Numerical inference of viscoelastic properties in tidal models of rubble pile asteroids

Tidal theory in binary asteroids is in a low state of development in comparison to that of planetary satellites. The dissipative processes within binary secondaries are divergent from the processes at work in planetary satellites, and also the evolutionary timescales are drastically shortened. To study the tidal torques and the spin evolution of the smaller secondaries in binary asteroid systems, it is believed to still be possible to apply a viscoelastic theory somewhat analogous to the models used for planetary satellites (see e.g. Murray & Dermott 1999). If this is true, then there should exist body-averaged “bulk” material properties, such as rigidity and viscosity, applicable for these models. Importantly, it should be possible to compute effective k2 (tidal potential Love number) and Q (quality factor) values for these tiny worlds.

Some pioneering works have derived first-order tidal laws for binary asteroids via analytic and semi-analytic methods. Nimmo & Matsuyama (Icarus 2019) derive a friction-driven effective quality factor Q which decreases (i.e. more dissipation) with stronger friction. Goldreich & Sari (The Astrophysical Journal 2009) argue that effective rigidity is dynamically dominant, deriving an important law for effective k2 for asteroids that scales linearly with the asteroid radius. They also argue that effective Q could be quite low for asteroids, lower than prior estimates of Q ~10². By contrast, Efroimsky (The Astronomical Journal 2015) argues that effective viscosity is dominant and rigidity doesn’t matter. Recently, Pou and Nimmo (Icarus 2024) showed that k2/Q values implied by the ages of some binary asteroids are much lower than the values predicted by Goldreich & Sari (2009), suggesting that the theory of the latter is incomplete.

In this work, we follow up on the aforementioned theoretical works with numerical experiments of binary asteroid tidal evolution, which have strong scientific motive to be carried out. This is accomplished using the massive N-body simulation architecture of GRAINS (Ferrari et al, MNRAS 2019), wherein gravitational, contact, and frictional effects are modeled in the interaction of thousands of non-spherical mass elements. We initialize a simplified binary system analogous somewhat to the scenarios employed in Agrusa et al (PSJ, 2022). To facilitate tidal locking, the static moment-of-inertia asymmetry is made sufficiently large, the secondary is initialized in a state of slightly super-synchronous rotation, and inter-element friction is enhanced, if needed, to yield a dissipation timescale in line with our numerical capabilities (a move inspired by the approach of Goldreich, The Astronomical Journal 1966). From our simulation results, we perform the following analysis:

• A simple regression analysis infers the effective k2/Q from the computed rotational energy dissipation rate, via parallel application of the classical 1D MacDonald tidal dissipation model.
• Previously derived scaling laws for effective k2 and Q are tested.
• The accuracy of the MacDonald tidal torque model for binary rubble pile asteroids is tested.
• With time permitting, if the (unimodal) MacDonald tidal model is shown to be inaccurate, we'll explore computation of an appropriate multimodal tidal potential, as in Darwin-Kaula theory.