Tidal Evolution of Asteroidal Binaries. Ruled by Viscosity. Ignorant of Rigidity

Abstract

The rate of tidal evolution of asteroidal binaries is defined by the dynamical Love numbers divided by quality factors. Common is the (often illegitimate) approximation of the dynamical Love numbers with their static counterparts. As the static Love numbers are, approximately, proportional to the inverse rigidity, this renders a popular fallacy that the tidal evolution rate is determined by the product of the rigidity by the quality factor: $k_l/Q\propto 1/(\mu Q)$. In reality, the dynamical Love numbers depend on the tidal frequency and all rheological parameters of the tidally perturbed body (not just rigidity). We demonstrate that in asteroidal binaries the rigidity of their components plays virtually no role in tidal friction and tidal lagging, and thereby has almost no influence on the intensity of tidal interactions (tidal torques, tidal dissipation, tidally induced changes of the orbit). A key quantity that determines the tidal evolution is a product of the effective viscosity $\eta$ by the tidal frequency $\chi$. The functional form of the torque's dependence on this product depends on who wins in the competition between viscosity and self-gravitation. Hence a quantitative criterion, to distinguish between two regimes. For higher values of $\eta \chi$ we get $k_l/Q\propto 1/(\eta \chi )$; while for lower values we obtain $k_l/Q\propto \eta \chi$. Our study rests on an assumption that asteroids can be treated as Maxwell bodies. Applicable to rigid rocks at low frequencies, this approximation is used here also for rubble piles, due to the lack of a better model. In the future, as we learn more about mechanics of granular mixtures in a weak gravity field, we may have to amend the tidal theory with other rheological parameters, ones that do not show up in the description of viscoelastic bodies.

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