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.

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