1. Rotation des satellites naturels Applications: Lune et Europe
Nicolas Rambaux1,2,
Jim Williams2, Dale Boggs2
Ozgur Karatekin3, Tim Van Hoolst3
UPMC-Paris 6 ; IMCCE-Observatoire de Paris
Jet Propulsion Laboratory, CalTech
Royal Observatory of belgium

2. Librations = departure from a uniform rotational motion
(geometric)
Mean uniform
rotation N W E S
(59 % surface observable)

3. — Scientific Context— * Diversity of natural satellites
The Moon’s core,
Europa certainly has a subsurface ocean,
Titan has a large atmosphere,
Enceladus has water geyser at S pole...
-Rotational observations more and more accurate
Lunar Laser Ranging observations (JPL,OCA)
VLBI, GPS, SLR measurements for the Earth
Radar Interferometry (Mercury + Venus)
Space missions (Messenger, BepiColombo, Laplace…)
Interplanetary optical telemetry (LATOR,TIPO)
-Limited knowledge of the core
Physical state, size, shape of core?
Geophysical parameters, circulation of the fluid core of the Earth? *

4. Investigation — Celestial Mechanics —
ROB
Idea from Hopkins (1889)
(movie from ROB)

5. Dynamical Equations — of the rotation —

6. Dynamical Equations — of the rotation —

7. — Comparative Planetology —

8. Dynamically active Moon 
(Kaguya mission, 2007)

9. — Scientific Context — The interior of the Moon is poorly constrained
Induced magnetic field (Hood etal 1999)
Seismology (Nakamura etal 1974)
Thermal history
- Lunar Laser Ranging Experiment (Williams et. al.) - Accuracy of 2 cm and 1 mas in rotation over 38 years. - Fundamental physics, geophysics, selenophysics and interior of the Moon.
385, ,100 sin(l) km

10. Rotation of the Moon is a complex dynamical system
3-body Problem
Orbit variation of several 1000s km
Solar torque
Earth’s torque
Lunar harmonics 2,3,4
Figure-figure effects
Rotation of the ecliptic plane
Planetary perturbations
Core-mantle
couplings
Rotation variation of several 100 as
Tidal deformation

11. — Lunar Laser Ranging data fitting —
Joint numerical integration of the orbits of the Moon, the Earth, and the planets, and of the lunar rotation.
The model contains
Relativistic Earth-Moon-planet interactions, gravitational harmonics for the Moon (up to deg 4), the Earth (zonal) and Sun (J2), the tides on Earth and Moon, and a fluid lunar core.
Dynamical partial derivatives
Numerical integration of partial derivatives of the orbits and lunar Euler angles with respect to solution parameters such as initial conditions, mass ratios, gravity coefficients, and tide, core, and relativity parameters.
Conservation of the angular momentum (Euler-Liouville Equations)
N, Nc external and internal couplings
I, Ic tensor of inertia for the whole Moon, core
, c rotational velocity for the mantle, the core

12. — Dynamical signature of the core —
Mean moment of inertia (Konopliv 1998)
I/MR2 = ±
k2 Love number (Williams etal 2008)
k2 = ± 0.003
Oblateness of the Core-Mantle Boundary
Dissipation in the Moon (Williams etal 2001)
Excitation of the free librations
(Williams etal 2001)

13. — The resonance makes the difference —
Figure axis
Mean Figure axis
Rotation axis
Three modes of libration
for a solid rigid-Moon:
One mode is the wobble PW = 75.1 years
One mode in latitude PN = 80.1 years One mode in longitude PL = 2.89 years
These modes are free modes of libration and their associated dissipation timescales are:
w = years
N = years
L = 104 years
x-axis
(pointing towards Earth)
In numerical model free librations are strongly sensitive to the initial conditions.

14. — Libration angles — Beat between two frequencies 400 as 400 as 200 as
Ephemeris DE421
(Eckhardt 1981; Moons 1982)

15. — Spectrum of frequencies is dense —
For example in the libration in longitude angle 
Ellipticity of the Earth’s orbit
Earth oblateness
Planetary perturbations
Free period
Moon’s orbit
Delaunay argument

16. — Spectrum analysis — I   Number of frequencies 84 78 67
- We fit the DE421 last ephemeris of the Moon’s rotation (1070 years) with the following test function (Fourier, Poisson, Polynomial and proper frequencies): I  
Number of frequencies 84 78 67
Amplitude minimum
6 mas
30 mas

17. — Free librations — Longitude mode Latitude mode Wobble mode Period
- We determine the free frequencies and their associated amplitudes in the formal solution.
Longitude mode
Latitude mode
Wobble mode
Period
days
81.5 years
75 years
Amplitude
1.8 as
0.03 as
8.2 x 3.3 as
 Since the free librations damp with time, the observational detection of free librations requires recent excitation or continuing stimulating mechanisms.

18. — Possible excitation mechanisms —
Forced periods (Delaunay arg)
1 Spectrum of forced period close to the free period of 2.9 years
Excitation in the past due to crossing of the resonance
(Eckhardt 1993)
Planetary periods
Free period
2 Precession-driven turbulence
by eddies at the CMB
(Yoder 1981)
Unlikely scenarios
3 Moonquake
4 Meteorite Impacts

19. Centrifugal libration and thickness  of the icy shell for Europa

20. Internal ocean? librations Indirect evidence:
induced magnetic field (Kivelson etal 2001)
theoretical models of the interior structure and thermal evolution (Sotin etal 2002; Hussman etal 2006)
Prime objective of plans for future space missions to the Jupiter system.
Measurements techniques:
tides (surface displacements + changes in the gravity field) (Wu etal 2001; Wahr 2006; Castillo etal 2000)
Magnetic field
librations

21. — Equation of the rigid rotation—
Proper frequency
Orbital Forced amplitude
Dominant parameter: (B-A)/C

22. — Three-layer Equations —
Orbital Forced amplitude 

23. — Librations of Europe —
Libration in longitude
of 3.55 days
Rigid case
Free period
Internal coupling
Study influence of a subsurface ocean: whole body libration or shell libration

24. — Interior models — 104 103 102 C/CS 10 1
Internal structure models with 3 or 4 layers
Satisfy mass, radius and moment of inertia

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

26. Libration in longitude — for uncoupled shell —
250 kilometers
Various interior models

27. Libration in longitude — for uncoupled shell —
250 kilometers
Resonance
between free
period and
orbital period
Various interior models

28. — Internal gravitational coupling —
Jupiter
Europa
: density,  equatorial oblateness,
A, B inertia moment, G gravitational constant.
(Van Hoolst, Rambaux, Karatekin, Dehant, Rivoldini, 2008, Icarus)

29. — Thickness of the icy shell —
Several internal structure models
25 m
~10 m
Rigid case
133 meters
Observations of libration amplitude can be used to constrain the thickness of the icy shell.
(Van Hoolst, Rambaux, Karatekin, Dehant, Rivoldini, 2008, Icarus)

30. — Smaller thickness shell —
Orbital period
2  n
(Rambaux, Van Hoolst, Karatekin, 2008, submitted)

31.  Conclusions 
Lunar Laser Ranging continues to provide new results because of improving range with new station (APOLLO, New Mexico, USA) and updated station (Grasse, OCA, France) and data analysis accuracies (DE418).
We have determined the amplitudes and phases of the free librations for the three lunar modes.
The accurate determination of the amplitudes of the free librations is a motivation to understand how the presence of a (turbulent) fluid core inside the Moon could excite free modes.
Future missions dedicated to the Moon (ALGEP...) will offer improved accuracies and new results by settled new retroreflectors at the surface of the Moon.

32.  Conclusions  Without ocean: libration amplitude ~ 134 m
An ocean increases libration amplitude by 8-25%
Resonant amplification of libration for very thin icy shell (1km)
The most important interior structure parameter is the thickness of the icy shell, then densities.
Observational libration accuracy ~ 10 m with orbiter, <1m with lander
Observation of libration amplitude can constrain the thickness of the icy shell of Europa