Exercise 3.5 Graphically describe the potential and attraction of a uniform, thick-walled shell (inner radius a\ and outer radius CI2) along a line
6.2 Zonal Harmonics
function, where 6 is colatitude and 0 is longitude (see Appendix A for a review of the spherical coordinate system), and consider just one circle of colatitude 0O. Along this colatitude, /(0o> <t>) is a function of <j> alone, has a fundamental period of 2TT, and can be represented as before,
f(0o,<t>) = ^2 («m cosm0 + 6m sinm0) .
m=0
A similar equation could be written for any colatitude on the sphere, each colatitude having its own set of coefficients. In other words, the coefficients are themselves functions of colatitude, and
/ (0, <t>) = ] T (am (6) cos nuj> + bm (6) sin mcj)) . (6.2)
ra=0
The question remains as to how best to formulate the coefficients am(0) and bm(0), and that is the subject of the next two sections.
6.2 Zonal Harmonics
The dependence of /(#, 0) on 6 is completely specified by the coeffi- cients am(0) and bm(0), but what form is most appropriate for these coefficients? We might expect that they could be approximated by a sum of weighted functions, just as the dependence on 0 was approxi- mated by a sum of weighted sinusoids. We also expect that the weighted functions should be orthogonal and periodic around any meridian.
The technique of least-squares analysis is a logical way to proceed.
We can approximate the dependence of /(#, </>) on 0 by a finite sum of weighted functions, requiring that the squared difference between /(#, 0) and the finite summation be a minimum when averaged over the surface of the sphere. (Fourier-series aficionados will recognize that a similar least-squares investigation of f(t) over period T will discover the "best"
weights for sines and cosines too.)
First let /(#, 0) be independent of (j) so that equation 6.2 becomes
and approximate f(0) by a weighted, finite sum of k 4- 1 orthogonal functions Pn(0),
fk(0) = CoPo(0) + CiPi(0) + • • • + CkPk(0). (6.3)
104
Fig. 6.2. Element of a spherical surface. The ribbon encircling the sphere has an area 2nr2 sin 0 dO.
We wish to minimize the squared difference between f(0) and /&(#) as averaged over the entire spherical surface (Figure 6.2). If the radius of the sphere is r, then the total area of the sphere is 47rr2, an element of area is 27rr2sin#d#, and the total squared error averaged over the spherical surface is given by
7T
f(O)-fk(O)\22irr2sinOd6
To simplify matters somewhat, we can make the substitutions and d\i — — sin 9 d6, and the integral becomes
= cos#
- 1
6.2 Zonal Harmonics 105
1
= \ J [f (/*) - 2
/(/X)f
k(I*) + ft (A*)] dn. (6.4)
- 1
Next we substitute the summation 6.3 into equation 6.4 and apply the orthogonality condition
/
An. if n = m:
0, if n 7= m,
where An is a constant. Then solving the least-squares condition
ocr
dE 0 for each C\ leads tol
Q =
Tj'/(M)^(A*)^- (6-5)
- 1
Exercise 6.1 Follow the previous instructions to derive equation 6.5 from equation 6.4.
Hence, by approximating f(0) as a weighted sum of orthogonal func- tions, we find that the best weights, in a least-squares sense, are given by equation 6.5, an integral expression involving f{0) itself. A remarkable property of orthogonal series is implicit in this result: The calculation of C\ is independent of k; that is, each C\ is the best possible coefficient regardless of how long the series is or how many terms are missing from the series.
At this point, any set of functions orthogonal over the interval — 1 <
/i < 1 would do. One well-known set of functions is particularly appro- priate: the Legendre polynomials, also known as Legendre functions or zonal functions. Legendre polynomials are given by Rodrigues 's formula
where n is the degree of the polynomial. The first five Legendre poly- nomials are given in Table 6.1, and Figure 6.3 shows several of these graphically. One can prove (Ramsey [235]) that
/
0, if n ^ n';
\ ., ; (6-7)
Table 6.1. Legendre polynomials of degree 0 through 5.
n 0 1 2 3 4 5
Pn(O) 1
COS0
|(3 cos 20 + 1)
|(5 cos 30+ 3 cos 0) -^(35 cos 4 0 + 20 cos 20 +
^ (63 cos 50 + 35 cos 30 -\
pn(.
i
fi
5(3/*
5(5/*
9) |(35, -30 cos 6») |(63,
• )
2-l)
3 - 3/x) M4 - 30/u2 M5 - 70p3
+ 3) + 15//)
which is a demonstration of the orthogonality of Legendre functions over the appropriate interval. The reasons for selecting these particular orthogonal functions will become clear later in this chapter.
Hence, the function /(/i) (or /(#)) can be represented as an infinite sum of weighted Legendre functions,
CnPn(n), (6.8)
n=0
called a zonal expansion. It should be clear from the previous discussion how to determine the coefficients in this summation. The orthogonality of Pn(l^) makes this quite simple, at least conceptually; if Cj is needed, for example, we simply multiply both sides of the preceding equation by Pj(fJ>) and integrate both sides of the equation over the interval — 1 <
\JL < 1. The jth term of the expansion is the only nonzero term, and this single term provides
•^ ' -*• / i V . . \ r > / . . \ J . . / £ ? Q \
- 1
Hence, the orthogonality property of Legendre functions allows the de- termination of each coefficient directly from the function being rep- resented, just as sines and cosines were successful in this regard for one-dimensional functions.
6.2 Zonal Harmonics 107
(a) +/
^m;xxx
-1 - 9 = 0
» = +!
(b)
Fig. 6.3. A few low-degree Legendre functions, (a) Functions PO((JL) through
PQ(H) are shown on the interval — 1 < /i < 1. (b) Function PQ(IJ) is shown along the circumference of a circle; gray and white zones indicate areas where the function would be positive or negative, respectively, if wrapped around a sphere.
6.2.1 Example
An example is appropriate at this point to put all of the previous math- ematics into perspective. Suppose /(#, (f>) is a function of position on a sphere given by
i f O < 0 < | ;
if f <6<7T,
as depicted in Figure 6.4. We can expect that such a discontinuous func- tion would be difficult to represent with nicely behaved functions such as those in Figure 6.3, but let's give it a try.
t=-i
Fig. 6.4. Example of discontinuous /(#, (f>) over surface of sphere.
First, we represent /(#,(/>) as an infinite sum of Legendre functions.
+
+ C2P2(fi) +The solution of equation 6.9 is simplified greatly by noting that /(/i) is an odd function over the interval — 1 < \x < 1, and the Legendre functions are even when n is even and odd when n is odd. Consequently, equation 6.9 becomes
Cn =
(2n + l ) j .
0
if n is even;
x, if n is odd.
In particular,