A new preprint turns a residual spatial integral in second post-Newtonian (2PN) equations for point-mass gravity into a form that can be evaluated numerically. In two controlled three-body benchmarks, the calculated acceleration was on the 10^-28 m/s^2 scale in the Sun-Jupiter-Saturn (SJS) system and the 10^-29 m/s^2 scale in the Sun-Mercury-Venus (SMV) system. The largest sampled finite-time distance perturbation was 2176.407 picometres for Jupiter-Saturn in SJS.
From a residual integral to a calculation
The paper's central aim is to make the harmonic-gauge 2PN equations of motion for a general point-mass N-body system computable. The formulation separates closed analytic pieces from a residual spatial integral that is not already reduced to a closed formula, then regularizes that term for numerical evaluation.
Different parts of the integral are handled differently. For the non-self sector, involving particles other than the one whose motion is being calculated, the construction uses centered-shell integration and an analytic contact term. In the self sector, it analytically removes the unique Taylor-series contribution that diverges linearly. The numerical evaluation completes the angular integration at each radius before radial integration and uses particle-centered patches for localized peaks near other particle positions.
What the two benchmarks showed
In SJS, the reported root-mean-square (RMS) non-closed acceleration values, a summary of typical size, were 2.954 x 10^-28, 4.202 x 10^-28 and 3.019 x 10^-28 m/s^2 for the Sun, Jupiter and Saturn, respectively. The sampled maxima were 7.970 x 10^-28, 1.468 x 10^-27 and 1.198 x 10^-27 m/s^2, respectively. The ratio of the non-closed term to the complete 2PN acceleration reached a sampled maximum of 2.698 x 10^-3 for the Sun, compared with 2.300 x 10^-7 for Jupiter and 1.531 x 10^-6 for Saturn.
In SMV, the corresponding RMS acceleration values were 1.434 x 10^-29, 2.008 x 10^-29 and 1.356 x 10^-29 m/s^2 for the Sun, Mercury and Venus, respectively. The sampled maximum ratios were 1.548 x 10^-5 for the Sun, 5.288 x 10^-13 for Mercury and 2.848 x 10^-12 for Venus. Neither benchmark showed an evident systematic increase or decrease in the instantaneous acceleration or its ratio to the complete 2PN acceleration, and both absolute and relative non-closed contributions were smaller in SMV than in SJS.
When the effect is propagated over time
To examine the residual over time, the finite-time perturbation calculation used the Newtonian linear operator and zero initial data. It isolates the correction from the non-closed source; it is not a replacement for the 1PN term or the remaining 2PN terms.
On the one-day output grid, SJS's sampled maximum absolute pairwise distance perturbations were 45.054 pm for Sun-Jupiter after 1785.714 years, 192.084 pm for Sun-Saturn after 1619.244 years and 2176.407 pm for Jupiter-Saturn after 1746.609 years.
For SMV, the sampled maxima on its one-day output grid were 0.01932 pm for Sun-Mercury after 49.555 years, 0.002475 pm for Sun-Venus after 47.729 years and 0.05245 pm for Mercury-Venus after 49.599 years.
Across the computed intervals, the perturbations remained oscillatory, with no evidence of monotonic or secular, or steady one-way, growth. Larger later sampled amplitudes were not accompanied by a systematic increase in the instantaneous forcing norms.
What the numbers do not show
These are controlled evaluations of two three-body benchmark configurations, not complete planetary ephemerides. The finite-time calculation isolates the non-closed-source correction rather than replacing the other post-Newtonian terms.
Because the maxima are sampled on one-day output grids, they are grid-based maxima rather than continuous-time extrema. The finite benchmark intervals also cannot establish unbounded long-term or secular behavior.
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