Global Instability in the Restricted Planar Elliptic Three Body Problem

Abstract

The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet or an asteroid) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum.The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold P + (resp. P − ). In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity P∞, which we call the manifold at parabolic infinity.On P∞ it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic both in the future and the past. Since the inner dynamics inside P∞ is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the RPETBP whose angular momentum varies in any predetermined fashion.2000 Mathematics Subject Classification: Primary 37J40, 70F15. Keywords: Elliptic Restricted Three Body problem, parabolic motions, Manifold of parabolic motions at infinity, Arnold diffusion, splitting of separatrices, Melnikov integral. Main result and methodologyThe restricted planar elliptic three body problem (RPETBP) describes the motion q of a massless particle (a comet) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter ) with mass ratio µ revolving around their center of mass on elliptic orbits with eccentricity J . In this paper we search for trajectories of motion which show a large variation of the angular momentum G = q ×q. In other words, we search for global instability ("diffusion"This is one of the reasons why we are going to restrict ourselves to the region G ≥ C large enough and J G ≤ c small enough along this paper to get the diffusive orbits.Among the harmonics L 0, of 0 order in s, by (42), the harmonic L 0,0 appears to be the dominant one, but we will also estimate L 0,1 to get information about the variable α, and bound the rest of harmonics L 0, for ≥ 2. Among the harmonics of first order L 1,k , again by (42), the and the error functions satisfyTo bound the integral (81) for m ≥ 0 we will consider two different cases: 0 ≤ q ≤ m and 0 ≤ m < q. Let us first consider the case 0 ≤ q ≤ m. By the analyticity and periodicity of the integral we change the path of integration from (E) = 0 to E = ln(2a 2 / J )so that e iE = e iu−ln(2a 2 / J) = J 2a 2 e iu and then, by (78), (79) and (83a), (83b), 23Therefore along the complex path...

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