Orbital Mechanics

Rotation Curves

Suppose the sun went dark one day, so that we could no longer see it - how could we tell if it was still there?  One way to check would be to keep track of the motions of the Earth and the other planets.  If the planets don't fly out of their orbits, then there must be something generating the gravitational pull that keeps them in their orbits!  In fact, if you know the approximate masses of the planets, you can do better than that.  Each planet's orbit and mass allow you to calculate the mass of the object the planet is orbiting.  By observing that the mass you get is roughly the same for all of the planets, you can conclude that the object the planets are orbiting must fit inside Mercury's orbit.  In other words, you can learn a lot about an object (in this case, the invisible sun) without ever seeing it!
Rotation curve
In cosmology, we can't quite do this, but we can do something analogous.  By studying the redshift of light from a distant spiral galaxy, we can determine how rapidly different parts of the galaxy are moving toward or away from us.  This lets us construct what's called a "rotation curve" - a plot giving the relationship between orbital speed and distance from the galactic center for the stars and gas that make up the galaxy.  This plot lets us map out the distribution of matter in the galaxy.

For a given galaxy, we can compare the rotation curve we actually measure to the rotation curve we would expect to get if the galaxy contained no dark matter at all - in other words, if the visible matter was the only matter in the galaxy.  It turns out that these two curves don't agree!  If only the stars were present, one would expect that the galaxy's mass was concentrated at its center, and that the material in the outer parts of the galaxy should be orbiting more slowly than the material in the inner parts.  Instead, we see that matter throughout the galaxy orbits at an almost constant speed - the rotation curve is said to be "flat"!
Halo image - Scientific American
From the observed flat rotation curve, astrophysicists have concluded that galaxies are immersed in large clouds of invisible matter.  This dark matter halo seems to have a diameter far larger than that of the visible part of the galaxy, and to have several times the mass.

The Virial Theorem

Even in blob-like systems where astronomers can't define a rotation curve (elliptical galaxies or galaxy clusters, for example), they can still use velocities to estimate masses.  In this case, astronomers treat these systems statistically and compute an average s for its members, and hence an approximate average kinetic energy.  A result of classical mechanics called the "virial theorem" relates the average kinetic and potential energies of a gravitational bound system, and the potential energy (since it is caused by gravity) is related to the total mass.  This analysis indicates that both elliptical galaxies and galaxy clusters reside in large dark matter haloes.

NASA/ChandraCluster X-Rays

In large galaxy clusters, the space between the galaxies is filled with a very thin cloud of ultra-hot gas (more than a million degrees).  These gas atoms are moving so fast that when they collide they release x-rays.  By studying these x-rays using instruments such as the Chandra space telescope, astronomers can estimate the velocity of this gas.  This in turn allows them to estimate the cluster mass using the virial theorem, as described above.  Again, these results support the idea of a massive dark matter halo.

For more detail, see pages by Martin White.

Jeff Filippini, UC Berkeley Cosmology Group (August 2005)
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