Rotation of a rigid satellite with a fluid component: a new light onto Titan’s obliquity

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Rotation of a rigid satellite with a fluid component: a new light onto Titan’s obliquity

Gwenaël Boué, Nicolas Rambaux, Andy Richard

To cite this version:

Gwenaël Boué, Nicolas Rambaux, Andy Richard. Rotation of a rigid satellite with a fluid component:

a new light onto Titan’s obliquity. Celestial Mechanics and Dynamical Astronomy, Springer Verlag,

2017, 129 (4), pp.449 - 485. �10.1007/s10569-017-9790-8�. �hal-01634864�

(will be inserted by the editor)

Rotation of a rigid satellite with a fluid component

A new light onto Titan’s obliquity

Gwena¨ el BOU ´ E ¹ · Nicolas RAMBAUX ¹ · Andy RICHARD ^1,2

Received: date / Accepted: date

Version: September 21, 2017

Abstract We revisit the rotation dynamics of a rigid satellite with either a liquid core or a global sub-surface ocean. In both problems, the flow of the fluid com- ponent is assumed inviscid. The study of a hollow satellite with a liquid core is based on the Poincar´ e-Hough model which provides exact equations of motion. We introduce an approximation when the ellipticity of the cavity is low. This simpli- fication allows to model both types of satellite in the same manner. The analysis of their rotation is done in a non-canonical Hamiltonian formalism closely related to Poincar´ e’s “forme nouvelle des ´ equations de la m´ ecanique”. In the case of a satellite with a global ocean, we obtain a seven-degree of freedom system. Six of them account for the motion of the two rigid components, and the last one is as- sociated with the fluid layer. We apply our model to Titan for which the origin of the obliquity is still a debated question. We show that the observed value is com- patible with Titan slightly departing from the hydrostatic equilibrium and being in a Cassini equilibrium state.

Keywords multi-layered body · spin-orbit coupling · Cassini state · synchronous rotation · analytical method · Io · Titan

1 Introduction

The spin pole of Titan, Saturn’s largest moon, is lying close to the plane defined by its orbit pole and the Laplace pole (Stiles et al., 2008, 2010). This observation,

Gwena¨ el Bou´ e

E-mail: Gwenael.Boue@obspm.fr Nicolas Rambaux

E-mail: Nicolas.Rambaux@obspm.fr Andy Richard

E-mail: Andy.Richard@universcience.fr

1

IMCCE, Observatoire de Paris, UPMC Univ. Paris 6, PSL Research University, Paris, France

2

Universcience, Palais de la d´ ecouverte, Paris, France

made by the RADAR instrument of the Cassini mission, suggests that Titan is in (or very close to) a Cassini state (Colombo, 1966; Peale, 1969). For a rigid body, the equilibrium obliquity is a function of its moments of inertia. Those of Titan have been deduced from its Stokes coefficients J 2 = (33.599 ± 0.332) × 10

⁻⁶

and C 22 = (10.121 ± 0.029) × 10

⁻⁶

and from the hydrostatic equilibrium hypothesis implying a mean moment of inertia I/(mR ² ) = 0.3431 (Iess et al., 2012, SOL1a), where m and R are the mass and radius of Titan, respectively. The assumed hydrostatic equilibrium is suggested by the ratio J 2 /C 22 ≈ 10/3 which is precisely the expected value for a hydrostatic body (e.g., Rappaport et al., 1997). Assuming these values, if Titan were rigid and in a Cassini equilibrium state, its obliquity would be 0.113 deg (Bills and Nimmo, 2011), i.e. about one third of the radiometric value 0.32 deg (Stiles et al., 2008, 2010; Meriggiola et al., 2016). To match the observations, the frequency of the free libration in latitude must be reduced by a factor 0.526 (Bills and Nimmo, 2011). In particular, this would be the case if I/(mR ² ) were increased to 0.45 (ibid.), a value exceeding 2/5 obtained for a homogeneous body, as if the mass of the satellite was concentrated toward the surface. This result leads to think that the observed obliquity is that of a thin shell partially decoupled from the interior by, e.g., a global ocean (ibid.).

The idea that the ice-covered satellites of the outer planets hold a global under- neath ocean has already been proposed based on models of their internal structures (e.g., Lewis, 1971). Even the dwarf planet Pluto is suspected to harbour a subsur- face ocean (Nimmo et al., 2016). In the case of Titan, the presence of the ocean is also revealed by laboratory experiments on the behaviour of water-ammonia compounds at high pressure and low temperature (Grasset and Sotin, 1996), by the detection of electromagnetic waves in its atmosphere (B´ eghin et al., 2012) and by the high value of its Love number k 2 (Iess et al., 2012).

A dynamical problem closely related to the present one is that of a hollow satellite with a liquid core as described by the Poincar´ e-Hough model (Poincar´ e, 1910; Hough, 1895). For this specific problem, Poincar´ e (1901) developed a new Lagrangian formalism, based on the properties of the Lie group acting on the configuration space, which allows to derive the equations of motion in a very simple and elegant manner. Such a system is characterised by four degrees of freedom, three of them being associated with the rotation of the rigid mantle and the last one being due to the motion of the liquid core (e.g., Henrard, 2008).

Applying this model to Jupiter’s satellite Io, Henrard (2008) observed that the frequency of the additional degree of freedom is close to the orbital frequency and should thus multiply the possibility of resonances. For Titan, we shall expect the same conclusion due to the presence of the ocean, but unfortunately, Poincar´ e’s model relies on the concept of a fluid simple motion which cannot be rigorously transposed to the case of a satellite with a global subsurface ocean.

In the case of Titan, the effect of an ocean on the rotation dynamics has

been studied numerically using Euler’s rotation equations taking into account the

gravitational interaction of Saturn on each layer, the pressure torques at the two

fluid-solid boundaries, and the gravitational coupling between the interior and the

shell (Baland et al., 2011, 2014; Noyelles and Nimmo, 2014). The elastic deforma-

tion of the solid layers and the atmospheric pressure have also been included in a

modelling of the libration in longitude (Richard et al., 2014) and in a modelling

of the Chandler polar motion (Coyette et al., 2016). Despite several arguments in

favour of an ocean, this model does not easily explain the tilt of Titan’s spin-axis.

Indeed, under the hydrostatic equilibrium hypothesis, Baland et al. (2011) and Noyelles and Nimmo (2014) found that the obliquity of the Cassini state remains bounded below 0.15 deg, i.e. about one half of the observed value. There thus seemed to be a need for a significant resonant amplification to bring the system out of the Cassini equilibrium (Baland et al., 2011; Noyelles and Nimmo, 2014).

However these studies do not invoke the same mode as the origin of the resonant amplification. In addition, this solution does not agree with extended observations of the spin-axis orientation (Meriggiola and Iess, 2012). The model has then been amended to allow the Cassini state obliquity to reach the observed 0.32 deg, but this has only been made possible after releasing the hydrostatic shape assumption leaving the ratio J 2 /C 22 ≈ 10/3 unexplained (Baland et al., 2014).

It should be stressed that models developed thus far discard the rotation of the ocean relative to the inertial frame. This is a valid assumption to reproduce libra- tions in longitude (e.g., Richard, 2014), but not anymore for precession motion.

By consequence, the associated dynamical system only has 6 degrees of freedom equally shared by the rigid interior and the shell (Noyelles and Nimmo, 2014). Yet, a comparison of this problem with that of a satellite with a liquid core strongly suggests that a three layered body must have 7 degrees of freedom, one of which being brought by the ocean. Here, we aim at building a new dynamical model accounting for the rotation of the liquid layer as done by Mathews et al. (1991) for the Earth. More recently, the latter model has been adapted to the study of the Moon (Dumberry and Wieczorek, 2016)

¹

and of Mercury (Peale et al., 2016).

Here we reconsider the problem with a Hamiltonian approach. In that scope, we first extend the Lagrangian formalism described in (Poincar´ e, 1901) to a non- canonical Hamiltonian formalism allowing to study relative equilibria in a very efficient manner as in (Maddocks, 1991; Beck and Hall, 1998). The method has proven its efficiency in the context of a rigid satellite in circular orbit (Beck and Hall, 1998), in the analysis of the two rigid body problem (Maciejewski, 1995), and in several studies of the attitude of a satellite with a gyrostat (e.g., Hall and Beck, 2007; Wang and Xu, 2012, and references therein). The approach is described in Sect. 2 and illustrated in the case of a rigid satellite in Sect. 3. We revisit the problem of a moon with a fluid core with this approach and we propose a sim- plification straightforwardly transposable to a three layered body in Sect. 4. The rotation dynamics of a satellite with a subsurface ocean is presented in Sect. 5. In the subsequent section 6, we test our model and our simplification on Io, a satel- lite with a liquid core, verifying that the derived eigenfrequencies are in very good agreement with those obtained in previous studies of the same problem made by Noyelles (2013, 2014). In this section, we also analyse the case of Titan showing that the additional degree of freedom makes the system highly sensitive to the in- ternal structure and that the observed obliquity can be easily reproduced. Finally, we discuss our model and conclude in Sect. 7. The notation used in this paper is explained in Tab. 1.

1

Dumberry and Wieczorek (2016) could only highlight 5 degrees of freedom because their

model of the Moon is axisymmetric and not triaxial.

2 Non-canonical Hamiltonian formalism

2.1 Equations of motion 2.1.1 General case

Let a dynamical system with n degrees of freedom described by a Lagrangian L. We denote by Q the configuration space and each point q ∈ Q is represented by a set of m ≥ n coordinates (q 1 , · · · , q m ). The number of coordinates is purposely allowed to be greater than the actual dimension of the manifold Q. As in (Poincar´ e, 1901), we assume that there exists a transitive Lie group G acting on Q. The transitivity of G means that for all q, q

⁰

∈ Q, there exists an element g of the group G such that q

⁰

= gq. In particular, given an initial condition q 0 , there exists g t ∈ G such that the configuration q(t) at time t reads q(t) = g t q 0 . In this work, G will be the rotation group SO(3), the translation group T(3), or some combinations of both.

Let g be the Lie algebra of G. By definition, there exists X ∈ g such that the generalised velocity reads ˙ q = X(q). Since the action of G on Q is transitive, the dimension of g is equal to the number n of degrees of freedom. Let B = (X 1 , · · · , X n ) be a basis of g and (X ij ) 1≤i≤n,1≤j≤m be the n × m functions of q defined as

X i =

m

X

j=1

X ij ∂

∂q j

. (1)

We denote by η = (η 1 , · · · , η n ) ∈ R ⁿ the coordinates of X in B such that

˙ q =

n

X

i=1

η i X i (q) . (2)

Because the term “generalised velocity” is already attributed to ˙ q, hereafter we call η the Lie velocity of the system. Given two configurations q and q

⁰

infinitely closed to each other, we also define the n-tuple δξ = (δξ 1 , · · · , δξ n ) such that

δq := q

⁰

− q =

n

X

i=1

X i (q)δξ i . (3)

Poincar´ e considers the Lagrangian as a function of (η, q) and writes its infinitesi- mal variation as

δL =

n

X

i=1

∂L

∂η i

δη i + X i (L)δξ i . (4)

The resulting equations of motion are (Poincar´ e, 1901) d

dt

∂L

∂η i

= X

j,k

c ^k _ij η j

∂L

∂η k

+ X i (L), (5)

where c ^k _ij , defined as

[X i , X j ] := X i X j − X j X i =

n

X

k=1

c ^k _ij X k , (6)

are the structure constants of g with respect to the chosen basis B.

To get the Hamiltonian equations equivalent to Eq. (5), we introduce a mo- mentum π associated with the Lie velocity η, and defined as

π := ∂L

∂η . (7)

Following the same nomenclature as for η, we call this momentum π the Lie mo- mentum of the system. The Hamiltonian H is constructed by means of a Legendre transformation as

H(π, q) := π · η − L(η, q). (8) Using Eqs. (4) and (7), the infinitesimal variation of H (Eq. 8) reads

δH =

n

X

i=1

η i δπ i − X i (L)δξ i . (9)

But since H is a function of π and q, we also have, as in Eq. (4), δH =

n

X

i=1

∂H

∂π i

δπ i + X i (H)δξ i . (10)

The identification of Eqs. (9) and (10) gives η i = ∂H

∂π i

and X i (H ) = −X _i (L). (11)

Using these identifications, the expression of ˙ q (Eq. 2), and Poincar´ e’s equation (5) where ∂L/∂η i is replaced by π i (Eq. 7), we get the non-canonical equations of motion associated with H, viz.,

˙ q i =

n

X

j=1

∂H

∂π j

X j (q i ) and π ˙ i = X

j,k

c ^k ij

∂H

∂π j

π k − X i (H ). (12)

Let us denote the state vector by y = (π, q) ∈ R ^n+m . The equations of motion (12) written in matrix form read

˙

y = −B(y)∇

y

H. (13)

The so-called Poisson matrix B(y) is B(y) =

C X

−X ^T 0

(14) where (·) ^T means the transpose of a vector or of a matrix. X is an n × m matrix and C an n × n matrix whose elements are

[X] ij = X ij and [C] ij = − X

k

c ^k _ij π k . (15)

2.1.2 Translation group

The simplest illustration of the above formalism is the case where G is the trans- lation group. In that case, η is the usual velocity vector v and π is the standard linear momentum, commonly denoted p. The vector fields of the tangent configu- ration space are X i = _∂q ^∂

i

. The associated structure constants c ^k ij are all nil. The Poisson matrix is then

B(y) = 0 1

−1 0

(16) and we retrieve the canonical equations of motion

˙

p i = − ∂H

∂q i

, q ˙ i = ∂H

∂p i

. (17)

2.1.3 Group SO(3) in the body-fixed frame

The group SO(3) naturally appears in studies of the rotation motion of solid bodies. For this problem, two choices can be made: vectors are expressed either in the body-fixed frame or in the “laboratory” frame. Here, we consider the first option where vectors are written in the body-fixed frame. The Lie velocity is the rotation vector designated by ω and the orientation of the body is parametrised by the coordinates in the body-fixed frame of the laboratory base vectors, i.e., q = (i, j, k). For any function f (i, j, k), we have

d

dt f (i, j, k) = −(ω × i) · ∂f

∂i − (ω × j) · ∂f

∂j − (ω × k) · ∂f

∂k

= −ω ·

i × ∂f

∂i + j × ∂f

∂j + k × ∂f

∂k

.

(18)

Thus, the vector field X = (X 1 , X 2 , X 3 ) is X = −i × ∂

∂i − j × ∂

∂j − k × ∂

∂k , (19)

with structure constants c ^k _ij = − ijk where ijk = 1 when (i, j, k) is a cyclic permu- tation of (1, 2, 3), -1 when (i, j, k) is a cyclic permutation of (3, 2, 1), 0 otherwise.

Hence, the Poisson matrix reads

B = −







 π ˆ ˆ ı ˆ  ˆ k

ˆ ı 0 0 0 ˆ  0 0 0 k 0 0 0 ˆ









(20)

where for any vector v, we have defined

ˆ v =





0 −v _z v y

v z 0 −v x

−v y v x 0



 . (21)

The corresponding equations of motion are dπ

dt = π × ∂H

∂π + i × ∂H

∂i + j × ∂H

∂j + k × ∂H

∂k , (22)

di

dt = i × ∂H

∂π , (23)

dj

dt = j × ∂H

∂π , (24)

dk

dt = k × ∂H

∂π (25)

with ∂H/∂π = ω.

2.1.4 Group SO(3) in the laboratory frame

Here we again consider the rotation motion of a solid body but now vector coordi- nates are written in the laboratory frame. The latter is the frame with respect to which the motion of the spinning body is described. Note that it does not have to be inertial. The generalised coordinates are the base vectors of the rotated frame q = (I, J, K) and the Lie momentum associated with the rotation vector is denoted Π. Applying the same method as above, we get

X = I × ∂

∂I + J × ∂

∂J + K × ∂

∂K . (26)

For this basis, the structure constants are c ^k ij = ijk and thus, the Poisson matrix is

B =









Π ˆ ˆ I J ˆ K ˆ ˆ I 0 0 0 J 0 0 0 ˆ K 0 0 0 ˆ









. (27)

The associated equations of motion are dΠ

dt = ∂H

∂Π × Π + ∂H

∂I × I + ∂H

∂J × J + ∂H

∂K × K, (28)

dI dt = ∂H

∂Π × I, (29)

dJ dt = ∂H

∂Π × J, (30)

dK dt = ∂H

∂Π × K (31)

where ∂H/∂Π still is the rotation vector, although expressed in the laboratory

frame.

2.2 Linearisation and driven solution

For the sake of completeness, we here recall the general method leading to the lin- earisation of the equations of motion in the non-canonical Hamiltonian formalism (Maddocks, 1991; Beck and Hall, 1998). We also present the criterion of nonlinear stability as described in ibid.

Let a non-autonomous Hamiltonian H (y, t) associated with an n degrees of freedom system expressed as a function of non-canonical variables y ∈ R ^p with p ≥ 2n. We assume that H(y, t) can be split as follows

H(y, t) = H 0 (y) + H 1 (y, t), (32) where H 0 (y) is the autonomous part of H(y, t) and H 1 (y, t) a small perturbation.

Let us skip the perturbation H 1 for a moment. The equations of motion associated with H 0 (y) are of the form

˙

y = −B(y)∇

y

H 0 (y). (33) The system has n degrees of freedom, its phase space Σ is thus a manifold of dimension 2n. Since y ∈ R ^p , there exists s = p − 2n Casimir functions C i (y) and s constants c i , 1 ≤ i ≤ s, such that

Σ = {y ∈ R ^p : C 1 (y) = c 1 , . . . , C s (y) = c s }. (34) We recall that Casimir functions are constants of the motion for any Hamiltonian because their gradients constitute a basis of the kernel of the Poisson matrix:

ker B(y) = span {∇

y

C 1 (y), . . . , ∇

y

C s (y)} , (35) and thus

C ˙ i (y) = (∇

y

H 0 ) ^T B(y)∇

y

C i = 0 (36) for all Hamiltonian H 0 .

Let y e be an equilibrium, i.e., a fixed point of H 0 . According to Eq. (33),

˙

y _e = 0 implies ∇

_y

H 0 (y e ) ∈ ker B(y e ). Thus, there exists s coefficients (µ i ) 1≤i≤s

such that

∇

_y

H 0 (y e ) =

s

X

i=1

µ i ∇

_y

C i (y e ). (37) Let

F (y) = H 0 (y) −

s

X

i=1

µ i C i (y). (38)

By construction, F satisfies ∇

_y

F (y e ) = 0. Coefficients µ i can be seen as Lagrange multipliers and functions C i (y) as constraints since we search for an extremum of H 0 (y) under the conditions C i (y) = c i . The p + s equations ∇

y

F (y e ) = 0 and C i (y e ) = c i allow to determine y e and the coefficients µ i .

Once y e and coefficients µ i are known, the linearisation of the equations of motion (Eq. 33) are given by

δ y ˙ = A(y e )δy (39)

I

K ω

Rigid satellite

I

K ω

ω

⁰_c

liquid core mantle Satellite with a liquid core

I

_s

K

s

I

c

K

c

ω

o

ω

c

ω

s

interior ocean

shell

Satellite with a subsurface ocean Fig. 1 Rigid satellites are characterised by their basis vectors (I, J, K) and their rotation vector ω with respect to the laboratory frame. The same vectors are used for satellites with a liquid core, but the angular speed ω

⁰_c

of the core with respect to the mantle is also specified.

In the case of a satellite with a global ocean, all vectors are expressed in the laboratory frame. These are the basis vectors of the shell (I

s

, J

s

, K

s

) and of the interior (I

c

, J

c

, K

c

), and the rotation vectors ω

c

, ω

o

, ω

s

associated with the central region, the ocean and the shell, respectively.

with δy = y − y e and (Maddocks, 1991)

A(y e ) = −B(y e )∇ ²

_y

F (y e ). (40) In a last step, the perturbation H 1 (y, t) is taken into account and the equations of motion become

δ y ˙ − A(y e )δy = z(t), (41) with

z(t) = −B(y _e )∇

_y

H 1 (y e , t). (42) Equation (41) is then solved using standard techniques.

The relative equilibria y = y e is said to be nonlinearly stable if the quadratic form (or Lyapunov function) N(y) = y ^T Ny, defined on the phase space Σ by its Hessian (below), is a strictly convex function (Beck and Hall, 1998). The Hessian of N(y) is given by (see ibid.)

N := ∇ ² N = Q(y e )∇ ² F (y e )Q(y e ), (43) where Q(y) is the orthogonal projection matrix onto the range of A(y),

Q(y) = 1 − K(y)(K ^T (y)K(y))

⁻¹

K ^T (y), (44) and where K(y) is a p × s matrix given by

K(y) =

∇C 1 (y) · · · ∇C s (y)

. (45)

Table 1 Notations

symbol definition

c

,

o

,

m

,

s

indices standing for Core, Ocean, Mantle, and Shell, respectively

rs

,

fc

,

go

indices standing for Rigid Satellite, Fluid Core, and Global Ocean F

in

= (i

0

, j

0

,k

0

) inertial frame

F

_lab

= (i, j,k) laboratory frame

F

_i

= (I

i

, J

i

,K

i

) frame associated with the layer i

Ω rotation vector of F

_lab

with respect to F

_in

expressed in F

_lab

ω

i

rotation vector of F

_i

with respect to F

_lab

expressed in F

_lab

ω

⁰_c

rotation vector of F

_c

with respect to F

_m

expressed in F

_m

Π

i

Lie momentum associated with ω

i

Π

⁰_c

Lie momentum associated with ω

⁰_c

I

i

, J

i

, K

i

basis vectors of F

_i

expressed in F

_lab

R

i

= [I

i

, J

i

, K

i

] rotation matrix of the layer i relative to F

_lab

y

i

= (Π

i

,I

i

,J

i

, K

i

) state vector of the layer i

y state vector of the whole system

T (y) kinetic energy

U(y, t) potential energy L(y, t) Lagrangian H(y, t) Hamiltonian

C

i

(y) Casimir functions

µ

i

Lagrange multipliers

F (y) Lagrangian associated with the minimisation of H

0

with constraints

B(y) Poisson matrix

A(y) matrix of the linearised system U

0

(y) constant part of U(y, t) U

1

(y, t) perturbation U(y, t) − U

0

(y)

U

self

(y) self gravitational energy of the satellite (u

ij

)

i,j∈{x,y,z}

constant parameters of U

self

H

0

(y) autonomous part of H(y, t) H

1

(y, t) perturbation H(y, t) − H

0

(y)

r, r(t) radius vector connecting the satellite barycenter to the planet S(t) GM

p

rr

^T

/r

⁵

S

0

constant part of S(t)

S

1

(t) S(t) − S

0

(σ

_uv⁰

)

u,v∈{x,y,z}

elements of the matrix S

0

(σ

_uv¹

(t))

u,v∈{x,y,z}

elements of the matrix S

1

(t)

G gravitational constant

M

p

mass of the central planet

α

i

, β

i

, γ

i

(C

i

− B

i

)/A

i

, (C

i

− A

i

)/B

i

, (B

i

− A

i

)/C

i

, respectively

ρ

i

density of the layer i

a

i

, b

i

, c

i

radii of the outer boundary of the layer i

ζ equatorial flattening (a − b)/a

I

i

inertia tensor of the layer i expressed in F

lab

I

⁰

ancillary inertia tensor

A

i

, B

i

, C

i

principal moments of inertia of the layer i A

⁰

, B

⁰

, C

⁰

ancillary moments of inertia

ω

u

frequency of libration in longitude ω

v

frequency of libration in latitude

ω

w

wobble frequency

3 Rigid satellite

Let a rigid satellite whose rotation is close to the synchronous state, i.e., whose mean rotation rate is equal to the orbital mean motion. The goal of this section is to compute the frequencies associated with the free modes of rotation, to evaluate the forced obliquity driven by the orbital precession, and eventually to check the nonlinear stability of the system in the vicinity of the equilibrium. The analysis is performed using the non-canonical Hamiltonian formalism described in Sect. 2. It turns out to be convenient to describe the problem in a laboratory frame rotating at constant angular speed Ω with respect to the inertial frame. Ω will then be chosen equal to the mean orbital motion. We denote by ω the rotation vector of the satellite with respect to the laboratory frame F _lab and by (I, J, K) its principal axes of inertia such that the matrix of inertia reads

I = R diag(A, B, C)R ^T , (46)

where R = [I, J, K] is the rotation matrix of the satellite with respect to the labo- ratory frame and where (.) ^T denotes the transpose operator. Note that the matrix of inertia can also be written in a equivalent form facilitating the computation of the gradient of the forthcoming Hamiltonian

I = AII ^T + BJJ ^T + C KK ^T . (47) The Lie velocity of the system is thus ω while (I, J, K) are the generalised coordi- nates. We also denote by (i, j, k) the basis vectors associated with the laboratory frame. The radius vector connecting the planet and the satellite barycenter is as- sumed to be a known function of time and is denoted either by r(t) or simply by r. G and M p are the gravitational constant and the mass of the planet, respec- tively. With these notations, the (non-autonomous) Lagrangian L rs (ω, I, J, K, t) governing the rotation of the rigid satellite is

L rs (ω, I, J, K, t) = (ω + Ω) ^T I (ω + Ω)

2 − 3GM p

2 r ^T I r

r ⁵ . (48) The Lie momentum Π associated with ω reads

Π = ∂L rs

∂ω = I (ω + Ω). (49)

We recognise the spin angular momentum of the satellite with respect to the iner- tial frame and expressed in the laboratory frame. The Hamiltonian H rs (Π, I, J, K, t) resulting from the Legendre transformation applied to L rs (ω, I, J, K, t) reads

H rs (Π, I, J, K, t) = Π ^T I

⁻¹

Π

2 − Ω ^T Π + 3GM p

2 r ^T I r

r ⁵ (50)

with

I

⁻¹

= II ^T A + JJ ^T

B + KK ^T

C . (51)

The Poisson matrix B rs (y) associated with y = (Π, I, J, K) is the one given in Eq. (27). The gradient of the Hamiltonian reads

∂H rs

∂Π = I

⁻¹

Π − Ω = ω, (52)

∂H rs

∂I = (I · Π)

A Π + 3 GM p

r ⁵ A(r · I)r, (53)

∂H rs

∂J = (J · Π)

B Π + 3 GM p

r ⁵ B(r · J)r, (54)

∂H rs

∂K = (K · Π)

C Π + 3 GM _p

r ⁵ C(r · K)r, (55) and thus the equations of motion are

Π ˙ = Π × Ω − 3 GM p

r ⁵ ( I r) × r, (56)

I ˙ = ω × I, (57)

J ˙ = ω × J, (58)

K ˙ = ω × K. (59)

Equations of motion (Eqs. 56-59) are those of the full Hamiltonian. Because r(t) is a function of time, the set of equations (56-59) has no fixed point. To proceed, we set Ω = Ωk with Ω equal to the mean orbital motion such that, in the laboratory frame (i, j, k),

S(t) := GM p

rr ^T

r ⁵ = S 0 + S 1 (t) (60) where S 0 is a constant matrix and S 1 (t) a small perturbation. Furthermore, the initial angle of the rotation is chosen such that S 0 is diagonal with components (σ _xx ⁰ , σ _yy ⁰ , σ _zz ⁰ ). Similarly, we denote by σ _uv ¹ , where u, v ∈ {x, y, z}, the elements of S 1 (t). The gravitational potential energy U (y, t) is then split into U 0 (y) + U 1 (y, t) with

U 0 (y) = 3 2

AI ^T S 0 I + BJ ^T S 0 J + CK ^T S 0 K

, (61)

U 1 (y, t) = 3 2

AI ^T S 1 (t)I + BJ ^T S 1 (t)J + CK ^T S 1 (t)K

. (62)

As a result, the Hamiltonian H rs (y, t) also get split into H _rs ⁰ (y) + H _rs ¹ (y, t) with H rs ⁰ (y) = Π ^T I

⁻¹

Π

2 − Ω ^T Π + U 0 (y), (63)

H rs ¹ (y, t) = U 1 (y, t). (64)

In the case of a Keplerian orbit with eccentricity e and inclination i with respect to the reference frame,

σ ⁰ _xx = GM _p a ³

X ₀

^−3,0

(e) + X ₂

^−3,2

(e)

2 cos ⁴

i 2

+ X ₀

^−3,0

(e) 2 sin ⁴

i 2

! , (65) σ ⁰ yy = GM p

a ³

X ₀

^−3,0

(e) − X ₂

^−3,2

(e)

2 cos ⁴

i 2

+ X ₀

^−3,0

(e) 2 sin ⁴

i 2

! , (66) σ ⁰ _zz = GM p

a ³

X ₀

^−3,0

(e)

2 sin ² i, (67)

where X _k ^n,m (e) are Hansen coefficients (Hansen, 1855) defined as Fourier coeffi- cients of the series

r a

n

e ^imv =

∞

X

k=−∞

X _k ^n,m (e)e ^ikM (68) with a, v, M being the semimajor axis, the true anomaly and the mean anomaly, respectively. Besides, in this study a single element of the matrix S 1 (t) plays a role in the tilting of the Cassini state, this is the term in σ ¹ xz (t) = σ ¹ zx (t) corresponding to the first harmonic of the orbital precession in inclination whose expression is

σ ¹ _xz (t) = GM _p a ³

X ₀

^−3,0

(e)

2 cos i + X ₂

^−3,2

(e) 2 cos ² i

2

!

sin i sin(Ωt − Φ) (69) where Φ is the longitude of the ascending node. The expression of the Hansen coefficients involved in S 0 and S 1 (t) are

X ₀

^−3,0

(e) = (1 − e ² )

^−3/2

, (70)

X ₂

^−3,2

(e) = 1 − 5 2 e ² + 13

16 e ⁴ − 35

288 e ⁶ + O(e ⁸ ). (71) Following the steps recalled in the previous section 2.2, we now skip the pertur- bation S 1 (t) for a while and only retain the autonomous part of the Hamiltonian H rs ⁰ (y). The gradient of the Hamiltonian H rs ⁰ (y) reads

∂H rs ⁰

∂Π = I

⁻¹

Π − Ω = ω, (72)

∂H rs ⁰

∂I = (I · Π)

A Π + 3AS 0 I, (73)

∂H rs ⁰

∂J = (J · Π)

B Π + 3BS 0 J, (74)

∂H _rs ⁰

∂K = (K · Π)

C Π + 3C S 0 K. (75)

Only ˙ Π (Eq. 56) is affected by the averaging process. Its new equation of motion reads

Π ˙ = Π × Ω + 3A(S 0 I) × I + 3B(S 0 J) × J + 3C(S 0 K) × K. (76)

3.1 Linearisation

To perform the linearisation of Eqs. (76, 57-59), we note that the phase space Σ rs of the system is a manifold of dimension 6 (associated with the 3 degrees of freedom of the group SO(3)) defined as

Σ rs = {y ∈ R ¹² : C _rs ¹ (y) = C _rs ² (y) = C _rs ³ (y) = 1/2,

C rs ⁴ (y) = C rs ⁵ (y) = C rs ⁶ (y) = 0}, (77) where the Casimir functions are

C _rs ¹ (y) = 1

2 I · I, C _rs ² (y) = 1

2 J · J, C _rs ³ (y) = 1 2 K · K,

C _rs ⁴ (y) = J · K, C _rs ⁵ (y) = K · I, C _rs ⁶ (y) = I · J. (78)

Indeed, it can be checked that

ker B rs (y) = span



 



 









 0 I 0 0







 ,







 0 0 J 0







 ,







 0 0 0 K







 ,







 0 0 K

J







 ,







 0 K

0 I







 ,







 0 J I 0











 



 



. (79)

Let F rs (y) = H _rs ⁰ (y) − P

i µ i C _rs ⁱ (y). The condition ∇

y

F rs (y e ) = 0 leads to I

⁻¹

Π e − Ω = ω e = 0, (80) (I e · Π e )

A Π e + 3AS 0 I e − µ 1 I e − µ 5 K e − µ 6 J e = 0, (81) (J e · Π e )

B Π e + 3BS 0 J e − µ 2 J e − µ 4 K e − µ 6 I e = 0, (82) (K e · Π e )

C Π e + 3CS 0 K e − µ 3 K e − µ 4 J e − µ 5 I e = 0, (83) whose a solution is

ω e = 0, Π e = CΩk, I e = i, J e = j, K e = k, µ 1 = 3Aσ ⁰ _xx , µ 2 = 3Bσ _yy ⁰ , µ 3 = 3Cσ _zz ⁰ + CΩ ² , µ 4 = µ 5 = µ 6 = 0. (84) The other solutions are equivalent to this one but with a permutation of the moments of inertia A, B, C. The matrix A rs (y e ) of the linearised system is given by Eq. (40). To simplify the result, we perform the change of variables δy = Pδy

^∗

with

δy

^∗

=

δΠ z , δI y , δΠ x , δΠ y , δI z , δJ z , δI x , δJ y , δK z , δI y + δJ x , δI z + δK x , δJ z + δK y

T

.

(85)

The first two components of δy

^∗

are associated with the libration in longitude, the next four components describe the wobble and the libration in latitude, and finally, the last six coordinates being in the kernel of B rs (y e ) remain identically equal to zero. Let A

^∗

_rs (y e ) be the matrix of the linear system in the new variables δy

^∗

, i.e., A

^∗

_rs = P

⁻¹

A rs P, and let A ¹ _rs and A ² _rs be the respective 2 × 2 and 4 × 4 matrices such that

A

^∗

rs (y e ) =





A ¹ _rs 0 · 0 A ² _rs · 0 0 0



 (86)

where the dots · represent arbitrary matrices not influencing the motion. We have A ¹ rs =

0 −3(B − A)(σ ⁰ _xx − σ _yy ⁰ )

1/C 0

, (87)

and

A ² _rs =













0 Ω 0 3(C − B)(σ ⁰ zz − σ yy ⁰ )

−Ω 0 3(C − A)(σ ⁰ xx − σ zz ⁰ ) 0 0 − 1

B 0 − C − B

B Ω

1

A 0 C − A

A Ω 0













. (88)

Hence, the frequency of libration in longitude ω rs,u , which is the eigenvalue of A ¹ _rs , reads

ω rs,u = p

3γ (κ 1 − κ 2 ), (89)

and the frequencies associated with the wobble ω rs,w and the libration in latitude ω rs,v , the eigenvalues of A ² rs , are given by

ω rs,w = p − p p ² − 4q 2

! 1/2

, ω rs,v = p + p p ² − 4q 2

! 1/2

(90) with

p = (1 + αβ) Ω ² + 3 (βκ 1 + ακ 2 ) , (91) q = αβ

Ω ⁴ + 3 (κ 1 + κ 2 ) Ω ² + 9κ 1 κ 2

, (92)

κ 1 = σ ⁰ xx − σ ⁰ zz , (93)

κ 2 = σ ⁰ _yy − σ _zz ⁰ , (94)

and

α = C − B

A , β = C − A

B , γ = B − A

C . (95)

Here we retrieve the well-known eigenfrequencies of a rigid satellite close to the synchronous equilibrium state (e.g., Rambaux et al., 2012). Let us nevertheless stress that Eqs. (89) and (90) are associated with the motion of the three vectors (I, J, K) in the rotating frame. By consequence, if we denote by ¯ ω rs,v ≈ 3βΩ/2 the frequency of libration in latitude associated with the motion of the sole vector K with respect to the inertial frame (as it is commonly defined for an axisymmetric body), we have ω rs,v = ¯ ω rs,v + Ω.

3.2 Stability

For this problem, the Lyapunov function N rs (y), as defined in Eq. (43), is N rs (y) = 1

2A

Π x + 1

2 (C − A)Ω(I z − K x ) 2

+ 1 2B

Π y + 1

2 (C − B)Ω(J z − K y ) 2

+ 1

2C Π _z ² + 1

2 n 1 (I y − J x ) ² + 1

2 n 2 (I z − K x ) ² + 1

2 n 3 (J z − K y ) ²

(96) with

n 1 = 3

4 (B − A)(κ 1 −κ 2 ), n 2 = 1

8 (C −A)(Ω ² + 3κ 1 ), n 3 = 1

8 (C − B)(Ω ² + 3κ 2 ).

(97)

We recall that the system is nonlinearly stable if N rs (y) is a strictly convex func-

tion. Coefficients A, B, and C are positive, as required. The nonlinear stability is

then achieved when n 1 , n 2 , and n 3 are all positive. Given that κ 1 > κ 2 > 0 at low

inclination i, the criterion implies C > B > A, which is the well-known stability

condition for this classical equilibrium where the longest axis points towards the

parent planet (e.g. Beck and Hall, 1998).

3.3 Driven solution

Here we look for the forced solution when the time-dependent perturbation H _rs ¹ (t) is taken into account. In the variables δy

^∗

(Eq. 85), and with the notation of Eq. (41), the perturbation δz

^∗

rs (t) is given by

δz

^∗

rs (t) = −P

⁻¹

B rs (y e )∇

y

H rs ¹ (y e , t). (98) To match the notation of the matrix A

^∗

_rs , let δy ¹ and δy ² be the first 2 and the next 4 components of δy

^∗

, idem for δz

^∗

_rs (t), such that the linear problem with perturbation reads

δ y ˙ ^k − A ^k rs δy ^k = δz ^k rs (t), k = 1, 2. (99) By definition,

δy ¹ = (δΠ z , δI y ) ^T , δy ² = (δΠ x , δΠ y , δI z , δJ z ) ^T , (100) and Eq. (98) implies

δz ¹ rs (t) =

3(B − A)σ ¹ _xy (t) 0

, δz ² rs (t) =









3(C − B)σ ¹ yz (t)

−3(C − A)σ xz ¹ (t) 0

0









. (101)

Note that the term σ ¹ yz (t) is present in the perturbation δz ² rs (t) but its effect on the orientation of the spin axis is very weak. For instance, according to the ephemeris of Titan in TASS1.6 (Vienne and Duriez, 1995), the amplitude associated with the angle (Ωt − Φ) in σ yz ¹ (t) is about 500 times lower than that in σ ¹ xz (t). In the numerical applications (Sect. 6), σ ¹ yz (t) is simply discarded.

4 Satellite with a liquid core

In this section we consider a satellite with a rigid mantle/crust layer surrounding a liquid core. In a first step, we analyse the problem using the Poincar´ e-Hough model which is valid for all eccentricities of the ellipsoidal cavity containing the fluid core (Poincar´ e, 1910; Hough, 1895). In a second one, we truncate the problem at the first order with respect to the equatorial and polar flattening of the cavity.

The same simplification will be used again in Sect. 5 where the case of a satellite with a subsurface ocean is treated. Here, the two models of the same problem are used to estimate the error made by the approximation.

4.1 Poincar´ e-Hough model

As in the previous model, A, B, C designate the principal moments of inertia of the

whole satellite. Those of the liquid core are denote by A c , B c , C c . We assume that

the axes of the core/mantle ellipsoidal boundary are aligned to those of the satellite

surface. Hence, the principal axes (I c , J c , K c ) of the core are aligned to those of the

mantle denoted (I m , J m , K m ) which are also aligned to those of the whole satellite

(I, J, K). The vector ω still represents the rotation vector of (I, J, K) with respect to the laboratory frame expressed in the laboratory frame. We add the rotation vector ω

⁰

c associated with the simple motion of the liquid core with respect to the mantle and expressed in the mantle-fixed frame (Poincar´ e, 1910). As in the rigid case, the laboratory frame rotates with respect to the inertial frame at the speed Ω. Let I , I

⁰

c and I

⁰

be the inertia matrices defined as

I = R diag(A, B, C)R ^T , (102)

I

⁰

c = diag(A c , B c , C c ), (103) I

⁰

= diag(A

⁰

, B

⁰

, C

⁰

)R ^T , (104) where R = [I, J, K] is the rotation matrix of the mantle relative to the laboratory frame. Furthermore, we have defined

A

⁰

= A c

p 1 − α ² c , B

⁰

= B c

p 1 − β ² c , C

⁰

= C c

p 1 − γ c ² , (105) with

α c = C c − B c

A c

, β c = C c − A c

B c

, γ c = B c − A c

C c

. (106)

For this problem, the Lie velocity is η = (ω, ω

⁰

c ) and the generalised coordinates are limited to q = (I, J, K). Coordinates associated with the simple motion of the liquid core do not appear in the equations of motion because the fluid is assumed to be incompressible and its volume is set by the mantle, thus the kinetic and the potential energies only depends on η and q. The kinetic energy T fc (η, q) of rotation of the satellite is (Poincar´ e, 1910; Hough, 1895)

T fc (η, q) = (ω + Ω) ^T I (ω + Ω)

2 + ω

^0T

c I

⁰

c ω

⁰

c

2 + ω

^0T

_c I

⁰

(ω + Ω), (107) The potential energy is the same as in the rigid satellite case (see sect. 3). Thus, the Lagrangian L fc (η, q) reads

L fc (η, q) = (ω + Ω) ^T I (ω + Ω)

2 + ω

^0T

c I

⁰

c ω

⁰

c

2 + ω

^0T

c I

⁰

(ω + Ω) − 3GM p

2 r ^T I r

r ⁵ . (108) The Lie momenta associated with ω and ω

⁰

c are respectively

Π = ∂L fc

∂ω = I (ω + Ω) + I

^0T

ω

⁰

c , (109) Π

⁰

_c = ∂L fc

∂ω

⁰

c

= I

⁰

c ω

⁰

_c + I

⁰

(ω + Ω), (110) with the inverse transformation,

ω = Q Π − Q

^0T

Π

⁰

c − Ω, (111)

ω

⁰

c = Q

⁰

c Π

⁰

c − Q

⁰

Π, (112) where

Q = R diag

A c

AA c − A

⁰²

, B c

BB c − B

⁰²

, C c

CC c − C

⁰²

R ^T , (113) Q

⁰

c = diag

A

AA c − A

⁰²

, B

BB c − B

⁰²

, C CC c − C

⁰²

, (114)

Q

⁰

= diag

A

⁰

AA c − A

⁰²

, B

⁰

BB c − B

⁰²

, C

⁰

CC c − C

⁰²

R ^T . (115)

The Hamiltonian of the problem is then H fc (y, t) = Π ^T Q Π

2 + Π

^0T

c Q

⁰

c Π

⁰

c

2 − Π

^0T

c Q

⁰

Π − Ω ^T Π + 3GM p

2 r ^T I r

r ⁵ , (116) with the state vector y = (Π

⁰

_c , Π, I, J, K). In these variables, the Poisson matrix reads

B fc (y) =















Π ˆ

⁰

c 0 0 0 0 0 Π ˆ ˆ I J ˆ K ˆ 0 ˆ I 0 0 0 0 J 0 0 0 ˆ 0 K 0 0 0 ˆ















(117)

and the equations of motion are

Π ˙

⁰

c = ω

⁰

_c × Π

⁰

_c , (118)

Π ˙ = Π × Ω − 3 GM p

r ⁵ ( I r) × r, (119)

I ˙ = ω × I, (120)

J ˙ = ω × J, (121)

K ˙ = ω × K. (122)

As in the rigid case (Sect. 3), we now split the Hamiltonian H fc (y, t) into its autonomous part H _fc ⁰ (y) and a perturbation H _fc ¹ (y, t) using the decomposition of the gravitational potential energy U 0 (y) and U 1 (y, t), Eqs. (61-62). There are seven Casimir functions given by

C fc ⁰ (y) = 1

2 Π

⁰

c · Π

⁰

c , C fc ¹ (y) = 1

2 I · I, C fc ² (y) = 1

2 J · J, C fc ³ (y) = 1 2 K · K, C _fc ⁴ (y) = J · K, C _fc ⁵ (y) = K · I, C _fc ⁶ (y) = I · J.

(123)

The equilibrium y e of H _fc ⁰ (y) is solution of

ω

⁰

_c,e − µ 0 Π

⁰

_c,e = 0, (124)

ω e = 0, (125)

A c (I e · Π e ) − A

⁰

(I e · Π

⁰

c,e )

AA c − A

⁰²

Π e + 3AS 0 I e − µ 1 I e − µ 5 K e − µ 6 J e = 0, (126) B c (J e · Π e ) − B

⁰

(J e · Π

⁰

c,e )

BB c − B

⁰²

Π e + 3BS 0 J e − µ 2 J e − µ 4 K e − µ 6 I e = 0, (127) C c (K e · Π e ) − C

⁰

(K e · Π

⁰

_c,e )

CC c − C

⁰²

Π e + 3CS 0 K e − µ 3 K e − µ 4 J e − µ 5 I e = 0. (128)

We stress that Π is written in the laboratory frame while Π

⁰

_c is expressed in the

mantle-fixed frame. Thus, in Eq. (126), (I · Π) = I x Π x + I y Π y + I z Π z whereas

(I · Π

⁰

c ) = Π c,x

⁰

. The same reasoning holds in Eqs. (127,128). The norm of the

angular velocity ω

⁰

c,e can be arbitrarily chosen. This is due to the conservation

of the Casimir C _fc ⁰ (y). Here, we assume that the fluid core has no mean angular

velocity with respect to the mantle and thus ω

⁰

_c,e = 0. Under this hypothesis, we get

ω e = 0, Π

⁰

_c,e = C

⁰

Ωk, Π e = CΩk, I e = i, J e = j, K e = k, µ 0 = 0, µ 1 = 3Aσ xx ⁰ , µ 2 = 3Bσ ⁰ yy , µ 3 = 3Cσ ⁰ zz + CΩ ² , µ 4 = µ 5 = µ 6 = 0 . (129) The linear system is expressed in the coordinates

δy

^∗

=

δΠ z , δI y , δΠ c,x

⁰

, δΠ c,y

⁰

, δΠ x , δΠ y , δI z , δJ z , δΠ c,z

⁰

, δI x , δJ y , δK z , δI y + δJ x , δI z + δK x , δJ z + δK y

T

.

(130)

Let A

^∗

_fc (y e ) be the matrix of the linear system evaluated at the equilibrium point and expressed in the coordinates δy

^∗

. As in the rigid case, we define the matrices A ¹ _fc and A ² _fc such that

A

^∗

fc (y e ) =





A ¹ _fc 0 · 0 A ² _fc · 0 0 0



 , (131)

where the dots · still denote arbitrary matrices. We get

A ¹ fc =





0 −3(B − A)(σ xx ⁰ − σ ⁰ yy ) C c

CC c − C

⁰²

0



 , (132)

and

A ² _fc =









































0 C

⁰

B Ω 0 − C

⁰

B

⁰

Ω 0 − C

⁰

B

⁰

CΩ ²

− C

⁰

A Ω 0 C

⁰

A

⁰

Ω 0 C

⁰

A

⁰

CΩ ² 0

0 0 0 Ω 0 −3(C − B)κ 2

0 0 −Ω 0 3(C − A)κ 1 0

0 1

B

⁰

0 − 1

B _c 0

1 − C

B _c

Ω

− 1

A

⁰

0 1

A _c 0 −

1 − C

A _c

Ω 0









































(133)

with

1

A = A

AA c − A

⁰²

, 1

A _c = A c

AA c − A

⁰²

, 1

A

⁰

= A

⁰

AA c − A

⁰²

, 1

B = B

BB c − B

⁰²

, 1

B _c = B c

BB c − B

⁰²

, 1

B

⁰

= B

⁰

BB c − B

⁰²

, 1

C = C

CC c − C

⁰²

, 1

C _c = C c

CC c − C

⁰²

, 1

C

⁰

= C

⁰

CC c − C

⁰²

.

(134)

The eigenfrequencies are

ω fc,u =

CC c

CC c − C

⁰²

1/2

ω rs,u , (135)

ω fc,v = ω rs,v + O(), (136)

ω fc,w = ω rs,w + O(), (137)

ω fc,z = C

⁰

√ A c B c

Ω + O() (138)

with being the mass of the core divided by the total mass of the satellite. ω rs,u , ω rs,v , and ω rs,w are the frequencies obtained in the rigid case (Eqs. 89,90). ω fc,z is the frequency of the additional degree of freedom induced by the presence of the liquid core. In the case where the fluid core represents a significant fraction of the total mass of the satellite, Eqs. (136-138) are no longer valid and eigenfrequencies should be directly computed from the matrix A ² _fc (Eq. 133).

The Lyapunov function (Eq. 43) associated with this problem is N fc (y) = 1

2A _c

Π x − A

⁰

A c

Π _c,x

⁰

+ 1

2 (C − A _c ) (I z − K x ) 2

+ 1 2B _c

Π y − B

⁰

B c

Π c,y

⁰

+ 1

2 (C − B _c ) (J z − K y ) 2

+ 1 2C _c Π z ²

+ 1 2A c

Π c,x

⁰

− 1

2 A

⁰

Ω(I z − K x ) 2

+ 1 2B c

Π c,y

⁰

− 1

2 B

⁰

Ω(J z − K y ) 2

+ 1

2 n 1 (I y − J x ) ² + 1

2 n 2 (I z − K x ) ² + 1

2 n 3 (J z − K y ) ² ,

(139) where n 1 , n 2 , and n 3 are the same as in the rigid case (see Eq. 97). Given that A _c , B _c , C _c , A c , and B c are all positive, the nonlinear stability criterion is identical to that of a rigid satellite, namely C > B > A. In particular, there is no restriction on the moments of inertia of the core (A c , B c , C c ).

The driven equations of motion of the satellite with a liquid core in the vicinity of the relative equilibrium y e are of the form

δ y ˙ ^k − A ^k fc δy ^k = δz ^k fc (t), k = 1, 2, (140) with

δy ¹ = (δΠ z , δI y ) ^T , δy ² = (δΠ _c,x

⁰

, δΠ _c,y

⁰

, δΠ x , δΠ y , δI z , δJ z ) ^T , (141) and

δz ¹ _fc (t) =

3(B − A)σ ¹ xy (t) 0

, δz ² _fc (t) =

















3(C − B)σ ¹ _yz (t)

−3(C − A)σ ¹ xz (t) 0

0 0 0

















. (142)

4.2 Quasi-spherical approximation

In this section, we reconsider the case of a satellite with a liquid core, but we assimilate A

⁰

, B

⁰

and C

⁰

to the moments of inertia of the core, i.e., we assume

A

⁰

≈ A c , B

⁰

≈ B c , C

⁰

≈ C c . (143) According to Eq. (105), this is equivalent to a first order approximation in α c , β c

and γ c . With this simplification, the kinetic energy (Eq. 107) can be rewritten as follows

T fc

⁰

(η, q) = (ω + Ω) ^T I m (ω + Ω)

2 + (ω

⁰

c + R ^T (ω + Ω)) ^T I

⁰

c (ω

⁰

c + R ^T (ω + Ω))

2 (144)

where

I ^m = I − R I

⁰

c R ^T

= R diag(A m , B m , C m )R ^T (145) is the inertia tensor of the mantle written in the laboratory frame (A m = A − A c , B m = B − B c , and C m = C − C c ). According to the expression (144), the problem behaves as if the liquid core were rotating rigidly relative to the mantle at the angular velocity ω

⁰

_c with a matrix of inertia I

⁰

c constant in the mantle-fixed frame.

Indeed, ω

⁰

c +R ^T (ω+Ω) is the rotation speed of the core with respect to the inertial frame written in the mantle-fixed frame. We here retrieve the approximation made by Mathews et al. (1991) who neglected the small departure of the fluid velocity field from a pure solid rotation. Following the same procedure as in Sect. 4.1, the two submatrices of the linearised system written in the set of variables δy

^∗

(Eq. 130) become

A ¹ _fc

0

=

"

0 −3(B − A)(σ xx ⁰ − σ ⁰ yy ) 1

C m

0

#

, (146)

and

A ² _fc

0

=



































0 BC c

B m B c

Ω 0 − C c

B m

Ω 0 − CC c

B m

Ω ²

− AC c

A m A c

Ω 0 − C c

A m

Ω 0 CC c

A m

Ω ² 0

0 0 0 Ω 0 −3(C − B)κ 2

0 0 −Ω 0 3(C − A)κ 1 0

0 1

B m

0 − 1

B m

0

1 − C

B m

Ω

− 1 A m

0 1

A m

0 −

1 − C

A m

Ω 0

































 .

(147)

Although we retrieve the eigenfrequencies obtained in section 4.1 within the ap-

proximation (Eq. 143) only, the second member δz fc

⁰

(t) of the driven system is

exactly the same as δz fc (t) (Eq. 142).

5 Satellite with a subsurface ocean

Here, we consider a satellite with a rigid central part c (also called interior) and a rigid shell s separated by a global ocean o. By assumption, the shell is ellipsoidal with inner radii a o , b o , c o and outer radii a s , b s , c s . The interior, an ellipsoid of radii a c , b c , c c , might be differentiated, i.e., it can be made of a succession of N concentric ellipsoidal layers with different densities (ρ i ) 1≤i≤N and outer radii a i , b i , c i . We have thus a N = a c , b N = b c and c N = c c . The ocean and the shell are assumed to be homogeneous with respective density ρ o and ρ s . Nevertheless, the results can easily be extended to the case of a stratified rigid shell. Because the simple motion introduced by Poincar´ e (1910) for a satellite with a liquid core cannot be applied in this case, we use the approximation described in Sect. 4.2. We could describe the evolution of the central region and of the ocean in the shell-fixed frame to remain close to the study made on the satellite with a liquid core, but equations are more symmetrical if all coordinates are given with respect to a same given frame which we chose to be the laboratory frame. In this frame, the configuration of the system is given by the coordinates of the principal axes of the interior and the shell, i.e., the generalised coordinates are q = (I c , J c , K c , I s , J s , K s ). The Lie velocities are the rotation vectors of the three layers with respect to the laboratory frame η = (ω o , ω c , ω s ). Within the approximation of Sect. 4.2, the kinetic energy of the satellite with a global ocean reads

T go (η, q) = (ω c + Ω) ^T I ^c (ω c + Ω)

2 + (ω s + Ω) ^T I ^s (ω s + Ω) 2

+ (ω o + Ω) ^T I ^o (ω o + Ω)

2 ,

(148)

with the inertia tensors

I c = R c diag(A c , B c , C c )R ^T c , (149) I ^s = R s diag(A s , B s , C s )R ^T s , (150) I o = R s diag(A

⁰

_s , B _s

⁰

, C _s

⁰

)R ^T _s − R c diag(A

⁰

_c , B _c

⁰

, C _c

⁰

)R ^T _c , (151) where R c = [I c , J c , K c ], R s = [I s , J s , K s ], and

A c =

N

X

i=1

4π 15 ρ i

a i b i c i (b ² i + c ² i ) − a i−1 b i−1 c i−1 (b ² i−1 + c ² i−1 )

, (152) A s = 4π

15 ρ s

a s b s c s (b ² s + c ² s ) − a o b o c o (b ² o + c ² o )

, (153)

A

⁰

c = 4π

15 ρ o a c b c c c (b ² c + c ² c ), (154) A

⁰

s = 4π

15 ρ o a o b o c o (b ² o + c ² o ). (155) In Eq. (152), we apply the convention a 0 = b 0 = c 0 = 0. The other quantities B, C are deduced from Eqs. (152-155) by circular permutation of a, b, c. Let us stress that the matrix of inertia of the whole satellite is simply

I = I c + I s + I o . (156)