The Equilibrium Tide Model for Tidal Friction

We derive from first principles equations governing (a) the quadrupole tensorof a star distorted by both rotation and the presence of a companion in apossibly eccentric orbit, (b) a functional form for the dissipative force oftidal friction, based on the concept that the rate of energy loss from atime-dependent tide should be a positive-definite function of the rate ofchange of the quadrupole tensor as seen in the frame which rotates with thestar, and (c) the equations governing the rates of change of the magnitude anddirection of the stellar rotation, and the orbital period and eccentricity,based on the concept of the Laplace-Runge-Lenz vector. Our analysis leadsrelatively simply to a closed set of equations, valid for arbitrary inclinationof the stellar spin to the orbit. The results are equivalent to classicalresults based on the rather less clear principle that the tidal bulge lagsbehind the line of centres by some time determined by the rate of dissipation;our analysis gives the effective lag time as a function of the dissipation rateand the quadrupole moment. We discuss briefly some possible applications of the formulation.Comment: LaTeX, 1 postscript figure included, ApJ, accepte