Love numbers

Definition

Let a planet be a spherically symmetric, non-rotating, linearly elastic and isotropic body of radius $R$ with surface gravity $g$. Let an external potential act on it, for example a term of the tide-generating potential. Assume its value on the surface $r = R$ is $W_n$, a spherical harmonic of degree $n \ge 2$. By linearity and symmetry, the static response at the surface has the form

1. radial displacement $u_r = h_n\, W_n / g$;
2. horizontal displacement $u_H = (l_n / g)\, R\, \nabla_H W_n$, where $\nabla_H$ is the horizontal gradient;
3. additional potential, due to the deformation, equal to $k_n W_n$ on the surface;

for constants $h_n, l_n, k_n$ that depend only on $n$ and the internal structure. These constants are the Love numbers of degree $n$; $l_n$ is also called the Shida number.