Geoscience Reference
In-Depth Information
Using Eqs.
9.13
and
9.14
, and the observed values,
d
D
3.844
10
8
m,
m
1
D
5.9736
10
24
kg,
m
2
D
7.349
10
22
kg,
R
e
D
6.3781
10
6
m,
R
m
D
1.7381
10
6
m (Allen
1973
; Williams
2010
),
r
1
and
r
2
are calculated,
r
1
D
3.8379
10
7
m,
r
2
D
3.4602
10
8
m. Using the spin period of the Moon
T
m
D
27.3217 da
y
(Williams
2010
), the spin period of the Earth is calculated:
T
e
D
23
h
39
m
2
s
Compare with the observation
T
e
D
23
h
56
m
4
s
(Williams
2010
).
To consider the different densities of the Earth along the radial distance, if a root-
Z
R
2
dm
=m
1=2
mean-square radius of the Earth defined as R
e
D
is used,
using the density distributions, 17 g/cm
3
for 1,300 km thick inner core, 10 g/cm
3
for
2,200 km thick outer core, 4.5585 g/cm
3
for 2,843.1 km thick mantle, and 3.3 g/cm
3
for 35 km thick shell (Smith and Jacobs
1973
), where the density and the thickness
for mantle were adjusted to match with the mass
an
d the radius of the Earth, the
root-mean-square radius of the Earth is calculated, R
e
D
4.6157
10
6
m. Using the
fact that the density of the Moon is uniform, approximat
el
y 3.35 g/cm
3
(Allen
1973
),
the root-mean-square radius of the Moon is calculated, R
m
D
1.3481
10
6
m. Refer
to Park (
2013
) for Mathcad calculations in detail. Using these radii,
T
e
is calculated,
T
e
D
24
h
3
m
5
s
, which is closer to the observation. If the density distribution of
the Moon is known accurately, the calculation will yield a closer value to the
observation. The root-mean-square radius is a radius of a spherical shell of which
mass is uniformly distributed on the shell and the moment of inertia to rotate the
shell is the same as that of the Earth.
9.3
Calculation of the Earth Spin Axis
The Earth spin axis which is inclined by 23.45
ı
with respect to the Earth orbit can
be derived from the gravitation of the Sun acting on the Earth. As shown in Fig.
9.5
,
the Sun acts the strongest gravitational force on the point A of the Earth and induces
the force
F
s
, which is exactly in the reverse direction of tangential spin velocity
v
e
.
The horizontal component
F
sx
of the force makes slower the horizontal compo-
nent
v
ex
of the spin velocity because the force
F
sx
is in the opposite direction to
the velocity
v
ex
. The vertical component
F
sy
, however, may increase the vertical
component
v
ey
of the spin velocity by tilting the Earth spin axis into larger angles.
If the vertical component
v
ey
becomes larger than the horizontal component
v
ex
,
then the same process will occur with the two components switched with each
other. Eventually, both components converge to the same magnitude. It may also
be explained by statistical equipartition of velocity components.
v
ex
v
ey
D
C; C
!
1
(9.15)
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