The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 131

Fig. 4.1 The attracting
mass m and the attracted
mass m 0

,

!

! ! !

m 0 ¼ ω ω ˁ ð 4 : 4 Þ

! denotes the vertical distance vector between the unit point mass and the
spin axis of the Earth and

where ˁ

! represents the angular velocity vector of the Earth
rotation, which can be determined precisely using astronomical methods. Its value
is

ω

10 5 rad/s. ! is perpendicular to the axis of rotation and is
directed against the spin axis (Fig. 4.2 ).
The force of gravity of the Earth ! is the resultant of the gravitational force
acting upon a unit point mass and the centrifugal force of the Earth, namely:

ω ¼ 7.292115

! ! ! ¼ þ : ð4:5Þ

Since the weight of a body is the product of its mass and the acceleration due to
gravity, for one unit point mass the force of gravity acting upon it is equal to the
value of its gravity acceleration. Therefore, in geodesy, the concepts of gravity
force and gravity acceleration are always used interchangeably. When we say “to
determine the force of gravity at a given point” we virtually mean to determine the
gravity acceleration at this given point, and the magnitude of the force of gravity at
a given point is actually the magnitude of its gravity acceleration. The gravity
acceleration is measured in centimeters per second squared (cm/s 2 ), known as gal
(after Galileo; symbol Gal) in geodesy. One thousandth of a gal is 1 mGal, and one
thousandth of 1 mGal is 1 μ Gal, as follows:

1Gal ¼ 1, 000mGal ¼ 1, 000, 000 μ Gal,

10 5 m s 2

1mGal ¼ = :

It is inconvenient to study the gravity vector directly. For any conservative
gravity vector, there exists a so-called potential function such that the gradient of
the function is the gravity vector. The partial derivatives of this function with