Introduction Non-resonant case Resonant case Exploration Application Conclusion A secular representation for the long-term resonant dynamics beyond Neptune Melaine Saillenfest Marc Fouchard, Giacomo Tommei, Giovanni Valsecchi IMCCE, Observatoire de Paris Dipartimento di Matematica, Università di Pisa 20/09/2016 Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Plan Introduction 1 Secular non-resonant theory 2 Secular theory for a single resonance 3 Exploration of the parameter space 4 Application to known objects 5 Conclusion 6 Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Introduction Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion A well-known but rather unexplored mechanism Mean-motion resonances with Neptune : origin of large orbital variations beyond the planetary region strong captures are relatively rare variety of possible behaviours yet to be explored Goal : general analysis of the resonant dynamics beyond Neptune (extensive exploration of what can be done by the known planets) A quasi-integrable dynamics : smooth long-term behaviour typical time-scales > 1 Gyr = ⇒ can be described by a secular theory Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion What is a secular theory ? ˙ M = 2 π/T The two-body problem is degenerate : ω = 0 ˙ ˙ Only one varying angle in a 3D space Ω = 0 Effect of a small perturbation : ω and Ω become slow angles 2 π ˙ M = 2 π/T + O ( ε ) ω (rad) π ω = O ( ε ) ˙ ˙ Ω = O ( ε ) 0 0 1 2 3 time (Gyrs) Secular theory : study of the slow (dominant) motion Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Secular non-resonant theory Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Hamiltonian of the problem Hamiltonian in heliocentric coordinates : µ = G M ⊙ N N H = − µ 2 � � 1 r i � � 2 L 2 + n i Λ i − G m i | r − r i | − r · | r i | 3 i =1 i =1 Canonic coordinates (Delaunay elements) : L = √ µa ℓ = M � µa (1 − e 2 ) g = ω G = et h = Ω � H = µa (1 − e 2 ) cos I λ 1 , λ 2 . . . λ N Λ 1 , Λ 2 . . . Λ N Secular Hamiltonian F (1 st order of the masses) : average of H with respect to the fast independent angles ℓ and λ 1 , λ 2 . . . λ N . Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Secular non-resonant Hamiltonian General form of the secular Hamiltonian : F = F ( L, G, H, g ) conservation of secular momenta L and H with L and H as parameters, the dynamics is described by the level curves of F in the ( g, G ) plane Simpler version of the parameters : � a = L 2 /µ C K = ( H/L ) 2 = (1 − e 2 ) cos 2 I ...and of the variables : ω = g � � � 1 − ( G/L ) 2 q = a 1 − Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Study of the lowest order terms C K > 1 / 5 C K < 1 / 5 55 50 q (AU) 45 40 35 0 π/ 2 π 3 π/ 2 2 π 0 π/ 2 π 3 π/ 2 2 π ω (rad) ω (rad) 0.2 10 6 0.18 10 5 0.16 Oscillation period (Gyrs) 10 4 0.14 0.12 10 3 C K 0.1 10 2 0.08 0.06 10 0.04 1 0.02 q < a N 0 0 . 1 100 200 300 400 500 600 700 800 900 1000 a (AU) Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Semi-analytical model Lowest order terms : only accurate for large semi-major axis and small eccentricity = ⇒ Numerical "exact" secular Hamiltonian : � 2 π � 2 π N 1 G m i � F ( L, G, H, g ) = − d λ i d ℓ � � 4 π 2 � r ( L, G, H, ℓ, g, h ) − r i ( λ i ) 0 0 � � i =1 � 0.26 no island for q > a N 0.24 single island at ω = 0 0.22 two islands at ω = 0 and π/ 2 0.2 single island at ω = π/ 2 C K 0.18 0.16 0.14 0.12 0.1 30 40 50 60 70 80 90 a (AU) Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Numerical exploration of the parameter space 25 20 width (AU) Maximum 16 . 4 AU 15 10 5 0 0.25 24 22 Width in perihelion (AU) 0.2 20 18 0.15 C K 16 0.1 14 12 0.05 10 0 8 30 50 100 200 1000 10000 a (AU) = ⇒ a non-resonant secular evolution allows a maximum perihelion excursion of about 16 . 4 AU Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Secular theory for a single resonance Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Coordinate change Principal resonant angle : σ = k λ − k p λ p − ( k − k p ) ̟ with k, k p ∈ N and k > k p matrices A and ( A T ) − 1 New canonical coordinates : ← − resonant angle M σ ← − fast angle λ p γ ω = u A Ω v ← − no change { λ i � = p } { λ i � = p } Three time-scales : • short periods < 10 3 years { λ i � = p } and γ • semi-secular periods ∼ 10 5 years σ • secular periods > 10 9 years ω and Ω Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Semi-secular Hamiltonian Semi-secular Hamiltonian K (1 st order of the masses) : average of H with respect to the fast angles γ and { λ i � = p } . general form : K = K (Σ , U, V, σ, u ) ⇒ two degrees of freedom � √ ⇒ semi-secular momentum conserved : V = √ µa � 1 − e 2 cos I − k p k Methods to describe the secular dynamics : 1) Poincaré sections 2) reduction to a one-degree-of-freedom system : = ⇒ use the adiabatic hypothesis with T u ≫ T σ Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion The adiabatic invariance Action-angle coordinates of K with ( U, u ) fixed : θ = mean angle along the trajectory J ∝ enclosed area = adiabatic invariant Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Resonant secular Hamiltonian Secular Hamiltonian : F = F 0 ( J, U, V, u ) + O ( ξ ) where ξ is related to the ratio of frequencies secular/semi-secular. conservation of secular momenta J and V with J and V as parameters, the dynamics is described by the level curves of F in the ( u, U ) plane Simpler version of the parameters (for a chosen a 0 ) : η 0 = V/ √ µa 0 + k p /k = � � e 2 cos ˜ 1 − ˜ I J (inchanged) ...and of the variables : ω = u � 1 − ( U/ √ µa 0 + k p /k ) 2 � � q = a 0 ˜ 1 − Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Separatrix crossing Stretching of the island during the secular evolution : ( U 1 , u 1 ) ( U 2 , u 2 ) ( U 3 , u 3 ) Σ = √ µa/k 0 π 2 π 0 π 2 π 0 π 2 π σ (rad) σ (rad) σ (rad) Corresponding secular model : trajectory integrable by parts (piecewise model) slow chaotic diffusion Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
Introduction Non-resonant case Resonant case Exploration Application Conclusion Double resonance island For resonances of type 1: k , a range of parameters allows two resonance islands = ⇒ three possible oscillating types for (Σ , σ ) Disappearance of one island during the secular evolution : ( U 1 , u 1 ) ( U 2 , u 2 ) ( U 3 , u 3 ) Σ = √ µa/k 0 π 2 π 0 π 2 π 0 π 2 π σ (rad) σ (rad) σ (rad) Corresponding secular model : frequent separatrix crossings long time-scale chaos Melaine Saillenfest A secular representation for the long-term resonant dynamics beyond Neptune
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