Asteroid Geophysics through a Tidal-BYORP Equilibrium

There exists a long-term stable orbital equilibrium for singly synchronous binary asteroids balancing the contractive BYORP (binary Yarkovsky-O'Keefe-Radzievskii-Paddack) effect and the expansive tidal torque from the secondary onto the primary [Jacobson & Scheeres 2011]. Observations of 1996 FG₃ determined that this object is consistent with occupying the predicted equilibrium [Scheirich, et al., 2012]. From the torque balance, the important tidal parameters of the primary and BYORP coefficient of the secondary can be directly determined for the first time, albeit degenerately. Singly synchronous systems consist of a rapidly spinning primary and a tidally locked secondary. Two torques evolve the mutual orbit of the system. First, the secondary raises a tidal torque on the primary, and this process expands the semi-major axis of the mutual orbit according to two parameters. The tidal Love number k is related to the strength (rigidity) of the body. The tidal dissipation number Q describes the mechanical energy dissipation. Second, the BYORP torque is the summed torques from all of the incident and exigent photons on the secondary acting on the barycenter of the system. Unless there is a spin-orbit resonance, the torques sum to zero. McMahon & Scheeres [2010] showed that showed that to first order in eccentricity the evolution of the semi-major axis and eccentricity depends only upon a single constant coefficient B determined by the shape of the secondary (size-independent). The BYORP torque can either contract or expand the mutual orbit, however it evolves the eccentricity with the opposite sign. Jacobson & Scheeres [2011] determined that when the BYORP torque is contractive, it can balance the expansive tidal torque. The system evolves to an equilibrium semi-major axis that is stable in eccentricity due to tidal decay overcoming BYORP excitation. If the singly synchronous population occupies this equilibrium, then the three unknown (i.e. unobserved) parameters: B_s Q_{p/k_p}, as shown in the figure. Since the BYORP coefficient is defined to be size independent, the tidal parameters Q_{p/k_p} ∝ R_p. This inverse dependence is different than the predicted dependencies of the classical tidal Love number k_p ∝ R_p² and the ``rubble-pile'' tidal Love number predicted in Goldreich & Sari [2009] k_p ∝ R_p. Calculated B_s Q_p/ k_p for each observed singly synchronous binary asteroid system. The circled system is 1996 FG₃. The solid line is the fit B_s Q_{p/k_p} = 2557 R_p and the dashed lines are a facto r of 10 and a factor of 0.01 different.