The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 134

G X

n

Gm 1
r 1 þ

Gm 2
r 2 þþ

Gm n
r n ¼

m i
r i

V ¼ ð 4 : 9 Þ

i¼1

The mass is continuously distributed within the body, and hence it only requires
conversion of the sum of ( 4.9 ) to an integral to obtain the formula for gravitational
potential of the body:

G ð

dm
r

V ¼ , ð 4 : 10 Þ

M

where dm denotes the differential mass element at the point ( ʾ , ʷ , ʶ ); it is a variable

q

x

2 2 2

of an integral (see Fig. 4.3 ); r is the distance

from dm to the attracted mass and the total mass of the integral area is M.
The centrifugal force or acceleration is given by:

¼ ð ʾ Þ þ ð y ʷ Þ þ ð z ʶ Þ

2

P ¼ ω ˁ , ð 4 : 11 Þ

where represents the vertical

distance from the point being studied to the axis of rotation (see Fig. 4.2 ). Assume
that the spin axis coincides with the z-axis of the rectangular coordinate system;
then for the point (x, y, z):

ω denotes the angular velocity of Earth rotation and ˁ

p

x 2

ˁ ¼ þ y 2 :

p

x 2 Inserting ˁ ¼ þ y 2 into ( 4.11 ):

p

x 2

2

P ¼ ω ð þ y 2 Þ :

Obviously its potential function is:

2

:

¼ ω x 2 y 2 Q þ ð 4 : 12 Þ

2

The force of gravity is the resultant of the gravitational force and the centrifugal
force, and thus the gravity potential W is equal to the sum of gravitational potential
V and the centrifugal potential Q, namely:

W ¼ V þ Q : ð 4 : 13 Þ

Hence the formula for gravity potential is expressed as: