On the foundations of general relativistic celestial mechanics

Towards the end of nineteenth century, Celestial Mechanics provided the most powerful tools to test Newtonian gravity in the solar system and also led to the discovery of chaos in modern science. Nowadays, in light of general relativity, Celestial Mechanics leads to a new perspective on the motion of satellites and planets. The reader is here introduced to the modern formulation of the problem of motion, following what the leaders in the field have been teaching since the nineties, in particular, the use of a global chart for the overall dynamics of N bodies and N local charts describing the internal dynamics of each body. The next logical step studies in detail how to split the N-body problem into two sub-problems concerning the internal and external dynamics, how to achieve the effacement properties that would allow a decoupling of the two sub-problems, how to define external-potential-effacing coordinates and how to generalize the Newtonian multipole and tidal moments. The review paper ends with an assessment of the nonlocal equations of motion obtained within such a framework, a description of the modifications induced by general relativity on the theoretical analysis of the Newtonian three-body problem, and a mention of the potentialities of the analysis of solar-system metric data carried out with the Planetary Ephemeris Program.

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Orbits of Trojan Asteroids

Symposium - International Astronomical Union ◽

10.1017/s007418090007039x ◽

1974 ◽

Vol 62 ◽

pp. 63-69 ◽

Cited By ~ 1

Author(s):

G. A. Chebotarev ◽

N. A. Belyaev ◽

R. P. Eremenko

Keyword(s):

Solar System ◽

Body Problem ◽

Equations Of Motion ◽

Orbital Evolution ◽

Restricted Three Body Problem ◽

Three Body Problem ◽

Trojan Asteroids ◽

Actual Motion ◽

Three Body

In this paper the orbital evolution of Trojan asteroids are studied by integrating numerically the equations of motion over the interval 1660–2060, perturbations from Venus to Pluto being taken into account. The comparison of the actual motion of Trojans in the solar system with the theory based on the restricted three-body problem are given.

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Orbital stability of high inclination asteroids

Symposium - International Astronomical Union ◽

10.1017/s0074180900127330 ◽

1996 ◽

Vol 172 ◽

pp. 187-192

Author(s):

N. A. Solovaya ◽

E. M. Pittich

Keyword(s):

Solar System ◽

Body Problem ◽

Equations Of Motion ◽

Orbital Stability ◽

Dynamical Model ◽

Restricted Three Body Problem ◽

Three Body Problem ◽

Three Body

The orbital evolutions of fictitious asteroids with high inclinations have been investigated. The selected initial orbits represent asteroids with movement, which corresponds to the conditions of the Tisserand invariant for C = C (L1) in the restricted three body problem. Initial eccentricities of the orbits cover the interval 0.0–0.4, inclinations the interval 40–80°, and arguments of perihelion the interval 0–360°. The equations of motion of the asteroids were numerically integrated from the epoch March 25, 1991 forward within the interval of 20,000 years, using a dynamical model of the solar system consisting of all planets. The orbits of the model asteroids are stable at least during the investigated period.

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Stability of the Moons Orbits in Solar System in the Restricted Three-Body Problem

Advances in Astronomy ◽

10.1155/2015/615029 ◽

2015 ◽

Vol 2015 ◽

pp. 1-7 ◽

Cited By ~ 6

Author(s):

Sergey V. Ershkov

Keyword(s):

Solar System ◽

Effective Mass ◽

Body Problem ◽

Equations Of Motion ◽

Centre Of Mass ◽

Three Body Problem ◽

Dynamical Parameter ◽

Lagrange Form ◽

Elliptic Orbits ◽

Three Body

We consider the equations of motion of three-body problem in aLagrange form(which means a consideration of relative motions of 3 bodies in regard to each other). Analyzing such a system of equations, we consider in detail the case of moon’s motion of negligible massm3around the 2nd of two giant-bodiesm1,m2(which are rotating around their common centre of masses on Kepler’s trajectories), the mass of which is assumed to be less than the mass of central body. Under assumptions of R3BP, we obtain the equations of motion which describe the relative mutual motion of the centre of mass of 2nd giant-bodym2(planet) and the centre of mass of 3rd body (moon) with additional effective massξ·m2placed in that centre of massξ·m2+m3, whereξis the dimensionless dynamical parameter. They should be rotating around their common centre of masses on Kepler’s elliptic orbits. For negligible effective massξ·m2+m3it gives the equations of motion which should describe aquasi-ellipticorbit of 3rd body (moon) around the 2nd bodym2(planet) for most of the moons of the planets in Solar System.

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On solar system dynamics in general relativity

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887817501171 ◽

2017 ◽

Vol 14 (09) ◽

pp. 1750117 ◽

Cited By ~ 3

Author(s):

Emmanuele Battista ◽

Giampiero Esposito ◽

Luciano Di Fiore ◽

Simone Dell’Agnello ◽

Jules Simo ◽

...

Keyword(s):

General Relativity ◽

Solar System ◽

System Dynamics ◽

Body Problem ◽

The Sun ◽

Three Body Problem ◽

Lagrangian Points ◽

The Earth ◽

Solar System Dynamics ◽

Three Body

Recent work in the literature has advocated using the Earth–Moon–planetoid Lagrangian points as observables, in order to test general relativity and effective field theories of gravity in the solar system. However, since the three-body problem of classical celestial mechanics is just an approximation of a much more complicated setting, where all celestial bodies in the solar system are subject to their mutual gravitational interactions, while solar radiation pressure and other sources of nongravitational perturbations also affect the dynamics, it is conceptually desirable to improve the current understanding of solar system dynamics in general relativity, as a first step towards a more accurate theoretical study of orbital motion in the weak-gravity regime. For this purpose, starting from the Einstein equations in the de Donder–Lanczos gauge, this paper arrives first at the Levi-Civita Lagrangian for the geodesic motion of planets, showing in detail under which conditions the effects of internal structure and finite extension get canceled in general relativity to first post-Newtonian order. The resulting nonlinear ordinary differential equations for the motion of planets and satellites are solved for the Earth’s orbit about the Sun, written down in detail for the Sun–Earth–Moon system, and investigated for the case of planar motion of a body immersed in the gravitational field produced by the other bodies (e.g. planets with their satellites). At this stage, we prove an exact property, according to which the fourth-order time derivative of the original system leads to a linear system of ordinary differential equations. This opens an interesting perspective on forthcoming research on planetary motions in general relativity within the solar system, although the resulting equations remain a challenge for numerical and qualitative studies. Last, the evaluation of quantum corrections to location of collinear and noncollinear Lagrangian points for the planar restricted three-body problem is revisited, and a new set of theoretical values of such corrections for the Earth–Moon–planetoid system is displayed and discussed. On the side of classical values, the general relativity corrections to Newtonian values for collinear and noncollinear Lagrangian points of the Sun–Earth–planetoid system are also obtained. A direction for future research will be the analysis of planetary motions within the relativistic celestial mechanics set up by Blanchet, Damour, Soffel and Xu.

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Chaotic Amplification in the Relativistic Restricted Three-body Problem

Zeitschrift für Naturforschung A ◽

10.1515/zna-2003-0102 ◽

2003 ◽

Vol 58 (1) ◽

pp. 13-22 ◽

Cited By ~ 1

Author(s):

Lucas F. Wanex

Keyword(s):

General Relativity ◽

Body Problem ◽

Equations Of Motion ◽

Restricted Three Body Problem ◽

Three Body Problem ◽

Orbital System ◽

The Earth ◽

The Difference ◽

Three Body ◽

Relativistic Equations

The relativistic equations of motion for the restricted three-body problem are derived in the first post-Newtonian approximation. These equations are integrated numerically for seven different trajectories in the earth-moon orbital system. Four of the trajectories are determined to be chaotic and three are not chaotic. Each post-Newtonian trajectory is compared to its Newtonian counterpart. It is found that the difference between Newtonian and post-Newtonian trajectories for the restricted three-body problem is greater for chaotic trajectories than it is for trajectories that are not chaotic. Finally, the possibility of using this Chaotic Amplification Effect as a novel test of general relativity is discussed.

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THE SCALAR ETHER-THEORY OF GRAVITATION AND ITS FIRST TEST IN CELESTIAL MECHANICS

International Journal of Modern Physics A ◽

10.1142/s0217751x0201323x ◽

2002 ◽

Vol 17 (29) ◽

pp. 4203-4208 ◽

Cited By ~ 3

Author(s):

MAYEUL ARMINJON

Keyword(s):

General Relativity ◽

Scalar Field ◽

Solar System ◽

Celestial Mechanics ◽

Equations Of Motion ◽

Alternative Interpretation ◽

Crucial Point ◽

Ether Theory ◽

Theory Of Gravitation ◽

Basic Equations

The motivations for investigating a theory of gravitation based on a concept of "ether" are discussed — a crucial point is the existence of an alternative interpretation of special relativity, named the Lorentz-Poincaré ether theory. The basic equations of one such theory of gravity, based on just one scalar field, are presented. To check this theory in celestial mechanics, an "asymptotic" scheme of post-Newtonian (PN) approximation is summarized and its difference with the standard PN scheme is emphasized. The derivation of PN equations of motion for the mass centers, based on the asymptotic scheme, is outlined. They are implemented for the major bodies of the solar system and the prediction for Mercury is compared with an ephemeris based on general relativity.

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The post-Newtonian approximation

10.1093/oso/9780198786399.003.0052 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

General Relativity ◽

Gravitational Field ◽

Body Problem ◽

Equations Of Motion ◽

Two Body Problem ◽

Newtonian Approximation ◽

Actual Motion ◽

Law Of Attraction ◽

Universal Law ◽

Two Bodies

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.

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On the Stability of Collinear Points of the RTBP with Triaxial and Oblate Primaries and Relativistic Effects

Current Journal of Applied Science and Technology ◽

10.9734/cjast/2021/v40i331285 ◽

2021 ◽

pp. 56-73

Author(s):

S. E. Abd El-Bar

Keyword(s):

Characteristic Equation ◽

Body Problem ◽

Equilibrium Points ◽

Equations Of Motion ◽

Relativistic Effects ◽

Three Body Problem ◽

Total Potential ◽

The Mean ◽

The Stability ◽

Three Body

Under the influence of some different perturbations, we study the stability of collinear equilibrium points of the Restricted Three Body Problem. More precisely, the perturbations due to the triaxiality of the bigger primary and the oblateness of the smaller primary, in addition to the relativistic effects, are considered. Moreover, the total potential and the mean motion of the problem are obtained. The equations of motion are derived and linearized around the collinear points. For studying the stability of these points, the characteristic equation and its partial derivatives are derived. Two real and two imaginary roots of the characteristic equation are deduced from the plotted figures throughout the manuscript. In addition, the instability of the collinear points is stressed. Finally, we compute some selected roots corresponding to the eigenvalues which are based on some selected values of the perturbing parameters in the Tables 1, 2.

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