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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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 [38]. 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.