A Possible Solution to a Relativistic Orbit for the 2-Body Celestial Mechanics Problem

The 'captured' 2-body Kepler problem considering relativity is solved by using an iterative integral equation. The rationale for this approach is to increase the accuracy of the limits for a satellite's motion and potentially provide a test to evaluate different gravitational laws. Moreover, this effort might provide additional insights to resolve other anomalies such as the flyby anomaly, the Faint Young Sun paradox, the Pioneer anomaly and other inconsistencies that potentially may be used to validate Einstein's Theory of Relativity. The mathematical solution solves a nonlinear Volterra integral equation using an iterative fashion, which reveals a correction factor for treating a given closed orbit. However. This correction factor is not a constant value but rather a function of the elliptical or circular orbit angular displacement. This function may be insignificant during portions of the trajectory, say at apogees or perigees. Nonetheless, these results are encouraging where relativity effects may or may not exist to understand and resolve several of the other variances or gravitational anomalies currently related to our solar system.