A preprint posted to arXiv on 20 Aug 2026 presents a way to calculate a difficult, non-closed part of the harmonic-gauge second post-Newtonian (2PN) equations, a higher-order correction in the gravity model, for a general point-mass N-body system. In two modeled three-body tests—the Sun–Jupiter–Saturn (SJS) and Sun–Mercury–Venus (SMV) systems—the term was a small share of the complete 2PN acceleration, while the associated pairwise-distance perturbations repeatedly changed direction instead of building steadily.
Making the residual term calculable
The work tackles a residual spatial integral that remains as an integral rather than a finished analytic expression. It sets out a semi-analytic, semi-numerical formulation under Hadamard regularization, separating the equations into closed analytic terms and non-closed integral terms. The non-closed integral is divided into D ≠ A and D = A sectors because their singular structures differ.
The local pieces were evaluated with centered spherical numerical quadrature, completing the angular integration at each radius. Auxiliary splitting radii were varied while the particle configuration was held fixed. Outer regions were treated as ordinary integrals, with the finite outer boundary increased to check convergence.
Both the Newtonian trajectories and the perturbation equations used the same sixth-order Runge–Kutta integration, with acceleration components linearly interpolated between evaluation-grid points.
Small forcing, different finite-time response
The tests were limited to two modeled three-body systems, not complete planetary ephemerides. SJS was integrated for 657,360 days—about 1,799.753 Julian years—with a one-day step and a uniform 180-day grid for evaluating the non-closed acceleration. SMV covered 18,260 days, about 49.993 Julian years, with a 0.125-day step and a 10-day evaluation grid.
The instantaneous contribution stayed small in both calculations. SJS had root-mean-square values of a few 10⁻²⁸ m s⁻², with sampled maxima at the 10⁻²⁷ m s⁻² level. SMV’s RMS values were a few 10⁻²⁹ m s⁻², and its sampled maxima stayed below 6 × 10⁻²⁹ m s⁻².
Relative to the complete 2PN acceleration, the largest solar norm ratio—the size of the non-closed term divided by the size of the full 2PN term—was 2.698 × 10⁻³, or 0.270%, in SJS. In SMV it was 1.548 × 10⁻⁵, or 0.001548%. Planetary ratios were several orders smaller. Newtonian and 1PN terms were excluded from this comparison.
The finite-time distance response showed a different scale. In SJS, the sampled maximum absolute perturbations were 45.054 picometres for Sun–Jupiter, 192.084 picometres for Sun–Saturn and 2,176.407 picometres for Jupiter–Saturn, the largest of the three.
In SMV, the corresponding values were 0.01932 picometres for Sun–Mercury, 0.002475 picometres for Sun–Venus and 0.05245 picometres for Mercury–Venus. All three remained below 0.1 picometres.
In both systems, distance perturbations repeatedly changed sign rather than accumulating monotonically. Later, larger perturbation amplitudes were not accompanied by increasing instantaneous forcing. Within the finite benchmark intervals, the calculations found no monotonic or secular growth, but they do not establish what happens over unbounded times.
A benchmark result, not an ephemeris
Only two representative three-body configurations were evaluated, so the numerical results do not establish general behavior across broader N-body systems. The reported peaks are sampled-grid maxima, not guaranteed continuous-time extrema, and the two benchmarks used different configurations and integration durations; their raw perturbation scales should not be treated as a duration-independent ranking.
The reported summaries are finite-interval sampled results, and no inferential uncertainty estimate is provided. Within that boundary, the preprint’s main contribution is computational: it supplies a regularized way to evaluate the non-closed term and shows how its finite-time response behaves in the two benchmark models.
Paper data and sources
Original title: The 2PN Point-Mass N-Body Equations of Motion in Harmonic Gauge: A Computable Formulation
Authors: Hongkun Huang, Jie Yang, Wei-Tou Ni
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text