than non-symplectic integrators
Summary
Precise long-term integration of planetary systems is still a challenge today. The numerical simulations must resolve the motion of the planets along their orbits, but the lifetime of a system is typically billions of years, resulting in computationally expensive simulations. High-order schemes permit a very good control of the numerical error by fully taking advantage of the hierarchical structure of the problem This has been used with success to carry out high-precision long-term integrations of the solar system (Farrés et al 2013). Dehnen & Hernandez 2017) It consists of a fixed time-step second-order symplectic integrator that treats every interaction between pairs of bodies as Keplerian steps. Another way to build a symplectic integrator that correctly regularises close encounters is time renormalisation. We show the results of long-term integration of six planet systems in the context of strong planet scattering (Sect. 7) and compare our method to a recent implementation of MERCURY described in Rein et al (2019), to SYMBA, and to the non-symplectic high-order integrator IAS15 (Sect. 8)
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