So the rotation curves of spiral galaxies are flat, and we have to invoke unseen dark matter to explain this phenomenon. But how can we learn anything about the properties of dark matter when we can't even see the stuff? One way is to use its influence on the rotation curve to determine how it is distributed in the galaxy.

For example, if rotation curves are flat this means that V=constant. Since

V=(GM/r) if V is constant then M/r needs to be constant, so if you go out twice as far in a galaxy, you find twice as much mass in dark matter.^{ ½ },^{}If M is proportional to r, then the density of dark matter must decline asr which is a much more gradual decline than the density of stars. So dark matter in galaxies is very extended (out to many tens or perhaps even hundreds of kiloparsecs) and we refer to the dark matter is being in "halos" surrounding the galaxy.^{-2}But we can actually get a bit more information out here with a technique known as "rotation curve modeling." If we know how much mass there is in the disk, we can then subtract that off and figure out how much mass there is in the halo (at least as far out in radius as we can observe in the galaxy). How do we do this?

We know how the starlight is distributed in spiral galay disks -- it is essentially an exponential distribution of light, where the luminosity density (known as the "surface brightness") is given by

where is the central surface brightness and h is a characteristic size of the disk. For example, our Milky Way has a scale length h = 3.5 kpc and a central surface brightness (in blue light) of

. We then assume a value known as the "disk mass-to-light ratio" which tells us how many solar masses of stuff there is for each solar luminosity of brightness. The mass-to-light ratio has units of solar units (ie the Sun has a mass-to-light ratio of 1, since its one solar mass produces one solar luminosity of brightness), and depends on both the mix of stars in the galaxy, and the wavelength of light you're measuring. But typical numbers fall in the range 0.5-2.0.

Once we know the properties of the disk and assume a value for the disk mass-to-light ratio, we can calculate the mass in the disk and the rotation curve that the disk mass should produce. An example of this is shown in the next figure, and we can see that the prediction fails to fit the rotation curve very well. Again, this is telling us we need to invoke dark matter to make the fit better. Even if we say we made a bad choice for the disk mass-to-light ratio, changing that value would move the predicted curve up and down, but would not make it fit better at large radius.

Figure 4 - NGC 2403 rotation curve and model. The model assumes that all the mass is in the galaxy disk, and that the disk has a mass-to-light ratio of 2. Now what? To make the curve fit, we add a model for the dark matter. Models for dark matter abound; for the purposes of this exercise, we choose a simple model known as the "isothermal sphere", which is characterized by a central density

rho and a core radius_{0}r . The task now is to vary these parameters until we get a good fit to the rotation curve. In practice, since disk mass-to-light ratios are not very precisely known, the mass-to-light ratio is often treated as a free parameter as well. With three parameters to play with, once can generally get a reasonable fit to the data._{core}

Figure 5 - NGC 2403 rotation curve and model. The model includes both a disk and halo. Once this fitting process is done, we have a model for the dark matter in the galaxy: its central density and radial distribution. If we do this for many galaxies, we can start to get a handle on the general properties of dark matter in galaxies, which has important implications for both the study of galaxies and the cosmology of the Universe.

: So what is the mass of a galaxy? Until we find a galaxy with a Keplerian-declining rotation curve we simply don't know. All we can say is the galaxies has at leastso much mass. (back)