The Equilibrium Tide Model for Tidal Friction

We derive from first principles the equations governing (a) the quadrupole tensor of a star distorted both by rotation and by the presence of a companion in a possibly eccentric orbit; (b) a functional form for the dissipative force of tidal friction, based on the concept that the rate of energy loss from a time-dependent tide should be a positive-definite function of the rate of change of the quadrupole tensor as seen in the frame that rotates with the star; and (c) the equations governing the rates of change of the magnitude and the direction of the stellar rotation, the orbital period and eccentricity, based on the concept of the Laplace-Runge-Lenz vector. Our analysis leads relatively simply to a closed set of equations, valid for arbitrary inclination of the stellar spin to the orbit. The results are equivalent to classical results based on the rather less clear principle that the tidal bulge lags behind the line of centers by some time determined by the rate of dissipation. Our analysis gives the effective lag time as a function of the dissipation rate and the quadrupole moment. We discuss briefly some possible applications of the formulation.