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Proterozoic Milankovitch cycles and the history of the solar system

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The geologic record of Milankovitch climate cycles provides a rich
conceptual and temporal framework for evaluating Earth system
evolution, bestowing a sharp lens through which to view our
planet’s history. However, the utility of these cycles for constraining
the early Earth system is hindered by seemingly insurmountable
uncertainties in our knowledge of solar system behavior
(including Earth–Moon history), and poor temporal control for validation
of cycle periods (e.g., from radioisotopic dates). Here we
address these problems using a Bayesian inversion approach to
quantitatively link astronomical theory with geologic observation,
allowing a reconstruction of Proterozoic astronomical cycles, fundamental
frequencies of the solar system, the precession constant,
and the underlying geologic timescale, directly from stratigraphic
data. Application of the approach to 1.4-billion-year-old rhythmites
indicates a precession constant of 85.79 ± 2.72 arcsec/year (2σ),
an Earth–Moon distance of 340,900 ± 2,600 km (2σ), and length of
day of 18.68 ± 0.25 hours (2σ), with dominant climatic precession
cycles of ∼14 ky and eccentricity cycles of ∼131 ky. The results
confirm reduced tidal dissipation in the Proterozoic. A complementary
analysis of Eocene rhythmites (∼55 Ma) illustrates how the
approach offers a means to map out ancient solar system behavior
and Earth–Moon history using the geologic archive. The method
also provides robust quantitative uncertainties on the eccentricity
and climatic precession periods, and derived astronomical timescales.
As a consequence, the temporal resolution of ancient Earth
system processes is enhanced, and our knowledge of early solar
system dynamics is greatly improved.

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Proterozoic Milankovitch cycles and the history of the solar system

  1. 1. Proterozoic Milankovitch cycles and the history of the solar system Stephen R. Meyersa,1 and Alberto Malinvernob a Department of Geoscience, University of Wisconsin–Madison, Madison, WI 53706; and b Lamont-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 conceptual and temporal framework for evaluating Earth system evolution, bestowing a sharp lens through which to view our planet’s history. However, the utility of these cycles for constrain- ing the early Earth system is hindered by seemingly insurmount- able uncertainties in our knowledge of solar system behavior (including Earth–Moon history), and poor temporal control for val- idation of cycle periods (e.g., from radioisotopic dates). Here we address these problems using a Bayesian inversion approach to quantitatively link astronomical theory with geologic observation, allowing a reconstruction of Proterozoic astronomical cycles, fun- damental frequencies of the solar system, the precession constant, and the underlying geologic timescale, directly from stratigraphic data. Application of the approach to 1.4-billion-year-old rhythmi- tes indicates a precession constant of 85.79 ± 2.72 arcsec/year (2σ), an Earth–Moon distance of 340,900 ± 2,600 km (2σ), and length of day of 18.68 ± 0.25 hours (2σ), with dominant climatic precession cycles of ∼14 ky and eccentricity cycles of ∼131 ky. The results confirm reduced tidal dissipation in the Proterozoic. A complemen- tary analysis of Eocene rhythmites (∼55 Ma) illustrates how the approach offers a means to map out ancient solar system behavior and Earth–Moon history using the geologic archive. The method also provides robust quantitative uncertainties on the eccentricity and climatic precession periods, and derived astronomical time- scales. As a consequence, the temporal resolution of ancient Earth system processes is enhanced, and our knowledge of early solar system dynamics is greatly improved. Milankovitch cycles | astrochronology | Bayesian inversion | Earth–Moon history | fundamental frequencies Quasiperiodic variations in insolation, known as Milankovitch cycles, serve as a primary control on climate change over timescales of 104 –106 y (1). Their expression in the stratigraphic record provides a powerful tool for reconstructing geologic timescales, or astrochronologies, and evaluating Earth history. Extending this astronomical metronome into the Precambrian, however, has proven challenging due to shortcomings in both theory and geologic data. From the perspective of the geologic archive, a major limitation is the lack of sufficient independent time control (e.g., radioisotopic dates) to unambiguously cali- brate the observed spatial rhythms to astronomical (temporal) periods. In terms of theory, the periods of Earth’s astronomical cycles also become more poorly constrained during the Pre- cambrian due to uncertainties in the evolution of the solar sys- tem (2). Although it is established that the dominant eccentricity and climatic precession cycles derive from fundamental fre- quencies associated with the orbits of the five innermost planets (g1 to g5; ref. 2) and the precession constant k, these values are not precisely determined because of the chaotic nature of the solar system (2, 3) and because the history of tidal dissipation of the Earth–Moon system is not well known (2, 4). In fact, the validity of theoretical astronomical solutions that underpin astrochronology are limited to the past 50 My (2, 5), although “floating” astrochronologies have been proposed for older intervals, and the 405-ky-long orbital eccentricity cycle is expected to be relatively stable with an uncertainty of 0.2% by 250 Ma (2). Recent advances in astrochronologic assessment yield a partial solution to the challenges noted above (6–8), in providing sta- tistical approaches that explicitly consider and evaluate timescale uncertainty in terms of the accumulation rate of a given sedi- mentary record. However, these methods require assumptions about the astronomical frequencies associated with the Earth’s orbital eccentricity, axial tilt, and climatic precession (the Milankovitch cycles). In the present study, we build upon prior work to formulate a Bayesian inversion approach that quanti- tatively links astronomical theory with geologic observation, thus overcoming limitations associated with each. At the core of this approach are three components: (i) the TimeOpt method (8), which explicitly considers timescale uncertainty, and utilizes multiple attributes of the astronomical signal to increase statis- tical reliability; (ii) the underlying astronomical theory, which links observed climatic precession and orbital eccentricity rhythms to fundamental frequencies of the solar system and Earth–Moon evolution (2, 4) (Table 1); and (iii) a Bayesian Markov Chain Monte Carlo approach that allows explicit ex- ploration of the data and model space and uncertainties. The result is a robust methodology for astrochronology that is suit- able for the Proterozoic, and greatly enhances the astronomical knowledge that we can obtain from younger strata (e.g., the early Significance Periodic variations in Earth’s orbit and rotation axis occur over tens of thousands of years, producing rhythmic climate changes known as Milankovitch cycles. The geologic record of these climate cycles is a powerful tool for reconstructing geo- logic time, for understanding ancient climate change, and for evaluating the history of our solar system, but their reliability dramatically decreases beyond 50 Ma. Here, we extend the analysis of Milankovitch cycles into the deepest stretches of Earth history, billions of years ago, while simultaneously reconstructing the history of solar system characteristics, in- cluding the distance between the Earth and Moon. Our results improve the temporal resolution of ancient Earth processes and enhance our knowledge of the solar system in deep time. Author contributions: S.R.M. initiated the project; S.R.M. and A.M. designed research; S.R.M. and A.M. performed research; S.R.M. and A.M. contributed new analytic tools; S.R.M. and A.M. analyzed data; and S.R.M. and A.M. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. Published under the PNAS license. 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 the package “astrochron.” 1 To whom correspondence should be addressed. Email: smeyers@geology.wisc.edu. This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1717689115/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1717689115 PNAS Latest Articles | 1 of 6 EARTH,ATMOSPHERIC, ANDPLANETARYSCIENCES
  2. 2. Cenozoic). We refer to this approach as TimeOptMCMC. We emphasize that although TimeOptMCMC provides a rigorous quantification of the uncertainties in astrochronologic results, the method does not by itself reduce these uncer- tainties. Ultimately, uncertainties in astrochronology can only be decreased by additional information provided by measured data. We apply the TimeOptMCMC method to evaluate two cyclostratigraphic records that are of special importance. The first is the 1.4-billion-year-old Xiamaling Formation from the North China Craton (9), one of the oldest proposed records of astronomical forcing (Fig. 1A). The second is the well-studied ∼55-million-year-old record from Walvis Ridge (ref. 10 and Fig. 1E), which is notable because it includes the Paleocene–Eocene Thermal Maximum, and it just exceeds the temporal limits of the available theoretical astronomical solutions [<50 Ma (2, 5)]. The methodology allows us to address two primary research objec- tives: (i) to provide well-constrained geologic estimates of the climatic precession and eccentricity periods for both the early Eocene and Proterozoic (including uncertainties); and (ii) to quantify length of day and Earth–Moon distance during the Proterozoic (via the precession constant k), at a time when ex- trapolation of the present-day rate of tidal dissipation would imply a condition near Earth–Moon collision (11). Results We focus our analysis of the Xiamaling Formation on a 2-m- thick section of rhythmically bedded black shale and chert (“unit 3” of ref. 9) that has been interpreted to reflect changes in up- welling and biological productivity. Paleogeographic recon- structions place this site in a subtropical/tropical marine environment that was under Hadley cell influence, suggesting an astronomical forcing scenario involving migration of the in- tertropical convergence zone (9). We investigate the published Cu/Al record (9), a proxy for productivity/redox state (12), which demonstrates high fidelity (SI Appendix, Fig. S6 and SI Appen- dix). Initial screening of the high-resolution dataset using the TimeOpt method with tentative (nominal) Proterozoic values for the climatic precession and eccentricity periods (2, 4) (SI Ap- pendix, Table S2) reveals a highly significant astronomical signal (r2 opt = 0.300; P < 0.005, 2,000 simulations; SI Appendix, Fig. S6) at a sedimentation rate of 0.33 cm/ky. This sedimentation rate is consistent with radioisotopic data in an overlying 52-m-thick interval (SI Appendix). The statistically significant TimeOpt re- sult is an important finding, as it overcomes the problem of false signal detection that complicates spectrum evaluation (13, 14) and provides an independent confirmation of the astronomical interpretation of Zhang et al. (9). Bayesian inversion of the Xiamaling Cu/Al record is con- strained by prior distributions for the fundamental frequencies g1 to g5, the precession constant k, and sedimentation rate (SI Appendix, Tables S3 and S4). Prior distributions for the funda- mental frequencies g1 to g5 are based on the full range of vari- ability in the model simulations of Laskar et al. (2) computed over 500 My. The prior distribution for the precession constant is derived from the recent study by Waltham (ref. 4; 78 ± 28 arcsec/y, 2σ), and sedimentation rate is permitted to vary across all plausible values for which it is possible to robustly identify a full astronomical signal, given the available data resolution. The posterior distribution from the TimeOptMCMC analysis indi- cates a precession constant of 85.79 ± 2.72 arcsec/y (2σ; Fig. 2B), consistent with an Earth–Moon distance of 340,900 ± 2,600 km (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 (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- centricity cycle, which has a duration of 405 ky in theoretical models for the Cenozoic (2), and is expected to be the most regular of the eccentricity cycles because it involves interaction 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 reconstructed Proterozoic short eccentricity periods (Fig. 2E) are consistent with those observed in the theoretical models for the Cenozoic (2) (95–131 ky), with a dominant period of ∼131.4 ky in the study interval (Fig. 1D). It is notable that the posterior distributions for sedimentation rate (Fig. 2A) and the precession constant (Fig. 2B) are much narrower than their prior distribu- tions, and the prior and posterior distributions of the funda- mental frequencies g1 to g5 are nearly identical (SI Appendix, Fig. S7). Notwithstanding little improvement in the posterior uncer- tainties of the fundamental frequencies, the coupled nature of the eccentricity and climatic precession cycles, which share common g terms (the climatic precession terms also share a common k term) allows resolution of the Proterozoic Milanko- vitch periods with low uncertainty (Fig. 2 D–F and Table 2). To provide a baseline assessment from the early Cenozoic, we investigate Eocene reflectivity data (a*, red/green) from the Walvis Ridge (10) (Ocean Drilling Program Site 1262; Fig. 1E). This dataset has been previously evaluated with the TimeOpt approach (8), and a statistically significant astro- nomical signal (P < 0.005) is identified at a sedimentation rate of 1.33 cm/ky (SI Appendix, Fig. S9). Application of the TimeOptMCMC algorithm allows a rigorous assessment of the uncertainty in the Eocene Milankovitch periods, in the un- derlying g and k terms, and in the sedimentation rate (Fig. 2, Table 2, and SI Appendix, Figs. S10 and S11). In this case, the posterior distributions for sedimentation rate (Fig. 2G) and for the fundamental frequencies g3 (Earth) and g4 (Mars) show the greatest change relative to their prior distributions (SI Appendix, Fig. S10 E and G). Most notably, the posterior mean value for g4 (Mars) is greater than the maximum value ob- served in the modeling study of Laskar et al. (2). This dis- crepancy in g4 is also expressed in the e2 (91.98 ky) and e3 (118.95 ky) eccentricity terms at Walvis Ridge (Fig. 2K and SI Appendix, Fig. S11 C and E), both of which share the g4 term and are notably shorter than those observed in the astro- nomical model simulations of Laskar et al. (2) (Table 2). A possible explanation for these differences in g3 and g4 is that the frequencies reported in figure 9 of Laskar et al. (2) are averaged over 20-My intervals, whereas the Walvis Ridge re- cord spans a shorter interval of about 1.7 My. Discussion Comparison of the Proterozoic and Eocene results highlights how the TimeOptMCMC approach combines cyclostratigraphic data and astronomical theory to improve model parameters. In Table 1. Source of the climatic precession and eccentricity periods, as derived from the fundamental frequencies and precession constant Parameter Source Period (ky)—Today* p1 k + g5 23.678 p2 k + g2 22.371 p3 k + g4 18.951 p4 k + g3 19.103 p5 k + g1 23.120 e1 g2 − g5 405.091 e2 g4 − g5 94.932 e3 g4 − g2 123.945 e4 g3 − g5 98.857 e5 g3 − g2 130.781 *Precession and eccentricity estimates from ref. 2. 2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1717689115 Meyers and Malinverno
  3. 3. the case of the Proterozoic, where the deviation of k from its present value is expected to be substantial due to the large un- certainties in the Earth–Moon history, the Xiamaling Cu/Al data provides strong constraints to improve our knowledge of the precession constant. In the case of the Eocene, the expected changes in k are much smaller, and the cyclostratigraphic data more strongly improve our knowledge of the fundamental fre- quencies g3 and g4. For both the Proterozoic and Eocene ex- amples, the sedimentation rate (and hence the duration of the stratigraphic interval) is highly constrained by the Bayesian inversion. Although stratigraphic-based estimates of the fundamental frequencies of the solar system are rare [g1 to g5; however, see Olsen and Kent (15)], numerous studies have attempted to reconstruct the precession constant and/or the Earth–Moon distance and length of day using geologic data. These approaches include inferences from the evaluation of tidal deposits and of growth patterns in marine invertebrate fossils and stromatolites throughout the past 2.5 billion years (16), and for the late Cenozoic (<25 Ma), the application of astronomical-based methods (1, 17, 18). In addition, a number of theoretical mod- eling exercises have been conducted to constrain the Earth– Moon history (2, 4, 16, 19, 20). In Fig. 3, we compare our astronomical-based results for the Proterozoic and Eocene to a number of Earth–Moon separation models, and also to two tida- lite datasets that are considered to be of high quality (16): rhythmites from the Big Cottonwood Formation (∼900 Ma; refs. 21 and 22), and the Elatina Formation and Reynell Siltstone 0 500 1000 1500 -2-1012 0 500 1000 1500 -201234 263.5 264.0 264.5 265.0 265.5 102030 A B Proterozoic Xiamaling Formation Eocene Walvis Ridge C Cu/Al D Cu/Al(standardized)Cu/Al(standardized) Height (m) Elapsed Time (ky) Elapsed Time (ky) 120 125 130 135 140 34567 E a*(red/green) Depth (mcd) F a*(standardized) Elapsed Time (ky) G a*(standardized) Elapsed Time (ky) H 0 100 200 300 400 500 600 -101234 0 100 200 300 400 500 600 -1.00.01.0 0.00 0.02 0.04 0.06 0.08 0.10 0.000.040.08 Power Frequency (cycles/ky) 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 0.000.040.08 Power Frequency (cycles/ky) 0.00 0.02 0.04 0.06 0.08 0.10 -1 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 pre- cession 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, and the 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. Meyers and Malinverno PNAS Latest Articles | 3 of 6 EARTH,ATMOSPHERIC, ANDPLANETARYSCIENCES
  4. 4. (∼620 Ma; refs. 16 and 23). It should be noted that the in- terpretation of tidalite (and also bioarchive) datasets in terms of Earth–Moon history remains a contentious issue due to problems with cycle recognition and the potential for missing laminations (4, 16), thus the TimeOptMCMC approach provides an independent means for their validation. To supplement our comparison, SI Appendix, Fig. S12 includes some additional more contro- versial estimates from Phanerozoic bioarchives, and a datum from the Weeli Wooli Formation rhythmite (∼2,450 Ma; refs. 11, 16, and 24). The TimeOptMCMC reconstructed Earth–Moon distance from the Xiamaling Formation (340,900 ± 2,600 km, 2σ; Table 2) is consistent with that derived from ocean models (20) that imply smaller torques, reduced tidal dissipation, and slower lunar retreat rates in the distant past, ultimately related to a less efficient excitation of the ocean’s normal modes by tidal forcing on an Earth with a faster rotation rate (Fig. 3). The Xiamaling result is also compatible with a model that employs an average tidal dissipation rate that is 60% of the present value (SI Appendix, Fig. S12). If the Elatina and Cottonwood tidalite data are taken at face value, either of these Earth– Moon separation models are possible, depending on the tidalite record considered. However, the small uncertainty of the Xiamaling estimate excludes a range of other potential models that are permitted by the tidalite data, such as one that employs a tidal dissipation rate that is 40% of the present value, and furthermore, the 60% model is inconsistent with estimates from the Weeli Wooli Formation (SI Appendix, Fig. S12). It should also be noted that the 40% and 60% rate models violate the constraint provided by modern observed Moon retreat rate, in contrast to the ocean model (20) and present rate model. Finally, the astronomical-based Bayesian reconstruction is consistent with a length of day (18.68 ± 0.25 h, 2σ), which is shorter than that of published Proterozoic estimates from geologic data (16), and is at the low end of existing model length of day estimates (4), but has a greatly reduced uncertainty (Fig. 2C). The methodology presented here is not affected by problems inherent in previous estimates of the precession constant, asso- ciated with ambiguity in the interpretation of bioarchives and F Length of Day (hours) 17 18 19 20 21 22 23 24 0.01.02.03.0 Precession Constant (arcsec/y) 20 40 60 80 100 120 0.000.100.200.30 k+g5 k+g2 k+g3k+g4 Precession Period (ky) 11 12 13 14 15 16 0.00.51.01.52.0 k+g1 Sedimentation Rate (cm/ky) 0.25 0.30 0.35 0.40 0.45 0.50 020406080 50 51 52 53 0.00.40.8 Long Eccentricity Period (ky) g2-g5 395 400 405 410 415 0.000.100.200.30 C Short Eccentricity Period (ky) g3-g2g4-g2 g3-g5 g4-g5 90 100 110 120 130 140 0.000.100.200.30 1.0 1.2 1.4 1.6 1.8 2.0 010203040 23.4 23.6 23.8 24.0 24.2 01234 395 400 405 410 415 0.00.10.20.30.40.5 90 100 110 120 130 140 0.00.10.20.30.40.5 18 19 20 21 22 23 24 0.00.51.01.52.02.5 Sedimentation Rate (cm/ky) Precession Constant (arcsec/y) Length of Day (hours) Long Eccentricity Period (ky) g2-g5 Short Eccentricity Period (ky) Precession Period (ky) g4-g5 g3-g5 g4-g2 g3-g2 k+g4 k+g3 k+g2 k+g1 k+g5 Proterozoic Xiamaling Formation Eocene Walvis Ridge I L D E A B G H J K 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. 4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1717689115 Meyers and Malinverno
  5. 5. tidal deposits (4, 16). Furthermore, the technique should be widely applicable, given the abundance of relatively continuous records of astronomically forced sedimentation. An important feature of this quantitative approach is a comprehensive treat- ment of uncertainties, facilitated by the explicit coupling of as- tronomical theory with geologic observation. The quantification of prior and posterior distributions allows for a rigorous treat- ment of astrochronologic uncertainties, addressing a major weakness in prior work and providing an objective way to inte- grate astrochronologies with radioisotopic data, from the Pro- terozoic to the Cenozoic. Application of this methodology to sedimentary records that span Earth history will facilitate im- provement in the calibration of the geologic time scale, will constrain the history of the Earth–Moon system in deep time, and shows promise of reconstructing the evolution of the fun- damental orbital frequencies of the solar system over billions of years. Materials and Methods We utilize the recently developed TimeOpt regression framework (8) to evaluate two features that are diagnostic of an astronomical fingerprint in strata: the concentration of spectral power at the proposed astronomical frequencies (25) (e.g., climatic precession and eccentricity), and the am- plitude modulation of climatic precession (26), which is caused by varia- tions in eccentricity. For the Bayesian inversion, TimeOpt is reformulated in terms of likelihood functions (27) (SI Appendix), and Markov Chain Monte Carlo (MCMC) simulation is utilized to sample values of the solar system secular frequencies g1 to g5, precession constant k, and sedimen- tation rate that are physically plausible and agree with the stratigraphic data. This allows evaluation of the five dominant eccentricity and five dominant climatic precession cycles that are observable in sedimentary strata, which depend on sums or differences of the g terms and/or k (Table 1). For example, the 405-ky-long eccentricity term of the Cenozoic origi- nates from the difference between g2 (Venus) and g5 (Jupiter), and one of the strongest climatic precession cycles (23.7 ky in the Cenozoic) corre- sponds to g5 + k. 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, km × 103 384.4§ 340.855800 1.293260 383.110800 0.822051 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. § Semimajor axis. { Sidereal day. -1500 -1000 -500 0 280300320340360380 Millions of Years Before Present Earth-MoonDistance(x1000km) Walvis Ridge Xiamaling Formation Today ElatinaCottonwood Ocean Model Present Rate of Dissipation Fig. 3. TimeOptMCMC reconstructed Earth–Moon distance, compared with two tidalite-based estimates and two models. Uncertainties in Earth–Moon distance are ±2σ and age uncertainties span minimum and maximum values. The Bayesian posterior TimeOptMCMC estimates for the Proterozoic Xiamaling Formation and Eocene Walvis Ridge are indicated with blue symbols. Note the significant improvement in precision between posterior (blue) and prior estimates for the Xiamaling data (prior = 319,743 to 380,309 km; 2σ). Age uncertainties for the Xiamaling and Walvis Ridge re- sults fall within the size of the blue symbols. Tidalite estimates from the Big Cottonwood Formation (∼900 Ma; ref. 22) and Elatina Formation and Rey- nell Siltstone (∼620 Ma; ref. 23) are shown with dark green symbols (based on the updated analyses of ref. 16). The light green Big Cottonwood For- mation estimate (364,192 km) is an alternative value reported by ref. 21 with uncertainties from the 348,884-km estimate (dark green symbol). The ocean model (red line), and a model using the present rate of tidal dissipation (black line), derive from ref. 20. Meyers and Malinverno PNAS Latest Articles | 5 of 6 EARTH,ATMOSPHERIC, ANDPLANETARYSCIENCES
  6. 6. A complete description of the approach, including evaluation and calibration with a synthetic astronomical test series, is in the SI Appendix. All analyses were conducted using the free software R (28), and an implementation of the TimeOptMCMC algorithm is available in the Astrochron package (29). ACKNOWLEDGMENTS. We thank the reviewers and the editor for their constructive remarks. This study was supported by NSF Grant EAR-1151438 (to S.R.M.), and by a sabbatical leave from the University of Wisconsin— Madison (S.R.M.) to conduct research at Lamont-Doherty Earth Observatory. 1. Hinnov LA (2013) Cyclostratigraphy and its revolutionizing applications in the Earth and planetary sciences. Geol Soc Am Bull 125:1703–1734. 2. Laskar J, et al. (2004) A long-term numerical solution for the insolation quantities of the Earth. Astron Astrophys 428:261–285. 3. Ma C, Meyers SR, Sageman BB (2017) Theory of chaotic orbital variations confirmed by Cretaceous geological evidence. 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