Orbit Determination with the two-body Integrals. III

TL;DR

This paper improves orbit determination methods using two-body integrals to link short observation arcs, introducing a new polynomial approach and simplified algebraic elimination, with numerical tests demonstrating effectiveness.

Contribution

It presents a novel univariate polynomial approach for orbit determination that simplifies previous algebraic elimination techniques and enhances computational efficiency.

Findings

A degree 9 polynomial in radial distance is derived.

Elementary calculations replace previous complex elimination methods.

Numerical tests confirm improved performance of the new algorithm.

Abstract

We present the results of our investigation on the use of the two-body integrals to compute preliminary orbits by linking too short arcs of observations of celestial bodies. This work introduces a significant improvement with respect to the previous papers on the same subject (see Gronchi et al. 2010, 2011). Here we find a univariate polynomial equation of degree 9 in the radial distance $\rho$ of the orbit at the mean epoch of one of the two arcs. This is obtained by a combination of the algebraic integrals of the two-body problem. Moreover, the elimination step, which in Gronchi et al. 2010, 2011 was done by resultant theory coupled with the discrete Fourier transform, is here obtained by elementary calculations. We also show some numerical tests to illustrate the performance of the new algorithm.