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to take for an initial velocity to decrease to a hundredth of its initial value. The
increased radius,
R
e
, of the Earth by the tidal force of the Moon can be inferred
by an equation:
Gm
2
d
R
e
;
Gm
1
.R
e
C
R
e
/
Gm
1
R
e
Gm
2
d
C
(9.35)
where the gravitational potential at the increased radius is equal to the gravitational
potential increased by the tidal force of the Moon. The increased radius is calculated
R
e
D
21.6 m. The volume of the increased radius is given by
V
(4
R
e
2
R
e
)
(1/3) (1/2), where the factor 1/3 comes from an increase in one direction among
three perpendicular directions and the factor 1/2 from a rough average of the
increased radius. The mass is given by
m
V
e
with the average density
e
of the Earth. For more accuracy, the radius of the Earth can be replaced by a
mass-averaged radius
R
0
e
D
[
R
r
e
(4
r
2
dr
)]/
m
(3/4)
R
e
to use the average density
e
of the Earth instead of the surface density of the Earth. As the Earth rotates,
gravitational potential of the mass
m
increases and decreases twice a rotation by
height
R
e
0
(3/4)
R
e
D
16.2 m, where is a constant representing the
difference between the real increased radius and the calculated one here. The whole
volume of increased radius per rotation may be approximated by 2
V, because
there are two perpendicular directions on the equator plane. The potential energy
loss per rotation is, approximately,
W
2
2
m
g
R
e
0
, which must be a loss
of the kinetic energy of the Earth per rotation. The constant is a nonconservative
potential energy loss rate as the mass m moves by the increased radius. The
number of rotations to lose all the kinetic energy of spin is
E
k
/
W
(1/2)
I
!
2
/
W
.
If the initial velocity
v
10
makes 100 rotations per year, then the number is 5.86
10
6
for
D
1and
D
1. It takes 2.7
10
5
years, many orders less than the Earth age,
for the initial spin to be damped to one rotation per year, where
R
D
R
0
exp(
Ct
),
N
0
-100
D
N
0
exp(-
Ct
0
),
R
0
D
100 rotations per year, and
R
D
1 rotation per year to
calculate the time,
N
0
D
5.86
10
6
and
t
0
D
1 year to calculate the constant
C
,are
used. Even for
D
0.1 and
D
0.1, it takes 2.7
10
8
years, still an order less than
the Earth age.
As the same calculations are made with the reduced two-body systems of planets,
for
D
1and
D
1, it takes 1.77
10
7
years for Jupiter, 4.55
10
7
years for Saturn,
6.98
10
7
years for Uranus, and 1.93
10
7
years for Neptune, for 100 rotation
per year of a initial spin to be damped to one rotation per year. For the Sun, it
takes 1.93
10
13
years, but if the imaginary planet were in a distance of Mercury
in early stage of solar system, it would take 5.63
10
7
years. For Mars, it takes
1.46
10
12
years calculated from its satellites. If the tidal force of the Sun was
taken into account in much closer distance at the early stage, it may take much
shorter time by many orders, which may apply to other planets. Although it is a
very rough calculation, the assumption that if they had not been driven by torques,
initial velocities were damped out to negligibly small values may be validated.
One further validity may be inferred from the spins of satellites in Table
9.3
where all major satellites but Hyperion are rotating synchronously to their orbital
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