Geoscience Reference
In-Depth Information
objects down inclined planes, which reduced their acceleration from
g
to
g
sin
δ
is the dip angle of the plane; thus, they moved more slowly and he
could time them more accurately.) In his honour a gravitational unit, the
gal
,was
named: 1 gal
,where
δ
10
−
2
ms
−
2
. The gravitational acceleration at the Earth's surface is
therefore about 981 gal. Nowadays gravity can be measured accurately by using
vibrating strings, spring balances and lasers (for details see a text such as Telford
et al
.(1990)).
If the Earth were perfectly spherical and not rotating, the gravitational accel-
eration would have the same value at every point on its surface. However, the
Earth is not a perfect sphere (it bulges at the equator and is flattened at the poles)
and it is rotating. The Earth's shape can be approximated by an
oblate spheroid
,
the surface that is generated by revolving an ellipse about its minor (shorter) axis.
The
ellipticity
,
2
or
polar flattening
,
f
of an ellipse is defined as
=
R
e
−
R
p
R
e
f
=
(5.16)
where
R
e
and
R
p
are the equatorial (longer) and polar (shorter) radii, respectively.
The oblate spheroid that best approximates the Earth's shape has an ellipticity of
1/298.247. The 'radius' of an oblate spheroid is given, to first order in
f
,by
f
sin
2
r
=
R
e
(1
−
λ
)
(5.17)
where
f
is the ellipticity,
the latitude and
R
e
the equatorial radius.
The fact that the Earth is spinning about its axis means that the value of
the gravitational acceleration on its surface is reduced.
Centrifugal acceleration
means that the gravitational acceleration on a sphere rotating with angular fre-
quency
λ
ω
,
g
rot
,isless than that on a non-rotating sphere,
g
, and is dependent on
latitude,
λ
:
2
R
e
cos
2
g
rot
=
g
−
ω
λ
(5.18)
The gravitational acceleration on a rotating oblate spheroid can also be calcu-
lated mathematically. The
reference gravity formula
adopted by the International
Association of Geodesy in 1967 is
sin
2
sin
4
g
(
λ
)
=
g
e
(1
+
α
λ
+
β
λ
)
(5.19)
where the gravitational acceleration at the equator
g
e
is 9.780 318 5 m s
−
2
and the
constants are
10
−
3
and
10
−
5
. (There are relations
α
=
5.278 895
×
β
=
2.3462
×
between
α
and
β
and between
f
and
ω
.) About 40% of this variation of gravity
with latitude
is a result of the difference in shape between the spheroid with the
best-fitting ellipticity and a perfect sphere; the remaining 60% of the variation is
due to the Earth's rotation. Gravity observations are expressed as deviations from
Eq. (5.19).
λ
2
The ellipticity of
an ellipse
should not be confused with the
eccentricity, e
,ofanellipse, which is
defined as
e
=
R
e
−
R
p
R
e
.
