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 ψc = (m + 1)λ − mλ − $ 0 ψL = (m + 1)λ − mλ − $ Maryame El Moutamid ESLAB-2012 — ESA/ESTEC Noordwijk Corotation Resonance - Aegaeon (7/6) : ψc = 7λMimas − 6λAegaeon − $Mimas - Methone (14/15) : ψc = 15λMethone − 14λMimas − $Mimas - Anthe (10/11) : ψc = 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

 χ˙ = − sin(ψ )  χ = 3 m a−ac  c c c  c 2 a where :  ˙   = 3m2 Ms a Ee ψc = 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

 h˙ = + ∂H = −(J − J + D)k  ∂k C L (1)  ˙ ∂H k = − ∂h = +(JC − JL + D)h + εL   h ∝ e cos(ψL) e=(h,k) : the vector eccentricity  k ∝ e sin(ψL) e2 JL ∝ 2

2 HAnd = JL − (Jc + D)JL − εLh

Maryame El Moutamid ESLAB-2012 — ESA/ESTEC Noordwijk Dynamical study

Equations of motions ⇒ The Coralin (CORotation And LINdblad) Model

0 0 ψc = (m + 1)λ − mλ − $ near ψ˙c = 0 0 ψL = (m + 1)λ − mλ − $ near ψ˙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

 ∂H J˙c = − = − sin(ψc )  ∂ψc    ∂H  ψ˙c = + = JC − [JL]  ∂Jc (2)  h˙ = + ∂H = −([J ] − J + D)k  ∂k C L    ˙ ∂H k = − ∂h = +([JC ] − JL + D)h + εL where the Hamiltonian H is given by : √ 1 2 H(Jc , ψC , JL, ψL) = 2 (Jc − JL) − DJL − cos(ψC ) − εL 3|m|e cos(ψL) where :   h ∝ e cos(ψL) e=(h,k) : the vector eccentricity  k ∝ e sin(ψL) 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 εL for a given value of the width of CER.

D εL 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  J˙ = − sin(ψ ) +˙a/2a  c c    ˙  ψc = JC − JL = χ (3)  ˙  h = −(JC − JL + D)k     k˙ = +(JC − JL + D)h + εL

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