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A classification of particle motions in the equatorial plane of a gravitational monopole-quadrupole field in Newtonian mechanics and general relativity

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Abstract

The motion of particles in a gravitational monopole-quadrupole field is considered both in Newtonian mechanics and the general theory of relativity. The Newtonian calculations are based on the first and third terms in the familiar multipole expansion of Laplace for axially symmetric matter; the relativistic ones are based on the Erez-Rosen metric, an exact solution of Einstein's vacuum field equations. Use is made of the mathematical equivalence between the Newtonian equations of motion in the plane of symmetry and the general relativistic equations in the spherically symmetric Schwarzschild field. By an extension of the method used by Morton and Leavitt to obtain the Schwarzschild geodesics, exact solutions for the Newtonian orbits are obtained in terms of Jacobian elliptic functions, and a complete classification of the orbits is given. The motions are all quasi-Keplerian, except for a curious subclass of orbits corresponding to a form of two-body capture. Approximate equatorial geodesics in the general relativistic case are also found and classified. The lowest order solutions, which are essentially a superposition of relativistic monopole (Schwarzschild) and Newtonian quadrupole contributions, are shown to give accurate descriptions for motions in the solar system. In higher order solutions, purely relativistic (but observationally insignificant) cross terms appear. The results are used to study the effect of a solar oblateness on the three classical tests of general relativity.

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Notes and References

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Author notes

  1. Angelo Armenti Jr.

    Present address: Dept. of Physics, Villanova University, 19085, Villanova, Pa.

Authors and Affiliations

  1. Dept. of Physics, Temple University, 19122, Philadelphia, Pa., U.S.A.

    Angelo Armenti Jr.

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Research supported by the National Science Foundation.

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Armenti, A. A classification of particle motions in the equatorial plane of a gravitational monopole-quadrupole field in Newtonian mechanics and general relativity. Celestial Mechanics 6, 383–415 (1972). https://doi.org/10.1007/BF01227754

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