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 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.
We revisit a classical perturbative approach to the Hamiltonian related to the motions of Trojan bodies, in the framework of the Planar Circular Restricted Three-Body Problem (PCRTBP), by introducing a number of key new ideas in the formulation. In some sense, we adapt the approach of [11] to the context of the normal form theory and its modern techniques. First, we make use of Delaunay variables for a physically accurate representation of the system. Therefore, we introduce a novel manipulation of the variables so as to respect the natural behavior of the model. We develop a normalization procedure over the fast angle which exploits the fact that singularities in this model are essentially related to the slow angle. Thus, we produce a new normal form, i.e. an integrable approximation to the Hamiltonian. We emphasize some practical examples of the applicability of our normalizing scheme, e.g. the estimation of the stable libration region. Finally, we compare the level curves produced by our normal form with surfaces of section provided by the integration of the non-normalized Hamiltonian, with very good agreement. Further precision tests are also provided. In addition, we give a step-by-step description of the algorithm, allowing for extensions to more complicated models. *
In the last decades a peculiar family of solutions of the circular restricted three body problem has been used to explain the temporary captures of small bodies and spacecrafts by a planet of the Solar System. These solutions, which transit close to the Lagrangian points L 1, L 2 of the CRTBP, have been classified using the values of approximate local integrals and of the Jacobi constant. The use for small bodies of the Solar System requires to consider a hierarchical extension of the model, from the CRTBP to the full N planetary problem. The elliptic restricted three body, which is the first natural extension of the CRTBP, represents already a challenge, since global first integrals such as the Jacobi constant are not known for this problem. In this paper we extend the classification of the transits occurring close to the Lagrangian points L 1, L 2 of the ERTBP using a combination of the Floquet theory and Birkhoff normalizations. Provided that certain non-resonance conditions are satisfied, we conjugate the Hamiltonian of the problem to an integrable normal form Hamiltonian with remainder, which is used to define approximate local first integrals and to classify the transits of orbits through a neighbourhood of the Lagrange equilibria according to the values of these integrals. We provide numerical demonstrations for the Earth–Moon ERTBP.
The dynamics near the Lagrange equilibria L 1 and L 2 of the Circular Restricted Three-body Problem has gained attention in the last decades due to its relevance in some topics such as the temporary captures of comets and asteroids and the design of trajectories for space missions. In this paper we investigate the temporary captures using the tube manifolds of the horizontal Lyapunov orbits originating at L 1 and L 2 of the CR3BP at energy values which have not been considered so far. After showing that the radius of convergence of any Hamiltonian normalization at L 1 or L 2 computed with the Cartesian variables is limited in amplitude by |1 − µ − x L 1 | (µ denoting the reduced mass of the problem), we investigate if regularizations allow us to overcome this limit. In particular, we consider the Hamiltonian describing the planar three-body problem in the Levi-Civita regularization and we compute its normalization for the Sun-Jupiter reduced mass for an interval of energy which overcomes the limit of Cartesian normalizations. As a result, for the largest values of the energy that we consider, we notice a transition in the structure of the tubes manifolds emanating from the Lyapunov orbit, which can contain orbits that collide with the secondary body before performing one full circulation around it. We discuss the relevance of this transition for temporary captures.
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