Continuing my discussion of dark matter, I will now explain another piece of evidence supporting its existence: galactic rotation curves.

How rotation curves work is easily understood using basic Newtonian gravitation and assuming spherical symmetry. In a spherically symmetric system (where the properties of the system depend only on the distance from the center (the radius) and not the direction.

Due to the inverse square force law, the force on an object in a spherically symmetric system is equivalent to the force of a point mass at the origin with the total mass at radii less than that of the object. So,

where *m* is the mass of the object, *ρ(r)* is the mass density at radius *r*, and *M(r)* is the mass of the galaxy held within a sphere of radius *r*. For a circular orbit, this requires that

as well. Combining these results, we see that the mass is related to the rotational velocity v by

.

We can use measurements of the rotational velocities of objects around the centers of galaxies to estimate the mass and compare these to other methods. In particular, we see that galaxies appear to have a fairly well-defined size. Above some radius there aren’t any more stars. We might then expect the mass to cease increasing, so the velocity will then fall with 1/r.

In the 1970s, Rubin and Ford (and a few other colleagues) made a number of measurements of the velocity curves for a number of spiral galaxies. Some example papers can be found here and here. These galaxies appear to have something more akin to cylindrical symmetry than spherical symmetry, so the above equations won’t be exactly correct, but similar behavior will be expected. The velocity curves were measured by carefully taking spectrographic measurements at different positions in the galaxies. The positions of absorption and emission lines will change slightly depending on the velocities with respect to us, so the positions of these lines allow astronomers to extract the velocities from the data.

Rather than finding this 1/*r* behavior far from the centers of galaxies, they found that the velocity curves actually leveled off or in some cases increased even far beyond where stars appeared to be. This kind of rotation curve suggests that the total mass is approximately proportional to the radius. Alternatively, this means that the density is proportional to 1/*r ^{2}*.

This gradual reduction in the mass density means that much of the mass of galaxies seems to be spread over a much larger volume than the gas and stars. This mass is not emitting or absorbing light, since otherwise it would be easily detected with photodetectors. A simple explanation of this phenomenon is that the masses of galaxies are dominated by diffuse clouds of nonluminous “dark” matter. This dark matter provides most of the gravitational effects of galaxies except right near the center where normal matter is concentrated.