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Unformatted text preview: Restricted four and five body problems in the Solar System Frederic Gabern and ` Angel Jorba CDS Seminar. October 8, 2003. Frederic Gabern Applied Mathematics and Analysis Control and Dynamical Systems University of Barcelona California Institute of Technology 1 Contents • Introduction. – The Trojan asteroids. – Review of the existing results. – Motivation for the present work. • Periodic models. – The Bicircular Coherent Problem. – The Elliptic Restricted Three Body Problem. • Quasiperiodic models. – The Bianular Problem. – The Tricircular Coherent Problem. • Conclusions. 2 The Trojan asteroids (I) Sun Jupiter 60 60 L5: Trojan asteroids L4: Greek asteroids L 4 : L 5 : Achilles Patroclus Hector Aeneas Nestor Memnon Agamemnon Paris 3 The Trojan asteroids (II) 4 The Trojan asteroids (III) 5 Previous Work The Restricted Three Body Problem • Analytic Tools. • Semianalytic tools. Theoretical Results m The Outer Solar System • Frequency analysis. • Lyapunov exponents. Numerical Results 6 The Restricted Three Body Problem The dynamics around the triangular points of the SunJupiter system has been studied using analytical and semianalytical tools (transformation of the Hamiltonian to a normal form , computation of some approximate first integrals ) by several authors: Giorgilli, Delshams, Fontich, Galgani & Sim´o , 1989; Sim´o , 1989; Celletti & Giorgilli , 1991; Jorba & Sim´o , 1994 (in the elliptic case); Giorgilli & Skokos , 1997; Skokos & Dokoumetzidis , 2000. 7 Realistic Model: OSS The dynamics of some jovian Trojan asteroids has been studied in several papers using the Outer Solar System : Bien & Schubart , 1987; Milani , 1993; Levison, Shoemaker & Shoemaker , 1997; PilatLohinger, Dvorak & Burger , 1999; Tsiganis, Dvorak & PilatLohinger , 2000. 8 Some New Models We want to build and study models more sophisticated than the RTBP. These models will try to simulate in a better way the relative SunJupiter motion and they will be written as a RTBP perturbation in order the semianalytical tools can be applied. In this work, we focus on the following models: • The Bicircular Coherent Problem ( BCCP ). • The Elliptic Restricted Three Body Problem ( ERTBP ). • The Bianular Problem ( BAP ). • The Tricircular Coherent Problem ( TCCP ). 9 The BCCP Model (I) • We look for a periodic solution of the planar SunJupiterSaturn Three Body Problem . • The period is chosen to be the relative period of Saturn in the SunJupiter RTBP system: T sat . • The osculating eccentricity of Jupiter is small (about 50 times smaller than the real one) but the semimajor axis and the period are quite well adjusted to the actual ones....
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 Fall '08
 Marsden,J
 Hamiltonian mechanics, body problem, Hamiltonians, symplectic quasiperiodic floquet

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