Tidal Dissipation in Rotating Giant Planets

Abstract

Many extrasolar planets orbit sufficiently close to their host stars that significant tidal interactions can be expected, resulting in an evolution of the spin and orbital properties of the planets. The accompanying dissipation of energy can also be an important source of heat, leading to the inflation of short-period planets and even mass loss through Roche lobe overflow. Tides may therefore play an important role in determining the observed distributions of mass, orbital period, and eccentricity of the extrasolar planets. In addition, tidal interactions between gaseous giant planets in the solar system and their moons are thought to be responsible for the orbital migration of the satellites, leading to their capture into resonant configurations. Traditionally, the efficiency of tidal dissipation is simply parameterized by a quality factor Q , which depends, in principle, in an unknown way on the frequency and amplitude of the tidal forcing. In this paper we treat the underlying fluid dynamical problem with the aim of determining the efficiency of tidal dissipation in gaseous giant planets such as Jupiter, Saturn, or the short-period extrasolar planets. Efficient convection enforces a nearly adiabatic stratification in these bodies, which may or may not contain solid cores. With some modifications, our approach can also be applied to low-mass stars with extended convective envelopes. In cases of interest, the tidal forcing frequencies are typically comparable to the spin frequency of the planet but are small compared to its dynamical frequency. We therefore study the linearized response of a slowly and possibly differentially rotating planet to low-frequency tidal forcing. Convective regions of the planet support inertial waves, which possess a dense or continuous frequency spectrum in the absence of viscosity, while any radiative regions support generalized Hough waves. We formulate the relevant equations for studying the excitation of these disturbances and present a set of illustrative numerical calculations of the tidal dissipation rate. We argue that inertial waves provide a natural avenue for efficient tidal dissipation in most cases of interest. In the presence of a solid core, the excited disturbance tends to be localized on a web of rays rather than resembling a smooth eigenfunction. The resulting value of Q depends, in principle, in a highly erratic way on the forcing frequency, but we provide analytical and numerical evidence that the frequency-averaged dissipation rate may be asymptotically independent of the viscosity in the limit of small Ekman number. For a smaller viscosity, the tidal disturbance has a finer spatial structure and individual resonances are more pronounced. In short-period extrasolar planets, tidal dissipation via inertial waves becomes somewhat less efficient once they are spun down to a synchronous state. However, if the stellar irradiation of the planet leads to the formation of a radiative outer layer that supports generalized Hough modes, the tidal dissipation rate can be enhanced, albeit with significant uncertainty, through the excitation and damping of these waves. The dissipative mechanisms that we describe offer a promising explanation of the historical evolution and current state of the Galilean satellites, as well as the observed circularization of the orbits of short-period extrasolar planets.