Orbit Determination with the two-body Integrals. II

TL;DR

This paper presents a novel algebraic method for preliminary orbit determination using two observed arcs, leveraging two-body integrals and polynomial equations, with an algorithm to verify if the arcs belong to the same celestial body.

Contribution

It introduces a new approach that employs the Laplace-Lenz vector to reduce computational complexity in orbit determination from short observational arcs.

Findings

Algorithm successfully links observed arcs to the same celestial body.

Polynomial equations are effectively solved using computational algebra tools.

The method reduces the degree of the system, improving efficiency.

Abstract

The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or radar observations. We write polynomial equations for this problem, that we can solve using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. In this paper we continue the research started in [Gronchi, Dimare, Milani, 'Orbit determination with the two-body intergrals', CMDA (2010) 107/3, 299-318], where the angular momentum and the energy integrals were used. A suitable component of the Laplace-Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half.