Global Positioning System Reference
In-Depth Information
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2
x
2
y
2
v
ρ
dv
s
1
2
ω
W(x,y,z)
=
V
+ Φ =
+
+
(2.53)
The gravity force vector
g
is the gradient of the gravity potential,
∂W
∂x
T
∂W
∂y
∂W
∂z
g
(x,y,z)
=
grad
W
=
(2.54)
and represents the total force acting at a point as a result of the gravitational and cen-
trifugal forces. The magnitude
g
=
g
is called gravity. It is traditionally measured
1 cm/sec
2
. The gravity increases as one moves from the
equator to the poles because of the decrease in centrifugal force. Approximate values
for gravity are
g
equator
=
in units of gals where 1 gal
=
978 gal and
g
poles
=
983 gal. The units of gravity are those
of acceleration, implying the equivalence of force per unit mass and acceleration. Be-
cause of this, the gravity vector
g
is often termed gravity acceleration. The direction
of
g
at a point and the direction of the plumb line or the vertical are the same.
Surfaces on which
W(x,y,z)
is a constant are called equipotential surfaces, or
level surfaces. These surfaces can principally be determined by evaluating (2.53)
if the density distribution and angular velocity are known. Of course, the density
distribution of the earth is not precisely known. Physical geodesy deals with theories
that allow estimation of the equipotential surface without explicit knowledge of the
density distribution. The geoid is defined to be a specific equipotential surface having
gravity potential
[32
Lin
—
1.4
——
Lon
PgE
W(x,y,z)
=
W
0
(2.55)
[32
In
practice this equipotential surface is chosen such that on the average it coincides
w
ith the global ocean surface. This is a purely arbitrary specification chosen for ease
of
the physical interpretation. The geoid is per definition an equipotential surface, not
so
me ideal ocean surface.
There is an important relationship between the direction of the gravity force and
eq
uipotential surfaces, demonstrated by Figure 2.9. The total differential of the grav-
ity
potential at a point is
∂W
∂x
∂W
∂y
∂W
∂z
dW
=
dx
+
dy
+
dz
(2.56)
=
grad
W
T
·
d
x
=
g
·
d
x
The quantity
dW
is the change in potential between two differentially separated
points
P(x,y,z)
and
P
(x
dz)
. If the vector,
d
x
is chosen such
that
P
and
P
occupy the same equipotential surface, then
dW
+
dx,y
+
dy, z
+
=
0 and
g
·
d
x
=
0
(2.57)
Expression (2.57) implies that the direction of the gravity force vector at a point is
normal or perpendicular to the equipotential surface passing through the point.
