Moon-Earth-Sun: The oldest three-body problem

Abstract

The daily motion of the Moon through the sky has many unusual features that a careful observer can discover without the help of instruments. The three different frequencies for the three degrees of freedom have been known very accurately for 3000 years, and the geometric explanation of the Greek astronomers was basically correct. Whereas Kepler's laws are sufficient for describing the motion of the planets around the Sun, even the most obvious facts about the lunar motion cannot be understood without the gravitational attraction of both the Earth and the Sun. Newton discussed this problem at great length, and with mixed success; it was the only testing ground for his Universal Gravitation. This background for today's many-body theory is discussed in some detail because all the guiding principles for our understanding can be traced to the earliest developments of astronomy. They are the oldest results of scientific inquiry, and they were the first ones to be confirmed by the great physicist-mathematicians of the 18th century. By a variety of methods, Laplace was able to claim complete agreement of celestial mechanics with the astronomical observations. Lagrange initiated a new trend wherein the mathematical problems of mechanics could all be solved by the same uniform process; canonical transformations eventually won the field. They were used for the first time on a large scale by Delaunay to find the ultimate solution of the lunar problem by perturbing the solution of the two-body Earth-Moon problem. Hill then treated the lunar trajectory as a displacement from a periodic orbit that is an exact solution of a restricted three-body problem. Newton's difficultly in explaining the motion of the lunar perigee was finally resolved, and the Moon's orbit was computed by a new method that became the universal standard until after WW II. Poincaré opened the 20th century with his analysis of trajectories in phase space, his insistence on investigating periodic orbits even in ergodic systems, and his critique of perturbation theory, particularly in the case of the Moon's motion. Space exploration, astrophysics, and the landing of the astronauts on the Moon led to a new flowering of celestial mechanics. Lunar theory now has to confront many new data beyond the simple three-body problem in order to improve its accuracy below the precision of 1 arcsecond; the computer dominates all the theoretical advances. This review is intended as a case study of the many stages that characterize the slow development of a problem in physics from simple observations through many forms of explanation to a high-precision fit with the data. [S0034-6861(98)00802-2] CONTENTS I. Introduction 590 A. The Moon as the first object of pure science 590 B. Plan of this review 591 II. Coordinates in the Sky 591 A. The geometry of the solar system 591 B. Azimuth and altitude-declination and hour angle 592 C. Right ascension-longitude and latitude 592 D. The vernal equinox 593 E. All kinds of corrections 593 F. The measurement of time 593 G. The Earth's rota...

Newton–Hooke-type symmetry of anisotropic oscillators

“…Hill's original aim has been to study the Moon-Earth-Sun system [8,9]. Later, his technique has been applied to stellar dynamics [10][11][12][13], and it is this second context that we have in mind here.…”

“…Remember that Hill originally devised his method for finding approximate solutions of the three-body problem, and in particular for the Moon-Earth-Sun system [8,9]. Further applications involve stellar dynamics [10][11][12][13]].…”

An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

“…The very first premises of analytical works involving a perturbative scheme dates back to the eighteenth century. The main developments have then been pushed forward from the end of the nineteenth century and the pioneering works of Poincaré, Lindstedt, Birkhoff and Lyapunov, among others, for solving difficult problems in celestial mechanics like the three-body problem [1][2][3]. They are now widely used in all fields of applied mathematics and have been very successful in providing a large amount of predictive results.…”

Astronomical forcing of sub-Milankovitch climate oscillations during the late Quaternary

“…Earth's velocity through its orbital path is not constant, sweeping out equal areas in equal intervals of time (Kepler's 2 nd law of planetary motion), thereby moving faster at perihelion and slower at aphelion. Conversely, lunar orbital speed increases at perihelion and slows at aphelion (Gutzwiller, 1998). The current Sun-Earth distance at perihelion is ~147.09 x 10 6 km, and at aphelion is ~152.1 x 10 6 km, a variation of around 3%.…”