REVIEW 2 major objections 5 minor 14 references
New general relativistic contributions to Mercury's orbital elements and their measurability
T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read General relativity's N-body contribution to Mercury's perihelion precession is 30.4 microarcseconds per century, five times smaller than the previously reported value.
desk verdict A small but solid self-correction: the 1pN N-body Mercury perihelion precession is 30 μas/cty, not 150, with new rates for other elements, all likely too small to measure soon. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central machinery is a set of three 1pN $N$-body accelerations, $A_{G2}$, $A_G$, and $A_{v_X}$ (Eqs. 1-3, from Will 2018), describing how a distant third body $X$ perturbs a test particle at first post-Newtonian order, applied through the Gauss perturbative equations that turn a perturbing acceleration into time derivatives of the Keplerian orbital elements $a,e,I,\Omega,\varpi,\epsilon$. The paper doubly averages these equations over the orbital periods of Mercury and of each perturbing planet to produce leading-order analytic rates, and verifies them with a one-century numerical integration. The load-bearing step is the corrected analytic formula for the perihelion rate produced by $A_G$: the correction to Eq. (B5) of Iorio (2018) removes the factor that had produced the earlier $150\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$ value.
What would settle it
Numerically integrate Mercury's orbit with the full 1pN N-body equations of motion of the Solar System, rather than the restricted three-body accelerations of Eqs. (1)-(3), with and without the direct 1pN third-body terms, and fit the difference in the perihelion longitude over one century; a slope significantly different from $30.4\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$ would invalidate the paper's correction.
Extended reading notes
Core claim
The core discovery is that the 1pN $N$-body perihelion precession of Mercury is $\dot\varpi_{1\mathrm{pN}}^\mathrm{X}=30.4\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$, not the $150\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$ reported in Iorio (2018), because a specific analytic expression for the velocity-dependent acceleration $A_G$ in that paper was wrong. For the same accelerations, the inclination, node, and mean-longitude-at-epoch rates are $\dot I=-4.3\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$, $\dot\Omega=18.2\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$, and $\dot\epsilon=271.4\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$. The corrected perihelion value is even smaller than the approximate $220\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$ of Will (2018), which strengthens the earlier conclusion that the effect is very hard to measure; the other rates are at or below the formal EPM2017 errors, and the largest signal, in the mean longitude at epoch, is masked by the much larger uncertainty in Mercury's mean motion arising from the solar gravitational parameter.
Load-bearing premise
The load-bearing premise is that the three post-Newtonian force formulas taken from Will (2018) correctly and completely describe, to first order, how a distant planet's gravity acts on a test particle; if those formulas are wrong or incomplete, every computed rate, including the perihelion correction, would change.
Editorial extensions
If this is right
- Mercury's 1pN N-body perihelion precession is $30.4\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$, roughly four times the formal EPM2017 error of $8\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$ but likely below realistic systematic errors, so direct detection with BepiColombo is doubtful.
- The inclination and node rates, $-4.3$ and $18.2\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$, are comparable to their formal uncertainties ($3$ and $24\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$), which makes them difficult to isolate as clean signals.
- The largest computed effect, $\dot\epsilon=271.4\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$, cannot currently be measured because the uncertainty in Mercury's mean motion from the solar gravitational parameter is about $20\,\mathrm{mas}\,\mathrm{cty}^{-1}$, roughly seventy times larger.
- If Mercury's orbit is later determined more accurately, linear combinations of the supplementary advances of several planets' elements, following Shapiro (1990), could cancel the competing Sun $J_2$ and Lense-Thirring precessions and extract the 1pN N-body rates.
- The analytic formulas apply to any test particle around a static, spherically symmetric primary with a distant massive third body, so the same rates can be obtained for other planets or artificial satellites without new integrations.
Reading between the lines
- The same doubly averaged machinery could be carried to the Earth-Moon system or to artificial satellites around the Earth, where range measurements are far more accurate than Mercury's astrometry, potentially turning this 1pN third-body acceleration into a measurable signal.
- Because the analytic formulas express each rate through the same orbital elements of the perturber, ratios such as $\dot\Omega/\dot\varpi$ or $\dot I/\dot\varpi$ for a given planet are fixed predictions; a future measurement of one rate could be checked against the others without an absolute calibration.
- A direct numerical test of the correction would integrate the full 1pN N-body equations of motion, rather than the restricted hierarchical approximation, and compare the extracted perihelion slope with $30.4\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$; disagreement would reopen the question.
- If BepiColombo does eventually deliver a perihelion residual after subtracting the standard Schwarzschild, $J_2$, and Lense-Thirring effects, that residual should be compared with $30\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$ rather than the earlier $150$ or $220\,\mu\mathrm{as}\,\mathrm{cty}^{-1}$ figures.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper computes the first-order post-Newtonian (1pN) orbital effects on Mercury's Keplerian elements that arise from the accelerations (1)–(3), adopted from Will (2018) as a 'particular case' of the full Einstein-Infeld-Hoffmann equations, with the perturbing planets taken to be Venus through Saturn. The author finds secular rates for the inclination, node, perihelion, and mean longitude at epoch (−4.3, 18.2, 30.4, and 271.4 μas cty−1, respectively), no secular rates for the semimajor axis and eccentricity, and a perihelion precession about five times smaller than the value reported in Iorio (2018), which is traced to an error in Eq. (B5) of that paper. The results are obtained both by numerical integration of the barycentric equations and by doubly averaged Gauss equations, and are compared with formal uncertainties derived from EPM2017 ephemerides, leading to a pessimistic conclusion about measurability with BepiColombo.
Significance. If the adopted acceleration model is accepted as complete, the paper is a valuable self-correction of a published value and provides the first quantitative estimates for all of Mercury's orbital elements under the Will (2018) accelerations. The numerical and analytical methods cross-validate each other, there are no fitted parameters, and the comparison with published ephemeris uncertainties is concrete. The paper also explicitly identifies the origin of the previous error, which is a strength for the literature. The main risk is that the three accelerations may not represent the complete 1pn N-body acceleration in the adopted test-particle limit, and the numerical slopes lack formal uncertainties; both issues affect the central quantitative claims.
major comments (2)
- [§1, Eqs. (1)–(3)] The paper adopts the three accelerations as 'a particular case' of the full 1PN Einstein-Infeld-Hoffmann equations, but it does not demonstrate that these three terms are the complete 1PN N-body acceleration in the Sun-centered test-particle limit. The full EIH equations contain additional contributions proportional to v_X^2, (r_X_hat · v_X)^2, and the acceleration of X; after subtracting the primary's acceleration, these yield terms of the same order as Eq. (1) for the actual configuration, e.g., r/r_X ≈ 0.5 for Venus and v_X^2/c^2 ≈ (GM_S/r_X)/c^2. The numerical integration (Section 2.1) and the analytical calculation (Section 2.2) both use exactly Eqs. (1)–(3), so their agreement is a check of internal consistency, not of completeness. Since the abstract and Section 3 present the results as the 1pN N-body precessions and compare them with EPM2017 errors, the quoted rates (30.4 μas cty−1 for ϖ, and the companion rates) are not yet established as the full 1pN N-body effect. Please either include the omitted terms or state explicitly that the results refer only to the subset in Eqs. (1)–(3) and justify numerically that the omitted terms are negligible at the μas cty−1 level.
- [§2.1 and Fig. 1] The secular rates are obtained by fitting a linear trend to difference time series, but the fitted slopes are reported without formal uncertainties or residual statistics. This matters because the rates are at the same level as the EPM2017 formal errors used in Section 3, and because the analytical formulas are only leading order in e (Mercury's e ≈ 0.21). Please report, for each element, the 1σ slope uncertainty from the fit, the rms residual of the difference time series, and the numerical-versus-analytical difference; this would make the 'excellent agreement' claim in Section 2.2 quantitative and support the three-significant-figure rates in the abstract.
minor comments (5)
- [§2.1] The description of the numerical integration is not sufficient for reproducibility; please specify the integrator, time step, tolerance, and the procedure used to compute the difference time series, and state whether the same HORIZONS initial conditions are used in both runs.
- [Title and text] The manuscript contains typographical artifacts from the LaTeX source ('orbit al' in the running title, 'pontlike' for 'pointlike', and several 'doteq' symbols); please clean these up in the published version.
- [Abstract and §3] The phrase 'at the same level of, or larger by one order of magnitude than' is too vague; for each element, the predicted rate should be compared numerically with the quoted EPM2017-derived uncertainty, e.g., 30.4 vs 8 μas cty−1 for ϖ and 18.2 vs 24 μas cty−1 for Ω.
- [§3] The estimate δn_b^obs = 20 mas cty−1 uses δμ_obs = 1e10 m^3 s−2 and δa_obs; please make the propagation formula explicit so that the reader can reproduce this factor.
- [§4] The proposed linear combinations of supplementary advances for several planets are only sketched; a concrete example, even for a simplified two-planet scenario, would help establish that the 1pN N-body effects can be separated from the J2 and Lense-Thirring precessions.
Circularity Check
No circularity: the paper computes 1pN N-body orbital rates from externally sourced equations of motion and compares them with independent ephemeris uncertainties.
full rationale
The central derivation is self-contained. The paper takes the 1pN N-body accelerations from Will (2018) and Poisson & Will (2014) as input, explicitly calling them 'a particular case of the full 1pN equations of motion' (Eqs. 1-3 and surrounding text), and then both numerically integrates the equations of motion and analytically doubly averages the same accelerations. The numerical and analytical routes share the identical acceleration input, so their agreement tests internal consistency, but neither route fits any parameter to the target rates or to the EPM2017 data. The quoted Mercury rates (-4.3, 18.2, 30.4, 271.4 microarcseconds per century) are direct consequences of those input accelerations. The comparison with formal uncertainties from Iorio (2019) is an external benchmark, not an input to the derivation. The paper's correction of Iorio (2018) is a disclosed error fix, and the self-citations are used either as the source of the incorrect value being corrected or as the source of independent uncertainty estimates; neither is load-bearing in a circular sense. The skeptical concern that Eqs. (1)-(3) might be incomplete or that the 'distant body' approximation is marginal is a correctness risk about the physical input, not a circularity, because the paper does not define its predictions in terms of the data it compares against.
Assumptions & free parameters
assumptions (4)
- domain assumption The 1pN N-body accelerations of Eqs. (1)-(3) (from Will 2018) are the correct and complete first-order post-Newtonian contributions of a distant third body to the test particle's acceleration.
- standard math The Gauss perturbative equations describe the secular evolution of Keplerian orbital elements under the small accelerations (1)-(3).
- domain assumption The solar system is modeled with a static, spherically symmetric Sun and point-like planets, and the difference of two numerical runs with and without the 1pN terms isolates the effect.
- domain assumption Initial conditions retrieved from JPL HORIZONS at epoch J2000 are sufficiently accurate for the 1-century integration difference.
Cite this review
Pith. "Pith review of New general relativistic contributions to Mercury's orbital elements and their measurability." pith.science (2026). https://pith.science/paper/ADW7LTCY
@misc{pith2026190809670,
author = {Pith},
title = {Pith review of: New general relativistic contributions to Mercury's orbital elements and their measurability},
year = {2026},
howpublished = {\url{https://pith.science/paper/ADW7LTCY}},
note = {Machine review of arXiv:1908.09670}
}
abstract
We numerically and analytically work out the first-order post-Newtonian (1pN) orbital effects induced on the semimajor axis $a$, the eccentricity $e$, the inclination $I$, the longitude of the ascending node $\Omega$, the longitude of perihelion $\varpi$, and the mean longitude at epoch $\epsilon$ of a test particle orbiting its primary, assumed static and spherically symmetric, by a distant massive third body X. For Mercury, the rates of change of the linear trends found are $\dot I_\mathrm{1pN}^\mathrm{X} = -4.3\,\mathrm{microarcseconds\,per\,century}\,\left(\mu\mathrm{as\,cty}^{-1}\right)$, $\dot\Omega_\mathrm{1pN}^\mathrm{X} = 18.2\,\mu\mathrm{as\,cty}^{-1}$, $\dot\varpi_\mathrm{1pN}^\mathrm{X} = 30.4\,\mu\mathrm{as\,cty}^{-1}$, $\dot\epsilon_\mathrm{1pN}^\mathrm{X} = 271.4\,\mu\mathrm{as\,cty}^{-1}$, respectively. Such values, which are due to the added actions of the other planets from Venus to Saturn, are essentially at the same level of, or larger by one order of magnitude than, the latest formal errors in the Hermean orbital precessions calculated with the EPM2017 ephemerides. The perihelion precession $\dot\varpi_\mathrm{1pN}^\mathrm{X}$ turns out to be smaller than some values recently appeared in the literature in view of a possible measurement with the ongoing BepiColombo mission. Linear combinations of the supplementary advances of the Keplerian orbital elements for several planets, if determined experimentally by the astronomers, could be set up in order to disentangle the 1pN $N$-body effects of interest from the competing larger precessions like those due to the Sun's quadrupole moment $J_2$ and angular momentum $\boldsymbol{S}$.
Figures
Reference graph
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Reviewed August 14, 2026 · model on record in the stance chip above.
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