Volume 628, August 2019
|Number of page(s)||13|
|Section||Numerical methods and codes|
|Published online||13 August 2019|
Frequency analysis and representation of slowly diffusing planetary solutions⋆
Purple Mountain Observatory, Chinese Academy of Sciences, 2 West Beijing Road, 210008 Nanjing, PR China
2 ASD/IMCCE, Observatoire de Paris, PSL University, CNRS, Sorbonne Université, 77 avenue Denfert-Rochereau, 75014 Paris, France
Accepted: 10 January 2019
Context. Over short time-intervals, planetary ephemerides have traditionally been represented in analytical form as finite sums of periodic terms or sums of Poisson terms that are periodic terms with polynomial amplitudes. This representation is not well adapted for the evolution of planetary orbits in the solar system over million of years which present drifts in their main frequencies as a result of the chaotic nature of their dynamics.
Aims. We aim to develop a numerical algorithm for slowly diffusing solutions of a perturbed integrable Hamiltonian system that will apply for the representation of chaotic planetary motions with varying frequencies.
Methods. By simple analytical considerations, we first argue that it is possible to exactly recover a single varying frequency. Then, a function basis involving time-dependent fundamental frequencies is formulated in a semi-analytical way. Finally, starting from a numerical solution, a recursive algorithm is used to numerically decompose the solution into the significant elements of the function basis.
Results. Simple examples show that this algorithm can be used to give compact representations of different types of slowly diffusing solutions. As a test example, we show that this algorithm can be successfully applied to obtain a very compact approximation of the La2004 solution of the orbital motion of the Earth over 40 Myr ([−35 Myr, 5 Myr]). This example was chosen because this solution is widely used in the reconstruction of the past climates.
Key words: methods: numerical / celestial mechanics / ephemerides / planets and satellites: dynamical evolution and stability / chaos
Full Tables A.3 and A.4 are only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (22.214.171.124) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/628/A84
© Y. N. Fu and J. Laskar 2019
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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