Coupling between the spin precession and polar motion of a synchronously rotating satellite: application to Titan

Abstract

We here develop, in an angular momentum approach, a consistent model that integrates all rotation variables and considers forcing both by the central planet and a potential atmosphere. Existing angular momentum approaches for studying the polar motion, precession, and libration of synchronously rotating satellites, with or without an internal global fluid layer (e.g., a subsurface ocean) usually focus on one aspect of rotation and neglect coupling with the other rotation phenomena. The model variables chosen correspond most naturally with the free modes, although they differ from those of Earth rotation studies, and facilitate a comparison with existing decoupled rotation models that break the link between the rotation motions. The decoupled models perform well in reproducing the free modes, except for the Free Ocean Nutation in the decoupled polar motion model. We also demonstrate the high accuracy of the analytical forced solutions of decoupled models, which are easier to use to interpret observations from past and future space missions. We show that the effective decoupling between the polar motion and precession implies that the spin precession and its associated mean obliquity are mainly governed by the external gravitational torque by the parent planet, whereas the polar motion of the solid layers is mainly governed by the angular momentum exchanges between the atmosphere (e.g., for Titan) and the surface. To put into perspective the difference between rotation models for a synchronously rotating icy moon with a thin ice shell and classical Earth rotation models, we also consider the case of the Moon, which has a thick outer layer above a liquid core. We also show that for non-synchronous rotators, the free precession of the outer layer in space degenerates into the tilt-over mode.

Appendices

Appendix 1: Cassini state and orientation of the different layers

Fig. 10

From Coyette et al. (2016). Angles between the inertial plane, the orbital plane, the BF equator, and the rotation equator for a solid satellite. The angle definitions also apply to any solid layer of a satellite with an internal liquid layer. For the ocean, we have chosen not to define a Body Frame, but the rotation frame is well defined

Let i and $\theta$ be the orbital inclination and inertial obliquity of a solid synchronous satellite, and $\chi ,\epsilon$ and $\phi$ the Euler angles between the rotation reference frame and the Body Frame (see Fig. 10). As can be seen from Fig. 10, since $i, \theta$ and $\epsilon$ are small angles, the synchronous rotation implies that (e.g., Peale 1969)

$$\begin{aligned}&\chi +\phi +\xi -\gamma \simeq \varOmega +\omega -\pi +\nu -s, \end{aligned}$$

(96)

where $\varOmega$ and $\xi$ are the longitudes of the ascending nodes of the orbital plane and of the equator of the rotation frame over the inertial plane, $\omega$ is the orbit pericenter node, $\nu$ is the true anomaly ($\nu =M+s$ with M the mean anomaly and $s=2 e \sin M$ the equation of the center), and $\gamma$ is the libration angle. Assuming that the synchronous satellite is in a Cassini state, the spin axis, the normal to the orbit and the normal to the Laplace/inertial plane are nearly coplanar, so that

$$\begin{aligned} \xi= & {} \varOmega +\sigma , \end{aligned}$$

(97)

$$\begin{aligned} \theta\simeq & {} i+\eta , \end{aligned}$$

(98)

with $\sigma$ a small difference between the two node longitudes and $\eta$ the orbital obliquity. The definition of the polar motion implies that

$$\begin{aligned} ( m_x, m_y)\simeq & {} -\epsilon \, ( \sin \chi , \cos \chi ), \end{aligned}$$

(99)

with $m_x$ and $m_y$ the equatorial components of the rotation normalized rotation vector of Eq. (1).

Following Eckhardt (1981), we define a unit vector $\hat{p}=(p_x,p_y,p_z)$ from the satellite center, along the direction of the Laplace/inertial pole (see Eq. 6). Taking advantage of the angles defined above, its components in the BF can be written, at first order in small quantities, as

$$\begin{aligned} \hat{p}=Rz[\chi ].Rx[-\epsilon ].R_z[\phi ].R_x[\theta ].\left( \begin{array}{c} 0 \\ 0 \\ 1 \\ \end{array} \right) \simeq \left( \begin{array}{c} m_x-\theta \sin (M+\omega )\\ m_y-\theta \cos (M+\omega )\\ 1 \\ \end{array} \right) \end{aligned}$$

(100)

and expresses the motion of the Laplace pole with respect to the body frame as a combination of the polar motion of the rotation axis in the BF and of the precession of the rotation axis in space with amplitude $\theta$. Since the difference between $\hat{p}$ (Laplace pole motion with respect to BF) and $\varvec{\Omega }$ (spin axis with respect to BF, see Eq. (1)) describes the difference between Laplace and spin poles, the inertial obliquity $\theta$ is simply computed, from $p_x,p_y,m_x,m_y$, as:

$$\begin{aligned} \theta\simeq & {} \sqrt{(p_x-m_x)^2+(p_y-m_y)^2}. \end{aligned}$$

(101)

The components of the unit vector $\hat{s}=(s_x,s_y,s_z)$ along the direction of the rotation axis, expressed in the Cartesian coordinates of the Laplace/inertial reference frame are given, at first order in $\theta$, by

$$\begin{aligned} \hat{s}=Rz[-\xi ].R_x[-\theta ].\left( \begin{array}{c} 0 \\ 0 \\ 1 \\ \end{array} \right) \simeq \left( \begin{array}{c} \theta \sin (\xi )\\ -\theta \cos (\xi )\\ 1 \\ \end{array} \right) \end{aligned}$$

(102)

and expresses the precession of the rotation axis in space, with amplitude $\theta$. The precession is slightly influenced by the polar motion through $\theta$, as can be seen from Eq. (101). The effect is about 5 m for the Titan toy model (see Sect. 2.4.3). The spin precession is also slightly influenced by the librations/LOD variations through $\xi$. The angle $\xi$ can be extracted from Eq. (96), where $\phi$ can be obtained from spheric trigonometry relations (the triangle formed by the ascending node of the rotation equator over the BF equator, the descending node of the Laplace plane over the BF equator and the ascending node of the rotation equator over the Laplace plane is fully determined by the knowledge of $\mathbf {m}$ and $\mathbf {p}$) and where $\gamma$ can be replaced by the solutions of Van Hoolst et al. (2009) for libration and semi-annual LOD variations, for example. The effect of $\gamma$ on the precession (not shown here) is a few meters, and can be safely neglected in front of the 1-km detection limit related to the position error of Cassini radar images (Meriggiola et al. 2016).

The unit vector $\mathbf {n}$ along the direction of the orbit pole expressed in the Cartesian coordinates of the Laplace/inertial reference frame is given, at first order in i, by

$$\begin{aligned} \hat{n}=Rz[-\varOmega ].R_x[-i].\left( \begin{array}{c} 0 \\ 0 \\ 1 \\ \end{array} \right) \simeq \left( \begin{array}{c} i \sin (\varOmega )\\ -i \cos (\varOmega )\\ 1 \\ \end{array} \right) \end{aligned}$$

(103)

and the exact expression for the orbital obliquity $\eta$ of Eq. (98) is given by

$$\begin{aligned} \cos \eta =\hat{n}.\hat{s}. \end{aligned}$$

(104)

The definitions of and the relations between the angles presented above also apply to any solid layer of a satellite with an internal liquid layer. We simply add a subscript or superscript to indicate to which layer each quantity is related. For instance, the inertial obliquity of Titan’s shell and interior are expressed as

$$\begin{aligned} \theta _s\simeq & {} \sqrt{(p^s_x-m^s_x)^2+(p^s_y-m^s_y)^2}, \end{aligned}$$

(105)

$$\begin{aligned} \theta _i\simeq & {} \sqrt{(p^i_x-m^i_x)^2+(p^i_y-m^i_y)^2}. \end{aligned}$$

(106)

As the orientation of the ocean masses depends on the orientation of the adjacent solid layers, which are misaligned to each other, we have chosen not to define a Body Frame for the ocean. However, the definition of an ocean rotation frame still stands. For instance, the inertial obliquity of Titan’s ocean is expressed as

$$\begin{aligned} \theta _o\simeq & {} \sqrt{(p^s_x-m^o_x)^2+(p^s_y-m^o_y)^2}. \end{aligned}$$

(107)

In both the solid and ocean cases, we have chosen to work with unit vector(s) from the satellite (solid layers) center, along the direction of the Laplace/inertial pole (see Eq. 6), as variables to be solved for, in order to properly take into account the coupling between polar motion and spin precession. This has consequences for some expressions (torques, transformation matrix) defined in the decoupled models of polar motion by Coyette et al. (2016,2018) where the spin precession was assumed to be known. These modifications are explained in the subsections below.

1.1 Consequence on the transformation matrix

The transformation matrix $R_{(i\rightarrow s)}$ from the interior BF to the shell BF is a composition of rotations, as detailed in Coyette et al. (2016). Adapting their Eq. (119) to our set of variables to be solved for in the ocean case of Sect. 3, and using relations of the form of Eq. (100) written for the two solid layers, the transformation matrix can be expressed as

$$\begin{aligned} R_{(i\rightarrow s)}=\left( \begin{array}{ccc} 1 &{} (\gamma _s-\gamma _i) &{} (p_x^s-p_x^i)\\ (\gamma _i-\gamma _s) &{} 1 &{} (p_y^s-p_y^i)\\ (p_x^i-p_x^s) &{} (p_y^i-p_y^s) &{} 1 \end{array}\right) . \end{aligned}$$

(108)

1.2 Consequence on the external and internal gravitational torques

Adapting Eq. (18) of Coyette et al. (2016) for the external torque on a solid body to our set of variables to be solved in the solid case, we easily obtain Eq. (5).

For the ocean case, the external torques on the solid layers, corrected for the effect of the hydrostatic pressure of Eqs. (85–86) become

$$\begin{aligned} \varvec{\Gamma }_{s,\tilde{\mathrm{ext}}}^s= & {} 3 n^2 \left( \begin{array}{c} 0\\ \left[ (A_s-C_s)+(A_{ot}-C_{ot})\right] [i \sin (\omega +M-\pi )-p_x^s] \\ \left[ (B_s-A_s)+(B_{ot}-A_{ot})\right] (s-\gamma _s) \\ \end{array} \right) , \end{aligned}$$

(109)

$$\begin{aligned} \varvec{\Gamma }_{i,\tilde{\mathrm{ext}}}^s= & {} 3 n^2 \left( \begin{array}{c} 0\\ \left[ (A_i-C_i)+(A_{ob}-C_{ob})\right] [i \sin (\omega +M-\pi )-p_x^i] \\ \left[ (B_i-A_i)+(B_{ob}-A_{ob})\right] (s-\gamma _i) \\ \end{array} \right) , \end{aligned}$$

(110)

whereas Eqs. (124–129) of Coyette et al. (2016) for the internal gravitational torque between the solid layers, here corrected for the effect of the hydrostatic pressure of Eqs. (85–86), become:

$$\begin{aligned} \varvec{\Gamma }_{\tilde{int}}^{s}= & {} \left( \begin{array}{c} -n \kappa _\mathrm{int}^x(p_y^s-p_y^i)\\ n \kappa _\mathrm{int}^y(p_x^s-p_x^i)\\ n \kappa _\mathrm{int}^z(\gamma _s-\gamma _i) \end{array} \right) =-\varvec{\Gamma }_{\tilde{int}}^{i}, \end{aligned}$$

(111)

with

$$\begin{aligned} \kappa _\mathrm{int}^x= & {} \frac{4\pi G}{5n}(C_i-B_i+C_{ob}-B_{ob})(\rho _s [2(\alpha _s-\alpha _o)-(\beta _s-\beta _o)]+\rho _o(2\alpha _o-\beta _o)), \end{aligned}$$

(112)

$$\begin{aligned} \kappa _\mathrm{int}^y= & {} \frac{4\pi G}{5n}(C_i-A_i+C_{ob}-A_{ob})(\rho _s [2(\alpha _s-\alpha _o)+(\beta _s-\beta _o)]+\rho _o(2\alpha _o+\beta _o)), \end{aligned}$$

(113)

$$\begin{aligned} \kappa _\mathrm{int}^z= & {} -\frac{8\pi G}{5n}(B_i-A_i+B_{ob}-A_{ob})(\rho _s (\beta _s-\beta _o)+\rho _o\beta _o). \end{aligned}$$

(114)

Appendix 2: A coupled system for a solid axisymmetric non-synchronous planet

Consider a non-synchronous planet rotating at the mean rate $\varOmega _o$, different from its revolution rate n, so that its rotation vector is written as

$$\begin{aligned} \varvec{\Omega }=\varOmega _o\left( \begin{array}{c} 0\\ 0\\ 1 \end{array}\right) +\varOmega _o\, \mathbf {m}=\varOmega _o\left( \begin{array}{c} m_x\\ m_y\\ 1+m_z \end{array}\right) , \end{aligned}$$

(115)

instead of Eq. (1) for the synchronous case. Neglecting librations (since we assume $A=B$) and assuming that the obliquity $\theta$ is a small angle (this assumption is not justified for planets like the Earth with a large obliquity, but is practical in the following), Eq. (96) is replaced by

$$\begin{aligned}&\chi +\phi +\xi \simeq \varOmega _o t. \end{aligned}$$

(116)

For simplicity, we have assumed that the phase of $\chi +\phi +\xi$ is zero at $t=0$. We also assume that the orbit is circular ($e=0$), that the orbital plane is the inertial plane ($i=0$), so that $\varOmega +\omega +M=L$ with L the mean longitude defined as $n t+L_o$.

Equation (99) for polar motion in the BF also applies in the non-synchronous case, whereas Eq. (100) for the Laplace pole motion changes to

$$\begin{aligned} \hat{p} \simeq \left( \begin{array}{c} m_x-\theta \sin (\varOmega _o t-\xi )\\ m_y-\theta \cos (\varOmega _o t-\xi )\\ 1 \\ \end{array} \right) . \end{aligned}$$

(117)

As a result, the components of the unit vector in the direction to the Sun in the coordinates of the planet BF can be written as

$$\begin{aligned} \left( \begin{array}{c} r_x\\ r_y\\ r_z \end{array}\right) =\left( \begin{array}{c} -\cos (L-\varOmega _o t)\\ -\sin (L-\varOmega _o t)\\ p_x \cos (L-\varOmega _o t) + p_y \sin (L-\varOmega _o t) \end{array}\right) \end{aligned}$$

(118)

and the components of the torque exerted by the Sun are given by

$$\begin{aligned} \varvec{\Gamma }_\mathrm{pb}= \frac{3}{2} n^2 (C-A) \left( \begin{array}{c} -p_y+p_y \cos 2(L-\varOmega _o t)-p_x \sin 2(L-\varOmega _o t)\\ p_x+p_x \cos 2(L-\varOmega _o t)+p_y \cos 2(L-\varOmega _o t) \\ 0 \\ \end{array} \right) . \end{aligned}$$

(119)

By dropping the terms of the torque which explicitly depend on the forcing frequency $2(n-\varOmega _o)$, the homogeneous system, formed by the angular momentum equation and the kinematic equation, to be solved to find the free latitudinal modes is written as

$$\begin{aligned}&A \varOmega _o \dot{m}_x+(C-A) \varOmega _o^2 m_y=-\frac{3}{2} n^2 (C-A) p_y, \end{aligned}$$

(120)

$$\begin{aligned}&A \varOmega _o \dot{m}_y-(C-A) \varOmega _o^2 m_x=\frac{3}{2} n^2 (C-A) p_x, \end{aligned}$$

(121)

$$\begin{aligned}&\dot{p}_x=\varOmega _o p_y-\varOmega _o m_y, \end{aligned}$$

(122)

$$\begin{aligned}&\dot{p}_y=-\varOmega _o p_x+\varOmega _o m_x, \end{aligned}$$

(123)

and is characterized by the following frequencies

$$\begin{aligned} \sigma _\mathrm{CW}= & {} \frac{3n^2+2\varOmega _o^2}{2\varOmega _o}\frac{(C-A)}{A}, \end{aligned}$$

(124)

$$\begin{aligned} \sigma _\mathrm{QDFW}= & {} \frac{3n^2}{2\varOmega _o}\frac{(C-A)}{A}+\varOmega _o. \end{aligned}$$

(125)

The main difference with respect to the homogeneous part of the system of Eqs. (15–18), besides the neglect of the atmosphere, is the existence of a term in $p_y$ in Eq. (120) and the modification of the right-hand side in the y-component of the angular momentum Eq. (121) by a factor 1 / 2.

Our expression for $\sigma _\mathrm{CW}$ slightly differs from the classical expression (see below) derived from a decoupled system for polar motion. In the IF, the QDFW translates to a free precession with

$$\begin{aligned} \sigma '_\mathrm{FP}=\frac{3n^2}{2\varOmega _o}\frac{(C-A)}{A}, \end{aligned}$$

(126)

similar to the classical expression for the precession rate of a non-synchronous planet, as derived from a decoupled equation for the spin precession in space (see below, Eq. 131), but without the factor $\cos \theta _o$, with $\theta _o$ the mean obliquity, as we have assumed that $\theta \ll 1$.

In studies dealing with the rotation of the Earth, it is customary to neglect the torque to find the free modes (e.g., page 47 of Moritz and Mueller (1987)). In that case, the angular momentum equation governing the Chandler Wobble can be solved independently from the kinematic equation, and the CW frequency reduces to the classical expression

$$\begin{aligned} \sigma _\mathrm{CW}= & {} \varOmega _o\frac{(C-A)}{A}. \end{aligned}$$

(127)

The CW is a purely prograde (see Eq. 2.10a of Smith (1977)) mode of $\mathbf {m}$. The effect of the external torque on the CW, obtained as the difference between Eq. (124) and Eq. (127), is only $0.001\%$ for the Earth, since $n\ll \varOmega _o$, and can safely be neglected.

Still neglecting the torque, but considering both the angular momentum and kinematic equations (note that the concept of kinematic equation here plays the same role as the concept of nutation frame introduced in Sect. 2.3.1 of Moritz and Mueller (1987), that is to say orienting the BF with respect to the IF), we also obtain the QDFW and FP frequencies which are given by

$$\begin{aligned} \sigma _\mathrm{QDFW}= & {} \varOmega _o, \end{aligned}$$

(128)

$$\begin{aligned} \sigma '_\mathrm{FP}= & {} 0. \end{aligned}$$

(129)

The QDFW is a purely retrograde (and diurnal) mode of $\mathbf {p}$. We see that when the external torque is neglected, the Free Precession degenerates into a mode called the Tilt-Over mode (TOM) and where the spin axis stays fixed in space. However, it is very well known that the Earth is precessing in space with a finite period (about 26, 000 years). The TOM and its infinite period is therefore not strictly speaking an existing rotation mode, but a mathematical degeneracy due to an extreme approximation ($n\ll \varOmega _o$) when dealing with equations written in the BF.

It is possible to write an angular momentum for the precession in the IF, by neglecting polar motion (see, e.g., Eq. (1) of Bills (2005)):

$$\begin{aligned} \frac{{\hbox {d}}\hat{s}}{{\hbox {d}}t}=\frac{1}{(1-e^2)^{3/2}}\frac{3}{2}\frac{C-A}{C} \frac{n^2}{\varOmega _o}(\hat{n}.\hat{s})(\hat{s}\wedge \hat{n}). \end{aligned}$$

(130)

Relaxing the assumption that the mean obliquity $\theta _o$ is a small angle, $\hat{s}=(s_x,s_y,s_z\simeq \cos \theta _o)$ and $\hat{n}=(n_x,n_y,n_z\simeq 1)$. The free frequency associated to Eq. (130), obtained by setting $n_x= n_y=0$ (or $i=0$) is

$$\begin{aligned} \sigma '_\mathrm{FP}=\frac{1}{(1-e^2)^{3/2}}\frac{3n^2}{2\varOmega _o} \cos {\theta _o}\frac{(C-A)}{A}, \end{aligned}$$

(131)

which is a generalization of Eq. (126) to an eccentric orbit and large obliquity.

The considerations above question the very definition of what is a free mode. Is it the solution obtained when the whole torque is neglected? Or is it the solution of the homogeneous system of equations? For the Earth, it is customary to assume that the homogeneous system of equations is indeed obtained by neglecting the torque. However, we have seen above that this is not entirely justified. For synchronous rotators in the Cassini state, the dependence of the torque on the orientation of the BF with respect to the IF is clearly apparent (see Eq. 5), and it is easy to understand how the torque affects the homogeneous system. For the Earth, it is not customary to attempt to write the torque as a function of the variable to be solved for, and the effect of the torque on the homogeneous system is usually overlooked.

Appendix 3: Solution for the periodic atmospheric coupling, in the solid rigid case

The solution corresponding to the periodic terms of the atmospheric forcing of Eq. (8) is given, by

$$\begin{aligned}&\left( \begin{array}{c} m_x\\ m_y\\ p_x\\ p_y \end{array}\right) _{\mathrm{atm}} \simeq \sum _{\varpi \ne 0}\left( \begin{array}{c} m_y^s(\varpi )\sin (\varpi t+\phi _y(\varpi ))+m_x^c(\varpi )\cos (\varpi t+\phi _x(\varpi ))\\ m_y^c(\varpi )\cos (\varpi t+\phi _y(\varpi ))+m_x^s(\varpi )\sin (\varpi t+\phi _x(\varpi ))\\ p_y^s(\varpi )\sin (\varpi t+\phi _y(\varpi ))+p_x^c(\varpi )\cos (\varpi t+\phi _x(\varpi ))\\ p_y^c(\varpi )\cos (\varpi t+\phi _y(\varpi ))+p_x^s(\varpi )\sin (\varpi t+\phi _x(\varpi )) \end{array}\right) ,\qquad \quad \end{aligned}$$

(132)

where

$$\begin{aligned} m_y^s(\varpi )= & {} \upsilon _y \left( n^2 \varpi \left( -\frac{h_z(0)}{n}+(4 A+B-6 C )\right) \right. \\&\left. +\,\varpi ^3\left( \frac{h_z(0)}{n}-(A+B-3 C) \right) \right) , \\ m_y^c(\varpi )= & {} \upsilon _y \left( n^3 \left( \frac{h_z(0)}{n}+4 (-A+C) \right) \right. \\&\left. +\,n \varpi ^2 \left( -\frac{h_z(0)}{n}+(A-B+C) \right) + \frac{\varpi ^4}{n}(B-2 C))\right) , \\ m_x^c(\varpi )= & {} \upsilon _x \left( n^3 \left( \frac{h_z(0)}{n} + (-B + C) \right) \right. \\&\left. +\, n \varpi ^2 \left( -\frac{h_z(0)}{n} + (-4 A + B + 4 C) \right) + \frac{\varpi ^4}{n} (A - 2 C)\right) ,\\ m_x^s(\varpi )= & {} \upsilon _x \left( n^2 \varpi \left( \frac{h_z(0)}{n}-(4 A+B-6 C) \right) \right. \\&\left. +\,\varpi ^3\left( -\frac{h_z(0)}{n}+(A+B-3 C) \right) \right) ,\\ p_y^s(\varpi )= & {} \upsilon _y \, (n^2-\varpi ^2) \varpi \, (B-2 C), \\ p_y^c(\varpi )= & {} \upsilon _y \left( n^3 \left( \frac{h_z(0)}{n}+4 (-A+C) \right) - n\varpi ^2 \left( \frac{h_z(0)}{n}+(-A+C) \right) \right) ,\\ p_x^c(\varpi )= & {} \upsilon _x \, (n^2-\varpi ^2) n \,\left( \frac{h_z(0)}{n}+(-B+C) \right) ,\\ p_x^s(\varpi )= & {} \upsilon _x \left( n^2 \varpi (-4 A+5 C) +\varpi ^3(A-2 C) \right) ,\\ \upsilon _x= & {} \frac{h_x(\varpi )}{C^2(\varpi ^2-\sigma _\mathrm{CW}^2)(\varpi ^2- \sigma _\mathrm{QDFW}^2)},\\ \upsilon _y= & {} \frac{h_y(\varpi )}{C^2(\varpi ^2-\sigma _\mathrm{CW}^2)(\varpi ^2- \sigma _\mathrm{QDFW}^2)}. \end{aligned}$$

For $\varpi \rightarrow 0$, $(m_y^s,m_x^s,p_y^s,p_x^s)\rightarrow 0$ and $(m_y^c,m_x^c)\rightarrow (p_y^c,p_x^c)$, so that $(m_x^{\mathrm{atm}},m_y^{\mathrm{atm}})\simeq (p_x^{\mathrm{atm}},p_y^{\mathrm{atm}})$.

The solution can also be written in elliptical form

$$\begin{aligned}&\left( \begin{array}{c} m_x\\ m_y\\ p_x\\ p_y \end{array}\right) _{\mathrm{atm}} \simeq \sum _{\varpi \ne 0}\left( \begin{array}{c} \tilde{m}_x(\varpi )\sin (\varpi t+\tilde{\phi }_x(\varpi ))\\ \tilde{m}_y(\varpi )\cos (\varpi t+\tilde{\phi }_y(\varpi ))\\ \tilde{p}_x(\varpi )\sin (\varpi t+\tilde{\varphi }_x(\varpi ))\\ \tilde{p}_y(\varpi )\cos (\varpi t+\tilde{\varphi }_y(\varpi )) \end{array}\right) \end{aligned}$$

(133)

or in prograde/retrograde circular form (see Eq. 47), after some trigonometric manipulations.

Appendix 4: Matrices coefficients for the coupled model

$$\begin{aligned} \mathbf {K}= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 &{} K_{1,2} &{} 0 &{} K_{1,4}&{} 0 &{} 0 &{} 0 &{} K_{1,8} &{} 0 &{}K_{1,10} \\ K_{2,1} &{} 0 &{} K_{2,3} &{} 0 &{} 0 &{} 0 &{} K_{2,7}&{} 0 &{} K_{2,9} &{} 0 \\ 0 &{} K_{3,2} &{} 0 &{} K_{3,4} &{} 0 &{}K_{3,6} &{} 0 &{} K_{3,8} &{} 0 &{} K_{3,10 } \\ K_{4,1}&{} 0 &{} +K_{4,3}&{} 0 &{}K_{4,5} &{} 0 &{} K_{4,7}&{} 0 &{}K_{4,9} &{} 0 \\ 0 &{} 0 &{} 0 &{} K_{5,4} &{} 0 &{} K_{5,6} &{} 0 &{}K_{5,8} &{} 0 &{}K_{5,10} \\ 0 &{} 0 &{} K_{6,3}&{} 0 &{}K_{6,5} &{} 0 &{} K_{6,7} &{} 0 &{} K_{6,9} &{} 0 \\ 0 &{} n &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -n &{} 0 &{} 0 \\ -n &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} n &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} n &{} 0 &{} 0 &{} 0 &{} -n \\ 0 &{} 0 &{} 0 &{} 0 &{} -n &{} 0 &{} 0 &{} 0 &{} n &{} 0 \\ \end{array} \right) \end{aligned}$$

(134)

$$\begin{aligned} K_{1,2}= & {} \frac{ [(C_s-B_s)n+h_{z}(0)]}{A_s} \end{aligned}$$

(135)

$$\begin{aligned} K_{1,4}= & {} \frac{(C_{ot}-B_{ot}) n}{A_s} \end{aligned}$$

(136)

$$\begin{aligned} K_{1,8}= & {} \frac{\kappa _\mathrm{int}^x}{A_s} \end{aligned}$$

(137)

$$\begin{aligned} K_{1,10}= & {} -\frac{\kappa _\mathrm{int}^x}{A_s } \end{aligned}$$

(138)

$$\begin{aligned} K_{2,1}= & {} \frac{ [(A_s-C_s)n-h_{z}(0)]}{B_s} \end{aligned}$$

(139)

$$\begin{aligned} K_{2,3}= & {} \frac{(A_{ot}-C_{ot}) n}{B_s} \end{aligned}$$

(140)

$$\begin{aligned} K_{2,7}= & {} \frac{3 n (A_{s+ot}-C_{s+ot})-\kappa _\mathrm{int}^y}{B_s} \end{aligned}$$

(141)

$$\begin{aligned} K_{2,9}&=\frac{\kappa _\mathrm{int}^y}{B_s} \end{aligned}$$

(142)

$$\begin{aligned} K_{3,2}= & {} n+\frac{(C_{ot}-A_{ot}) n}{A_o} \end{aligned}$$

(143)

$$\begin{aligned} K_{3,4}= & {} -n-\frac{(C_o-A_o) n}{A_o} \end{aligned}$$

(144)

$$\begin{aligned} K_{3,6}= & {} K_{3,8}=-K_{3,10}=-\frac{(A_{ob}-C_{ob}) n}{A_o} \end{aligned}$$

(145)

$$\begin{aligned} K_{4,1}= & {} -n-\frac{(C_{ot}-B_{ot}) n}{B_o} \end{aligned}$$

(146)

$$\begin{aligned} K_{4,3}= & {} n+\frac{n (C_o-B_o)}{B_o} \end{aligned}$$

(147)

$$\begin{aligned} K_{4,5}= & {} K_{4,7}=-K_{4,9}=\frac{(B_{ob}-C_{ob}) n}{B_o} \end{aligned}$$

(148)

$$\begin{aligned} K_{5,4}= & {} \frac{C_{ob}-B_{ob}}{A_i} n \end{aligned}$$

(149)

$$\begin{aligned} K_{5,6}= & {} \frac{(C_i-B_i) n}{A_i} \end{aligned}$$

(150)

$$\begin{aligned} K_{5,8}= & {} -K_{5,10}=-\frac{(C_{ob}-B_{ob}) n+\kappa _\mathrm{int}^x}{A_i} \end{aligned}$$

(151)

$$\begin{aligned} K_{6,3}= & {} \frac{(A_{ob}-C_{ob}) n}{B_i} \end{aligned}$$

(152)

$$\begin{aligned} K_{6,5}= & {} \frac{(A_i-C_i) n}{B_i} \end{aligned}$$

(153)

$$\begin{aligned} K_{6,7}= & {} -\frac{(A_{ob}-C_{ob}) n-\kappa _\mathrm{int}^y}{B_i} \end{aligned}$$

(154)

$$\begin{aligned} K_{6,9}= & {} \frac{4(A_{ob}-C_{ob}) n+3 (A_i-C_i) n-\kappa _\mathrm{int}^y}{B_i} \end{aligned}$$

(155)

For $\mathbf {K}_{ns}$, replace $K_{2,7}$ by $\frac{-\kappa _\mathrm{int}^y}{B_s}$ and $K_{6,9}$ by $\frac{(A_{ob}-C_{ob}) n-\kappa _\mathrm{int}^y}{B_i}$.

Appendix 5: A precession model decoupled from the polar motion

We transform the angular momentum Eqs. (54–56) expressed in the coordinates of the shell and interior BF into three equations expressed in the coordinates of the IF attached to the Laplace plane (hence the mention (IF) in the subscripts below):

$$\begin{aligned} \frac{d \mathbf {H}_{s}}{d t}+\varvec{\Omega }_s\wedge \mathbf {H}_s=\varvec{\Gamma }_s\Rightarrow & {} \frac{d \mathbf {H}_{s(IF)}}{d t}=\varvec{\Gamma }_{s(IF)}, \end{aligned}$$

(156)

$$\begin{aligned} \frac{d \mathbf {H}_{o(s)}}{d t}+\varvec{\Omega }_s\wedge \mathbf {H}_{o(s)}=\varvec{\Gamma }_{o(s)}\Rightarrow & {} \frac{d \mathbf {H}_{o(IF)}}{d t}=\varvec{\Gamma }_{o(IF)}, \end{aligned}$$

(157)

$$\begin{aligned} \frac{d \mathbf {H}_i}{d t}+\varvec{\Omega }_i\wedge \mathbf {H}_i=\varvec{\Gamma }_i\Rightarrow & {} \frac{d \mathbf {H}_{i(IF)}}{d t}=\varvec{\Gamma }_{i(IF)}. \end{aligned}$$

(158)

Note that we do not consider the periodic tidal deformations of the satellite and the effects of atmosphere and lakes here.

By analogy with the angular momentum of a solid satellite (Eq. A.13 of Baland et al. (2012)), the angular momentum of a layer k can simply be written as

$$\begin{aligned} \mathbf {H}_{k(IF)}=n C_k \hat{s}_k, \end{aligned}$$

(159)

with $\hat{s}_k=(s_x^k,s_y^k,s_z^k\simeq 1)$ the unit vector along the rotation axis.

The torque $\varvec{\Gamma }_{k(IF)}$ on layer k is the sum of the external torque by the parent planet and of the internal gravitational torque exerted by the other layers, both corrected for the effect of the hydrostatic pressure, and of the hydrodynamic pressure torque.

In the coordinates of the shell or interior BF, the expressions of the torque are given in Eqs. (109–111) and Eqs. (88–89). We transform them to the IF thanks to the following transformation matrices, defined by analogy with the transformation from a solid satellite BF to the IF (Eq. A.8 of Baland et al. (2012)):

$$\begin{aligned} R_{(s\rightarrow IF)}= & {} R_z(-\xi _{s}).R_x(-\theta _s).R_z(-\phi _s), \end{aligned}$$

(160)

$$\begin{aligned} R_{(i\rightarrow IF)}= & {} R_z(-\xi _{i}).R_x(-\theta _i).R_z(-\phi _i). \end{aligned}$$

(161)

Note that the polar motion and longitudinal librations of the solid layers are neglected in the matrices since we consider a decoupled model. As the precession is a slow motion, the torques are averaged over the orbit period, and we obtain

$$\begin{aligned} \varvec{\Gamma }_{s(IF)}= & {} n\kappa _{s} (\hat{s}_{s} \wedge \hat{n})- n K (\hat{s}_{s} \wedge \hat{s}_{i})+\varvec{\Gamma }_{s(IF),\mathrm{phd}}, \end{aligned}$$

(162)

$$\begin{aligned} \varvec{\Gamma }_{o(IF)}= & {} \varvec{\Gamma }_{o(IF),\mathrm{phd}}, \end{aligned}$$

(163)

$$\begin{aligned} \varvec{\Gamma }_{i(IF)}= & {} n\kappa _{i} (\hat{s}_{i} \wedge \hat{n}) +n K (\hat{s}_{s} \wedge \hat{s}_{i})+\varvec{\Gamma }_{i(IF),\mathrm{phd}}, \end{aligned}$$

(164)

with $\hat{n}$ defined as in Eq. (103). The coupling constants of the external and internal torques, $\kappa _{s/i}$ and K, respectively, are given by

$$\begin{aligned} \kappa _s= & {} \frac{3}{2}n (C_s-A_s+C_{ot}-A_{ot}), \end{aligned}$$

(165)

$$\begin{aligned} \kappa _i= & {} \frac{3}{2}n (C_i-A_i+C_{ob}-A_{ob}), \end{aligned}$$

(166)

$$\begin{aligned} K= & {} -\frac{8\pi G}{5n}\left( \left( C_{i+ob}-\frac{A_{i+ob} +B_{i+ob}}{2}\right) (\rho _s(\alpha _s-\alpha _o)+ \rho _o\alpha _o)\right. \nonumber \\&\left. +\left( \frac{B_{i+ob}-A_{i+ob}}{4}\right) (\rho _s(\beta _s- \beta _o)+\rho _o\beta _o)\right) , \end{aligned}$$

(167)

(see also Baland et al. 2012, 2016 for the terms in $\kappa _{s,i}$ and K).

Expressions for the hydrodynamic pressure torques in the IF at the top and bottom of the liquid layer have been derived by Peale et al. (2014, 2016) for a biaxial planet (simply called pressure torque therein). Here we present a full demonstration for a triaxial satellite. As we have seen above, in the shell BF, the hydrodynamic torque on the shell $\varvec{\Gamma }_{s,\mathrm{phd}}$ reads as Eq. (88). We need to express this torque in the variables to be solved for in the decoupled model, which are $(s_x^s=\theta _s\cos (\xi _s-\pi /2), s_y^s=\theta _s\sin (\xi _s-\pi /2),s_x^o=\theta _o\cos (\xi _o-\pi /2),$ and $s_y^o=\theta _o\sin (\xi _o-\pi /2))$, instead of the variables ($m_x^o,m_y^o$) of the coupled model. This can be done by noting that in the shell BF, neglecting librations, the ocean rotation vector is given by (see Eq. 63)

$$\begin{aligned} \varvec{\Omega }_{o(s)}=n\left( \begin{array}{c} m^o_x\\ m^o_y\\ 1 \end{array}\right) \end{aligned}$$

(168)

but can also be written by transforming the ocean rotation vector in the reference frame related to the rotation equator (0, 0, n) to the BF of the shell:

$$\begin{aligned} \varvec{\Omega }_{o(s)}= & {} R_z[\phi _s].R_x[\theta _s].R_z[\xi _s-\xi _o].R_x[-\theta _o].\left( \begin{array}{c} 0\\ 0\\ n \end{array}\right) , \end{aligned}$$

(169)

with $\phi _s=\,-\,\xi _s+\varOmega +\omega -\pi +M$, so that

$$\begin{aligned} m^o_x= & {} \theta _o \sin \left( M+\omega -\xi _o+\varOmega \right) -\theta _s \sin \left( M+\omega -\xi _s+\varOmega \right) , \end{aligned}$$

(170)

$$\begin{aligned} m^o_y= & {} \theta _o \cos \left( M+\omega -\xi _o+\varOmega \right) -\theta _s \cos \left( M+\omega -\xi _s+\varOmega \right) . \end{aligned}$$

(171)

We stress that, although we have neglected the polar motion $\mathbf {m}_s$ and $\mathbf {m}_i$ of the solid layers in order to obtain a decoupled precession model, we do not neglect the motion of the ocean rotation axis with respect to the shell BF $\mathbf {m}_o$, which is not an ocean polar motion, as it is defined with respect to the BF of the shell. We then express the torque in the IF thanks to the appropriate rotations:

$$\begin{aligned} \varvec{\Gamma }_{s(IF),\mathrm{phd}}=R_{(s\rightarrow IF)}.\varvec{\Gamma }_{s,\mathrm{phd}} \end{aligned}$$

(172)

and average it over an orbit period with the slowly varying angles $\varOmega , \xi _s$ and $\xi _o$ held constant, to get

$$\begin{aligned} \varvec{\Gamma }_{s(IF),\mathrm{phd}}= & {} n\left( C_{ot}-\frac{A_{ot}+ B_{ot}}{2}\right) (\hat{s}_s \wedge \hat{s}_o). \end{aligned}$$

(173)

This torque has the same form as in the axially symmetric case, as $(A_{ot}+B_{ot})/2$ is the mean equatorial moment of inertia of the top ocean.

Similarly, it is possible to show that the hydrodynamic pressure torque on the interior, expressed in the IF and averaged over the orbit period, is given by

$$\begin{aligned} \varvec{\Gamma }_{i(IF),\mathrm{phd}}= & {} n\left( C_{ob}-\frac{A_{ob}+ B_{ob}}{2}\right) (\hat{s}_i \wedge \hat{s}_o) \end{aligned}$$

(174)

and that the corresponding torque on the ocean can be written as

$$\begin{aligned} \varvec{\Gamma }_{o(IF),\mathrm{phd}}=-\varvec{\Gamma }_{s(IF),\mathrm{phd}}-\varvec{\Gamma }_{i(IF),\mathrm{phd}}. \end{aligned}$$

(175)

Making use of Eqs. (162–164) and of Eqs. (173–175), the system of angular momentum equations (156–158) becomes

$$\begin{aligned} \dot{\mathbf {u}}_d+\mathbf {K_d} . \mathbf {u}_d = n\left( \begin{array}{c} \kappa _s N \\ 0\\ \kappa _i N \\ \end{array}\right) , \end{aligned}$$

(176)

with

$$\begin{aligned} \mathbf {u}_d = \left( \begin{array}{c} S_s\\ S_o\\ S_i \end{array}\right) ,\quad \mathbf {K_d}=\left( \begin{array}{ccc} \frac{K-\kappa _s-\kappa _\mathrm{phd}^s}{C_s} &{} \frac{\kappa _\mathrm{phd}^s}{C_s} &{} -\frac{K}{C_s} \\ \frac{\kappa _\mathrm{phd}^s}{C_o} &{} \frac{-\kappa _\mathrm{phd}^i-\kappa _\mathrm{phd}^s}{C_o} &{} \frac{\kappa _\mathrm{phd}^i}{C_o} \\ -\frac{K}{C_i} &{} \frac{\kappa _\mathrm{phd}^i}{C_i} &{} \frac{K-\kappa _i-\kappa _\mathrm{phd}^i}{C_i} \\ \end{array} \right) . \end{aligned}$$

(177)

$\kappa ^{s/i}_\mathrm{phd}$ are the coupling constants of the hydrodynamic pressure torques

$$\begin{aligned} \kappa _\mathrm{phd}^s= & {} n\left( C_{ot}-\frac{A_{ot}+B_{ot}}{2}\right) , \end{aligned}$$

(178)

$$\begin{aligned} \kappa _\mathrm{phd}^i= & {} n\left( C_{ob}-\frac{A_{ob}+B_{ob}}{2}\right) . \end{aligned}$$

(179)

$S_j=s_x^j+Is_y^j$ and $N=n_x+In_y$ are the projections onto the Laplace plane of the spin and orbit unit vectors, respectively.

For a uniformly precessing satellite, at first order in small orbital inclination i, the forcing is given by

$$\begin{aligned} N\simeq i\, e^{I (\varOmega -\pi /2)}, \end{aligned}$$

(180)

with $\varOmega$ the longitude of the orbital ascending node. By substituting the projected spin vectors in the explicit form, correct up to the first order in i and in constant over time orbital obliquities $\eta _0^j$, of

$$\begin{aligned} S_j\simeq (i+\eta _0^j)e^{I (\varOmega -\pi /2)} \end{aligned}$$

(181)

into Eq. (176), we find that

$$\begin{aligned} \eta _0^{j}= & {} -\frac{i\, \dot{\varOmega }\, n_j}{C_sC_oC_i(d_0+d_1\dot{\varOmega } +d_2\dot{\varOmega }^2 + \dot{\varOmega }^3)}, \end{aligned}$$

(182)

with

$$\begin{aligned} n_{s}= & {} \left[ C_s \left( \kappa _\mathrm{phd}^i (\dot{\varOmega } (C_i+C_o)+\kappa _{i}+\kappa _\mathrm{phd}^s)+(C_i \dot{\varOmega }+\kappa _{i})\right. \right. \nonumber \\&\left. \left. (C_o \dot{\varOmega }+\kappa _\mathrm{phd}^s)-K (C_o \dot{\varOmega }+\kappa _\mathrm{phd}^i+\kappa _\mathrm{phd}^s)\right) \right. \end{aligned}$$

(183)

$$\begin{aligned}&\left. +\,C_o \left( \kappa _\mathrm{phd}^s (C_i \dot{\varOmega }+\kappa _{i}+\kappa _\mathrm{phd}^i)-K (\kappa _\mathrm{phd}^i+\kappa _\mathrm{phd}^s)\right) \right. \end{aligned}$$

(184)

$$\begin{aligned}&\left. -\,C_i \left( K (C_o \dot{\varOmega }+\kappa _\mathrm{phd}^i+\kappa _\mathrm{phd}^s)-\kappa _\mathrm{phd}^i \kappa _\mathrm{phd}^s\right) \right] , \end{aligned}$$

(185)

$$\begin{aligned} n_{o}= & {} \left[ C_o \left( (C_i \dot{\varOmega }+\kappa _{i}+\kappa _\mathrm{phd}^i) (C_s \dot{\varOmega }+\kappa _{s}+\kappa _\mathrm{phd}^s)\right. \right. \end{aligned}$$

(186)

$$\begin{aligned}&\left. \left. -\,K (\dot{\varOmega } (C_i+C_s)+\kappa _{i}+\kappa _\mathrm{phd}^i+\kappa _{s}+\kappa _\mathrm{phd}^s)\right) \right. \end{aligned}$$

(187)

$$\begin{aligned}&\left. +\,C_i \left( C_s \dot{\varOmega } (\kappa _\mathrm{phd}^i+\kappa _\mathrm{phd}^s)+\kappa _\mathrm{phd}^i (\kappa _{s}+\kappa _\mathrm{phd}^s)-K (\kappa _\mathrm{phd}^i+\kappa _\mathrm{phd}^s)\right) \right. \end{aligned}$$

(188)

$$\begin{aligned}&\left. +\,C_s \left( \kappa _\mathrm{phd}^s (\kappa _{i}+\kappa _\mathrm{phd}^i)-K (\kappa _\mathrm{phd}^i+\kappa _\mathrm{phd}^s)\right) \right] , \end{aligned}$$

(189)

$$\begin{aligned} n_i= & {} \left[ C_i \left( \dot{\varOmega } (C_o (\kappa _{s}+\kappa _\mathrm{phd}^s)+C_s (\kappa _\mathrm{phd}^i+\kappa _\mathrm{phd}^s))+C_o C_s \dot{\varOmega }^2\right. \right. \end{aligned}$$

(190)

$$\begin{aligned}&\left. \left. -\,K (C_o \dot{\varOmega }+\kappa _\mathrm{phd}^i+\kappa _\mathrm{phd}^s)+\kappa _\mathrm{phd}^i \kappa _{s}+\kappa _\mathrm{phd}^i \kappa _\mathrm{phd}^s+\kappa _\mathrm{phd}^i+\kappa _{s} \kappa _\mathrm{phd}^s\right) \right. \end{aligned}$$

(191)

$$\begin{aligned}&\left. +\,C_o \left( \kappa _\mathrm{phd}^i (C_s \dot{\varOmega }+\kappa _{s}+\kappa _\mathrm{phd}^s)-K (\kappa _\mathrm{phd}^i+\kappa _\mathrm{phd}^s)\right) \right. \end{aligned}$$

(192)

$$\begin{aligned}&\left. -\,C_s \left( K (C_o \dot{\varOmega }+\kappa _\mathrm{phd}^i+\kappa _\mathrm{phd}^s)-\kappa _\mathrm{phd}^i \kappa _\mathrm{phd}^s\right) \right] , \end{aligned}$$

(193)

$$\begin{aligned} d_0= & {} \frac{\kappa _{i} \kappa _\mathrm{phd}^i \kappa _{s}+\kappa _{i} \kappa _\mathrm{phd}^i \kappa _\mathrm{phd}^s+\kappa _{i} \kappa _{s} \kappa _\mathrm{phd}^s+\kappa _\mathrm{phd}^i \kappa _{s} \kappa _\mathrm{phd}^s}{C_i C_o C_s} \end{aligned}$$

(194)

$$\begin{aligned}&+\frac{-\kappa _{i} \kappa _\mathrm{phd}^i K-\kappa _{i} \kappa _\mathrm{phd}^s K-\kappa _\mathrm{phd}^i \kappa _{s} K-\kappa _{s} \kappa _\mathrm{phd}^s K}{C_i C_o C_s}, \end{aligned}$$

(195)

$$\begin{aligned} d_1= & {} \frac{\kappa _i \kappa _\mathrm{phd}^i+\kappa _i \kappa _\mathrm{phd}^s+\kappa _\mathrm{phd}^i \kappa _\mathrm{phd}^s-\kappa _\mathrm{phd}^i K-\kappa _\mathrm{phd}^s K}{C_i C_o} \end{aligned}$$

(196)

$$\begin{aligned}&+\,\frac{\kappa _i \kappa _s+\kappa _i \kappa _\mathrm{phd}^s+\kappa _\mathrm{phd}^i \kappa _s+\kappa _\mathrm{phd}^i \kappa _\mathrm{phd}^s}{C_i C_s} \end{aligned}$$

(197)

$$\begin{aligned}&+\,\frac{-\kappa _i K-\kappa _\mathrm{phd}^i K-\kappa _s K-\kappa _\mathrm{phd}^s K}{C_i C_s} \end{aligned}$$

(198)

$$\begin{aligned}&+\,\frac{\kappa _\mathrm{phd}^i \kappa _s+\kappa _\mathrm{phd}^i \kappa _\mathrm{phd}^s+\kappa _s \kappa _\mathrm{phd}^s-\kappa _\mathrm{phd}^i K-\kappa _\mathrm{phd}^s K}{C_o C_s}, \end{aligned}$$

(199)

$$\begin{aligned} d_2= & {} \frac{\kappa _{i}+\kappa _\mathrm{phd}^i-K}{C_i}+\frac{\kappa _\mathrm{phd}^i+\kappa _\mathrm{phd}^s}{C_o}+\frac{\kappa _{s}+\kappa _\mathrm{phd}^s-K}{C_s}. \end{aligned}$$

(200)

By analogy with the constant over time obliquity $\eta _0$ from Eq. (44) for the solid case, we use the subscript 0 to indicate that the quantities $\eta _0^j$ correspond to the solution of a precession model decoupled from the polar motion of the solid layers.