Two birds with one stone: simultaneous realization of both Lunar Coordinate Time and lunar geoid time by a single orbital clock

Chong Yang; Jing Zhang; Ren-Fang Geng; Tian-Ning Yang; Yi Xie; Yong Huang

arxiv: 2512.23098 · v1 · submitted 2025-12-28 · 🌌 astro-ph.IM · astro-ph.EP· gr-qc

Two birds with one stone: simultaneous realization of both Lunar Coordinate Time and lunar geoid time by a single orbital clock

Tian-Ning Yang , Ren-Fang Geng , Jing Zhang , Chong Yang , Yong Huang , Yi Xie This is my paper

Pith reviewed 2026-05-16 20:04 UTC · model grok-4.3

classification 🌌 astro-ph.IM astro-ph.EPgr-qc

keywords Lunar Coordinate Timeselenoid proper timetime aligned orbitorbital clocklunar timekeepingproper timelunar geoid

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The pith

A clock in a time-aligned lunar orbit can simultaneously realize both Lunar Coordinate Time and selenoid proper time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper identifies special orbits around the Moon where an ideal clock's reading matches the proper time experienced on the lunar geoid. These same readings convert to Lunar Coordinate Time through a known linear transformation, letting one clock serve both reference systems. Simulations that include realistic lunar gravity show the orbital clock drifts from the geoid time by no more than 190 nanoseconds after a full year, with a frequency offset of 6 times 10 to the minus 15. This offset remains only 3.75 percent of the frequency variation already introduced by lunar surface topography. The method therefore offers a practical route to lunar timekeeping without separate clocks or coordinate rescaling.

Core claim

There exist time aligned orbits around the Moon with semi-major axis of about 1.5 lunar radii, slightly dependent on inclination, such that the proper time of a clock on these orbits equals the selenoid proper time. These readings can be converted to Lunar Coordinate Time by a known linear transformation. Numerical simulations in a realistic lunar environment show the proper time desynchronizes from selenoid proper time by up to 190 ns after one year with a frequency offset of 6E-15, amounting to 3.75 percent of the frequency difference caused by surface topography. These errors could drop to 13 ns and 4E-16 if mean orbits match the nominal ones more closely.

What carries the argument

Time aligned orbit: an orbital path whose clock proper time equals selenoid proper time and converts linearly to Lunar Coordinate Time.

If this is right

A single orbital clock suffices to realize both Lunar Coordinate Time and selenoid proper time without steering.
The method avoids any need to rescale spatial coordinates or the lunar mass parameter when adopting geoid time.
The achieved performance of 190 ns annual drift is only 3.75 percent of the topography-induced variation in selenoid time.
The same orbital construction scales directly to other terrestrial planets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Precise lunar gravity mapping would be required to keep real orbits close enough to the nominal paths over long durations.
A similar set of time-aligned orbits could be derived for Mars to unify its coordinate and surface time definitions.
Such clocks could supply a common time base for lunar navigation networks and surface operations without separate systems.

Load-bearing premise

The actual orbit must stay close enough to the nominal time-aligned path that the reported desynchronization errors continue to apply.

What would settle it

A deployed lunar orbiter clock whose time difference from selenoid proper time exceeds 190 nanoseconds after one year would show the simulation performance does not hold in practice.

Figures

Figures reproduced from arXiv: 2512.23098 by Chong Yang, Jing Zhang, Ren-Fang Geng, Tian-Ning Yang, Yi Xie, Yong Huang.

**Figure 1.** Figure 1: (a) shows that the desynchronization ∆ ∗ grow with the increment of the initial inclination, from ∆ ∗ = 40 ns for ip,0 = 0 to ∆ ∗ = 190 ns for ip,0 = 85◦ after a year. It suggests that the frequency offset ∆f ∗ is at the level of . 6 × 10−15 (see [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

read the original abstract

Context. Among options for definition of the lunar reference time, the option taking Lunar Coordinate Time (O1) has its simplicity but cannot be realized by any clock without steering, while another option adopting the lunar geoid (selenoid) proper time (O2) has its convenience for users on the lunar surface but would bring a new scaling of spatial coordinates and mass parameter of the Moon. Aims. We propose a ''time aligned orbit'' that the readings of an ideal clock in this orbit could equal to the selenoid proper time in O2 and these readings could be converted to Lunar Coordinate Time in O1 by a known linear transformation. Methods. We show that there exist the time aligned orbit around the Moon with its semi-major axis of about 1.5 lunar radius slightly depending on its inclination. We conduct a set of numerical simulations to assess to what extent a clock on these orbits could realize O2 in a more realistic lunar environment. Results. We find that the proper time in our simulations would desynchronize from the selenoid proper time up to 190 ns after a year with a frequency offset of 6E-15, which is solely 3.75% of the frequency difference in O2 caused by the lunar surface topography. These numbers might be further reduced to 13 ns and 4E-16, if we could account for the deviation of the mean orbits in our simulations from the nominal ones. Conclusions. One might simultaneously realize O1 and O2 by deployment of a single clock in the time aligned orbit. This approach also has its scalability for other terrestrial planets beyond the Earth-Moon system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proposes a time-aligned lunar orbit for dual time scale realization, but the quoted performance depends on an orbit fidelity assumption that is not fully demonstrated in the results.

read the letter

The key point here is that the authors have found a lunar orbit where a clock's reading can directly give the selenoid proper time, while a straightforward linear shift turns it into Lunar Coordinate Time. This avoids the need for either steering or coordinate rescaling. The new element is the identification of these time-aligned orbits with semi-major axis near 1.5 lunar radii. The derivation comes from matching the proper time rates in the two definitions, and they show the dependence on inclination is mild. The numerical work then tests how close a real clock on such an orbit stays to the ideal selenoid time in a modeled lunar gravity field. This approach has the advantage of being concrete: they give specific numbers from the simulations, like a maximum desync of 190 nanoseconds over a year and a frequency offset around 6 times 10 to the minus 15. They also compare it to the effect of lunar topography, which puts the size of the error in perspective. The main limitation is that these numbers assume the actual mean orbit in the integration stays very close to the analytically chosen nominal orbit. The paper points out that including the deviations would improve things to 13 nanoseconds and 4 times 10 to the minus 16, but it does not include that adjusted calculation or explain the mapping from element differences to time error. Without seeing that step, the best-case performance remains a claim rather than a demonstrated result. Details on the exact lunar gravity model and the integrator are also thin. The paper is relevant for anyone working on lunar reference frames, navigation systems, or mission planning that needs consistent time across orbit and surface. It offers a practical idea worth checking. I recommend sending it to referees so they can verify the orbit derivation and ask for the full simulation outputs including the correction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes 'time aligned orbits' around the Moon with semi-major axis approximately 1.5 lunar radii (weakly inclination-dependent) such that an ideal clock on the orbit realizes selenoid proper time (O2) directly; these readings can be converted to Lunar Coordinate Time (O1) by a known linear transformation. Analytical derivation identifies such orbits, and numerical simulations in a realistic lunar field report proper-time desynchronization from selenoid time of up to 190 ns after one year with frequency offset 6×10^{-15} (3.75 % of the topographic redshift difference). The authors note that correcting for mean-orbit deviations from nominal values could reduce these figures to 13 ns and 4×10^{-16}.

Significance. If the performance numbers are confirmed, the approach simultaneously realizes both O1 and O2 with a single orbital clock, avoiding steering for O1 and coordinate rescaling for O2. The work applies standard general-relativistic proper-time calculations to the lunar case, supplies explicit orbit parameters, and includes quantitative numerical assessment against lunar models; these elements strengthen its relevance for lunar navigation and timing infrastructure and indicate scalability to other terrestrial bodies.

major comments (2)

[Results] Results section: The central performance claim (190 ns/yr desynchronization and 6×10^{-15} frequency offset) rests on the unverified premise that the integrated mean orbital elements remain sufficiently close to the analytically derived nominal time-aligned orbits. The manuscript states that explicitly accounting for the observed deviations would improve the figures to 13 ns and 4×10^{-16}, yet supplies neither the corrected integration results nor a quantitative mapping from element mismatch to proper-time error. This assumption is load-bearing for the headline claim that the orbit realizes O2 to the stated precision.
[Methods] Methods section: The numerical simulations are cited with specific desynchronization numbers, but the text provides no details on the lunar gravity model employed, the numerical integration scheme, step size, or error-budget analysis. Without these elements the robustness of the 190 ns and 6×10^{-15} figures cannot be evaluated.

minor comments (2)

[Abstract] Abstract: '6E-15' should be rendered as 6×10^{-15} for standard scientific notation.
[Abstract] Abstract: The 3.75 % comparison to topographic redshift would benefit from an explicit statement of the absolute topographic frequency difference used in the percentage calculation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the two major comments below and will revise the manuscript to incorporate the requested clarifications and additional results.

read point-by-point responses

Referee: [Results] Results section: The central performance claim (190 ns/yr desynchronization and 6×10^{-15} frequency offset) rests on the unverified premise that the integrated mean orbital elements remain sufficiently close to the analytically derived nominal time-aligned orbits. The manuscript states that explicitly accounting for the observed deviations would improve the figures to 13 ns and 4×10^{-16}, yet supplies neither the corrected integration results nor a quantitative mapping from element mismatch to proper-time error. This assumption is load-bearing for the headline claim that the orbit realizes O2 to the stated precision.

Authors: We agree that the headline performance figures require stronger validation. In the revised manuscript we will add the results of new integrations that explicitly correct the mean orbital elements to the analytically derived nominal values, together with a quantitative mapping (derived from the first-order proper-time perturbation formula already used in the paper) showing how semi-major-axis and eccentricity mismatches translate into accumulated proper-time error. This will confirm or adjust the projected 13 ns and 4×10^{-16} figures under realistic conditions. revision: yes
Referee: [Methods] Methods section: The numerical simulations are cited with specific desynchronization numbers, but the text provides no details on the lunar gravity model employed, the numerical integration scheme, step size, or error-budget analysis. Without these elements the robustness of the 190 ns and 6×10^{-15} figures cannot be evaluated.

Authors: We accept that the current Methods section lacks the necessary reproducibility information. In the revision we will specify the lunar gravity model (including the degree and order of the spherical-harmonic expansion employed), the numerical integrator and its order, the fixed step size used, and a concise error-budget analysis for the proper-time integration (truncation, round-off, and model truncation contributions). revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard GR proper-time formulas and external lunar models

full rationale

The paper analytically identifies time-aligned orbits from general-relativistic proper-time expressions equating orbital and selenoid clocks, then performs numerical integration against independent lunar gravity models. No equation reduces a claimed prediction to a fitted parameter by construction, no self-citation supplies a load-bearing uniqueness theorem, and the reported 190 ns / 6E-15 figures are direct simulation outputs rather than tautological renamings. The noted orbit-deviation caveat is an explicit modeling limitation, not a hidden self-definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard general relativity for proper time and a conventional lunar gravity field model; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

domain assumption General relativity governs lunar proper time calculations
Invoked for all clock rate derivations in the abstract.
domain assumption Lunar gravity field and topography models are sufficiently accurate for the simulation
Required to produce the quoted 190 ns and 6E-15 figures.

pith-pipeline@v0.9.0 · 5626 in / 1187 out tokens · 37641 ms · 2026-05-16T20:04:07.765235+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

TCL = (1 + LL)(τs − τs0) + ... with LL ≡ c^{-2} WM0 (Eq. 2–3); LP ≡ (3/2) GM_M / (c^2 ā_p) [...] (Eq. 6); time-aligned orbit when LP = LL yields ā_p ≈ 1.5 R_M (Eq. 8–10)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Numerical simulations include point-mass Moon + Sun + planets + 100th-degree harmonics; desynchronization ≤ 190 ns/yr, frequency offset ≤ 6×10^{-15}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

to establish a standard Lunar Ce lestial Reference System (LCRS) and Lunar Coordinate Time (TCL)

Ardalan, A. A. & Karimi, R. 2014, Celest. Mech. Dyn. Astron., 118, 75 Bourgoin, A., Defraigne, P ., & Meynadier, F. 2025, arXiv e- prints [ arXiv:2507.21597], Lunar Reference Timescale, https://arxiv.org/abs/2507.21597 Formichella, V ., Galleani, L., Signorile, G., & Sesia, I. 2021, GPS Solut., 25, 56 IAU. 2024a, Resolution II: “to establish a standard Lu...

work page arXiv 2014
[2]

The resulting and corrected frequency o ﬀsets are also listed.Number ¯ ap [km] ¯ ep ¯ip ¯a∗ p [km] ¯ e∗ p ¯i∗ p ∆f ∗ ∆f ∗ c 1 2606.2658 0 0 2606.1094 0.0039 1

Comparison of nominal mean orbital elements ¯σp and mean orbital elements ¯σ∗ p from our numerical simulations with σ= {a, e, i}. The resulting and corrected frequency o ﬀsets are also listed.Number ¯ ap [km] ¯ ep ¯ip ¯a∗ p [km] ¯ e∗ p ¯i∗ p ∆f ∗ ∆f ∗ c 1 2606.2658 0 0 2606.1094 0.0039 1 . 907◦

work page arXiv
[3]

6 ×10−15 −2 ×10−16 2 2606.1186 0 25 ◦ 2605.9063 0.0056 23 . 395◦

work page arXiv
[4]

736◦ 2605.3468 0.0046 53

3 ×10−15 −4 ×10−16 3 2605.7163 0 54 . 736◦ 2605.3468 0.0046 53 . 475◦

work page arXiv
[5]

4 ×10−15 −3 ×10−16 4 2605.4477 0 85 ◦ 2604.9803 0.0040 84 . 715◦

work page arXiv

[1] [1]

to establish a standard Lunar Ce lestial Reference System (LCRS) and Lunar Coordinate Time (TCL)

Ardalan, A. A. & Karimi, R. 2014, Celest. Mech. Dyn. Astron., 118, 75 Bourgoin, A., Defraigne, P ., & Meynadier, F. 2025, arXiv e- prints [ arXiv:2507.21597], Lunar Reference Timescale, https://arxiv.org/abs/2507.21597 Formichella, V ., Galleani, L., Signorile, G., & Sesia, I. 2021, GPS Solut., 25, 56 IAU. 2024a, Resolution II: “to establish a standard Lu...

work page arXiv 2014

[2] [2]

The resulting and corrected frequency o ﬀsets are also listed.Number ¯ ap [km] ¯ ep ¯ip ¯a∗ p [km] ¯ e∗ p ¯i∗ p ∆f ∗ ∆f ∗ c 1 2606.2658 0 0 2606.1094 0.0039 1

Comparison of nominal mean orbital elements ¯σp and mean orbital elements ¯σ∗ p from our numerical simulations with σ= {a, e, i}. The resulting and corrected frequency o ﬀsets are also listed.Number ¯ ap [km] ¯ ep ¯ip ¯a∗ p [km] ¯ e∗ p ¯i∗ p ∆f ∗ ∆f ∗ c 1 2606.2658 0 0 2606.1094 0.0039 1 . 907◦

work page arXiv

[3] [3]

6 ×10−15 −2 ×10−16 2 2606.1186 0 25 ◦ 2605.9063 0.0056 23 . 395◦

work page arXiv

[4] [4]

736◦ 2605.3468 0.0046 53

3 ×10−15 −4 ×10−16 3 2605.7163 0 54 . 736◦ 2605.3468 0.0046 53 . 475◦

work page arXiv

[5] [5]

4 ×10−15 −3 ×10−16 4 2605.4477 0 85 ◦ 2604.9803 0.0040 84 . 715◦

work page arXiv