Historical Background

The motion of the Planets around the Sun is fairly accurately described by Kepler's laws (see Chapter 5). Likewise, to a first approximation, the motion of the Moon around the Earth can also be accounted for via these laws. However, unlike the planetary orbits, the deviations of the lunar orbit from a Keplerian ellipse are sufficiently large that they are easily apparent to the naked eye. Indeed, the largest of these deviations, which is generally known as evection, was discovered in ancient times by the Alexandrian astronomer Claudius Ptolemy (90-168 AD). Moreover, the next largest deviation, which is called variation, was first observed by Tycho Brahe (1546-1601) without the aid of a telescope. Another non-Keplerian feature of the lunar orbit, which is sufficient obvious that it was known to the ancient Greeks, is the fact that the lunar perigee (i.e., the point of closest approach to the Earth) precesses (i.e., orbits about the Earth in the same direction as the Moon) at such a rate that it completes a full circuit every $8.85$ years. The ancient Greeks also noticed that the lunar ascending node (i.e., the point at which the Moon passes through the fixed plane of the Earth's orbit around the Sun from south to north) regresses (i.e., orbits about the Earth in the opposite direction to the Moon) at such a rate that it completes a full circuit every 18.6 years. Of course, according to standard two-body orbit theory, both the lunar perigee and ascending node should be stationary (see Chapter 5).

Newton demonstrated, in Book III of his Principia, that the deviations of the lunar orbit from a Keplerian ellipse are due to the gravitational influence of the Sun, which is sufficiently large that it is not completely negligible compared with the mutual gravitational attraction of the Earth and the Moon. However, Newton was not able to give a complete account of these deviations, due to the complexity of the equations of motion which arise in a system of three mutually gravitating bodies (see Chapter 13). In fact, Clairaut (1713-1765) is generally credited with the first reasonably accurate and complete theoretical explanation of the Moon's orbit. His method of calculation makes use of an expansion of the lunar equations of motion in terms of various small parameters. Clairaut, however, initially experienced difficulty in accounting for the precession of the lunar perigee. Indeed, his first calculation overestimated the period of this precession by a factor of about two, leading him to question Newton's inverse-square law of gravitation. Later on, he realized that he could indeed account for the precession, in terms of standard Newtonian dynamics, by continuing his expansion in small parameters to higher order. After Clairaut, the theory of lunar motion was further elaborated in major works by D'Alembert (1717-1783), Euler (1707-1783), Laplace (1749-1827), Damoiseau (1768-1846), Plana (1781-1864), Poisson (1781-1840), Hansen (1795-1874), De Pontécoulant (1795-1824), J. Herschel (1792-1871), Airy (1801-1892), Delaunay (1816-1872), G.W. Hill (1836-1914), and E.W. Brown (1836-1938). The fact that so many celebrated mathematicians and astronomers devoted so much time and effort to lunar theory is a tribute to its inherent difficulty, as well as its great theoretical and practical interest. Indeed, for a period of about one hundred years (between 1767 and about 1850) the so-called method of lunar distance was the principal means used by mariners to determine longitude at sea. This method depends crucially on a precise knowledge of the position of the Moon in the sky as a function of time. Consequently, astronomers and mathematicians during the period in question were spurred to make ever more accurate observations of the Moon's orbit, and to develop lunar theory to greater and greater precision. An important outcome of these activities were various tables of lunar motion (e.g., those of Mayer, Damoiseau, Plana, Hansen, and Brown), most of which were published at public expense. Finally, it is worth noting that the development of lunar theory gave rise to many significant advances in mathematics and dynamics, and was, in many ways, a precursor to the discovery of chaotic motion in the 20th century (see Chapter 15).

This chapter contains an introduction to lunar theory in which approximate expressions for evection, variation, the precession of the perigee, and the regression of the ascending node are derived from the laws of Newtonian dynamics.