Planetary Spin and Keppler's Laws


In the proof of Kepler's second and third law, why spin angular momentum of the planet is not considered ?


It seems that planetary systems are formed by the accretion of matter in the form of small particles, under the influence of gravity. The probability that all of the particles that go to make up a star and its planets would have zero angular momentum with respect to the center of gravity of the whole collection is vanishingly small. Consequently when a star and its planets are formed, conservation of angular momentum requires that the entire mass to be rotating about the center of gravity.

As the radius of this mass of small particles is reduced by gravity the angular velocity increases and local variations in density of the mass may cause planets to be formed as well as the central star. Each of the formed objects carries its share of the angular momentum of the whole, resulting in a rotating central star and perhaps several rotating planets. Once the star and planets are formed the principle interaction among them is the gravitational attraction which is a central force and therefore conservative. The conservation of momentum then causes the planets to continue to spin about their own axis.

Among the closest neighbors there may be a non-conservative interaction in the form of tidal forces. This interaction will have the effect of eventually slowing the rotation of the participants until their rotation period matches their orbital period so that they present the same face to each other all the time. This interaction has already locked the moon up with the same face toward the earth and will eventually cause the earth to present always the same face to the moon. This tidal interaction is too small to have any significant effect on more distant neighbors each orbiting the sun rather than each other.

Non-destructive collisions between a planet and another object like a comet or a large asteroid may cause the planet's axis to be tilted away from the normal to the plane of its orbit as well as possible tilt its orbit or change the orbit's eccentricity.

Regarding the angular momentum conservation invoked in proving Keppler's laws, because the angular momentum of the planet's spin about its axis is uncoupled from the angular momentum of its orbital motion, each must be separately conserved. This uncoupling is the result of the fact that the gravitational interaction of the planet and its star, after each has become well condensed, is primarily the central force not the tidal interaction.