Axially symmetric mass distributions

Consider an *axially symmetric* mass distribution; that is, a
that 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 (3.10). In fact,
given that
in spherical coordinates, this equation yields

with the right-hand side evaluated at . However, because is independent of , Equation (3.27) can also be written

where denotes an average over the azimuthal angle.

Now,

(3.29) |

and

(3.30) |

where (at )

(3.31) |

Hence,

(3.32) |

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

(3.33) |

Let us now average this expression over the azimuthal angle, . Because , , and , it is easily seen that

(3.34) | ||||||

and | ||||||

(3.35) |

Hence,

Now, the well-known *Legendre polynomials*,
, are defined (Abramowitz and Stegun 1965b) as

(3.37) |

for . It follows that

and so on. The Legendre polynomials are mutually orthogonal:

(Abramowitz and Stegun 1965b). Here, is 1 if , and 0 otherwise. The Legendre polynomials also form a complete set: any function of that is well behaved in the interval can be represented as a weighted sum of the . Likewise, any function of that is well behaved in the interval can be represented as a weighted sum of the .

A comparison of Equation (3.36) and Equations (3.38)-(3.40) makes it reasonably clear that, when , the complete expansion of is

Similarly, when , we can expand in powers of to obtain

It follows from Equations (3.28), (3.43), and (3.44) that

where

Given that the form a complete set, we can always write

This expression can be inverted, with the aid of Equation (3.42), to give

(3.48) |

Hence, Equation (3.46) reduces to

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