Orbit determination with the Keplerian integrals

TL;DR

This paper reviews two methods for initial orbit determination using Kepler's laws, capable of linking short observational arcs separated by long time intervals, with proven optimal polynomial properties and demonstrated numerical tests.

Contribution

Introduces two novel algorithms, Link2 and Link3, for linking short observational arcs using Keplerian integrals, with proven polynomial optimality and practical numerical validation.

Findings

Link2 results in a degree 9 polynomial for two arc linkage.

Link3 results in a degree 8 polynomial for three arc linkage.

Numerical tests validate the effectiveness of both methods.

Abstract

We review two initial orbit determination methods for too short arcs (TSAs) of optical observations of a solar system body. These methods employ the conservation laws of Kepler's problem, and allow to attempt the linkage of TSAs referring to quite far epochs, differing by even more than one orbital period of the observed object. The first method ({\tt Link2}) concerns the linkage of 2 TSAs, and leads to a univariate polynomial equation of degree 9. An optimal property of this polynomial is proved using Gr\"obner bases theory. The second method ({\tt Link3}) is thought for the linkage of 3 TSAs, and leads to a univariate polynomial equation of degree 8. A numerical test is shown for both algorithms.