Classical Physics to Calculate Rotation Periods of Planets and the Sun - Planetary Exploration and Science: Recent Results and Advances - page 257

Tabl e 9. 2 Densities to give the exact periods of spin

Density of
volume
A

Mass of
volume
A

Radius of
core sphere
(radius of
planet)
(

B
(density of
planets)
(g/cm 3 )

C

Density of
core sphere
(g/cm 3 )

B(mass
of planet)
(

C

Periods of
spin (hours) 10 3 ) 10 4 m)

Mars 24.623 15.289 3.968 (3.933) 6.402 (6.418) 3.732 (339.6)

Jupiter 9.925 3.563 1.326 (1.326) 10,020 (18,990) 2,516 (7,149.2)

Saturn 10.657 1.827 0.6 (0.687) 2,263 (5,685) 2,513 (6,028.6)

Uranus 17.245 3.1 1.27 (1.27) 524 (868.3) 717.5 (2,555.9)

Neptune 16.11 3.224 1.638 (1.638) 531.9 (1,024) 840.4 (2476.4)

observed mean density. Neptune is orbiting with Triton as if it is a two-body system,
because other satellites of Neptune are so small to be ignored. The spin period of
Neptune, T N D 16.11 h, is obtained by Eq. 9.12 , and the densities of Neptune are
3.677 g/cm 3 for core and 1.638 g/cm 3 from the observed mean density. The numbers
are summarized in Table 9.2 . Core densities of the three planets, Jupiter, Uranus,
and Neptune, are closely distributed around 3.5 g/cm 3 , though the spin periods are
largely distributed.
The spin periods of Saturn and Mars are also obtained by the same way as
above with small changes of their mean densities. For Saturn, the mass, 2 10 26 kg,
of volumes A and B in Fig. 9.9 is smaller than the half of the Saturn mass,
5.685 10 26 kg, which means that the density of volumes A and B can be smaller
than the mean density, 0.687 g/cm 3 , of Saturn. A combination to give the spin
period, 10.656 h, of Saturn can be chosen 0.6 g/cm 3 for volumes A and B and
1.827 g/cm 3 for core. If the mean density 0.687 g/cm 3 for volumes A and B is
taken into account, then the density for core to give the spin period is 1.55 g/cm 3 .
The procedure to calculate the spin of Mars is in the opposite way to Saturn. The
mass of the volumes A and B, 6.402 10 23 , is close to the mass, 6.418 10 23 ,of
Mars, which means that the density of volumes A and B can be a little larger than
the mean density of Mars, because the core is included in the volumes A and B.
The best combination to give the rotation period, 24.623 h, of Mars may be chosen
3.968 g/cm 3 for volumes A and B and 15.289 g/cm 3 for core. If the mean density,
3.933 g/cm 3 , of Mars is used in the density of volumes A and B, then the density for
core to give the spin period is unreasonably large, 119.5 g/cm 3 .
The masses of the volumes A and B for Jupiter, Uranus, and Neptune are a little
larger than half of their masses, for Jupiter 1.002 10 28 kg/1.899 10 28 kg, for
Uranus 5.24 10 25 kg/8.683 10 25 kg, and for Neptune 5.319 10 25 kg/10.24
10 25 kg, so that the mean densities of the planets can be used for volumes A and
B. As summarized in Table 9.2 , rotation periods of planets are calculated accurately
in a very consistent way using the densities for volumes A and B and densities
for cores, which proves that the theory and equations to obtain spins of the planets
including the Sun are valid and universal.