Consider a spherical planet whose unperturbed surface corresponds to
. Suppose that this planet is
subject to a small, externally generated, tidal potential
, where
is a solid harmonic of degree 2.
The planet is modeled as a solid body of uniform mass density
whose perturbed surface lies at
![$\displaystyle r = a + \delta_2(\theta,\phi),$](img4228.png) |
(C.8) |
where
is a surface harmonic of degree 2. It is assumed that
.
The net gravitational acceleration in the vicinity of the planet takes the form
![$\displaystyle {\bf g}({\bf r}) = -\nabla {\mit\Phi},$](img4231.png) |
(C.9) |
where
is the gravitational potential.
According to standard gravitational theory,
![$\displaystyle \nabla^{\,2}{\mit\Phi} = 4\pi\,G\,\rho,$](img4232.png) |
(C.10) |
where
is the mass density distribution.
Thus, we can write
![$\displaystyle {\mit\Phi}(r,\theta,\phi) = {\mit\Phi}_0(r) + {\mit\Phi}_2(r,\theta,\phi),$](img4233.png) |
(C.11) |
where
![\begin{displaymath}\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.,\end{displaymath}](img4234.png) |
(C.12) |
and
![$\displaystyle \nabla^{\,2}{\mit\Phi}_2 = 4\pi\,G\,\rho\,\delta_2\,\delta(r-a).$](img4235.png) |
(C.13) |
[See Equations (E.2) and (E.3).]
Here,
is a Dirac delta function (Riley 1974b).
The physical boundary conditions are
as
.
As is well known, the solutions to Laplace's equation,
,
take the general form
and
. Moreover, the
axisymmetric solution to
that is well behaved at the origin is
.
(Riley 1974c.)
It follows that
and
where
![$\displaystyle g = \frac{4\pi}{3}\,G\,\rho\,a$](img4255.png) |
(C.20) |
is the gravitational acceleration at the planet's surface. Note that
is a solid harmonic of degree 2 inside the planet (i.e.,
).