Geoscience Reference
In-Depth Information
Fig. 9.11
Forces acting on
Jupiter and the imaginary
satellite
m
o
f
1
r
o
f
2
r
J
CMI
D
D
m
J
To proceed as in Eq.
9.5
, the inertial mass
m
0
is replaced by m
0
. The inertial
mass
m
J
has to be replaced not by
m
J
but by the mass inside the radius of
r
J
,
because the inertial mass is bound to Jupiter by the gravitational force of the mass
inside the radius of
r
J
. It is easy to prove that the gravitational force of the shell
measured at the point CMI vanishes, where the inner and outer radii of the shell are
r
J
and
R
J
, respectively. Now it is ready to use Eq.
9.12
. Since the sphere D has the
radius of
R
D
D
r
J
,Eq.
9.12
does not work. Instead of
R
D
, root-mean-square radius
Z
R
2
dm
=m
1=2
R
D
D
giving an accurate value of the Earth spin is used.
D
.3=5/
1=2
r
J
For
a
uniform density, the root-mean-square radius is related by R
D
and R
0
D
.3=5/
1=2
R
0
.
The sphere D is a core volume of Jupiter and its density is much higher than the
mean density of Jupiter. Since the core density of Jupiter is not known, it is more
meaningful to obtain the core density that provides the exact value of the Jupiter
spin using Eq.
9.12
rather than guessing a core density. The obtained core density
should not only be within a reasonable boundary, but core densities of all five planets
should have a consistency within their boundaries.
To find the rotational period of the imaginary satellite of Jupiter, Eq.
9.28
is
used to obtain
T
0
D
8.773 day with the fact that the four satellites of Jup
ite
r are
rotating synchr
on
ously. Putting
r
J
D
2.516
10
7
m,
r
0
D
1.201
10
9
m, R
D
D
1:949
10
7
m, R
0
D
2:628
10
6
m, and
m
0
D
3.931
10
23
kg into Eq.
9.12
,
m
0
r
0
R
0
R
0
m
D
r
J
R
D
R
D
T
J
D
T
0
;
the exact number of the Jupiter spin,
T
J
D
9.925 h, is obtained from the core
density of the sphere,
D
D
3.563 g/cm
3
and
m
D
D
4
R
D
3
D
/3
D
2.377
10
26
kg.
Considering the densities of the Earth, 17 g/cm
3
for core and 5.515 g/cm
3
for mean,
a reasonable boundary of the core density would be around 3 times the mean density
of a planet. The number, 3.563 g/cm
3
as a core density of Jupiter, is definitely within
a reasonable boundary, considering the mean density of Jupiter,
J
D
1.326 g/cm
3
.
The spin periods of Uranus and Neptune can also be calculated by the same
way as that of Jupiter. The spin period of Uranus,
T
U
D
17.24 h, is obtained by
Eq.
9.12
, and the densities of Uranus are 3.1 g/cm
3
for core and 1.27 g/cm
3
from the
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