Proterozoic Milankovitch Cycles and the History of the Solar System

Proterozoic Milankovitch cycles and the history of the solar system

Stephen R. Meyersa,1 and Alberto Malinvernob

aDepartment of Geoscience, University of Wisconsin–Madison, Madison, WI 53706; and bLamont-Doherty Earth Observatory, Columbia University, Palisades, NY 10964-1000

Edited by Paul E. Olsen, Columbia University, Palisades, NY, and approved March 30, 2018 (received for review October 9, 2017) The geologic record of Milankovitch climate cycles provides a rich intervals, and the 405-ky-long orbital eccentricity cycle is conceptual and temporal framework for evaluating Earth system expected to be relatively stable with an uncertainty of 0.2% by evolution, bestowing a sharp lens through which to view our 250 Ma (2). planet’s history. However, the utility of these cycles for constrain- Recent advances in astrochronologic assessment yield a partial ing the early Earth system is hindered by seemingly insurmount- solution to the challenges noted above (6–8), in providing sta- able uncertainties in our knowledge of solar system behavior tistical approaches that explicitly consider and evaluate timescale (including Earth–Moon history), and poor temporal control for val- uncertainty in terms of the accumulation rate of a given sedi- idation of cycle periods (e.g., from radioisotopic dates). Here we mentary record. However, these methods require assumptions address these problems using a Bayesian inversion approach to about the astronomical frequencies associated with the Earth’s quantitatively link astronomical theory with geologic observation, orbital eccentricity, axial tilt, and climatic precession (the allowing a reconstruction of Proterozoic astronomical cycles, fun- Milankovitch cycles). In the present study, we build upon prior damental frequencies of the solar system, the precession constant, work to formulate a Bayesian inversion approach that quanti- and the underlying geologic timescale, directly from stratigraphic tatively links astronomical theory with geologic observation, thus data. Application of the approach to 1.4-billion-year-old rhythmi- overcoming limitations associated with each. At the core of this tes indicates a precession constant of 85.79 ± 2.72 arcsec/year (2σ), approach are three components: (i) the TimeOpt method (8), an Earth–Moon distance of 340,900 ± 2,600 km (2σ), and length of which explicitly considers timescale uncertainty, and utilizes day of 18.68 ± 0.25 hours (2σ), with dominant climatic precession multiple attributes of the astronomical signal to increase statis- EARTH, ATMOSPHERIC, cycles of ∼14 ky and eccentricity cycles of ∼131 ky. The results AND PLANETARY SCIENCES ii confirm reduced tidal dissipation in the Proterozoic. A complemen- tical reliability; ( ) the underlying astronomical theory, which tary analysis of Eocene rhythmites (∼55 Ma) illustrates how the links observed climatic precession and orbital eccentricity rhythms to fundamental frequencies of the solar system and approach offers a means to map out ancient solar system behavior – and Earth–Moon history using the geologic archive. The method Earth Moon evolution (2, 4) (Table 1); and (iii) a Bayesian also provides robust quantitative uncertainties on the eccentricity Markov Chain Monte Carlo approach that allows explicit ex- and climatic precession periods, and derived astronomical time- ploration of the data and model space and uncertainties. The scales. As a consequence, the temporal resolution of ancient Earth result is a robust methodology for astrochronology that is suit- system processes is enhanced, and our knowledge of early solar able for the Proterozoic, and greatly enhances the astronomical system dynamics is greatly improved. knowledge that we can obtain from younger strata (e.g., the early

Milankovitch cycles | astrochronology | Bayesian inversion | Earth–Moon Significance history | fundamental frequencies Periodic variations in Earth’s orbit and rotation axis occur over uasiperiodic variations in insolation, known as Milankovitch tens of thousands of years, producing rhythmic climate Qcycles, serve as a primary control on climate change over changes known as Milankovitch cycles. The geologic record of timescales of 104–106 y (1). Their expression in the stratigraphic these climate cycles is a powerful tool for reconstructing geo- record provides a powerful tool for reconstructing geologic logic time, for understanding ancient climate change, and for timescales, or astrochronologies, and evaluating Earth history. evaluating the history of our solar system, but their reliability Extending this astronomical metronome into the Precambrian, dramatically decreases beyond 50 Ma. Here, we extend the however, has proven challenging due to shortcomings in both analysis of Milankovitch cycles into the deepest stretches of Earth history, billions of years ago, while simultaneously theory and geologic data. From the perspective of the geologic reconstructing the history of solar system characteristics, in- archive, a major limitation is the lack of sufficient independent cluding the distance between the Earth and Moon. Our results time control (e.g., radioisotopic dates) to unambiguously cali- improve the temporal resolution of ancient Earth processes brate the observed spatial rhythms to astronomical (temporal) and enhance our knowledge of the solar system in deep time. periods. In terms of theory, the periods of Earth’s astronomical cycles also become more poorly constrained during the Pre- Author contributions: S.R.M. initiated the project; S.R.M. and A.M. designed research; cambrian due to uncertainties in the evolution of the solar sys- S.R.M. and A.M. performed research; S.R.M. and A.M. contributed new analytic tools; tem (2). Although it is established that the dominant eccentricity S.R.M. and A.M. analyzed data; and S.R.M. and A.M. wrote the paper. and climatic precession cycles derive from fundamental fre- The authors declare no conflict of interest. quencies associated with the orbits of the five innermost planets This article is a PNAS Direct Submission. Published under the PNAS license. (g1 to g5; ref. 2) and the precession constant k, these values are not precisely determined because of the chaotic nature of the Data deposition: The function “timeOptMCMC” has been deposited in the Comprehen- sive R Archive Network (CRAN) repository (https://cran.r-project.org), as a component of solar system (2, 3) and because the history of tidal dissipation of the package “astrochron.” – the Earth Moon system is not well known (2, 4). In fact, the 1To whom correspondence should be addressed. Email: smeyers@geology.wisc.edu. validity of theoretical astronomical solutions that underpin This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. astrochronology are limited to the past 50 My (2, 5), although 1073/pnas.1717689115/-/DCSupplemental. “floating” astrochronologies have been proposed for older

www.pnas.org/cgi/doi/10.1073/pnas.1717689115 PNAS Latest Articles | 1of6 Downloaded by guest on September 25, 2021 Table 1. Source of the climatic precession and eccentricity Bayesian inversion of the Xiamaling Cu/Al record is con- periods, as derived from the fundamental frequencies and strained by prior distributions for the fundamental frequencies g1 precession constant to g5, the precession constant k, and sedimentation rate (SI Parameter Source Period (ky)—Today* Appendix, Tables S3 and S4). Prior distributions for the fundamental frequencies g1 to g5 are based on the full range of vari- + p1 k g5 23.678 ability in the model simulations of Laskar et al. (2) computed + p2 k g2 22.371 over 500 My. The prior distribution for the precession constant is + p3 k g4 18.951 derived from the recent study by Waltham (ref. 4; 78 ± 28 arcsec/y, + p4 k g3 19.103 2σ), and sedimentation rate is permitted to vary across all + p5 k g1 23.120 plausible values for which it is possible to robustly identify a full − e1 g2 g5 405.091 astronomical signal, given the available data resolution. The − e2 g4 g5 94.932 posterior distribution from the TimeOptMCMC analysis indi- − e3 g4 g2 123.945 cates a precession constant of 85.79 ± 2.72 arcsec/y (2σ; Fig. 2B), − e4 g3 g5 98.857 consistent with an Earth–Moon distance of 340,900 ± 2,600 km − e5 g3 g2 130.781 (2σ) and length of day of 18.68 ± 0.25 h (2σ; Fig. 2C and Table 2). Climatic precession periods range between 12.5 and 14.4 ky *Precession and eccentricity estimates from ref. 2. (Fig. 2F and Table 2), with a dominant cycle of ∼14 ky in the study interval (Fig. 1D). The Proterozoic analog of the long ec- Cenozoic). We refer to this approach as TimeOptMCMC. centricity cycle, which has a duration of 405 ky in theoretical We emphasize that although TimeOptMCMC provides a models for the Cenozoic (2), and is expected to be the most rigorous quantification of the uncertainties in astrochronologic regular of the eccentricity cycles because it involves interaction results, the method does not by itself reduce these uncer- between the very stable Jupiter and relatively stable Venus, has a duration of 405.1 ky (401.3–408.9 ky, 2σ; Fig. 2D). Finally, the tainties. Ultimately, uncertainties in astrochronology can only be reconstructed Proterozoic short eccentricity periods (Fig. 2E) are decreased by additional information provided by measured consistent with those observed in the theoretical models for the data. Cenozoic (2) (95–131 ky), with a dominant period of ∼131.4 ky We apply the TimeOptMCMC method to evaluate two in the study interval (Fig. 1D). It is notable that the posterior cyclostratigraphic records that are of special importance. The distributions for sedimentation rate (Fig. 2A) and the precession first is the 1.4-billion-year-old Xiamaling Formation from the constant (Fig. 2B) are much narrower than their prior distribu- North China Craton (9), one of the oldest proposed records of tions, and the prior and posterior distributions of the funda- astronomical forcing (Fig. 1A). The second is the well-studied mental frequencies g to g are nearly identical (SI Appendix, Fig. ∼ 1 5 55-million-year-old record from Walvis Ridge (ref. 10 and Fig. S7). Notwithstanding little improvement in the posterior uncer- 1E), which is notable because it includes the Paleocene–Eocene tainties of the fundamental frequencies, the coupled nature of Thermal Maximum, and it just exceeds the temporal limits of the the eccentricity and climatic precession cycles, which share available theoretical astronomical solutions [<50 Ma (2, 5)]. The common g terms (the climatic precession terms also share a methodology allows us to address two primary research objec- common k term) allows resolution of the Proterozoic Milanko- tives: (i) to provide well-constrained geologic estimates of the vitch periods with low uncertainty (Fig. 2 D–F and Table 2). climatic precession and eccentricity periods for both the early To provide a baseline assessment from the early Cenozoic, we Eocene and Proterozoic (including uncertainties); and (ii)to investigate Eocene reflectivity data (a*, red/green) from the quantify length of day and Earth–Moon distance during the Walvis Ridge (10) (Ocean Drilling Program Site 1262; Fig. Proterozoic (via the precession constant k), at a time when ex- 1E). This dataset has been previously evaluated with the trapolation of the present-day rate of tidal dissipation would TimeOpt approach (8), and a statistically significant astro- imply a condition near Earth–Moon collision (11). nomical signal (P < 0.005) is identified at a sedimentation rate of 1.33 cm/ky (SI Appendix,Fig.S9). Application of the Results TimeOptMCMC algorithm allows a rigorous assessment of the We focus our analysis of the Xiamaling Formation on a 2-m- uncertainty in the Eocene Milankovitch periods, in the un- thick section of rhythmically bedded black shale and chert (“unit derlying g and k terms, and in the sedimentation rate (Fig. 2, 3” of ref. 9) that has been interpreted to reflect changes in up- Table 2, and SI Appendix,Figs.S10and S11). In this case, the welling and biological productivity. Paleogeographic recon- posterior distributions for sedimentation rate (Fig. 2G)and structions place this site in a subtropical/tropical marine for the fundamental frequencies g3 (Earth) and g4 (Mars) show environment that was under Hadley cell influence, suggesting an the greatest change relative to their prior distributions (SI astronomical forcing scenario involving migration of the in- Appendix,Fig.S10E and G). Most notably, the posterior mean g tertropical convergence zone (9). We investigate the published value for 4 (Mars) is greater than the maximum value observed in the modeling study of Laskar et al. (2). This dis- Cu/Al record (9), a proxy for productivity/redox state (12), which crepancy in g is also expressed in the e (91.98 ky) and e demonstrates high fidelity (SI Appendix, Fig. S6 and SI Appen- 4 2 3 (118.95 ky) eccentricity terms at Walvis Ridge (Fig. 2K and SI dix). Initial screening of the high-resolution dataset using the Appendix,Fig.S11C and E), both of which share the g4 term TimeOpt method with tentative (nominal) Proterozoic values for and are notably shorter than those observed in the astro- the climatic precession and eccentricity periods (2, 4) (SI Ap- nomical model simulations of Laskar et al. (2) (Table 2). A pendix, Table S2) reveals a highly significant astronomical signal possible explanation for these differences in g and g is that 2 = < 3 4 (r opt 0.300; P 0.005, 2,000 simulations; SI Appendix, Fig. S6) the frequencies reported in figure 9 of Laskar et al. (2) are at a sedimentation rate of 0.33 cm/ky. This sedimentation rate is averaged over 20-My intervals, whereas the Walvis Ridge re- consistent with radioisotopic data in an overlying 52-m-thick cord spans a shorter interval of about 1.7 My. interval (SI Appendix). The statistically significant TimeOpt result is an important finding, as it overcomes the problem of false Discussion signal detection that complicates spectrum evaluation (13, 14) Comparison of the Proterozoic and Eocene results highlights and provides an independent confirmation of the astronomical how the TimeOptMCMC approach combines cyclostratigraphic interpretation of Zhang et al. (9). data and astronomical theory to improve model parameters. In

2of6 | www.pnas.org/cgi/doi/10.1073/pnas.1717689115 Meyers and Malinverno Downloaded by guest on September 25, 2021 Proterozoic Xiamaling Formation Eocene Walvis Ridge A E 7 56 Cu/Al 10 20 30 a* (red/green) 34

263.5 264.0 264.5 265.0 265.5 120 125 130 135 140 B Height (m) F Depth (mcd) 4 3 012 -1 -2 a* (standardized) -1 0 1 2 3 4 Cu/Al (standardized) Cu/Al

0 100 200 300 400 500 600 0 500 1000 1500 Elapsed Time (ky) Elapsed Time (ky) C G 2 1 -1.0 0.0 1.0 a* (standardized) Cu/Al (standardized) Cu/Al -2 -1 0 EARTH, ATMOSPHERIC,

0 100 200 300 400 500 600 0 500 1000 1500 AND PLANETARY SCIENCES Elapsed Time (ky) Elapsed Time (ky) D H 0.08 0.04 Power Power 0.00 0.00 0.04 0.08 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 Frequency (cycles/ky) Frequency (cycles/ky)

Fig. 1. TimeOptMCMC results for the ∼1.4 Ga Proterozoic Xiamaling Formation Cu/Al data (9) and the ∼55 Ma Eocene Walvis Ridge a* (red/green) data (10). (A) Xiamaling Cu/Al data versus stratigraphic height. (B) Astronomically tuned Xiamaling Cu/Al data, using the TimeOptMCMC derived posterior mean sedimentation rate (Table 2). The data series has been standardized to unit variance, and a linear trend has been removed. (C) Xiamaling Cu/Al data precession envelope (red line) for the posterior mean sedimentation rate, and precession filter output (blue line). The black line illustrates the TimeOpt reconstructed eccentricity model. (D) Xiamaling Cu/Al data power spectrum (squared Fourier transform) using the posterior mean sedimentation rate. The vertical dashed red lines indicate the reconstructed target periods (mean posterior values in Table 2) for climatic precession and eccentricity, andthe blue line illustrates the frequency response of the bandpass filter for precession modulation evaluation. (E) Eocene Walvis Ridge a* data versus meters composite depth (mcd). (F) Astronomically tuned Walvis Ridge a* data, using the TimeOptMCMC derived posterior mean sedimentation rate (Table 2). The data series has been standardized to unit variance, and a linear trend has been removed. (G) Walvis Ridge a* data precession envelope (red line) for the posterior mean sedimentation rate, and precession filter output (blue line). The black line illustrates the TimeOpt reconstructed eccentricity model. (H) Walvis Ridge a* data power spectrum (squared Fourier Transform) using the posterior mean sedimentation rate. The vertical dashed red lines indicate the reconstructed target periods for climatic precession and eccentricity (Table 2), and the blue line illustrates the frequency response of the bandpass filter for precession modulation evaluation.

the case of the Proterozoic, where the deviation of k from its reconstruct the precession constant and/or the Earth–Moon present value is expected to be substantial due to the large un- distance and length of day using geologic data. These approaches certainties in the Earth–Moon history, the Xiamaling Cu/Al data include inferences from the evaluation of tidal deposits and of provides strong constraints to improve our knowledge of the growth patterns in marine invertebrate fossils and stromatolites precession constant. In the case of the Eocene, the expected throughout the past 2.5 billion years (16), and for the late changes in k are much smaller, and the cyclostratigraphic data Cenozoic (<25 Ma), the application of astronomical-based more strongly improve our knowledge of the fundamental fre- methods (1, 17, 18). In addition, a number of theoretical mod- – quencies g3 and g4. For both the Proterozoic and Eocene ex- eling exercises have been conducted to constrain the Earth amples, the sedimentation rate (and hence the duration of the Moon history (2, 4, 16, 19, 20). In Fig. 3, we compare our stratigraphic interval) is highly constrained by the Bayesian astronomical-based results for the Proterozoic and Eocene to a inversion. number of Earth–Moon separation models, and also to two tida- Although stratigraphic-based estimates of the fundamental lite datasets that are considered to be of high quality (16): frequencies of the solar system are rare [g1 to g5; however, see rhythmites from the Big Cottonwood Formation (∼900 Ma; refs. Olsen and Kent (15)], numerous studies have attempted to 21 and 22), and the Elatina Formation and Reynell Siltstone

Meyers and Malinverno PNAS Latest Articles | 3of6 Downloaded by guest on September 25, 2021 Proterozoic Xiamaling Formation AB C 0 20406080 0.0 1.0 2.0 3.0 0.00 0.10 0.20 0.30 0.25 0.30 0.35 0.40 0.45 0.50 20 40 60 80 100 120 17 18 19 20 21 22 23 24 Sedimentation Rate (cm/ky) Precession Constant (arcsec/y) Length of Day (hours)

g4-g5 k+g4 k+g3 DEg3-g5 F k+g2 k+g1 k+g5 g2-g5 g4-g2 g3-g2 0.0 0.5 1.0 1.5 2.0 0.00 0.10 0.20 0.30 0.00 0.10 0.20 0.30 395 400 405 410 415 90 100 110 120 130 140 11 12 13 14 15 16 Long Eccentricity Period (ky) Short Eccentricity Period (ky) Precession Period (ky) Eocene Walvis Ridge GH I 01234 0 10203040 0.0 0.4 0.8

1.0 1.2 1.4 1.6 1.8 2.0 50 51 52 53 23.4 23.6 23.8 24.0 24.2 Sedimentation Rate (cm/ky) Precession Constant (arcsec/y) Length of Day (hours)

JKg4-g5 g3-g5 L k+g4 k+g3 k+g2 k+g1 k+g5 g4-g2 g3-g2 g2-g5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

395 400 405 410 415 90 100 110 120 130 140 18 19 20 21 22 23 24 Long Eccentricity Period (ky) Short Eccentricity Period (ky) Precession Period (ky)

Fig. 2. Summary of TimeOptMCMC prior and posterior distributions for the ∼1.4 Ga Proterozoic Xiamaling Formation Cu/Al data (9), and the ∼55 Ma Eocene Walvis Ridge Site 1262 a* (red/green) data (10). (A–F) Prior (red line) and posterior (histogram) probability distributions for Xiamaling Formation sedimentation rate, precession constant, length of day, long eccentricity period, short eccentricity periods, and climatic precession periods (C). (G–L) Prior (red line) and posterior (histogram) probability distributions for Eocene Walvis Ridge Site 1262 sedimentation rate, precession constant, length of day, long eccentricity period, short eccentricity periods, and climatic precession periods (I). See Table 1 for the relationship between g, k, and observed astronomical periods. See Table 2 for the mean posterior values associated with each distribution shown in this figure.

(∼620 Ma; refs. 16 and 23). It should be noted that the in- tidalite data are taken at face value, either of these Earth– terpretation of tidalite (and also bioarchive) datasets in terms of Moon separation models are possible, depending on the Earth–Moon history remains a contentious issue due to problems tidalite record considered. However, the small uncertainty of with cycle recognition and the potential for missing laminations (4, the Xiamaling estimate excludes a range of other potential 16), thus the TimeOptMCMC approach provides an independent models that are permitted by the tidalite data, such as one that means for their validation. To supplement our comparison, SI employs a tidal dissipation rate that is 40% of the present Appendix,Fig.S12includes some additional more contro- value, and furthermore, the 60% model is inconsistent with versial estimates from Phanerozoic bioarchives, and a datum estimates from the Weeli Wooli Formation (SI Appendix,Fig. from the Weeli Wooli Formation rhythmite (∼2,450 Ma; refs. 11, S12). It should also be noted that the 40% and 60% rate 16, and 24). models violate the constraint provided by modern observed The TimeOptMCMC reconstructed Earth–Moon distance Moon retreat rate, in contrast to the ocean model (20) and from the Xiamaling Formation (340,900 ± 2,600 km, 2σ;Table present rate model. Finally, the astronomical-based Bayesian 2) is consistent with that derived from ocean models (20) that reconstruction is consistent with a length of day (18.68 ± imply smaller torques, reduced tidal dissipation, and slower 0.25 h, 2σ), which is shorter than that of published Proterozoic lunar retreat rates in the distant past, ultimately related to a estimates from geologic data (16), and is at the low end of less efficient excitation of the ocean’s normal modes by tidal existing model length of day estimates (4), but has a greatly forcing on an Earth with a faster rotation rate (Fig. 3). The reduced uncertainty (Fig. 2C). Xiamaling result is also compatible with a model that employs The methodology presented here is not affected by problems an average tidal dissipation rate that is 60% of the present inherent in previous estimates of the precession constant, asso- value (SI Appendix,Fig.S12). If the Elatina and Cottonwood ciated with ambiguity in the interpretation of bioarchives and

4of6 | www.pnas.org/cgi/doi/10.1073/pnas.1717689115 Meyers and Malinverno Downloaded by guest on September 25, 2021 Table 2. TimeOptMCMC reconstructed sedimentation rate, precession and eccentricity frequencies, fundamental frequencies (g terms), precession constant (k), and Earth–Moon distance, for the Proterozoic Xiamaling Formation and the Eocene Walvis Ridge Parameter Today* Xiamaling Formation† ±σ Walvis Ridge‡ ±σ

Sedimentation rate (cm/ky) — 0.357 0.005 1.316 0.011

p1, ky 23.678 14.392 14.178–14.613 23.335 23.121–23.554

p2, ky 22.371 13.899 14.407–13.983 22.066 21.874–22.261

p3, ky 18.951 12.497 12.673–12.327 18.613 18.448–18.781 p4, ky 19.103 12.569 12.744–12.398 18.848 18.685–19.013

p5, ky 23.120 14.192 14.407–13.983 22.827 22.608–23.050

e1, ky 405.091 405.077 406.971–403.201 405.613 403.969–407.270

e2, ky 94.932 94.912 96.028–93.821 91.975 91.099–92.867 e3, ky 123.945 123.955 125.868–122.100 118.946 117.483–120.447

e4, ky 98.857 99.211 100.386–98.064 98.009 97.161–98.871

e5, ky 130.781 131.392 133.467–129.380 129.236 127.752–130.755 k, arcsec/y 50.475838 85.790450 1.362320 51.280910 0.515371 Earth–Moon distance, 384.4§ 340.855800 1.293260 383.110800 0.822051 km × 103 { h/d 23.93447 18.684750 0.126679 23.804850 0.123398

g1, arcsec/y 5.579378 5.531285 0.129246 5.494302 0.128076 g2, arcsec/y 7.456665 7.456848 0.014886 7.452619 0.013001

g3, arcsec/y 17.366595 17.320480 0.152805 17.480760 0.115325

g4, arcsec/y 17.910194 17.912240 0.158683 18.348310 0.135346 g5, arcsec/y 4.257564 4.257456 0.000020 4.257451 0.000020

*Precession, eccentricity, g, and k estimates from ref. 2. † Results from the Xiamaling Formation are based on 50 MCMC simulation chains of length 1 × 106 each. ‡Results from the Walvis Ridge are based on 150 MCMC simulation chains of length 2 × 105 each. EARTH, ATMOSPHERIC, § Semimajor axis. AND PLANETARY SCIENCES {Sidereal day.

tidal deposits (4, 16). Furthermore, the technique should be widely applicable, given the abundance of relatively continuous Today records of astronomically forced sedimentation. An important feature of this quantitative approach is a comprehensive treat- Cottonwood Elatina Walvis ment of uncertainties, facilitated by the explicit coupling of as- Ridge tronomical theory with geologic observation. The quantification Xiamaling of prior and posterior distributions allows for a rigorous treat- Formation ment of astrochronologic uncertainties, addressing a major weakness in prior work and providing an objective way to inte- Ocean Model Present Rate of Dissipation grate astrochronologies with radioisotopic data, from the Pro- terozoic to the Cenozoic. Application of this methodology to 320 340 360 380 sedimentary records that span Earth history will facilitate improvement in the calibration of the geologic time scale, will constrain the history of the Earth–Moon system in deep time,

Earth-Moon Distance (x1000 km) and shows promise of reconstructing the evolution of the fundamental orbital frequencies of the solar system over billions 280 300 of years.

-1500 -1000 -500 0 Materials and Methods Millions of Years Before Present We utilize the recently developed TimeOpt regression framework (8) to evaluate two features that are diagnostic of an astronomical fingerprint in Fig. 3. TimeOptMCMC reconstructed Earth–Moon distance, compared with strata: the concentration of spectral power at the proposed astronomical two tidalite-based estimates and two models. Uncertainties in Earth–Moon frequencies (25) (e.g., climatic precession and eccentricity), and the am- distance are ±2σ and age uncertainties span minimum and maximum values. plitude modulation of climatic precession (26), which is caused by varia- The Bayesian posterior TimeOptMCMC estimates for the Proterozoic tions in eccentricity. For the Bayesian inversion, TimeOpt is reformulated Xiamaling Formation and Eocene Walvis Ridge are indicated with blue symbols. Note the significant improvement in precision between posterior in terms of likelihood functions (27) (SI Appendix), and Markov Chain (blue) and prior estimates for the Xiamaling data (prior = 319,743 to Monte Carlo (MCMC) simulation is utilized to sample values of the solar 380,309 km; 2σ). Age uncertainties for the Xiamaling and Walvis Ridge re- system secular frequencies g1 to g5, precession constant k,andsedimen- sults fall within the size of the blue symbols. Tidalite estimates from the Big tation rate that are physically plausible and agree with the stratigraphic Cottonwood Formation (∼900 Ma; ref. 22) and Elatina Formation and Rey- data. This allows evaluation of the five dominant eccentricity and five nell Siltstone (∼620 Ma; ref. 23) are shown with dark green symbols (based dominant climatic precession cycles that are observable in sedimentary on the updated analyses of ref. 16). The light green Big Cottonwood For- strata, which depend on sums or differences of the g terms and/or k (Table mation estimate (364,192 km) is an alternative value reported by ref. 21 with 1). For example, the 405-ky-long eccentricity term of the Cenozoic origi- uncertainties from the 348,884-km estimate (dark green symbol). The ocean nates from the difference between g2 (Venus) and g5 (Jupiter), and one of model (red line), and a model using the present rate of tidal dissipation the strongest climatic precession cycles (23.7 ky in the Cenozoic) corre-

(black line), derive from ref. 20. sponds to g5 + k.

Meyers and Malinverno PNAS Latest Articles | 5of6 Downloaded by guest on September 25, 2021 A complete description of the approach, including evaluation and calibration ACKNOWLEDGMENTS. We thank the reviewers and the editor for their with a synthetic astronomical test series, is in the SI Appendix. All analyses constructive remarks. This study was supported by NSF Grant EAR-1151438 were conducted using the free software R (28), and an implementation of (to S.R.M.), and by a sabbatical leave from the University of Wisconsin— the TimeOptMCMC algorithm is available in the Astrochron package (29). Madison (S.R.M.) to conduct research at Lamont-Doherty Earth Observatory.

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