High order normal form stability estimates for co-orbital motion

Abstract

We present estimates of the size of the analytic domain of stability for co-orbital motions obtained by a high order normal form in the framework of the elliptic restricted three body problem. As a demonstration example, we consider the motion of a Trojan body in an extrasolar planetary system with a giant planet of mass parameter $\mu =0.005$ and eccentricity $e^{\prime }=0.1$. The analysis contains three basic steps: (i) derivation of an accurate expansion of the Hamiltonian, (ii) computation of the normal form up to an optimal order (in the Nekhoroshev sense), and (iii) computation of the optimal size of the remainder at various values of the action integrals (proper elements) of motion. We explain our choice of variables as well as the method used to expand the Hamiltonian so as to ensure a precise model. We then compute the normal form up to the normalisation order $r=50$ by use of a computer-algebraic program. We finally estimate the size $||R||$ of the remainder as a function of the normalization order, and compute the optimal normalization order at which the remainder becomes minimum. It is found that the optimal value $\log (||R_{opt}||)$ can serve in order to construct a stability map for the domain of co-orbital motion using only series. This is compared to the stability map found by a purely numerical approach based on chaotic indicators.