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Analysis of the Stability of a Planetary System on Cosmogonic Time Scales

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Abstract

We consider the dynamical evolution of planetary systems whose structure is nearly circular and coplanar. The analysis is performed by the Hori–Deprit averaging method within the theory of the first order in planetary masses. A convenient set of canonical elements and a rarely employed variety of astrocentric coordinates are used to derive the equations of motion. Owing to the use of the chosen system of canonical elements, the expansions of the right-hand sides of the averaged equations contain a relatively small number of terms. Compared to other widespread coordinate systems, the astrocentric coordinates used by us allow a more convenient representation of the disturbing function to be obtained and do not require its expansion into a series in powers of a small parameter. On time scales \({\sim}10^{5}{-}10^{7}\) years we have studied the long-term evolution of the planetary systems HD 12661, \(\upsilon\) Andromedae, and some model systems by numerical integration of the averaged equations. Possible secular resonances have been revealed in the systems considered.

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Notes

  1. Recall that a particular solution of a system of ordinary differential equations is called Lagrange-stable if it remains definite and uniformly bounded at all \(t\geqslant t_{0}\), where \(t_{0}\) is the initial time. The Lagrangian triangular solutions in the general three-body problem can serve as an example of a Lagrange-stable system. A precise definition of the Lagrange stability can be found in the textbooks by Grebenikov (1976) and Demidovich (1967).

  2. Strictly speaking, small denominators appear even in the first order, see the first paragraph of Section 5.

  3. By the two-planet Sun–Jupiter–Saturn system Kuznetsov and Kholshevnikov (2006) mean a model two-planet system in which two planets with the masses and orbital parameters of Jupiter and Saturn revolve around a solar-mass star.

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ACKNOWLEDGMENTS

I thank K.V. Kholshevnikov for supervising the work and V.S. Shaidulin for his help in preparing the figures. I also express my gratitude to the anonymous referee for the fruitful comments and useful suggestions on the manuscript. All our computations were performed using the equipment of the Computing Center in the scientific park of the St. Petersburg State University. This work was financially supported by the Russian Science Foundation (project no. 19-72-10023).

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  1. St. Petersburg State University, Universitetskii pr. 28, Petrodvoretz, St. Petersburg, 198504, Russia

    D. V. Mikryukov

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Translated by V. Astakhov

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Mikryukov, D.V. Analysis of the Stability of a Planetary System on Cosmogonic Time Scales. Astron. Lett. 46, 344–358 (2020). https://doi.org/10.1134/S1063773720050059

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