# Rotation Curves

Evidence of dark matter has been confirmed through the study of rotation curves. To make a rotation curve one calculates the rotational velocity of e.g. stars along the length of a galaxy by measuring their Doppler shifts, and then plots this quantity versus their respective distance away from the center.

The rotation curve for the galaxy NGC3198 from Begeman 1989

## Stellar Motions

Dutch astronomer Jan Oort first discovered the presence of dark matter in the 1930's when studying stellar motions in the local galactic neighborhood. By observing the Doppler shifts of stars moving near the galactic plane, Oort was able to calculate how fast the stars were moving. Since he observed that the galaxy was not flying apart he reasoned that there must be enough matter around that the gravitational pull kept the stars from escaping, much as the sun's gravitational pull keeps the planets in the solar system in orbit. He was able to determine that there must be three times as much mass as is readily observed in the form of visible light. Hence, Oort's calculations yielded an M/L ratio of 3 for the region of the immediate galactic neighborhood. The M/L ratio increases by several orders of magnitude as larger astro-physical phenomena come under similar scrutiny.

## Galactic Rotation Curves

When studying other galaxies it is invariably found that the stellar rotational velocity remains constant, or "flat", with increasing distance away from the galactic center. This result is highly counterintuitive since, based on Newton's law of gravity, the rotational velocity would steadily decrease for stars further away from the galactic center. Analogously, inner planets within the Solar System travel more quickly about the Sun than do the outer planets (e.g. the Earth travels around the sun at about 100,000 km/hr while Saturn, which is further out, travels at only one third this speed). One way to speed up the outer planets would be to add more mass to the solar system, between the planets. By the same argument the flat galactic rotation curves seem to suggest that each galaxy is surrounded by significant amounts of dark matter. It has been postulated,and generally accepted, that the dark matter would have to be located in a massive, roughly spherical halo enshrouding each galaxy.

The first real surprise in the study of dark matter lay in the outermost parts of galaxies, known as galaxy halos. Here there is negligible luminosity, yet there are occasional orbiting gas clouds which allow one to measure rotation speeds and distances. The rotation speed is found not to decrease with increasing distance from the galactic center, implying that the mass distribution of the galaxy cannot be concentrated, like the light distribution. The mass must continue to increase: since the rotation speed satisfies v^2=GM/r, where M is the mass within radius r, we infer that M increases proportionally to r. This rise appears to stop at about 50kpc, where halos appear to be truncated. We infer that the mass-to-luminosity ratio of the galaxy, including its disk halo, is about 5 times larger than estimated for the luminous inner region, or equal to about 50.

## Galaxy Clusters

While Oort was carrying out his observations of stellar motions, Fritz Zwicky of Caltech discovered the presence of dark matter on a much larger scale through his studies of galactic clusters. A galactic cluster is an group of galaxies which are gravitationally bound. Our own galaxy, the Milky Way, is a member of a small cluster known as the Local Group. Using the same method employed by Oort, Zwicky determined the Doppler shifts of individual galaxies in one particular system, the Coma cluster--about 300 million light years away. Zwicky found nearly 10 times as much mass as observed in the form of visible light was needed to keep the individual galaxies within the cluster gravitationally bound. It was clear to Zwicky, as it had been to Oort, that a large sum of mass was extant which was simply not visible. At the time, astronomers referred to the material as "missing mass". However, this was deemed a misnomer as the mass was clearly present, but simply not visible. Hence, the more appropriate term "dark matter" came to supercede "missing mass". Since Zwicky's efforts, more recent measurements have found that certain galaxy clusters (and binary galaxies) have M/L ratios up to 300.

The mass-to-light ratio can also be evaluated by studying galaxy pairs, groups, and clusters. In each case, one measures velocities and length-scales, leading to a determination of the total mass required to provide the necessary self-gravity to stop the system from flying apart. The inferred ratio of mass-to-luminosity is about 100 in galaxy pairs, which typically have separations of about 100 kpc, and increases to 300 for groups and clusters of galaxies over a length scale of about 1 Mpc.

## Superclusters

The largest scale on which the mass density has been measured with any precision is that of superclusters. A supercluster is an aggregate of several clusters of galaxies, extending over about 10 Mpc. Our local supercluster is an extended distribution of galaxies centered on the Virgo cluster, some 10-20 Mpc distant, and our Milky Way galaxy together with the Andromeda galaxy forms a small group (the Local Group) that is an outlying member of the Virgo Supercluster. The mass between us and Virgo tends to decelerate the recession of our galaxy, as expected according to Hubble's law by about ten percent. This effect is seen as a deviation from the uniform Hubble expansion of the galaxies and provides a measure of the mean density within the Virgo Supercluster. One again finds a ratio of mass-to-luminosity equal to 300 over this scale, which amounts to about 20Mpc.