Lie-series for orbital elements -- II. The spatial case

TL;DR

This paper extends Lie-series methods to the spatial N-body problem, providing recurrence relations for high-accuracy orbital element computation over long timescales, with derivations for both first and higher orders.

Contribution

It introduces a set of recurrence relations for Lie-integration of spatial orbital elements, generalizing previous planar case results to three dimensions.

Findings

Recurrence relations enable high-precision long-term orbital calculations.

Series vanish in absence of perturbations, confirming consistency.

Method applicable to arbitrary order for enhanced accuracy.

Abstract

If one has to attain high accuracy over long timescales during the numerical computation of the N-body problem, the method called Lie-integration is one of the most effective algorithms. In this paper we present a set of recurrence relations with which the coefficients needed by the Lie-integration of the orbital elements related to the spatial N-body problem can be derived up to arbitrary order. Similarly to the planar case, these formulae yields identically zero series in the case of no perturbations. In addition, the derivation of the formulae has two stages, analogously to the planar problem. Namely, the formulae are obtained to the first order, and then, higher order relations are expanded by involving directly the multilinear and fractional properties of the Lie-operator.