Surface Harmonics and Solid Harmonics

A surface harmonic of degree $n$ (where $n$ is a non-negative integer), denoted ${\cal S}_n(\theta,\varphi)$ , is defined as a well-behaved solution to

$$r^{\,2}\,\nabla^{\,2} {\cal S}_n + n\,(n+1)\,{\cal S}_n = 0$$ (12.36)

on the surface of a sphere (i.e., $r=$ constant). Here, $r$ , $\theta$ , $\varphi$ are standard spherical coordinates, and $\nabla^{\,2}$ is the Laplacian operator. It follows that (Love 1927)

$${\cal S}_n(\theta,\varphi) = \sum_{m=0,n} \left[c_n^{\,m}\,P_n^{\... ...\,\cos(m\,\varphi) + d_n^{\,m}\,P_n^{\,m}(\cos\theta)\,\sin(m\,\varphi)\right],$$ (12.37)

where the $c_n^{\,m}$ and $d_n^{\,m}$ are arbitrary coefficients, and the $P_n^{\,m}(x)$ associated Legendre functions.

A solid harmonic of degree $n$ (where $n$ is a non-negative integer), denoted ${\cal R}_n(r,\theta,\varphi)$ , is defined as a well-behaved solution to

$$\nabla^{\,2} {\cal R}_n = 0$$ (12.38)

in the interior of a sphere (i.e., the region $r<$ constant). It follows that (Love 1927)

$${\cal R}_n(r,\theta,\varphi) \propto r^{\,n}\,{\cal S}_n(\theta,\varphi).$$ (12.39)

In particular, the functions ${\cal R}_n^{\,(\pm m)}(r,\theta,\varphi)$ and ${\cal R}_n^{\,(0)}(r,\theta)$ , introduced in Section 12.3, are solid harmonics of degree $n$ . Note that the Cartesian coordinates $x_i$ (where $i$ runs from $1$ to $3$ ) are solid harmonics of degree $1$ . Moreover, $\partial {\cal R}_n/\partial x_i$ is a solid harmonic of degree $n-1$ . Finally, ${\mit\Phi}_{\rm tide}({\bf r})$ , specified in Equation (12.35), is a solid harmonic of degree 2.

The following results regarding solid harmonics are useful (Love 1927):

$$x_i\,\frac{\partial {\cal R}_n}{\partial x_i} = r\,\frac{\partial {\cal R}_n}{\partial r} = n\,{\cal R}_n,$$ (12.40)

$$\nabla^{\,2}(x_i\,{\cal R}_n) = \nabla\cdot({\cal R}_n\,\nabla x_i + x_i\,\nabla{\cal R}_n) = 2\,\nabla x_i\cdot\nabla{\cal R}_n = 2\,\frac{\partial {\cal R}_n}{\partial x_i},$$ (12.41)

$$\nabla^{\,2}(r^{\,m}\,{\cal R}_n) = \nabla^{\,2}(r^{\,m+n}\,{\cal S}_n) = \frac{1}{r^{\,2}}\,\frac{... ...,\frac{d}{dr}(r^{\,m+n}\,{\cal S}_n)\right] - n\,(n+1)\,r^{\,m+n-2}\,{\cal S}_n$$

$$= m\,(m+2\,n+1)\,r^{\,m-2}\,{\cal R}_n.$$ (12.42)

Here, use has been made of the Einstein summation convention (Riley 1974).