Kirkwood Gaps hundreds of thousands of asteroids orbit the sun within the asteroid belt, which extends from about 3 X 108 k/n to about 5 X 108 k/n from the sun.
(a) Find the orbital period (it years) of (i) an asteroid at the inside of the belt and (ii) an asteroid at the outside of the belt. Assume circular orbits.
(b) In 1867 the American astronomer Daniel Kirkwood pointed out that several gaps exist it the asteroid belt where relatively few asteroids are found. It is now understood that these Kirkwood gaps are caused by the gravitational attraction of Jupiter, the largest planet, which orbits the sun once every 11.86 years. As an example if an asteroid has an orbital period half that of Jupiter, or 5.93 years, on every other orbit this asteroid would be at it’s closest to Jupiter and feel a strong attraction toward the planet. This attraction, acting over and over on successive orbits, could sweep asteroids out of the Kirkwood gap. Use this hypothesis to determine the orbital radius for this Kirkwood gap.
(c) One of several other Kirkwood gaps appears at a distance from the sun where the orbital period is 0.400 that of Jupiter. Explain why this happens, and find the orbital radius for this Kirkwood gap.