The advance of perihelion, in particular for Mercury, is regarded as a classical test of general relativity, but a number of other (in some cases much larger) contributions to this phenomenon are seldom discussed in detail in textbooks. This paper presents a unified framework for evaluating the advance of perihelion due to (a) general relativity, (b) the solar quadrupole moment, and (c) planetary perturbations, the last in a ring model where the mass of each perturbing planet is “smeared out” into a coplanar circular orbit. The exact solution of the ring model agrees to within 4% with the usually quoted figure. Time-dependent contributions beyond the ring model contain some surprising features: they are not small, and some with long periods could mimic a secular advance.
We are grateful to K.-F. Li, L. M. Lin, and B. Sheen for advice on astronomical coordinates, and to C. M. Will for advice on the PPN formalism and for pointing us to the literature.20 P. Goldreich and K.-F. Li have kindly commented on the manuscript.
Article outline: I. INTRODUCTION II. GENERAL FORMALISM A. Kepler problem B. Time-independent perturbations III. GENERAL RELATIVITY IV. QUADRUPOLE MOMENT V. PLANETARY PERTURBATIONS A. Ring model B. Discussion VI. TIME-DEPENDENT CONTRIBUTIONS A. Derivation B. Comparison of terms C. Time averaging VII. CONCLUSION
36.Whether Pluto is regarded as a planet is irrelevant.
37.I. S. Gradyshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York and London, 1965).
38.This is evidenced by the first-order expansion in in arriving at their Eq. (10).
39.In our formalism and in that of Stewart,22 it does not matter whether the expansion is carried out around or , since all terms are kept.
40.When only one planet is discussed, there is the freedom to change , , so only enters but not . But when is already defined by the orbit of one planet, this freedom is no longer available to change the sign of for another planet, so corrections will enter to .
41.A sum over different planets i is understood.
42.See, e.g., J. D. Jackosn, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 92, Eq. (3.38) and p. 102, Eq. (3.70).
43.Heliocentric longitudes on 1 January 1960 are found from <http://cohoweb.gsfc.nasa.gov/helios/planet.html>: for the perihelion of Mercury, and , respectively for the longitudes of Uranus and Neptune. Within the approximation of circular orbits for the perturbing planets, the latter two are calculated at other times by assuming a uniform rate of change according to the known periods.
44.This situation is not axiomatic. One can imagine a hypothetical system where for Venus.
45.Replacing the finite difference by a derivative in N is justified only for those terms with .
46.O. I. Chashchina and Z. K. Silagadze, “ Remark on orbital precession due to central-force perturbations,” Phys. Rev. D77, 107502 (2008).