Symplectic integrators in the shearing sheet

Abstract

The shearing sheet is a model dynamical system that is used to study the small-scale dynamics of astrophysical disks. Numerical simulations of particle trajectories in the shearing sheet usually employ the leapfrog integrator, but this integrator performs poorly because of velocity-dependent (Coriolis) forces. We describe two new integrators for this purpose; both are symplectic, time-reversible and second-order accurate, and can easily be generalized to higher orders. Moreover, both integrators are exact when there are no small-scale forces such as mutual gravitational forces between disk particles. In numerical experiments these integrators have errors that are often several orders of magnitude smaller than competing methods. The first of our new integrators (SEI) is well-suited for disks in which the typical inter-particle separation is large compared to the particles' Hill radii (e.g., planetary rings), and the second (SEKI) is designed for disks in which the particles are on bound orbits or the separation is smaller than the Hill radius (e.g., irregular satellites of the giant planets).

9 pages, 6 figures, accepted for publication in MNRAS, v2: discussion/tests for symmetrized and modified leapfrog integrators added

References in corpus (4)

• Implementation of the Shearing Box Approximation in Athena
• Algorithmic regularization with velocity-dependent forces
• Migration of a moonlet in a ring of solid particles : Theory and application to Saturn's propellers
• Stochastic orbital migration of small bodies in Saturn's rings

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• Switching integrators reversibly in the astrophysical $N$-body problem
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• The gravitational instability of a stream of co-orbital particles
• Radial Dependence of the Proto-Globular Cluster Contribution to the Milky Way Formation
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• Thermal hysteresis and front propagation in dense planetary rings