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arxiv: 2512.02431 · v2 · pith:5Q6OTX55new · submitted 2025-12-02 · 🌌 astro-ph.IM

High-Precision Amplitude-Modulated Continuous-Wave Lunar Laser Ranging

Slava G. Turyshev This is my paper

Pith reviewed 2026-05-21 19:09 UTC · model grok-4.3

classification 🌌 astro-ph.IM
keywords lunar laser rangingamplitude-modulated continuous wavephase observablesrange precisionretroreflectorsatmospheric effectssignal-to-noise ratio
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The pith

A dedicated lunar laser ranging station using amplitude-modulated continuous-wave transmission can reach 0.08 mm absolute range precision.

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

This paper develops the metrology and covariance framework for high-power amplitude-modulated continuous-wave lunar laser ranging. It centers on radio-frequency envelope phase observables, multi-tone ambiguity removal, range and range-rate estimators, Doppler derotation, and observation covariances. Calculations show that with photon rates from a 1 kW 1064 nm transmitter on a 1-2 m telescope to 10 cm retroreflectors, the photon-statistical range floor reaches 0.08-0.14 mm in generic high-power operation and 30-60 micrometers in a dedicated case. Adding representative residual atmosphere and instrument effects, the analysis concludes that 0.08 mm absolute range precision is plausible under favorable conditions. This precision would tighten existing millimeter-level constraints on relativistic gravity and the Moon's interior structure.

Core claim

For a GHz-class precision tone, c/(4 pi f_m) equals 2.38567 cm/rad, so 0.10 mm photon-limited range precision requires signal-to-noise ratio around 240. With detected photon rates appropriate to a 1 kW, 1064 nm transmitter on a 1-2 m class telescope ranging to 10 cm corner-cube retroreflectors, the photon-statistical range floor is 0.08-0.14 mm in a generic high-power case, 30-60 micrometers in a dedicated AM-CW case, and less than 30 micrometers in a photon-rich case. With representative residual atmosphere and instrument allocations, a dedicated station can plausibly reach 0.08 mm absolute range precision under favorable conditions. Range-rate precision below 1 micrometer per second needs

What carries the argument

RF-envelope phase observables from amplitude-modulated continuous-wave signals, with multi-tone ambiguity removal and full covariance analysis

If this is right

  • Range-rate precision below 1 micrometer per second requires observation windows of several hundred seconds, or shorter windows only in photon-rich operation.
  • Differential lunar laser ranging between nearby reflectors suppresses common-mode station and atmospheric terms but leaves independent photon noise unaffected.
  • Robust design precision bands are 45-90 micrometers for the dedicated AM-CW case and 35-60 micrometers in photon-rich excellent-seeing operation.
  • The approach imposes concrete requirements on link signal-to-noise ratio, Doppler derotation, detector mode, instrument stability, and multi-tone linearity.

Where Pith is reading between the lines

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

  • Sub-0.1 mm ranging would allow tighter discrimination among lunar interior density models that remain degenerate at current millimeter precision.
  • The same phase-based metrology could be adapted to ranging other solar-system targets equipped with corner-cube retroreflectors.
  • Hybrid operation that interleaves AM-CW and existing pulsed observations could reduce overall systematic uncertainty through cross-checks.

Load-bearing premise

The optical link budget and detected photon rates for a 1 kW 1064 nm transmitter on a 1-2 m telescope to 10 cm retroreflectors are taken as given from companion analysis.

What would settle it

A field measurement of actual range residuals obtained with a prototype high-power AM-CW system would test whether the predicted 0.08 mm floor is reached or whether unmodeled effects produce larger scatter.

Figures

Figures reproduced from arXiv: 2512.02431 by Slava G. Turyshev.

Figure 1
Figure 1. Figure 1: FIG. 1. Photon-limited two-way range uncertainty [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Expected one-way line-of-sight range-rate uncerta [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Simplified schematic of the RF-coherent high-power A [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Representative absolute two-way range error budget [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
read the original abstract

Lunar laser ranging (LLR) currently delivers mm-class tests of relativistic gravity and the lunar interior, but further gains are limited by photon-starved pulsed systems, array-induced pulse broadening, and atmospheric variability. This paper develops the metrology and covariance layer for high-power amplitude-modulated continuous-wave (AM-CW) LLR. The optical link budget and kW-class CW architecture are taken from the companion high-power CW LLR analysis; here the focus is on RF-envelope phase observables, multi-tone ambiguity removal, range and range-rate estimators, detector requirements, Doppler derotation, and observation-level covariances. For a GHz-class precision tone, \(c/(4\pi f_m)\) =2.38567 cm/rad, so 0.10 mm photon-limited range precision requires SNR ~ 240. With detected photon rates appropriate to a 1 kW, 1064 nm transmitter on a 1-2 m class telescope ranging to 10 cm corner-cube retroreflectors, the T ~ 100 s photon-statistical range floor is 0.08-0.14 mm in a generic high-power case, 30-60 um in a dedicated AM-CW case, and <30 um in a photon-rich case. With representative residual atmosphere and instrument allocations, a dedicated station can plausibly reach 0.08 mm absolute range precision under favorable conditions. Range-rate precision below 1 um/s requires several-hundred-second windows, or shorter windows only in photon-rich operation. Differential LLR between nearby lunar reflectors suppresses common-mode station and atmospheric terms, but it cannot suppress independent photon noise. Robust design bands are ~45-90 um for the dedicated AM-CW case and ~35-60 um in photon-rich excellent-seeing operation. The resulting requirements on link SNR, Doppler derotation, detector mode, instrument PSD/Allan stability, oscillator slew, multi-tone nonlinearity, and differential CONOPS are presented.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops the metrology and covariance framework for high-power amplitude-modulated continuous-wave (AM-CW) lunar laser ranging. It focuses on RF-envelope phase observables, multi-tone ambiguity removal, range and range-rate estimators, Doppler derotation, detector requirements, and observation-level covariances. Using detected photon rates taken from a companion high-power CW LLR analysis for a 1 kW 1064 nm transmitter on a 1-2 m telescope to 10 cm retroreflectors, it reports a photon-statistical range floor of 0.08-0.14 mm (generic) or 30-60 µm (dedicated) for T ~ 100 s, and concludes that a dedicated station can plausibly reach 0.08 mm absolute range precision under favorable conditions with representative residual atmosphere and instrument allocations. Range-rate precision below 1 µm/s and robust design bands of 45-90 µm are also derived.

Significance. If the central projections hold, the work could enable a meaningful advance in LLR precision beyond current mm-class limits, supporting improved tests of relativistic gravity and lunar interior models. The detailed treatment of phase observables, multi-tone methods, and covariances is a clear strength and provides concrete design requirements on link SNR, oscillator stability, and differential CONOPS that future stations could use. The significance is reduced, however, by the wholesale dependence on external photon-rate inputs whose independent validation is not shown here.

major comments (2)
  1. [Abstract] Abstract: the headline precision floors (0.08-0.14 mm generic, 30-60 µm dedicated, and the 0.08 mm absolute claim) are obtained by converting the photon-statistical range floor via c/(4π f_m) = 2.38567 cm/rad and requiring SNR ≈ 240 for 0.10 mm. These numbers rest directly on 'detected photon rates appropriate to a 1 kW, 1064 nm transmitter...' taken from the companion analysis, with no independent link-budget derivation, atmospheric transmission model, or efficiency budget appearing in this manuscript. This makes the central claim load-bearing on unexamined external assumptions.
  2. [Abstract] Abstract: the statement that 'with representative residual atmosphere and instrument allocations, a dedicated station can plausibly reach 0.08 mm absolute range precision under favorable conditions' is presented as a conclusion, yet the manuscript provides only summary results for the SNR and T ~ 100 s floor; full error propagation from phase observable to range, including any covariance contributions from Doppler derotation or multi-tone nonlinearity, is not shown and cannot be checked against the companion link budget.
minor comments (2)
  1. [Abstract] The conversion factor is given as c/(4π f_m) =2.38567 cm/rad; confirm that the numerical value is exact for the chosen GHz-class modulation frequency and that it is used consistently in all subsequent range-rate and covariance expressions.
  2. Ensure every reference to photon rates or link budget explicitly cites the companion paper's section or equation number so readers can trace the inputs without ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. This manuscript is a companion paper focused on the metrology, phase observables, and covariance framework for AM-CW lunar laser ranging, with the optical link budget and photon rates taken from the companion high-power CW LLR analysis. We respond point by point to the major comments and will revise the manuscript to improve clarity and self-containment.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline precision floors (0.08-0.14 mm generic, 30-60 µm dedicated, and the 0.08 mm absolute claim) are obtained by converting the photon-statistical range floor via c/(4π f_m) = 2.38567 cm/rad and requiring SNR ≈ 240 for 0.10 mm. These numbers rest directly on 'detected photon rates appropriate to a 1 kW, 1064 nm transmitter...' taken from the companion analysis, with no independent link-budget derivation, atmospheric transmission model, or efficiency budget appearing in this manuscript. This makes the central claim load-bearing on unexamined external assumptions.

    Authors: We acknowledge that the precision projections rely directly on detected photon rates and link parameters from the companion high-power CW LLR analysis. This manuscript is deliberately scoped to the RF-envelope phase observables, multi-tone ambiguity removal, range and range-rate estimators, Doppler derotation, detector requirements, and observation-level covariances. To address the concern about external assumptions, we will add a concise summary section or table in the revised manuscript that lists the key link parameters (transmitter power, wavelength, telescope aperture, retroreflector size, and resulting photon rates) drawn from the companion work, together with explicit cross-references. This will allow the central claims to be traced without requiring immediate consultation of the companion paper. revision: yes

  2. Referee: [Abstract] Abstract: the statement that 'with representative residual atmosphere and instrument allocations, a dedicated station can plausibly reach 0.08 mm absolute range precision under favorable conditions' is presented as a conclusion, yet the manuscript provides only summary results for the SNR and T ~ 100 s floor; full error propagation from phase observable to range, including any covariance contributions from Doppler derotation or multi-tone nonlinearity, is not shown and cannot be checked against the companion link budget.

    Authors: The covariance framework developed in the manuscript converts the RF phase observable to range using the factor c/(4π f_m), incorporates Doppler derotation to maintain phase stability, accounts for multi-tone nonlinearity in ambiguity resolution, and combines these with the photon-statistical floor to produce the reported range and range-rate precisions as well as the robust design bands. The 0.08 mm figure for a dedicated station already folds in representative residual atmosphere and instrument allocations. Nevertheless, we agree that the explicit step-by-step propagation and breakdown of individual covariance terms could be presented more transparently. In revision we will expand the relevant methods or results section (and add an appendix if needed) to show the full error budget from phase observable through to final range estimate, including the separate contributions from Doppler derotation, multi-tone effects, and the allocated atmosphere/instrument terms, so that the numbers can be verified against the assumed photon rates. revision: yes

Circularity Check

1 steps flagged

Precision claim rests on photon rates and link budget taken from companion paper without re-derivation here

specific steps
  1. self citation load bearing [Abstract]
    "The optical link budget and kW-class CW architecture are taken from the companion high-power CW LLR analysis; here the focus is on RF-envelope phase observables... With detected photon rates appropriate to a 1 kW, 1064 nm transmitter on a 1-2 m class telescope ranging to 10 cm corner-cube retroreflectors, the T ~ 100 s photon-statistical range floor is 0.08-0.14 mm in a generic high-power case, 30-60 um in a dedicated AM-CW case... a dedicated station can plausibly reach 0.08 mm absolute range precision under favorable conditions."

    The 0.08 mm precision floor is computed by converting the imported photon-statistical rates via the paper's SNR formula (c/(4π f_m) =2.38567 cm/rad, SNR ≈ 240 for 0.10 mm) and adding residual atmosphere/instrument allocations. Since the rates and architecture are taken directly from the companion paper without independent derivation or external benchmark shown here, the central numerical prediction reduces to those external assumptions.

full rationale

The paper develops independent metrology for RF-envelope phase observables, multi-tone ambiguity removal, range/rate estimators, Doppler derotation, and covariances, including the explicit conversion c/(4π f_m) = 2.38567 cm/rad requiring SNR ~240 for 0.10 mm precision. These steps are self-contained. However, the headline numerical results (0.08 mm absolute range precision, 0.08-0.14 mm generic floor, 30-60 µm dedicated floor) are obtained by applying those formulas to detected photon rates and link budget imported wholesale from the companion high-power CW LLR analysis. No independent optical budget, atmospheric model, or efficiency derivation appears here, so the strongest claim reduces to the companion's assumptions. This matches self-citation load-bearing because the cited source overlaps in authorship and supplies the load-bearing inputs without re-verification in the present manuscript.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The precision claims rest on hardware performance assumptions drawn from a companion study together with standard optical-metrology relations; no new physical entities are postulated.

free parameters (2)
  • integration time T
    Chosen as ~100 s to compute the photon-statistical range floor of 0.08-0.14 mm
  • residual atmosphere and instrument error allocations
    Representative values added to photon noise to reach the final 0.08 mm total precision
axioms (2)
  • domain assumption RF-envelope phase provides a direct range observable with scale c/(4 pi f_m)
    Central metrology relation used to convert phase precision into range precision
  • domain assumption Photon rates achievable with 1 kW 1064 nm transmitter and 1-2 m telescope to lunar retroreflectors
    Foundation for all statistical floor calculations

pith-pipeline@v0.9.0 · 5891 in / 1565 out tokens · 68768 ms · 2026-05-21T19:09:34.677558+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. High-Power AM-CW Lunar Laser Ranging as a $\mu$Hz SGWB Detector

    gr-qc 2026-05 unverdicted novelty 6.0

    AM-CW lunar laser ranging achieves μHz SGWB sensitivity of 5.29×10^{-9} D_cov (80 μm range uncertainty) or 2.07×10^{-9} D_cov (50 μm) over 5 years, with discovery possible if covariance degradation stays below ~3.6-13.7.

  2. Fundamental Physics in 2025: Status, Decisive Targets, and Path Forward

    gr-qc 2025-12 unverdicted novelty 2.0

    The review summarizes the baseline SM+GR+Lambda-CDM framework, lists major anomalies and missing pieces, surveys theoretical and experimental approaches, and outlines a staged roadmap organized by decision points.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages · cited by 2 Pith papers · 1 internal anchor

  1. [1]

    For a tone near 50 MHz this interval is of order a few metres

    Use the lowest-frequency tone to obtain a coarse estimate of t he unwrapped phase and hence of the two-way range modulo the corresponding ambiguity interval. For a tone near 50 MHz this interval is of order a few metres

    work page

  2. [2]

    Form a synthetic-wavelength observable from the close freque ncy pair at 50 and 50 . 1 MHz. The associated synthetic two-way wavelength Λ 2w is of order kilometres, and the measured phase difference between the two tones constrains the admissible set of integer ambiguities for the co arse range solution

    work page

  3. [3]

    200 MHz) to refine t he range estimate within the remaining synthetic- wavelength bins

    Use the intermediate-frequency tone (e.g. 200 MHz) to refine t he range estimate within the remaining synthetic- wavelength bins. The shorter ambiguity interval at this frequency restricts the allowed integer combinations further, still subject to consistency with the dynamical light-time model

    work page

  4. [4]

    1 GHz) as the prec ision carrier

    Finally, use the highest-frequency tone (e.g. 1 GHz) as the prec ision carrier. The admissible integer for this tone is selected such that the corresponding range solution is simult aneously consistent with the coarse and intermediate-frequency constraints and with the predicted roun d-trip light time

    work page

  5. [5]

    photon-rich

    Reject any integer combination for which the implied range differs f rom the modelled light time by more than the allocated synthetic-wavelength tolerance or for which the multi-to ne residuals indicate unmodelled frequency- dependent path delays. In this scheme the internal metrology and calibration keep the freq uency-dependent instrumental path differenc...

    work page

  6. [6]

    treats the detected photon stream as a stationary Poisson pro cess with constant mean rate ˙Nγ ; atmospheric scintillation then enters only through slow fluctuations of ˙Nγ and, hence, of SNR AM(T ). For a 1 m aperture at 1064 nm on an Earth–Moon path the expected scintillation index is modest, so that amplitude noise at the modulation frequency is small c...

    work page

  7. [7]

    two-color

    Atmosphere The relevant quantity for LLR is the variation of the neutral-atmos phere delay over an integration window T , not its absolute value. Under Kolmogorov turbulence with frozen flow, t he optical-path structure function obeys the usual (τ /τ 0)5/ 3 scaling, and for T ≫ τ0 the variance of the time-averaged path scales as σ 2 R, atm(T ) ∝ τ0/T , as ...

    work page

  8. [8]

    To keep instrument below ∼ 5 × 10− 5 m over 100 s at 1 GHz, residual instrument phase must remain ≪ 2 × 10− 3 rad

    Instrument A continuous internal reference through the same RF and ADC cha in removes most drift. To keep instrument below ∼ 5 × 10− 5 m over 100 s at 1 GHz, residual instrument phase must remain ≪ 2 × 10− 3 rad. Bench stability near 0.1 K, co–located short fibers/RF cables, low–CTE mounts, and car eful thermal design can achieve this; the overall behavior...

    work page

  9. [9]

    × 103, (3–5) × 104, and ∼ 105 s− 1; (ii) the three turbulence regimes of Table III, with σR, atm(T ) in the 300–500, 50–150, and 30–80 µ m bands on T ≃ 30–100 s windows; and (iii) a common instrumental allocation σR, inst ≃ 40 µ m from Sec. V A 2. Case A couples the lowest–flux regime to the generic mid-latitude at mosphere (Regime A) and therefore reprodu...

    work page

  10. [10]

    It synthesizes the modulation tones {fm} in the 50 MHz–1 GHz range and provides timing for the ADCs

    Frequency reference A hydrogen maser (or equivalent ultra–stable oscillator) [ 30]. It synthesizes the modulation tones {fm} in the 50 MHz–1 GHz range and provides timing for the ADCs. Short–term f ractional frequency instability (Allan deviation) at the level of 10 − 14 on 1–10 s and better than 10 − 15 on 10 2 s ensures that stochastic oscillator noise ...

    work page

  11. [11]

    precision tone

    Laser and modulation A single–frequency 1064 nm master oscillator feeding a master–osc illator power–amplifier (MOPA) chain. The modulation is imposed at the seed using a LiNbO 3 Mach–Zehnder modulator with tone set {fm,i }; typical amplitude– modulation indices am ∼ 0. 3–0. 7 are feasible. The MOPA is operated well below saturation to preser ve AM index a...

    work page

  12. [12]

    A fast steering mirror provides milli– arcsecond pointing control

    Transmit/receive optics A 1–2 m telescope, used in a monostatic or near–monostatic configu ration. A fast steering mirror provides milli– arcsecond pointing control. A narrow optical passband (on the or der of 1–3 nm) and spatial filtering in the focal plane suppress lunar background, especially during bright phases. Aper ture shape and central obscuration ...

    work page

  13. [13]

    The back end performs digital I/Q demodulation at each tone, producin g z(t) = A(t) exp[iφ(t)] at Hz–kHz rates

    Detector and back-end A photon–counting superconducting nanowire single–photon dete ctor (SNSPD) at 1064 nm, with system detection efficiency of a few tenths and dark count below ∼ 10 s− 1, or a low–noise InGaAs photodiode at higher flux. The back end performs digital I/Q demodulation at each tone, producin g z(t) = A(t) exp[iφ(t)] at Hz–kHz rates. Phase and...

    work page

  14. [14]

    Its phase φ inst(t) monitors instrument drift: laser/mixer phase noise, RF path length , ADC timing, and residual AM–to–PM conversion

    Internal reference A short, stable reference path is measured continuously through the same RF and ADC chain. Its phase φ inst(t) monitors instrument drift: laser/mixer phase noise, RF path length , ADC timing, and residual AM–to–PM conversion. Subtracting φ inst(t) from the lunar phase per tone removes the bulk of instrument pat h and electronics drift. ...

    work page

  15. [15]

    The laser room and RF bench are therefore designed with several kilowatts of heat reje ction and active temperature control that keeps bulk temperature excursions within ∼ 0

    Thermal and electrical budget A 1 kW optical transmitter with overall wall–plug efficiency of order 20 –30% implies an electrical draw of ∼ 3–5 kW for the laser system alone, plus additional load for chillers, RF electr onics, and cryogenics. The laser room and RF bench are therefore designed with several kilowatts of heat reje ction and active temperature ...

    work page

  16. [16]

    Vibration from cryocoolers and HV AC systems is isolated t o keep induced path–length noise below the atmospheric floor on the relevant time scales

    Mechanical and vibrational environment The RF/optical bench and beam transport to the telescope are mo unted on a low–vibration pier, with the amplifier chain, modulator, and internal reference arranged to minimize diffe rential path motion between the lunar and reference channels. Vibration from cryocoolers and HV AC systems is isolated t o keep induced pa...

    work page

  17. [17]

    Environmental sensors (temperature, pressure, hum idity, wind) and a seeing monitor (e.g

    Telescope enclosure and site monitoring The telescope sits in a dome or roll–off enclosure designed to minimize loc al seeing and thermal gradients across the primary. Environmental sensors (temperature, pressure, hum idity, wind) and a seeing monitor (e.g. MASS/DIMM) provide per–night estimates of Fried parameter r0, coherence time τ0, and isoplanatic ang...

    work page

  18. [18]

    The control system enforce s night–time operation only, checks for aircraft and satellite exclusion zones, and closes shutters in the event of any int erlock violation

    High-power safety and beam control Operation of a 1 kW, 1064 nm CW beam requires a safety system with h ardwired shutters, interlocked doors, and software limits on telescope pointing. The control system enforce s night–time operation only, checks for aircraft and satellite exclusion zones, and closes shutters in the event of any int erlock violation. Nom...

    work page

  19. [19]

    Residual timing jitter at the ADC is r equired to be ≲ 0

    Timing and frequency distribution The hydrogen maser reference is distributed to the modulation sou rce, local oscillators, and ADC clocks over phase–stable RF or optical links. Residual timing jitter at the ADC is r equired to be ≲ 0. 5 ps rms, corresponding to ≪ 0. 1 mm in two–way range, and any differential delays between the lunar and internal–referenc...

    work page

  20. [20]

    Geophysical co-location and ancillary sensors For long-term interpretation of sub-mm normal points it is advanta geous to embed the AM–CW LLR station in a broader geodetic and gravimetric environment. Co-located contin uous Global Navigation Satellite System (GNSS), a superconducting gravimeter, tiltmeters or strainmeters beneat h the telescope pier, and ...

    work page

  21. [21]

    J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L . Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple , J. R. Wiant, J. G. Williams, and C. F. Yoder, Lunar Laser Rangi ng: A Continuing Legacy of the Apollo Program, Science 265, 482 (1994)

    work page 1994

  22. [22]

    T. W. Murphy Jr., E. G. Adelberger, J. B. R. Battat, L. N. Ca rey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, The Apache Point Observatory Lunar Laser- ranging Operation: Instrument Description and First Detec tions, PASP 120, 20 (2008)

    work page 2008

  23. [23]

    T. W. Murphy, Lunar laser ranging: the millimeter challe nge, Rep. Progr. Phys. 76, 076901 (2013)

    work page 2013

  24. [24]

    J. G. Williams, S. G. Turyshev, and D. H. Boggs, Progress i n Lunar Laser Ranging Tests of Relativistic Gravity, Phys. Rev. Lett. 93, 261101 (2004)

    work page 2004

  25. [25]

    J. G. Williams, S. G. Turyshev, and D. H. Boggs, Lunar lase r ranging tests of the equivalence principle with the Earth and Moon, CQG 29, 184004 (2012)

    work page 2012

  26. [26]

    S. G. Turyshev, Lunar laser ranging with high-power cont inuous-wave lasers, Phys. Rev. Applied 23, 064066 (2025) , arXiv:2502.02796 [astro-ph.IM]

    work page arXiv 2025

  27. [27]

    S. G. Turyshev, High-Precision Lunar Corner-Cube Retro reflectors: A Wave-Optics Perspective (2025), arXiv:2504.06409 [physics.optics]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  28. [28]

    S. G. Turyshev, M. Shao, and I. Hahn, Fundamental physics and lunar science investigations with advanced lunar laser ranging (2021), NAS Decadal Survey on Biological and Physical Scien ces (BPS) Research in Space 2023-2032. 26

    work page 2021

  29. [29]

    Zhang, J

    M. Zhang, J. M¨ uller, L. Biskupek, and V. V. Singh, Charac teristics of differential lunar laser ranging, Astron. Astrophys. 659, A148 (2022)

    work page 2022

  30. [30]

    Zhang, J

    M. Zhang, J. M¨ uller, and L. Biskupek, Advantages of com bining Lunar Laser Ranging and Differential Lunar Laser Ranging, Astron. Astrophys. 681, A5 (2024)

    work page 2024

  31. [31]

    Zhang, Characteristics and Benefits of Differential Lunar Laser Rang ing, Ph.D

    M. Zhang, Characteristics and Benefits of Differential Lunar Laser Rang ing, Ph.D. thesis , Leibniz University, Hannover (2023)

    work page 2023

  32. [32]

    Briaud, C

    A. Briaud, C. Ganino, A. Fienga, A. M´ emin, and N. Rambau x, The lunar solid inner core and the mantle overturn, Nature 617, 743 (2023)

    work page 2023

  33. [33]

    Blas and A

    D. Blas and A. C. Jenkins, Bridging the µ Hz Gap in the Gravitational-Wave Landscape with Binary Reso nances, PRL 128, 101103 (2022)

    work page 2022

  34. [34]

    Blas and A

    D. Blas and A. C. Jenkins, Detecting stochastic gravita tional waves with binary resonance, Phys. Rev. D 105, 064021 (2022)

    work page 2022

  35. [35]

    D. Blas, A. C. Jenkins, and S. G. Turyshev, Gravitational-wave discovery with lunar and satellite las er ranging , White paper (NASA Fundamental Physics and Gravitation (FunPAG), 2024) dated 4 March 2024

    work page 2024

  36. [36]

    S. G. Turyshev, J. G. Williams, D. H. Boggs, and R. S. Park , Relativistic Time Transformations Between the Solar Syst em Barycenter, Earth, and Moon, Astrophys. J. 985, 140 (2025) , arXiv:2406.16147 [astro-ph.EP]

    work page arXiv 2025

  37. [37]

    S. G. Turyshev, High-Precision Relativistic Time Scal es for Cislunar Navigation (2025), arXiv:2507.22145 [gr-qc]

    work page arXiv 2025

  38. [38]

    P. E. Ciddor, Refractive index of air: new equations for the visible and near infrared, Applied Optics 35, 1566 (1996)

    work page 1996

  39. [39]

    V. B. Mendes and E. C. Pavlis, High-accuracy zenith dela y prediction at optical wavelengths, Geophys. Res. Lett. 31, L14602 (2004)

    work page 2004

  40. [40]

    A. E. Niell, Global mapping functions for the atmospher e delay at radio wavelengths, JGR: Solid Earth 101, 3227 (1996)

    work page 1996

  41. [41]

    B¨ ohm, A

    J. B¨ ohm, A. E. Niell, P. Tregoning, and H. Schuh, Global Mapping Function (GMF): A new empirical mapping function based on numerical weather model data, Geophys. Res. Lett. 33, L07304 (2006)

    work page 2006

  42. [42]

    J. W. Goodman, Statistical Optics (Wiley, New York, 1985)

    work page 1985

  43. [43]

    S. M. Kay, Fundamentals of Statistical Signal Processing, Volume I: E stimation Theory (Prentice Hall, Englewood Cliffs, NJ, 1993)

    work page 1993

  44. [44]

    D. C. Rife and R. R. Boorstyn, Single-tone parameter est imation from discrete-time observations, IEEE Transact. Inform. Theory 20, 591 (1974)

    work page 1974

  45. [45]

    V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw–Hill, New York, 1961)

    work page 1961

  46. [46]

    L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media , 2nd ed. (SPIE Press, Bellingham, W A, 2005)

    work page 2005

  47. [47]

    Roddier, The effects of atmospheric turbulence in opt ical astronomy, in Progr

    F. Roddier, The effects of atmospheric turbulence in opt ical astronomy, in Progr. in Optics, Volume 19 , edited by E. Wolf (North-Holland, Amsterdam, 1981) pp. 281–376

    work page 1981

  48. [48]

    J. G. Williams, S. G. Turyshev, and T. W. Murphy, Improvi ng LLR Tests of Gravitational Theory, IJMPD 13, 567 (2004)

    work page 2004

  49. [49]

    J. G. Williams, S. G. Turyshev, and D. H. Boggs, Lunar Las er Ranging Tests of the Equivalence Principle with the Earth and Moon, IIJMPD 18, 1129 (2009)

    work page 2009

  50. [50]

    D. W. Allan, Statistics of Atomic Frequency Standards, Proc. of the IEEE 54, 221 (1966)

    work page 1966

  51. [51]

    J. J. Degnan, Millimeter accuracy laser ranging: a revi ew, Proc. of the IEEE 81, 1833 (1993)

    work page 1993

  52. [52]

    Currie, S

    D. Currie, S. Dell’Agnello, and G. O. D. Monache, A Lunar Laser Ranging Retroreflector Array for the 21st Century, Acta Astronautica 68, 667 (2011) . Appendix A: Atmospheric turbulence and time averaging We model residual atmospheric path fluctuations as a zero-mean r andom process δRatm(t) with a Kolmogorov optical-path structure function DR(τ) ≡ ⟨[ δRatm(...

    work page 2011