The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 139

, ,

A þ B B A

a 20 ¼ G C a 22 ¼ G

2 4

where A, B, and C indicate the moments of inertia of the Earth with respect to the
X-axis, Y-axis, and Z-axis, respectively, namely:

ð ð ð

dm, B dm, C dm

y 1 þ z 1 x 1 þ z 1 x 1 þ y 1

A ¼ ¼ ¼ :

Earth Earth Earth

The other three coefficients are:

G ð G ð

2 G ð
Earth

1

a 21 ¼ z 1 x 1 dm, b 21 ¼ y 1 z 1 dm, b 22 ¼ x 1 y 1 dm :

Earth Earth

The above three integrals are the products of inertia about the X-axis, Y-axis, and
Z-axis. Therefore, the second-degree term is dependent on the moments of inertia
and products of inertia of the Earth with respect to the coordinate axes.
The coefficients of the terms of the third degree or higher are rather complex and
are not discussed here.
Substituting the above nine coefficients into ( 4.17 ), putting the origin of the
coordinate system at the Earth's center of mass, and making the coordinate axes
coincide with the Earth's principal axis of inertia, the coefficients of the first-degree
terms, a 21 , b 21 , and b 22 in the second-degree terms are all zero, and then the
expansion of the Earth's gravitational potential is:

0 1

A 3

3

2 þ

ʸ þ

GM
ˁ

G
ˁ

A þ B

2

1 3 B ð A Þ

@

2 cos 2

sin 2 V ˁ;ʸ;ʻ

Þ ¼

þ C ʸ cos 2 ʻ

ð

4

1

1 X

n

1

þ ð a nk cos k ʻ þ b nk sin k ʻ Þ P nk cos ð ʸ Þ :

ˁ

n þ

n¼3 k¼0

In practice, the spherical harmonics expansion of the gravitational potential of
the Earth is written as:

" 1

n n

1 1 X

n

GM
ˁ

a
ˁ

a
ˁ

V p;ʸ;ʻ

Þ ¼

J n P n cos ð ʸ Þþ

ð

n ¼ 2 n ¼ 2 k ¼1

# ,

P nk cos

J nk cos k ʻ þ S nk sin k ʻ ð ʸ Þ ð 4 : 19 Þ

where a denotes the semimajor axis of the Earth ellipsoid and P nk cos is the fully

normalized associated Legendre polynomials, which differ from the associated
Legendre polynomials only by a constant factor:

ð ʸ Þ