Lunar theory

theoretical description of motion of Earth's moon

In physics and mathematics, lunar theory is the physical and mathematical theory of the motion of the Earth's Moon as deduced from the principles of gravitational theory (especially the Newtonian, non-relativistic theory).

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  • The general problem of Celestial Mechanics consists in the determination of the relative motions of p bodies attracting one another according to the Newtonian law. This problem is not able to be solved directly: in order to deal with it, certain limitations must be made.
    Again, owing to the conditions under which the bodies of our solar system move, we are further able to divide the problem of p bodies into several others, each of which may be treated as a case of the problem of three particles, or, as it is generally called, the Problem of Three Bodies.
    The greater part of the Lunar Theory is a particular case of the Problem of Three Bodies; it involves the determination of the motion of the Moon relative to the Earth, when the mutual attraction of the Earth, Moon and Sun, considered as particles, are the only forces under consideration. When this has been found, the effects produced by the actions of the planets, the non-spherical forms of the bodies etc., can be be exhibited as small corrections to the coordinates.
  • In Book I, Prop. LXVI of the first edition of Philosophiae naturalis principia mathematica, Newton (1687) discussed the dynamical problem of three bodies in a general way, and then in Book III he asserted that the vagaries of the Moon's motion could be accounted for by the gravitational attraction of the Sun. He recognized that he needed to develop the theory further, and summarized his later results in The theory of the Moon's motion of 1702 (Cohen 1975). He continued to refine his treatment up to the publication of the second edition of Principia (Newton 1712), some sections of which differ greatly from the first edition. He made almost no further changes of his own in the third edition, but added a scholium by Machin (1726) on the motion of the nodes. The published account of the rotation of the apse line, much the same in all versions, was seriously wrong, but even before 1690 Newton had developed a somewhat more satisfactory treatment, with which, however, he remained dissatisfied and never published (Whiteside 1976). (Since this article was prepared, the new English translation of the Principia by Cohen and Whitman (1999) has appeared. It is a translation of the third edition of 1726, which differs significantly in a few places from the first and second editions, as will be indicated.)
  • There are three essentially different types of lunar theory — that of de Pontécoulant, that of Delaunay, and that first developed by Hill, to which may perhaps be added that of Hansen as containing many features of more or less importance different from the others. That of de Pontécoulant and most of his predecessors consists in developing certain coordinates in periodic series of assumed form with the time or true anomaly as argument and determining the coefficients step by step as powers of the small parameters involved ; that of Delaunay consists in applying the method of the variation of parameters in the canonical form over and over in such a way as to remove the most important parts of the perturbative function ; that of Hill consists in finding very accurate particular solutions of the differential equations after the parts depending on the parallax of the sun, the eccentricity of the earth's orbit, and the latitude of the moon have been neglected, and then finding the deviations from this orbit due to general initial conditions and the neglected part of the perturbative function.

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