A simple procedure to extend the Gauss method of determining orbital parameters from three to N points

Abstract

A simple procedure is developed to determine orbital elements of an object orbiting in a central force field which contribute more than three independent celestial positions. By manipulation of formal three point Gauss method of orbit determination, an initial set of heliocentric state vectors ${\bf r}_i$ and $\dot{\bf{r}}_i$ is calculated. Then using the fact that the object follows the path that keep the constants of motion unchanged, I derive conserved quantities by applying simple linear regression method on state vectors ${\bf r}_i$ and $\dot{\bf{r}}_i$. The best orbital plane is fixed by applying an iterative procedure which minimize the variation in magnitude of angular momentum of the orbit. Same procedure is used to fix shape and orientation of the orbit in the plane by minimizing variation in total energy and Laplace Runge Lenz vector. The method is tested using simulated data for a hypothetical planet rotating around the sun.