Briefly explain all answers.
(Mathematically, the expression for the mass enclosed within an orbit of radius r is M = v^{2}r/G, where G is Newton's gravitational constant (numerically equal to 4.30 x 10^{-6}(kpc/M_{ })·(km/s)^{2}) and v is the orbital speed of a star at distance r. This formula is essentially another way of writing Kepler's Law P_{orb}^{2} = constant * r^{3}.)
This concept works equally well for the orbits of stars and gas within spiral galaxies. By looking at the mass inside the orbit of stars or gas at different distances from the center of the galaxy, the mass of a galaxy as a funcion of radial distance from the center (the mass of a galaxy INSIDE radius r) can be obtained from the rotation curve of the galaxy.
Basically, both when discussing the orbits of planets around a star, or the orbits of stars around the center of the galaxy, the mass inside the orbit determines, via gravity, the properties of the orbit. The difference here is that if you look at a bigger orbit (farther from the center), the mass inside that orbit is bigger than the mass inside the smaller orbit (something that doesn't happen with different-sized orbits around stars - remember that the mass of the planets is insignificant compared to the mass of the star).
Measuring the Mass
Shown above are two possible rotation curves of the Milky Way Galaxy. (Neither one is necessarily correct.)
Dark Matter!
The rotation curve of the Milky Way Galaxy is more like curve B than curve A! So there is matter that we cannot see: Dark Matter! In fact, only about 1% to 10% of our Galaxy's mass is luminous.
We have also applied this method to other spiral galaxies. We can estimate the the expected mass of a galaxy based on its brightness and distance. This gives us an estimate of the mass of the luminous matter in the galaxy. The rotation curve method tells us the mass based on the gravitational influence of all the mass in the galaxy. For most galaxies, the luminous mass is in the neighbourhood of 10 times less than the mass determined from the rotation curve method. Therefore, a significant fraction of the mass of the galaxy must be in dark matter.
Hubble's Law says that the expansion of space between the galaxies is simply v = H_{o}d, where v is the apparent recessional velocity of a galaxy, H_{o} is the Hubble constant, and d is the distance of the galaxy.