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

$$r = a + \delta_2(\theta,\phi),$$ (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

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

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

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

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

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

where

$$\nabla^{\,2}{\mit\Phi}_0 \simeq\left\{ \begin{array}{lll} 4\p... ...mbox{\hspace{0.5cm}}&r\leq a\\ [0.5ex] 0r>a \end{array}\right.,$$ (C.12)

and

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

$${\mit\Phi}_0 \rightarrow 0,$$ (C.14)

$${\mit\Phi}_2 \rightarrow {\mit\chi}_2$$ (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

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

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

and

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

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

where

$$g = \frac{4\pi}{3}\,G\,\rho\,a$$ (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., $r\leq a$).