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high degree of cancellation (Lissauer et al.
2000
; Ida and Nakazawa
1990
), and only
the driven components of spins remain to current speeds.
r
0
1
m
1
v
1
D
r
0
2
m
2
v
2
:
(9.5)
Since the Earth and the Moon consist of rigid materials, the tangential relative
velocities of the mass increments in the points, A
1
and A
2
, can determine the spin
velocities of the whole bodies. A planet consisting of gas materials, however, may
have different spin velocities along the radial distance or along the latitude. Since
radial vectors are perpendicular to the velocities
v
1
and
v
2
,Eq.
9.5
can be written
as, with a relation of
m
1
/
m
2
D
m
1
/
m
2
,
m
1
r
0
1
v
1
D
m
2
r
0
2
v
2
(9.6)
From Fig.
9.3
, the velocity
v
2
is
r
0
2
v
2
D
T
;
(9.7)
where
T
is a period of point A
2
if the point A
2
rotates around CMI with the tangential
velocity of
v
2
. To calculate the spin period of the Moon, the velocity
v
2
must be
expressed by the variables,
R
m
, '
m
,
T
m
, which are not independent of
r
2
0
, ',
T
,where
T
m
is the spin period of the Moon,
R
m
is the equatorial radius of the
Moon, and '
m
is the angle shown in Fig.
9.3
. The relation,
r
2
0
'
D
R
m
'
m
,is
obvious from Fig.
9.3
. The period,
T
m
, must be inversely proportional to '
m
and
T
because the spin period is small as '
m
is large. As the point A
2
moves from
CMI to O
2
, the period
T
m
decreases, while
T
increases with
v
2
fixed. It leads to a
relation:
r
0
2
'
v
2
D
T
/
R
m
'
m
T
m
;
(9.8)
v
2
D
CR
m
'
m
T
m
;
(9.9)
where
C
is a proportional constant.
For an average velocity during one period,
v
2
D
C2R
m
T
m
:
(9.10)
By the same way,
v
1
D
C2R
e
T
e
;
(9.11)
where
T
e
is the spin period of the Earth and
R
e
is the equatorial radius of the Earth.
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