Global Positioning System Reference
In-Depth Information
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45
k
2
m
s
2
k
2
m
s
2
− ξ
s
∂V
∂x
∂
∂x
∂s
∂x
=−
x
k
2
m
=
=
=−
F
x
(2.47)
Similar expressions can be written for
F
y
and
F
z
. Thus, the gradient
V
is
∂V
∂x
T
=
F
x
F
z
T
∂V
∂y
∂V
∂z
grad
V
≡
F
y
(2.48)
From (2.45) it is apparent that the gravitational potential is a function only of the
separation of the masses and is independent of any coordinate system used to de-
scribe the position of the attracting mass and the direction of the force vector
F
. The
gravitational potential, however, completely characterizes the gravitational force at
any point by use of (2.48).
Because the potential is a scalar, the potential at a point is the sum of the individual
potentials,
[31
V
i
=
k
2
m
i
s
i
Lin
—
1
——
No
*PgE
V
=
(2.49)
Co
nsidering a solid body
M
rather than individual masses, a volume integral replaces
th
e discrete summation over the body,
k
2
k
2
dm
s
v
ρ
dv
s
V (x,y,z)
=
=
(2.50)
M
[31
where
ρ
denotes a density that varies throughout the body and
v
denotes the mass
volume.
When deriving (2.50), we assumed that the body is at rest. In the case of the earth,
we
must consider the earth's rotation. Let the vector
f
denote the centrifugal force
ac
ting on the unit mass. If the angular velocity of the earth's rotation is
ω
, then the
ce
ntrifugal force vector can be written
=
ω
0
T
2
p
2
x
2
y
f
= ω
ω
(2.51)
Th
e centrifugal force acts parallel to the equatorial plane and is directed away from
th
e axis of rotation. The vector
p
is the distance from the rotation axis. Using the
de
finition of the potential and having the
z
axis coincide with the rotation axis, we
ob
tain the centrifugal potential:
2
x
2
y
2
1
2
ω
Φ =
+
(2.52)
Equation (2.52) can be verified by taking the gradient to get (2.51). Note again that
the centrifugal potential is a function only of the distance from the rotation axis and
is not affected by a particular coordinate system definition. The potential of gravity
W
is the sum of the gravitational and centrifugal potentials: