The 2PN Point-Mass N-Body Equations of Motion in Harmonic Gauge: A Computable Formulation

Abstract

We develop a semi-analytic and semi-numerical formulation of the harmonic-gauge second post-Newtonian (2PN) equations of motion for a general point-mass N-body system within Hadamard regularization. The equations of motion are separated into a closed analytic contribution and a non-closed integral contribution. We analyze the singular structure of the latter and further regularize it into a numerically evaluable representation. We apply the formulation to the Sun-Jupiter-Saturn and Sun-Mercury-Venus systems, evaluating the instantaneous non-closed 2PN acceleration along Newtonian trajectories and its leading finite-time relative-distance response through the corresponding perturbation equations. In both benchmarks, the non-closed acceleration remains a small fraction of the complete 2PN acceleration, while the induced relative-distance perturbation remains oscillatory and can reach larger oscillation amplitudes at later times.

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BibTeX

@misc{huang2026pointmass,
  title = {{The 2PN Point-Mass N-Body Equations of Motion in Harmonic Gauge: A Computable Formulation}},
  author = {Hongkun Huang and Jie Yang and Wei-Tou Ni},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.20193},
  eprint = {2608.20193},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.20193},
}

RIS

TY  - UNPB
TI  - The 2PN Point-Mass N-Body Equations of Motion in Harmonic Gauge: A Computable Formulation
AU  - Hongkun Huang
AU  - Jie Yang
AU  - Wei-Tou Ni
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2608.20193
UR  - https://arxiv.org/abs/2608.20193
AB  - We develop a semi-analytic and semi-numerical formulation of the harmonic-gauge second post-Newtonian (2PN) equations of motion for a general point-mass N-body system within Hadamard regularization. The equations of motion are separated into a closed analytic contribution and a non-closed integral contribution. We analyze the singular structure of the latter and further regularize it into a numerically evaluable representation. We apply the formulation to the Sun-Jupiter-Saturn and Sun-Mercury-Venus systems, evaluating the instantaneous non-closed 2PN acceleration along Newtonian trajectories and its leading finite-time relative-distance response through the corresponding perturbation equations. In both benchmarks, the non-closed acceleration remains a small fraction of the complete 2PN acceleration, while the induced relative-distance perturbation remains oscillatory and can reach larger oscillation amplitudes at later times.
ER  -

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