Tidal effects and the motion of a satellite

Abstract

The effect of resonant planetary perturbations on the evolution of the orbit of a satellite driven by tidal forces is studied in this paper. The basic equations that govern it are similar to the equations found in orbit-orbit and in spin-orbit couplings. The general form of these equations is:

A general treatment of such equations, proposed earlier (J. Kovalevsky, in Dynamical Trapping and Evolution of the Solar system, IAU Colloquium n^o74, V. V. Markellos and Y. Kozai, eds., 1983) is sketched.

In particular, the effects of the large long periodic variations of the excentricity e' of the planet are analysed on an example taken from the lunar theory and the Earth's general theory due to Bretagnon.

The argument of the well known planetary term θ=18 V-16Tℓ due to the tidal friction and quasi-periodic variations due to the presence of e' in the expression of the mean motion of the Moon. Their joint effect, has been to produce in the past resonant situations for this argument that repeated more than 100 times. Every such situation can be treated by equation (1).

Numerical integration, using conditions that might have occurred while θ or similar other arguments were quasi resonant, have produced the following results: (a) In some cases, the argument becomes temporarily resonant. Between the capture to and the escape from the resonance, the semi-major axis undergoes oscillations, but the tidal secular evolution is stopped. (b) In other cases, the argument is not trapped into a resonant conditions, but the semi-major axis undergoes a quick change while dθ/dt is close to zero.

A number of arguments that have been quasi resonant in the past history of the Earth-Moon system has been identified from the Chapront and Chapront-Touzé Lunar Theory. It appears that the phenomena described are frequent features in the evolution of the Lunar orbit.