Tidal friction
Authors: MacDonald Gordon J. F. 
Published: February 2008

The perturbing potential due to the tidal interaction between the earth, sun, and moon is obtained without restrictive assumptions on the internal constitution of the earth. The derived potential is used in Gauss's equations for the time rate of change of the eccentricity, semimajor axis, and inclination of the moon's orbit. The tidal forces perturb the elements describing the earth's motion about the sun by a negligibly small amount. The variation in the earth's angular momentum due to the solar and lunar tides is described by Euler's equations. It is assumed that the moment of inertia of the earth corresponds to that appropriate for a rotating fluid with the density distribution of the present earth.

The rate of change of the moon's orbital elements is determined by the phase lag in the elastic component of the tidal bulge raised by the sun and moon. Astronomical data give a current value of the lag of the lunar tidal bulge of 2.16°. The present phase lag corresponds to a rate of increase of the semimajor axis of 3.2 cm yr−1, a rate of decrease of the inclination of the moon's orbital plane of 9.35 × 10−12 rad yr−1, and a rate of increase in the eccentricity of 1.2 × 10−10 yr−1. The obliquity of the earth's equator to the ecliptic and the period of rotation are increasing at present, owing to the combined effects of the lunar and solar tides. The effects of the solar tides are small at present but become important in the future.

Numerical integrations of the coupled Gauss‐Euler equations are used to describe both the past history and the future evolution of the earth‐moon system. If the moon traveled a circular orbit, a backward tracing of the history shows the moon approaching the earth, reaching a minimum distance of 2.72 present earth radii. During the time of closest approach, the inclination and obliquity change rapidly, with the moon's orbital plane passing over the pole and the moon's motion becoming retrograde. If the orbit is eccentric, the eccentricity increases rapidly at the time of close approach. The perigee height remains nearly constant, while the apogee increases without bounds over a time of about 1000 years. If the current phase lag remains constant, the time of closest approach is only 1.78 × 109 years ago. Thus, the current phase lag is not consistent with the hypothesis that the earth‐moon system has existed throughout geologic time.

The rotational parameters of Mars, Venus, and Mercury are discussed in terms of the dynamical theory. The distribution of rotational angular momentum of the solar system is described, and it is proposed that the major planets and Mars have lost only a very small proportion of their initial rotational angular momentum. The observed dependence of rotational angular momentum on planetary mass yields an estimate of the initial rotational period of the earth of between 9 and 13 hours.

The mechanisms by which the earth's rotational energy can be dissipated are reviewed. It is argued that the phase lag in the tides may have remained constant over geologic time or may have been somewhat larger. This conclusion raises the problem of the age of the earth‐moon system.

Theories of the origin of the moon are discussed against the requirements imposed by dynamical considerations. It is concluded that the theories of the fission of the earth to form the moon must be discarded, because the moon, once placed in the equatorial orbit, would remain there. Dynamics places severe constraints on the initial conditions for either a capture origin or the development of a binary system. The theory of several initial moons is briefly described, as are the climatological and tectonic problems raised by the consideration of the past history of the earth‐moon system.