Simple and reliable analytic approximation to the numerical solution of the relativistic Binet’s equation: an application to Mercury

The nonlinear trajectory equation (Binet’s equation) for a particle in a relativistic force field can only be solved numerically or, alternatively, by using a perturbational solution scheme. The latter approach was successfully applied by Albert Einstein in 1915 to deduce the celebrated formula that explains the anomalous precession of the perihelion of Mercury. In this article, Binet’s equation for Mercury is solved numerically to a high degree of accuracy (16 decimal digits). This is a necessary comparison basis for the main goal of this work – to deduce a simple analytic formula that perfectly reproduces the real relativistic trajectory. Several analytical models are proposed, and the main goal has been indeed achieved. Moreover, the fitting parameters for model D described in Section 3 can be obtained independently of the solution of Binet’s equation. Thus, we can say that the highly accurate relativistic trajectory (the largest discrepancy being about 30 cm) can be obtained without actually solving the nonlinear differential equation for this trajectory.