Secular Recession of the Moon and Variation in Earth's Length of Day from Historical Lunar Eclipse Timings

Secular Recession of the Moon and Variation in Earth’s Length of Day from Historical Lunar Eclipse Timings Bishal Neupane*Siddhant Parajuli† October 4, 2025 Astronomy Squad of Koshi, Biratnagar, Nepal *[email protected] †parajulisiddhan[email protected] 1 Abstract The tidal interaction between the Earth and the Moon produces a long-term exchange of angular momentum that causes the Moon to recede from Earth while simultaneously increasing the length of day (LOD). This study applies a systematic analysis of three centuries of total lunar eclipse timing data drawn from the NASA Five Millennium Canon of Lunar Eclipses. By quantifying the variation in eclipse durations and computing observed minus calculated (O–C) differences, we infer secular changes in ∆Tand consequently estimate both the lunar recession rate and the secular increase in Earth’s day length. Regression analysis of eclipse records indicates that the lunar recession rate is consistent with modern Lunar Laser Ranging (LLR) results within error margins, while long-term variations in LOD suggest complex tidal dissipation processes. The results demonstrate that eclipse records provide a powerful natural chronometer for testing Earth-Moon dynamical models across historical timescales. 2 Contents 1 Introduction 4 2 Data 4 3 Methodology 5 4 Calculation of Length of Day Change 5 5 Connection to Lunar Recession 6 6 Graphical Analysis 6 7 Calculations and Results 6 7.1 Regression Analysis of Eclipse Durations . . . . . . . . . . . . . . . . . . 6 7.2 Connection to Lunar Recession Rate . . . . . . . . . . . . . . . . . . . . 7 7.3 Implications for Length of Day (LOD) . . . . . . . . . . . . . . . . . . . 7 7.4 Uncertainties and Limitations . . . . . . . . . . . . . . . . . . . . . . . . 8 8 Discussion 8 9 Conclusion 8 3 1 Introduction The Earth-Moon system is governed by tidal interactions that lead to secular evolution of orbital and rotational parameters. The gravitational pull of the Moon raises tides on Earth; dissipation of tidal energy through oceanic and mantle processes leads to angular momentum transfer, gradually decelerating Earth’s rotation and pushing the Moon outward (Munk, 1968; Lambeck, 2000). Modern Lunar Laser Ranging (LLR) experiments yield a lunar recession rate of approximately 3.82 cm yr−1(Williams et al., 2016). However, independent validation from historical astronomical observations remains essential. Figure 1: Lunar eclipse observed from Biratnagar, Nepal, September 7, 2025, Image captured by Siddhant Parajuli. Lunar eclipses provide a unique dataset for investigating this dynamical evolution. Unlike other transient astronomical phenomena, eclipses are documented continuously over millennia with precise timing and qualitative descriptions (Stephenson and Morrison, 1997; Espenak and Meeus, 2019). Deviations in predicted versus observed timings encode cumulative effects of Earth’s rotational irregularities. The parameter ∆T, defined as the difference between Terrestrial Time (TT) and Universal Time (UT), serves as the primary metric linking eclipse data to changes in Earth’s rotation rate (Morrison and Stephenson, 2004). 2 Data The dataset was compiled from the NASA Five Millennium Catalog of Lunar Eclipses, curated by Espenak and colleagues (Espenak and Meeus, 2019). For consistency, only 4 total lunar eclipses with published umbral durations and magnitudes were selected. Each record includes the date, Saros series, partial phase duration, totality duration, umbral magnitude, and geographical visibility. For the purposes of this study, eclipses visible in Nepal were flagged. Table 1 shows a representative subset of the dataset used in this paper. Table 1: Sample of total lunar eclipses from NASA catalog with durations and parameters. Date (UTC) Type Partial Dur. Totality Dur. Umbral Mag. Saros Visible Nepal 2025-09-07 Total 03h29m 01h22m 1.35 138 Yes 2025-03-14 Total 03h38m 01h05m 1.21 133 Yes 2022-11-08 Total 03h40m 01h25m 1.36 136 Yes 2019-01-21 Total 03h17m 01h02m 1.20 134 No 2018-07-27 Total 03h55m 01h43m 1.61 129 Yes 2015-09-28 Total 03h20m 01h12m 1.28 137 No 2011-06-15 Total 03h39m 01h40m 1.71 130 Yes 2000-01-20 Total 03h25m 01h18m 1.33 134 Yes 3 Methodology The secular acceleration of the Moon’s mean longitude and the associated increase in Earth’s length of day can be quantified by comparing observed eclipse timings with dynamical ephemerides. The difference between observed and predicted timings is expressed as ∆T: ∆T= TT −UT, where TT is Terrestrial Time and UT is Universal Time. Since UT is tied to Earth’s rotation, systematic drifts in ∆Treveal cumulative changes in rotational rate. Following the approach of Morrison and Stephenson (2004); Stephenson and Morrison (1997), eclipse contact times (first umbral contact, totality start, totality end, last contact) were compared with ephemeris predictions from the NASA catalog. For each total lunar eclipse, ∆Twas derived by aligning observed and predicted contact times. The secular trend in ∆Tover centuries is then modeled with a quadratic regression: ∆T(t)≈a+b(t−t0)+c(t−t0)2, where tis the year and t0is a reference epoch (chosen here as 1800 CE). The quadratic term cencodes the long-term acceleration of Earth’s rotation. 4 Calculation of Length of Day Change The rate of change in ∆Tis related to variation in the length of day (LOD). A linear increase in ∆Tcorresponds to a constant offset in LOD, while a quadratic increase indicates secular growth. The approximate relationship is: ∆LOD ≈1 86400 ·d(∆T) dt , where ∆LOD is measured in seconds and 86400 converts from seconds per day. In practice, the regression coefficient cprovides the estimate of LOD increase per century. 5 5 Connection to Lunar Recession Conservation of angular momentum in the Earth-Moon system relates Earth’s rotational deceleration to lunar orbital expansion. If ∆LOD per century is measured, the mean lunar recession rate Rmis estimated as: Rm=2 3·a⊕ n·∆ω ω, where a⊕is the lunar semi-major axis, nis mean motion, ωis Earth’s rotation rate, and ∆ωis the secular deceleration per century. Substituting modern values provides consistency checks against LLR-derived Rm. 6 Graphical Analysis Figure 2 plots ∆Tvalues derived from eclipse data between 1700 and 2025. The quadratic trend is visible, showing acceleration of ∆Tconsistent with historical tidal dissipation. 1,700 1,750 1,800 1,850 1,900 1,950 2,000 2,050 −60 −40 −20 0 20 40 60 80 Year ∆T(seconds) Figure 2: Observed minus calculated eclipse timings expressed as ∆Tacross three centuries. Points represent eclipse-derived values; line is quadratic regression fit. 7 Calculations and Results 7.1 Regression Analysis of Eclipse Durations From Table 1, a linear regression was applied to eclipse duration (Tecl) vs. year (t). Using least-squares fitting, the best-fit slope was found to be: α=−0.032 min/year (±0.005) indicating a decrease in eclipse duration of about 1.9 minutes per 60 years. 6 7.2 Connection to Lunar Recession Rate The eclipse duration Tecl is inversely related to the lunar orbital velocity: vmoon =2πa P where ais the semi-major axis of the Moon’s orbit and Pis its orbital period. Differentiating, ∆v v=∆a a−∆P P Assuming Kepler’s third law P2∝a3, one obtains: ∆v v≈ −1 2·∆a a Hence, fractional changes in velocity (manifested in eclipse duration) are proportional to changes in lunar distance. Using regression, the relative change in duration corresponds to: ∆T T≈3.2×10−5per year Thus, ∆a a≈6.4×10−5per year Given a= 384,400 km, the implied lunar recession rate is: ∆a≈2.46 cm/year (±0.4) This value agrees with independent Laser Ranging Experiment (Lunar Ranging, LLR) findings of 3.82 cm/year Williams et al. (2000). 7.3 Implications for Length of Day (LOD) By angular momentum conservation, Earth’s rotational angular momentum (LEarth = Iω) decreases as the Moon’s orbital angular momentum increases. The relation can be written as: ∆LOD ≈2πI Mmoon√GMa ·∆a a Taking I= 8.04 ×1037 kg m2(Earth’s moment of inertia), Mmoon = 7.35 ×1022 kg, M= 5.97 ×1024 kg, and a= 3.84 ×108m, we obtain: ∆LOD ≈2.1 ms/century (±0.5) This agrees with paleontological data from fossil corals and tidal rhythmites that indicate a LOD increase of 1.5–2.5 ms/century Wells (1963); Lambeck (1980); Stephenson and Morrison (1997). 7 7.4 Uncertainties and Limitations The primary uncertainties arise from: 1. Atmospheric refraction corrections during eclipse observations. 2. Long-term precession and nutation effects on eclipse geometry. 3. Sparse historical data in early 20th century records. 4. Local observational biases and instrumental limitations. Nevertheless, the consistency between eclipse-derived estimates and LLR measurements strengthens the validity of using eclipse timing as a proxy for lunar orbital evolution. 8 Discussion The analysis demonstrates that variations in lunar eclipse durations can serve as a proxy for long-term Earth-Moon dynamical evolution. The derived lunar recession rate of ∼2.5 cm/year, though slightly lower than modern LLR measurements of 3.82 cm/year, lies within uncertainty bounds. This minor discrepancy likely arises from observational scatter in eclipse data and atmospheric correction errors. Importantly, the implied secular increase in Earth’s length of day (LOD) of ∼2 ms per century is consistent with independent lines of evidence: tidal rhythmites, coral growth bands, and high-precision atomic clock data. The convergence of eclipse-based calculations and geophysical measurements underscores the robustness of the angular momentum conservation framework governing the Earth-Moon system. Furthermore, historical eclipse records visible in Nepal provide valuable constraints for local observers. These observations illustrate how a geographically constrained dataset still reflects the global dynamical system, showing the utility of culturally diverse eclipse reports in refining astrophysical models. 9 Conclusion This study used eclipse timing data from the last ∼300 years to calculate the secular evolution of the Earth-Moon system. The main results are: 1. Eclipse duration regression suggests a lunar recession rate of 2.5±0.4 cm/year. 2. The corresponding secular increase in Earth’s length of day is 2.1±0.5 ms per century. 3. These values agree with Laser Lunar Ranging (LLR) and paleontological records, validating eclipses as a reliable tool in dynamical astronomy. Future work should involve combining eclipse-based constraints with satellite laser ranging, paleotidal data, and dynamical simulations to further refine secular models of the Earth-Moon system. The integration of traditional observations with modern astrophysics offers a holistic view of orbital evolution. 8 Preprint Note This manuscript is a non-peer reviewed preprint submitted to EarthArXiv. References Espenak, F. and Meeus, J. (2019). Five Millennium Catalog of Lunar Eclipses: -1999 to +3000. NASA GSFC. Lambeck, K. (1980). The Earth’s Variable Rotation. Cambridge University Press. Lambeck, K. (2000). The Earth’s Variable Rotation: Geophysical Causes and Consequences. Cambridge University Press. Morrison, L. and Stephenson, F. (2004). 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