Quadrature solution for the general relativistic motion of a satellite or a planet

Abstract

The Schwarzschild field of a central massM is used to derive the general relativistic motion of a particle in a bounded orbit aroundM. A quadrature gives the central angle ϕ as a quasi-periodic function ϕ (f) of an effective true anomalyf. The linear term is an infinite series, whose second term yields the usual rate of advance of pericenter. For an artificial satellite this may be as large as 17″ of arc per year. The periodic part is a sine series, with coefficients containing the small parameter β≡2GM/c ² p, wherep is closely approximated by the classical semi-latus rectum. The radius vectorr is a Kepler-like function off.

The essentially new features of the calculation are the appropriate factoring of a certain cubic polynomialF(p/r), the use of the above effective true anomalyf, and the introduction of an effective eccentric anomalyE. The latter serves to reduce the differential equation forf as a function of the timet, obtained by combining the solution for ϕ(f) with the relativistic integrals of motion, to a Kepler equation forE.

Knowing the constants of the motion, one can then solve successively forE(t), f(t), r(t), and ϕ(t). This is best done as a variational calculation, comparing the relativistic orbiter with a classical orbiter having the same initial conditions. The resulting variations agree with those of Lass and Solloway, but the present method is quite different from theirs and results in a simpler algorithm. The results show that the radial and transverse corrections, δr andr δϕ, arising from the Schwarzschild field, may be of the order of a kilometer for 1000 revolutions of an Earth satellite of orbital eccentricitye ₀≈0.6.

For bounded motion, the cubic polynomialF(p/r) has three positive real zeros, the two smaller ones corresponding to apocenter and pericenter. The third and apparently non-physical one occurs forr≈Schwarzschild radius. It may thus correspond to the incipient fall of the orbiter into the central body, if the latter is a black hole.