NoteTidal end states of binary asteroid systems with a nonspherical component
Introduction
Recent studies have examined energy, stability, and orbital relative equilibria in the planar two-body problem for a non-rotating sphere and an arbitrary, rotating ellipsoid (Scheeres, 2007, Bellerose and Scheeres, 2008) and approximately for two arbitrary, rotating ellipsoids (Scheeres, 2009). Here, we examine the special case of a rotating sphere interacting with a rotating oblate or prolate spheroid and provide exact, tractable analytical results for the locations of the fully synchronous end states of tidal evolution. The terms fully synchronous tidal end state and orbital relative equilibrium can be used interchangeably to describe a zero-eccentricity binary system that has ceased tidally evolving because the spin rates of both components have synchronized to the mean motion of the components about the center of mass of the system.
This note is organized as follows. In Section 2, we review fully synchronous tidal end states of a binary system consisting of two spheres. Section 3 extends the discussion to a sphere interacting with an ellipsoid and explores the specific cases of oblate and prolate spheroids with applications to real asteroid systems. Comparisons to previous work in Sections 3 Fully synchronous orbits with a nonspherical component, 4 Discussion place this work in context and possible avenues for contact-binary formation are suggested.
Section snippets
Fully synchronous orbits with spherical components
The locations of the fully synchronous end states of tidal evolution for binary asteroids with spherical components were discussed by Taylor and Margot (2011) and are summarized here. For components of equal, uniform density with radii and and mass ratio separated by a distance a in their circular mutual orbit, the sum of the orbital and spin angular momentum J upon full synchronization, scaled by , where is the effective radius of a sphere with the
Fully synchronous orbits with a nonspherical component
Let component 1 of the binary system be a uniform-density ellipsoid with principal semi-axes such that the equivalent radius of the ellipsoid is . For rotation about the shortest principal axis, the ratio of the moment of inertia of the ellipsoid to that of its equivalent-volume sphere with radius is the nonsphericity parameter (Descamps and Marchis, 2008):where , and . The nonsphericity parameter is always larger than unity because
Discussion
We have presented analytical formulae for contours of constant angular momentum for ensembles of binary systems consisting of an oblate/prolate component and a spherical component that, in turn, determine the fully synchronous tidal end states for specific binary systems according to their mass fractions. In retaining the spin angular momentum of the sphere, we extend the results of Bellerose and Scheeres (2008). In general, the presence of a nonspherical component breaks the symmetry about
Acknowledgement
This material is based upon work supported by the National Aeronautics and Space Administration under Grant No. NNX12AF24G issued through the Near-Earth Object Observations Program.
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