Abstract
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.
Highlights
Long-term solutions for planetary orbits were derived by perturbation methods, and were obtained in the form of the sum of periodic terms
The first such solution was obtained by Lagrange (1782) and was later on improved by LeVerrier (1840, 1841), who took Uranus into account as well. These long-term solutions have been found to be of fundamental importance for the understanding of the past climate of the Earth, when it was understood that the changes of the orbit of the Earth induce some change in its obliquity and in the insolation at the Earth surface (Milankovitch 1941; for a detailed review, see Laskar et al 2004)
Laskar (1985, 1986, 1988) developed a mixed strategy in which an analytical averaging of the planetary equations by perturbation methods was obtained with dedicated computer algebra, followed by numerical integration of the averaged system
Summary
Long-term solutions for planetary orbits were derived by perturbation methods, and were obtained in the form of the sum of periodic terms. One outcome of these computations was to demonstrate that the solar system motion is chaotic (Laskar 1989, 1990). We represent the eccentricity and inclination of the Earth as given by the long-term numerical solution La2004 (Laskar et al 2004) For such a realistic solution, it is natural to take restrictions on the representation model from previous results into account, such as the important libration frequencies from numerical analysis (e.g., Laskar et al 2004), so that the resulting representations can be as similar as possible to the physical model
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