Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.ipa.nw.ru/conference/2002/sovet/PS/BRUMBERG.PS Äàòà èçìåíåíèÿ: Mon Aug 19 15:47:01 2002 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 07:41:39 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: massive stars

IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Relativistic celestial mechanics--2002: results and
prospects
V. A. Brumberg
Institute of Applied Astronomy, St. Petersburg, Russia
1. Introduction
The aim of this paper is to review the present status of relativistic celestial
mechanics in its broadest sense. Nowadays many physicists and astrophysicists do
researches in the field of relativistic celestial mechanics. Due to these researches
the subject field of celestial mechanics has been significantly extended beyond the
scope of motion of the solar system bodies (weak gravitational fields). The main
attention is focused now on investigating motion of astrophysical and cosmolog­
ical objects such as black holes, gravitational waves, neutron stars, inspiralling
compact binaries, etc. (strong gravitational fields). This review covers four main
aspects of relativistic celestial mechanics, i.e. GRT and observations, GRT and
astrophysics, GRT and ephemeris astronomy, and GRT and classical celestial
mechanics problems.
2. GRT and observations
The detailed analysis of the current status of comparison of general relativity
(weak field and strong field effects) with observations is given in very compact
and elucidative form by Sch¨afer (2000). In spite of the present agreement between
experiment and theory new theoretical developments in astrophysics and cosmol­
ogy provide new motivations for pursuing the experimental tests of GRT. From
this point of view the confrontation between GRT and observations is analysed
by Damour (see relevant papers in http://arXiv.org, gr­qc series). restrict here
by exposing the main tests of GRT as described in (Sch¨afer, 2000 with references
therein).
GRT is based on the Equivalence Principle. A distinction is made between
the Weak Equivalence Principle (WEP), Einstein Equivalence Principle (EEP)
and Strong Equivalence Principle (SEP). WEP, i.e. the identity of inertial and
gravitational mass, is checked now experimentally within a precision of 1 \Theta 10 \Gamma12 .
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EEP, i.e. the local equivalence of gravitational and inertial fields involving the
universality of gravitational redshift, is verified with a precision of 2 \Theta 10 \Gamma4 . SEP,
i.e. the extension of EEP for self--gravitating test systems, is characterized by
Nordtvedt parameter j G = 4fi \Gamma fl \Gamma 3, fi and fl being the main PPN (Parametrized
Post--Newtonian) parameters equal to 1 in GRT. The present SEP tests result
in j G = \Gamma0:0007 \Sigma 0:0010. On the other hand, the LLR and VLBI deflection
measurements give fl = 0:9996 \Sigma 0:0017; fi = 0:9997 \Sigma 0:0005.
The LLR measurements enable also to check the effect of the geodetic pre­
cession. The presently highest relative precision amounts to 5 \Theta 10 \Gamma3 . Radar
measurements to planets and satellites result in an upper bound of the possible
variation of the Newtonian gravitational constant of j —
G=Gj ! 0:6 \Theta 10 \Gamma11 yr \Gamma1 .
Quite recently, the precession of the orbital planes of the Earth's artificial satel­
lites caused by the Earth's rotation (Lense--Thirring precession) was verified with
a precision about 20%.
All these effects are characteristic for the weak gravitational fields. The most
important GRT test for strong gravitational fields is related with binary pulsar
motion. Two close binary radio pulsars with neutron--star companions are used
now for testing strong--field effects of GRT: PSR B1913+16 and PSR B1534+12.
The consistent solution for masses and orbital elements of PSR B1913+16 proves
the correctness of GRT effects including the existence of gravitational waves. The
precision of this test amounts presently 0.35%. In the nearest future the analysis
of observations of PSR B1534+12 may be even more important permitting to
measure the corresponding geodetic precession and some strong--field effects of
alternative gravitation theories. The planned future space missions and ground
observatories designed for direct investigation of gravitational waves will result
in further tests for black holes and the big band.
3. GRT and astrophysics
Practically until two last decades of the XXth century all problems of rel­
ativistic celestial mechanics have been treated in the first post--Newtonian ap­
proximation (1PNA), i.e. within c \Gamma2 accuracy with respect to the Newtonian
terms. In this respect relativistic celestial mechanics of that time was simpler
mathematically than high--accuracy Newtonian celestial mechanics with its sub­
sequent approximations far beyond the first order. The situation changed with
the discovery of binary pulsar PSR B1913+16. To study its motion in taking into
account gravitational radiation it is necessary to derive and to solve the equations
of motion in 2.5PNA, i.e. within c \Gamma5 accuracy. It turns out that in 2PNA (c \Gamma4
accuracy) the N body problem does not differ qualitively from the corresponding
Newtonian problem (conservative dynamical system). The qualitative difference
reveals in 2.5PNA due to the loss of the energy of the system by gravitational
radiation. The system becomes non--conservative and irreversible (in time). The
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consistent solution for the binary pulsar problem in 2.5PNA obtained by celes­
tial mechanics techniques proved implicitly the existence of gravitational waves.
To get insight into further evolution of the binary when the distance between
bodies becomes less than the radius of the innermost stable circular orbit one
needs to proceed further approximations with respect to the GRT small parame­
ters. Present investigations (Jaranowski and Sch¨afer, 1997, 1998; Damour et al.,
2000a,b, 2001) deal with 3PNA and even 3.5PNA ( c \Gamma6 and c \Gamma7 accuracy, respec­
tively). Such higher--order approximations are necessary for understanding the
processes of coalescing inspiralling galaxies and gravitational wave emission. The
analytical techniques applied in these investigations extend the arsenal of existing
methods of celestial mechanics (as an effective one--body approach to two--body
dynamics developed by Buonanno and Damour, 1999).
On the other hand, these techniques are complemented by numerical relativity
methods for the regions deeply inside the innermost stable circular orbit.
4. GRT and ephemeris astronomy
First GRT--based IAU resolutions on reference systems and time scales were
adopted by the IAU in 1991. This event may be regarded as recognition of rel­
ativistic character of modern ephemeris astronomy both with respect to its the­
oretical accuracy and observational precision. The latest IAU resolutions were
adopted in 2000 (IAU, 2001). Much remains to be done for realization of these
resolutions (Brumberg and Groten, 2001). IAU resolutions demand to consider
two principal astronomical reference systems ICRS and ITRS as relativistic four--
dimensional systems with TCB and TCG, respectively, as their time scales. In
practice, these systems are often used as three--dimensional Newtonian systems
in combination to TDB and TT, respectively. This fact causes a lot of confusion.
IAU(2000) resolutions involve one more system, GCRS, to be served as an
intermediary between ICRS and ITRS. To avoid any GRT ambiguities in inter­
preting ephemeris astronomy concepts one needs even more reference systems at
the barycentric and geocentric level.
Accurate analytical expression for the difference TDB--TT (in the geocentre)
has been given in (Fairhead and Bretagnon, 1990 ; see also Irwin and Fukushima,
1999). Guinot (2000) pointed out the necessity to take into account the constant
value of the mixed and trigonometric terms in this difference for the rigorous
fulfillment of the IAU resolutions (both the time scales difference and the geodetic
rotation vector are determined by differential equations and one should specify
initial conditions in solving these equations).
New advances in solving the equations of light propagation are made in
(Kopeikin et al.,1999) and (Blanchet et al., 2001).
The most accurate algorithms of relativistic reduction of astronomical obser­
vations are proposed by Klioner (2001) for space astrometry, by Kopeikin and
39

Ozernoy (1999) for binary star observations and by Klioner (1991) for VLBI
observations.
Numerical planetary theories have been analysed in (Pitjeva, 2001 and refer­
ences therein).
5. GRT and classical celestial mechanics problems
GRT planetary equations used now in ephemeris astronomy represent the
well--known EIH (Einstein--Infeld--Hoffman) equations for the point masses. Vari­
ous generalizations of these equations for the rotating extended masses (including
the equations for rotational motion) were proposed in the second half of the last
century but the physical structure of the bodies was often considered there rather
formally (not violating mathematical correctness of these equations). Only recent­
ly the physically reliable equations for the binaries with consideration of spin and
quadrupole moments were derived in (Xu et al., 1997). Various ways to derive
the explicit rotational equations of motion of celestial bodies are discussed in
(Klioner and Soffel, 1998, 1999).
Earth's satellite equations were analysed in (Damour and Esposito--Far`ese,
1994) with respect to the major GRT effects. Klioner (2001) gave these equations
in explicit form with accuracy more than enough for present practical purposes.
All these equations intended to study the motion of the solar system bodies
are derived in 1PNA. Using the 2.5PNA equations for binary pulsar it is possi­
ble to study the motion of a test particle in the binary pulsar gravitational field
(Brumberg, 2002). As a simple example of a non--conservative and irreversible (in
time) type of motion one may consider the relativistic restricted quasi--circular
three--body problem with gravitational radiation taken into account. In the sim­
plest approximation the equations of motion of such problem have formally the
Newtonian form with coordinates of the binary (K = 1; 2, i = 1; 2)
x i
K = \Gamma(\Gamma1) K MK
M R i ; R i = R
`
cos u
sin u
'
; R = A(1 \Gamma 2k\Lambda); u = \Lambda + 3
2
k\Lambda 2 ;
with
\Lambda = N t + \Lambda 0 ; k = 32
5
c \Gamma5 N 5 A 5 ¯; ¯ = M 1 M 2
M 2
;
N; A; M being mean motion, semi--major axis and sum of mass of the binary,
respectively. The simplest quasi--circular plane solution for a test particle ar large
distance from the binary reads
`
x
y
'
= a
i
1 + 3k¯ A 2
a 2
\Lambda
j `
cos '
sin '
'
; ' = – \Gamma 3k¯ A 2
a 2
n
N
\Lambda 2 ; – = nt + '';
n; a being unperturbed values for the mean motion and semi--major axis of a test
particle.
40

6. Conclusion
It is of interest to see the list of the most important unsolved problems in
astrophysics given by Wesson (2001). The list contains 20 fundamental problems
of modern astrophysics. It seems that at least a quarter of them should be treated
by methods of relativistic celestial mechanics underlying its role in setting closer
relationship between astrophysics and celestial mechanics.
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