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Trojan resonant dynamics, stability, and chaotic diffusion, for parameters relevant to exoplanetary systems

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Abstract

The possibility that giant extrasolar planets could have small Trojan co-orbital companions has been examined in the literature from both viewpoints of the origin and dynamical stability of such a configuration. Here we aim to investigate the dynamics of hypothetical small Trojan exoplanets in domains of secondary resonances embedded within the tadpole domain of motion. To this end, we consider the limit of a massless Trojan companion of a giant planet. Without other planets, this is a case of the elliptic restricted three body problem (ERTBP). The presence of additional planets (hereafter referred to as the restricted multi-planet problem, RMPP) induces new direct and indirect secular effects on the dynamics of the Trojan body. The paper contains a theoretical and a numerical part. In the theoretical part, we develop a Hamiltonian formalism in action-angle variables, which allows us to treat in a unified way resonant dynamics and secular effects on the Trojan body in both the ERTBP or the RMPP. In both cases, our formalism leads to a decomposition of the Hamiltonian in two parts, \(H=H_b+H_{sec}\). \(H_b\), called the basic model, describes resonant dynamics in the short-period (epicyclic) and synodic (libration) degrees of freedom, while \(H_{sec}\) contains only terms depending trigonometrically on slow (secular) angles. \(H_b\) is formally identical in the ERTBP and the RMPP, apart from a re-definition of some angular variables. An important physical consequence of this analysis is that the slow chaotic diffusion along resonances proceeds in both the ERTBP and the RMPP by a qualitatively similar dynamical mechanism. We found that this is best approximated by the paradigm of ‘modulational diffusion’. In the paper’s numerical part, we then focus on the ERTBP in order to make a detailed numerical demonstration of the chaotic diffusion process along resonances. Using color stability maps, we first provide a survey of the resonant web for characteristic mass parameter values of the primary, in which the secondary resonances from 1:5 to 1:12 (ratio of the short over the synodic period), as well as their transverse resonant multiplets, appear. We give numerical examples of diffusion of weakly chaotic orbits in the resonant web. We finally make a statistics of the escaping times in the resonant domain, and find power-law tails of the distribution of the escaping times for the slowly diffusing chaotic orbits. Implications of resonant dynamics in the search for Trojan exoplanets are discussed.

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Notes

  1. The variables \(u,v\) proposed here are similar to the variables \(X,Y\) defined in equations (24) or (25) of Beaugé and Roig (2001), via a so-called ‘Jupp transformation’. Note, however, the following difference: Our Hamiltonian \(\overline{H_b}\), analogous to Beaugé and Roig’s \(F_0\), is, nevertheless, labeled also by \(e_0'\) (or, simply \(e'\) in the ERTBP). Thus, it is not the averaged (over fast angles) Hamiltonian of the circular RTBP. From a physical point of view, this expresses the possibility to find an integrable approximation to synodic motions even when \(e'\ne 0\). Accordingly, \(Y_p\) in our formalism, which is analogous to Beaugé and Roig’s \(W\), provides a label for the proper eccentricity rather than simply the eccentricity of the test particle. Again, this expresses the fact that the former, but not the latter, is nearly constant even when \(e'\ne 0\).

  2. A comparison between the terminology for resonances employed here and in Robutel and Gabern (2006, referring to the case of Jupiter’s Trojan asteroids), is in order. Robutel and Gabern distinguish four ‘families’ of resonances. Two of these, however, (Families II and IV) involve the frequencies \(\nu _{1,2}=2n_{Saturn}-n_{Jupiter}\) and \(\nu _{2,5}=5n_{Saturn}-2n_{Jupiter}\), which, due to the Great Inequality, both play an important role in the dynamics of Jupiter’s Trojan swarm. Also, Family III involves the asteroids’ secular frequency \(s\), thus it applies only to inclined motions. Here we consider systems with a rather planar geometry and far from mean motion resonances, for which only Family I-type resonances of Robutel and Gabern are relevant. Their \(\nu \) corresponds to our synodic frequency \(\omega _s\). Also, they use \(n_5\) (the mean motion of Jupiter \(=\) 1 in our units), while we use \(\omega _f=n_5-g=1-g\), which, as explained above, is the frequency of epicyclic oscillations of the Trojan body. Thus, The Family I of Robutel and Gabern corresponds to our definition (25) if we set \(m_f=-1\), \(m_s\equiv p\), \(m\equiv q-1\), \(m'\equiv q_5\), \(m_1\equiv q_6\). We emphasize, however, that due to the particular values of the frequencies for the Trojan swarm, Family I of Robutel and Gabern has the restrictions \(m_f=-1\), \(m+m'+m_1=0\). On the contrary, such restriction is not present in our computations. On the other hand, our ’secular’ resonances require both \(m_s\) and \(m_f\) to be equal to zero. Thus, from a dynamical point of view, they are qualitatively closer to their Family III.

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Acknowledgments

The authors would like to thank Professors Z. Knezevic and U. Locatelli for their useful remarks, and the anonymous referees for helping to improve the original manuscript. R.I.P. is supported by the Astronet-II Marie Curie Training Network (PITN-GA-2011-289240). C.E. acknowledges support by the Grant 200/815 of the Research Committee of the Academy of Athens.

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  1. Department of Mathematics, University of Rome “Tor Vergata”, Rome, Italy

    Rocío Isabel Páez

  2. Research Center for Astronomy and Applied Mathematics, Academy of Athens, Athens, Greece

    Christos Efthymiopoulos

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  1. Rocío Isabel Páez
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Correspondence to Rocío Isabel Páez.

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Páez, R.I., Efthymiopoulos, C. Trojan resonant dynamics, stability, and chaotic diffusion, for parameters relevant to exoplanetary systems. Celest Mech Dyn Astr 121, 139–170 (2015). https://doi.org/10.1007/s10569-014-9591-2

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