By A. Lemaître (auth.), Jean J. Souchay, Rudolf Dvorak (eds.)
This publication bargains an updated evaluation of present examine on Dynamics of Small sunlight method our bodies and Exoplanets. In course-tested large chapters the authors conceal issues of theoretical celestial mechanics, physics and dynamics of asteroids, comets, balance of exoplanets and numerical integration codes utilized in dynamical astronomy.
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Extra info for Dynamics of Small Solar System Bodies and Exoplanets
We assume that this step has been performed and we finally obtain the averaged Hamiltonian (33). In other words, the original Hamiltonian contains periodic terms with linear combinations of the angles σ , λ, and λ3 + Ω; we average over the short periods, which means over λ. The remaining Hamiltonian (for a circular orbit and neglecting the terms of fourth order in ξ and η) becomes H = nΓ − nS + S S2 + 2C 2C γ1 − γ2 γ1 + γ2 ξ2 + η2 1 − γ1 − γ2 1 − γ1 + γ2 (32) + n 2 C δ1 (x 2 + y 2 ) + δ2 (x 2 − y 2 ) , with x 2 + y 2 = F0 + F1 cos ν + F2 cos 2ν − (ξ 2 + η2 ) [G 0 + G 1 cos ν + G 2 cos 2ν] 5 − (ξ 2 − η2 ) Bi cos (2σ + iν) i=0 5 + 2ξ η Ai sin (2σ + iν), i=0 5 x 2 − y 2 = (2 − ξ 2 − η2 ) Ci cos (2σ + iν) i=0 − (ξ 2 − η2 ) [H0 + H1 cos ν + H2 cos 2ν] 5 Di sin (2σ + iν), + 2ξ η i=0 (33) Resonances: Models and Captures 51 where ν = λ3 + Ω.
Of course, in non-averaged models, the orbital period is also present as the fourth period. This formalism has been developed not only for the Galilean satellites, Europa and Io [13–15, 17], but also for Titan . 8 The Case of Mercury The case of Mercury is slightly different from these mentioned above: it is blocked in a 3:2 spin-orbit resonance, which means that the basic (kernel) model depends on the eccentricity, which is not the case for the 1:1 commensurability. The influence of ˙ is much less important, the equilibrium obliquity K the precession of the orbit (Ω) is moved by a quantity of the order of 2 from the inclination of Mercury, which is about 7◦ with respect to the ecliptic.
3 The Phase Space The Hamiltonian (19) is a function of x and y and of the parameter δ; for each value of δ, we can draw the curves K = constant, in the Cartesian phase space (x, y). Different cases are represented in Fig. 13: δ = −3, δ = −1, δ = 2, and δ = 5. The first two cases correspond to negative values of δ, with only one stable real equilibrium. The level curves are almost ellipses for δ = −3, far from the resonance, giving a target-like global picture; for δ = −1 this is not the case anymore and the curves are different, their behavior already showing the proximity of the resonance.