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: [email protected]. 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 funda- mental 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.