Bifurcation sequences in the secular 3D planetary 3-body problem

Background

A classical problem of Celestial Mechanics is the planetary three-body problem, which deals with the motion of two planets of non-negligible mass under the gravitational attraction of a star, and including the gravitational perturbation of each planet on the other. In recent years, new interest in the problem has emerged owing to its close relevance to understanding planetary motions in extrasolar planetary systems. Most recent studies stem from the planar case, i.e., where the orbits of all three bodies are co-planar. A basic known configuration of planetary orbits in such systems is the one of apsidal corotation resonance (ACR). While the planar secular planetary three body problem is well understood, several of its phase space features are already non trivial and give rise to substantial complication in the analysis when passing from the 2D to the full 3D problem. In the latter, the planetary orbits are allowed to have non-zero mutual inclination, whose value becomes an additional parameter of the problem. Several planetary systems have been observed in a state of mutual inclination of several degrees (see [2] for a review). The main change from 2D to the 3D case is that the 3D secular Hamiltonian is non-integrable.

Project goal

In the near-integrable regime, arising at moderate values of the mutual inclination, numerical experiments show that, while regular, the structure of the phase space in this regime exhibits a substantial departure from the structure of the phase space in the corresponding planar case. Most notably, in the 3D case we witness the birth of a rich variety of new possible periodic states with features quite distinct from those of the ACR states of the planar problem.

The purpose of the project is to show that such states can be efficiently predicted and/or classified through an analysis of the sequences of bifurcations of periodic orbits, which stem from the ACR states (A or B) as we gradually increase the level of mutual orbital inclination. Various normal form models are examined regarding the extent to which they lead regarding phase space dynamics qualitatively similar as that in the complete system. Moreover, through a geometric method ([3,4,5]) we analytically predict the sequence of bifurcations leading to a change of stability and/or the appearance of new periodic orbits in the secular 3D planetary three body problem. In some cases, it is possible to find critical values of the second integral giving rise to pitchfork and saddle-node bifurcations of new periodic orbits in the system.

This analysis renders possible to predict the most important structural changes in the phase space, as well as the emergence of new possible stable periodic planetary orbital configurations which can take place as the mutual inclination between the two planets is allowed to increase.

References

1. Mastroianni R, Efthymiopoulos C. The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination. Celestial Mech Dyn Astron 2023;135(3):22 https://doi.org/10.1007/s10569-023-10136-5

2. Naoz S. The eccentric Kozai-Lidov effect and its applications. Annu Rev Astron Astrophys 2016;54:441–89 https://doi.org/10.1146/ annurev-astro-081915-023315

3. Kummer M. On resonant non linearly coupled oscillators with two equal frequencies. Commun Math Phys 1976;48:53–79. https://doi.org/10.1007/BF01609411

4. Marchesiello A, Pucacco G. Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance. Int J Bifurcation Chaos 2016;26(04):1630011 https://doi.org/10.1142/S0218127416300111

5. Cushman RH, Bates LM. Global aspects of classical integrable systems; vol. 94. Springer; 1997.

Outcome