Long Period Tidal Force Variations and Regularities in Orbital Motion of the Earth-Moon Binary Planet System

We have studied long period, 206 and 412 day, variations in tidal sea level corresponding to various moon phases collected from five observatories in the Northern and Southern hemispheres. Variations in sea level in the Bay of Fundy, on the eastern Canadian seaboard, with periods of variation 206 days, and 412 days, have been discovered and carefully studied by C. Desplanque and D. J. Mossman (2001, 2004). The current manuscript focuses on analyzing a larger volume of observational sea level tide data as well as on rigorous mathematical analysis of tidal force variations in the Sun-Earth-Moon system. We have developed a twofold model, both conceptual and mathematical, of astronomical cycles in the Sun-Earth-Moon system to explain the observed periodicity. Based on an analytical solution of the tidal force variation in the Sun-Earth-Moon system, it is shown that the tidal force can be decomposed into two components: the Keplerian component and the Perturbed component. The Perturbed component of the tidal force variation was calculated, and it was shown that the observed periodicity, 206 and 412 days, of atmospheric and hydrosphere tides results from variations of the Perturbed component of tidal force. The amplitude of the Perturbed component of tidal force is . It is the same order of magnitude as the amplitude of the Keplerian component of tidal force: . It follows that the Perturbed component of the variation of a tidal force must always be taken into consideration along with the Keplerian component in geodynamical constructions involving tides.

💡 Research Summary

The paper investigates a long‑period tidal signal with periods of approximately 206 days (≈7 months) and 412 days (≈14 months) that appears in sea‑level records from five geographically dispersed observatories (Murmansk and Magadan in Russia, Lerwick in Scotland, Puerto Williams in Chile, and Suva in Fiji). The authors first present observational evidence: tide gauge data spanning the late 1970s to the early 1990s show that the difference between syzygial (full‑moon or new‑moon) and quadrature tides follows a sinusoidal pattern with a dominant 14‑month cycle. By fitting a simple sine function to the full‑moon–new‑moon height difference, they obtain a period of about 412 days, and the zero‑crossings of this function line up with the minima of the lunar anomalistic month (the time between successive perigees), which recur every ≈206 days. The same pattern is observed at all stations, suggesting a global phenomenon independent of local bathymetry or climate.

To explain the observed periodicity, the authors develop a theoretical model of the Sun–Earth–Moon three‑body system. They decompose the motion of the Earth–Moon binary into two components: (1) the Keplerian motion of the Earth‑Moon barycenter around the Sun, and (2) the perturbed motion of the Earth (and Moon) around their common barycenter caused by the mutual gravitational influence of the Sun and Moon. The perturbed motion corresponds to the well‑known lunar inequality (a 6.44‑arc‑second variation in the argument of perigee). Using Lagrange’s equations, they derive an expression for the total tidal force acting on a point inside the Earth as the sum of a Keplerian term K and a perturbed term P.

The Keplerian term (Eq. 3) has the familiar 1/r³ dependence of the tidal tensor, while the perturbed term (Eq. 4) contains additional sinusoidal factors involving the Sun’s and Moon’s longitudes. The authors implement a numerical routine that evaluates P at two‑week intervals over several decades. The resulting envelope (Fig. 6) exhibits clear oscillations with periods of ≈206 days and ≈412 days. Moreover, the amplitude of the 206‑day component is about 1.8 times smaller than that of the 412‑day component, matching the amplitude ratio observed in the tide‑gauge records.

A striking claim of the paper is that the magnitude of the perturbed component is of the same order as the Keplerian component: both are quoted as ~10⁻⁸ kg · N⁻¹ (the units are unconventional, but the authors intend to convey that the two forces differ by only a factor of a few). Consequently, they argue that any geodynamical model that includes tidal forcing must retain the perturbed term; neglecting it would omit a contribution comparable to the primary tidal force.

The discussion extends the implications of the 206/412‑day signal beyond ocean tides. The authors cite lunar seismicity studies (Apollo mission data) that show similar periodicities, as well as atmospheric temperature records from Moscow that display 206‑day cycles. They suggest that the perturbed tidal force could be a common driver for these phenomena, linking oceanic, atmospheric, and lithospheric processes through a single astronomical mechanism.

While the paper presents an interesting synthesis of observational and theoretical work, several methodological issues limit the strength of its conclusions. The tide‑gauge analysis lacks rigorous statistical treatment: no spectral analysis, confidence intervals, or detrending procedures are described, making it difficult to assess the significance of the identified periods against red‑noise backgrounds typical of sea‑level records. The claim of global synchrony is based on a small number of stations and relatively short records; longer, higher‑resolution datasets (e.g., satellite altimetry) would be needed to confirm the universality of the signal.

On the theoretical side, the derivation of the perturbed term appears to rely on a first‑order expansion that discards terms of order (R₀/R)², yet the resulting amplitude is claimed to be comparable to the Keplerian term, which raises questions about the consistency of the approximation. Moreover, the use of “kg · N⁻¹” as a unit for tidal force is non‑standard; tidal forces are usually expressed as accelerations (m s⁻²) or as potential gradients (N m⁻²). The numerical values presented (10⁻⁸ kg · N⁻¹ and 10⁵⁸ · 10⁻⁸ kg · N⁻¹) contain typographical errors that obscure the actual magnitude of the forces. A proper dimensional analysis would be essential to verify that the perturbed term indeed contributes at the claimed level.

Finally, the causal link between the perturbed tidal force and atmospheric temperature or lunar seismicity is speculative. Correlation does not imply causation, and the paper does not provide a mechanistic pathway (e.g., how a 10⁻⁸ N m⁻³ tidal acceleration could modulate atmospheric heat transport or trigger moonquakes). Future work should incorporate climate and seismological models that explicitly couple tidal stresses to the relevant physical processes.

In summary, the study identifies a plausible 206‑day and 412‑day modulation in global tide‑gauge records and proposes a perturbed tidal component arising from the Sun‑Earth‑Moon three‑body dynamics as its source. The concept that this perturbed term is of comparable magnitude to the primary Keplerian tide is intriguing and merits further investigation, but the current analysis would benefit from more rigorous statistical validation, clearer physical units, and a quantitative assessment of the term’s impact on oceanic, atmospheric, and lithospheric systems.