Librations of Outer Solar System Satellites

1. Librations of Outer Solar System Satellites O. Karatekin, T. Van Hoolst, N. Rambaux, Royal Observatory of Belgium AOGS, 2 AUG 2007

2. Introduction Major satellites are in synchronous spin-orbit resonance Mean uniform rotation Eccentricity (e) > 0 (radial and optical tides) Eccentricity (e) = 0

3. Librations Librations are oscillations around the mean uniform rotation

4. Librations • Physical librations in longitude (Torques in the equatorial plane & changes in the spin rate) • Libration amplitude is • directly proportional to equatorial flattening, (B-A) (~ C22) & eccentricity, e. • inversely proportional to polar moment of inertia, C. (Solid interior (g~1/C), decoupled mantle/shell (g~1/Cs)) Observed Librations of: Moon (Eckhardt 1981) Earth (Chao et al.1991) Mercury (Margot et al. 2007) Internal structure

5. Outer Solar System Satellites • Prime objective of future space missions to the Jupiter and Saturn system is to characterize internal structure of their satellites . • Measurements of: • Gravity field • Surface displacements • Magnetic field • Librations What are the expected Libration amplitudes? How the presence of subsurface oceans will affect the librations?

6. Librations of Europa Libration in longitude of 3.55 days Free period Two frequencies : forced frequency (2 /orbital period = n) free frequency

7. Libration in longitude for a rigid Europa (no ocean) Analytical solution of longitudinal librations for a solid body (linearized equations, Van Hoolst et al. 2007 submitted, Comstocks and Bills 2003) From C22 measurements

8. 104 103 102 10 1 Subsurface Ocean of Europa:1- Uncoupled Shell relative moment of inertia of the icy shell: C/Cs C/Cs

9. Libration of Europa for an uncoupled shell 250 kilometers s~134m Various interior models

10. Librations of Europafor an uncoupled shell 250 kilometers Resonance between free period and orbital period Various interior models

11. Subsurface Ocean of Europa:2- Internal gravitational coupling Jupiter Europa Differential librations of internal layers

12. Interior models for EuropaInterior flattening • Internal structure models with 3 or 4 layers • Satisfy mass, radius and moment of inertia • Solve Clairaut’s equation

13. Librations of Europa Amplitude depends on ice shell thickness h. It is 6% to 11% larger than for models without ocean,much smaller than de-coupled models Several internal structure models ~7-8 m ~10 m s~134 m

14. Resonant Librations of Europa Resonant amplification of the librations occurs when the free libration period is close to the orbital period

15. Outer Solar System Satellites Amplitudes of solid libration (No Ocean) Arbitrary

16. Enceladus Amplitudes of shell librations with/without ocean (gravitational coupling is on)

17. Callisto Amplitudes of shell librations with/without ocean (gravitational coupling is on)

18. Titan Amplitudes of shell librations with/without ocean (gravitational coupling is on)

19. Conclusions • Librations of outer solar system satellites show similarities. • Libration amplitudes depends on internal mass distribution (B-A)/C. • A subsurfaceoceanwouldincreases libration amplitude only by 10% (not >100 %) unless there is resonance. • The most important interior structure parameter is the thickness of the icy shell. • To constrain the thickness of the icy shell observational libration accuracy depends on the satellite <10 m is needed for Europa (possible with an orbiter, <1m with lander), <1m for Callisto. • Resonant amplification of libration possible (The dissipation due to librations may become significant for resonant case). • Other factors that will cause oscillations around mean rotation: Dynamical Tides (direct & indirect ), Ocean dynamics (LOD, non-equilibrium tides, internal waves).

20. Dissipation What about resonance ??

21. Surface Displacement (No resonance)

22. Interior Structure • Tides stretch satellites in the direction of the planet, and rotation flattens it in the poles. • Mass and Radius are known • If J2 and/or C22 are measured --> (B-A)/C • If J2 and/or C22 are NOT measured --> Equilibrium figure (C22) under the perturbing tidal and rotational potentials for a assumed density variation. • Internal flattening--> Clairaut’s equation.

23. RIGID ROTATIONS

24. Future Missions librations

25. Libration in longitude for a rigid Europa (no ocean) 133.92 +/- 0.25 meters Various interior models

26. Librations in longitude Amplitude 8% to 25% larger than for models without ocean Exponential Linear Rotational reaction to Jupiter’s torque on icy shell  (Bs-As)/Cs Resonance with free mode Resonance with free mode

27. Thickness of the icy shell 25 m Several internal structure models ~10 m Observations of libration amplitude can be used to constrain the thickness of the icy shell.

28. 104 103 102 10 1 Dynamical parameters of Europa • Equatorial flattening: B-A (C22) • Gravitational torque that causes the libration is proportional to that flattening • Equatorial flattening due to tides • Polar moment of inertia (rotational reaction  1/C) Main parameter: (B-A)/C Inverse relative moment of inertia of the icy shell: C/Cs

29. Implications for Future Measurements • Librations can be determined from gravity field, altimetry measurements • Accuracy to constrain the thickness of the ice shell depends on the satellite (<10 m for Europa, <1 m Callisto). But there are also other factors that will cause surface displacement (about same order of magnitude !): Zonal Tides (direct & indirect ), Ocean (LOD, Ocean dynamics (non-equilibrium tides, internal waves))