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

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 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.

Drift and Its Mediation in Terrestrial Orbits

“…In the previous sections, we computed FLIs and diameters in various sections nearly-integrable Hamiltonian system, symplectic Maps and in Dynamical Astronomy [16,18,19,40,42]. For a recent overview specifically around the FLIs and their applications, we advise the reader to consult [20] for a pedagogical introductory note.…”

Secondary resonances and the boundary of effective stability of Trojan motions

“…Such an effect is possible to visualize already in the Elliptic Restricted Three-Body Problem, namely the simplest model with non trivial H sec . The reader is refered to Figures 5 to 15 of [33] which show in detail the statements below, by exemplifying the outcome of the modulation effects for the secondary resonances 1:5 up to 1:12, when the modulus of the eccentricity vector e 0 of the primary companion varies from e 0 = 0 to just a moderate value e 0 = 0.1. By inspecting the stability maps in the space of the Trojan body's proper elements, one sees that, for e 0 slightly larger than zero, the separatrix pulsation for the secondary resonances becomes large enough so as to wipe out nearly completely the domain of stable motions occupied by such resonances.…”

“…For some combinations of physical parameters, these resonances occupy a large fraction of the domain of stability and rule the dynamics within the stable tadpole region. In this work, we present an application of a recently introduced 'basic Hamiltonian model' H b for Trojan dynamics [33], [35]: we show that the inner border of the secondary resonance of lowermost order, as defined by H b , provides a good estimation of the region in phase-space for which the orbits remain regular regardless the orbital parameters of the system. The computation of this boundary is straightforward by combining a resonant normal form calculation in conjunction with an 'asymmetric expansion' of the Hamiltonian around the libration points, which speeds up convergence.…”

Moser’s Quadratic, Symplectic Map

“…Symplectic maps naturally arise as Poincaré maps of Hamiltonian flows and therefore are relevant for many different applications, for example in the dynamics of the solar system and galaxies [4][5][6][7][8][9], beam dynamics of particle accelerators [10][11][12][13], plasma physics [14], and chemistry [15][16][17][18][19][20]. Investigating the simplest prototypical examples, such as the area-preserving Hénon map or Chirikov's standard map in the two-dimensional case, helps to give a good understanding of the way in which dynamical properties depend on parameters in typical maps.…”