Bodily Tides

Rosemary.A Mardling

doi:10.1051/epjconf/20101103002

Abstract

The equilibrium tide model in the weak friction approximation is used by the binary star and exoplanet communities to study the tidal evolution of short-period systems. However, each uses a slightly diﬀerent approach which potentially leads to diﬀerent conclusions about the timescales on which various processes occur. Here we present an overview of these two approaches, and show that for short-period planets the circularization timescales they predict diﬀer by at most a factor of a few. A discussion of the timescales for orbital decay, spin-orbit synchronization and spin-oribt alignment is also presented.

Highlights

The effects on an orbit due to tidal distortion and rotation can be divided into two categories: those associated with the viscosity of the fluid, and those associated with the non-spherical shape of the distorted body
The shape of the tidal bulge is obtained by assuming hydrostatic equilibrium within the distorted body, that is, the fluid’s internal pressure gradients and viscous stresses are exactly balanced by the gravitational force exerted both by the fluid body itself and the companion, implying there is no relative motion of neighbouring fluid elements and no viscous heat loss
As long as the orbit is noncircular, and/or the fluid body rotates at a rate different to the mean motion, and/or the rotation axis points in a different direction to the orbit normal, the assumption of hydrostatic equilibrium is only approximate

Summary

Two-body tidal evolution

The effects on an orbit due to tidal distortion and rotation can be divided into two categories: those associated with the viscosity of the fluid (and rheology in the case of a rigid component), and those associated with the non-spherical shape of the distorted body. Perhaps the most succinct and elegant derivation of the equations governing the variation of the orbital elements due to tidal friction in stars is given by Hut (1981) who follows Darwin in assuming a constant time lag and weak friction, but unlike Darwin and most others, uses energy and angular momentum arguments instead of a Fourier decomposition of the tidal field He shows that, in this approximation, one is. The sign of the rate of change of eccentricity is decided by the sign of the dominant contributing tidal component, this being the one with forcing frequency 2Ω − 3n,3 where Ω and n are the spin frequency and the mean motion respectively (Goldreich 1963, p261) As such the relevant forcing frequency for a synchronously rotating body (Ω = n) in a non-circular orbit is the mean motion itself (since 2Ω − 3n = −n in this case).. That if such a hot Jupiter is inflated its Love number will be reduced (as is the case if the planet has a substantial core), and this will further increase the circularization timescale

Timescales

Conclusion