# celestial mechanics

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## celestial mechanics,

the study of the motions of astronomical bodies as they move under the influence of their mutual gravitation**gravitation,**

the attractive force existing between any two particles of matter.

**The Law of Universal Gravitation**

Since the gravitational force is experienced by all matter in the universe, from the largest galaxies down to the smallest particles, it is often called

**.....**Click the link for more information. . Celestial mechanics analyzes the orbital motions of planets, dwarf planets, comets, asteroids, and natural and artificial satellites within the solar system as well as the motions of stars and galaxies. Newton's laws of motion and his theory of universal gravitation are the basis for celestial mechanics; for some objects, general relativity is also important. Calculating the motions of astronomical bodies is a complicated procedure because many separate forces are acting at once, and all the bodies are simultaneously in motion. The only problem that can be solved exactly is that of two bodies moving under the influence of their mutual gravitational attraction (see ephemeris

**ephemeris**

(pl., ephemerides), table listing the position of one or more celestial bodies for each day of the year. The French publication

*Connaissance de Temps*is the oldest of the national astronomical ephemerides, founded in 1679.

**.....**Click the link for more information. ). Since the sun is the dominant influence in the solar system, an application of the two-body problem leads to the simple elliptical orbits as described by Kepler's laws

**Kepler's laws,**

three mathematical statements formulated by the German astronomer Johannes Kepler that accurately describe the revolutions of the planets around the sun. Kepler's laws opened the way for the development of celestial mechanics, i.e.

**.....**Click the link for more information. ; these laws give a close approximation of planetary motion. More exact solutions, which consider the effects of the planets on each other, cannot be found in a straightforward way. However, methods accounting for these other influences, or perturbations

**perturbation**

, in astronomy and physics, small force or other influence that modifies the otherwise simple motion of some object. The term is also used for the effect produced by the perturbation, e.g., a change in the object's energy or path of motion.

**.....**Click the link for more information. , have been devised; they allow successive refinements of an approximate solution to be made to almost any degree of precision. In computing the motions of stars and the rotations of galaxies, statistical methods are often used. Columbia astronomer Wallace Eckert was the first to use a computer for orbit calculations; now computers are used for this work almost exclusively.

## celestial mechanics

The study of the motions and equilibria of celestial bodies subjected to mutual gravitational forces, usually by the application of Newton's law of gravitation and the general laws of mechanics, based on Newton's laws of motion. Satellite and planetary motions, tides, precession of the Earth's axis, and lunar libration are all described by these laws within the limits of accuracy of measurements. Newtonian mechanics is much simpler to use than the more accurate general theory of relativity. Despite fundamental differences, the equations of relativistic mechanics, based on the concepts of relativity, reduce in a first approximation to those of Newtonian mechanics, observational and predicted Newtonian values normally being very close.## Celestial Mechanics

the branch of astronomy that deals with the motion of bodies of the solar system in a gravitational field. In the solution of certain problems in celestial mechanics—for example, in the theory of cometary orbits—nongravitational effects are also considered; instances of such effects are reactive forces, resistance of the medium, and variation of mass. An important branch of modern celestial mechanics is astrodynamics, which studies the motion of artificial celestial bodies. The methods developed in celestial mechanics can also be used to study other celestial bodies. However, in modern astronomy, such problems as the study of the motions of systems of binary and multiple stars and statistical investigations of regularities in the motion of stars and galaxies are dealt with in stellar astronomy and extragalactic astronomy.

The term “celestial mechanics” was first introduced in 1798 by P. Laplace, who included within this branch of science the theory of the equilibrium and motion of solid and liquid bodies comprising the solar system (and similar systems) under the action of gravitational forces. In the Russian scientific literature, the branch of astronomy devoted to these problems has long been called theoretical astronomy. In the English literature, the term “dynamic astronomy” is also used.

* Problems in celestial mechanics*. The problems that are resolved by celestial mechanics fall into four large groups:

**(1)** the solution of general problems involving the motion of celestial bodies in a gravitational field (the η-body problem, particular cases of which are the three-body problem and the two-body problem);

**(2)** the construction of mathematical theories of the motion of specific celestial bodies—both natural and artificial—such as planets, satellites, comets, and space probes;

**(3)** the comparison of theoretical studies with astronomical observations leading to the determination of numerical values for fundamental astronomical constants (orbital elements, planetary masses, constants that are connected with the earth’s rotation and characterize the earth’s shape and gravitational field);

**(4)** the compilation of astronomical almanacs (ephemerides), which (a) consolidate the results of theoretical studies in celestial mechanics, as well as in astrometry, stellar astronomy, and geodesy, and (b) fix at each moment of time the fundamental space-time coordinate system necessary for all branches of science concerned with the measurement of space and time.

Since the general mathematical solution of the n-body problem is very complicated and cannot be used in concrete problems, celestial mechanics considers particular problems whose solution can be based on certain special properties of the solar system. Thus, to a first approximation, the motion of planets or comets may be assumed to take place in the gravitational field of the sun alone. In this case, the equations of motion permit a solution in closed form (the two-body problem). The differential equations of motion of the system of major planets can be solved by expansion in mathematical series (analytical methods) or by numerical integration. The theory of satellite motion is in many respects similar to the theory of the motion of the major planets, but with one important difference: the mass of the planet, which in the case of satellite motion is the central body, is much smaller than the mass of the sun, whose attraction causes a significant perturbation of the satellite’s motion. The deviation of a planet’s shape from spherical also has a large effect on the motion of satellites close to the planet. A distinctive feature of the moon’s motion is the fact that its orbit lies entirely outside the sphere of influence of the earth’s gravity, that is, beyond the limits of the region in which the attraction of the earth predominates over that of the sun. Thus, in setting up a theory of the moon’s motion, it is necessary to carry out a greater number of successive approximations than is necessary for planetary problems. In the modern theory of the moon’s motion, as a first approximation we consider, not the two-body problem, but the Hill problem (a special case of the three-body problem), whose solution gives an intermediate orbit that is more convenient than an ellipse for carrying out successive approximations.

In the application of analytical methods to the theory of the motion of comets and asteroids, numerous difficulties arise because of the marked eccentricities and inclination of the orbits of these celestial bodies. Moreover, certain ratios (commen-surabilities) between the mean orbits of the asteroids and the orbit of Jupiter greatly complicate the motion of the asteroids. Therefore, numerical methods are widely used in the study of the motion of comets and asteroids. In the motion of comets, non-gravitational effects have been observed, that is, deviations of their orbits from the orbits computed according to the law of universal gravitation. These anomalies in cometary motion are apparently connected with reactive forces arising as a result of evaporation of the material of the comet’s nucleus as the comet approaches the sun, as well as with a number of less-studied factors, such as resistance of the medium, decrease in the comet’s mass, solar wind, and gravitational interaction with streams of particles ejected from the sun.

A special branch of celestial mechanics deals with the study of the rotation of planets and satellites. The theory of the earth’s rotation is especially important, since the fundamental systems of astronomical coordinates are linked with the earth.

The theory of planetary figures arose in celestial mechanics; however, in modern science the study of the earth’s figure is a subject of geodesy and geophysics, while astrophysics is occupied with the structure of the other planets.The theory of the figures of the moon and planets has become especially relevant since the launching of artificial satellites of the earth, moon, and Mars.

The problem of the stability of the solar system is a classical problem of celestial mechanics. This problem is closely connected with the existence of secular (aperiodic) changes in the semimajor axes, eccentricities, and inclinations of planetary orbits. The question of the stability of the solar system cannot be completely solved by the methods of celestial mechanics, since the mathematical series used in problems in celestial mechanics are applicable only for a limited interval of time. Moreover, the equations of celestial mechanics do not contain such small factors as, for example, the continuous loss of mass by the sun; these small factors can, nevertheless, play a significant role over large intervals of time. Nevertheless, the absence of secular perturbations of the first and second orders on the semimajor axes of planetary orbits permits us to assert that the solar system’s configuration will remain the same over several million years.

* History*. Celestial mechanics is one of the most ancient sciences. As early as the sixth century B.C.,the peoples of the ancient East possessed considerable knowledge about the motion of celestial bodies. But for many centuries, this knowledge consisted only of the empirical kinematics of the solar system. The foundations of modern celestial mechanics were laid by I. Newton in his

*Philosophiae naturalis principia mathematica*(1687). Newton’s law of gravitation did not immediately receive general acceptance. However, it had already become apparent by the middle of the 18th century that this law well explained the most characteristic features of the motion of the bodies in the solar system (J. D’Alembert, A. Clairaut). The classical methods of perturbation theory were developed by J. Lagrange and P. Laplace. The first modern theory of planetary motion was formulated by U. Leverrier in the mid-19th century. To this day, this theory remains the basis for the French national astronomical almanac or ephemeris. Leverrier first indicated the secular precession of Mercury’s perihelion, which cannot be explained by Newton’s law and which for 70 years has been the most important experimental confirmation of the general theory of relativity.

Planetary theory was further developed at the end of the 19th century (1895–98) by the American astronomers S. Newcomb and G. Hill. The works of Newcomb opened up a new stage in the development of celestial mechanics. He was the first to analyze series of observations extending over long periods of time, and, on this basis, he obtained a system of astronomical constants that differs only slightly from the system accepted in the 1970’s. In order to reconcile theory with the observed motion of Mercury, Newcomb resorted to a hypothesis proposed by A. Hall (1895); this hypothesis involved changing the value of the exponent in Newton’s law of gravitation in order to explain certain discrepancies in planetary motion. Newcomb took this exponent to equal 2.00000016120. Hall’s law was retained in astronomical almanacs until 1960, when it was finally replaced by relativistic corrections resulting from the general theory of relativity *(see below)*.

Continuing the tradition of Newcomb and Hill, the American Bureau of Ephemerides (of the US Naval Observatory) under the direction of D. Brouwer and G. Clemence carried out extensive work during the 1940’s and 1950’s on a revision of planetary theories. In particular, this work led to the publication in 1951 of *Coordinates of the Five Outer Planets*, which marked an important step in the study of the orbits of the outer planets. This work was the first successful application of electronic computers to a basic astronomical problem. An analytical theory of the motion of Pluto was worked out in 1964 in the USSR. The modern theory of planetary motion has such high accuracy that comparison of theory with observation has confirmed the precession of planetary perihelia predicted by the general theory of relativity not only for Mercury but also for Venus, the earth, and Mars (see Table 1).

Table 1. Secular precession of planetary perihelia | ||
---|---|---|

Observed precession (sec of arc) | Calculated precession^{1} (sec of arc) | |

^{1}Calculated using the general theory of relativity | ||

Mercury ............... | 43.11 ±0.45 | 43.03 |

Venus ............... | 8.4 ±4.8 | 8.6 |

Earth................ | 5.0 ±1.2 | 3.8 |

Mars ................ | 1.1 ± 0.3 | 1.4 |

The first theories of lunar motion were developed by Clairaut, D’Alembert, L. Euler, and Laplace. The theory of the German astronomer P. Hansen (1857) was preferable from a practical viewpoint, and it was used in ephemerides from 1862 to 1922. In 1867 an analytical theory of the moon’s motion was published; this theory had been developed by the French astronomer C. Delaunay. The modern theory of the moon is based on the works of G. Hill (1886). The construction of lunar tables on the basis of Hill’s method was begun in 1888 by the American astronomer E. Brown. Three volumes of tables were published in 1919, and the ephemerides for 1923 were the first to contain a lunar ephemeris based on Brown’s tables. In order to reconcile theory and observation, Brown (as well as Hansen) was forced to include in the coordinate expansion an empirical term, which could not in any way be explained by a gravitational theory of lunar motion. Not until the 1930’s was it finally clarified that this empirical term reflects the effect of the earth’s nonuniform rotation on the motion of celestial bodies. Since 1970, lunar ephemerides have been computed directly from Brown’s trigonometric series without the help of tables.

The theory of the motion of planetary satellites, especially of the moons of Mars and Jupiter, has gained importance at present. The theory of the motion of the four largest satellites of Jupiter had already been worked out by Laplace. In the theory advanced by W. de Sitter in 1918, which is used in astronomical ephemerides, the oblateness of Jupiter, solar perturbations, and the mutual perturbations of the moons are all taken into account. The outer moons of Jupiter have been studied at the Institute of Theoretical Astronomy of the Academy of Sciences of the USSR. Ephemerides for these moons up to the year 2000 have been computed by the American astronomer P. Herget (1968) with the aid of numerical integration. A theory for the motion of Saturn’s moons based on classical methods was constructed by the German astronomer G. Struve (1924–33). The stability of satellite systems was considered by the Japanese astronomer Y. Hagihara in 1952. The Soviet mathematician M. L. Lidov, analyzing the evolution of orbits of artificial planetary satellites, obtained results that are also of interest in the study of natural satellites. He was the first to demonstrate (1961) that if the orbit of the moon were inclined at 90° to the plane of the ecliptic, then it would crash onto the earth’s surface after only 55 revolutions, that is, after approximately four years.

In addition to the development of a theory that has a high degree of accuracy but is applicable for only relatively short time intervals (hundreds of years), celestial mechanics is also concerned with investigations of the motion of bodies in the solar system on a cosmogonical time scale, that is, over hundreds of thousands of or millions of years. For a long time, attempts to solve this problem did not give satisfactory results. The advent of high-speed computers, which revolutionized celestial mechanics, has led to new attempts at solving this fundamental problem. In the USSR and abroad, effective methods have been developed for constructing an analytical theory of planetary motion, opening up the possibility of studying the motion of the planets over very long intervals of time.

In the USSR in the 1940’s, in connection with the development of the cosmogonical hypothesis of O. Iu. Shmidt, numerous studies were conducted on the final motions in the three-body problem; the results of these studies are important for an infinite interval of time. In the USA in 1965, a numerical method was used to investigate the evolution of the orbits of the five outer planets for a time interval of 120, 000 years. The most interesting result of this work was the discovery of the libration of Pluto relative to Neptune; because of this the minimum distance between these planets cannot be less than 18 astronomical units, although the orbits of Pluto and Neptune intersect when projected on the plane of the ecliptic. In the USSR, considerable work was done (1967) on the application of the Lagrange-Brouwer theory of secular perturbations to the study of the evolution of the earth’s orbit over the course of millions of years. This work is very important for understanding the changes in the earth’s climate in the various geological epochs.

The beginning of the 20th century was marked by significant progress in the development of mathematical methods in celestial mechanics. This progress was connected, in the first place, with the work of the French mathematician J. H. Poincaré, the Russian mathematician A. M. Liapunov, and the Finnish astronomer K. Sundmann. Sundmann succeeded in solving the general three-body problem by using infinite convergent power series. However, his series have proved to be completely unsuitable for practical use because of their extremely slow convergence. Series convergence in celestial mechanics is closely connected with the problem of small divisors. The mathematical difficulties of this problem have been overcome to a large extent by mathematicians of the A. N. Kolmogorov school.

The development of celestial mechanics in the USSR has been closely connected with the activity of two scientific centers that arose immediately after the Great October Socialist Revolution: the Institute of Theoretical Astronomy of the Academy of Sciences of the USSR in Leningrad and the subdepartment of celestial mechanics at Moscow University.

The Leningrad and Moscow schools, built up at these centers, have determined the development of celestial mechanics in the USSR. In Leningrad, questions of celestial mechanics have been treated chiefly in connection with practical problems such as the compilation of ephemerides and the computation of asteroid ephemerides. At Moscow, cosmogonical problems and astrodynamics have been the main fields of research for many years.

The leading foreign scientific institutions that conduct research in celestial mechanics include the US Naval Observatory, the Royal Greenwich Observatory, the Bureau of Longitudes in Paris, and the Astronomical Institute at Heidelberg.

* Relativistic celestial mechanics*. In the mid-20th century, the calculation of relativistic effects in the motion of bodies of the solar system is acquiring increasing importance as a result of increased precision of optical observations of celestial bodies, the development of new observational methods (Doppler-shift observations, radar, and laser ranging), and the possibility of conducting experiments in celestial mechanics with the help of space probes and artificial satellites. These problems are solved by relativistic celestial mechanics, which is based upon Einstein’s general theory of relativity. The role of the general theory of relativity in celestial mechanics is not limited to the computation of small corrections to theories of motion of celestial bodies. The emergence of the general theory of relativity has led to an explanation of the phenomenon of gravitation, and thus celestial mechanics as the science dealing with the gravitational motion of celestial bodies is becoming by its very nature relativistic.

According to the fundamental idea of the general theory of relativity, the properties of the space of real-world events are determined by the motion and distribution of masses; the motion and distribution of masses, in turn, are determined by the space-time metric. This interconnection is reflected in the field equations—nonlinear partial differential equations—which determine the metric of the field. In Newton’s theory of gravitation, the equations of motion (Newton’s laws of mechanics) are postulated separately from the field equations (the linear equations of Laplace and Poisson for the Newtonian potential). But in the general theory of relativity, the equations of motion of bodies are contained in the field equations. However, a rigorous solution of the field equations, which is of interest in celestial mechanics, and the form of the rigorous equations of motion for the n-body problem, have not been obtained in the general theory of relativity, even for n = 2. Only for *n =* 1 have rigorous solutions of the field equations been found: the Schwarzschild solution for a spherically symmetric stationary body and the Kerr solution, which describes the field of a rotating body having spherical structure. In order to solve the n-body problem (n > 2), it is necessary to resort to approximate methods and seek a solution in the form of power series in small parameters. In the case of the motion of bodies in the solar system, one such parameter may be the ratio of the square of the characteristic orbital velocity to the square of the velocity of light. Because this ratio is so small (approximately 10^{-8}), it is sufficient for all practical purposes to take account only of terms containing this parameter to the first power in the equations of motion and their solutions.

Relativistic effects in the motion of the major planets in the solar system can be obtained with sufficient accuracy on the basis of the Schwarzschild solution. The main effect in this case is a secular motion of the perihelia of the planets. In the Schwarzschild solution there is also a relativistic secular term in the motion of the orbital nodes, but this effect cannot be isolated in explicit form in the observations. This secular term partially accounts for the radar effect in the radar determination of the distance of Mercury and Venus from the earth (the radar effect is a delay in the return of a signal to earth in excess of the Newtonian delay. This effect has been experimentally confirmed. It is fairly certain that relativistic effects will appear in the motion of comets and asteroids, although they have not yet been detected because of the lack of a well-developed Newtonian theory for the motion of these objects and because of an insufficient number of accurate observations.

Relativistic effects in the moon’s motion have been obtained on the basis of the solution of the relativistic three-body problem; these effects are primarily caused by the action of the sun. They consist of secular motions of the nodes and perigee of the moon’s orbit at a rate of 1.91 sec of arc per century (geodesic precession), as well as periodic perturbations of the moon’s coordinates. These effects can apparently be detected by laser ranging to the moon. In order to refine the theories of motion of other natural planetary satellites, it is sufficient to add relativistic, secular terms to the orbital elements in the Newtonian theory. The first group of these terms is caused by the Schwarzschild precession of the pericenter. The second group consists of secular terms involving the longitude of the pericenter and the ascending node; these terms are due to the rotation of the planet itself. Finally, the motion of the planet around the sun also leads to secular terms in these elements (geodesic precession). All these terms may reach significant magnitudes for certain satellites (especially for the inner moons of Jupiter), but the lack of accurate observations inhibits their detection. The determination of relativistic effects in the motion of artificial earth satellites also does not give positive results because of the impossibility of accurately calculating the effects of the atmosphere and the anomalies in the earth’s gravitational field on the motion of these satellites. Relativistic corrections to the rotation of celestial bodies are of considerable theoretical interest, but many difficulties are still associated with their detection. The only real possibility of actual detection of these relativistic effects lies apparently in the study of the precession of gyroscopes on the earth and on earth satellites.

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