Numerical algorithms for time-delayed Newtonian gravity

Abstract

The anomalous precession of the perihelion of Mercury troubled the brilliant scientists of the time when it was first discovered. With ideas ranging from the possibilities of a new planet to interstellar dust, it was a mystery until Einstein correctly described it using his theory of general relativity in the early 20th century. This project revisits the idea of a finite propagation speed of gravity, using time-delayed Newtonian gravity as a means to investigate if the anomalous precession can be explained in a simpler way. The delay-differential equations that arise are solved using numerical analysis, with a modified Runge-Kutta-Fehlberg integrator. When solving delay-differential equations, either constant or state dependent time delays typically must be considered. For the time-delayed gravity it is shown that the time delays must instead be implicit and history-dependent, and an efficient method of solving these are proposed. Finally, it is shown that time-delayed gravity is not sufficient to describe the anomalous precession and that the systems lose energy over time, dependent on the speed of gravity. This verifies the work done analytically in the 18th and 19th century.