The Exact Solution to the General Relativistic Differential Equation describing Planetary Orbits

Abstract

The exact analytical solution to the differential equation describing a planetary orbit, including General Relativistic effects, is derived and expressed in terms of a Jacobi elliptic function. Moreover, the exact expression for the perihelion precession rate is derived. The perihelion precession rate for the planet Mercury, estimated from the exact expression, is 42.98 arcseconds/century (the observed value is $$43.1 \pm 0.5$$ arcseconds/century), and agrees with values found by perturbation methods. After one million years the perihelion advance of Mercury is about 120 degrees, and the analytical solution agrees exactly with this result. Values for the perihelion advance of several other planets and asteroids, estimated by the present method, are presented and compared with observed values, and the agreement is exact as compared to the values obtained by perturbation theory, and consistent with the observed ones. Values calculated by the present method are based on three parameters: the distance at aphelion, the distance at perihelion, and the number of revolutions per century.