Bayesian Estimation of Past Astronomical Frequencies, Lunar Distance, and Length of Day From Sediment Cycles

Astronomical cycles recorded in stratigraphic sequences offer a powerful data source to estimate Earth's axial precession frequency k, as well as the frequency of rotation of the planetary perihelia (gi) and of the ascending nodes of their orbital planes (si). Together, these frequencies control the insolation cycles (eccentricity, obliquity and climatic precession) that affect climate and sedimentation, providing a geologic record of ancient Solar system behavior spanning billions of years. Here, we introduce two Bayesian methods that harness stratigraphic data to quantitatively estimate ancient astronomical frequencies and their uncertainties. The first method (TimeOptB) calculates the posterior probability density function (PDF) of the axial precession frequency k and of the sedimentation rate u for a given cyclostratigraphic data set, while setting the Solar system frequencies gi and si to fixed values. The second method (TimeOptBMCMC) applies an adaptive Markov chain Monte Carlo algorithm to efficiently sample the posterior PDF of all the parameters that affect astronomical cycles recorded in stratigraphy: five gi, five si, k, and u. We also include an approach to assess the significance of detecting astronomical cycles in cyclostratigraphic records. The methods provide an extension of current approaches that is computationally efficient and well suited to recover the history of astronomical cycles, Earth-Moon history, and the evolution of the Solar system from geological records. As case studies, data from the Xiamaling Formation (N. China, 1.4 Ga) and ODP Site 1262 (S. Atlantic, 55 Ma) are evaluated, providing updated estimates of astronomical frequencies, Earth-Moon history, and secular resonance terms. Earth's transit through our Solar system is ever evolving, and so are such seemingly unwavering planetary characteristics as the number of hours in a day. For example, it is well known that the length of the day generally increases with time as Earth's rotation rate decreases from tidal interactions with our orbiting Moon. But the ability to chart out this evolution over the history of the Solar system has been hampered by limitations of both data and theoretical models. This study presents a computational approach to map out the history of Solar system motions and the history of the Earth-Moon system, including the length of a day, by leveraging geological data and astronomical theory within a statistical framework that fully accounts for uncertainties. As such, the approach provides a means to use the geological archive as an astronomical observatory, allowing us to explore Solar system and Earth-Moon dynamics throughout their long history. We present two updated methods for Bayesian astrochronology: TimeOptB and TimeOptBMCMC TimeOptB simultaneously estimates the Earth's axial precession frequency and the sedimentation rate from cyclostratigraphic data In addition, TimeOptBMCMC simultaneously estimates Solar system g-frequencies and s-frequencies from cyclostratigraphic data