docRotation des corps et déformations de marée
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Rotation des corps et déformations de marée
Librations 3 layers model Application to Titan Rotation des corps et déformations de marée Andy Richard Direction: N. Rambaux, J. Laskar Elbereth 2013 25 novembre 2013 logo A. Richard Rotation des corps Librations 3 layers model Application to Titan How to destroy a planet with a Death Star? logo A. Richard Rotation des corps Librations 3 layers model Application to Titan Synchronous orbit Satellite evolving on a circular orbit Synchronous rotation: ω = n ω: rotation frequency n: orbital frequency logo A. Richard Rotation des corps Librations 3 layers model Application to Titan Synchronous orbit Satellite evolving on a circular orbit Synchronous rotation: ω = n ω: rotation frequency n: orbital frequency Consequence: Orientation with respect to the planet is conserved logo A. Richard Rotation des corps Librations 3 layers model Application to Titan Eccentric orbit Ψ = f − θ : Direction of the planet in the satellite frame f : true anomaly, θ: rotation angle of the satellite Libration: Variations around the uniform rotation of the satellite Synchronous rotation: θ = M + γ (M: mean anomaly, γ: libration angle) Then: Ψ = f − M − γ ∼ 2e sin M + 0(e2 ) with e the eccentricity Restoring torque proportional to Ψ induces a librational response γ (Goldreich 1966) A. Richard Rotation des corps logo Librations 3 layers model Application to Titan Many parameters influence the libration: Internal structure of the satellite Orbital perturbations Surface forcing (atmosphere) ⇓ Tool to study the internal structure of icy satellites logo A. Richard Rotation des corps Librations 3 layers model Application to Titan Icy satellites logo A. Richard Rotation des corps Librations 3 layers model Application to Titan Elastic case Elastic response: the surface deforms instantaneously under the influence of tidal potential Restoring torque is maximal Restoring torque is null Radial elastic deformation ur of the surface with Love formalism (1911): ur = h2 V2 g(R) where h2 is the Love number, g(R) the surface gravity and V2 the tidal potential A. Richard Rotation des corps logo Librations 3 layers model Application to Titan Obtained librations High amplitudes librations: Low frequencies (∼ 1 km), diurnal librations (∼ 500 m in the rigid case, ∼ 50 m in the elastic case) A. Richard Rotation des corps logo Librations 3 layers model Application to Titan Libration spectrum Frequencies contained in the librational motion: Quasi-periodic approximation of the orbital P motion (frequency analysis: Laskar, 1988): f − M = j Hj sin(ωj t + αj ). Titan’s orbit eccentricity / Perturbations: Diurnal frequencies Saturn’s orbital motion: Seasonal frequencies Elasticity: Low frequencies A. Richard Rotation des corps logo Librations 3 layers model Application to Titan Conclusion Summary: Librations: Oscillations around the uniform rotation of the satellite Importance of the rheology to modelize the restoring torque exerted by the planet on the satellite figure At low frequencies, the libration follows the orbital forcing (rigid or elastic cases). The diurnal libration is strongly reduced for an elastic satellite compared to a rigid one: informations on the internal structure Some librations at low frequencies containing informations on the internal structure appears for an elastic satellite. logo A. Richard Rotation des corps Librations 3 layers model Application to Titan Libration amplitudes logo A. Richard Rotation des corps Librations 3 layers model Application to Titan Couplage atmosphérique Echange et transfert de moment angulaire avec la surface: couplage atmosphérique Global Circulation Model (Tokano 2005) Titan IPSL GCM (Lebonnois et al. 2012, Charnay et al. 2012) Thomson Higher Education, 2007 Insolation de l’atmosphère entraîne des mouvements convectifs: cellules de Hadley logo A. Richard Rotation des corps Librations 3 layers model Application to Titan Cas d’un satellite rigide ~ dH dt = ~Γ ~ = [I]~ H ω : Moment cinétique, ~Γ: Couple de force appliqué au système Satellite à 3 couches (s: shell, i: inner core) Equations de la libration linéarisées pour un satellite rigide (Van Hoolst et al., 2008): Cs γ¨s + Kint,s γs − Kint γi = Ks (f − M) + ΓA , Ci γ̈i + Kint,i γi − Kint γs = Ki (f − M) , Rouge: Couple gravitationnel exercé par la planète sur les couches solides Bleu: Couplage gravitationnel interne dû au déphasage des couches solides Vert: Couple atmosphérique exercé sur la surface (Charnay et al. 2012) logo A. Richard Rotation des corps Librations 3 layers model Application to Titan Effet de l’élasticité Variation périodique du potentiel de marée à la fréquence orbitale n= Forme du corps périodique & Amplitude des couplages périodique A la fréquence orbitale, au 1er ordre en eccentricité: Cs γ¨s + Kint,s γs − Kint γi = 2e(Ks − ∆Ks ) sin M , Ci γ̈i + Kint,i γi − Kint γs = 2e(Ki − ∆Ki ) sin M , p/s p avec ∆Ks = 12 32 Ks + 13 Kint − nĊs et par exemple p/s Kint = 24πG 5 R Z ρ(r ) ri d (d̃(r ))dr [(Bi − Ai ) − (Bi0 − A0i )] dr Effet de l’élasticité: A la fréquence orbitale, le couplage gravitationnel exercé sur les couches solides est réduit. Dans un cas pratique: La libration est-elle détectable? Quelles informations sur la structure peut-on en tirer? A. Richard Rotation des corps logo Librations 3 layers model Application to Titan Influence de la coquille Libration diurne Paramètre dominant pour un modèle rigide: épaisseur de la coquille de glace Pas vrai pour un modèle élastique Excentricité de l’orbite de Titan / Perturbations: Fréquences diurnes Mouvement orbital de Saturne: Fréquences saisonnières Elasticité: Basses fréquences A. Richard Rotation des corps logo
