Classical Physics to Calculate Rotation Periods of Planets and the Sun - Planetary Exploration and Science: Recent Results and Advances - page 243

result in full-sized 10 27 -10 28 g terrestrial planets (Kortenkamp et al. 2000 ; Dones
and Tremaine 1993 ; Wetherill 1990 ; Lissauer 1993 ; Lissauer and Stewart 1993 ).
Until almost 1900, cosmogonist believed that planets accumulated from the
material in Keplerian orbits would have retrograde rotation, because the material
closer to the Sun moves faster in Keplerian orbits. As the rotation directions of
most planets were known to be prograde, various “solutions” to this dilemma were
proposed. Chamberlin ( 1897 ) was apparently the first to realize that the retrograde
rotation depended on the assumption of circular orbits and that accumulation of
material on eccentric orbits could produce prograde rotation (Lissauer et al. 2000 ;
Chamberlin 1897 ).
If the planet and planetesimals are all on circular orbits prior to approaching one
another, the planet accumulates enough spin angular momentum to rotate at roughly
the rates of Earth and Mars, but in the retrograde direction (Giuli 1968 ). There is
a small range of eccentricities that produce prograde rotation at a comparable rate,
but for a realistic distribution of planetesimal eccentricities, a planet accreting from
either a uniform or nonuniform surface density disk of planetesimals rotates much
slower than does Earth and Mars, mainly because a high degree of cancellation
occurs between prograde and retrograde accumulations (Ida and Nakazawa 1990 ;
Greenberg et al. 1997 ; Ohtsuki and Ida 1998 ; Lissauer and Kary 1991 ). It suggests
that the random component of planetary spin imparted by large impactors must
provide most of the spin angular momentum of the terrestrial planets. Cosmogonists
generally accept that the final stage of terrestrial planet formation is the giant impact
stage, where terrestrial planets obtain spin angular momentum from the relative
motions of colliding protoplanets (Lissauer et al. 2000 ; Dones and Tremaine 1993 ;
Kokubo and Ida 2007 ).
Six of the nine planets in our solar system have an obliquity (the angle between
the rotational angular momentum vector and orbital angular momentum vector) of
less than 30 ı , a distribution that would have only a 10 5 probability of occurring
randomly (Lissauer and Kary 1991 ). The spin angular momentum imparted by giant
impacts should produce an isotropic distribution of obliquities with both prograde
and retrograde rotations, which has been used to argue against giant impacts as the
source of the terrestrial planet's spin (Lissauer et al. 2000 ; Kokubo and Ida 2007 ).
Cosmogonists believe that primordial rotation periods of planets have been
changed significantly by tidal frictions between planet and satellites and Sun. The
Earth spun faster in the past, and the Moon was closer to the Earth. Obliquities
of planets have also changed by spin-orbit coupling to make the planet's spin axis
precess like the Earth's precession. For the Earth, a torque to precess it is exerted
not only by the Sun but also by the Moon. Orbits of the planets also precess as a
result of their mutual gravitational interactions (Lissauer et al. 2000 ).
Four gas giant planets rotate faster than terrestrial planets, the fastest being
Jupiter with a spin period of only 0.387 days. The origin of gas giant rotation has
been less studied and far from general acceptance, more interested in the formation
scenarios of the gas giants. One scenario, the core accretion hypothesis (Podolak
2007 ; Pollack et al. 1996 ), argues that the planetesimals in the outer solar system