Geoscience Reference
In-Depth Information
If more point masses are present, it is possible to sum up the individual forces. If the
number of point masses becomes infinite the sum can be replaced by the integral,
leading for a body
v
to
G
v
G
v
d
m
r
r
d
v
V
=
=
,
(6)
where d
m
is a mass element, d
v
a volume element,
r
the distance between the mass
element and the attracted point
P
and
ρ
describes the density
d
m
d
v
.
ρ
=
(7)
The potential
V
with the unit m
2
s
−
2
is continuous and finite and vanishes following
1
which allows to approximate a body at large distances as a point
mass. Also the first derivatives are continuous and finite in interior and exterior space.
But at points where the density changes discontinuously, i.e. at the bounding surface
or at density jumps in the interior, the second derivatives show discontinuities. This
becomes evident when looking at
Poisson's equation
, which has to be satisfied by
the potential
V
/
r
for
r
→∞
Δ
V
=−
4
π
G
ρ,
(8)
with
2
V
δ
2
V
δ
2
V
δ
=
δ
+
δ
+
δ
Δ
V
z
2
,
(9)
x
2
y
2
where
Δ
is the so called Laplace operator. In exterior space, where
ρ
=
0, this
equation becomes the
Laplace equation
:
Δ
V
=
0
.
(10)
1.2 Gravity field of the Earth
In a rotating system, such as the Earth, the total force acting on a resting mass on the
Earth's surface is the sum of the
gravitational force
and the
centrifugal force
due to
the rotation. This quantity is called gravity vector:
g
=
F
+
F
z
.
(11)
In a rectangular coordinate system with its origin in the Earth center, its z-axis
coinciding with the rotation axis and assuming that the x-axis points to the Greenwich
meridian, the components of the centrifugal
force a
cting on
P
are given by the vector
of Earth rotation
x
2
y
2
to the rotation axis (see Fig.
1
).
ω
and the distance
p
=
+
