Geography Reference
In-Depth Information
Fig. 1.6
Relationship between the true gravitation
vector
g*
and gravity
g
. For an idealized
homogeneous spherical earth,
g*
would
be directed toward the center of the earth.
In reality,
g*
does not point exactly to the
center except at the equator and the poles.
Gravity,
g
, is the vector sum of
g*
and
the centrifugal force and is perpendicular
to the level surface of the earth, which
approximates an oblate spheroid.
surfaces slope upward toward the equator (see Fig. 1.6). As a consequence, the
equatorial radius of the earth is about 21 km larger than the polar radius.
Viewed from a frame of reference rotating with the earth, however, a geopo-
tential surface is everywhere normal to the sum of the true force of gravity,
g
∗
,
and the centrifugal force
2
R
(which is just the reaction force of the centripetal
acceleration). A geopotential surface is thus experienced as a level surface by an
object at rest on the rotating earth. Except at the poles, the weight of an object
of mass
m
at rest on such a surface, which is just the reaction force of the earth
on the object, will be slightly less than the gravitational force m
g
∗
because, as
illustrated in Fig. 1.6, the centrifugal force partly balances the gravitational force.
It is, therefore, convenient to combine the effects of the gravitational force and
centrifugal force by defining
gravity
g
such that
g
∗
+
2
R
g
≡−
g
k
≡
(1.7)
where
k
designates a unit vector parallel to the local vertical. Gravity,
g
, sometimes
referred to as “apparent gravity,” will here be taken as a constant (g
9.81 ms
−
2
).
Except at the poles and the equator,
g
is not directed toward the center of the earth,
but is perpendicular to a geopotential surface as indicated by Fig. 1.6. True gravity
g
∗
, however, is not perpendicular to a geopotential surface, but has a horizontal
component just large enough to balance the horizontal component of
2
R
.
Gravity can be represented in terms of the gradient of a potential function ,
which is just the geopotential referred to above:
=
=−
g
∇
However, because
g
=−
g
k
where g
≡|
g
|
, it is clear that
=
(z) and
d/dz
g. Thus horizontal surfaces on the earth are surfaces of constant geopo-
tential. If the value of geopotential is set to zero at mean sea level, the geopotential
=
