Geoscience Reference
In-Depth Information
For the average velocities during one period, Eq.
9.6
becomes
m
1
.
r
1
R
e
/R
e
T
e
D
m
2
.
r
2
R
m
/R
m
T
m
;
(9.12)
where
r
1
and
r
2
are radial distances from CMI to the centers of the Earth and the
Moon. Since
T
m
is the same as the period of orbital motion, the spin period of the
Earth can be calculated by Eq.
9.12
.FromEq.
9.1
, the position of CMI of two bodies
is easily found by two equations:
m
1
r
1
D
m
1
r
2
;
(9.13)
d
D
r
1
C
r
2
(9.14)
Although the Earth spin rate decreases due to the tidal force of the Moon, which
makes the orbital period and the distance between the Earth and the Moon increase
to conserve the total angular momentum, the ratio of spin periods given by Eq.
9.12
must always be true, because the tidal force of the Moon leads to the same equation,
Eq.
9.12
, by the torque and the reaction torque acted by the Moon.
In Figs.
9.2
and
9.3
, it is assumed that the planes of equators of the Earth and the
Moon are in the same plane as their orbits, which leads to a subsequent assumption
that the velocities
v
1
and
v
2
of points A
1
and A
2
are in the same directions as the
forces
f
1
and
f
2
. If the planes of equators of the Earth and the Moon are inclined by
1
and
2
from their orbital planes as shown in Fig.
9.4
, then the points A
1
and A
2
are accelerated toward the directions different, by the angles,
1
and
2
, from the
forces that are always in the plane of orbit. Since the directions of accelerations are
different from the forces, the inertial masses,
m
1
and
m
2
, at the points A
1
and A
2
should be taken larger by cosine factors,
m
1
!
m
1
/cos
1
,
m
2
!
m
2
/cos
2
.
Since the velocity shown in Fig.
9.4
is
v
1
!
v
1
cos
1
, the factor in Eq.
9.8
becomes
R
e
'
e
!
R
e
'
e
cos
1
,
R
m
'
m
!
R
m
'
m
cos™
2
. It is easy to prove that Eq.
9.12
is
unchanged. Although the planes of equators of the Earth and the Moon are inclined
by
1
and
2
from their orbital planes, these angles do not affect the periods of their
spins. If
1
and
2
are large to approach to 90
ı
, however, then they might affect the
calculation of periods.
ʔ˕
e
v
1
R
e
v
1
cosʸ
1
O
1
x
A
1
ʸ
1
CMI
Fig. 9.4
The rotation axis of
the Earth inclined by
Earth
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