Celestial mechanics: Past, present, future

ISSN 00380946, 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.

347

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

notsodistant 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

highlyaccurate 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 threebody 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

highprecision observations, computer generation,

development of spatial dynamics, and progress in mathe

matics and theoretical physics. Celestial mechanics

Celestial Mechanics: Past, Present, Future

1

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

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SOLAR SYSTEM RESEARCH Vol. 47 No. 5 2013

BRUMBERG

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 presentday 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, preNewtonian 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 abovementioned 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 timeintervals).

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 Vol. 47 No. 5 2013

CELESTIAL MECHANICS: PAST, PRESENT, FUTURE 349

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 precomputation 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 highprecision

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 highaccu

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

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 threedimen

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

threedimensional Euclidean space. The abovemen

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 twobody problem, the problem of two fixed cen

ters, the restricted threebody problem, the three

body problem and the problem of

n

(

n

> 3) bodies.

(1) The twobody 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

twobody 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 socalled 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 twobody 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 nonattracting 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 threebody 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 threebody

problem with finite mass bodies moving on circular

orbits and the restricted elliptical threebody problem

with finite mass bodies moving on elliptical orbits.

Next to the twobody problem, the restricted circular

threebody 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 threebody 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

threebody problem goes beyond the solar system,

e.g., to the problem of the existence of the planets

around the massive binary systems.

(4) The threebody 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 twobody problem) turned out to be

extremely inefficient for real applications. Contrary to

widespread 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

threebody 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 threebody

problem, etc.). In general, the character of motion in

the threebody 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

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CELESTIAL MECHANICS: PAST, PRESENT, FUTURE 351

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

n

body

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 onetype operations and, consequently, is

very laborintensive 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

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 quasiperiodic 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 firstclass 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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SOLAR SYSTEM RESEARCH Vol. 47 No. 5 2013

BRUMBERG

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 highaccuracy

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 oddman

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

n

body 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 nonrigid 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 wellknown expression “the new is the wellfor

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., highprecision 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 shortperiod and longperiod

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 Vol. 47 No. 5 2013

CELESTIAL MECHANICS: PAST, PRESENT, FUTURE 353

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.

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 threedimensional space and

onedimensional 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 fourdimen

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 fourdimensional 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

354

SOLAR SYSTEM RESEARCH Vol. 47 No. 5 2013

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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 fourdimensional 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 quasiGalilean 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 socalled 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 Vol. 47 No. 5 2013

CELESTIAL MECHANICS: PAST, PRESENT, FUTURE 355

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

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 wellknown 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 quasiGalilean 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 nonphysical 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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SOLAR SYSTEM RESEARCH Vol. 47 No. 5 2013

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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 twobody 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 onebody problem. In

the Newtonian case (Kepler problem) the solution is

described by means of three linear parameters charac

terizing the size of the orbit (semimajor 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 onebody 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

v

2

/

c

2

. 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

firstorder terms with respect to this parameter (post

Newtonian approximation). Even within the second

order of accuracy with respect to this parameter (post

postNewtonian 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

v

5

/

c

5

are taken into account (an

approximation following the postpostNewtonian

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 (noninfinitesimal) 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 Vol. 47 No. 5 2013

CELESTIAL MECHANICS: PAST, PRESENT, FUTURE 357

(2) to use any welldefined 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 coordinateindependent 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 fourdimensional

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 fourdimensional coordinates

and the coordinateindependent measurable quanti

ti es ( int erval s of the proper tim e of an o bse rver, an gul ar

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 (nonspherical 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 fourdimensional 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 highlyaccurate theories of motion of

the major planets and the Moon, as well as the Earth’s

rotation theory have been constructed with account

358

SOLAR SYSTEM RESEARCH Vol. 47 No. 5 2013

BRUMBERG

ing for the main relativistic terms (postNewtonian

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

(postNewtonian 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

postNewtonian 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 highprecision 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.