Incursive Algorithms for Newtonian and Relativistic Gravitations, and Simulation of the Mercury Orbit

This paper firstly recalls the mathematical model of the Newtonian gravitation law applied to the solar system dealing with the Sun and the Mercury planet. The orbits of all the planets are given by closed ellipses in the Newton paradigm. This paper continues with the introduction of the relativistic correction to this Newton gravitation law, for exhibiting the precession of orbits and, more particularly, the advance of the perihelion of the Mercury planet. The following section gives the Euler and the First Incursive algorithms for the classical Newton gravitation law and the relativistic correction of this Newtonian law. The last section of this paper gives the result of the numerical simulations of the Mercury orbit around the Sun. It is shown that the simulation with the Euler algorithm is not stable and does not give a closed ellipse to the Mercury orbit with the Newtonian law. The simulation with the First Incursive algorithm gives a perfect simulation of the Newtonian orbit in 88 days. Then, the simulation with this First Incursive algorithm of the relativistic Newtonian gravitation shows the correct Mercury precession angle that is equal to 80 = 180 degrees, after 15,000 centuries, in agreement with the experimental data and Einstein relativity.