Abstract
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.
Highlights
In celestial mechanics, a classical problem is to model the dynamics of two rigid, extended bodies under mutual gravitational attraction
We explore the application of a surface integration method to compute the force and mutual gravitational potential between two extended, rigid bodies
By assuming that the gravitational potential of one body can be analytically expressed as a spheroid (MacMillan 1930), and integrating over a spheroid assumed to be the second body, we solve the equations of motion to test the method in a few simple planar cases
Summary
A classical problem is to model the dynamics of two rigid, extended bodies under mutual gravitational attraction. In the most general case, the bodies have arbitrary shapes and can have both translational and rotational motion, yielding twelve degrees of freedom. To model such a system is computationally expensive, and simplifications and approximations are commonly made. During the last 20 years, there has been renewed interest in the extended two-body problem in astronomy as binary asteroids have been discovered and studied in detail (e.g., Margot et al 2002). To describe the dynamics of such a system requires the full two-body problem to be solved
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