Axially Symmetric Mass Distributions

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

(872) |

where denotes an average over the azimuthal angle, .

Now,

(874) |

(875) |

(876) |

(877) |

Suppose that . In this case, we can expand
as a convergent power series in , to give

(878) |

(879) | |||

(880) |

Hence,

Now, the well-known *Legendre polynomials*, , are defined

(882) |

Here, is 1 if , and 0 otherwise. The Legendre polynomials also form a

A comparison of Equation (881) and Equations (883)-(885) makes it reasonably clear that, when , the complete expansion
of
is

It follows from Equations (873), (887), and (888) that

where

Now, given that the
form a complete set, we can always
write

(892) |

Thus, we now have a general expression for the gravitational potential, , generated by any axially symmetric mass distribution, .