Equations for the Keplerian elements: Hidden symmetry as an unexpected source of numerical error

Abstract

We revisit the Lagrange's system of equations for the six osculating elements, in the context of long-term planetary-orbit integration. An accurate re-examination of the derivation of Lagrange's system shows that, in fact, the orbit is always located not in the 6-dimensional space of the osculating elements, but in a certain 3-dimensional submanifold. If an analytic solution to Lagrange's system were available, it would obey this demand. However, whatever numerical integrator will cause drift away from this submanifold. This will result in a new type of accumulating numerical error that will be especially significant at long time spans. We point out an adjustment to be instilled in the integrator, that would eliminate this error. We point out that the choice of the said submanifold is mathematically equivalent to fixing a gauge in field theory. The existing freedom of subminifold choice (~=~freedom of gauge fixing) reveals a symmetry (and a fibre bundle structure) hiding behind Lagrange's system. Just as a choice of the convenient gauge simplifies calculations in electrodynamics, the freedom in choice of the submanifold may, potentially, lead to simpler schemes of orbit integration.