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\begin{document}

\title{Bayesian estimation of past astronomical frequencies, lunar distance,
and length of day from sediment cycles}



\author[1]{Alberto Malinverno}%
\author[2]{Stephen R. Meyers}%
\affil[1]{Lamont-Doherty Earth Observatory of Columbia University}%
\affil[2]{University of Wisconsin - Madison}%


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  \date{May 26, 2024}


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\begin{abstract}
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.%
\end{abstract}%



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