Main

Tidal forces of an orbiting satellite control the evolution of a planet’s rotation1,2. Because Earth rotates faster than the orbital angular velocity of the Moon, Earth’s oceanic tidal bulge is pushed ahead of the Moon. This offset exerts a torque on the Moon, which—transferring angular momentum from Earth to the Moon—boosts the Moon to a farther orbit and slows Earth’s rotation, increasing its length of day (LOD) over time. Theoretical lunar recession models have generally been characterized by a steady trend over the past 3 or 4 billion years3,4,5. In contrast with traditional models that propose continuously evolving conditions is the speculation of a temporarily long time interval of constant day length during Precambrian time potentially arising from a stabilization due to resonance with the atmospheric thermal tide6,7,8. There is therefore a need for a sufficiently abundant number of Precambrian LOD constraints that can be used to further understand the evolution of the Earth–Moon system in high temporal resolution.

In contrast with their oceanic counterpart gravitationally excited by the pull of the Moon, atmospheric tides are thermally excited by the absorption of sunlight by water vapour and ozone, the largest of which is semidiurnal. Due to the arrangement of the three-body system and Earth’s sense of rotation, the lunar semidiurnal oceanic tide applies a decelerative torque and the solar semidiurnal atmospheric tide applies an accelerative torque on Earth’s rate of rotation (Fig. 1). A resonance-stabilized day length is the unique condition where, at the point of resonance, the lunar oceanic torque can be cancelled by the solar atmospheric torque. This might have been possible in the Precambrian when the lunar torque is thought to have been one-fourth of its present value due to weaker frictional coupling between Earth and the Moon at faster palaeorotation4,6,7. If the opposing atmospheric torque became equal to the diminished lunar torque, Earth’s long-term rotational deceleration could have been temporarily halted at a constant day length for some interval of Precambrian time6.

Fig. 1: Opposing torques of semidiurnal oceanic and atmospheric tides acting on Earth’s rotation.
figure 1

At the point of resonance, the oceanic and atmospheric tidal torques would balance, stabilizing Earth’s rotation rate at a constant day length. The images are not to scale. Figure adapted with permission from ref. 6, Elsevier. Earth and Moon images were generated using GoogleEarth. Earth data/image sources: data (SIO, NOAA, U.S. Navy, NGA, GEBCO) and image (Landsat/Coperincus and IBCAO) and Moon image source: NASA/USGS/JAXA/SELENE. Sun image is from NASA/SDO.

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The geological record has multiple means of preserving an archive of LOD over time and, by inference, the history of Earth–Moon distance. Biologically mediated means of recording LOD, such as tree rings and coral growth bands, preserve reliable annual banding, but are restricted to the Phanerozoic eon, since 539 million years ago (Ma), when animal life evolved9. For Precambrian time, before the Phanerozoic, other LOD records must be used. When the resonance-stabilized day length hypothesis was first proposed, only a small handful of Precambrian LOD constraints were available. These came from the seasonal sinusoidal growth of stromatolites (following the annual Sun angle), but such constraints require high sediment accumulation rates, which are rare in such depositional settings. The only other constraints at the time came from tidal rhythmites, but such depositional settings are rarely preserved and/or are difficult to prove in the absence of independent geochronology, which is often lacking in Precambrian successions. For example, conflicting interpretations of LOD have been attained for the tidal rhythmites of the ~900 Ma Big Cottonwood Formation10,11.

Critically, LOD can also be calculated from cyclostratigraphic studies of Milankovitch orbital cycles because the astronomical precession frequency (k) is a function of Earth’s rotation rate12. This study was prompted by a recent proliferation of new cyclostratigraphic-based LOD constraints having become available for the Precambrian, with 12 of the 22 Precambrian LOD constraints having been generated in the past 7 years (Extended Data Table 1). Our statistical change-point analysis of the most recent, much more abundant Precambrian LOD compilation indeed yields, as theorized, a flatlining of day length between 2 and 1 billion years ago (Ga) during the mid-Proterozoic (Fig. 2). This observation thus empirically vindicates the modelling-based hypothesis of the emergence of resonance between oceanic (lunar) and atmospheric (solar) tides acting on Earth’s rotation during the Precambrian6,7.

Fig. 2: Precambrian LOD constraints and trends.
figure 2

The LOD compilation is detailed in the Methods and provided in Extended Data Table 1. Following ref. 21, LOD error bars are defined by the difference between the average of all precession/obliquity-based LOD estimates for a rock unit and the estimate with the largest difference from the mean. The red linear regressions were automatically assigned by the change-point algorithm without any a priori data binning (Methods). The grey band encapsulates numerous Phanerozoic LOD constraints9, with two recent cyclostratigraphic constraints from the Palaeozoic19 and the present day length shown. Non-cyclostratigraphic constraints from tidal rhythmites and stromatolites (triangles) with fine-scale layering possibly containing cryptic discontinuities may represent maximum LOD estimates18,22. Some data lack uncertainties because insufficient information was provided in the original studies (a sensitivity test is provided in Extended Data Fig. 1). Arch, Archaean; δ13C, carbon isotope.

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Naturally, when data and theory are compared, even when there is general agreement, there is also something that can be learned from the slight mismatch. Given their vertical wavelength, atmospheric tides (also known as Lamb waves) are measurable phenomena in Earth’s atmosphere, with a 10.4-h period in the present atmosphere13. Hypotheses for the potential for a Precambrian resonance-stabilized day length6,7 have previously assumed that atmospheric tides have always had a constant 21-h period (~10.5-h resonance). However, the Precambrian LOD data indicate a shorter ~19-h period (~9.5-h resonance). There is uncertainty in the ~19-h estimate from several sources, including uncertainties in each of the LOD estimates, differences in the methods used to estimate LOD and still a limited number of Precambrian LOD data, particularly spanning the critical Proterozoic interval. A mid-Proterozoic flatlining is well supported by five modern cyclostratigraphic constraints with relatively small uncertainties, but more such studies have to be conducted to further test and more accurately determine the resonance period. As the LOD data indicate that the putative resonance would have been achieved sometime between 2.46 and 2.00 Ga, the question then becomes whether the atmospheric conditions during this time were amenable to Lamb waves being ~10% faster than they are today.

The stabilizing of day length appears to have occurred shortly after two of the most notable fluctuations in atmospheric conditions that may have caused a fundamental change in the atmospheric tide, thus inducing resonance with the oceanic tide. These two intervals—the Great Oxidation Event14 (GOE) and the exit from oxygen15 (Oxit)—stand in stark contrast, with the GOE representing the addition of oxygen and Oxit its subtraction. Only low oxygen levels, plus hotter temperatures, relative to today may make Lamb waves the requisite 10% faster than they are today, approaching the ~19-h mid-Proterozoic day length observed (Fig. 3 of ref. 6). Furthermore, as the absorption of sunlight by ozone occurs at both great depths and higher altitudes than that of water vapour, ozone is more important in the excitation of atmospheric tides16; thus, its accumulation following the GOE (by O2 molecules being photolysed by ultraviolet radiation and recombining with O atoms to create O3) could have sufficiently increased the magnitude of the opposing solar tide to rival that of the smaller Precambrian lunar tide, thereby stabilizing LOD. Therefore, the succession of the GOE creating an ozone layer and the Oxit then decreasing oxygen levels may have played iterative roles in Earth entering resonance, with the most plausible surface temperatures at a resonance of ~19 h of day length being achieved with reduced O2 and modern O3 (ref. 6). The warm surface temperatures implied by such a model may have been achieved by the reduced O2 following Oxit having caused surface warming17. There are multiple possible explanations for how escape from the resonance might have occurred6,7. Based on timing, this was probably somehow related to changes in atmospheric temperature or composition during the onset of severe climatic variability in the Neoproterozoic (Fig. 2).

Day length remaining constant at ~19 h for 1 billion years carries implications for the Earth–Moon system, as well as for Earth history. One consequence would be an increase in the angular momentum of the overall Earth–Moon system due to the extraction of additional angular momentum from Earth’s orbit during the resonance6. One current limitation is that cyclostratigraphic studies assume conservation of angular momentum when converting from k to LOD18,19; therefore, proposals should be considered for how to estimate LOD without such an assumption. On Earth, the prolonged interval of stalled day length corresponds closely with the interval of similarly stalled biologic evolution, and geochemical and tectonic quiescence known as the boring billion (Fig. 2). Tectonically, temporal changes in the kinetic energy of rotation due to LOD can be mechanistically considered a contributing force to energies required to power plate tectonics20. Biologically, our work is consistent with the idea8 that both the delayed rise of photosynthetically produced atmospheric oxygen and the rise of complex life, both which occurred only late in the Neoproterozoic, were delayed until the resonance was broken and longer days could provide photosynthetic bacteria with sufficient sunlight to raise oxygen levels high enough to support large Metazoan life.

Methods

Precambrian LOD constraints

LOD constraints from Precambrian time are compiled in Extended Data Table 1 Refs. 30,31,32,33,34,35,36,37,38,39,40,41,42,43. A recent compilation of Precambrian LOD constraints21 provided a basis for our compilation. Additions to and exclusions from the LOD compilation of ref. 21 are discussed.

The ~2,450 Ma Weeli Wolli Formation of the Pilbara craton of Western Australia has yielded two tidalite-based LOD estimates (17.1 ± 1.1 and 18.8 ± 0.6 h) based on different tidal interpretations of the same layering9,23, one of which must be incorrect, and no independent constraint on sedimentation is available. As we were unable to pick between them, and since a reliable cyclostratigraphic study24 of the underlying and only slightly older ~2,460 Ma Joffre Member of the Brockman Iron Formation recently became available to use instead for this age, we excluded the ambiguous estimates from the Weeli Wolli Formation. Also in Pilbara, the ~2,465 Ma Dales Gorge Member of the Brockman Iron Formation directly underlying the Joffre Member was studied by ref. 25, and its LOD estimate, including uncertainty, has been updated by ref. 19. Ref. 19 also provides an additional LOD estimate for the ~655 Ma Datangpo Formation of South China. An oldest LOD estimate at ~3.2 Ga comes from the Moodies Group tidal rhythmites of the Barberton Greenstone Belt of Kaapvaal craton in South Africa, but is typically excluded from most compilations18,19,21 due to the ambiguities detailed in ref. 24, and is also excluded here. A LOD estimate from the ~1,450 Ma Hongshuizhuang Formation of North China26, overlooked by the compilation of ref. 21, is included here.

One additional constraint not included in previous compilation comes from the ~1,890 Ma Rocknest Formation27,28 from the Coronation Basin of Wopmay Orogen of the Slave craton. Multiple stratigraphic sections of the carbonate parasequences were measured along strike and in great detail. Two scales of cyclicity were identified: a high-frequency bandwidth with parasequence cycles of an average thickness of 10 m; and grand cycles of 75–100 m. Interpreting the former as precession and the latter as short eccentricity (consistent with independent geochronological estimates of the sediment accumulation rate) yields a bundling of ~8.75 precession cycles per short eccentricity cycle. Assuming that the eccentricity metronome is more or less stable through geologic time29 and taking the arithmetic mean of 109.4 thousand years (kyr) of the two strongest components of short eccentricity (94.9 and 123.9 kyr) thus yields a period for precession at this age of 12.5 kyr. As only one average bandwidth of precession is identified (of the five main components of the climatic precession index), we convert to LOD for both the shortest and the longest precessional components (LOD = 17.30 and 18.41 h, respectively) and take the average of this range (LOD = 17.9 h).

Using all of the identified obliquity and precession periods to calculate LODs, ref. 21 regards the mean value of the largest and smallest values as the LOD and the difference between the average and the largest value as the uncertainty. A MATLAB script modified from ref. 21 for converting measured Milankovitch periods of precession and/or obliquity to LOD is provided online. See also refs. 18,19 for further discussion on how climatic precession, which is measured by cyclostratigraphy and is derived from the fundamental frequencies associated with the orbits of the five innermost planets and the precession constant k, is used to constrain LOD. In light of our evidence for LOD stabilization, however, the conventional approach may be insufficient. Conversion of cyclostratigraphy-based k to LOD assumes conservation of angular momentum in the Earth–Moon system. However, constant day length invalidates this assumption, where there would be an increase in the angular momentum of the Earth–Moon system during the period of resonance. Zahnle and Walker6 estimated that this increase in angular momentum being extracted from Earth’s orbit would be 10–20% over the lifetime of the solar system. Therefore, over the ~1 billion year-long resonance interval, the increase was probably «10%. Nonetheless, whether this increase is still potentially large enough not to be negligible should be considered in future work.

Change-point analysis

The MATLAB algorithm findchangepts was used to conduct a change-point analysis30,31 and detect any abrupt changes in LOD signal. Lines of free slopes and ages of change points were automatically assigned by the algorithm without any user input. The number of change points and their associated slopes are defined such that the sum of the residual error is minimized.

The statistical resampling approach shown in Extended Data Fig. 1 was next used to test the robustness of our results. For LOD estimates without associated estimates of uncertainty, an error of 1 h was assigned related to the largest observed uncertainties. Then, each error was used as standard deviation in a normally distributed random value that was added (subtracted) to each respective LOD value. This resampling was done 1,000 times. For each of the 1,000 repetitions, the findchangepts analysis was carried out. To investigate the stability of the middle interval of the LOD evolution, all linear regression lines of the middle interval of the 1,000 analyses were plotted and a mean slope of all regression lines was calculated. We also plot a histogram of the ages of the resulting change points, where for analyses with two change points blue marks the older and green the younger of the two change points, with red representing analyses for which only one change point resulted. The conservative scenario of 1-h errors assigned to LOD values without known uncertainties yields a shallow mean slope indicating a resonance LOD value of between 18 and 20 h (Extended Data Fig. 1). The change points at ~2 and 1.2 Ga in our analysis (Fig. 2) therefore seem very robust, even when such attentional sources of potential uncertainty are considered. Overall, our statistical analysis reveals that two change points seem evident using the current dataset, but our results still need to be treated with caution and more data are urgently needed to confirm our conclusions.