History of the lunar orbit

A method of calculating the past states of the earth‐moon system is developed. The method is based on the existence of three distinct time scales for dynamical change. The short time scale is determined by the revolution periods of the sun and moon about the earth or, equivalently, by the year and current month. The intermediate time scale is set by the precessional motions of the lunar orbit plane and the earth's equator plane. The rate at which tidal friction alters the state of the earth‐moon system defines the long time scale. The equations of motion governing the earth‐moon system are successively averaged over the short and then the intermediate time scales. These averaged equations are then integrated back a short interval on the long time scale. The equations of motion appropriate to this new state of the earth‐moon system are then re‐averaged on the short and intermediate time scales, and once again the averaged eqations are stepped back on the tidal time scale. The first step in this procedure (i.e. averaging on the short time scale) is performed analytically, whereas the calculations on the intermediate and long time scales require the use of a large computer. At present, the inclination of the lunar orbit plane to the ecliptic remains nearly constant during the precessional motion. On the other hand, if the moon's semimajor axis were ever less than 10 R ⨁, the inclination of the lunar orbit plane would have maintained a fixed value with respect to the earth's equator plane. The current investigation shows that this inclination could never have been less than 10° and, therefore, that the moon could never have moved on an equatorial orbit. This result contradicts theories that postulate fission of the earth to form the moon and also those which propose that the moon formed by accretion within 10 R ⨁