7. MASSES OF GALAXIES AND CLUSTERS:

Astronomers have four basic methods to measure the masses of galaxies and clusters: rotation curves, random velocities, X-ray emission, and gravitational lensing.

Rotation curves: In astronomy, we often infer masses from orbits. Thus, we infer the mass of the Earth by applying Kepler's 3rd Law (as understood by Newton) to observations of the orbit of the Moon. Likewise, we infer the mass of Jupiter from the orbits of its moons; the mass of the Sun from the orbit of the Earth; the mass of stars from the orbits of binary stars about each other.

By exactly the same logic, astronomers can infer the mass of the Milky Way galaxy from the (more-or-less) circular orbits of its stars and gas, as we described in Lesson 8. But when they did, they found that the Milky Way has about 10 times as much mass as the mass in stars and gas that astronomers can detect through radiation. Astronomers call this unseen matter dark matter, and they call the ratio of dark matter to visible matter the Mass-to-Light Ratio (M/L). (More precisely, M/L is the mass of the system in solar units divided by its light in solar units. Thus, the Sun has M/L = 1.) The Milky Way has M/L = 50. Likewise, other spiral galaxies have values of M/L ranging from 5 to 50.

In fact, by measuring the rotation curves of a spiral galaxies we can not only infer its total mass, we can infer how this mass is distributed throughout the galaxy. This concept is demonstrated in the page Stronger Evidence: Rotation Curves of Galaxies. Be sure to play with the Java experiment there, and notice how the motions of the stars change when you change the distribution of mass within the galaxy. If the mass of the galaxy is concentrated toward its center, the interior stars move faster. (Just as the inner planets of the solar system move faster than the outer planets.) But if the mass of the galaxy is spread out, the orbital velocities of the stars can actually increase with radius, as illustrated below.

The most accurate method to measure the rotation curves of galaxies is to observe the Doppler shift of the 21-cm line emitted by the interstellar hydrogen gas with radio telescopes. We find that rotation velocities of spiral galaxies at large distances from the center of the galaxy are much greater than they would be if the mass were distributed in the same fashion as the luminous stars. Indeed, we find that the dark matter in spiral galaxies must be distributed in a more-or-less spherical shape that extends far beyond the distribution of stars.

Random velocities: Unlike spiral galaxies, elliptical galaxies don't rotate rapidly. The stars move in random orbits, not circular orbits. Even so, astronomers can measure the masses of elliptical galaxies by a technique that is similar to measuring rotation curves of spiral galaxies. They measure the Doppler shifts of the spectral lines of many stars to calculate average random velocities and distances from the center of the galaxy. Then they can infer the mass of the galaxy from an equation very similar to Kepler's Third Law. The result is similar to that for spiral galaxies: elliptical galaxies have M/L = 5 to 50.

By exactly the same logic, astronomers can infer the masses of clusters of galaxies by measuring average random velocities and orbital distances of many galaxies from the center of the cluster. In fact, the first evidence for dark matter in the universe was found by this technique in 1933 by the astronomer Fritz Zwicky. This method is described here, by Curt Higgins. Rich clusters have even higher mass-to-light ratios than galaxies -- in some cases as high as M/L = 300. That fact implies that there is more dark matter between the galaxies in the clusters than there is in the galaxies themselves.

X-ray emission: Another way to infer the masses of elliptical galaxies is to observe the X-ray emitting gas in the galaxy. Since this gas is very hot, it would naturally tend to expand and escape from the galaxy. But it doesn't because it is bound by the galaxy's gravity, which depends on the galaxy's mass. Therefore, by measuring the temperature and radial distribution of the X-ray emitting gas, we can infer the mass. Actually, the method depends on the same physical principle as the method of measuring random velocities of stars; the main difference is that we are measuring the random velocities of the atoms in the hot gas when we measure the X-ray temperature. The method is described in a little more detail here, by Robin Elliott.

We see a fine example of X-ray emission from a cluster of galaxies above. On the left is an optical image of the Virgo cluster of galaxies, with the giant elliptical M87 in the center. On the right is an X-ray image of the same cluster, showing a cloud of X-ray emitting gas centered on M87 that is much larger than the optical galaxy. The visible M87 about 10 times as massive as the Milky Way. By analyzing the X-ray image, astronomers can see that the halo of dark matter around M87 has five times as much mass as M87 itself. In the few cases where we can compare the mass of dark matter in clusters of galaxies as inferred from the random velocities of the galaxies with that inferred from the X-ray emission, the results agree. That fact provides a good check that the results are reliable, at least within the error margins of the measurements (about a factor of 2).

Up to now, the technique of measuring the masses of clusters of galaxies through their X-ray emission has been limited by the inability of X-ray telescopes to measure their images and spectra simultaneously. But recently launched X-ray telescopes such as NASA's Chandra and the European Space Agency's XMM are far more powerful than their predecessors, so we expect soon to know a lot more about the distribution of matter in clusters of galaxies.

Gravitational lensing: Astronomers have found yet a third way of measuring the distribution of dark matter. That is by measuring the deflection of light rays by gravity. This phenomenon is described in the next section.

The most important conclusions from these studies are: (1) galaxies and clusters of galaxies -- indeed, the universe itself -- contain approximately 10 times as much dark matter as visible matter; and (2) the dark matter in galaxies and clusters extends over much greater distances than the luminous matter.

We don't know what this dark matter is. We discussed some possibilities in Lesson 8. But here, we see that the dark matter is everywhere, not just in the Milky Way. As you will see, the amount and distribution of dark matter controls the evolution of the universe -- not only the development of galaxies in the universe, but also its ultimate fate. Therefore, understanding the nature of the dark matter is one of the most important questions in all science.

For some more details and links to a few more excellent sites, see Dark Matter by Martin White.

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Last modified March 30, 2002
Copyright by Richard McCray