On the construction of the Kolmogorov normal form for the Trojan asteroids

In this paper we focus on the stability of the Trojan asteroids for the planar Restricted Three-Body Problem (RTBP), by extending the usual techniques for the neighbourhood of an elliptic point to derive results in a larger vicinity. Our approach is based on the numerical determination of the frequencies of the asteroid and the effective computation of the Kolmogorov normal form for the corresponding torus. This procedure has been applied to the first 34 Trojan asteroids of the IAU Asteroid Catalog, and it has worked successfully for 23 of them. The construction of this normal form allows for computer-assisted proofs of stability. To show it, we have implemented a proof of existence of families of invariant tori close to a given asteroid, for a high order expansion of the Hamiltonian. This proof has been successfully applied to three Trojan asteroids. PACS: 05.10.-a, 05.45.-a, 45.10.-b, 45.20.Jj, 45.50.Pk, 95.10.Ce.