The planar two-body problem for spheroids and disks
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2021Metadata
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Wold, M. Conway, J. T. (2021). The planar two-body problem for spheroids and disks. Celestial mechanics & dynamical astronomy, 133. https://doi.org/10.1007/s10569-021-10023-xAbstract
We outline a new method suggested by Conway (CMDA 125:161–194, 2016) for solving the
two-body problem for solid bodies of spheroidal or ellipsoidal shape. The method is based on
integrating the gravitational potential of one body over the surface of the other body. When
the gravitational potential can be analytically expressed (as for spheroids or ellipsoids), the
gravitational force and mutual gravitational potential can be formulated as a surface integral
instead of a volume integral and solved numerically. If the two bodies are infinitely thin disks,
the surface integral has an analytical solution. The method is exact as the force and mutual
potential appear in closed-form expressions, and does not involve series expansions with
subsequent truncation errors. In order to test the method, we solve the equations of motion
in an inertial frame and run simulations with two spheroids and two infinitely thin disks,
restricted to torque-free planar motion. The resulting trajectories display precession patterns
typical for non-Keplerian potentials. We follow the conservation of energy and orbital angular
momentum and also investigate how the spheroid model approaches the two cases where the
surface integral can be solved analytically, i.e., for point masses and infinitely thin disks.