Full asymptotic expansion of the relativistic orbit of a test particle under the exact Schwarzschild metric

Do-Nhat, T.

doi:10.1016/s0375-9601(97)00883-9

Cited by 8 publications

(11 citation statements)

References 17 publications

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“…This formula agrees with the results obtained with the Poincaré-Lindstedt method and some of its equivalent modifications 8,15 . By denoting with P the anomalistic period, that is the time that elapses between two passages of the object at its perihelion expressed in days, the average advance rate is…”

Section: The Iteration Methodssupporting

confidence: 90%

“…In particular, if we consider a geodesic which at τ = 0 starts within the symmetry hyperplane and is tangent to it, it must coincide with the transformed geodesic, since the initial values of position and velocity determine a geodesic uniquely. These considerations are confirmed by the analysis of the Euler-Lagrange equation for θ θ + 2ṙ r θ − φ2 2 sin 2θ = 0, (8) which admits the solution θ = π/2 satisfying the initial conditions θ 0 = π/2, θ0 = 0. If we reorient the coordinate system so that these conditions are met, the motion takes place in the equatorial plane, and we can simplify the Lagrangian (2) assuming sin θ = 1, θ = 0…”

Section: Introductionmentioning

confidence: 64%

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Higher-order corrections to the relativistic perihelion advance and the mass of binary pulsars

D’Eliseo¹

2010

Astrophys Space Sci

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We study the general relativistic orbital equation and using a straightforward perturbation method and a mathematical device first introduced by d'Alembert, we work out approximate expressions of a bound planetary orbit in the form of trigonometrical polynomials and the first three terms of the power series development of the perihelion advance. The results are applied to a more precise determination of the total mass of the double pulsar J0737-3039.

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Section: The Iteration Methodssupporting

confidence: 90%

Section: Introductionmentioning

confidence: 64%

Higher-order corrections to the relativistic perihelion advance and the mass of binary pulsars

D’Eliseo¹

2010

Astrophys Space Sci

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“…(1.9) to 2PN order. Do-Nhat [14] used an asymptotic series method to solve the geodesic equation in Schwarzschild coordinates and expressed the pericenter advance in terms of E and L, in agreement with Eq. (1.9) to 3PN order.…”

Section: Discussionmentioning

confidence: 99%

Pericenter advance in general relativity: comparison of approaches at high post-Newtonian orders

Tucker

Will

2019

Class. Quantum Grav.

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The advance of the pericenter of the orbit of a test body around a massive body is a standard textbook topic in general relativity. It can be calculated in a number of ways. In one method, one studies the geodesic equation in the exact Schwarzschild geometry and finds the angle between pericenters as an integral of a certain radial function between turning points of the orbit. This can be done in several different coordinate systems. In another method, one describes the orbit using osculating orbit elements, and analyzes the "Lagrange planetary equations" that give the evolution of the elements under the perturbing effects of post-Newtonian (PN) corrections to the motion. After separating the perturbations into periodic and secular effects, one obtains an equation for the secular rate of change of the pericenter angle. While the different methods agree on the leading post-Newtonian contribution to the advance, they do not agree on the higher-order PN corrections. We show that this disagreement is illusory. When the orbital variables in each case are expressed in terms of the invariant energy and angular momentum of the orbit and when account is taken of a subtle difference in the meaning of "pericenter advance" between the two methods, we show to the third post-Newtonian order that the different methods actually agree perfectly. This illustrates the importance of expressing results in general relativity in terms of invariant or observable quantities.

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“…The problem of calculating at the second post-Newtonian (2PN) order of general relativity (Debono & Smoot 2016) the secular 1 precession ω2PN of pericentre ω of a full two-body system made of a pair of detached, non-rotating masses of comparable sizes and of a restricted two-body system characterized by a test particle orbiting its massive primary has been analytically tackled several times so far with a variety of calculational approaches (Hoenselaers 1976;Damour & Schaefer 1987;Damour & Schafer 1988;Ohta & Kimura 1989;Schäfer & Wex 1993a,b;Kopeikin & Potapov 1994;Wex 1995;Do-Nhat 1998;Memmesheimer, Gopakumar & Schäfer 2004;Königsdörffer & Gopakumar 2005;Heng & Zhao 2009;D'Eliseo 2011;Bagchi 2013;Blanchet 2014;Gergely & Keresztes 2015;Will & Maitra 2017;Marín & Poveda 2018;Mak, Leung & Harko 2018;Tucker & Will 2019;Walters 2018;. In spite of their formal elegance, it is not always easy to extract from them quickly understandable formulas, ready to be read and used in practical calculations in view of possible confrontation with actual data from astronomical and astrophysical scenarios of potential experimental interest.…”

Section: Introductionmentioning

confidence: 99%

Revisiting the 2PN Pericenter Precession in View of Possible Future Measurements

Iorio

2020

Universe

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At the second post-Newtonian (2PN) order, the secular pericentre precessionω 2PN of either a full two-body system made of well detached non-rotating monopole masses of comparable size and a restricted two-body system composed of a point particle orbiting a fixed central mass have been analytically computed so far with a variety of approaches. We offer our contribution by analytically computingω 2PN in a perturbative way with the method of variation of elliptical elements by explicitly calculating both the direct contribution due to the 2PN acceleration A 2PN , and also an indirect part arising from the self-interaction of the 1PN acceleration A 1PN in the orbital average accounting for the instantaneous shifts induced by A 1PN itself. Explicit formulas are straightforwardly obtained for both the point particle and full two-body cases without recurring to simplifying assumptions on the eccentricity e. Two different numerical integrations of the equations of motion confirm our analytical results for both the direct and indirect precessions. The values of the resulting effects for Mercury and some binary pulsars are confronted with the present-day level of experimental accuracies in measuring/constraining their pericentre precessions. The supermassive binary black hole in the BL Lac object OJ 287 is considered as well. A comparison with some of the results appeared in the literature is made.

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Full asymptotic expansion of the relativistic orbit of a test particle under the exact Schwarzschild metric

Cited by 8 publications

References 17 publications

Higher-order corrections to the relativistic perihelion advance and the mass of binary pulsars

Higher-order corrections to the relativistic perihelion advance and the mass of binary pulsars

Pericenter advance in general relativity: comparison of approaches at high post-Newtonian orders

Revisiting the 2PN Pericenter Precession in View of Possible Future Measurements

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