Discovery of Neptune

Abstract: This project is an analysis of how a planet can be found in space with the aid of mathematics. This is based on the fact that in the 19th century two mathematicians John C. Adams and Urbain Le Verrier both independent of each other found Neptune, the 8th planet in the solar system, by calculating its location based on discrepancies between theoretical and observed longitudes. We recreate Adams’ problem and solve it with numerical analysis to see how one could improve this method of finding a planet using mathematics. We created a model of the solar system using Runge-Kutta 4 (RK4) to solve ODE’s explaining how the planets affect each other. We then created an inverse problem where we pretended that Neptune did not exist and tried to find its position and data using Gauss-Newton’s algorithm. Our method gives a better result than those of Adams, although we use a better start guess for the position of Neptune than he did. The important parameter to find is at what direction to look for the planet, also called the longitude angle. Both Adams and us get close to the correct longitude—Adams’ being 2:5_ off and us within 1_. This is especially interesting since without getting this parameter correct they would never have found the planet at that time.