Relativistic Epicycles : another approach to geodesic deviations

Details Relativistic Epicycles : another approach to geodesic deviations Relativistic Epicycles : another approach to geodesic deviations

2001-02-22

arxiv.org/abs/gr-qc/0102099v2

Class.Quant.Grav. 18 (2001) 4725-4742

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

23 pages, LaTeX, 2 Postscript figures. Version substantially improved to be published in Classical and Quantum Gravity, with :

Scientific paper

10.1088/0264-9381/18/22/302

We solve the geodesic deviation equations for the orbital motions in the Schwarzschild metric which are close to a circular orbit. It turns out that in this particular case the equations reduce to a linear system, which after diagonalization describes just a collection of harmonic oscillators, with two characteristic frequencies. The new geodesic obtained by adding this solution to the circular one, describes not only the linear approximation of Kepler's laws, but gives also the right value of the perihelion advance (in the limit of almost circular orbits). We derive also the equations for higher-order deviations and show how these equations lead to better approximations, including the non-linear effects. The approximate orbital solutions are then inserted into the quadrupole formula to estimate the gravitational radiation from non-circular orbits.

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