Total Gravitational Potential

The planet is modeled as a solid body of uniform mass density $\rho$ whose surface lies at

$$r = a + \zeta_b(\theta,\phi),$$ (12.45)

where

$$\zeta_b(\theta,\phi) = \sum_{n=1,\infty}\zeta_{b\,n}(\theta,\phi),$$ (12.46)

and $\zeta_{b\,n}$ is a surface harmonic of degree $n$ . Suppose that the planetary surface is covered by an ocean of uniform mass density $\skew{3}\bar{\rho}$ whose surface lies at

$$r = a+ d+\zeta_t(\theta,\phi),$$ (12.47)

where

$$\zeta_t(\theta,\phi)=\sum_{n=1,\infty}\zeta_{t\,n}(\theta,\phi),$$ (12.48)

and $\zeta_{t\,n}$ is a surface harmonic of degree $n$ . Here, $d$ is the constant unperturbed depth of the ocean, whereas $\zeta_b$ and $\zeta_t$ are the radial displacements of the ocean's bottom and top surfaces, respectively, generated by planetary rotation and tidal effects. It is assumed that $\vert\zeta_b\vert$ , $\vert\zeta_t\vert \ll d\ll a$ .

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

$${\bf g}({\bf r}) = -\nabla {\mit\Phi},$$ (12.49)

where ${\mit\Phi}({\bf r})$ is the total gravitational potential. In the limit $d/a\ll 1$ , we can write

$${\mit\Phi}(r,\theta,\phi) = {\mit\Phi}_0(r) +\sum_{n=1,\infty}{\mit\Phi}_n(r,\theta,\phi) + {\mit\Phi}_{\rm tide}(r,\theta,\phi),$$ (12.50)

where

$$\nabla^{\,2}{\mit\Phi}_0 =\left\{ \begin{array}{lll} 4\pi\,G\,\rh... ...,\delta(r-a)&\mbox{\hspace{0.5cm}}&r\leq a\\ [0.5ex] 0&&r>a \end{array}\right.,$$ (12.51)

and

$$\nabla^{\,2}{\mit\Phi}_{n>0} = 4\pi\,G\left(\rho\,\zeta_{b\,n}+ \skew{3}\bar{\rho}\,\zeta_n\right)\delta(r-a).$$ (12.52)

Here,

$$\zeta(\theta,\phi)= \zeta_{t}(\theta,\phi)-\zeta_{b}(\theta,\phi),$$ (12.53)

is the net change in ocean depth due to planetary rotation and tidal effects, and $\zeta_n=\zeta_{t\,n}-\zeta_{b\,n}$ . Moreover, $\delta(x)$ is a Dirac delta function (Riley 1974). The boundary conditions are that the ${\mit\Phi}_n$ be well behaved as $r\rightarrow 0$ , and

$${\mit\Phi}_n({\bf r})\rightarrow 0\\ [0.5ex]$$ (12.54)

as $r\rightarrow \infty$ . It follows that

$${\mit\Phi}_0(r\leq a) = -\frac{g}{2\,a}\,(3\,a^{\,2}-r^{\,2}) - 3\,g\,d\,\frac{\skew{3}\bar{\rho}}{\rho},$$ (12.55)

$${\mit\Phi}_0(r>a) =- \frac{g\,a^{\,2}}{r}\left(1 + 3\,\frac{d}{a}\,\frac{\skew{3}\bar{\rho}}{\rho}\right),$$ (12.56)

and

$${\mit\Phi}_{n>0}(r\leq a,\theta,\phi) = -g\left(\frac{3}{2\,n+1}\right)\left(\zeta_{b\,n} + \frac{\skew{3}\bar{\rho}}{\rho}\,\zeta_n\right)\left(\frac{r}{a}\right)^n,$$ (12.57)

$${\mit\Phi}_{n>0}(r>a,\theta,\phi) =-g\left(\frac{3}{2\,n+1}\right)\left(\zeta_{b\,n} + \frac{\skew{3}\bar{\rho}}{\rho}\,\zeta_n\right)\left(\frac{r}{a}\right)^{-(n+1)},$$ (12.58)

where

$$g = \frac{G\,m}{a^{\,2}}= \frac{4\pi}{3}\,G\,\rho\,a$$ (12.59)

is the mean gravitational acceleration at the planet's surface. Note that, inside the planet (i.e., $r\leq a$ ), ${\mit\Phi}_n({\bf r})$ is a solid harmonic of degree $n$ . We can identify the three terms appearing on the right-hand side of Equation (12.50) as the equilibrium gravitational potential generated by the planet and the ocean, the potential generated by non-spherically-symmetric radial displacements of the planet and ocean surfaces, and the tide generating potential, respectively.