Tidal De-spin of Planets and Satellites

Abstract

RECENTLY, Peale and Gold¹ have shown that the non-synchronous rotation of Mercury is likely to be a consequence of tidal friction. They point out that in an eccentric orbit the spin of an axially symmetric planet will not relax to the orbital mean motion, but instead will approach a final value which is somewhat larger. The final spin rate will be somewhere between the mean orbital angular velocity and the orbital angular velocity at perihelion. The precise value for the final spin is determined by the condition that the net tidal torque on the planet around each orbit be equal to zero. The spin rate at which this condition is satisfied is determined by the frequency and amplitude dependence of the planet's ‘Q’ (1/Q is the specific dissipation function²). According to Peale and Gold: “The condition discussed here is based on the supposition that the solar torque exerted on the tidal bulge exceeds that exerted on any permanent deformation from axial symmetry. In the converse case a period of 88 days for the rotation would indeed result.” This last statement, if true, would imply that Mercury's principal moments of inertia, in the plane perpendicular to its spin axis, differ by less than a few parts in 10⁷. This value is very small when compared with the values known for the Moon—another solid, slowly rotating body.