**Annals of the New York Academy of Sciences**

# Gauge Freedom in Orbital Mechanics

*Ann. N.Y. Acad. Sci.*1065: 346–374 (2005).

Address for correspondence: Michael Efroimsky, U.S. Naval Observatory, Washington DC 20392, USA. efroimsky.michael@usno.navy.mil

## Abstract

**Abstract: ** Both orbital and attitude dynamics employ the method of variation of parameters. In a non-perturbed setting, the coordinates (or the Euler angles) are expressed as functions of the time and six adjustable constants called elements. Under disturbance, each such expression becomes ansatz, the “constants” being endowed with time dependence. The perturbed velocity (linear or angular) consists of a partial time derivative and a convective term containing time derivatives of the “constants.” It can be shown that this construction leaves one with a freedom to impose three arbitrary conditions on the “constants” and/or their derivatives. Out of convenience, the Lagrange constraint is often imposed. It nullifies the convective term and thereby guarantees that under perturbation the functional dependence of the velocity upon the time and “constants” stays the same as in the undisturbed case. “Constants” obeying this condition are called osculating elements. The “constants” chosen to be canonical, are called Delaunay elements, in the orbital case, or Andoyer elements, in the spin case. (Because some of the Andoyer elements are time dependent, even in the free-spin case, the role of “constants” is played by the initial values of these elements.) The Andoyer and Delaunay sets of elements share a feature not readily apparent: in certain cases the standard equations render these elements non-osculating. In orbital mechanics, elements calculated via the standard planetary equations turn out to be non-osculating when perturbations depend on velocities. To keep elements osculating under such perturbations, the equations must be amended with additional terms that are *not* parts of the disturbing function (Efroimsky and Goldreich 2003, 2004). For the Kepler elements, this merely complicates the equations. In the case of Delaunay parameterization, these extra terms not only complicate the equations, but also destroy their canonicity. So under velocity-dependent disturbances, osculation and canonicity are incompatible. Similarly, in spin dynamics the Andoyer elements turn out to be non-osculating under angular-velocity-dependent perturbation (a switch to a noninertial frame being one such case). Amendment of the dynamical equations only with extra terms in the Hamiltonian makes the equations render nonosculating Andoyer elements. To make them osculating, more terms must enter the equations (and the equations will no longer be canonical). It is often convenient to deliberately deviate from osculation by substituting the Lagrange constraint with an arbitrary condition that gives birth to a family of nonosculating elements. The freedom in choosing this condition is analogous to the gauge freedom. Calculations in nonosculating variables are mathematically valid and sometimes highly advantageous, but their physical interpretation is nontrivial. For example, nonosculating orbital elements parameterize instantaneous conics *not* tangent to the orbit, so the nonosculating inclination will be different from the real inclination of the physical orbit. We present examples of situations in which ignorance of the gauge freedom (and of the unwanted loss of osculation) leads to oversights.