Gravitational potential theory

Consider a spherical planet whose unperturbed surface corresponds to $r=a$. Suppose that this planet is subject to a small, externally generated, tidal potential $\chi_2(r,\theta,\phi)$, where $\chi_2$ is a solid harmonic of degree 2. The planet is modeled as a solid body of uniform mass density $\rho $ whose perturbed surface lies at

$\displaystyle r = a + \delta_2(\theta,\phi),$ (C.8)

where $\delta_2$ is a surface harmonic of degree 2. It is assumed that $\vert\delta_2\vert\ll a$.

The net gravitational acceleration in the vicinity of the planet takes the form

$\displaystyle {\bf g}({\bf r}) = -\nabla {\mit\Phi},$ (C.9)

where ${\mit\Phi}({\bf r})$ is the gravitational potential. According to standard gravitational theory,

$\displaystyle \nabla^{\,2}{\mit\Phi} = 4\pi\,G\,\rho,$ (C.10)

where $\rho({\bf r})$ is the mass density distribution. Thus, we can write

$\displaystyle {\mit\Phi}(r,\theta,\phi) = {\mit\Phi}_0(r) + {\mit\Phi}_2(r,\theta,\phi),$ (C.11)


\begin{displaymath}\nabla^{\,2}{\mit\Phi}_0 \simeq\left\{
\begin{array}{lll} 4\p...
...mbox{\hspace{0.5cm}}&r\leq a\\ [0.5ex]
\end{array}\right.,\end{displaymath} (C.12)


$\displaystyle \nabla^{\,2}{\mit\Phi}_2 = 4\pi\,G\,\rho\,\delta_2\,\delta(r-a).$ (C.13)

[See Equations (E.2) and (E.3).] Here, $\delta(r-a)$ is a Dirac delta function (Riley 1974b). The physical boundary conditions are

$\displaystyle {\mit\Phi}_0$ $\displaystyle \rightarrow 0,$ (C.14)
$\displaystyle {\mit\Phi}_2$ $\displaystyle \rightarrow {\mit\chi}_2$ (C.15)

as $r\rightarrow \infty$. As is well known, the solutions to Laplace's equation, $\nabla^{\,2}{\mit\Phi}=0$, take the general form $r^{\,n}\,{\cal S}_n(\theta,\phi)$ and $r^{-(n+1)}\,{\cal S}_n(\theta,\phi)$. Moreover, the axisymmetric solution to $\nabla^{\,2}{\mit\Phi}=c$ that is well behaved at the origin is ${\mit\Phi}=c\,r^{\,2}/6$. (Riley 1974c.) It follows that

$\displaystyle {\mit\Phi}_0(r\leq a)$ $\displaystyle =-\frac{g}{2\,a}\,(3\,a^{\,2}-r^{\,2}),$ (C.16)
$\displaystyle {\mit\Phi}_0(r>a)$ $\displaystyle = - \frac{g\,a^{\,2}}{r},$ (C.17)


$\displaystyle {\mit\Phi}_2(r\leq a,\theta,\phi)$ $\displaystyle = {\mit\chi}_2-\frac{3}{5}\,g\,\delta_2\left(\frac{r}{a}\right)^{2},$ (C.18)
$\displaystyle {\mit\Phi}_2(r>a,\theta,\phi)$ $\displaystyle ={\mit\chi}_2-\frac{3}{5}\,g\,\delta_2\left(\frac{r}{a}\right)^{-3},$ (C.19)


$\displaystyle g = \frac{4\pi}{3}\,G\,\rho\,a$ (C.20)

is the gravitational acceleration at the planet's surface. Note that ${\mit\Phi}_2({\bf r})$ is a solid harmonic of degree 2 inside the planet (i.e., $r\leq a$).