Next: Potential Due to a
Up: Gravitational Potential Theory
Previous: Gravitational Potential
Axially Symmetric Mass Distributions
At this point, it is convenient to adopt standard spherical coordinates,
, aligned along the
-axis. These coordinates are related to
regular Cartesian coordinates as follows (see Section A.17):
Consider an axially symmetric mass distribution: i.e., a
which is independent of the azimuthal angle,
. We would expect
such a mass distribution to generated an axially symmetric gravitational
potential,
. Hence, without loss of generality, we can
set
when evaluating
from Equation (868). In fact,
given that
in spherical coordinates, this equation yields
![\begin{displaymath}
\Phi(r,\theta) = - G\int_0^\infty\int_0^\pi\int_0^{2\pi}
\fr...
...in\theta'}{\vert{\bf r}-{\bf r}'\vert}\,dr'\,d\theta'\,d\phi',
\end{displaymath}](img2078.png) |
(872) |
with the right-hand side evaluated at
. However, since
is independent of
, the above equation
can also be written
![\begin{displaymath}
\Phi(r,\theta) = - 2\pi\,G\int_0^\infty\int_0^\pi
r'^{\,2}\,...
...,\langle\vert{\bf r}-{\bf r}'\vert^{-1}\rangle\,dr'\,d\theta',
\end{displaymath}](img2081.png) |
(873) |
where
denotes an average over the azimuthal angle,
.
Now,
![\begin{displaymath}
\vert{\bf r}'-{\bf r}\vert^{-1} = (r^{2}-2\,{\bf r}\cdot{\bf r}' + r'^{\,2})^{-1/2},
\end{displaymath}](img2083.png) |
(874) |
and
![\begin{displaymath}
{\bf r}\cdot{\bf r}' = r\,r'\,F,
\end{displaymath}](img2084.png) |
(875) |
where (at
)
![\begin{displaymath}
F = \sin\theta\,\sin\theta'\,\cos\phi' + \cos\theta\,\cos\theta'.
\end{displaymath}](img2085.png) |
(876) |
Hence,
![\begin{displaymath}
\vert{\bf r}'-{\bf r}\vert^{-1} = (r^{2}-2\,r\,r'\,F + r'^{\,2})^{-1/2}.
\end{displaymath}](img2086.png) |
(877) |
Suppose that
. In this case, we can expand
as a convergent power series in
, to give
![\begin{displaymath}
\vert{\bf r}'-{\bf r}\vert^{-1}= \frac{1}{r}\left[
1 + \left...
...ght)^2(3\,F^2-1)
+ {\cal O}\left(\frac{r'}{r}\right)^3\right].
\end{displaymath}](img2090.png) |
(878) |
Let us now average this expression over the azimuthal angle,
. Since
,
, and
, it is easily seen that
Hence,
Now, the well-known Legendre polynomials,
, are defined
![\begin{displaymath}
P_n(x) = \frac{1}{2^n\,n!}\,\frac{d^n}{dx^n}\!\left[(x^2-1)^n\right],
\end{displaymath}](img2103.png) |
(882) |
for
.
It follows that
etc.
The Legendre polynomials are mutually
orthogonal: i.e.,
![\begin{displaymath}
\int_{-1}^1 P_n(x)\,P_m(x)\,dx = \int_0^\pi P_n(\cos\theta)\...
...(\cos\theta)\,\sin\theta\,d\theta = \frac{\delta_{nm}}{n+1/2}.
\end{displaymath}](img2111.png) |
(886) |
Here,
is 1 if
, and 0 otherwise. The Legendre polynomials also form a complete set: i.e., any well-behaved function
of
can be represented as a weighted sum of the
. Likewise,
any well-behaved (even) function of
can be represented as a weighted
sum of the
.
A comparison of Equation (881) and Equations (883)-(885) makes it reasonably clear that, when
, the complete expansion
of
is
![\begin{displaymath}
\left\langle\vert{\bf r}'-{\bf r}\vert^{-1}\right\rangle = \...
...\left(\frac{r'}{r}\right)^n P_n(\cos\theta)\,P_n(\cos\theta').
\end{displaymath}](img2116.png) |
(887) |
Similarly, when
, we can expand in powers of
to obtain
![\begin{displaymath}
\left\langle\vert{\bf r}'-{\bf r}\vert^{-1}\right\rangle = \...
...\left(\frac{r}{r'}\right)^n P_n(\cos\theta)\,P_n(\cos\theta').
\end{displaymath}](img2119.png) |
(888) |
It follows from Equations (873), (887), and (888)
that
![\begin{displaymath}
\Phi(r,\theta) = \sum_{n=0,\infty} \Phi_n(r)\,P_n(\cos\theta),
\end{displaymath}](img2120.png) |
(889) |
where
Now, given that the
form a complete set, we can always
write
![\begin{displaymath}
\rho(r,\theta) = \sum_{n=0,\infty} \rho_n(r)\,P_n(\cos\theta).
\end{displaymath}](img2124.png) |
(891) |
This expression can be inverted, with the aid of Equation (886), to
give
![\begin{displaymath}
\rho_n(r) = (n+1/2)\int_0^\pi\rho(r,\theta)\,P_n(\cos\theta)\,\sin\theta\,d\theta.
\end{displaymath}](img2125.png) |
(892) |
Hence, Equation (890) reduces to
![\begin{displaymath}
\Phi_n(r) = -\frac{2\pi\,G}{(n+1/2)\,r^{n+1}}\int_0^r r'^{\,...
...,\pi\,G\,r^n}{n+1/2}\int_r^\infty r'^{\,1-n}\,\rho_n(r')\,dr'.
\end{displaymath}](img2126.png) |
(893) |
Thus, we now have a general expression for the gravitational potential,
,
generated by any axially symmetric mass distribution,
.
Next: Potential Due to a
Up: Gravitational Potential Theory
Previous: Gravitational Potential
Richard Fitzpatrick
2011-03-31