The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 130

4.1 Gravity Potential of the Earth and Geoid

4.1.1 Gravity and Gravity Potential

In light of Newton's law of universal gravitation, any two bodies in the universe,
possessing mass, exert gravitational attraction on each other, which will thus create
a gravitational field around the point mass. ! is used to represent this attractive
force. The force ! is directly proportional to the product of their masses m and m 0
and inversely proportional to the square of the distance r between them, and can be
expressed as:

!
r ,

Gmm 0
r 2

!

¼ ð 4 : 1 Þ

where G is the scale factor, referred to as the gravitational constant, which can be
obtained through experiment. Its value is 6.67428 10 11 m 3 kg 1 s 2 (Pent and

Luzum 2010). The direction of ! is from the attracting mass toward the attracted
mass (Fig. 4.1 ).
In geodesy, the particle of mass m is referred to as the attracting mass, while the
other particle of mass m 0 is the attracted mass, the mass of which is used as a unit,
i.e., m 0 ¼ 1. Thus:

!
r :

Gm
r 2

!

¼ ð 4 : 2 Þ

The Earth can be regarded as a body constituted by infinite number of continuous
point masses. The attraction that the Earth has exerted on the unit point mass ! is the
integral:

G ð
Earth

!
r dm,

1
r 2

!

¼ ð 4 : 3 Þ

where dm is the differential mass element of the Earth and ! represents the position
vector between dm and the attracted mass, which is a variable of integration; the
integral area is the total mass of the Earth. The direction of the gravitational
attraction is toward the center of the Earth.
Due to the rotation of the Earth, every point on the Earth experiences an inertial
centrifugal force ! :