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S (cos4« - icos»». S V ' 35/

^ SIN0^COS3d - 2 COSdj

¿^SIN^OS2* -

106 SIN3d COS &

106 SIN4 0

* Because Eq. (G-10) is a homogeneous equation in P, it does not define the normalization of P-Equations (G-ll) and (G-13) define the conventional Neumann normalization, but other normalizations are used (see Appendix H or Chapman and Bartels [1940]).

* Because Eq. (G-10) is a homogeneous equation in P, it does not define the normalization of P-Equations (G-ll) and (G-13) define the conventional Neumann normalization, but other normalizations are used (see Appendix H or Chapman and Bartels [1940]).

Using Eq. (G-13), the functions Pnm can be shown to be orthogonal; that is,

where 8* is the Kronecker delta.

It is now possible to write the complete solution to Laplace's equation as

U{r.0,<(,)= J t [C„mcosm<f>+ S„msmm<f>]/>„m(cos0) (G-16)

describing the potential exterior to a spherical surface of radius a. Customarily, Eq. (G-16) is written in the form

+ ! i ( 7 )" + ' [ C„mcosm<t> + S„msin m<t>]P„m(cos0) (G-17)

where J„ = Cn0. Terms for which wi = 0 are called zonal harmonics and the J„ are zonal harmonic coefficients. Nonzero m terms are called tesseral harmonics or. for the particular case of n = m, sectoral harmonics.

Visualizing the different harmonics geometrically makes the origin of the names clear. The zonal harmonics, for example, are polynomials in cos0 of degree n. with n zeros, meaning a sign change occurs n times on the sphere (0° < 6 < 180°). and the sign changes are independent of <f>. Figure G-l shows the "zones" (analogous to the temperate and tropical zones on the Earth) for the case of

Fig. G-l. Zones for />((cos0) Spherical Harmonics

P6(cos0). The tesseral and sectoral harmonics have n-m zeros for 0" <0< 180°, and 2m zeros for 0° <$<360°. Figure G-2, the representation of P63(cos0)cos3<f>, illustrates the division of the sphere into alternating positive and negative tesserae. Hie word "tessera" is Latin for tiles, such as would be used iri^a mosaic. When n = m, the tesseral pattern reduces to the "sector" pattern in Fig. G-3.

Fig. G-2. Pjjicosff)cos34 Showing Alternating Positive and Negative Tesseral Harmonics

Fig. G-3. Pm(cos0)cos6$ Showing Tesseral Pattern Reduced to Sectoral Pattern

Fig. G-2. Pjjicosff)cos34 Showing Alternating Positive and Negative Tesseral Harmonics

For a more detailed discussion of spherical harmonics, see Hobson [1931].

References

1. Arfken, G., Mathematical Methods for Physicists. New York: Academic Press, Inc., 1970.

2. Chapman, Sydney, and Julius Bartels, Geomagnetism. Oxford: Clarendon Press, pp. 609-611, 1940.

3. Fitzpatrick, P. M., Principles of Celestial Mechanics. New York: Academic Press, Inc., 1970.

4. Hieskanen, W., and H. Moritz, Physical Geodesy. San Francisco: W. H. Freeman, 1967.

5. Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics. New York: Chelsea Publishing Co., 1931.

6. Jackson, John David, Classical Electrodynamics. New York: John Wiley & Sons, Inc., 1962.

7. Yevtushenko, G., et al., Motion of Artificial Satellites in the Earth's Gravitational Field, NASA, TTF-539, June 1969.

Fig. G-3. Pm(cos0)cos6$ Showing Tesseral Pattern Reduced to Sectoral Pattern

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