   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):   (869)   (870)   (871)

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)

with the right-hand side evaluated at . However, since is independent of , the above equation can also be written (873)

where denotes an average over the azimuthal angle, .

Now, (874)

and (875)

where (at ) (876)

Hence, (877)

Suppose that . In this case, we can expand as a convergent power series in , to give (878)

Let us now average this expression over the azimuthal angle, . Since , , and , it is easily seen that   (879)     (880)

Hence,   (881) Now, the well-known Legendre polynomials, , are defined (882)

for . It follows that   (883)   (884)   (885)

etc. The Legendre polynomials are mutually orthogonal: i.e., (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 (887)

Similarly, when , we can expand in powers of to obtain (888)

It follows from Equations (873), (887), and (888) that (889)

where    (890)

Now, given that the form a complete set, we can always write (891)

This expression can be inverted, with the aid of Equation (886), to give (892)

Hence, Equation (890) reduces to (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