The luminous matter we observe in our Milky Way galaxy is only about 5\% of the galaxy's total mass: The rest is called "dark matter," which seems to act upon all matter gravitationally but in no other way. As a rough approximation, we can therefore neglect luminous matter entirely as a source of gravity in understanding the dynamics of the galaxy. Along with many other stars, both much closer and much farther from the galactic center, we all circle about the center of the galaxy with about the same velocity \(220 \mathrm{~km} / \mathrm{s}\). Our solar system in particular is 8.5 kiloparsecs from the galactic center \((1 \mathrm{psc}=3.26\) light years.)

(a) From this information, how must the dark-matter density \(ho\) for this range of orbital radii depend upon \(r\), the distance of an orbiting star from the galactic center?

(b) The dark-matter density in the vicinity of the sun is thought to be \(ho_{0} \simeq 0.3 \mathrm{GeV} / \mathrm{c}^{2}\) per \(\mathrm{cm}^{3}\). Assuming now that the radial dependence of density \(ho(r)\) found in part (a) is valid all the way to the center of the galaxy, what is the total mass of dark matter within the orbit of our sun as a multiple of one solar mass, where \(M_{\text {sun }}=2 \times 10^{30} \mathrm{~kg}\) ?

(c) Suppose several rogue stars are in highly non-circular orbits around the galactic center, perhaps as a result of collisions with one another. Which (if any) of Kepler's laws would then still be correct for these stars? Explain.

(d) Consider a proposal that the radial dependence of dark-matter density as found in part (a) might still be valid for arbitrarily large distances from the center. Show that in fact this is not possible, and explain why.