The earth and moon are considered as a two‐body system in gravitational isolation from the sun and other planets. The lunar orbit is taken as circular, and the solid earth is assumed to be a rigid sphere (with no tidal deformation) so that there are no precessional torques other than those arising from the tidally deformed ocean. Numerical solutions to Laplace's tidal equations, with dissipation by linear bottom friction, are used to obtain ocean‐wide distributions of tidal amplitude for two idealized continentalities: a single circular continent (spherical cap) centered at the north pole, and a single spherical cap centered on the equator. Calculations are made for two different values of the frictional resistance coefficient, thus giving rise to four sets of solutions. The computed tidal amplitudes are used to calculate the oceanic tidal torque, which in turn is used to integrate the orbital equations backward in time for solution of the two‐body problem. The Coriolis parameter and the tidal frequencies change with time, thus requiring that the tidal equations be solved several times during the course of each orbital integration. In this manner, the earth's rotational and the moon's orbital histories are determined on a geologic time scale for each of the four models. The calculated position of the lunar orbit at 4.5 billion years ago is found to range from 38 to 53 earth radii in the four models and corresponds to a sidereal month of 330 to 550 hours. The sidereal day would have been 12 to 18 hours, with a relative inclination of 3° to 22° between the terrestrial and lunar poles. These results are in sharp contrast to those from previous studies of the earth‐moon system, most of which indicated a Roche limit approach of the two bodies roughly 1 to 2 billion years ago and presented therefore a time scale difficulty in theories of lunar origin. This contrast arises mainly from the fact that previous modelers avoided solution of Laplace's tidal equations by prescribing a constant frictional phase lag angle between the angular position of the moon and the major axis of the second‐degree harmonic of the tidally deformed surface of the earth. The amount of phase lag was established from the present astronomically deduced rate of tidal dissipation, but this precluded dynamic variations in tidal torque over geologic time, which are critical for determination of the orbital time scale. The present rate of oceanic tidal dissipation is evidently anomalously high because of the near resonance of the oceanic response in both frequency and shape to the tidal forcing. For the earth and moon to have evolved to their present separation from a distance of less than about 35 earth radii solely on the basis of oceanic tidal dissipation would have required the M2 response to have been near resonance for a significant portion of geologic history, a rather implausible scenario. The calculations reported here show that, on the contrary, frictional coupling between the earth and moon was much weaker than at present throughout most of the orbital history, and consequently the moon's closest approach could not have been as small as 35 earth radii. Although idealized continentalities were used in these calculations, the general conclusion of a weaker frictional coupling in the past is believed to be relatively insensitive to the continental configuration. This is because at faster paleorotation, the semidiurnal tidal frequencies would be resonant only with smaller‐scale normal modes of the ocean rather than with global scale modes as at present. The time scale difficulty heretofore present in models of orbital evolution is thereby eliminated. It should be noted that the orbital history developed here supports the theory of binary planetary accretion for lunar origin as opposed to fission or tidal capture.
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Open Access
Jul 1, 1996
Milan Burša 
