CELESTIAL MECHANICS

in the broadest sense, the application of classical mechanics to the motion of celestial bodies acted on by any of several types of forces. By far the most important force experienced by these bodies, and much of the time the only important force, is that of their mutual gravitational attraction. But other forces can be important as well, such as atmospheric drag on artificial satellites, the pressure of radiation on dust particles, and even electromagnetic forces on dust particles if they are electrically charged and moving in a magnetic field. The term celestial mechanics is sometimes assumed to refer only to the analysis developed for the motion of point mass particles moving under their mutual gravitational attractions, with emphasis on the general orbital motions of solar system bodies. The term astrodynamics is often used to refer to the celestial mechanics of artificial satellite motion. Dynamic astronomy is a much broader term, which, in addition to celestial mechanics and astrodynamics, is usually interpreted to include all aspects of celestial body motion (e.g., rotation, tidal evolution, mass and mass distribution determinations for stars and galaxies, fluid motions in nebulas, and so forth). branch of astronomy that deals with the mathematical theory of the motions of celestial bodies. The foundation was laid by Sir Isaac Newton with the publication in 1687 of his Philosophiae Naturalis Principia Mathematica, usually referred to as the Principia. Here he published the three laws of motion that express the principles of mechanics, consolidating progress begun with the pioneer work of Galileo earlier in the 17th century. Newton also formulated the universal law of gravitation, which states that any two particles of mass in the universe attract each other with a force that varies directly as the product of the masses and inversely as the square of the distance between them. Newton's fundamental principles permit the statement of a problem in celestial mechanics in the form of a set of equations of motion, ordinary differential equations of the second order. An important triumph of Newton's was that Johannes Kepler's three laws of planetary motion were obtained as a consequence of the law of gravitation in conjunction with the laws of motion, applied to the problem of two bodies. The next in order of difficulty is the case in which three bodies are considered. The solar system, consisting of the Sun and nine known principal planets, all but three surrounded by one or more satellites, constitutes a problem of many bodies. The significant circumstance that the mass of the Sun is about 1,000 times that of the most massive planet, Jupiter, makes the Sun's gravitational attraction far outweigh the mutual attractions of the planets. This suggests a process of successive approximations that has become the standard procedure in the mathematical theory of planetary motion, the deviations from elliptic motion being called the perturbations. In the case of the Moon's motion, the Earth produces the principal attraction. Notwithstanding the very great mass of the Sun, the effect of the Sun's attraction is a small fraction of that of the Earth owing to the close proximity of the latter. In the case of some satellites, however, the perturbations produced by the Sun's attraction may become very sizable. During the 18th century, powerful analytical methods, made possible by the development of differential and integral calculus, were applied to the problems of celestial mechanics. These methods were generally successful in accounting for the observed motions of bodies in the solar system. The Moon's motion was an apparent exception until this problem was finally solved during the second quarter of the 20th century. The observed deviations between observations and theory were shown to be caused not by defects in the theory but by lack of uniformity of the Earth's rotation. This led, in 1950, to the introduction of ephemeris time, which is independent of the Earth's rotation but based on the observed motions of the Moon and the Sun. Ephemeris time may thus be regarded as the independent variable of Newtonian mechanics. It is now recognized that the Newtonian laws of motion and law of gravitation are approximations to the true laws governing the motions of celestial bodies. In the motion of the perihelion of the innermost planet, Mercury, and in a very few other cases, relativity effects are large enough to be revealed by the most precise observations. A comparison between observations and theory, in which the perturbations are properly taken into account, confirms the excess of the motion of the perihelion in the amount of 43 seconds of arc per century, as required by the theory of relativity. This is one of the most convincing observational proofs of that theory. A branch of celestial mechanics deals with the gravitational theory of rotating liquid or gaseous masses, with applications to the Earth and the other large planets. Newton explained the ocean tides as caused by the gravitational attraction of the Moon and the Sun. Sir George Howard Darwin, in addition to developing modern methods of tidal analysis and tidal prediction, also treated the cosmogonic aspect of tidal theory in his work on the development of the EarthMoon system. An important method for the treatment of planetary perturbations was introduced during the 1770s by Joseph-Louis Lagrange. In an elliptic orbit the six orbital elements have constant values, completely determined by the three coordinates and the three components of the velocity at any time. Since the attractions by other planets cause a planet to follow a path differing from a fixed ellipse, the elements of its orbit so determined will necessarily vary with the time. Hence one may describe the perturbed orbit of a planet by giving the elements as functions of the time. Lagrange's method provides a process for deriving analytical expressions for the derivatives of the varying elements. An accomplishment that demonstrated strikingly the power of the theory of planetary motions was the discovery of the planet Neptune in 1846. Its presence and location in the sky had been predicted with astonishing accuracy by J.C. Adams and by U.-J.-J. Le Verrier from deviations in the motion of the planet Uranus. Attempts were made to discover planets beyond Neptune by a similar procedure, but the discovery of Pluto at the Lowell Observatory in 1930 must be ascribed to perseverance in systematic search rather than accuracy of prediction by mathematical theory. Additional reading Modern introductory treatments and discussions of some advanced techniques and classic developments include J.M.A. Danby, Fundamentals of Celestial Mechanics, 2nd ed., rev. and enlarged (1988); Dirk Brouwer and Gerald M. Clemence, Methods of Celestial Mechanics (1961); and Henry Crozier Keating Plummer, An Introductory Treatise on Dynamical Astronomy (1918, reprinted 1960). Orbital resonances are discussed in two review articles by S.J. Peale: Orbital Resonances in Solar-System, Annual Review of Astronomy and Astrophysics, 14:215246 (1976), and Orbital Resonances, Unusual Configurations, and Exotic Rotation States Among Planetary Satellites, in Joseph A. Burns and Mildred Shapley Matthews (eds.), Satellites (1986), pp. 159223. Current practice in solving the n-body problem on computers is given in the introduction to a paper by Lars Hernquist, Performance Characteristics of Tree Codes, The Astrophysical Journal: Supplement Series, 64(4):715734 (August 1987). An introduction to modern dynamics involving chaos and an introduction to algebraic maps is given by Michel Henon, Numerical Exploration of Hamiltonian Systems, in Grard Iooss, Robert H.G. Helleman, and Raymond Stora (eds.), Chaotic Behaviour in Deterministic Systems (1983), pp. 54170. Readable accounts of examples of chaotic dynamics in celestial mechanics are found in two articles by Jack Wisdom: Chaotic Dynamics in the Solar-System, Icarus, 72(2):241275 (1987), and Chaotic Behaviour in the Solar System, in M.V. Berry, I.C. Percival, and N.O. Weiss (eds.), Dynamical Chaos (1987), pp. 109129. A simple discussion of tides and tidal evolution is given by S.J. Peale, Consequences of Tidal Evolution, in Margaret G. Kivelson (ed.), The Solar System: Observations and Interpretations (1986), pp. 275288. Advanced discussions of tidal evolution analysis as applied to the Earth are given by Kurt Lambeck, The Earth's Variable Rotation: Geophysical Causes and Consequences (1980). Stanton J. Peale