On the secondary’s rotation in a synchronous binary asteroid

Abstract

This paper studies the secondary's rotation in a synchronous binary asteroid system in which the secondary enters the 1:1 spin-orbit resonance. The model used is the planar full two-body problem composed of a spherical primary plus a tri-axial ellipsoid secondary. Compared with classical spin-orbit work, there are two differences: (1) Influence on the mutual orbit from the secondary's rotation is considered; (2) Instead of the Hamiltonian approach, the approach of periodic orbits is adopted. Our studies find: (1) Genealogy of the two families of periodic orbits is same as that of the families around triangular libration points in the restricted three-body problem. That is, the long-period family terminates onto a short-period orbit travelling N times; (2) In the limiting case where the secondary's mass is negligible, our results can be reduced to the classical spin-orbit theory, by equating the long-period orbit with the free libration, and by equating the short-period orbit with the forced libration caused by orbit eccentricity. However, the two models show obvious differences when the secondary's mass is non-negligible. (3) By studying the stability of periodic orbits, for a specific binary asteroid system, we are able to obtain the maximum libration amplitude of the secondary (which is usually less than 90°), and the maximum mutual orbit eccentricity which does not break the secondary's synchronous state. We also find the anti-correlation between the secondary's libration amplitude and the orbit eccentricity. The (65803) Didymos system is taken as an example to show the results.

Breakup of the Synchronous State of Binary Asteroid Systems

“…The dynamical model used in this article is described in this section. The current work is a continuation of the authors' previous work (Wang & Hou 2020) which studied the correlation between the mutual orbit eccentricity and secondary's synchronous rotation state. Compared with the previous work, there are two differences in the force model used in the current work: (1) Tidal torque and BYORP torque are considered, which cause the migration of the binary asteroid system (BAS); (2) Non-spherical terms of the primary are also considered.…”

“…Moreover, we assume that the two asteroids are rotating along their shortest axis and their mutual orbital planes coincides with their rotation plane. This is the well-known planar two-ellipsoid model used in many previous studies (Ureche 1974(Ureche , 1978Bellerose & Scheeres 2008;Hou & Xin 2017;Hou et al 2020;Wang & Xu 2018). Fig.…”

“…This equation has the same form as the sphere-ellipsoid model used in our previous work (Wang & Hou 2020), but the coefficients contain the averaged terms from the primary's gravity.…”

“…Stability of the equilibrium points can be determined by eigenvalues of the variation matrix associated with the equilibrium points. The variational matrix takes the form of Stability of the equilibrium points can be different depending on the sign of 𝑠 1 and 𝑠 2 (see (Wang & Hou 2020) for more details). For the synchronous BAS in which the primary is usually much larger than the secondary, the long-axis mode is usually stable while the shortaxis mode is usually unstable, i.e., the secondary in the synchronous BAS usually has its long axis pointing at the primary.…”

“…A natural question is the how large the stable region of the synchrous state is for a synchronous BAS. In a previous paper by the authors (Wang & Hou 2020), using the sphere-ellipsoid (primary-secondary) model, this problem is studied via the approach of periodic orbits. In the body-fixed frame of the ellipsoid secondary, the synchronous rotation of the secondary can be interpretated as quasi-periodic orbits of the primary in this frame.…”