The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 138

Fig. 4.5 Integral area is the
entire Earth

The first-degree term has three coefficients, i.e., a 10 , a 11 , and b 11 . For P 1 (cos

ʸ 1 ) ʸ 1 , it follows from ( 4.18 ) along with expres-

sions of the spherical coordinates and the rectangular coordinates that:

¼ cos ʸ 1 , and P 11 (cos ʸ 1 ) ¼ sin

G ð G ð a 10 ¼

Earth ˁ 1 cos

ʸ 1 dm ¼ z 1 dm,

G ð G ð

Earth

Earth

a 11 ¼

Earth ˁ 1 sin

ʸ 1 cos ʻ 1 dm ¼ x 1 dm,

G ð G ð

b 11 ¼

Earth ˁ 1 sin

ʸ 1 sin ʻ 1 dm ¼ y 1 dm :

Earth

Assume the Cartesian coordinates of the Earth's center of mass are x 0 , y 0 , z 0 ;in
the light of physics, its follows that:
ð

ð ð

x 1 dm

y 1 dm z 1 dm

Earth

¼ x 0 , ¼ y 0 ,

Earth Earth

M ¼ z 0

M M

Hence the three coefficients of the first-degree term are:

a 10 ¼ fMz 0 , a 11 ¼ fMx 0 , b 11 ¼ fMy 0

It is thus clear that the first-degree terms are related to the center-of-mass
coordinates of the Earth. If we place the origin of the coordinate system at the
Earth's center of mass, then the numerical values of these terms are zero.
The second-degree term has five coefficients, i.e., a 20 , a 21 , a 22 , b 21 , b 22 . Inte-
grating ( 4.18 ) gives: