Dynamics of Saturn's small moons in coupled first order planar resonances Maryame El Moutamid Bruno Sicardy and St´efanRenner LESIA/IMCCE | Paris Observatory 26 juin 2012 Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Saturn system Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Very small moons Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk New satellites : Anthe, Methone and Aegaeon (Cooper et al., 2008 ; Hedman et al., 2009, 2010 ; Porco et al., 2005) Very small (0.5 km to 2 km) Vicinity of the Mimas orbit (outside and inside) The aims of the work A better understanding : - of the dynamics of this population of news satellites - of the scenario of capture into mean motion resonances Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Dynamical structure of the system µ µ´ Mp We consider only : - The resonant terms - The secular terms causing the precessions of the orbit When µ ! 0 ) The symmetry is broken ) different kinds of resonances : - Lindblad Resonance - Corotation Resonance D'Alembert rules : 0 0 c = (m + 1)λ − mλ − $ 0 L = (m + 1)λ − mλ − $ Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Corotation Resonance - Aegaeon (7/6) : c = 7λMimas − 6λAegaeon − $Mimas - Methone (14/15) : c = 15λMethone − 14λMimas − $Mimas - Anthe (10/11) : c = 11λAnthe − 10λMimas − $Mimas Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Corotation resonances Mean motion resonance : n1 = m+q n2 m Particular case : Lagrangian Equilibrium Points Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Adam's ring and Galatea Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Corotation resonances : Pendular motion 8 χ_ = − sin( ) 8 χ = 3 m a−ac < c c c < c 2 a where : : _ : = 3m2 Ms a Ee c = χc c Mp ac s 3 2 1 c J 0 -1 -2 -3 -200 -150 -100 -50 0 50 100 150 200 sc (deg) Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Lindblad Resonance 8 h_ = + @H = −(J − J + D)k < @k C L (1) : _ @H k = − @h = +(JC − JL + D)h + "L 8 < h / e cos( L) e=(h,k) : the vector eccentricity : k / e sin( L) e2 JL / 2 2 HAnd = JL − (Jc + D)JL − "Lh Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Dynamical study Equations of motions ) The Coralin (CORotation And LINdblad) Model 0 0 c = (m + 1)λ − mλ − $ near _c = 0 0 L = (m + 1)λ − mλ − $ near _L = 0 The coupling leads to a chaotic motion Asymptotic cases General solutions based on adiabatic invariance arguments Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Hamiltonian formalism : The Coralin Model 8 @H J_c = − = − sin( c ) > @ c > > > @H > _c = + = JC − [JL] <> @Jc (2) > h_ = + @H = −([J ] − J + D)k > @k C L > > :> _ @H k = − @h = +([JC ] − JL + D)h + "L where the Hamiltonian H is given by : p 1 2 H(Jc ; C ; JL; L) = 2 (Jc − JL) − DJL − cos( C ) − "L 3jmje cos( L) where : 8 < h / e cos( L) e=(h,k) : the vector eccentricity : k / e sin( L) e2 JL / 2 Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk To simplify 3 Jc-JL 2 1 2 -D 0 sc -1 -2 -3 -4 -3 -2 -1 0 1 2 3 4 This study depends essentially of D which gives the distance between the two resonances, and the forcing value "L for a given value of the width of CER. D "L Anthe 0.256 -0.133 Aegaeon -1.34 0.132 Methone 0.122 -0.115 Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Case Anthe Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Probability of capturing into the Corotation Eccentric Resonance (CER) - add migration in the CoraLin model to estimate the capture probabilities into CER perturbed by LER Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Hamiltonian formalism with migration 8 J_ = − sin( ) +_a=2a > c c > > > _ <> c = JC − JL = χ (3) > _ > h = −(JC − JL + D)k > > > : k_ = +(JC − JL + D)h + "L Tav =a _=2a = α + βχ Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk constant torque : No capture Tav = α 80 60 40 20 c J 0 -20 -40 -60 -80 -200 -150 -100 -50 0 50 100 150 200 sc (deg) Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk torque with gradient : possible capture Tav = α + βχ 80 60 40 20 c J 0 -20 -40 -60 -200 -150 -100 -50 0 50 100 150 200 sc (deg) Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Torque with gradient Tav = α + βχ 50 0.005 40 0.0045 30 0.004 20 0.0035 10 0.003 0 0.0025 -10 Eccentricity 0.002 Semi-major axis -20 0.0015 -30 -40 0.001 -50 0.0005 -60 0 0 100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06 0 100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06 Time (days) Time (days) a = 197655.4 km e = 0.003 Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Case Anthe Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Case Anthe Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk Conclusions and prospects - CoraLin : simple model to investigate the dynamics of a satellite on CER/LER - Capture probabilities into CER + applications to our satellites - Anthe crosses a chaotic region favors the capture - The capture is sure in the second scenario - For Aegaeon : need a migration in the other direction. Maryame El Moutamid ESLAB-2012 | ESA/ESTEC Noordwijk.