From geodesics of the multipole solutions to the perturbed Kepler problem

Abstract

A static and axisymmetric solution of the Einstein vacuum equations with a finite number of Relativistic Multipole Moments (RMM) is written in MSA coordinates up to certain order of approximation, and the structure of its metric components is explicitly shown. From the equation of equatorial geodesics we obtain the Binet equation for the orbits and it allows us to determine the gravitational potential that leads to the equivalent classical orbital equations of the perturbed Kepler problem. The relativistic corrections to Keplerian motion are provided by the different contributions of the RMM of the source starting from the Monopole (Schwarzschild correction). In particular, the perihelion precession of the orbit is calculated in terms of the quadrupole and 2${}^4$-pole moments. Since the MSA coordinates generalize the Schwarzschild coordinates, the result obtained allows measurement of the relevance of the quadrupole moment in the first order correction to the perihelion frequency-shift.

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Cited by in corpus (9)

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• Source integrals of multipole moments for static space-times
• Event Horizon of the Monopole-Quadrupole solution: geometric and thermodynamic properties
• Whom actually do multipole moments belong to?
• Optical properties of null geodesics emerging from dynamical systems
• A quadrupolar generalization of the Erez-Rosen coordinates
• Gravitational fields of axially symmetric compact objects in 5D space-time-matter gravity
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