# Exact and Approximate Relativistic Corrections to the Orbital Precession of Mercury

Exact and Approximate Relativistic Corrections to the Orbital Precession of Mercury

Accounting for the anomaly in the perihelion precession of Mercury provided early support for Einstein's General Theory of Relativity [1]. Very conveniently, accurate astronomical data was already available. Schwarzschild's solution to Einstein's gravitational field equations can be approximated by Newtonian gravity perturbed by a short-range term of magnitude proportional to the Schwarzschild radius =2GM/, where is the universal gravitational constant, is the attracting mass (the Sun) and is the speed of light [2, 3].

r

s

2

c

G

M

c

Assuming negligible gravitational radiation and using the Schwarzschild spacetime geometry, the precessing orbit of a test mass (the planet Mercury) can be solved exactly, in terms of the Weierstrass function [4, 5]. The same solution pertains to a star orbiting a black hole [6]. On many astronomical scales, is a small number, so that perturbation theory is applicable. Using the methods of [7] and Approximating the Jacobian Elliptic Functions, we calculate an approximation to the exact solution as an expansion in a parameter up to order [8]. This Demonstration shows that in many cases the complicated Weierstrass function is not necessary for high-precision analysis. An approximation of sufficient precision can usually be obtained in a relatively short time.

m

r

s

β∝

E

N=10