Within the solar system itself it is often thought that the density of unseen dark matter is quite uniform, with mass density \(ho_{0} \simeq 0.3 \mathrm{GeV} / \mathrm{c}^{2}\) per \(\mathrm{cm}^{3}\) (the mass equivalent of about 1 proton per three cubic centimeters.) The sun itself has mass \(M_{0}=\) \(2 \times 10^{30} \mathrm{~kg}\).

(a) What fraction of a solar mass within the radius of earth's orbit might one expect in the form of dark matter? (The average radius of earth's orbit is \(150 \times 10^{6} \mathrm{~km}\).)

(b) Would Kepler's second law still be valid for orbits of comets within the solar system? Explain.

(c) Would Kepler's third law still be valid for the planets?