Pith. sign in

REVIEW 3 major objections 4 minor 69 references

This paper sets the relativistic clock-drift numbers a lunar positioning system will need: TCL runs 58.7 µs/day ahead of Earth time, surface clocks vary by ±15 ns/day, and frozen-orbit navigators run 1.99 µs/day fast relative to the surface

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-04 20:21 UTC pith:RSBHIZ5E

load-bearing objection Useful cross-check and concrete clock-budget numbers for lunar PNT, but the surface redshift map is too approximate near the south pole to serve as a hard requirement. the 3 major comments →

arxiv 2509.08871 v1 pith:RSBHIZ5E submitted 2025-09-10 astro-ph.EP

Relativistic Time Modeling for Lunar Positioning Navigation and Timing

Yan Seyffert This is my paper
classification astro-ph.EP
keywords Lunar Coordinate Time (TCL)relativistic timekeepinglunar PNTgravitational redshiftpost-Newtonian 1PNlunar surface clockselliptical lunar frozen orbitTCL-TT secular drift
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper aims to give future lunar navigation a reliable relativistic time budget. It formalizes Lunar Coordinate Time (TCL) — the Moon's analogue of Earth's Geocentric Coordinate Time — in two published formulations, runs them with independent planetary ephemerides, and shows they agree: TCL runs about 58.7 µs/day faster than Terrestrial Time, with periodic terms at the 0.5 µs/day level tied to the Moon's orbital arguments. It then maps the gravitational redshift a stationary clock experiences across the lunar surface, finding ±15 ns/day variation (about 28.7 ns/day from lowest to highest terrain), and simulates clocks aboard the elliptical lunar frozen orbits planned for the lunar navigation constellation, finding a secular drift of −1.986 µs/day relative to surface time with sub-0.1 µs/day periodic variations. If these numbers hold, they become the working clock-budget inputs for lunar Positioning, Navigation and Timing: nanosecond-level time offsets translate directly into meter-level range errors.

Core claim

The thesis establishes that two independent routes to Lunar Coordinate Time (TCL) give one consistent answer. The first route defines TCL through an explicit transformation from barycentric coordinate time, evaluated over a 10-year span; the second route integrates a differential-rate equation for TCL-TT over a multi-decade window as part of a planetary-ephemeris integration. After removing the linear drift, both waveforms agree at the 0.05 µs level, and the principal periodic terms match luni-solar arguments. The thesis then attaches numbers to the clock environment: TCL-TT drifts 58.7 µs/day; the lunar surface gravitational redshift varies by about ±15 ns/day around an arbitrary reference

What carries the argument

The load-bearing object is the 1st post-Newtonian relation between coordinate time and proper time, translated to a lunar-centered frame. TCL is defined by the same integral that defines TCG — the Moon's barycentric velocity and the external gravitational potential of every body except the Moon — and the thesis shuttles between its explicit transformation form and a differential-rate form that is numerically integrated with planetary ephemerides. For the surface, the carrying device is the orthometric height h_orth = h_topo − h_geoid and the radial potential V_surface = u0 r0/(r0 + h_orth), which turns a gravity model and a topography model into a fractional time-dilation map V/c^2. For orbi

Load-bearing premise

The surface redshift maps assume the Moon's gravity at a clock's location can be obtained by evaluating a global gravity model at that location's height above a reference surface; this breaks down where mass sits beside the clock rather than below it, such as the crater walls that dominate the lunar south pole.

What would settle it

Compute the TCL-TCG time difference using the DE440 ephemeris with the same sampling, detrending, and 2020-2022 window used for the INPOP21a series; if the residual between the two computed waveforms exceeds the ~0.05 µs level shown in the thesis, the claimed consistency of the two TCL formulations fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • A lunar navigation system that distributes Earth-referenced time must apply a rate offset near 58.7 µs/day to TCL; without it, accumulated timing errors grow to tens of kilometres per day.
  • Surface users must correct for elevation-dependent redshift up to ~28.7 ns/day between extremes; cm-level positioning needs clock or signal models at that resolution.
  • ELFO navigation satellites can use a single frequency offset of about −1.99 µs/day plus a small harmonic model (sub-0.1 µs/day) instead of continuous clock steering.
  • The agreement of the two TCL formulations means the IAU-style definition can be realized numerically either as an explicit transformation or as a differential-rate integration, giving implementers a cross-check.

Where Pith is reading between the lines

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

  • Beyond the paper: the crater-wall limitation in the surface potential maps is most acute exactly where future missions concentrate — the south pole — so the ±15 ns/day map likely needs recomputation with direct multipole evaluation at varying radii before it can anchor operational clock budgets.
  • Beyond the paper: the ~28.7 ns/day surface spread corresponds to fractional clock stabilities around 3×10⁻¹³ over a day, meaning optical-clock-class hardware (not rubidium) would be needed to turn surface redshift into sub-metre vertical geodesy on the Moon.
  • Beyond the paper: the sub-1 ns harmonic lines in the ELFO clock signal, tied to external perturbations, suggest clock comparisons along frozen orbits could serve as a weak probe of lunar orientation or gravity-field variations — an inverse problem the paper flags but does not solve.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript (arXiv:2509.08871) is a Master's thesis that develops and tests a 1PN relativistic time framework for lunar PNT. It formalizes TCL analogously to TCG, compares the Kopeikin-Kaplan and Fienga et al. formulations, and finds the secular TCL-TT drift of 58.7 µs/day consistent between them (Sec. 4.2.1). It then builds gravitational-redshift maps for stationary lunar surface clocks from LDEM128 topography and GRGM900C gravity, reporting ±15 ns/day variation and a 28.7 ns/day maximum (Sec. 5.1.2), derives clock stability requirements, and estimates orientation/Sagnac effects (Sec. 5.3). Finally, it propagates four ELFO satellites with GODOT and evaluates their proper time relative to a lunar surface clock using three formula forms, obtaining a secular drift of -1.986 µs/day and periodic variations below 0.1 µs/day (Sec. 6.3).

Significance. If correct, this work provides cross-validated reference numbers for emerging lunar timekeeping standards: the TCL-TT secular drift, a surface clock-budget map, and ELFO clock behavior for Moonlight. The author deserves credit for re-deriving the two TCL formulations rather than fitting parameters, for using independent ephemerides (DE440 vs. INPOP21a) and tools (pyshtools, GODOT), for making interactive 3D maps publicly available, and for candidly stating limitations. The secular drift consistency is arithmetic and robust. However, the surface redshift map rests on an approximation whose error is unquantified and that is acknowledged in the manuscript itself; this limits the reliability of one of the three headline results.

major comments (3)
  1. [Sec. 5.1.2, Eq. (5.2), and Sec. 5.4] The surface clock drift map is generated from Vsurface = u0·r0/(r0 + horth), i.e. a spherical 1/r potential. The manuscript itself states this is 'possibly limited to heights near the selenoid' and inaccurate at crater walls. The lunar south pole, the target for Moonlight/NovaMoon, is dominated by such topography. Since the claimed signal is only ±15 ns/day (max 28.7 ns/day), an unquantified error from lateral mass distributions could be a nontrivial fraction. I request an error estimate for representative south-polar sites, e.g. by evaluating U(r,θ,λ) directly from the degree-900 SH coefficients at the actual topographic radius (the method identified in Sec. 5.4) and comparing with Eq. (5.2). Without this, the headline surface budget is not established to the stated accuracy.
  2. [Sec. 4.2.2 and Sec. 4.3] The periodic-term comparison of TCL is based on digitized curves from Kopeikin Figs. 3-4, not the underlying time series, and covers only 2020-2022. The author acknowledges in Sec. 4.3 that a definitive test would require regenerating the DE440 dataset with identical processing. This is acceptable for a preliminary consistency check, but the claim that the two formulations agree 'within our uncertainties' for periodic terms would be considerably stronger if the DE440 series were regenerated with matched processing or if the underlying data were shared. Please either supply the comparison data/processing chain or explicitly restrict the consistency claim to the secular drift and the main spectral lines.
  3. [Sec. 6.3, Fig. 6.4, and Sec. 6.4] The residual between the Lander-Like formula and the Cartesian/Keplerian orbital formulas accumulates to ~0.7 µs over 250 days with a near-linear slope, about 10^-3 of the 500 µs main signal. The text's suspicion that 'problems lie within the evaluation ... maybe because both use the semi-major axis a approximated from the inertial state vectors' is not a demonstration. Since Eq. (6.5) is an analytic Keplerian result while the real trajectory is non-Keplerian, the residual should be explained (e.g. by re-deriving the closed form with osculating elements, or by showing a pure Keplerian test case yields zero residual). This does not change the -1.986 µs/day secular value, but it determines which formula is quoted for periodic behavior.
minor comments (4)
  1. [Appendix B] The text gives the mean lunar surface contribution as '3.1383 × 10^-11 (−2.712 ns/day)'; this should be µs/day, not ns/day. The factor-1000 error could confuse readers comparing with Sec. 5.1.2.
  2. [Sec. 4.2] Typos: 'Kopeikin et el.' should be 'Kopeikin et al.'; 'Kopkeikin' appears misspelled. Eq. (4.2) is also hard to parse; please check formatting and signs.
  3. [Sec. 6.3, Fig. 6.4] The caption uses 'TLL - TKO' while the text refers to the Lander-Like and Cartesian/Keplerian orbital formulas. Please make the notation consistent and define TLL/TKO in the caption.
  4. [Sec. 5.4] The statement that the method is 'possibly limited to heights near the selenoid' seems to conflict with the maps using topography up to ±10 km from the mean radius. Please clarify the intended domain of validity.

Circularity Check

0 steps flagged

No circularity: all claimed outputs are forward computations from independent ephemerides, gravity/topography models, and published formulations; the acknowledged surface-potential caveat is an accuracy limitation, not a circular reduction.

full rationale

I walked the derivation chain. Chapter 4 defines TCL following the IAU/TCG analogy and the published Kopeikin and Fienga formulations; the numerical comparison is a cross-check between two independent implementations (DE440 vs INPOP21a), and the arithmetic 'recovery' of 58.7 µs/day is a path-consistency check around Fig. 4.1, not a fitted parameter renamed as a prediction. Chapter 5 computes surface redshift maps by applying a standard Newtonian 1/r potential scaling to independent LDEM128 topography and GRGM900C gravity data, with the arbitrarily chosen u0 subtracted; the resulting ±15 ns/day and 28.7 ns/day numbers are benchmarked against the external Bourgain et al. result in Appendix B. The manuscript explicitly flags in Sec. 5.4 that the radial-only potential may be inaccurate near crater walls; this is an acknowledged modeling limitation, not a circular step. Chapter 6 propagates published ELFO orbital elements in GODOT and evaluates three equivalent 1PN proper-time forms; the secular drift of −1.986 µs/day is dominated by the analytic GM/c²(3/2a − 1/r0) term, and the residual between the Lander-like and orbital formulas is a numerical consistency check rather than an input. No uniqueness theorem is imported from the authors, no ansatz is smuggled via citation, and the only self-citation (Fienga et al. 2024, whose data were supplied by the supervisor) is non-load-bearing because the central claims are independently verified against Kopeikin et al. and the external literature. The thesis therefore contains no significant circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central numbers rest on the IAU/IERS 1PN time transformation framework, on published spherical harmonic gravity/topography models, and on ephemerides and orbit propagation tools; the thesis adds no new physical entities. The main hand-chosen input is the reference potential u0 used to define the selenoid zero for the surface maps.

free parameters (2)
  • Lunar selenoid reference potential u0 = 2.82100 × 10^6 m²/s²
    Chosen 'arbitrarily' (Sec 5.1.1) close to the value at the mean lunar radius; it sets the zero-drift surface for the ±15 ns/day map. The relative variation is insensitive to this choice, but the absolute map depends on it.
  • Lunar time scale rate LL = ΦL/c² = 2.7128 µs/day
    Taken from Kopeikin [23], where ΦL = 2.822336927×10^6 m²/s² was obtained by least-squares fitting to lunar surface features [44]; the thesis uses it to define LT and the TCL-TT drift of 58.7 µs/day.
axioms (5)
  • standard math 1PN metric and IERS time transformation framework (Eqs. 3.2-3.12, 4.1)
    The thesis derives and uses the IAU/IERS 1PN relation between proper time and coordinate time; this is standard post-Newtonian gravity.
  • domain assumption TCL is defined analogously to TCG with external potential from all bodies except the Moon (Eq. 4.1, Eq. 4.3)
    Adopts the proposed IAU definition of lunar coordinate time; the thesis does not derive it from first principles but cross-checks two published implementations.
  • domain assumption Lunar gravity field is described by spherical harmonic model GRGM900C and topography by LDEM128, with potential evaluated at r0 + horth (Eq. 5.2)
    Underlies the surface redshift maps; the author flags validity limits near the selenoid and crater walls in Sec. 5.4.
  • domain assumption 2PN and gravitomagnetic terms are negligible at current clock accuracies
    Stated in Sec 3.2.2 with references [23][24]; the thesis computes at 1PN only.
  • domain assumption Orbit propagation with GODOT using 120×120 lunar gravity field and DE431 constants, with non-gravitational forces deactivated
    The ELFO proper time results depend on this dynamical setup; the paper notes SRP and station-keeping are not included (Sec. 6.4).

pith-pipeline@v1.3.0-alltime-deepseek · 37796 in / 12062 out tokens · 107531 ms · 2026-08-04T20:21:31.565727+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Relativistic Time Modeling for Lunar Positioning Navigation and Timing." pith.science (2026). https://pith.science/paper/RSBHIZ5E

@misc{pith2026250908871,
  author       = {Pith},
  title        = {Pith review of: Relativistic Time Modeling for Lunar Positioning Navigation and Timing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RSBHIZ5E}},
  note         = {Machine review of arXiv:2509.08871}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Future lunar missions will depend on an internationally agreed upon timescale that remains accurate under the Moon's unique gravitational environment and its orbital dynamics. This thesis investigates the proposed Lunar Coordinate Time (TCL), derived analogously to Geocentric Coordinate Time (TCG) and thus aligned with current IAU proposals. We first formalise the TCL transformation and quantify its characteristics from solar system simulations. Next, we compute stationary surface-clock drifts caused by gravitational redshift and the Moon's changing orientation parameters, evaluating how accurate atomic clocks deployed on the surface of the Moon (much like for ESA's proposed NovaMoon mission) would have to be to measure these effects. Finally, we simulate relativistic proper time for ESA's Moonlight navigation satellites, identifying average drift and harmonic variations, to better understand the system that will comprise and enable a Lunar PNT (Positioning, Navigation and Timing) architecture. These kinds of investigations are an essential step toward a sustained internationally cooperative operation at the lunar south pole and beyond.

Figures

Figures reproduced from arXiv: 2509.08871 by Yan Seyffert.

Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p031_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows a couple of views of this interactive representation [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p034_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p036_5.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p039_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: illustrates these ELFO orbits [PITH_FULL_IMAGE:figures/full_fig_p042_6.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p043_6.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p044_6.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p046_6.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p047_6.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p048_6.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

69 extracted references · 7 canonical work pages

  1. [1]

    NovaMoon: A New Paradigm in Lunar Exploration

    InsideGNSS. NovaMoon: A New Paradigm in Lunar Exploration . Apr. 11, 2025. url: https : / / insidegnss . com / novamoon - a - new - paradigm - in - lunar - exploration/ (visited on 08/31/2025)

    2025

  2. [2]

    Argonaut: Europe’s lunar lander programme

    European Space Agency (ESA). Argonaut: Europe’s lunar lander programme . 2025. url: https : / / www . esa . int / Science _ Exploration / Human _ and _ Robotic _ Exploration / Exploration/Argonaut_Europe_s_lunar_lander_programme (visited on 08/31/2025)

    2025

  3. [3]

    What is ESA’s Moonlight initiative? ESA Multimedia video (duration 3:21)

    European Space Agency. What is ESA’s Moonlight initiative? ESA Multimedia video (duration 3:21). Licence: ESA Standard Licence. Nov. 9, 2022. url: https://www.esa. int / ESA _ Multimedia / Videos / 2022 / 11 / What _ is _ ESA _ s _ Moonlight _ initiative (visited on 08/31/2025)

    2022

  4. [4]

    Temperature, Reflectance Point to Frost near the Moon ’s Poles

    NASA Scientific Visualization Studio. Temperature, Reflectance Point to Frost near the Moon ’s Poles. https://svs.gsfc.nasa.gov/4574/. 2017. (Visited on 08/23/2025)

    2017

  5. [5]

    National Aeronautics and Space Administration (NASA). Artemis. Program landing page. NASA. 2025. url: https : / / www . nasa . gov / humans - in - space / artemis/(visited on 08/31/2025)

    2025

  6. [6]

    Initial Results of the Lunar GNSS Receiver Experiment (LuGRE)

    J. Parker et al. “Initial Results of the Lunar GNSS Receiver Experiment (LuGRE)”. In: Proceedings of the 38th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2025) . Session B3: Future of Space, Lunar, and Ex- traterrestrial Navigation 1; presentation abstract. The Institute of Navigation. Baltimore, MD,...

    2025

  7. [7]

    Project Olympus

    ICON. Project Olympus . url: https : / / www . iconbuild . com / projects / project - olympus (visited on 09/04/2025)

    2025

  8. [8]

    Project Olympus

    BIG — Bjarke Ingels Group. Project Olympus. Client: ICON; collaborators: NASA; SEArch+ (Space Exploration Architecture). BIG — Bjarke Ingels Group. url: https://big.dk/ projects/nasa-olympus-7493 (visited on 09/04/2025)

    2025

  9. [9]

    Schauer and D

    K. Schauer and D. Baird. LunaNet: Empowering Artemis with Communications and Navi- gation Interoperability. https://www.nasa.gov/humans-in-space/lunanet-empowering- artemis- with- communications- and- navigation- interoperability/. NASA, Space Communications and Navigation (SCaN) program office. Oct. 2021

    2021

  10. [10]

    Thales Alenia Space to Build ESA Moonlight Satellites for Telespazio

    Parsonson, Andrew. Thales Alenia Space to Build ESA Moonlight Satellites for Telespazio . Eurpean Spaceflight Online Article. Mar. 13, 2025. url: https://europeanspaceflight. com/thales-alenia-space-to-build-esa-moonlight-satellites-for-telespazio/ (visited on 08/31/2025)

    2025

  11. [11]

    Moonlight: Thales Alenia Space to develop the space segment of the navigation system orbiting around the Moon

    Thales Alenia Space. Moonlight: Thales Alenia Space to develop the space segment of the navigation system orbiting around the Moon . Press release. Mar. 12, 2025. url: https: / / www . thalesaleniaspace . com / en / press - releases / moonlight - thales - alenia - space-develop-space-segment-navigation-system-orbiting (visited on 08/31/2025). 48 Bibliography 49

    2025

  12. [12]

    Moonlight : qu’est-ce que c’est ? Includes artist’s impression by ESA Pierre Carril

    Futura-Sciences. Moonlight : qu’est-ce que c’est ? Includes artist’s impression by ESA Pierre Carril. 2021. url: https://www.futura-sciences.com/sciences/actualites/ exploration- lunaire- moonlight- nous- voulons- fournir- large- gamme- services- communication-navigation-lies-exploration-lune-87636/ (visited on 08/23/2025)

    2021

  13. [13]

    Resolutions adopted at the General Assemblies

    International Astronomical Union. Resolutions adopted at the General Assemblies . https: / / iauarchive . eso . org / administration / resolutions / general _ assemblies/. IAU Archive (hosted by ESO). Accessed 2025-09-10

    2025

  14. [14]

    Manchester, August 2000

    Resolutions of the 24th International Astronomical Union General Assembly. Manchester, August 2000. Final version hosted by SYRTE, Observatoire de Paris. International Astro- nomical Union. Aug. 2000. url: https://syrte.obspm.fr/IAU_resolutions/Resol- UAI.htm (visited on 09/10/2025)

    2000

  15. [15]

    The IAU 2000 Resolutions for Astrometry, Celestial Mechanics, and Metrology in the Relativistic Framework: Explanatory Supplement

    M. Soffel et al. “The IAU 2000 Resolutions for Astrometry, Celestial Mechanics, and Metrology in the Relativistic Framework: Explanatory Supplement”. In:The Astronomical Journal 126.6 (Dec. 2003), p. 2687. doi: 10.1086/378162. arXiv: astro-ph/0303376

    Pith/arXiv arXiv 2000

  16. [16]

    English version

    IAU 2006 Resolution B2: Supplement to the IAU 2000 Resolutions on reference sys- tems. English version. International Astronomical Union. Aug. 24, 2006. url: https : / / iauarchive . eso . org / static / resolutions / IAU2006 _ Resol2 . pdf(visited on 09/10/2025)

    2006

  17. [17]

    English version

    IAU 2006 Resolution B3: Re-definition of Barycentric Dynamical Time (TDB) . English version. International Astronomical Union. Aug. 24, 2006. url: https://iauarchive. eso.org/static/resolutions/IAU2006_Resol3.pdf (visited on 09/10/2025)

    2006

  18. [18]

    Adopted at the XXXII IAU General Assembly, Cape Town

    IAU 2024 Resolution II: to establish a standard Lunar Celestial Reference System (LCRS) and Lunar Coordinate Time (TCL) . Adopted at the XXXII IAU General Assembly, Cape Town. International Astronomical Union (IAU), Commission A3: Fundamental Standards. Aug. 2024. url: https://iauarchive.eso.org/static/resolutions/IAU2024_Resol2_ English.pdf (visited on 0...

    2024

  19. [19]

    Adopted at the XXXII IAU General Assembly, Cape Town

    IAU 2024 Resolution III: on the establishment of a coordinated lunar time standard by international agreement. Adopted at the XXXII IAU General Assembly, Cape Town. In- ternational Astronomical Union (IAU), Commission A3: Fundamental Standards. Aug

    2024

  20. [20]

    Petit and B

    G. Petit and B. L. (eds.) IERS Conventions (2010). IERS Technical Note 36. Frankfurt am Main: Verlag des Bundesamts f¨ ur Kartographie und Geod¨ asie, 2010.url: https://iers- conventions.obspm.fr/content/tn36.pdf

    2010

  21. [21]

    Why new time scales?

    P. K. Seidelmann and T. Fukushima. “Why new time scales?” In: Astronomy and Astro- physics 265.2 (Nov. 1992), pp. 833–838. url: https://ui.adsabs.harvard.edu/abs/ 1992A&A...265..833S

    1992

  22. [22]

    Relativistic Transformations for Time Synchronization and Dissemination in the Solar System

    R. A. Nelson and T. Ely. “Relativistic Transformations for Time Synchronization and Dissemination in the Solar System”. In: Proceedings of the 38th Annual Precise Time and Time Interval (PTTI) Meeting . Dec. 2006, pp. 305–317

    2006

  23. [23]

    Lunar time in general relativity

    S. M. Kopeikin and G. H. Kaplan. “Lunar time in general relativity”. In: Physical Review D 110.8 (Oct. 2024). issn: 2470-0029. doi: 10.1103/physrevd.110.084047

    doi:10.1103/physrevd.110.084047 2024

  24. [24]

    Fienga, N

    A. Fienga, N. Rambaux, and K. Sosnica. Lunar References Systems, Frames and Time- scales in the context of the ESA Programme Moonlight. 2024. arXiv: 2409.10043 [astro-ph.EP]. url: https://arxiv.org/abs/2409.10043

    Pith/arXiv arXiv 2024

  25. [25]

    Testing theories of gravity with planetary ephemerides

    A. Fienga and O. Minazzoli. “Testing theories of gravity with planetary ephemerides”. In: Living Reviews in Relativity 27.1 (2024). Review article; open access, pp. 1–99. doi: 10.1007/s41114-023-00047-0 . 49 Bibliography 50

    work page doi:10.1007/s41114-023-00047-0 2024

  26. [26]

    The JPL Planetary and Lunar Ephemerides DE440 and DE441

    R. S. Park et al. “The JPL Planetary and Lunar Ephemerides DE440 and DE441”. In: The Astronomical Journal 161.3 (2021), p. 105. doi: 10.3847/1538-3881/abd414

    doi:10.3847/1538-3881/abd414 2021

  27. [27]

    Fienga et al

    A. Fienga et al. INPOP19a planetary ephemerides . Notes Scientifiques et Techniques de l’IMCCE S109. IMCCE, Observatoire de Paris, Dec. 14, 2019. url: https://www.imcce. fr/content/medias/recherche/equipes/asd/inpop/inpop19a_20191214.pdf (visited on 08/31/2025)

    2019

  28. [28]

    Development of planetary ephemerides EPM and their applications

    E. V. Pitjeva and N. P. Pitjev. “Development of planetary ephemerides EPM and their applications”. In: Celestial Mechanics and Dynamical Astronomy 119.3-4 (Aug. 2014), pp. 237–256. doi: 10.1007/s10569-014-9569-0

    doi:10.1007/s10569-014-9569-0 2014

  29. [29]

    A comparison of the high accuracy planetary ephemerides DE421, EPM2008, and INPOP08

    J. Hilton and C. Hohenkerk. “A comparison of the high accuracy planetary ephemerides DE421, EPM2008, and INPOP08”. In: Parameters 156.228 (2010), p. 260

    2010

  30. [30]

    Orbit1.svg Diagram illustrating and explaining various terms in relation to Orbits of Celestial bodies

    Lasunncty. Orbit1.svg Diagram illustrating and explaining various terms in relation to Orbits of Celestial bodies. Wikimedia Commons. Licensed under CC BY-SA 3.0 and GFDL 1.2 or later. Oct. 10, 2007. url: https://commons.wikimedia.org/wiki/File:Orbit1. svg (visited on 08/31/2025)

    2007

  31. [31]

    GODOT documentation. 2022. url: https://godot.io.esa.int/godotpy/ (visited on 08/21/2025)

    2022

  32. [32]

    Development of a High Precision Simulation Tool for Gravity Recovery Missions like GRACE

    F. W¨ oscke et al. “Development of a High Precision Simulation Tool for Gravity Recovery Missions like GRACE”. In: Advances in the Astronautical Sciences . Vol. 158. Conference paper. 2016, 2-s2.0-85007365474, pp. 2445–2457

    2016

  33. [33]

    XHPS: A modular approach for the high precision modeling of satel- lite formations for geodetic and fundamental physics applications

    B. Rievers et al. “XHPS: A modular approach for the high precision modeling of satel- lite formations for geodetic and fundamental physics applications”. In: 42nd COSPAR Scientific Assembly . Vol. 42. July 2018, H0.5-9-18

    2018

  34. [34]

    NIST: How UTC(NIST) Works . Nov. 2020. url: https://www.nist.gov/pml/time- and-frequency-division/time-realization/utcnist-time-scale-0/how-utcnist- works (visited on 08/21/2025)

    2020

  35. [35]

    The time scales TAI and EAL

    Physikalisch-Technische Bundesanstalt (PTB). The time scales TAI and EAL . PTB web- site. 2023. url: https://www.ptb.de/cms/en/ptb/fachabteilungen/abt4/fb-44/ag- 441/realisation-of-legal-time-in-germany/the-time-scales-tai-and-eal.html (visited on 08/27/2025)

    2023

  36. [36]

    P. J. G. Teunissen and O. Montenbruck, eds. Springer Handbook of Global Navigation Satellite Systems . Cham: Springer, 2017. isbn: 978-3-319-42927-4. doi: 10.1007/978-3- 319-42928-1

    doi:10.1007/978-3- 2017

  37. [37]

    Time and frequency dissemination: Advances in GPS transfer techniques

    T. Parker and D. Matsakis. “Time and frequency dissemination: Advances in GPS transfer techniques”. In: GPS World (Nov. 2004), pp. 32–38. url: https://www.researchgate. net / publication / 291614271 _ Time _ and _ frequency _ dissemination _ Advances _ in _ GPS_transfer_techniques

    2004

  38. [38]

    Travagnin

    M. Travagnin. Chip-Scale Atomic Clocks: Physics, technologies, and applications . JRC Technical Report JRC125394. European Commission, Joint Research Centre (JRC), Aug

  39. [39]

    Optimization of Time Synchronization Techniques on Computer Networks

    F. Mkacher. “Optimization of Time Synchronization Techniques on Computer Networks”. Ph.D. dissertation. Grenoble, France: Universit´ e Grenoble Alpes, June 2020.url: https: //www.researchgate.net/publication/345357290_Optimization_of_Time_Synchronization_ Techniques_on_Computer_Networks

    arXiv 2020

  40. [40]

    Atomic clocks for geodesy

    T. E. Mehlst¨ aubler et al. “Atomic clocks for geodesy”. In: Reports on Progress in Physics 81.6 (Apr. 2018), p. 064401. issn: 1361-6633. doi: 10.1088/1361-6633/aab409. 50 Bibliography 51

    doi:10.1088/1361-6633/aab409 2018

  41. [41]

    A Relativistic Framework to Estimate Clock Rates on the Moon

    N. Ashby and B. R. Patla. “A Relativistic Framework to Estimate Clock Rates on the Moon”. In: The Astronomical Journal 168.3 (Aug. 2024), p. 112. doi: 10.3847/1538- 3881/ad643a

    doi:10.3847/1538- 2024

  42. [42]

    INPOP06: a new planetary ephemeris

    A. Fienga et al. “INPOP06: a new planetary ephemeris”. In: IAU Joint Discussion. Vol. 16. Prague, Czech Republic, Aug. 2006. url: https : / / ui . adsabs . harvard . edu / abs / 2006IAUJD..16E..15F

    2006

  43. [43]

    INPOP06: a new numerical planetary ephemeris

    A. Fienga et al. “INPOP06: a new numerical planetary ephemeris”. In: Astronomy and Astrophysics 477.1 (2008), pp. 315–327. doi: 10.1051/0004-6361:20066607

    work page doi:10.1051/0004-6361:20066607 2008

  44. [44]

    Effect of topographic bias on geoid and reference ellipsoid of Venus, Mars, and the Moon

    A. A. Ardalan and R. Karimi. “Effect of topographic bias on geoid and reference ellipsoid of Venus, Mars, and the Moon”. In: Celestial Mechanics and Dynamical Astronomy 118 (2014), pp. 75–88. doi: 10.1007/s10569-013-9523-6

    work page doi:10.1007/s10569-013-9523-6 2014

  45. [45]

    Bourgoin, P

    A. Bourgoin, P. Defraigne, and F. Meynadier. Lunar Reference Timescale . 2025. arXiv: 2507.21597 [gr-qc]. url: https://arxiv.org/abs/2507.21597

    arXiv 2025

  46. [46]

    https://shtools.github.io/SHTOOLS/

    SHTOOLS: Spherical Harmonic Tools. https://shtools.github.io/SHTOOLS/. Version: v4.13.1. 2024. (Visited on 08/20/2025)

    2024

  47. [47]

    SHTools: Tools for Working with Spherical Har- monics

    M. A. Wieczorek and M. Meschede. “SHTools: Tools for Working with Spherical Har- monics”. In: Geochemistry, Geophysics, Geosystems 19.8 (May 2018), pp. 2574–2592. doi: 10.1029/2018GC007529

    doi:10.1029/2018gc007529 2018

  48. [48]

    Introduction to Gravity and Magnetic Field Classes

    SHTOOLS Developers. Introduction to Gravity and Magnetic Field Classes . Jupyter note- book in the SHTOOLS examples repository. url: %5Curl % 7Bhttps : / / github . com / SHTOOLS / SHTOOLS / blob / master / examples / notebooks / gravity - and - magnetic - fields.ipynb%7D (visited on 08/20/2025)

    2025

  49. [49]

    plotly.py: The interactive graphing library for Python

    Plotly Technologies Inc. plotly.py: The interactive graphing library for Python . https : //github.com/plotly/plotly.py. Version: v6.1.2. 2025

    2025

  50. [50]

    Wieczorek

    M. Wieczorek. Spherical harmonic models of the shape of the Moon (principal axis coordi- nate system) [LDEM128] . Zenodo. Dataset containing five lunar shape spherical harmonic models in a principal axis reference frame, highest degree 11519. 2024. doi: 10 . 5281 / zenodo.11533783

    2024

  51. [51]

    Elson et al

    P. Elson et al. SciTools/cartopy: REL: v0.24.1. Version v0.24.1. Oct. 2024. doi: 10.5281/ zenodo.13905945. url: https://doi.org/10.5281/zenodo.13905945

    work page doi:10.5281/zenodo.13905945 2024

  52. [52]

    GRGM900C: A degree 900 lunar gravity model from GRAIL primary and extended mission data

    F. G. Lemoine et al. “GRGM900C: A degree 900 lunar gravity model from GRAIL primary and extended mission data”. In: Geophysical Research Letters 41.10 (2014), pp. 3382–3389. doi: 10.1002/2014GL060027

    doi:10.1002/2014gl060027 2014

  53. [53]

    Earth2014: 1 arc-min shape, topography, bedrock and ice-sheet models – available as gridded data and degree-10,800 spherical harmonics

    C. Hirt and M. Rexer. “Earth2014: 1 arc-min shape, topography, bedrock and ice-sheet models – available as gridded data and degree-10,800 spherical harmonics”. In: Interna- tional Journal of Applied Earth Observation and Geoinformation 39 (July 2015), pp. 103–

    2015

  54. [54]

    The development and evaluation of the Earth Gravitational Model 2008 (EGM2008)

    N. K. Pavlis et al. “The development and evaluation of the Earth Gravitational Model 2008 (EGM2008)”. In: Journal of Geophysical Research: Solid Earth 117 (Apr. 2012), B04406. doi: 10.1029/2011JB008916

    doi:10.1029/2011jb008916 2008

  55. [55]

    Long-distance chronometric leveling with a portable optical clock

    J. Grotti et al. “Long-distance chronometric leveling with a portable optical clock”. In: Physical Review Applied 21 (6 June 2024), p. L061001. doi: 10.1103/PhysRevApplied. 21.L061001

    doi:10.1103/physrevapplied 2024

  56. [56]

    Expanded Uncertainty and Cov- erage Factor

    National Institute of Standards and Technology (NIST). Expanded Uncertainty and Cov- erage Factor. NIST Reference on Constants, Units and Uncertainty (CUU). url: https: //physics.nist.gov/cuu/Uncertainty/coverage.html (visited on 08/31/2025). 51 Bibliography 52

    2025

  57. [57]

    Riley and D

    W. Riley and D. Howe. Handbook of Frequency Stability Analysis. July 2008. url: https: //tsapps.nist.gov/publication/get_pdf.cfm?pub_id=50505

    2008

  58. [58]

    Datasheet

    High-Performance Space-Qualified Rubidium Atomic Frequency Standard (RAFS). Datasheet. Excelitas Technologies Corp. 2011. url: https://www.excelitas.com/file-download/ download/public/59176?filename=___Space_Qualified_Rubidium_Atomic_Frequency_ Standard__RAFS_.pdf (visited on 08/29/2025)

    2011

  59. [59]

    Cacciapuoti et al

    L. Cacciapuoti et al. Atomic Clock Ensemble in Space . 2024. arXiv: 2411.02912 [gr-qc]. url: https://arxiv.org/abs/2411.02912

    Pith/arXiv arXiv 2024

  60. [60]

    Relativity in the Global Positioning System

    N. Ashby. “Relativity in the Global Positioning System”. In: Living Reviews in Relativity 6, 1 (Jan. 2003). Accepted 08 January 2003, Published 28 January 2003.doi: 10.12942/lrr- 2003-1

    doi:10.12942/lrr- 2003

  61. [61]

    Derivation of the Sagnac (Earth-rotation) correction and analysis of its accuracy for GNSS applications

    Wang Hu and J. A. Farrell. “Derivation of the Sagnac (Earth-rotation) correction and analysis of its accuracy for GNSS applications”. In: Journal of Geodesy 98.1 (2024), p. 102. doi: 10.1007/s00190-024-01914-6

    work page doi:10.1007/s00190-024-01914-6 2024

  62. [62]

    Effect of Irregularities in the Earth’s Rotation on Relativistic Shifts in Frequency and Time of Earthbound Atomic Clocks

    V. F. Fateev, S. M. Kopeikin, and S. L. Pasynok. “Effect of Irregularities in the Earth’s Rotation on Relativistic Shifts in Frequency and Time of Earthbound Atomic Clocks”. In: Measurement Techniques 58.6 (Sept. 2015), pp. 647–654. issn: 1573-8906. doi: 10.1007/ s11018-015-0769-0

    2015

  63. [63]

    phys.org

    A New Paradigm for Lunar Orbits . phys.org. Source: Science@NASA Trudy E. Bell. Dec

  64. [64]

    Design and station-keeping strategies for robust lunar navigation con- stellations

    M. Ceresoli et al. “Design and station-keeping strategies for robust lunar navigation con- stellations”. In: Acta Astronautica 236 (2025), pp. 20–31. issn: 0094-5765. doi: https: //doi.org/10.1016/j.actaastro.2025.06.025 . url: https://www.sciencedirect. com/science/article/pii/S0094576525003777

    work page doi:10.1016/j.actaastro.2025.06.025 2025

  65. [65]

    Iess et al

    L. Iess et al. ATLAS — Fundamental Techniques, Models and Algorithms for a Lunar Radio Navigation System. Concept Document Part 1: ODTS and Reference Frames . Tech- nical Report CRAS/2021/ATLAS CD-001. ESA AO/1-10712/21/NL/CRS, Version 1.7. July 2023. 52 Acronyms 1PN 1st post-Newtonian order. 8–10, 16, 17, 31, 36, 40, 43, 45 2PN 2nd post-Newtonian order. ...

    2021

  66. [112]

    doi: 10.1016/j.jag.2015.03.001

    work page doi:10.1016/j.jag.2015.03.001 2015

  67. [2006]

    url: https://phys.org/news/2006-12-paradigm-lunar-orbits.html (visited on 08/30/2025)

    2006

  68. [2021]

    2760 / 278540

    doi: 10 . 2760 / 278540. url: https : / / publications . jrc . ec . europa . eu / repository/handle/JRC125394

  69. [2024]

    url: https : / / iauarchive . eso . org / static / resolutions / IAU2024 _ Resol3 _ English.pdf (visited on 09/10/2025)

    2025