Tidal synchronization of close-in satellites and exoplanets. A rheophysical approach

Abstract

This paper presents a new theory of the dynamical tides of celestial bodies. It is founded on a Newtonian creep instead of the classical delaying approach of the standard viscoelastic theories and the results of the theory derive mainly from the solution of a non-homogeneous ordinary differential equation. Lags appear in the solution but as quantities determined from the solution of the equation and are not arbitrary external quantities plugged in an elastic model. The resulting lags of the tide components are increasing functions of their frequencies (as in Darwin's theory), but not small quantities. The amplitudes of the tide components depend on the viscosity of the body and on their frequencies; they are not constants. The resulting stationary rotations (pseudo-synchronous) have an excess velocity roughly proportional to $6ne^2/(\chi^2+1/\chi^2)$ ($\chi$ is the mean-motion in units of one critical frequency - the relaxation factor - inversely proportional to the viscosity) instead of the exact $6ne^2$ of standard theories. The dissipation in the pseudo-synchronous solution is inversely proportional to $(\chi+1/\chi)$. In the inviscid limit it is roughly proportional to the frequency (as in standard theories), but that behavior is inverted when the viscosity is high and the tide frequency larger than the critical frequency. For free rotating bodies, the dissipation is given by the same law, but now $\chi$ is the frequency of the semi-diurnal tide in units of the critical frequency. This approach fails to reproduce the actual tidal lags on Earth. In this case, to reconcile theory and observations, we need to assume the existence of an elastic tide superposed to the creeping tide. The theory is applied to several Solar System and extrasolar bodies and currently available data are used to estimate the relaxation factor $\gamma$ (i.e. the critical frequency) of these bodies.