Geoscience Reference
In-Depth Information
5.3.2 Orbits of satellites
The orbit of a small object about a point mass is an ellipse. Consider a satellite
orbiting the Earth, and for simplicity consider a circular orbit with radius
r
(many
satellite orbits are almost circular; a circle is just an ellipse with an ellipticity of
zero). The gravitational force of the Earth acting on the satellite is
GM
E
m
r
2
/
2
r
(from Eq. (5.15)), and this is balanced by the outward centrifugal force
m
ω
(where
ω
is the angular velocity and
m
the mass of the satellite). Thus,
GM
E
m
r
2
2
r
=
m
ω
(5.20)
ω
Rearranging this equation gives the angular velocity
:
GM
E
r
3
1
/
2
ω
=
(5.21)
Alternatively, the period of the satellite orbit,
T
=
2
π/ω
,isgivenby
4
π
1
/
2
2
r
3
GM
E
T
=
(5.22)
Equations (5.21) and (5.22) are Kepler's third law. Johann Kepler (1571-1630)
spent years studying the motions of the planets and discovered that the square of
the period of the planet's orbit was proportional to the cube of the orbit's radius.
This relationship was explained later when Isaac Newton developed his laws of
motion and gravitation, which led to the exact relation derived here and given in
Eq. (5.22).
5.4 The shape of the Earth
The Earth is neither a perfect sphere nor a perfect oblate spheroid. Clearly, moun-
tains and deep oceanic trenches are deviations of several kilometres. Geodesists
use the surface of the oceans as the reference surface, which is sensible since
a liquid surface is necessarily an equipotential
3
(if it were not, the liquid would
adjust until the surface became an equipotential). The Earth's reference surface
is called the
geoid
.Over the oceans the geoid is the mean sea level, and over
the continents it can be visualized as the level at which water would lie if imagi-
nary canals were cut through the continents. All navigation and all surveying are
referenced to the geoid. The surveyor's plumb bob, for example, does not point
'down', it points perpendicular to the local equipotential surface which, if not
too far above sea level, means perpendicular to the geoid.
3
An equipotential is a surface over which the potential has a constant value. The gradient of the
potential is therefore perpendicular to this surface (see Appendix 1). Thus, gravity (
−
grad
V
)is
always normal to the mean sea-level surface.
