Abstract
We study analytically and experimentally certain symplectic and time-reversible N-body integrators which employ a Kepler solver for each pair-wise interaction, including the method of Hernandez & Bertschinger (2015). Owing to the Kepler solver, these methods treat close two-body interactions correctly, while close three-body encounters contribute to the truncation error at second order and above. The second-order errors can be corrected to obtain a fourth-order scheme with little computational overhead. We generalise this map to an integrator which employs a Kepler solver only for selected interactions and yet retains fourth-order accuracy without backward steps. In this case, however, two-body encounters not treated via a Kepler solver contribute to the truncation error.
Highlights
The gravitational N-body problem has been studied ever since Newton first wrote down his universal gravitational law of attraction
The N-body problem appears often in dynamical astronomy, for example planetary systems, stellar associations, star clusters, galaxies, dark matter haloes and even the Universe as a whole can be modelled to good approximation as N-body problems (Heggie & Hut 2003), other, typically less-accurate, alternative models are possible in some cases
A major problem arises from the dynamical stiffness of these systems in the sense that the relevant time-scales differ by orders of magnitude: already a simple elliptic or hyperbolic orbit poses problems for numerical integration owing to the large variation of angular speed, i.e. of the local orbital time-scale
Summary
The gravitational N-body problem has been studied ever since Newton first wrote down his universal gravitational law of attraction. We are instead concerned with the collisional N-body problem, which emerges for example when modelling the planetary systems including our own, planetesimals in a circum-stellar disc, or a globular cluster In this case, the accurate long-term time integration of the unsoftened gravitational forces poses a formidable problem. Switching methods, which change between different symplectic integrators, have been proposed for the study of single-star planetary systems (Duncan, Levison & Lee 1998; Chambers 1999; Kvaerno & Leimkuhler 2000) Tests indicate these methods may break time-reversibility and symplecticity (Hernandez 2016). The resulting method involves the solution of a large implicit system of equations requiring an excessive amount of computational effort and has not been used in practice Because of these complications, contemporary methods for the integration of planetary systems employ a fixed global time step. The appendices provide some detailed calculations and discuss implementation details
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