Rotation curves and dark matter

If the rotation curve is flat, $V_c(R)=constant$, it means there must be more mass in the outskirts of spirals than we've assumed so far. Let's compute how much more.

Assume the density $\rho$ of the MW is $\rho=\rho_0(R_0/R)^{\alpha}$, where $\rho_0$ and $R_0$ are constants. For such a density distribution, the mass within a given radius $R$, $M(<R)=\int_0^R 4\pi \rho(R) R^2dR\propto R^{3-\alpha}$. Computing $V_c(R)$, you easily find that a flat rotation curve requires $\alpha=2$, hence $\rho(R)\propto 1/R^2$ is required for a flat rotation curve. And so, although the light density drops significantly in the outer parts of the MW, the mass density drops much slower with increasing $R$.

More recently, rotation curves have been measured using the 21-cm line in many other spiral galaxies, with always the same result: although the rotation curve rises from the centre outward, at sufficiently large radius $R_f$ it invariably becomes flat, even though most of the light is enclosed within $R_f$. So why don't we see this mass? Two solutions seem possible. Firstly, for some strange reason, the mass-to-light ratio of the stars outside $R_f$ increases very much, so that the little amount of light we see, actually represents much more mass than we think. Or secondly, there is some type of invisible mass - dark matter - that dominates the mass-density in the outer parts of spiral galaxies.^5.5 Both seem rather contrived, yet evidence for dark matter seems to reoccur again and again in other places as well. Note that this dark matter needs to be (1) massive, (2) invisible, i.e. it does not absorb light, nor does it emit light, it simply does not interact with photons.

Already in the 1930s, the Swiss astronomer Fritz Zwicky said that the galaxies in a cluster of galaxies move too fast for the amount of matter accounted for in the galaxies themselves. In fact, they moved so fast that gravity could not contain them to remain inside the cluster. So Zwicky postulated that there must be some type of invisible matter inside galaxy clusters. We'll discuss evidence for dark matter in elliptical galaxies, in clusters, and finally in the Universe as a whole. The case is sufficiently strong that experiments have been set-up all over the world, to try to detect this mysterious type of mass in the laboratory.