Librations of a body composed of a deformable mantle and a fluid core

Abstract

We present fully three-dimensional equations to describe the rotations of a body made of a deformable mantle and a fluid core. The model in its essence is similar to that used by INPOP19a (Integration Planétaire de l’Observatoire de Paris) Fienga et al. (INPOP19a planetary ephemerides. Notes Scientifiques et Techniques de l’Institut de Mécanique Céleste, vol 109, 2019), and by JPL (Jet Propulsion Laboratory) (Park et al. The JPL Planetary and Lunar Ephemerides DE440 and DE441. Astron J 161(3):105, 2021. doi:10.3847/1538-3881/abd414), to represent the Moon. The intended advantages of our model are: straightforward use of any linear-viscoelastic model for the rheology of the mantle; easy numerical implementation in time-domain (no time lags are necessary); all parameters, including those related to the “permanent deformation”, have a physical interpretation. The paper also contains: (1) A physical model to explain the usual lack of hydrostaticity of the mantle (permanent deformation). (2) Formulas for free librations of bodies in and out-of spin-orbit resonance that are valid for any linear viscoelastic rheology of the mantle. (3) Formulas for the offset between the mantle and the idealised rigid-body motion (Peale’s Cassini states). (4) Applications to the librations of Moon, Earth, and Mercury that are used for model validation.

Appendices

The value of the coefficients $c_1,c_2,c_3$ in Eq.  (2.23) for a body in a Cassini state

The theoretical determination of the almost equilibrium orientation of the spin axis of an extended body, or the guiding motion as described in Sect. 2, is a difficult task that requires some knowledge about the rheological behaviour of the body. An important situation, which will be the one considered in this appendix, is that of a rigid body under the influence of a single point mass that moves according to a Keplerian precessing orbit. In this case the guiding motion is called “a Cassini state” Peale (1969). We remark that different Cassini states can be obtained under different rheological hypotheses (see, for instance, Boué (2020) for bodies with a rigid mantle and a fluid core).

Let $\kappa$ be the inertial frame and ${\mathrm {K}}_g$ be the guiding frame described in Sect. 2. We recall that the common origin of both frames is the centre of mass of the extended body. The position of the point mass with respect to the extended body in the frame $\kappa$ is given by

$$\begin{aligned} {\mathbf {r}}={\mathbf {R}}_{\mathbf {3}}(\varOmega _p)\mathbf {R_1}(\iota _p){\mathbf {R}}_{\mathbf {3}}(\omega _p) \begin{bmatrix} r \cos (f_p) \\ r \sin (f_p) \\ 0 \end{bmatrix}, \end{aligned}$$

(A.1)

where: $\iota _p$, $f_p$, $\omega _p$, and $\varOmega _p$ are the inclination with respect to an invariable plane (the “Laplace plane”), the true anomaly, the argument of the periapsis and the longitude of the ascending node, respectively, of a classical Keplerian orbit that has semi-major axis $a_p$, eccentricity e, and mean anomaly $M_p$ (the index p stands for point-mass).

We recall that the three axes of ${\mathrm {K}}_g$ are those of the body’s principal moments of inertia, $\overline{I}_{1}< \overline{I}_{2}< \overline{I}_{3}$, in the absence of deformation. The orientation of the guiding frame $\mathbf {R_g}:{\mathrm {K}}_g\rightarrow \kappa$ is given by three Euler angles

$$\begin{aligned} \mathbf {R_g}={\mathbf {R}}_{\mathbf {3}}(\psi _g)\mathbf {R_1}(\theta _g){\mathbf {R}}_{\mathbf {3}}(\phi _g), \end{aligned}$$

(A.2)

where: $\theta _g$ is the inclination of the body equator to the Laplace plane²⁷, $\psi _g$ is the longitude of the ascending node of the body equator, and $\phi _g$ is the angle between the ascending node and Axis 1. There are two possible orientations of Axis 1, so $\varvec{R}_g$ is not uniquely defined. In the case of the Moon it is usual to choose the positive orientation of Axis 1 as that pointing towards the Earth (see, e.g. Eckhardt 1981). To change from one orientation to the other it is enough to change the signs of both unit vectors $\varvec{e}_1$ and $\varvec{e}_2$.

The orientation of the slow frame is given by ${\mathbf {R}}_s(t)= {\mathbf {R}}_g(t){\mathbf {R}}_{\mathbf {3}}^{-1}(\omega \, t):{\mathrm {K}}_s\rightarrow \kappa$, Eq. (2.14), that in the present case becomes

$$\begin{aligned} {\mathbf {R}}_s={\mathbf {R}}_{\mathbf {3}}(\psi _g)\mathbf {R_1}(\theta _g){\mathbf {R}}_{\mathbf {3}}(\zeta )\quad \text {where}\quad \zeta =\phi _g-\omega t\,. \end{aligned}$$

(A.3)

The identity $\langle \varvec{\omega }_{s,g}\,,\varvec{e}_3\rangle =0$ implies

$$\begin{aligned} {{\dot{\zeta }}}=- {{\dot{\psi }}}_g\cos \theta _g\,. \end{aligned}$$

(A.4)

The position of the point mass in the guiding frame ${\mathrm {K}}_g$ is given by $\mathbf {r_g}=\mathbf {R_g}^{-1}{\mathbf {r}}$ and Eqs. (A.1) and (A.2) imply

$$\begin{aligned} \mathbf {r_g}=r \Big ( \cos \big (f_p\big ) \varvec{u} + \sin \big (f_p\big ) \varvec{v} \Big ), \end{aligned}$$

(A.5)

where $\varvec{u}$ and $\varvec{v}$ are the first and second columns, respectively, of the matrix $\mathbf {R_g}^{-1}{\mathbf {R}}_{\mathbf {3}}(\varOmega _p)\mathbf {R_1}(\iota _p){\mathbf {R}}_{\mathbf {3}}(\omega _p)$. The force matrix in the guiding frame ${\mathbf {J}}_{g}$, as defined in Sect. 2, is given by

$$\begin{aligned} \begin{aligned}&J_{gij}=\bigg (\frac{3\mathcal{G}m_p}{|\mathbf {r_g}|^5}\mathbf {r_g}\otimes \mathbf {r_g} \bigg )_{ij}\\&\ \quad = \frac{3\mathcal{G}m_p}{2r^3}\bigg (\big (u_iu_j+v_iv_j\big )+ \cos \big (2f_p\big )\big (u_iu_j-v_iv_j\big )+ \sin \big (2f_p\big )\big (u_iv_j+v_iu_j\big )\bigg )\, .\end{aligned} \end{aligned}$$

(A.6)

Notice that the vectors $\varvec{u}$ and $\varvec{v}$ in Eq. (A.5) are the rows of an orthogonal matrix, which implies $|\varvec{u}|=|\varvec{v}|=1$ and $\langle \varvec{u},\varvec{v}\rangle =0$. So,

$$\begin{aligned} {{\,\mathrm{Tr}\,}}\varvec{J}_g=\frac{3\mathcal{G}m_p}{r^3}\, . \end{aligned}$$

(A.7)

Now we impose that the extended body is in a Cassini state that is characterised by three laws Colombo (1966), Peale (1969), Peale (1974), which are generalisations of the original laws of Cassini (1693) (aimed to describe the motion of the Moon, see Eckhardt (1981)).

• 1- The body spin is in s-to-2 resonance with the orbital mean motion, where s is a positive integer (for the Moon $s=2$ and for Mercury $s=3$);

• 2- The inclination of the body equator $\theta _g$ with respect to the Laplace plane is constant;

• 3- Either the ascending node (state 1) or the descending node (state 2) of the body equator on the Laplace plane precesses in coincidence with the ascending node of the orbit on the Laplace plane (the spin axis, orbit normal, and Laplace plane normal are coplanar).

The third law implies that either

$$\begin{aligned} \begin{array}{rll} \psi _g&{}=\varOmega _p \quad &{}\text {state 1 (e.g. Mercury) or}\\ \psi _g&{}=\varOmega _p+\pi \quad &{}\text {state 2 (e.g. Moon),} \end{array} \end{aligned}$$

(A.8)

then the second law implies that the inclination of the body spin axis with respect to the normal to the orbital plane, given by ($\theta _g\ge 0$, $\iota _p\ge 0$)

$$\begin{aligned} \begin{array}{rll} \chi _p&{}=\theta _g-\iota _p\ge 0\quad &{} \text {state 1}\\ \chi _p&{}=\theta _g+\iota _p\quad &{}\text {state 2}\end{array} \end{aligned}$$

(A.9)

is constant (the inequality in state 1 is of dynamical origin, see Peale (1969) paragraph Eq. (18)); and then the first law implies that $s M_p=2(\phi _g-\omega _p)$²⁸$^{,}$²⁹.

For a body in a Cassini state 2 (e.g. Moon) Eq. (A.5) becomes³⁰

$$\begin{aligned} \mathbf {r_g}=r \cos f_p\underbrace{\begin{bmatrix} -\cos \phi _g \cos \omega _p - \cos \chi _p \sin \phi _g \sin \omega _p\\ \cos \omega _p \sin \phi _g - \cos \phi _g \cos \chi _p \sin \omega _p\\ \sin \chi _p \sin \omega _p \end{bmatrix}}_{ \varvec{u}}+ r \sin f_p\underbrace{\begin{bmatrix} -\cos \chi _p \cos \omega _p \sin \phi _g + \cos \phi _g \sin \omega _p\\ -\cos \phi _g \cos \chi _p \cos \omega _p - \sin \phi _g \sin \omega _p\\ \cos \omega _p \sin \chi _p \end{bmatrix}}_{\varvec{v}} \qquad \end{aligned}$$

(A.10)

and for a body in a Cassini state 1 we must replace $\varvec{u}\rightarrow -\varvec{u}$ and $\varvec{v}\rightarrow -\varvec{v}$. Since $J_{gij}$ given in Eq. (A.6) depends only on the products of components of $\varvec{u}$ and $\varvec{v}$, the matrix $\varvec{J}_g$ has the same expression for both states 1 and 2. Therefore, the following computation of $c_1$, $c_2$ and $c_3$ holds in both cases.

The coefficients $c_1,c_2,$ and $c_3$ we aim to compute are determined by the constant term in the Fourier expansion of $J_{gij}$ given in Eq. (A.6). In order to compute the Fourier expansion of the factors involving r and $f_p$ we use the Hansen coefficients defined by³¹

$$\begin{aligned} \bigg (\frac{r}{a_p}\bigg )^n\mathrm{e} ^{i m f}=\sum _{k=-\infty }^\infty X^{n,m}_k(e) \mathrm{e} ^{i k M}. \end{aligned}$$

(A.13)

The computation of the coefficients $c_1,c_2,$ and $c_3$ in Eq. (2.23) can be done in the following way. At first we use Eq. (A.7) to obtain

$$\begin{aligned} c_3=\frac{\mathcal{G}m_p}{\omega ^2 a_p^3} X_0^{-3,0}(e)\, . \end{aligned}$$

(A.14)

The term $\left( \frac{3\mathcal{G}m_p}{\omega ^2|\mathbf {r_g}|^5}\mathbf {r_g}\otimes \mathbf {r_g}\right) _{33}$ can be easily computed using Eqs. (A.6) and (A.12) and the result is $\frac{3\mathcal{G}m_p}{2 \omega ^2 a_p^3} X_0^{-3,0}(e)\sin ^2(\chi _p)=c_3-2c_1/3$. So, we obtain that

$$\begin{aligned} c_1=\frac{3\,\mathcal{G}m_p}{2\,\omega ^2 a_p^3} X_0^{-3,0}(e)\bigg (1-\frac{3}{2}\sin ^2(\chi _p)\bigg )\,. \end{aligned}$$

(A.15)

In order to compute the constant $c_2$ associated with an $s-to-2$ spin-orbit resonance it is enough to compute the constant term in the Fourier expansion of $\left( \frac{3\mathcal{G}m_p}{\omega ^2|\mathbf {r_g}|^5}\mathbf {r_g}\otimes \mathbf {r_g}\right) _{11}$ and then to use that this term is equal to $c_1/3+c_2+c_3$, Eq. (2.23). Assuming that the precession of the periapsis is different from zero a computation gives

$$\begin{aligned} c_2=\frac{3\,\mathcal{G}m_p}{2\,\omega ^2 a_p^3} X_s^{-3,2}(e) \cos ^4(\chi _p/2). \end{aligned}$$

(A.11)

Another case of interest is that of a point mass that moves in a precessing Keplerian orbit, as that parameterised in Eq. (A.1), with constant inclination, $\iota _p=$constant, and with no spin-orbit resonance. By no spin-orbit resonance we mean that the constant frequencies ${{\dot{\psi }}}_g$, ${{\dot{\varOmega }}}_p$, $\omega$, ${{\dot{\omega }}}_p$, and $\dot{M}$ are noncomensurable. Equation (A.7) implies that $c_3$ is again given by equation (A.13) and the condition of no spin-orbit resonance implies $c_2=0$. As before, to compute $c_1$ we look for a term in $J_{g33}$ that is constant in time. Using that $X^{-3,\pm 2}_0=0$ and the no spin-orbit hypothesis we obtain that

$$\begin{aligned} \frac{3\mathcal{G}m_p}{2r^3}\bigg (\cos \big (2f_p\big )\big (u_3^2-v_3^2\big )+ \sin \big (2f_p\big )2 u_3v_3\bigg ) \end{aligned}$$

does not contain any term that is constant. Using that $-\dot{\psi }_g+{{\dot{\varOmega }}}_p\ne 0$, an analysis of the remaining term, $\frac{3\mathcal{G}m_p}{2r^3}\big (u_3^2+v_3^2\big )$, shows that the constant term of $J_{g33}$ is

$$\begin{aligned} \frac{3\mathcal{G}m_p}{2 a_p^3} X^{-3,0}_0 \frac{1}{8} \Big (-\cos (2 \theta _g)-(3 \cos (2 \theta _g)+1) \cos (2 \iota _p)+5\Big )= \omega ^2\left( c_3-\frac{2}{3}c_1\right) . \end{aligned}$$

So, using Eq. (A.13) for $c_3$ we obtain that for no spin-orbit resonance:

$$\begin{aligned} \begin{aligned} c_1&=\frac{3\mathcal{G}m_p}{\omega ^2 a_p^3} X^{-3,0}_0 \frac{1}{32} \Big (3 \cos (2 \theta _g)+1\Big ) \Big (3 \cos (2 \iota _p)+1\Big )\\ c_2&=0 \end{aligned}. \end{aligned}$$

(A.16)

If there are several point masses, $\beta =1,2,\ldots$, orbiting the extended body, each one in a precessing Keplerian orbit, with constant inclination, $\iota _{\beta }=$constant, and with no spin-orbit resonance; as in the case where the Earth is the extended body and the Moon and the Sun represent the point masses; then $c_2=0$ and (the index p was omitted in several constants)

$$\begin{aligned} c_1\!=\!\bigg (\sum _\beta s_\beta \bigg )\frac{1+3 \cos (2 \theta _g)}{4}\quad \text {where}\quad s_\beta \!=\! \frac{3\mathcal{G}m_\beta }{\omega ^2 a_\beta ^3} (1-e_\beta ^2)^{-3/2}\, \frac{1+3 \cos (2 \iota _\beta )}{8}\,. \qquad \end{aligned}$$

(A.17)

In this case the average rate of precession ${{\dot{\psi }}}_g$ of the spin axis of the extended body is Williams (1994):

$$\begin{aligned} \frac{{{\dot{\psi }}}_g}{\omega }=-\big (\sum _\beta s_\beta \big ) \frac{\overline{I}_3-\overline{I}_e}{\overline{I}_3} \cos \theta _g\approx -\big (\sum _\beta s_\beta \big ) \overline{\alpha }_e\, \cos \theta _g\,; \end{aligned}$$

(A.18)

and the motion of the slow frame is determined by $\varvec{R}_s$ as given in Eqs. (A.3) and (A.4) with $\theta _g=$constant.

A simple model: Considerations about the guiding frame

In this appendix, we show by means of an example how the inertial forces that appear due to the non-inertial character of the guiding frame are mostly cancelled out by external torques. This example will be used in Appendix E.

The problem is to describe the axial precession of an extended rigid body of mass m and moment of inertia $\varvec{I}$ under the gravitational force of a point of mass $m_p$ that moves in a circular orbit of radius $a_p$ and with angular frequency n. We assume that the body is axisymmetric with $\overline{I}_1=\overline{I}_2=\overline{I}_e<\overline{I}_3$. We define a body frame ${\mathrm {K}}$ such that the $\varvec{e}_3$-axis is the $\overline{I}_3$-principal axis and an inertial frame $\kappa$ such that the orbit of the point mass has components $(a_p\cos (nt),a_p\sin (nt),0)$.

Let $\omega >0$ be the initial spin angular speed of the body that is defined as the projection of the initial angular velocity vector $\varvec{\omega }$ on the $I_3$-principal axis. We assume that the angular velocity is almost aligned with the $I_3$-principal axis and that inertial forces prevail over the gravitational torque, namely

$$\begin{aligned} \omega /\Vert \varvec{\omega }\Vert \approx 1\quad \text {and}\quad \frac{3}{2} \frac{G m_p}{\omega ^2 a^3}=s \ll 1. \end{aligned}$$

(B.1)

In this case the torque-free inertial motion dominates and the body rotates almost steadily about the spin-axis. Note that s is the quantity that we denoted as $s_\beta$ in Eq. (A.17).

The tidal-force operator $\varvec{J}= \frac{3\mathcal{G}m_p}{r^5}{\mathbf {r}}\otimes {\mathbf {r}}$ can be decomposed into two terms

$$\begin{aligned} \varvec{J}= \underbrace{s \, \omega ^2\left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{array} \right) }_{= \varvec{J}_0} +\underbrace{s\,\omega ^2 \left( \begin{array}{ccc} \cos (2 n t) &{} \sin (2 n t) &{} 0 \\ \sin (2 n t) &{} -\cos (2 n t) &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{array} \right) }_{=\varvec{J}_1}\,. \end{aligned}$$

(B.2)

The axial precession is determined by the constant part $\varvec{J}_0$, so in the following we consider the problem of the motion of the extended body only under tidal-force operator $\varvec{J}_0$, namely

$$\begin{aligned} \begin{aligned}&\dot{\varvec{{\widehat{\pi }}}} =[\varvec{I}\, ,{\mathbf {J}}_0] \quad \text {where}\quad \varvec{R}:{\mathrm {K}}\rightarrow \kappa \\&\varvec{{\widehat{\pi }}}=\mathrm{\!\ Tr\!\ } \big (\varvec{I}\big ) \varvec{\widehat{\omega }}- \varvec{I} \varvec{{\widehat{\omega }}}- \varvec{\widehat{\omega }}\varvec{I}\,,\quad \varvec{{\widehat{\omega }}}= \dot{\varvec{R}}\varvec{R}^{-1}\,.\\ \end{aligned} \end{aligned}$$

(B.3)

A computation shows that $\varvec{R}=\varvec{R}_3({{\dot{\psi }}} \, t)\varvec{R}_1(\theta ) \varvec{R}_3(-{{\dot{\psi }}} \, t\cos \theta )\varvec{R}_3(\omega \, t)$ is a solution to this equation, with ${{\dot{\theta }}}=\ddot{\psi }=0$, if

$$\begin{aligned} \left( s\frac{I_3-I_e}{I_3} -\frac{I_e}{I_3} \frac{{{\dot{\psi }}}^2}{\omega ^2}\right) \cos \theta +\frac{\dot{\psi }}{\omega }=0. \end{aligned}$$

(B.4)

This equation has two solutions: one with ${{\dot{\psi }}}/\omega >0$ (prograde) and another with ${{\dot{\psi }}}/\omega <0$ (retrograde). Up to leading order in the small parameter s the prograde solution

$$\begin{aligned} \frac{{{\dot{\psi }}}}{\omega }=\frac{I_3}{I_e} \sec \theta \end{aligned}$$

(B.5)

is the fast torque-free precession and the retrograde solution

$$\begin{aligned} \frac{{{\dot{\psi }}}}{\omega }= -s\frac{I_3-I_e}{I_3} \cos \theta \approx -s\overline{\alpha }_e \cos \theta \end{aligned}$$

(B.6)

is the slow precession that exists due to the gravitational interaction.

We will use the retrograde solution to define the guiding-frame:

$$\begin{aligned} \varvec{R}_g=\underbrace{\varvec{R}_3({{\dot{\psi }}}_g \, t)\varvec{R}_1(\theta _g) \varvec{R}_3(-{{\dot{\psi }}}_g \, t\cos \theta _g)}_{=\varvec{R}_s}\varvec{R}_3(\omega \, t)\,, \end{aligned}$$

(B.7)

where ${{\dot{\psi }}}_g<0$ is the solution to Eq. (B.6) with $\theta =\theta _g$. Notice that if $\theta _g=0$ then $\varvec{R}_s=$Identity.

The angular velocities of the guiding frame $\varvec{\widehat{\omega }}_{g,g}=\varvec{R}_g^{-1} \dot{\varvec{R}}_g:{\mathrm {K}}_g\rightarrow {\mathrm {K}}_g$ and of the slow frame $\varvec{{\widehat{\omega }}}_{s,s}=\varvec{R}_s^{-1} \dot{\varvec{R}}_s:{\mathrm {K}}_s\rightarrow {\mathrm {K}}_s$ are:

$$\begin{aligned} \varvec{\omega }_{g,g}=\omega \varvec{e}_3+\underbrace{\varvec{R}_3^{-1}(\omega t) \varvec{\omega }_{s,s}}_{{\varvec{\omega }}_{s,g}}\,,\quad \varvec{\omega }_{s,s}= {{\dot{\psi }}}_g \sin \theta _g\left( \begin{array}{c} - \sin (t{{\dot{\psi }}}_g \cos \theta _g) \\ \ \ \cos (t{{\dot{\psi }}}_g \cos \theta _g)\\ 0 \end{array} \right) . \end{aligned}$$

(B.8)

Since for the problem considered in this section the guiding motion is a solution to the equations of motion, the linearised equations about the guiding motion are just the ordinary linearised equations about a solution. These equations can be obtained directly from Eq. 5.16), for the motion of the mantle, with the following simplifications: $\varvec{\alpha }_m\rightarrow \varvec{a}$ (${\mathbb {I}} +\varvec{{{\widehat{a}}}}:{\mathrm {K}}\rightarrow {\mathrm {K}}_g$), $\varvec{I}_m=\varvec{I}$ and $\varvec{I}_c=0$ (there is no fluid core), $\varvec{\delta }\varvec{B}_T=0$ (the body is rigid), and $c_2=0\Longrightarrow \xi _1=\xi _2=1=c_1$ (there is no spin-orbit resonance) with

$$\begin{aligned} c_1= s\frac{1+3 \cos (2 \theta _g)}{4} \quad \Big (\text {consequence of Eq.} (A.17)\Big )\,. \end{aligned}$$

(B.9)

There are two forcing terms that appear in the right-hand side of Eq. (5.16): the “true-torque” that comes from $\varvec{J}_0$ and the “fictitious-torque” that comes from the non-inertial nature of the guiding frame. The true-torque term is given by:

$$\begin{aligned}{}[\overline{\varvec{I}},\varvec{R}_g^{-1}\varvec{J}_0\varvec{R}_g]^\vee =[\overline{\varvec{I}},{\varvec{\delta }}\varvec{J}_{0,g}]^\vee = s\,\omega ^2\,(\overline{I}_3-\overline{I}_e)\cos \theta _g\sin \theta _g \begin{bmatrix} -\cos \phi _g\\ \ \ \, \sin \phi _g\\ 0 \end{bmatrix}\,, \end{aligned}$$

(B.10)

where we used that: $[\overline{\varvec{I}},\overline{\varvec{J}}]=0$, the check map $^\vee$ is the inverse of the hat map, and $\phi _g=t(\omega -{{\dot{\psi }}}_g\cos \theta _g)$ (see Eqs. (A.3) and (A.4)). The fictitious torque is

$$\begin{aligned} \begin{bmatrix} \overline{I}_{e}\dot{{\omega }}_{s,g1} +\omega \big (\overline{I}_3-\overline{I}_e\big ){\omega }_{s,g2} \\ \overline{I}_{e}\dot{{\omega }}_{s,g2} -\omega \big (\overline{I}_3-\overline{I}_e\big ){\omega }_{s,g1} \\ 0 \end{bmatrix}=\sin \theta _g\Big (\overline{I}_e(\omega -{{\dot{\psi }}}_g\cos \theta _g){{\dot{\psi }}}_g+\omega (\overline{I}_3-\overline{I}_e) {{\dot{\psi }}}_g\Big ) \begin{bmatrix} -\cos \phi _g\\ \ \ \, \sin \phi _g\\ 0 \end{bmatrix}\,,\nonumber \\ \end{aligned}$$

(B.11)

where we used equation ${\varvec{\omega }}_{s,g}$ as given in Eq. (B.8). If we use that ${{\dot{\psi }}}_g$ is a solution to Eq. (B.4), then we obtain that the fictitious torque is equal to minus the true torque. So, the true torque cancels out the fictitious torque.

In general the guiding motion is not a particular solution of the dynamical equations and so, we cannot expect the full cancellation of the fictitious torque. If the guiding motion is a good approximation for the real motion, then the residue after the partial cancellation of the fictitious torque must be of the order of the small terms in the equation.

In conclusion, for Eq. (B.3) the linearised equation about the guiding motion is

$$\begin{aligned} \begin{aligned}&\overline{I}_{e}\ddot{a}_{1}-\omega (2\overline{I}_{e}-\overline{I}_{3}){\dot{a}}_{2} +\omega ^2\xi _1(\overline{I}_3-\overline{I}_e)a_{1}=0\\&\overline{I}_{e}\ddot{a}_{2} +\omega (2\overline{I}_e-\overline{I}_{3}){\dot{a}}_{1} +\omega ^2\xi _2(\overline{I}_3-\overline{I}_e)a_{2}=0\\&\overline{I}_{3}\ddot{a}_{3}=0 \,. \end{aligned} \end{aligned}$$

(B.12)

The classification of the roots of Eq. (6.21) into NDFW and FLL eigenvalues

In this Appendix, we classify the two roots of the equation (6.21), i.e. $x^2-x(f_0+1) (y+z )+(f_0+1)z\,y=0$.

If the discriminant $\varDelta$ of Eq. (6.21) is different from zero, then the equation has two solutions. If the core is negligible $f_0=\frac{\mathrm{I}_{\circ c}}{\mathrm{I}_{\circ m}}=0$, then one of the root is $x_{dw}=y$ and is related to the Nearly Diurnal Free Wobble (NDFW) and the other $x_{\ell a}=z$ is related to the Free Libration in Latitude (FLL). In order to classify a given solution one must deform $f_0>0$ from its current value to $f_0=0$ keeping (y, x) constant. If the function $f_0\rightarrow \varDelta$ remains different from zero during the deformation, then the root will move in the complex plane as a smooth function of $f_0$ and it will eventually become either y, and the root will be classified as $x_{nd}$, or z, and the root will be classified as $x_{\ell a}$.

The discriminant of Eq. (6.21) becomes simpler if we use the variables

$$\begin{aligned}&p:=\frac{1}{2}(y+ z)\,,\quad q:=\frac{1}{2}( y-z)\Longrightarrow y=p+q\,,\quad z=p-q\,,\nonumber \\&\text {and}\quad \varDelta =4 (f_0+1) \left( f_0 \,p^2+q^2\right) \,. \end{aligned}$$

(C.1)

The two solutions to Eq. (6.21) are $(f_0+1)p\pm \sqrt{1+f_0}\sqrt{f_0 \,p^2+q^2}$, where denotes the principal value of the square root, which is not continuous on the non-positive real axis and satisfies Re $\sqrt{z}\ge 0$ with $\sqrt{-1}=i$. The lack of continuity of the principal value of the square root leads to some difficulties in the classification of the roots because $f_0\rightarrow \varDelta$ can cross the non-positive real axis during the variation of $f_0$. The eigenvalues are given by the following algorithm obtained essentially from Fig. 15.

Notation: (1) Given (y, z) we define $q:=q_r+iq_i=(y_r-z_r)/2+i(y_i-z_i)/2$ and $p:=p_r+ip_i=(y_r+z_r)/2+i(y_i+z_i)/2$, as in Eq. (C.1), and $p^2:=p_{2r}+i p_{2i}=(p^2_r-p^2_i)+i2 p_rp_i$ and $=q^2:=q_{2r}+i q_{2i}=(q^2_r-q^2_i)+i2 q_rq_i$. (2) We define the set of inequalities

$$\begin{aligned} V=\bigg \{q_{2r}<0\,,q_{2i}<0\,, \frac{q_{2i}}{q_{2r}}<\frac{p_{2i}}{p_{2r}}\,, \ \text {and} \ f_0>-\frac{q_{2r}}{p_{2r}}\bigg \}\,. \end{aligned}$$

(C.2)

All the inequalities in V hold if $p^2$, $q^2$ and $f_0$ are arranged as illustrated in Fig. 15.

Assumptions: (1) To simplify the analysis we assume $p_i/p_r<1$, which implies that $p^2$ is in the positive quadrant of the complex plane, $p_{2r}>0$ and $p_{2i}>0$ (the analysis could also be made without this hypothesis). (2) We also assume that $|y|\ne |z|$ in order to avoid $\varDelta =0$ during a variation of $f_0$.

• If $y_r>z_r$, then either all inequalities in V are true and Eq. (C.4) holds or at least one inequality in V is not true and Eq. (C.3) holds.

• If $y_r<z_r$, then either all inequalities in V are true and Eq. (C.3) holds or at least one inequality in V is not true and Eq. (C.4) holds.

$$\begin{aligned}&x_{dw}=(1+f_0)p+\frac{\sqrt{\varDelta }}{2}\,,\quad x_{\ell a}=(1+f_0)p-\frac{\sqrt{\varDelta }}{2}\,; \end{aligned}$$

(C.3)

$$\begin{aligned}&x_{dw}=(1+f_0)p-\frac{\sqrt{\varDelta }}{2}\,,\quad x_{\ell a}=(1+f_0)p+\frac{\sqrt{\varDelta }}{2}\,; \end{aligned}$$

(C.4)

where $\varDelta =4 (f_0+1) \left( f_0 \,p^2+q^2\right)$.

Fig. 15

Diagram used in the classification of the eigenvalues of NDFW and FLL. Note that $y_r>z_r$ ( $y_r<z_r$) implies: $q_r>0$ ($q_r<0$), $\sqrt{q^2}=q$ ($\sqrt{q^2}=-q$). The inequality $\{q_{2r}<0\,,q_{2i}<0\,, \frac{q_{2i}}{q_{2r}}<\frac{p_{2i}}{p_{2r}}\}$ is represented by the shaded areas in the figure and $f_0>-\frac{q_{2r}}{p_{2r}}$ implies $\mathrm{Re}\,(f_0p^2+q^2)>0$

The two eigenfrequencies coincide, $x_{dw}=x_{\ell a}$, when $\varDelta =0$, what happens for $(y,z,f_0)$ in the set

$$\begin{aligned} \bigg \{ |z|=|y|\,, f_0=\tan ^2(\psi /2)\bigg \}\,, \end{aligned}$$

(C.5)

where $\psi$ is the angle between y and z, $y=\mathrm{e} ^{i\psi }z$.

The NDFW and the FLL eigenfrequencies (this property does not extend to the eigenvectors) are dual to each other in the sense that if two bodies have the same $f_0$ but one has $(y,z)=(a_1,a_2)$ while the second $(y,z)=(a_2,a_1)$, then the value of $x_{dw}$ ($x_{\ell a}$) of the first is equal to the value of $x_{\ell a}$ ($x_{dw}$) of the second. This is a consequence of the symmetry of Eq. (6.21) with respect to the permutation of y and z.

In the limit as the mantle becomes negligible, i.e. $f_0=\frac{\mathrm{I}_{\circ c}}{\mathrm{I}_{\circ m}}\rightarrow \infty$, we find two solutions to Eq. (6.21)

$$\begin{aligned} x_c=\frac{yz}{y+z} \end{aligned}$$

(C.6)

and $(1+f_0)(y+z)$. These solutions represent the limit of $x_{dw}(f_0)$ and $x_{\ell a}(f_0)$ as $f_0\rightarrow \infty$. In order to decide whether $x_{dw}$ or $x_{\ell a}$ is asymptotic to $x_c$, we solve Eq. (6.21) with $y=0$, the solutions being $x_{dw}(f_0)=0=x_c$ and $x_{\ell a}(f_0)=(f_0+1)z$, and with $z=0$, the solutions being $x_{\ell a}(f_0)=0=x_c$ and $x_{dw}(f_0)=(f_0+1)y$. Since $\varDelta \ne 0$, by continuity the same result hold for |y| small in the first case and |z| small in the second. This fact, the duality of $x_{dw}$ and $x_{\ell a}$ discussed in the previous paragraph, and the invariance of $x_c$ and $(1+f_0)(y+z)$ with respect to the permutation $y\leftrightarrow z$, lead us to:

$$\begin{aligned} \begin{aligned}&|y|<|z| \Longrightarrow x_{dw}(f_0)\rightarrow x_c\ \ \text {and} \ \ x_{\ell a}(f_0)\rightarrow (f_0+1)z\ \ \text {as}\ \ f_0\rightarrow \infty \\&|z|<|y| \Longrightarrow x_{dw}(f_0)\rightarrow (f_0+1)z \ \ \text {and} \ \ x_{\ell a}(f_0)\rightarrow x_c \ \ \text {as}\ \ f_0\rightarrow \infty \,.\end{aligned} \end{aligned}$$

(C.7)

The Poincaré-Hough flow

Following Poincaré (see Lamb (1932) paragraphs 146 and 384), let $\varvec{x}_0\in \kappa$ be the position at time $t=0$ of a fluid particle inside a triaxial ellipsoid with semi-axes $A_1>A_2>A_3$. The mean radius of the ellipsoid is $R_c=(A_1A_2A_3)^{1/3}$. For any time the center of the ellipsoid coincides with the origin of $\kappa$ and at $t=0$ the axes of the ellipsoid are aligned with the axes of $\kappa$. At $t=0$ the operator $\varvec{A}:=$Diagonal$\{A_1,A_2,A_3\}$ maps the ball of radius unity onto the ellipsoid. The ellipsoid is fixed in the frame of the mantle ${\mathrm {K}}_m$ and rotates inside the inertial frame $\kappa$ according to $\varvec{R}_m(t):{\mathrm {K}}_m\rightarrow \kappa$, with $\varvec{R}_m(0)={\mathbb {I}}$. The motion of the fluid particle initially at $\varvec{x}_0$ is assumed to be:

$$\begin{aligned} \varvec{x}(t)=\varvec{R}_m(t)\varvec{A}\varvec{R}_f(t)\varvec{A}^{-1}\varvec{x}_0\,, \end{aligned}$$

(D.1)

where $t\rightarrow \varvec{R}_m(t)$ is known. The unknown rotation matrix $\varvec{R}_f(t)$ must be determined from the fluid dynamic equations.

The velocity field associated with the motion in Eq. (D.1) is

$$\begin{aligned} \varvec{v}(\varvec{x}, t)=\dot{\varvec{x}}=\underbrace{\big (\varvec{{\widehat{\omega }}}_m+ \varvec{R}_m\varvec{A}\varvec{{\widehat{\omega }}}_f\varvec{A}^{-1} \varvec{R}_m^{-1}\big )}_{:=\varvec{P}}\varvec{x}=\varvec{P} \varvec{x}. \end{aligned}$$

(D.2)

It is convenient to split the operator $\varvec{A}\varvec{\widehat{\omega }}_f\varvec{A}^{-1}$ into symmetric and anti-symmetric parts

$$\begin{aligned} \varvec{A}\varvec{{\widehat{\omega }}}_f\varvec{A}^{-1}=\underbrace{\frac{1}{2}\Big ( \varvec{A}\varvec{\widehat{\omega }}_f\varvec{A}^{-1}+ \big (\varvec{A}\varvec{{\widehat{\omega }}}_f\varvec{A}^{-1}\big )^T\Big )}_{:=\varvec{S}_A}+ \underbrace{\frac{1}{2}\Big ( \varvec{A}\varvec{{\widehat{\omega }}}_f\varvec{A}^{-1}- \big (\varvec{A}\varvec{{\widehat{\omega }}}_f\varvec{A}^{-1}\big )^T\Big )}_{:=\varvec{{\widehat{\tau }}}}\,, \end{aligned}$$

(D.3)

such that

$$\begin{aligned} \varvec{P}= \varvec{{\widehat{\omega }}}_m+ \varvec{R}_m\varvec{{\widehat{\tau }}}\varvec{R}_m^{-1}+ \varvec{R}_m\varvec{S}_A \varvec{R}_m^{-1}\,. \end{aligned}$$

(D.4)

The expressions for $\varvec{S}_A$ and $\varvec{\tau }$ are

$$\begin{aligned} \varvec{S}_A=\frac{1}{2}\left( \begin{array}{ccc} 0 &{} \left( \frac{A_2}{A_1}-\frac{A_1}{A_2}\right) \omega _{f3} &{} \left( \frac{A_1}{A_3}-\frac{A_3}{A_1}\right) \omega _{f2} \\ \left( \frac{A_2}{A_1}-\frac{A_1}{A_2}\right) \omega _{f3} &{} 0 &{} \left( \frac{A_3}{A_2}-\frac{A_2}{A_3}\right) \omega _{f1} \\ \left( \frac{A_1}{A_3}-\frac{A_3}{A_1}\right) \omega _{f2} &{} \left( \frac{A_3}{A_2}-\frac{A_2}{A_3}\right) \omega _{f1} &{} 0 \\ \end{array} \right) \end{aligned}$$

(D.5)

and

$$\begin{aligned} \varvec{{\widehat{\tau }}}=\frac{1}{2}\left( \begin{array}{ccc} 0 &{} -\left( \frac{A_2}{A_1}+\frac{A_1}{A_2}\right) \omega _{f3} &{} \left( \frac{A_1}{A_3}+\frac{A_3}{A_1}\right) \omega _{f2} \\ \left( \frac{A_2}{A_1}+\frac{A_1}{A_2}\right) \omega _{f3} &{} 0 &{}- \left( \frac{A_3}{A_2}+\frac{A_2}{A_3}\right) \omega _{f1} \\ - \left( \frac{A_1}{A_3}+\frac{A_3}{A_1}\right) \omega _{f2} &{} \left( \frac{A_3}{A_2}+\frac{A_2}{A_3}\right) \omega _{f1} &{} 0 \\ \end{array} \right) . \end{aligned}$$

(D.6)

Using that $\varvec{v}=\varvec{P} x= \varvec{ \omega }_m\times \varvec{x} + (\varvec{R}_m\varvec{\tau })\times \varvec{x} + \varvec{R}_m\varvec{S}_A \varvec{R}_m^{-1}\varvec{x}$ we obtain the vorticity field

$$\begin{aligned} \varvec{w}:=\varvec{curl}\, \varvec{v}= 2 \big (\varvec{ \omega }_m+\varvec{R}_m\varvec{\tau }\big )\,, \end{aligned}$$

(D.7)

which implies

$$\begin{aligned} \varvec{P}= \frac{\varvec{{{\widehat{w}}}}}{2}+ \varvec{R}_m\varvec{S}_A \varvec{R}_m^{-1}\quad \text {and}\quad \varvec{v}= \frac{\varvec{w}}{2}\times \varvec{x} + \varvec{R}_m\varvec{S}_A \varvec{R}_m^{-1}\varvec{x}\,. \end{aligned}$$

(D.8)

The equations for the motion of an inviscid, volume preserving, isentropic fluid are (Euler):

$$\begin{aligned} \begin{array}{lll} &{} \partial _t \varvec{v}+\varvec{v}\cdot \nabla \varvec{v}=-\nabla h-\nabla \varPhi \quad &{} (\text {dynamical equation})\,,\\ &{} \varvec{div}\, \varvec{v}=0\quad &{}(\text {incompressibility})\,,\\ &{}\partial _t \rho + \varvec{div}\,(\rho \varvec{v})=0\quad &{}(\text {conservation of mass})\,,\end{array} \end{aligned}$$

(D.9)

where: $\rho$ is the density, h is the enthalpy ($dh=\frac{dp}{\rho }$ where p is the pressure), and $\varPhi$ is the external gravitational potential.

The divergence of $\varvec{v}=\varvec{P} x$ is zero because ${{\,\mathrm{Tr}\,}}(\varvec{P})=0$. In order to fulfil the equation of continuity we impose that,

$$\begin{aligned} \rho (\varvec{x},t)=\rho _\circ (r)\,,\ \ \text {where}\ \ r= \Vert \varvec{A}^{-1} \varvec{R}_m^{-1}(t) \varvec{x}\Vert \,, 0\le r\le R_c. \end{aligned}$$

(D.10)

If the fluid cavity would be slowly deformed to a round shape, then the density of the fluid inside the round core would be the radial function $\rho _\circ (r)$. Using that $\varvec{div}\, \varvec{v}=0$ the equation for conservation of mass can be written as $\partial _t \rho + \varvec{v}\cdot \nabla \rho =\frac{d}{dt}\rho (\varvec{x}(t),t)=0$. Equation (D.1) implies $\rho (\varvec{x}(t),t)=\rho _\circ ( \Vert \varvec{A}^{-1} \varvec{R}_m^{-1}(t) \varvec{x}(t)\Vert )= \rho _\circ ( \Vert \varvec{A}^{-1} \varvec{x}_0\Vert )$, so $\frac{d}{dt}\rho (\varvec{x}(t),t)=0$ and the equation for conservation of mass is verified.

The last equation in (D.9) to be verified is the dynamical one. If we take the curl of this equation, then we obtain $\partial _t \varvec{w}+\varvec{v}\cdot \nabla \varvec{w}-\varvec{w}\cdot \nabla \varvec{v}=0$, which is the dynamical equation for the vorticity. If the equation for the vorticity is verified, then the enthalpy can be determined integrating the dynamical equation for $\varvec{v}$. Since $\varvec{w}$ does not depend on $\varvec{x}$, $\varvec{v}\cdot \nabla \varvec{w}=0$ and, since $\varvec{v}=\varvec{P} x$ depends linearly on $\varvec{x}$, $\varvec{w}\cdot \nabla \varvec{v}=\varvec{P}\varvec{w}$, so the equation for the vorticity reduces to

$$\begin{aligned} \dot{\varvec{w}}=\varvec{P} \varvec{w}\,. \end{aligned}$$

(D.11)

This equation yields a simple ordinary differential equation for the unknown function $\varvec{\omega }_f$ from which we determine the Poincaré-Hough flow.

Equation (D.8) implies that the Tisserand angular velocity ${\varvec{\omega }}_c$ associated with the Poincaré-Hough flow satisfies

$$\begin{aligned} {\varvec{\pi }}_c= \varvec{I}_c {\varvec{\omega }}_c=\int _{\mathcal B(t)}\rho (\varvec{x}\times \varvec{v})d x^3= \varvec{I}_c \frac{\varvec{w}}{2} + \int _{{\mathcal {B}}(t)}\rho \, \varvec{x}\times \big (\varvec{R}_m\varvec{S}_A \varvec{R}_m^{-1}\varvec{x})dx^3,\quad \end{aligned}$$

(D.12)

where ${\mathcal {B}}(t)$ is the set of points inside the cavity at time t. Since the density function is carried by the flow, $\varvec{I}_c(t)=\varvec{R}_m(t) \varvec{I}_c(0)\varvec{R}_m^{-1}(t)$.

In order to compute the moment of inertia of the fluid at $t=0$ we first compute the components of the density tensor $\varvec{M}(0)$

$$\begin{aligned} M_{ij}(0)=\int _{\Vert \varvec{A}^{-1}\varvec{x}\Vert \le 1}x_ix_j\rho _\circ (\Vert \varvec{A}^{-1}\varvec{x}\Vert )dx^3= \frac{ A_iA_j}{R_c^2}\int _{\Vert \varvec{y}\Vert \le R_c}y_iy_j\rho _\circ (\Vert \varvec{y}\Vert )dx^3\,, \end{aligned}$$

(D.13)

where we did the change of variables $x_i=(A_i/R_c) y_i$, $i=1,2,3$. Parity arguments imply that $M_{ij}(0)=0$ if $i\ne j$. Now, consider a round cavity of radius $R_c$ filled with a fluid with spherical density $\rho _\circ (r)$. The moment of inertia about any axis, say $\varvec{e}_3$, of the spherical mass of fluid is

$$\begin{aligned} \mathrm{I}_{\circ c}=\int _{\Vert \varvec{y}\Vert \le R_c}(y_1^2+y_2^2)\rho _\circ (\Vert \varvec{y}\Vert )dx^3 \end{aligned}$$

(D.14)

So, by symmetry $\int _{\Vert \varvec{y}\Vert \le R_c}y_i^2\rho _\circ (\Vert \varvec{y}\Vert )dx^3= \mathrm{I}_{\circ c}/2$ for any $i=1,2,3$ and

$$\begin{aligned} M_{ii}(0)= \frac{ A_i^2}{R_c^2}\frac{\mathrm{I}_{\circ c}}{2}\,, \quad i=1,2,3\,. \end{aligned}$$

(D.15)

Using that $\varvec{I}_c={{\,\mathrm{Tr}\,}}\big (\varvec{M}\big ){\mathbb {I}} -\varvec{M}$ we obtain

$$\begin{aligned} \varvec{I}_c(0)=\frac{\mathrm{I}_{\circ c}}{2 R_c^2}\left( \begin{array}{ccc} A_2^2+A_3^2&{}0&{}0\\ 0&{} A_1^2+A_3^2&{}0\\ 0&{}0&{} A_1^2+A_2^2\end{array}\right) . \end{aligned}$$

(D.16)

Note that ${{\,\mathrm{Tr}\,}}\varvec{I}_c(0)= \mathrm{I}_{\circ c}\frac{A_1^2+A_2^2+A_3^2}{R_c^2}$, so, $\mathrm{I}_{\circ c}$ may not have the usual meaning $\frac{1}{3}{{\,\mathrm{Tr}\,}}\varvec{I}_c(0)$.

In order to compare the results obtained from the Poincaré-Hough flow with ours we have to assume that the fluid cavity is slightly aspherical, namely

$$\begin{aligned} A_1=R_c(1+\epsilon _1)\quad A_2=R_c(1+\epsilon _2)\quad A_3=R_c(1+\epsilon _3)\,, \end{aligned}$$

(D.17)

where $|\epsilon _1|,|\epsilon _2|,$ and $|\epsilon _3|$ are small. Since $A_1A_2A_3=R_c^3$, $\epsilon _1+\epsilon _2+\epsilon _3=0$ and equation (D.16) implies

$$\begin{aligned} \varvec{I}_c(0)=\mathrm{I}_{\circ c} {\mathbb {I}} + \mathrm{I}_{\circ c}\left( \begin{array}{ccc} \epsilon _2+\epsilon _3&{}0&{}0\\ 0&{} \epsilon _1+\epsilon _3&{}0\\ 0&{}0&{} \epsilon _1+\epsilon _2\end{array}\right) +{\mathcal {O}}(|\epsilon |^2)\,, \end{aligned}$$

(D.18)

so, $\frac{1}{3}{{\,\mathrm{Tr}\,}}\varvec{I}_c(0)=\mathrm{I}_{\circ c}$ holds up to second order in the ellipticity³². Equation (D.5) implies

$$\begin{aligned} \varvec{S}_A=\left( \begin{array}{ccc} 0 &{} \left( \epsilon _2-\epsilon _1\right) \omega _{f3} &{} \left( \epsilon _1-\epsilon _3\right) \omega _{f2} \\ \left( \epsilon _2-\epsilon _1\right) \omega _{f3} &{} 0 &{} \left( \epsilon _3-\epsilon _2\right) \omega _{f1} \\ \left( \epsilon _1-\epsilon _3\right) \omega _{f2} &{} \left( \epsilon _3-\epsilon _2\right) \omega _{f1} &{} 0 \\ \end{array} \right) +{\mathcal {O}}(|\epsilon |^3)\,, \end{aligned}$$

(D.19)

and equation (D.6) implies $\varvec{ \tau }={\varvec{\omega }}_f+{\mathcal {O}}(|\epsilon |^2)$.

A computation shows that the last term in the right-hand side of equation (D.12) is of the order ${\mathcal {O}}(|\epsilon |^2)$ and, therefore

$$\begin{aligned} {\varvec{\pi }}_c\!= \!\varvec{I}_c {\varvec{\omega }}_c= \varvec{I}_c \frac{\varvec{w}}{2} \!+\!{\mathcal {O}}(|\epsilon |^2) \Longrightarrow {\varvec{\omega }}_c\!=\! \frac{\varvec{w}}{2} +{\mathcal {O}}(|\epsilon |^2)= \varvec{\omega }_m+\varvec{R}_m \varvec{\omega }_f +{\mathcal {O}}(|\epsilon |^2) \end{aligned}$$

(D.20)

The fact that $\varvec{\omega }_c\approx \varvec{w}/2$ for small ellipticities has been noted in Henrard (2008).

For small ellipticities we can write $\varvec{\omega }_c= \varvec{w}/2$ and equation (D.11) implies

$$\begin{aligned} \dot{\varvec{\omega }}_c=\varvec{P} \varvec{\omega }_c\,. \end{aligned}$$

(D.21)

Therefore

$$\begin{aligned} \dot{{\varvec{\pi }}}_c=\dot{\varvec{I}}_c\varvec{\omega }_c+{\varvec{I}}_c\dot{\varvec{\omega }}_c= {\varvec{\omega }}_m\times \varvec{I}_c \varvec{\omega }_c+ \varvec{I}_c(-\varvec{{\widehat{\omega }}}_m+\varvec{P}) \varvec{\omega }_c. \end{aligned}$$

(D.22)

Equation (D.2) implies that the last term in the right-hand side of this equation can be written as

$$\begin{aligned} \varvec{I}_c(-\varvec{{\widehat{\omega }}}_m+\varvec{P}) \varvec{\omega }_c=\varvec{R}_m \varvec{I}_c(0)\varvec{A}\varvec{\widehat{\omega }}_f\varvec{A}^{-1} \varvec{R}_m^{-1} \varvec{\omega }_c\,. \end{aligned}$$

A computation using equation (D.18) shows that $\varvec{I}_c(0)\varvec{A}\varvec{{\widehat{\omega }}}_f\varvec{A}^{-1}= \varvec{{\widehat{\omega }}}_f \varvec{I}_c(0)+ {\mathcal {O}}(|\epsilon |^2)$, so up to second order in the ellipticities

$$\begin{aligned} \dot{{\varvec{\pi }}}_c= {\varvec{\omega }}_m\times \varvec{I}_c \varvec{\omega }_c+ \varvec{R}_m \varvec{{\widehat{\omega }}}_f \varvec{I}_c(0) \varvec{R}_m^{-1} \varvec{\omega }_c= {\varvec{\omega }}_m\times \varvec{I}_c \varvec{\omega }_c+ \varvec{R}_m \varvec{{\widehat{\omega }}}_f \varvec{R}_m^{-1} \varvec{I}_c \varvec{\omega }_c \end{aligned}$$

and using equation (D.20) we obtain

$$\begin{aligned} \dot{{\varvec{\pi }}}_c= {\varvec{\omega }}_c\times \varvec{I}_c \varvec{\omega }_c, \end{aligned}$$

(D.23)

that is exactly the equation for the angular momentum of the core obtained in (5.1) with $k_c=0$.

We remark that the hypotheses we used in this Appendix to obtain the general results in Eq. (5.1) from the Poincaré-Hough flow, namely: small asphericity, density stratification along concentric ellipsoidal shells and the volume preserving property of the fluid flow; are the same hypotheses we have assumed since Ragazzo and Ruiz (2017). In this Appendix, any rotational motion $\varvec{R}_m(t)$ of the ellipsoid is allowed and there is no reason to assume that the principal axes of the mantle are aligned with those of the fluid cavity (a hypothesis commonly assumed). Finally, it is crucial for the results in this Appendix that the centre of mass of the mantle coincides with that of the fluid core for all time.

The inertial offset of the core rotation axis

In this appendix, we analyse the effect of the inertial torque studied in Sect. 7 upon the fluid core. Two different situations will be considered: one without spin-orbit resonance and another with spin-orbit resonance.

1.1 Precession under no spin-orbit resonance, the Tisserand angular velocity of the fluid core, and the vorticity of Robert and Stewartson

In this section, we assume that both the mantle and the core are axisymetric with $\overline{\gamma }=0$, $\overline{\alpha }=\overline{\beta }=\overline{\alpha }_e$, and $I_{c1}=I_{c2}=I_{c3}(1-f_c)$, the mantle is rigid, and the precession rate is retrograde, ${{\dot{\psi }}}_g<0$. In this section, we further assume that $\eta _c/(\omega f_c)\ll 1$. The goal is to study the angular velocity of the core produced exclusively by the inertial torque. The axisymmetry of the problem implies that in the precessional frame (equation (7.2)) both ${\varvec{\omega }}_{m,pr}$ and ${\varvec{\omega }}_{c,pr}$ are stationary, and according to equations (7.14) and (7.17) are given by

$$\begin{aligned} \begin{aligned} {\varvec{\omega }}_{m,pr}&= \omega \varvec{e}_3+{{\dot{\psi }}}_g\sin \theta _g\varvec{e}_2+ \omega \, {\varvec{\delta }}_m\\ {\varvec{\omega }}_{c,pr}&= \underbrace{ \omega \varvec{e}_3+{{\dot{\psi }}}_g\sin \theta _g\varvec{e}_2}_{\varvec{\omega }_{g,pr}}+ \omega \, {\varvec{\delta }}_c\end{aligned}. \end{aligned}$$

(E.1)

In order to write ${\varvec{\delta }}_c$ we follow the same steps we did to obtain ${\varvec{\delta }}_m$ in equation (7.22). After doing this and using that the complex compliance is null (the mantle is rigid) and $\eta _c/(\omega f_c)\ll 1$ ($\Rightarrow y\approx f_c$ in equation (7.22)) we obtain

$$\begin{aligned} \begin{aligned} \varvec{\delta }_m&= \frac{\mathrm{I_{\circ c}}}{{\,\mathrm I}_\circ }\frac{ {{\dot{\psi }}}_g}{\omega f_c} \frac{2 \cos ^2 \theta _g}{ \sin \theta _g} \left( \frac{ \mathrm{I}_{\circ m}}{{\,\mathrm I}_\circ } \frac{\eta _c}{f_c\,\omega }\, \varvec{e}_1-\varvec{e}_2 \right) \\ \varvec{\delta }_c&=\frac{{{\dot{\psi }}}_g}{\omega f_c} \sin \theta _g \left( 1- 2 \frac{\mathrm{I}_{\circ c}}{{\,\mathrm I}_\circ }\cot ^2\theta _g\right) \left( \varvec{e}_2- \frac{ \mathrm{I}_{\circ m}}{{\,\mathrm I}_\circ } \frac{\eta _c}{f_c\,\omega }\varvec{e}_1\right) \\&=\varvec{\delta }_m+ \frac{ {{\dot{\psi }}}_g\,\sin \theta _g}{\omega f_c}\left( \varvec{e}_2- \frac{ \mathrm{I}_{\circ m}}{{\,\mathrm I}_\circ } \frac{\eta _c}{f_c\,\omega }\varvec{e}_1\right) \end{aligned} \end{aligned}$$

(E.2)

Roberts and Stewartson studied the motion of an incompressible fluid of constant density inside an ellipsoidal shell of revolution that precesses in the same way as the guiding motion used to obtain the angular velocities in equations (E.1) and (E.2). In Stewartson and Roberts (1963) and Roberts and Stewartson (1965) the authors solved the Navier-Stokes equations by perturbation methods (for a numerical study in the case of a spherical shell, see Tilgner and Busse (2001)) assuming the two hypotheses

$$\begin{aligned} \frac{f_c \,\omega }{|{{\dot{\psi }}}_g|}\gg 1\quad \text { and}\quad f_c \,\omega \frac{R_c^2}{\nu }=f_c\,\frac{\omega }{\eta _c}\gg 1\,, \end{aligned}$$

(E.3)

where: $R_c$ is the core mean radius, $\nu$ is kinematic viscosity of the fluid, and $\eta _c= R_c^2/\nu$ is the viscosity coefficient defined in equation (3.22). These are the same hypotheses we assumed to obtain equation (E.2). In the following we show that the difference ${\varvec{\omega }}_{c,pr}-{\varvec{\omega }}_{m,pr}=\omega (\varvec{\delta }_c-\varvec{\delta }_m)$ is equal to the average vorticity of the flow inside the cavity, computed by Robert and Stewartson, minus the vorticity of the flow induced by a rigid rotation.

The velocity field of the fluid inside the cavity with respect to the precessional frame ${\mathrm {K}}_{pr}$ is Stewartson and Roberts (1963)

$$\begin{aligned} \varvec{u}={{\dot{\phi }}}_g\varvec{e}_3\times \varvec{x}+ 2{{\dot{\psi }}}_g\sin \theta _g \frac{a^2}{a^2-b^2}\left( x_3\varvec{e}_1-\frac{b^2}{a^2} x_1\varvec{e}_3\right) , \end{aligned}$$

(E.4)

where $b<a$ are the semi-axis of the cavity³³. The vorticity associated with this velocity field is

$$\begin{aligned} \mathrm{curl}\,\varvec{u}=2{{\dot{\phi }}}\varvec{e}_3+ 2{{\dot{\psi }}}_g\sin \theta _g \frac{a^2+b^2}{a^2-b^2}\varvec{e}_2. \end{aligned}$$

(E.5)

If the fluid were moving as a rigid body attached to the mantle, then its vorticity would be $\mathrm{curl} {{\dot{\phi }}}_g\varvec{e}_3\times \varvec{x}= 2{{\dot{\phi }}}\varvec{e}_3$. Recalling that vorticity is a measure of rotation of the vector field, as we have already mentioned in Appendix D,

$$\begin{aligned} 2{{\dot{\psi }}}_g\sin \theta _g \frac{a^2+b^2}{a^2-b^2}\varvec{e}_2= \underbrace{\mathrm{curl}\, \varvec{u}}_{rot. fluid}- \underbrace{2{{\dot{\phi }}}\varvec{e}_3}_{rot. mantle}. \end{aligned}$$

(E.6)

In order to relate equation (E.6) to the difference ${\varvec{\omega }}_{c,pr}-{\varvec{\omega }}_{m,pr}$ we use that the moments of inertia of an ellipsoid of revolution of constant density $\rho$ are $\overline{I}_{ce}=\frac{4\pi }{15}\rho (a^2+b^2)a^2b$ and $\overline{I}_{c3}=\frac{8\pi }{15}\rho a^4b$ that implies

$$\begin{aligned} f_c= \frac{\overline{I}_{c3}-\overline{I}_{ce}}{\overline{I}_{c3}}\approx \frac{\overline{I}_3-\overline{I}_{ce}}{\overline{I}_{ce}}=\frac{a^2-b^2}{a^2+b^2}. \end{aligned}$$

(E.7)

Therefore, using the second hypothesis of Robert and Stewartson $\frac{\eta _c}{f_c\omega }\ll 1$ (equation (E.3)) equations (E.1), (E.2), (E.6), and (E.7) imply the claimed result:

$$\begin{aligned} {\varvec{\omega }}_{c,pr}-{\varvec{\omega }}_{m,pr}\approx \frac{1}{2}\bigg (\mathrm{curl}\, \varvec{u}- 2{{\dot{\phi }}}\varvec{e}_3\bigg )= \frac{{{\dot{\psi }}}_g}{f_c}\sin \theta _g\varvec{e}_2\,. \end{aligned}$$

(E.8)

We could also have computed the total angular momentum of the fluid with respect to the precessional frame and arrived at the same result.

1.2 The resonant case and the Cassini states of G. Boué

The Cassini states of a body made of a rigid mantle and a fluid core with no viscous coupling ($k_c=\eta _c=0$) were computed in Boué (2020). In ibid, the difference between the angular velocities of mantle and core was not supposed small and many possible Cassini states were found. Our goal in the rest of this Appendix is to compare the inertial offsets ${\varvec{\delta }}_m$ and ${\varvec{\delta }}_c$ with some of the Cassini states found in Boué (2020).

The expressions for ${\varvec{\delta }}_m$ and ${\varvec{\delta }}_c$ in equations (7.24) simplify when we assume that the moment of inertia of the core is not much smaller than that of the mantle is rigid and $\eta _c=0$:

$$\begin{aligned} \begin{aligned}&\delta _m= \delta _{m2}= - \frac{\frac{\mathrm{I}_{\circ c}}{\mathrm{I}_{\circ m}} \frac{{{\dot{\psi }}}_g}{\omega } \cos \theta _g}{\left( \frac{{{\dot{\psi }}}_g}{\omega } \cos \theta _g\right) ^2 +\frac{{{\dot{\psi }}}_g}{\omega } \cos \theta _g\frac{{\,\mathrm I}_\circ }{\mathrm{I}_{\circ m}}(y+z)+\frac{{\,\mathrm I}_\circ }{\mathrm{I}_{\circ m}} y z } \frac{{{\dot{\psi }}}_g}{\omega }\sin \theta _g\,, \\&\delta _c=\delta _{c2} = \frac{\frac{{\,\mathrm I}_\circ }{\mathrm{I}_{\circ m}} z+ \frac{{{\dot{\psi }}}_g}{\omega } \cos \theta _g}{\left( \frac{{{\dot{\psi }}}_g}{\omega } \cos \theta _g\right) ^2 +\frac{{{\dot{\psi }}}_g}{\omega } \cos \theta _g\frac{{\,\mathrm I}_\circ }{\mathrm{I}_{\circ m}}(y+z)+\frac{{\,\mathrm I}_\circ }{\mathrm{I}_{\circ m}} y z }\frac{{{\dot{\psi }}}_g}{\omega }\sin \theta _g\,, \end{aligned} \end{aligned}$$

(E.9)

where $z=c_1 \overline{\alpha }_e +\frac{c_2}{2}\overline{\gamma }$ and $y=f_c$. According to equation (7.17) all the three vectors: the normal to the invariable plane $\varvec{e}_3\in \kappa$, the angular velocity of the mantle $\langle {\varvec{\omega }}_{m}\rangle _{pr}\in \kappa$, and the angular velocity of the core $\langle {\varvec{\omega }}_{c}\rangle _{pr}\in \kappa$, both averaged in the precessional frame; are contained in the same plane, which is denoted by $\varSigma$ in Fig. 8 LEFT.

The Cassini states in Boué (2020) are determined by the following two equations (Equations (34a) and (34b) in ibid.)³⁴

$$\begin{aligned} \begin{aligned}&f_c{{\dot{\phi }}}_g\cos (\delta _m-\delta _c)\sin (\delta _m-\delta _c)+ {{\dot{\psi }}}_g\sin ({{\tilde{\theta }}}_g-\delta _c)=0 \\&\overline{I}_{c3}\Big ({{\dot{\psi }}}_g\sin ({{\tilde{\theta }}}_g-\delta _m)+f_c{{\dot{\phi }}}_g \cos (\delta _m-\delta _c)\sin (\delta _m-\delta _c)\Big )-\frac{\overline{I}_3\omega ^2}{{{\dot{\phi }}}_g}P_e(\delta _m)=0 \end{aligned} \end{aligned}$$

(E.10)

where

$$\begin{aligned} \begin{aligned} P_e(\delta _m)=&\frac{3}{2}\frac{G m_p}{a_p^3\omega ^2}\bigg (\overline{\alpha }_e X^{-3,0}_0\cos ({{\tilde{\chi }}}_p-\delta _m)\sin ({{\tilde{\chi }}}_p-\delta _m)\\ {}&+\frac{\overline{\gamma }}{4}X^{-3,2}_s \big (1+\cos ({{\tilde{\chi }}}_p-\delta _m)\big )\sin ({{\tilde{\chi }}}_p-\delta _m)\bigg ) +\frac{{{\dot{\psi }}}_g{{\dot{\phi }}}_g}{\omega ^2}\sin ({{\tilde{\theta }}}_g-\delta _m) \,.\end{aligned} \end{aligned}$$

(E.11)

The identity $P_e(0)=0$ holds due to Peale’s equation (7.6).

Since our inertial offset was computed by means of a perturbation of a rigid-body motion, for which $\delta _m=\delta _c=0$, it is natural to look for solutions $(\delta _m,\delta _c)\approx (0,0)$ to equations (E.10). For $(\delta _m,\delta _c)=(0,0)$, the first equation in (E.10) implies ${{\dot{\psi }}}_g\sin ({{\tilde{\theta }}}_g)=0$ and then the second equation implies

$$\begin{aligned} P_e(0)=0\Longrightarrow \bigg (\overline{\alpha }_e X^{-3,0}_0\cos ({{\tilde{\chi }}}_p) +\frac{\overline{\gamma }}{4}X^{-3,2}_s \big (1+\cos ({{\tilde{\chi }}}_p)\big )\bigg )\sin ({{\tilde{\chi }}}_p)=0\,. \end{aligned}$$

(E.12)

This equation has several solutions ${{\tilde{\chi }}}_p=0$, ${{\tilde{\chi }}}_p=\pi$, and $\cos {{\tilde{\chi }}}_p= -\frac{\overline{\gamma }X^{-3,2}_s}{4 \overline{\alpha }_e X^{-3,0}_0+\overline{\gamma }X^{-3,2}_s}$, where ${{\tilde{\chi }}}_p$ is the obliquity of the body spin to the orbital plane. Among these solutions only ${{\tilde{\chi }}}_p=0$ has obliquity smaller then $\pi /2$ (except $\cos {{\tilde{\chi }}}_p= -\frac{\overline{\gamma }X^{-3,2}_s}{4 \overline{\alpha }_e X^{-3,0}_0+\overline{\gamma }X^{-3,2}_s}$ with s=1, for which the obliquity is smaller but close to $\pi /2$). We will only consider solutions to Eq. (E.10) with obliquities smaller than $\pi /2$ that originate from $\tilde{\chi }_p=0$.

In order to show that the small solutions $(\delta _m,\delta _c)$ to Eqs. (E.10) are given by Eq. (E.9), it is enough to expand the functions in the left-hand side of equations (E.10) up to first order in $(\delta _m,\delta _c)$ and then to solve the linear system. This was done using the algebraic manipulator Mathematica.

The Hessian determinant of equation (E.10) is proportional to $(x_{\ell a}+\frac{{{\dot{\psi }}}_g}{\omega } \cos \theta _g) (x_{dw}+\frac{{{\dot{\psi }}}_g}{\omega } \cos \theta _g)$, where $\omega x_{dw}$ and $\omega x_{\ell a}$ are the eigenfrequencies of the nearly diurnal free wobble (NDFW) and the free libration in latitude (FLL) in the inertial space. This shows that the critical flattening of the core $f_c\approx 0.005 \overline{\alpha }_c$ found for Mercury in Section 6.1 of Boué (2020) (see also Fig. 3 in ibid.) is indeed a resonance of the precession angular speed $-\dot{\psi }_d\cos \theta _g\approx$330 kyr with the NDFW eigenfrequency, which decreases to zero as $f_c\rightarrow 0$. The presence of dissipation attenuates the singularity, as illustrated in Fig. 9.