Analysis of the PPN two-Body Problem using non-osculating orbital elements

The parameterised post-Newtonian (PPN) formalism is a weak-field and slow-motion approximation for both General Relativity (GR) and some of its viable generalisations. Within this formalism, the motion can be approached using various parameterisations, among which are the Lagrange-type and Gauss-type orbital equations. Often, these equations are developed under the premise of the Lagrange constraint. This constraint makes the evolving orbital elements parameterise the instantaneous conics always tangent to the actual trajectory. Arbitrary mathematically, this choice of a constraint is convenient under perturbations dependent only on positions. However, under perturbations dependent also on velocities (like in the relativistic celestial mechanics) the Lagrange constraint unnecessarily complicates solutions that can be simplified by relaxing the constraint and introducing a freedom in the orbit parameterisation, which is analogous to the gauge freedom in electrodynamics and gauge field theories. Geometrically, this freedom is the freedom of non-osculation, i.e., of the degree to which the instantaneous conics are permitted to be non-tangent to the actual orbit. Under the same perturbation, all solutions with different degrees of non-osculation look mathematically different, though describe the same physical orbit. While non-intuitive, the modeling of an orbit with a sequence of nontangent instantaneous conics can at times simplify calculations. The appropriately generalised (“gauge-generalised”) Lagrange-type equations, and their applications, appeared in the literature heretofore. We, in this paper, derive the gauge-generalised Gauss-type equations and apply them to the PPN two-body problem. Fixing the gauge freedom in three different ways (i.e., modeling an orbit with non-osculating elements of three different types), we find three parameterisations of the PPN two-body dynamics. These parameterisations render orbits with either a fixed non-osculating semimajor axis, or with a fixed non-osculating eccentricity, or with a fixed non-osculating argument of periastron. We also develop a transformation from non-osculating to classical osculating orbital elements, and illustrate the new solutions using numerical simulations.