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(-90
8
, 156.513
8
)
160
140
120
100
80
8
8
(0
, 90
)
60
40
20
8
8
(90
, 23.487
)
-100 -80
-60
-40
-20
0
20
40
60
80
100
Angle (
j
8
)
Fig. 9.7
A curve of Earth equator obtained theoretically
The graph in Fig.
9.7
shows a curve obtained from Eq.
9.19
, which represents
a trace of Earth equator inclined by 23.487
ı
from the orbital plane of the Earth.
Observed angles of planets inclined from their orbits are listed in Table
9.1
(Williams
2010
). The angles are dominantly dispersed near 23.5
ı
and 0
ı
which are
gravitationally stable or equilibrium angles as expected from Fig.
9.5
. The vertical
component of
F
s
vanishes at 0
ı
so that the gravitation of the Sun to tilt the spin axis
of the planet might be negligible. As shown in Table
9.1
, the Earth is very close to
the case of
C
1. Uranus and Pluto might be regarded as exceptional cases.
9.4
The Rotation Period of the Sun
The Earth and the Moon are an ideal two-body system where the spin period and the
spin axis of the Earth can be calculated by an accuracy of almost exact numbers. A
system involving many bodies such as the solar system or Jupiter and its satellites,
however, implies very complicacy. The spin of the Sun must be affected by various
motions of all the planets and satellites in very complicate ways. The calculation
of the spin may be approximated by reducing the many-body system into a two-
body system on the basis that the weighted collective contributions from planets
determine the spin of the Sun. It is assumed that the period of spin of the Sun can
be calculated by an imaginary planet which rotates synchronously as the case of the
Earth.
Figure
9.8
shows an imaginary planet which has the same mass and the same
moment of inertia as those of total sums of all planets. The Sun is denoted by
m
s
,
the imaginary planet by
m
o
, and planets by
m
i
. CMI means the center of moment of
inertia of the solar system. The distance from CMI to the Sun or planets is denoted
by
r
s
or
r
i
.
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