Reference (2)

Solid-Phase

To describe tidal and rotational deformations of a spherically symmetric body, Obliqua considers the spheroidal displacement–stress–gravity system. For each harmonic degree ($\ell$) and order ($m$), the spheroidal perturbed state is represented by the 6-vector

\[ \mathbf{y}_{\ell m} = \begin{pmatrix} U_{\ell m} \ V_{\ell m} \ R_{\ell m} \ S_{\ell m} \ \Phi_{\ell m} \ Q_{\ell m} \end{pmatrix},\]

where

• $U_{\ell m}$: radial displacement
• $V_{\ell m}$: tangential displacement
• $R_{\ell m}$: radial stress
• $S_{\ell m}$: tangential stress
• $\Phi_{\ell m}$: gravitational potential perturbation
• $Q_{\ell m}$: “potential stress,” defined as

\[Q_{\ell m} = \frac{\partial \Phi_{\ell m}}{\partial r} + \frac{\ell+1}{r}\Phi_{\ell m} + 4\pi G \rho_0 U_{\ell m}.\]

The spheroidal vector satisfies the first-order ODE system

\[\frac{d\mathbf{y}_{\ell m}}{dr} = \mathbf{A}_{\ell}(r)\,\mathbf{y}_{\ell m}(r).\]

Here, the coefficient matrix $\mathbf{A}_{\ell}(r)$ represents the responds of the mantle to deformations, and is given by

\[A(r) = \begin{pmatrix} -\frac{2\lambda}{r\beta} & \frac{\ell(\ell+1)\lambda}{r\beta} & \frac{1}{\beta} & 0 & 0 & 0 \\[1.2em] -\frac{1}{r} & \frac{1}{r} & 0 & \frac{1}{\mu} & 0 & 0 \\[1.2em] \frac{4}{r}\!\left( \frac{3\kappa\mu}{r\beta} - \rho_{0}g \right) - \rho_{0}\omega^{2} & \frac{\ell(\ell+1)}{r}\!\left(\rho_{0}g - \frac{6\kappa\mu}{r\beta}\right) & -\frac{4\mu}{r\beta} & \frac{\ell(\ell+1)}{r} & -\frac{\rho_{0}(\ell+1)}{r} & \rho_{0} \\[1.2em] \frac{1}{r}\!\left(\rho_{0}g - \frac{6\mu\kappa}{r\beta}\right) & \frac{2\mu}{r^{2}}\!\left[\ell(\ell+1)\!\left(1+\frac{\lambda}{\beta}\right)-1\right] - \rho_{0}\omega^{2} & -\frac{\lambda}{r\beta} & -\frac{3}{r} & \frac{\rho_{0}}{r} & 0 \\[1.2em] -4\pi G \rho_{0} & 0 & 0 & 0 & -\frac{\ell+1}{r} & 1 \\[1.2em] -\frac{4\pi G \rho_{0} (\ell+1)}{r} & \frac{4\pi G \rho_{0} \ell(\ell+1)}{r} & 0 & 0 & 0 & \frac{\ell-1}{r} \end{pmatrix}.\]

The material parameters satisfy

\[\beta = \lambda + 2\mu\]

and

\[\lambda = \kappa - \frac{2}{3}\mu.\]

Core–Mantle Boundary

In order to solve the system a general solution is constructed through propagation from the core-mantle boundary outwards. At the CMB radius $r_C$, the spheroidal solution satisfies

\[\mathbf{y}_\ell(r_C^+) = \mathbf{I}_C \mathbf{C},\]

where $\mathbf{C} = (C_1, C_2, C_3)^T$ is a vector of integration constants determined by surface boundary conditions. The CMB Interface Matrix is given as

\[\mathbf{I}_C = \begin{pmatrix} -\psi_\ell(r_C)/g(r_C) & 0 & 1 \\[1.2em] 0 & 1 & 0 \\[1.2em] 0 & 0 & g(r_C)/\rho_0(r_C^-) \\[1.2em] 0 & 0 & 0 \\[1.2em] \psi_\ell(r_C) & 0 & 0 \\[1.2em] q_\ell(r_C) & 0 & 4\pi G \rho_0(r_C^-) \end{pmatrix}.\]

Once the constants $\mathbf{C}$ are determined, the full perturbed state of the solid mantle is known.

(Visco)elastic Solution

We propagate the solution using the so-called propagator matrix ($\pmb{\Pi}_\ell$). The propagator matrix solves the homogeneous differential system

\[\frac{d\pmb{\Pi}_\ell(r, r')}{dr} = \pmb{A}_\ell(r)\,\pmb{\Pi}_\ell(r, r'),\]

at radius $r$ w.r.t. the solution at the previous layer $r'$, this is also know as the Cauchy data at radius ($r'$). If $r = r'$ we have

\[\pmb{\Pi}_\ell(r', r') = \pmb{1}.\]

Each column of the propagator matrix is one of the six linearly independent solutions of

\[\frac{d\pmb{y}_{\ell m}}{dr} = \pmb{A}_\ell(r)\,\pmb{y}_{\ell m}.\]

We impose continuity:

\[\pmb{\Pi}_\ell(r_j^+, r') = \pmb{\Pi}_\ell(r_j^-, r'),\]

and apply CMB boundary conditions:

\[\pmb{y}_{\ell m}(r_C^+) = \pmb{y}_0 = \pmb{I}_C\,\pmb{C}.\]

Therefore,

\[\pmb{y}_{\ell m}(r) = \pmb{\Pi}_\ell(r, r_C^+)\,\pmb{I}_C\,\pmb{C}.\]

This equation can be solved iteratively, up till the surface to yield the general responds of the interior to any form of tidal- or load induced deformation.

Surface Boundary Conditions

Finally, to find the specific solution for the case where a planet is orbiting around a star-like object, we can specify the surface ($r=a^-$) boundary condition. In particular we need to constrain the 3rd, 4th, and 6th $y$-values. To do this employ the projector on the 3rd, 4th, and 6th components given by

\[\pmb{P}_1\,\pmb{y}(a^-) = \pmb{P}_1 \left[ \pmb{\Pi}_\ell(a^-, r_C^+)\,\pmb{I}_C\,\pmb{C} \right] = \pmb{b},\]

where

\[\pmb{b} = \sigma_{\ell m}^L\,\pmb{b}^L + \left(\Phi_{\ell m}^T(a) + \Phi_{\ell m}^C(a)\right)\pmb{b}^T,\]

with

\[\pmb{b}^L = \begin{pmatrix} -\dfrac{(2\ell+1)g(a)}{4\pi a^2} \\[1em] 0 \\[1em] -\dfrac{(2\ell+1)G}{a^2} \end{pmatrix} \qquad\text{(Load)},\]

\[\pmb{b}^T = \begin{pmatrix} 0 \\[1em] 0 \\[1em] \dfrac{2\ell+1}{a} \end{pmatrix} \qquad\text{(Tidal)}.\]

Incase the solid part of the mantle extends up to the surface of the planet, then the surface is stress-free and the toroidal part has $\sigma_{\ell m}^L = \pmb{0}$, and the solution no longer depends on $\pmb{b}^L$. However, as in our formalism the solid is often loaded by a liquid magma ocean, we cannot ignore this term.

The integration constants follow from

\[\pmb{C} = \left(\pmb{P}_1 \pmb{\Pi}_\ell(a, r_C)\,\pmb{I}_C\right)^{-1}\pmb{b}.\]

Thus,

\[\pmb{y}_{\ell m}(r) = \pmb{\Pi}_\ell(r, r_C)\, \pmb{I}_C\, \left(\pmb{P}_1 \pmb{\Pi}_\ell(a, r_C)\,\pmb{I}_C\right)^{-1} \pmb{b}.\]

To solve this system we thus need only provide $\pmb{P}_1\,\pmb{y}(a^-)$, we will provide some examples in Chapter 7 at the end of this component.