Journal of Conference Abstracts

Newtonian Description of Tidal Force and Laplace Formula

Boris Levin (levin@rfbr.ru) & Jurij Avsjuk (levin@rfbr.ru)

P.P. Shirshov Institute of Oceanology, RAS, Nakhimovsky prosp., 36, Moscow, 117851, Russia

The description of the tidal force is a problem of dynamics and not statics. Therefore it is of principal importance to discuss Laplace remark: "... Newton has argued that in order to calculate the maximum tide raise, the Sun's action must be multiplied by a cosine of twice the synodic Moons motion over appropriate time period. But this correction is not valid..." (Krylov, 1936, p.589).

Newton speaks of the "Sun's action", and this additional term is nothing else but one of the most significant terms, describing the perturbations of the Moon's orbital motion within the Sun's field. "If we consider that the Earth and the Moon rotate around a common center of the mass, than the Earth's motion perturbations are governed by the same forces" [Sentence XXV, Problem VI]. For this reason, the tidal forcing, leading to actual motion of the object under study (the Earth), must be treated as a sum of several terms: the first one corresponds to the unperturbed (Kepler) characteristic of the tidal force and other terms define the perturbations. Newton described the tidal force in step by step fashion. Firstly, to demonstrate the relation of the tides to Law of Universal Gravitation, Newton investigated the case of the external object motion in Kepler (unperturbed) orbit [Book I, Sentence LXV, Theorem XXV]. According to that time traditions his analyses was based on geometric constructions.

Specifically for this part of Newtonian description Laplace has produced an analytical formulation. Laplace formula characterizes the tidal forcing for the case of gravitational interaction of two objects moving in Kepler (unperturbed) orbit. Therefore the Newton's correction is valid and is an important contribution to tidal forcing description.

This presentation deals with a variety of natural processes, that can be explained only by a full description of Newtonian tidal force and for which Laplace formula proves to be insufficient. For example, Earth-Moon system in respect to the ratio of perturbed and unperturbed parts of the tidal force (which equals to 0.34) is unique in the Solar system. For Jupiter-Io system the ratio is -4 power of 10. The tidal evolution of planet-satellite system is different from the evolution of planet with no satellite system. For the Earth-Moon system the evolution can be oscillatory. The detailed description of some geophysical processes, described in terms of full Newtonian tidal forcing theory can be found in several papers (Avsjuk, 1996; Levin, 1996).

Avsjuk JN, Tidal forces and natural processes, M., (1996), p.189 (in russian)

Krylov AN, Proceedings of academician A.N.Krylov. M.-L., AS USSR, (1936).

Levin BW, CHAOS, 6(3),405-413, (1996)