By A. Lemaître (auth.), Jean J. Souchay, Rudolf Dvorak (eds.)
This publication bargains an up to date review of present examine on Dynamics of Small sun method our bodies and Exoplanets. In course-tested huge chapters the authors conceal themes 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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Additional info for Dynamics of Small Solar System Bodies and Exoplanets
The first set (h, K , g) locates the position of the angular momentum vector G in an inertial frame of reference (the ecliptic plane at some epoch, for example); the second Euler’s set (g, J, l) locates the body frame (the axis of inertia) in the Fig. 26 The linked sets of Euler angles (l, K , g) and (g, J, h) from which the Andoyer’s angular variables are defined. They locate the body frame (f1 , f2 , f3 ) with respect to the inertial frame (e1 , e2 , e3 ) Resonances: Models and Captures 47 previous frame tied to the angular momentum.
1 Formulation and Scaling The first model of resonance is the pendulum, as already mentioned in (11); in celestial mechanics cases (development of the third body perturbation, spin–orbit 22 A. Lemaˆıtre resonance, geostationary resonances), it is generally given first in the following Hamiltonian form: H(σ, S) = αS 2 + β S + cos(σ − σ0 ). We introduce a change of phase to make σ0 disappear (r = σ − σ0 ) and a translation on S to get rid of the linear term: R = S − S0 . H becomes H(r, R) = α (R + S0 )2 + β (R + S0 ) + cos r = α R 2 + 2α R S0 + αS02 + β R + β S0 + cos r = α R 2 + cos r + C0 , β .
T The Hamiltonian writes K(Ψ, J, δ) = H (σ (Ψ, J, δ), S(Ψ, J, δ), δ) ∂F = K (−, J, δ) + ∂t ∂F (Ψ, J, δ) = K (−, J, δ) + δ˙ ∂δ and is associated to the dynamics: ∂K ∂K = + O(η) Ψ˙ = ∂J ∂J and ∂K J˙ = − = O(η). ∂Ψ We can conclude that for small η the area J is quasi-constant, as long as we avoid the separatrices regions, and for times smaller than η1 . We follow the behavior of the dynamical system by the help of a guiding trajectory, the area of which is quasi-constant; δ is slowly changing with time, but at each time we fix its value and calculate the enclosed area, we get a quasi-constant quantity (with variations smaller than η for times smaller than η1 ).
Dynamics of Small Solar System Bodies and Exoplanets by A. Lemaître (auth.), Jean J. Souchay, Rudolf Dvorak (eds.)