prevail in celestial mechanics. Actually, celestial mechanics became a science about the motion of the solar system bodies under Newton's law of gravitation.
ISSN 00380946, Solar System Research, 2013, Vol. 47, No. 5, pp. 347–358. © Pleiades Publishing, Inc., 2013. Published in Russian in Astronomicheskii Vestnik, 2013, Vol. 47, No. 5, pp. 376–389.
Celestial Mechanics: Past, Present, Future1 V. A. Brumberg Institute of Applied Astronomy, Russian Academy of Sciences, nab. Kutuzova 10, St. Petersburg, 191187 Russia Received December 20, 2012
Abstract—Over all steps of its development celestial mechanics has played a key role in solar system researches and verification of the physical theories of gravitation, space and time. This is particularly charac teristic for celestial mechanics of the second half of the 20th century with its various physical applications and sophisticated mathematical techniques. This paper is attempted to analyze, in a simple form (without math ematical formulas), the celestial mechanics problems already solved, the problems that can be and should be solved more completely, and the problems still waiting to be solved. DOI: 10.1134/S0038094613040011 1
1. INTRODUCTION
The domain of astronomy discussed below is not too popular nowadays. Modern astronomy, with its prevailing astrophysical topics, answers mainly the questions about the structure of celestial bodies and their evolution. Here, we consider more applied prob lems related to the motion of celestial bodies in the solar system and accurate determination of their posi tion in space and time. Historically, these problems were the subjects of celestial mechanics and astrome try, which once covered the contents of astronomy as a whole. Now the situation has been changed so drasti cally that one may seriously ask if the people of the notsodistant future will be able themselves to com pute the motion of the planets, the Moon, planetary satellites, etc., and to determine their positions, or if it will simply be just a routine procedure of specialized computer specialized software? The theoretical base of modern celestial mechanics and astrometry is the gen eral relativity theory (GRT). Therefore, this essay concerns also the applied aspects of GRT demonstrat ing the use of GRT for constructing highly accurate theories of the motion of celestial bodies and discuss ing very precise observations. 2. CELESTIAL MECHANICS 2.1. Methodology of Celestial Mechanics In very brief terms, celestial mechanics is a science of studying the motion of celestial bodies. This laconic and nevertheless very broad definition involves many ambiguities. What is to be meant by celestial bodies? Does this term include both the actually existing nat ural bodies as well as model mathematical objects? In the case of artificial celestial bodies (satellites, space probes, etc.), do the problems of guidance motion lie in the scope of celestial mechanics? Celestial mechan 1 The article was translated by the author.
ics is, without a doubt, one of the most ancient sci ences, but from the antique times until the Newtonian epoch, it managed to describe only the kinematical aspects of the motion of celestial bodies (Ptolemaeus’ theory of the motion of planets, the Sun and the Moon, Kepler’s laws). Only since the Newtonian epoch have the dynamical aspects of motion begun to prevail in celestial mechanics. Actually, celestial mechanics became a science about the motion of the solar system bodies under Newton’s law of gravitation. In the 18th–19th centuries, celestial mechanics was advancing with permanent success in developing highlyaccurate theories of the motion of the planets and the Moon. This advance resulted in the triumphal discovery of Neptune based on the analysis of pertur bations caused by Neptune in the motion of Uranus. In the end of 19th century, Poincaré, who contributed so much to the development of celestial mechanics, formulated the aim of celestial mechanics to be the solution of the question whether Newton’s law of gravitation alone is sufficient to explain all of the observed motions of celestial bodies. Poincaré has indeed received general recognition in pure mathe matics and theoretical physics; however, this formula tion of the aim of celestial mechanics demonstrates that Poincaré has contributed a crucial part to the agreement of astronomical observations with the results of mathematical and physical theories. The first half of the 20th century was a period of comparative stagnation for celestial mechanics. The only brilliant exclusion was Sundman’s finding of the general solution of the threebody problem in 1912. Even the development of the general relativity theory by Einstein (1915) had no essential influence on celes tial mechanics of that period. Drastic changes began in the middle of the 20th century. The new advances of celestial mechanics were stimulated by new techniques of highprecision observations, computer generation, development of spatial dynamics, and progress in mathe matics and theoretical physics. Celestial mechanics
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became much more versatile than before. It lost the title of theoretical astronomy (historical title when astronomy was restricted only by astrometry and celes tial mechanics representing its observational and the oretical parts, respectively) but became related much closer to physics and mathematics. Actually, celestial mechanics of the second half of the 20th century dealt with four interrelated groups of topics, as follows: (1) Physics of motion, i.e., investigation of the physical nature of forces affecting the motion of celes tial bodies and formulation of a physical model for a specific celestial mechanics problem. The final aim in this domain is to derive the differential equations of motion of celestial bodies and of light propagation. The global physical model underlying contemporary celestial mechanics is Einstein’s general relativity the ory (GRT). Within presentday physics, Newtonian celestial mechanics is regarded as a completed science since the equations of motion for any Newtonian problem are known and the problem is reduced to the mathematical investigation of these equations. As it was already stated above, preNewtonian celestial mechanics was in fact a purely empirical science. Even nowadays it is practically possible to develop purely empirical theories of the motion of celestial bodies based only on observations (e.g., such theories are suf ficient to predict lunar–solar eclipses). But the poor accuracy of such theories and the rather short time interval of their validity make them noncompetitive as compared with the dynamical theories of motion that have arisen since the development of Newtonian mechanics combined with Newton’s gravitation law. Newtonian theories of motion of the major planets and the Moon were purely dynamic with the exception of some empirical terms introduced for better agree ment with observations. At the same time, the physical substance of the gravitation law remained unknown. The essence of gravitation was explained only by Ein stein’s general relativity theory. Since then, celestial mechanics in its broad meaning became relativistic. Presently, relativistic theories of motion of the major planets and the Moon without any additive empirical terms are in complete agreement with observational data. By updating the abovementioned question by Poincaré, the aim of relativistic celestial mechanics can be formulated as the solution of the question whether the Einstein general relativity theory alone is sufficient to explain all observed motions of celestial bodies; (2) Mathematics of motion, i.e., investigation of the mathematical characteristics of the solutions of the differential equations of motion of celestial bodies (various forms of solution representation, asymptotic behavior, stability, convergence, etc.). Within this domain a problem of celestial mechanics is considered solved if the general solution form and qualitative pic ture of motion are known. Celestial mechanics of the 18th and 19th centuries has developed in close relation with the classical branches of mathematics (mathe
matical analysis, higher algebra, differential equa tions, special functions, and so on). Many results were obtained at first in solving specific celestial mechanics problems to be generalized later as purely mathemati cal results. Many mathematicians of that period made remarkable contributions to celestial mechanics. No doubt, celestial mechanics of the 18th–19th centuries was the most mathematized amongst all natural sci ences. But along with the evident merits, such early mathematization had its drawbacks. In particular, due to the highly developed techniques based on classical mathematics, new mathematical trends of the 20th century were implemented in celestial mechanics less efficiently as was done earlier; (3) Computation of motion, i.e., the actual deter mination of the quantitative characteristics of motion. In many natural sciences this subject presents no diffi culty and is not treated separately. This is not so in celestial mechanics. For instance, if it is known that some problem may be solved in the form of a power/trigonometric series of many variables, then the actual determination of the necessary number of the terms of such a series and its summation is not a trivial problem when the number of terms ranges to hundreds or even thousands. Numerical integration of the equa tions of motion of celestial bodies over a long interval of time is also not a trivial problem. Analytical and numerical techniques of celestial mechanics have been permanently improved over the history of celestial mechanics. In its turn, it was a stimulatory for many branches of mathematics (the theory of special func tions, linear algebra, differential equations, theory of approximation, etc.). Representation of analytical or numerical solutions of the celestial mechanics equa tions in the form suitable for actual computation has always been an independent and complicated task. Indeed, demands for the accuracy of the celestial mechanics solutions were always ahead of the time of the existing technical computational possibilities. That is why it is no wonder that the first sufficiently accurate methods of numerical integration of the ordi nary differential equations have been elaborated just for application in celestial mechanics problems (high accuracy integration over very long timeintervals). The advent of computer facilities in the second half of the 20th century has resulted in revolutionary changes both in numerical and analytical techniques of celes tial mechanics. It is to be noted that the first (special ized) systems to perform symbolic (analytical) opera tions by computer were developed in celestial mechanics. Later on there appeared the universal methods of numerical integration of the ordinary dif ferential equations and universal computer algebra systems (CAS) for symbolic operations. The actual task became to combine this general software with spe cific features of celestial mechanics problems. How ever, these facilities may have negative influences if the modern supercomputers with their practically unlim SOLAR SYSTEM RESEARCH
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ited memory and processing speed are used for a too straightforward approach to solve a problem; (4) Astronomy of motion, i.e., application of the mathematical solution to a problem of a specific celes tial body, comparison with the results of observations, determination of initial values and parameters of motion, and precomputation of motion for the future. By comparing the theoretical (computed) and observational results, one may make conclusions about the adequacy of physical and mathematical models to the observed picture of motion. If this ade quacy is not satisfactory, the investigation of the prob lem returns to one of the previous steps (improvement of the physical model and mathematical solution). The increase in precision of astronomical observations is doubtless the main factor stimulating the advance of celestial mechanics. The use of the highprecision observations enables one to improve the accuracy of the computation of the motion of natural and artificial celestial bodies, increasing the applied role of celestial mechanics. On the other hand, the comparison between the computed and the observed characteris tics of motion permits one to estimate the validity of the physical model used far beyond the scope of celes tial mechanics. Over the whole period of its develop ment there was talk about the completeness of celestial mechanics. But each time the further increases of the observational precision have opened new challenges for celestial mechanics. In the first three items, celestial mechanics acts as a fundamental science. The fourth section character izes celestial mechanics as an applied science, although eventually just the results of the fourth sec tion’s investigations (agreement or disagreement with observations) are crucial for the development of celes tial mechanics as a whole. Needless to say, this classi fication of the philosophy of celestial mechanics is rather conventional, but in general it is a characteristic for celestial mechanics of the second half of the 20th century. 2.2. Components of Newtonian Celestial Mechanics As stated above, contemporary celestial mechanics is relativistic both for its physical basis and highaccu racy applications. However, in no way does it diminish the value of Newtonian celestial mechanics as the mathematical foundation of relativistic celestial mechanics. Mutually independent components of Newtonian celestial mechanics are based on the fol lowing concepts: (1) Absolute time, i.e., one and the same time inde pendent of the reference system of its actual measure ment. A reference system can be intuitively meant as a laboratory equipped by clocks and some devices to measure linear spatial quantities (a local physical ref erence system) or angular quantities at the background of distant reference celestial objects (a global astro nomical reference system). Within this concept the SOLAR SYSTEM RESEARCH
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time interval between two events has the same value in any reference system (invariance of time). More sim ply, the clock rate does not depend on the velocity of motion of a clock and its location in the gravitational field of the celestial bodies; (2) Absolute space described by the threedimen sional Euclidean geometry. This space has maximal homogeneity (no distinguished privileged points) and maximal isotropy (no distinguished privileged direc tions). In particular, the distance between two points has the same value (invariance of length) independent of the reference system of its actual measurement. More simply, the linear sizes of a body and the dis tances between bodies do not depend on the velocity of motion of bodies and the gravitational field at their location; (3) Newtonian mechanics. The first of three basic laws of Newtonian mechanics is the law of inertia. A reference system providing its validity is called an iner tial system. Any reference system moving uniformly and rectilinearly relative to a given inertial system is also inertial as well. The laws of Newtonian mechanics are valid in any inertial system in accordance with Galileo’s principle of relativity. Mathematically, this principle manifests itself as the invariance of the equa tions of Newtonian mechanics (ordinary differential equations) under the Galilean transformation describ ing the relationship between two inertial systems in threedimensional Euclidean space. The abovemen tioned features of absolute time (homogeneity) and absolute space (both homogeneity and isotropy) reflect the characteristics of the inertial systems; (4) Newton’s law of universal gravitation. Mathe matically, this law is formulated as the solution of the linear equation in partial derivatives (Poisson equa tion) describing the gravitational field of material bod ies (Newtonian potential). Newtonian equations of body motion (ordinary differential equations) and equations of gravitational fields (linear equations in partial derivatives) are absolutely independent. The combination of Newton’s law of universal gravitation and the laws of motion of Newtonian mechanics within the concepts of absolute time and absolute space defines the essence of Newtonian celestial mechanics. 2.3. Classical Problems of Celestial Mechanics Ranging in increasing order of complexity, the typ ical problems of Newtonian celestial mechanics are the twobody problem, the problem of two fixed cen ters, the restricted threebody problem, the three body problem and the problem of n (n > 3) bodies. (1) The twobody problem is usually treated as the problem of the motion of two material points mutually attracted in accordance with Newton’s gravitation law. Mathematically, this problem is reduced to the one body problem, i.e., the problem of the motion of a test particle (a particle of zero mass) in the Newtonian
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gravitation field of a central body with mass equal to the sum of the masses of the two original bodies. Depending on the initial conditions (initial position and velocity), the test particle can move on an ellipse, parabola or hyperbola, the central body being located at the focus of this conic section. Particular (degener ated) cases are circular motion and rectilinear motion (this last case involves collision with the central body). Elliptic motion presents the most important case for practical applications. The solution of the elliptical twobody problem is presented most often in one of the following forms: (a) the closed form, where the coordinates and velocity components of the particle are expressed by the closed expressions in terms of the auxiliary variable (of the type of arc length), called anomaly and related with the physical time by the transcendent equation (in addition to true and eccentric anomalies of classi cal celestial mechanics, the socalled elliptic anomaly has recently come into use); (b) infinite trigonometric series in terms of the mean anomaly (representing some linear function of time); (c) series in powers of time (contrary to the first two forms the solution in this form is valid generally only for limited time intervals and just this form is used in many numerical integration techniques). Since in the solar system the mass of the Sun exceeds by three orders of magnitude the total mass of all of the planets, the twobody problem is an adequate initial approximation in constructing the theories of motion of many bodies of the Solar System. (2) The problem of two fixed centers represents a purely mathematical model problem of the motion of a test particle in the gravitational field of two motion less mutually nonattracting bodies (material points). This problem admitting the solution in a closed form (with the aid of elliptic functions) has played an important role in the development of celestial mechanics. In the second half of the 20th century, this problem turned out to be useful in constructing some theories of the motion of Earth’s artificial satellites. (3) The restricted threebody problem deals with the motion of a test particle in the gravitational field of two mutually attracting bodies (material points). Of the most interest are the restricted circular threebody problem with finite mass bodies moving on circular orbits and the restricted elliptical threebody problem with finite mass bodies moving on elliptical orbits. Next to the twobody problem, the restricted circular threebody problem is the most investigated problem of celestial mechanics. This problem is incapable of being solved in the closed form and has always been an object of application of various techniques of celestial mechanics. In particular, just this problem stimulated the development of qualitative techniques of celestial mechanics (and mathematics generally) aimed to investigate the features of the solutions without explic
itly obtaining the solutions themselves. In astronomy, the restricted threebody problem is of great practical importance in studying the motion of the natural sat ellites of the planets (in the first instance the motion of the Moon under the attraction of the Earth and the Sun), minor planets (motion of asteroids in the field of the Sun and the Jupiter) and comets. Each of these cases, i.e., satellite, asteroid and comet, demands its own specific techniques. Applicability of the restricted threebody problem goes beyond the solar system, e.g., to the problem of the existence of the planets around the massive binary systems. (4) The threebody problem is mathematically the best known celestial mechanics problem to study the motion of three material points under the action of Newton’s law of gravitation. Many outstanding spe cialists in celestial mechanics and mathematics have contributed to its investigation. But the question of whether or not this problem has been solved may be both positive and negative. One knows the general solution of this problem potentially permitting com putation with known initial values (the positions and velocities of the bodies at the initial epoch) the posi tions and velocities of the bodies at any arbitrarily far moment of time in the past or future (excepting initial values making possible the triple collision of the bod ies). But this solution found in 1912 by Finnish math ematician Sundman in form of the power series in terms of some auxiliary variable (of the type of an anomaly of the twobody problem) turned out to be extremely inefficient for real applications. Contrary to widespread opinion, the matter does not consist of only the astronomical number of terms of the Sund man series required to obtain the result within any acceptable accuracy. This drawback can be overcome purely mathematically by replacing the power series by a more effective series of polynomials. The actual problem is that this power series form of solution, like as all numerical integration solutions of the equations of celestial mechanics, does not permit to have any insight into the features of the solution. Other tech niques not claiming to be a general solution of the threebody problem are more effective in different particular cases of this problem that are important in the astronomical respect (the Sun and two planets, the Sun–Earth–Moon problem, the stellar threebody problem, etc.). In general, the character of motion in the threebody problem can be regarded as known suf ficiently well, enabling one to speak about its solution rather optimistically. At the same time this problem, as a purely mathematical problem, continues to be a challenge to mathematicians and remains open for further research. (5) The problem of many bodies, i.e., the problem of motion of n (n > 3) material points under the action of Newton’s law of gravitation. No doubt this is a cen tral problem of celestial mechanics. One knows in this problem some rigorous particular solutions, as well as the main types of motion and a set of theorems of gen SOLAR SYSTEM RESEARCH
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eral character. When applied to the motion in the Solar System, this problem is treated as a problem of motion of (n – 1) bodies of small masses and one body of large mass (the Sun). With such a statement, the solution of this problem is developed by different tech niques of subsequent approximations. The nbody problem, when all masses are of the same order, is an example of an unsolved problem of Newtonian celes tial mechanics. From the viewpoint of astronomers, the role of celestial mechanics has been estimated not so much by its advances in the problem of three or n bodies (these researches have been regarded as more related to mathematics), as by its efficiency in constructing the theories of motion of the specific bodies of the Solar System. One may note several interesting features in this “astronomical” part of celestial mechanics. First, there were also a variety of techniques used to solve a specific problem. Many of these different tech niques, which are rather sophisticated mathematically, remained practically unrealized. The fact is that the construction of a theory of the motion of a specific celestial body generally demands a great number of repetitive onetype operations and, consequently, is very laborintensive of human time. For various rea sons such a work has not been available for all special ists in celestial mechanics. Secondly, throughout the entire period of contem porary celestial mechanics there has been a competi tion between analytical and numerical solution tech niques (between analytical and numerical theories of motion speaking in terms of final results). This com petition has often resulted into implacable antagonism between supporters of these two trends. However, there should not be any contrast between these trends. The either/or decision should be replaced by the option of both. Indeed, the analytical solution of a celestial mechanics problem retaining all or a part of the initial values and problem parameters in the literal form acts as a general solution of the mathematical problem. A numerical solution where all initial condi tions and parameters have specific numerical values represents a particular solution of the mathematical problem. Both of these types of solutions are used in contemporary celestial mechanics. They complement each other and have different purposes. Analytical theories are necessary in investigating the dependence of a solution on the change of the initial values and parameters, in using a given theory in other problems and in studying the general characteristics of the solution. Numerical theories are generally more effective in obtaining the solution of maximum accuracy with spe cific values for the initial conditions and parameters. The third feature of the historical development of celestial mechanics is the permanent search for a com promise between the form of an analytical solution and the time interval of the validity of this solution. Purely theoretically, it was supposed that an ideal con figuration of an analytical solution is provided by the SOLAR SYSTEM RESEARCH
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trigonometric form with the coordinates and compo nents of velocity of celestial bodies represented by a trigonometric series in some linear functions of time. With application to the problem of the motion of the major planets of the Solar System, the theory ensuring such a form that is also valid, at least formally, for an the infinite time interval has been called the general planetary theory. Laplace was the first to propose solving the equa tions of planetary motion in a trigonometric form, but technical difficulties of such a solution forced him to develop another form of planetary theory, which has since become classic and admits secular and mixed terms (with respect to time) as well. For the major planets of the Solar System, the classical theories are valid for the intervals of the order of several hundred years. The next attempt to find efficient methods for constructing the general planetary theory was under taken by Le Verrier. Not being successful in this direc tion, Le Verrier developed his famous theories of motion of the major planets in the form indicated by Laplace. A mathematical form of the general plane tary theory was rigorously proved for the first time by Newcomb in 1876. It is of interest that Newcomb con sidered his technique to be only an existence theorem for such a solution, but he actually used the Newton type quadratic convergence iterations underlying the contemporary KAM theory (Kolmogorov–Arnold– Moser theory) concerning the existence and construc tion of quasiperiodic solutions of the celestial mechanics equations. At the end of the 19th century and the beginning of the 20th century, the general planetary theory was advanced by Dziobek, Poincaré and Charlier. Gyldén most closely approached the practical construction of the general planetary theory. He created therewith his own world of the art of celes tial mechanics (the theory of periplegmatic orbits). Finally, Hill, who created several firstclass classic type planetary theories, considered them as only a temporary compromise solution until the develop ment of more efficient methods for constructing the general planetary theory. The development of the general planetary theory continued in the second half of the 20th century. By this time it became evident that the trigonometric form of the solution is not efficient because of the great number of trigonometric terms with practically identi cal periods (the slow motions of the perihelia and nodes of the planetary orbits have a small influence on the periods due to the fast angular variables, i.e., the mean longitudes of the planets). An alternative form of the general planetary theory is provided by a normal izing transformation of the planetary coordinates by means of the trigonometric series in fast angular vari ables with the coefficients dependent on slowly chang ing variables. These slow variables satisfy an autono mous system of differential equations (the secular sys tem). With dependence on the analytical form of the solution of the secular system (including the trigono
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metric form only as a particular case) one can obtain different explicit expressions of the final series for the planetary coordinates. The solution of the secular sys tem can be found numerically as well, underlying once again the possibility and feasibility of the combination of analytical and numerical techniques. General planetary theory in this form can be expanded for the rotation of the planets, also resulting into a unified general theory of the motion and rota tion of the planets of the Solar System. This theory avoids the fictitious secular terms inherent in classical theories of the planetary motion and rotation enabling one to use it for the time intervals of the order of many thousands of years at least. Actual construction of the general planetary theory being performed in the 70s of the 20th century in the Institute of Theoretical Astronomy (Leningrad) and Bureau des Longitudes (Paris) was not developed to the stage of comparison with observations. Within the level of the available computation facilities of that time it was necessary to solve purely technical problems related to inadequate computer memory, insufficient processing speed, etc. Today, the solution of this prob lem that once was a challenge for celestial mechanics is technically quite feasible. However, there is no longer any interest in this problem. Even the contem porary analytical theories of major planets’ motion and the Earth’s rotation elaborated in the Bureau des Longitudes by Bretagnon in advancing the theories by Laplace and Le Verrier give way to numerical theories, when it comes to practical needs in highaccuracy ephemerides. In this competition of efficiency between classical analytical theories and numerical integration over time intervals of the order of hundreds of years the general planetary theory is the oddman out. But for intervals of the order of thousands of years, the general planetary theory is beyond any competi tion and thanks to it, one may still hope for its eventual completion. Generally speaking, in spite of its completeness from the viewpoint of physicists, Newtonian celestial mechanics, even in its classical form, still has many unsolved and interesting problems. First of all, one may note that the investigation of the evolution of motion in the nbody problem, most particularly in the general case of comparable masses. Even in the case of one dominant mass (the case of the Solar Sys tem), the problem of the presentation of a solution valid for long time intervals still remains timely. Inter esting possibilities for compact presentation of the analytical solutions (e.g., using the compact expan sions for the elliptic functions) also remain unex plored. Finally, beyond the model of point masses, the motion of the nonrigid bodies, taking into account their proper rotation, represents an immense field of research. It is true that celestial mechanics nowadays has lost its former relevance, but this is the general fate of each science and does not signal the completeness of the mathematical and astronomical content of
celestial mechanics. It should be noted therewith that the wellknown expression “the new is the wellfor gotten old” fully concerns contemporary celestial mechanics because, very regretfully, many of the tech niques and results of classical celestial mechanics obtained still by Laplace, Le Verrier and its other founders, turned out to be forgotten and are only now being rediscovered again (sometimes in a worse ver sion). 2.4. Trends of Contemporary Celestial Mechanics At present, Newtonian celestial mechanics is char acterized by two features making it cardinally different from classical celestial mechanics, i.e., new objects of research and new types of motion. New objects are provided by exoplanets (planets beyond the Solar Sys tem), new families of satellites of the major planets, and minor planets of the Solar System with orbits located outside the Neptune orbit (Kuiper belt). The new types of motion are primarily embrace the chaotic motions. Some people believe that celestial mechanics has discovered new horizons, becoming much more extensive than classical celestial mechanics with its narrow class of objects (mainly major planets and their satellites) and deterministic motions. But one forgets therewith that “new” celestial mechanics takes on the risk of losing its chief distinguished merit as compared with all other sciences, i.e., highprecision observa tions and the high accuracy of its mathematical theo ries. As a result, celestial mechanics may lose its mean ing for physics as a tool to verify the physical gravita tion theories and its stimulating influence for applied and computational mathematics. Indeed, along with the exclusive interest of exoplanets for astronomy, it is unlikely that someday their motion will be observed and computed with the accuracy characteristic for the Solar System bodies. Statistical techniques applied in investigating the motion of exoplanets and Kuiper belt asteroids have very little in common with classical celestial mechanics methods. As far as presently pop ular chaotic celestial mechanics is concerned, it deals with cosmogony time intervals where there is no case of observations at all. In terms of deterministic (pre dictable) and nondeterministic (unpredictable) motions one may separate three time zones as follows: (1) predictable near zone (small time intervals of the order of hundreds of years for the planetary prob lems) available for using classical planetary theories with the secular and mixed terms; (2) predictable intermediate zone (large time inter vals of the order of thousands of years for the planetary problems) suitable for using general planetary theory with separation of the shortperiod and longperiod terms (with the potential possibility of the purely trig onometric form); (3) unpredictable far zone (overlarge time intervals of the order of millions of years for the planetary problems) with chaotic motions (in virtue of the KAM theory this SOLAR SYSTEM RESEARCH
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does not exclude the existence of the deterministic solu tions of the type of general planetary theory). Chaotic behavior in dynamical systems is of great interest in its mathematical arespect. The supporters of the chaos theory speak about the chaotic state of the Solar System in the infinite past and infinite future. Their opponents argue that the very existence of the mankind enables one to hope for the evolution of the Solar System within the KAM theory. In any case, there is no relation to any astronomical observations. Moreover, this model has little to do with physics. Any sophistication of the model, e.g., by replacing the material points with more complicated objects in Newtonian theory or by replacing the Newtonian gravitation theory with the general relativity theory, all the results of the chaos theory may be radically changed. Therefore, “old” celestial mechanics as an organic part of mathematics, physics and astronomy should not be regarded as a science of the past. The development of any science has been always accompanied by a conflict of opinions. From the viewpoint of some physicists, a physical theory that cannot be confirmed or refuted by experiment (obser vations) has no interest and cannot be regarded as a physical theory at all. On the other hand, there are mathematicians claiming that any mathematical model is of interest for the natural sciences with no relation to any experiments. These are two polar view points. With application to celestial mechanics these two viewpoints represent not the mutually exclusive directions, but just different aspects of its methodology mentioned above. 3. RELATIVISTIC CELESTIAL MECHANICS 3.1. Special Relativity Theory (SRT) One of the greatest scientific achievements to open the 20th century was the creation of the special relativ ity theory by Albert Einstein in 1905. Nowadays, it is even difficult to imagine the astonishment and admi ration of the intellectual’s mankind caused by the SRT. In its further development the 20th century generated so much novelty into human life (both positively and negatively), that people seemed to have lost the capa bility to be surprised by anything. But, in the beginning of the 20th century, the SRT and the resulting revolu tionary change of the physical description of the world was met by mankind in a quite adequate manner. Indeed, for two preceding centuries, Newtonian mechanics and the Newtonian gravitation theory had successfully advanced in the description of the observed world phenomena and the prediction of observable effects. Therefore, the concepts of Newto nian physics seemed to be absolutely true. As it was mentioned above, these concepts include absolute time, absolute space, the laws of Newtonian mechan ics and Newton’s law of universal gravitation. SOLAR SYSTEM RESEARCH
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Newton’s law of universal gravitation and Newto nian mechanics, within the concepts of absolute time and absolute space, were fully consistent with to satisfy the scientific and technical demands of human society during these two centuries. The difficulties that arose in the middle of the 19th century resulted in the crisis of Newtonian physics at the beginning of the 20th cen tury in attempting to explain the observed data in elec trodynamics and optics of the moving bodies (Max well’s electromagnetic theory and wave light theory). These experimental data have led to the four position statements: (1) all points of space and all moments of time are alike (homogeneity of space and time); (2) all directions in space are alike (isotropy of space); (3) all laws of nature are the same in all inertial ref erence systems (special principle of relativity); (4) the velocity of light in a vacuum is the same constant in all inertial reference systems (postulate of the constancy of the velocity of light). The first two statements are common both for Newtonian mechanics and SRT. The latter two state ments specific for SRT were formulated in the famous paper by Einstein “On the electrodynamics of moving bodies” published in September 1905, in the journal “Annalen der Physik”. The adoption of the special principle of relativity and the postulate of the light velocity constancy dras tically changed the Newtonian conceptions of space and time. Instead of threedimensional space and onedimensional time, SRT deals with a single four dimensional space–time. The usual Euclidean geom etry is valid in such a space provided that complemen tary to three spatial coordinates, a quantity ict is added as a fourth coordinate, t being the physical time and i being the imaginary unit whose square is equal to –1. The transformations between two inertial fourdimen sional reference systems of SRT are called Lorentz transformations. These transformations generalizing the Galileo transformations of Newtonian mechanics reflect mathematically the special principle of relativ ity. If the Galileo transformations retain invariant a time interval and a spatial length measured in some inertial system, then the Lorentz transformations retain invariant a fourdimensional interval calculated by Euclidean geometry with the condition indicated above. Lorentz transformations involve a set of kinemati cal consequences that demonstrate the relativity of the space–time observational data, in dependentce of a reference system of actual measurements. One should remember that Lorentz transformations imply that inertial systems are to be considered as a special class of all possible systems (justifying the name of SRT). SRT is now not only a theory experimentally veri fied in all of its aspects; it represents also a working theory used in many domains of applied science and
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technology from astronavigation (by means of naviga tion satellites) to the physics of elementary particles. In the distant future, described now in science fiction, SRT might play a major role as a scientific base for interstellar flights with the use of photon rockets. Nowadays, there are no explicit opponents of SRT (although at the present time with the broad activity of pseudoscience, one can often hear, from time to time, about sensational “discoveries” claiming to argue against the postulate of the light velocity constancy). For Einstein, SRT was of importance not only as a theory of space and time in the absence of gravitation but, also, as a starting point to elaborate a theory of space, time and gravitation. This new theory, com pleted in 1915 and called general relativity theory, is the physical foundation for contemporary celestial mechanics. From a purely operational point of view general relativity theory extends SRT demonstrating that all space–time characteristics at the point of observation in some reference system depend not only on the velocity of this point but also on the value of the gravitational potential (and its higher moments) at this point. 3.2. General Relativity Theory (GRT) The decade after 1905, when the SRT was created, was significant. While a large part of Europe was living in anticipation of the first world war and related social changes, the scientists (physicists mainly) mastered the SRT. Einstein, who considered the SRT as the first step towards a more universal physical theory, tried to generalize it to include gravitation. In 1915 Einstein managed to formulate the general relativity theory. His final summing paper on the foundations of the GRT was published in 1916. Some physicists believe that it might currently be possible to develop the main idea of the GRT just from experimental results. Yet Einstein derived the basic statements of the GRT by purely logical consider ations proceeding from the SRT and the fundamental law of equality of gravitational and inertial mass. Having completed the SRT, Einstein successfully put forward the principle of equivalence and the prin ciple of general covariance. According to the principle of equivalence, all physical processes follow the same pattern both in an inertial system under the action of the homogeneous gravitational field and in a non inertial uniformly accelerated system in the absence of gravitation. The principle of equivalence is strictly local in contrast to the law of identity of the gravita tional and inertial mass underlying it. The principle of general covariance, being of a purely mathematical character, implies that equations of physics should have the same form in all reference systems, i.e., all systems should be equivalent. Combination of these two principles enabled Einstein to formulate the prin ciple of general relativity as a generalization of the spe cial principle of relativity.
Following this, Einstein came to the conclusion that in the presence of gravitation, the space–time relations correspond not to the flat (Euclidean) four dimensional space of events of the SRT, but to a curved (Riemannian) space. The curvature of the space is caused by the presence of the gravitatingional masses. The most important characteristic of the Riemannian space is its metric, i.e., the square of the infinitely small fourdimensional distance between two points of this space. According to the basic idea of the GRT, the properties of space and time, i.e., the space–time met ric, are determined by the motion and distribution of masses and, conversely, the motion and distribution of masses are governed by the field metric. This interrela tion is revealed in the field equations for determining the metric coefficients in terms of the gravitating masses. The equations for the motion of mass and the light propagation of light follow from the field equa tions. The GRT is distinguished by its logical simplicity and perfection. Newton’s gravitation theory consists of four mutually independent parts with their own postulates (absolute time, absolute space, Newtonian mechanics laws, Newton’s law of universal gravita tion) giving therewith no physical explanation of grav itation. GRT is based on the field equations written in the covariant form valid for any reference systems. The SRT permits one, if desirable, to write all equations in the covariant form and to use any reference systems. But the space–time of the SRT represents the flat Euclidean space (without curvature) admitting the existence of privileged distinguished systems (inertial systems) defined up to the Lorentz transformation. The corresponding mathematical coordinates of these systems are called Galilean. Physically, they are ade quate for time and three spatial coordinates. There are no Galilean coordinates in the GRT. But the GRT admits the quasiGalilean coordinates. In terms of these coordinates, the Riemannian metric of the GRT differs little from the Euclidean metric of the SRT. However, this distinction caused by the gravitating masses looks different for each reference system. Moreover, at every point of the GRT space–time, one may introduce the socalled local geodesic coordi nates such that in the infinitesimal region of the given point one has (in neglecting by the small quantities of at least of second order) the SRT space–time. All SRT relations will be valid in this infinitesimal region. This possibility of introducing the local geodesic coordi nates is due to the principle of equivalence valid only locally. The most amazing fact in the history of the creation of the GRT creation is the absence of any experimen tal reasons. A new physical theory often arises when an old theory comes into contradiction with the corre sponding experimental data. There was nothing of the kind in the case of the GRT. Indeed, even since the second half of the 19th century, one knew the disagree ment between the observed value of the secular SOLAR SYSTEM RESEARCH
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advance of the perihelion of the orbit of Mercury and its theoretical value calculated by the Newtonian the ory of its motion. But it has not bothered physicists, especially as there were some other (less significant) disagreements in the problem of the major planets motion, e.g., in the motion of the perihelion of Mars and in the motion of the node of Venus (only in the middle of the 20th century a more rigorous analysis of observations removed these disagreements). There fore, when in 1916, Schwarzschild derived a rigorous solution for the GRT motion of Mercury in the gravi tational field of the Sun and obtained the missing cor rection contribution to the Newtonian value, this first experimental confirmation of GRT seemed rather unexpected. On the other hand, the test of the effect of the deflection of light in the Sun’s gravitational field, predicted by Einstein, was greatly anticipated. The observation performed during the total solar eclipse on May 29, 1919, confirmed this effect. Quite reasonably, it was regarded as a triumph for the GRT. Not stopping at the laboratory experiments on measuring the GRT effect in spectroscopy (mainly the Mössbauer effect), let’s consider the “global” applications of the GRT in astronomy. In fact, the origin of the GRT has led to three new domains of astronomy, i.e., (1) relativistic cosmology; (2) relativistic astrophysics; (3) relativistic celestial mechanics. The most significant astronomical prediction of GRT is doubtless the theory of the expanding universe developed by A.A. Friedmann on the basis of the solu tion of the Einstein equations. The phenomenon of the expanding universe was discovered from observa tions in 1929. Relativistic cosmology nowadays pre sents an intensively developing branch of astronomy based on the GRT, on the one hand, and on the vast quantities of observational data, on the other hand. As far as astrophysics is concerned, the GRT enables one to analyze phenomena completely incon sistent with Newtonian theory. Two examples are characteristic. The GRT predicts the existence of qualitatively new objects, e.g., the black holes with such a strong gravitational field that no emission can escape into the external space. The GRT has permit ted the accurate computation of the binary pulsar motion (as a problem of relativistic celestial mechan ics). Binary pulsar observations confirmed the GRT conclusion about the loss of binary system energy due to gravitational radiation. The coincidence of the the oretical and observational results relative to the binary pulsar systems demonstrates implicitly the existence of the gravitational waves predicted by the GRT, although so far there are no direct results from the gravitational wave detectors. In general, the GRT plays quite an extraordinary role for celestial mechanics. Relativistic celestial mechanics does not deal with such impressive and SOLAR SYSTEM RESEARCH
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unusual events as intrinsic to cosmology and astro physics. However, relativistic celestial mechanics has one irrefutable merit, i.e., its exceptionally high preci sion of observations absolutely unattainable in cos mology and astrophysics. Just this feature makes celestial mechanics and the related astrometry so important in verifying the effects of the GRT. Para phrasing the wellknown saying by Poincaré concern ing Newtonian celestial mechanics, the final goal of relativistic celestial mechanics is to answer the ques tion whether GRT alone is capable of explaining all observed motions of celestial bodies and the propaga tion of light. Currently, celestial mechanics answers this question positively. GRT is used therewith not only as a theoretical basis of celestial mechanics, but also as a working framework for increasing the accu racy of celestial mechanics and astrometry solutions. The problem of the comparison of the theoretical and observational data is here of fundamentally new prin cipal novelty as compared with Newtonian astronomy. In Newtonian astronomy this problem is simply solved by introducing the inertial systems with all quantities having physical meaning. In relativistic astronomy the solution of the equations of the motion of bodies and the light propagation depends on the employed four dimensional quasiGalilean coordinates close to the SRT Galilean coordinates (only the most significant secular effects such as the Mercury perihelion advance of Mercury and the angle of the light deflection near the solar limb do not depend on these coordinate con ditions). Comparison of theoretical and experimental data is based on the description of the observational procedure (by means of the equations of the light propagation) in the same space–time as is used for the presentation of the motion of the bodies, enabling one to exclude eventually all nonphysical immeasurable quantities characteristic of Newtonian mechanics (distances, coordinates, etc.). The contemporary the ories of motion of the major planets of the Solar Sys tem, lunar motion and the Earth’s rotation have been developed in the GRT framework. The space astron omy projects planned for the first quarter of the 21th century and designed for the observational preci sion of one microarcsecond in the mutual angular dis tances between celestial objects demand the intensive use of the GRT analysis of observations. 3.3. Relativistic Celestial Mechanics and Astrometry As indicated above, relativistic celestial mechanics represents a science to study the motion of celestial bodies within the framework of the GRT. Just this change of the physical basis (GRT instead of Newto nian mechanics and Newton’s gravitation law) speci fies the qualitative difference between relativistic and Newtonian celestial mechanics. But from purely the operational point of view, i.e., in obtaining the practi cal results to be compared with observational data (within the domain “astronomy of motion” according
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to our classification) the difference between relativis tic and Newtonian treatment of the problem of celes tial bodies’ motion is revealed in two aspects, as fol lows: (1) mathematically, i.e., the difference in the equa tions of the gravitational field and in the equations of motion of bodies resulting in the differences in the solutions of these equations (this is the object of rela tivistic celestial mechanics in the narrow meaning of the phrase); (2) physically, i.e., the difference in the manners to compare the calculated and observed data and in the reduction itself of the calculated data to the measur able quantities (the object of relativistic astrometry as a part of relativistic celestial mechanics in the broad sense). The mathematical distinction is not essentially new for celestial mechanics. Even in Newtonian celestial mechanics all actually important problems beyond the scope of the twobody problem cannot be solved in the closed form, which demands the application of the method of consecutive approximations (iterations) for their approximate solution. Indeed, in Newtonian celestial mechanics, the equations of motion of the bodies can be formulated rigorously and only their solution is to be found by approximations. In relativis tic celestial mechanics only the equations of the one body problem can be formulated rigorously. A good example is provided by the Schwarzschild problem dealing with the motion of a test particle in the spher ically symmetrical gravitational field of one body. In all more complicated cases, even for the problem of the motion of two bodies of finite mass, the equations of motion may be derived only in an approximate form. It does not signify a significant obstruction in practical work since, in any case, these equations can be solved only by iterations, but this distinction is of importance for theoretical studies. The theoretical distinction between the solutions of the Newtonian problem and its relativistic counterpart can be seen even in the simplest case of the onebody problem. In the Newtonian case (Kepler problem) the solution is described by means of three linear parameters charac terizing the size of the orbit (semimajor axis), its form (eccentricity) and its positions in the space (inclina tion), as well as by means of three angular parameters determining the position of a moving particle in orbit (anomaly or longitude), orientation of the orbit in the plane of motion (longitude of the pericenter) and in space (longitude of the node). Only the first of these angular parameters varies in time whereas the two other parameters remain constant (degenerate case). In the relativistic case (Schwarzschild problem), not only the first angular parameter, but also the second one varies in time (this feature is used in the relativistic discussion of observations of binary pulsars). In more complicated problems, this distinction is not signifi cant because all three angular quantities generalizing the angular parameters of the onebody problem vary
in time. In the practical case of the motion of the Solar System bodies, the smallness of the relativistic terms with respect to the Newtonian terms is characterized by a small parameter of the order v2/c2. With v being the characteristic velocity of the motion of the bodies (30 km/s in case of the motion of the Earth around the Sun) and c being the velocity of light in vacuum (300000 km/s) it gives the order 10–8 for this parame ter. For Solar System dynamics, it is generally suffi cient to know these relativistic equations of motion and their solutions with taking into account only the firstorder terms with respect to this parameter (post Newtonian approximation). Even within the second order of accuracy with respect to this parameter (post postNewtonian approximation) the solution of the actual problems in the GRT framework is certainly more complicated than in the Newtonian case, but there are no qualitative distinctions. The significant difference between Newtonian problems of motion and the GRT problems of motion is revealed when the terms of the order v5/c5 are taken into account (an approximation following the postpostNewtonian one). This approximation involves the gravitational radiation from the system of bodies resulting in the loss of the energy in the system. The evolution of the sys tem in this case qualitatively differs from the Newto nian case. This approximation is not needed for the Solar System dynamics. But, as mentioned above, just this approximation applied to the binary pulsar motion has enabled one to prove indirectly the exist ence of the gravitational waves. Any solution of the GRT equations of motion of celestial bodies by itself has nothing to do with the real relativistic effects valid for comparison with observa tions. Contrary to the inertial coordinates of Newto nian mechanics and SRT, no GRT coordinates for finite (noninfinitesimal) domain of the space–time have physical meaning and can be directly compared with observational data. The solutions of the equations of motion in different coordinates are inevitably dif ferent from each other. This is in no way a drawback of the GRT, as was sometimes believed by its opponents. It is simply a demonstration that relativistic four dimensional coordinates are nothing more than a con venient mathematical tool to obtain a purely mathe matical solution. That is why the problem of compar ing the theoretical and observed data is so important for contemporary relativistic astronomy. There is no such problem in Newtonian mechanics or SRT since the introduction of the inertial coordinates from the very start or at the final step (if a solution was derived in some curvilinear coordinates) immediately results into the solution in terms of the measurable quantities. In principle, there are three main possibilities for solving the problem to compare the theoretical and observed data: (1) Eliminate coordinates completely by con structing the solutions for motion of Solar System bodies motion in terms of measurable quantities; SOLAR SYSTEM RESEARCH
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(2) to use any welldefined coordinates mathemat ically suitable for a specific problem potentially solv ing the equations of the light propagation in the same coordinates, enabling one to combine the solution of the dynamical problem (body motion) and the solu tion of the kinematic problem (light propagation) to obtain the coordinateindependent quantities; (3) for treating the actual problems directly or indi rectly related with observations to use one specific type of coordinate conditions adopted by some conven tional agreements. The first approach that might be tentatively consid ered physical is often used in physical studies on local GRT effects (in a sufficiently small space–time region). Theoretical physicists have developed the techniques potentially adequate for global astronomi cal problems as well, but so far they have not found application in astronomical practice. Nevertheless, it should be noted that the antique (purely kinematical) planetary theory by Ptolemaeus was constructed just in terms of measurable quantities (mutual angular dis tances between celestial bodies). The second approach is rather mathematical, giv ing primary consideration to how well different coor dinates are suitable for the mathematical solution of dynamical problems. Very regretfully, within this approach one sometimes forgets the necessity of reducing the employed coordinates to measurable quantities. However, for the study of theoretical prob lems of relativistic celestial mechanics, this approach is the most flexible. The third approach, widely used nowadays in prac tical astronomy, is to avoid deliberately the GRT arbi trariness in coordinate conditions for the sake of prag matic simplicity. Positional astronomy deals with a set of observational results obtained by different observers at different moments of time rather than with a single result at one space–time point. In discussing the observations one has to use also the theoretical results relating to the body motion (dynamical problem) and light propagation (kinematic problem). For astronom ical applications, there is no difference, which coordi nates are used in these problems. However, it is very important that both problems be treated in the same coordinates. For the sake of actual convenience, the specific coordinate option is used by the resolutions of the International Astronomical Union (IAU). But in so doing there is a danger of the too straightforward “engineering” application of GRT in celestial mechanics. The immense theoretical potentialities of GRT are substituted therewith by a narrow set of prac tical recipes adopted by IAU. The coordinate method in relativistic celestial mechanics is realized by means of fourdimensional reference systems (three spatial coordinates and one time coordinate). A reference system (RS) represents a purely mathematical construction to facilitate math ematical solution of astronomical problems. The rela tionship between the fourdimensional coordinates SOLAR SYSTEM RESEARCH
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and the coordinateindependent measurable quanti ties (intervals of the proper time of an observer, angular distances between celestial bodies reduced to the infi nitely far distance, etc.) is determined by formulating an observational procedure with the aid of the light propagation solution found in the same RS. For exam ple, as already mentioned, in the Solar System bary centric RS, the relativistic terms in the equations of body motion are of the order 10–8 with respect to the Newtonian terms. In introducing a relativistic geocen tric RS where the Earth is the main attracting body and the action of all other celestial bodies (the Sun, the Moon, the major planets) is revealed only in the form of the tidal force terms, then the ratio of the GRT terms to the Newtonian ones will be less by two orders of magnitude. Moreover, all parameters characterizing the Earth (nonspherical figure of the Earth, angular velocity of the axial rotation of the Earth, and so on), will be, in the geocentric RS, in much better corre spondence with the measurable quantities than in the barycentric RS. Therefore, the problems such as the motion of the Earth’s artificial satellites or the rotation of the Earth are better examined in a more adequate geocentric RS. Similarly, in investigating the motion of a celestial body in the vicinity of any planet it is rea sonable to use the corresponding planetocentric RS. The fourth coordinate of such relativistic systems rep resents the scale of the corresponding coordinate time (barycentric or geocentric or planetocentric time), used as an argument in the corresponding theories of motion or rotation. Relativistic transformations gen eralizing the Lorentz transformations of the SRT enable one to reduce the fourdimensional coordi nates of these systems (including the coordinate time) to measurable quantities. Practical realization of RSs (“materialization”) is realized in astronomy by attributing the coordinate values to some reference astronomical objects. In such a way, the RS as mathematical construction is trans formed to an astronomical reference frame (RF). In modern positional astronomy two RFs are constantly maintained, International Celestial Reference Frame (ICRF) and International Terrestrial Reference Frame (ITRF). The first RF is given by the positions of quasars in the International Celestial Reference System (ICRS), representing a specific barycentric RS. The second RF is given by the positions of the ground reference stations in the International Terres trial Reference System (ITRS), representing a specific geocentric RS rotating with the Earth. The relation ship between these systems is derived theoretically from solving the GRT Earth’s rotation equations, on the one hand, and is determined from observations, on the other hand. The absence of any discrepancies between these data can be regarded presently as one more convincing verification of the GRT in astronomy. The present highlyaccurate theories of motion of the major planets and the Moon, as well as the Earth’s rotation theory have been constructed with account
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ing for the main relativistic terms (postNewtonian approximation). The agreement of these theories with observations enables one to conclude that currently the GRT completely satisfies the available observa tional data. It should be noted that the discussion of observations performed now in many institutions involves also the determination of the parameters of the alternate gravitation theories competing with GRT (postNewtonian formalism). This discussion demon strates that there are no data now demanding for inclusion of any empirical parameters to the GRT framework as a physical basis of relativistic celestial mechanics. Relativistic celestial mechanics is a rather young science with many problems waiting to be solved. In addition to the problems of Newtonian celestial mechanics requiring a relativistic generalization in a postNewtonian approximation (sufficient for the most actual applications), there are specific problems of great theoretical interest, such as the investigation of the general form of the GRT equations of motion, orbital evolution under the gravitational radiation, the general relativistic treatment of the body rotation, the motion of bodies in the background of the expanding universe (combination of the solar system dynamics and cosmology problem), and many other problems. Relativistic celestial mechanics is awaiting its new researchers. One should not forget therewith that being based physically on the GRT, relativistic celestial mechanics mathematically is based on Newtonian celestial mechanics with its extensive abundance of mathemat
ical techniques. Disregarding this inheritance and the present trend of some physicists, astronomers and space dynamics specialists to treat relativistic celestial mechanics aside from Newtonian celestial mechanics may negatively affect the whole level of celestial mechanics. 4. CONCLUSION Investigation of the Solar System has been always, and hopefully will be long further in the focus of celes tial mechanics for a long time. Just in this investigation one has the synthesis of highprecision observations, the most sophisticated mathematical techniques (numerical and analytical ones), and physical theories of gravitation, space and time. The performed analysis of the tendencies and problems of celestial mechanics (already solved or still waiting to be solved) is aimed at attracting the attention to the present goals of celestial mechanics. For the first time in the author’s practice this paper contains no formulas. In mathematical lan guage, much of the above can be found in the latest papers of the author indicated in the References. REFERENCES Brumberg, V.A., Relativistic celestial mechanics on the verge of its 100 year anniversary (Brouwer Award lec ture), Celest. Mech. Dyn. Astron., 2010, vol. 106, pp. 209–234. Brumberg, V.A. and Ivanova, T.V., On constructing the gen eral Earth’s rotation theory, Celest. Mech. Dyn. Astron., 2011, vol. 109, pp. 385–408.
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