Reference (7)

Tidal potentials

Different sources can provide different tidal potentials, first we need to complete the solid-phase by providng the relevant surface boundary condition. Here we list some examples.

Loaded Surface

For an impermeable load,

\[y_2 = - \frac{(2\ell + 1) g(R)}{4 \pi G R}, \qquad y_4 = 0, \qquad y_6 = \frac{2\ell + 1}{R}, \qquad y_8 = 0\]

where $g(R)$ is the gravitational acceleration at the surface (e.g., Saito, 1974).

Lunar Tide

For a fully solid mantle, the boundary conditions at the surface are

\[y_3 = y_4 = 0, \qquad y_6 = - \frac{(2\ell + 1) g_s}{4} \frac{M_M}{M_E} \left(\frac{R_E}{a}\right)^3.\]

(see Takeuchi et al. 1972).

The final condition is

\[\pmb{P}_1\,\pmb{y}(a^-) = \begin{pmatrix} 0 \\[1em] 0 \\[1em] -\dfrac{(2\ell+1) g_s}{4} \dfrac{M_M}{M_E} \left(\dfrac{R_E}{a}\right)^3 \end{pmatrix}.\]

After having solved and generated the $k_2 (\sigma)$ spectrum, we can find the power input from tides.

Tidal Dissipation

The mode amplitude of the external tidal potential is

\[U_{22k} = \frac{G M_\star}{a} \left(\frac{R}{a}\right)^2 \sqrt{\frac{6\pi}{5}}\,X_{22k}(e),\]

The tidal power per mode is

\[P_{T,k} = \frac{5 R \sigma}{8\pi G} \,\Im\!\bigl[k_{2}(\sigma)\bigr] \,|U_{22k}|^2.\]

Such that the total tidal dissipation:

\[P_{\mathrm{tidal}} = -\sum_k P_{T,k}.\]