Uniform formulation for orbit computation: the intermediate elements

TL;DR

This paper introduces a novel uniform orbital element formulation for perturbed two-body problems, enabling accurate and consistent orbit propagation across all energy levels using an intermediate reference frame.

Contribution

It presents a new set of orbital elements based on Sundman's transformation and the Burdet-Ferrándiz linearisation, applicable uniformly for any total energy value.

Findings

Effective orbit propagation for comets C/2003 T4 and C/1985 K1.

Explicit derivatives for orbit determination and uncertainty analysis.

Demonstrated numerical stability and accuracy of the method.

Abstract

We present a new method for computing orbits in the perturbed two-body problem: the position and velocity vectors of the propagated object in Cartesian coordinates are replaced by eight orbital elements, i.e., constants of the unperturbed motion. The proposed elements are uniformly valid for any value of the total energy. Their definition stems from the idea of applying Sundman's time transformation in the framework of the projective decomposition of motion, which is the starting point of the Burdet-Ferr\'andiz linearisation, combined with the Stumpff functions. In analogy with Deprit's ideal elements, the formulation relies on a special reference frame that evolves slowly under the action of external perturbations. We call it the intermediate frame, hence the name of the elements. Two of them are related to the radial motion, the next four, given by Euler parameters, fix the orientation of the intermediate frame. The total energy and a time element complete the state vector. All the necessary formulae for extending the method to orbit determination and uncertainty propagation are provided. For example, the partial derivatives of the position and velocity with respect to the intermediate elements are obtained explicitly together with the inverse partial derivatives. Numerical tests are included to assess the performance of the proposed special perturbation method when propagating the orbit of comets C/2003~T4 (LINEAR) and C/1985~K1 (Machholz).