An application of symplectic integration for general relativistic planetary orbitography subject to non-gravitational forces

Abstract

Spacecraft propagation tools describe the motion of near-Earth objects and interplanetary probes using Newton’s theory of gravity supplemented with the approximate general relativistic n-body Einstein–Infeld–Hoffmann equations of motion. With respect to the general theory of relativity and the long-standing recommendations of the International Astronomical Union for astrometry, celestial mechanics and metrology, we believe modern orbitography software is now reaching its limits in terms of complexity. In this paper, we present the first results of a prototype software titled General Relativistic Accelerometer-based Propagation Environment (GRAPE). We describe the motion of interplanetary probes and spacecraft using extended general relativistic equations of motion which account for non-gravitational forces using end-user supplied accelerometer data or approximate dynamical models. We exploit the unique general relativistic quadratic invariant associated with the orthogonality between four-velocity and acceleration and simulate the perturbed orbits for Molniya, Parker Solar Probe and Mercury Planetary Orbiter-like test particles subject to a radiation-like four-force. The accuracy of the numerical procedure is maintained using a 5-stage, $$10^\mathrm{th}$$ 10 th -order structure-preserving Gauss collocation symplectic integration scheme. GRAPE preserves the norm of the tangent vector to the test particle worldline at the order of $$10^{-32}$$ 10 - 32 .

Reconstructing the cruise-phase trajectory of deep-space probes in a general relativistic framework: An application to the Cassini gravitational wave experiment

“…As discussed at length in Ref. [31], the treatment of non-gravitational perturbations in general relativity has not received the same attention in the literature and requires further elucidation. The nonlinear field equations of general relativity [38,39] can be decomposed according to contributions from the Einstein tensor G µν and the energy momentum tensor T µν .…”

“…If, for example, e µ (ν) denoted the components of the so-called natural tetrad, then a further Lorentz transformation is required to account for the relative motion (see Ref. [31,Section 2.4] for further details).…”

“…[46], where the reader is referred to Refs. [31,46,47] for the specific Butcher tableau coefficients required for practical implementation. In this section, we review implicit Runge-Kutta algorithms and present a derivation of their structure-preserving properties for quadratic invariants.…”

“…Finally, we note that this section is based on Ref. [31,Section 4] and remains largely unchanged from its original presentation.…”

“…In a recent article [31], we presented the first results of a prototype software, namely General Relativistic Accelerometer-based Propagation Environment (GRAPE), which describes the perturbed motion of Parker Solar Probe [32], Mercury Planetary Orbiter [33], and Molniya-like [34] interplanetary probes and spacecraft subject to a radiation-like non-gravitational four-force using the complete framework of general relativity. GRAPE employs extended relativistic equations of motion that account for non-gravitational forces using either end user supplied accelerometer data or approximate dynamical models in the local frame of the spacecraft.…”