Abstract
Numerical integrations of the Solar System reveal a remarkable stability of the orbits of the inner planets over billions of years, in spite of their chaotic variations characterized by a Lyapunov time of only 5 million years and the lack of integrals of motion able to constrain their dynamics. To open a window on such long-term behavior, we compute the entire Lyapunov spectrum of a forced secular model of the inner planets. We uncover a hierarchy of characteristic exponents that spans 2 orders of magnitude, manifesting a slow-fast dynamics with a broad separation of timescales. A systematic analysis of the Fourier harmonics of the Hamiltonian, based on computer algebra, reveals three symmetries that characterize the strongest resonances responsible for the orbital chaos. These symmetries are broken only by weak resonances, leading to the existence of quasi-integrals of motion that are shown to relate to the smallest Lyapunov exponents. A principal component analysis of the orbital solutions independently confirms that the quasi-integrals are among the slowest degrees of freedom of the dynamics. Strong evidence emerges that they effectively constrain the chaotic diffusion of the orbits, playing a crucial role in the statistical stability over the Solar System lifetime.
- Received 18 November 2022
- Accepted 3 March 2023
DOI:https://doi.org/10.1103/PhysRevX.13.021018
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Viewpoint
ViewpointTackling the Puzzle of Our Solar System’s Stability
Published 3 May 2023
A new theory explains why our planets avoid collisions for far longer times than standard theories of planetary stability predict.
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Popular Summary
The long-term motion of the planets in the Solar System is a long-standing issue, going back to Newton. Numerical simulations reveal a remarkable stability of the orbits of the inner planets (Mercury, Venus, Earth, and Mars). While their motion is chaotic and unpredictable beyond 60 million years, the typical time to wait for catastrophic events (close encounters, collisions, or ejections of planets) is larger than the age of the Universe. This timescale separation is fascinating, as there is no way to constrain a priori the chaotic variations of the orbits. So, what makes the inner planets so stable over the age of the Solar System? Here, we provide a physically intuitive framework that justifies this stability in terms of quasiconserved quantities.
We show that the long-term motion of the inner planets is described by a slow-fast dynamical system, with fundamental timescales ranging from 5 to 500 million years. Moreover, a set of symmetries characterizes the strongest planet interactions responsible for the orbital chaos. They are broken only by weak interactions, leading to the existence of quasiconserved quantities that represent the slowest degrees of freedom of the dynamics. This is independently confirmed by a principal component analysis of the orbital solutions. The statistical stability of the inner planets over the lifetime of the Solar System naturally emerges from the adiabatic constraints exerted by the slowly varying quasiconserved quantities.
This work opens a window on the dynamical mechanisms that shape the architecture of exoplanetary systems and govern their long-term stability.
