New Hamiltonian expansions adapted to the Trojan problem

Abstract

A number of studies, referring to the observed Trojan asteroids of various planets in our Solar System, or to hypothetical Trojan bodies in extrasolar planetary systems, have emphasized the importance of so-called secondary resonances in the problem of the long term stability of Trojan motions. Such resonances describe commensurabilities between the fast, synodic, and secular frequency of the Trojan body, and, possibly, additional slow frequencies produced by more than one perturbing bodies. The presence of secondary resonances sculpts the dynamical structure of the phase space. Hence, identifying their location is a relevant task for theoretical studies. In the present paper we combine the methods introduced in two recent papers (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez and Locatelli in MNRAS 453(2):2177, 2015) in order to analytically predict the location of secondary resonances in the Trojan problem. In Páez and Efthymiopoulos (2015), the motion of a Trojan body was studied in the context of the planar Elliptic Restricted Three Body or the planar Restricted Multi-Planet Problem. It was shown that the Hamiltonian admits a generic decomposition $H=H_b+H_{sec}$. The term $H_b$, called the basic Hamiltonian, is a model of two degrees of freedom characterizing the short-period and synodic motions of a Trojan body. Also, it yields a constant ‘proper eccentricity’ allowing to define a third secular frequency connected to the body’s perihelion precession. $H_{sec}$ contains all remaining secular perturbations due to the primary or to additional perturbing bodies. Here, we first investigate up to what extent the decomposition $H=H_b+H_{sec}$ provides a meaningful model. To this end, we produce numerical examples of surfaces of section under $H_b$ and compare with those of the full model. We also discuss how secular perturbations alter the dynamics under $H_b$. Secondly, we explore the normal form approach introduced in Páez and Locatelli (2015) in order to find an ‘averaged over the fast angle’ model derived from $H_b$, circumventing the problem of the series’ limited convergence due to the collision singularity at the 1:1 MMR. Finally, using this averaged model, we compute semi-analytically the position of the most important secondary resonances and compare the results with those found by numerical stability maps in specific examples. We find a very good agreement between semi-analytical and numerical results in a domain whose border coincides with the transition to large-scale chaotic Trojan motions.

Appendix

Variables corresponding to the three degrees of freedom appearing in the expression of the Basic Hamiltonian $H_b$ in Eq. (5), (u, v), $(Y_f,\phi _f)$ and $(Y_p,\phi _p)$, in terms of the orbital elements:

$$\begin{aligned} u= & {} \lambda - \lambda ' - \frac{\pi }{3}, \end{aligned}$$

(40)

$$\begin{aligned} v= & {} \sqrt{a} - 1, \end{aligned}$$

(41)

$$\begin{aligned} \beta= & {} \omega - \phi ',\nonumber \\ y= & {} \sqrt{a} \left( \sqrt{1-e^2} -1 \right) ,\nonumber \\ V= & {} \sqrt{-2y} \sin \beta - \sqrt{-2y_0} \sin \beta _0,\nonumber \\ W= & {} \sqrt{-2y} \cos \beta - \sqrt{-2y_0} \cos \beta _0,\nonumber \\ Y= & {} - \left( \frac{W^2 + V^2}{2} \right) \nonumber \\ \phi= & {} \arctan \left( \frac{V}{W} \right) \end{aligned}$$

(42)

$$\begin{aligned} \phi _f= & {} \lambda ' - \phi , \end{aligned}$$

(43)

$$\begin{aligned} Y_f= & {} \int \frac{\partial E}{\partial \lambda '} \mathrm {d} t + v, \end{aligned}$$

(44)

$$\begin{aligned} Y_p= & {} Y - Y_f, \end{aligned}$$

(45)

where $\lambda$, $\omega$, a and e are the mean longitude, the longitude of the perihelion, the major semiaxis and eccentricity of the Trojan body, $\lambda '$ and $\phi ' = \omega '$ are the mean longitude and longitude of the perihelion of the perturber, $\beta _0 = \pi /3$, $y_0 = \sqrt{1-e'^2} -1$, and E represents the total energy of the Trojan as computed from Eq. (4) (see Páez and Efthymiopoulos 2015 for further details in the construction).