The evolution of the lunar orbit revisited. I

Abstract

After recalling the contribution of Halley, J. Kepler, and G. Darwin to our understanding of the secular acceleration of the Moon, we establish a set of differential equations for the variation of the semi-major axis, and the inclination of the Moon on the maximum area plane. These equations are obtained without expanding the disturbing function, due to the tidal bulge, in term of the elliptic elements. The equations thus obtained are simple enough to allow us a qualitative discussion of the solution, followed by a numerical integration.

The results obtained show the Moon was in the distant past in a retrograde orbit, approaching the Earth, its inclination increasing towards 90°; once after a closer approach to the Earth, the Moon receeded and it will finally reach an equilibrium point, the orbital and the equatorial planes being blended.

The solution of the equations appears as a fascicle of curves, becoming extremely dense as we come nearer to the present. Owing to the high sensitivity of the solution to the initial conditions, a weak disturbance added to our modeled forces may lead to a past situation very different from the conclusion drawn by Goldreich (1966) and MacDonald (1964); the minimal approach distance could be greater than 10 Earth's radii.