Tidal friction in close-in satellites and exoplanets: The Darwin theory re-visited

Abstract

This report is a review of Darwin's classical theory of bodily tides in which we present the analytical expressions for the orbital and rotational evolution of the bodies and for the energy dissipation rates due to their tidal interaction. General formulas are given which do not depend on any assumption linking the tidal lags to the frequencies of the corresponding tidal waves (except that equal frequency harmonics are assumed to span equal lags). Emphasis is given to the cases of companions having reached one of the two possible final states: (1) the super-synchronous stationary rotation resulting from the vanishing of the average tidal torque; (2) capture into the 1:1 spin-orbit resonance (true synchronization). In these cases, the energy dissipation is controlled by the tidal harmonic with period equal to the orbital period (instead of the semi-diurnal tide) and the singularity due to the vanishing of the geometric phase lag does not exist. It is also shown that the true synchronization with non-zero eccentricity is only possible if an extra torque exists opposite to the tidal torque. The theory is developed assuming that this additional torque is produced by an equatorial permanent asymmetry in the companion. The results are model-dependent and the theory is developed only to the second degree in eccentricity and inclination (obliquity). It can easily be extended to higher orders, but formal accuracy will not be a real improvement as long as the physics of the processes leading to tidal lags is not better known.

Tidal Venuses: Triggering a Climate Catastrophe via Tidal Heating

“…As described below, this approach leads to a set of six coupled, nonlinear differential equations, but note that the model is, in fact, linear in the sense that there is no coupling between the surface waves which sum to the equilibrium shape. A substantial body of research is devoted to tidal theory (e.g., Hut, 1981;Ferraz-Mello et al, 2008;Wisdom, 2008;Efroimsky and Williams, 2009;Leconte et al, 2010), and the reader is referred to these studies for a more complete description of the derivations and nuances of tidal theory. For this investigation, we will use the models and nomenclature of Heller et al (2011), which are presented below.…”

“…If the orbits are noncircular, as for many exoplanets , then tidally evolved planets may reach an equilibrium state where they rotate faster than synchronous with an ''equilibrium'' or ''pseudo-synchronous'' period. This aspect of tidal theory has been known for decades (e.g., Goldreich, 1966;Greenberg and Weidenschilling, 1984) but has only recently been pointed out for the case of exoplanets (Barnes et al, 2008;Correia et al, 2008;Ferraz-Mello et al, 2008). Therefore, some exoplanets, such as Gl 581 d with an eccentricity of 0.38 , may be ''tidally locked'' but rotate about twice per orbit (Barnes et al, 2008;Heller et al, 2011).…”

Asymptotic Behavior of an Elastic Satellite with Internal Friction

“…Take now the derivative of such a quantity with respect to x a , one gets the componentwise equation 11) or using the invertibility of the matrix ∂ξ k ∂x ȧ…”

“…Darwin's work was subsequently generalized by Kaula [13] and many other authors (for instance, [2,12,17,19]). Critical reviews of the work by Darwin, Kaula and followers can be found in [9][10][11]. However, the Darwin-Kaula procedure is heuristic and, from a mathematical point of view, its range of validity is far from being clear.…”

Uncertainties in tidal theory: Implications for bloated hot Jupiters

“…This quasi circular approximation, developed initially to study the tidal evolution of the solar system planets (Goldreich & Soter 1966;Ferraz-Mello, Rodríguez & Hussmann Uncertainties in tidal theory 249 2008), which have negligible eccentricities, is valid only in this very limit, e 1. In the context of exoplanetary systems, current high eccentricities are common and initial high eccentricities are very likely, as inferred from non-transiting planets observed by radial velocity.…”