Gravity field of the earth - Physical Geodesy - page 58

the same dimensions, then the series for V would indeed still converge at the
surface of the earth. Owing to the mass irregularities, however, the series of
the actual potential V of the earth can be divergent or also convergent at the
surface of the earth. Theoretically , this makes the use of a harmonic expan-
sion of V at the earth's surface somewhat dicult; practically ,itisalways
safe to regard it as convergent. For a detailed discussion see Moritz (1980 a:
Sects. 6 and 7) and Sect. 8.6 herein.
It need hardly be pointed out that the spherical-harmonic expansion,
always expressing a harmonic function, can represent only the potential out-
side the attracting masses, never inside.

2.6 Harmonics of lower degree

It is instructive to evaluate the coecients of the first few spherical harmonics
explicitly. For ready reference, we first state some conventional harmonic
functions R nm and S nm , using (1-60), (1-66), and (1-82):

R 00 =1 , S 00 =0 ,

R 10 =cos ϑ, S 10 =0 ,

R 11 =sin ϑ cos λ, S 11 =sin ϑ sin λ,

(2-81)

R 20 = 2 cos 2 ϑ

1

−

2 ,

S 20 =0 ,

R 21 =3sin ϑ cos ϑ cos λ, S 21 =3sin ϑ cos ϑ sin λ,

R 22 =3sin 2 ϑ cos 2 λ, S 22 =3sin 2 ϑ sin 2 λ.

The corresponding solid harmonics r n R nm and r n S nm are simply homoge-
neous polynomials in x, y, z . For instance,

r 2

22 =6 r 2 sin 2 ϑ sin λ cos λ =6( r sin ϑ cos λ )( r sin ϑ sin λ )=6 xy . (2-82)

S

In this way, we find

R 00 =1 , S 00 =0 ,

r R 10 = z, rS 10 =0 ,

r R

11 = x,

r S

11 = y,

(2-83)

r 2

1

2 x 2

1

2 y 2 + z 2 ,

r 2 R 20 = − − S 20 =0 ,

r 2 r 2 R 21 =3 xz, S 21 =3 yz,

r 2 22 =3 x 2 3 y 2 , r 2 R − S 22 =6 xy.