by George Rhee
The method has its pitfalls. As we have mentioned, Leverrier believed he had found a new planet, which was perturbing the motion of Mercury. In this case, it was Newton’s laws that were in error. Unfortunately there is no recipe for telling whether we should prefer to believe in the existence of dark matter or the falsity of Newton’s laws. Newton’s laws have not been tested on the scales on which we observe galaxies and clusters of galaxies. Most (but not all) astronomers today believe in the existence of dark matter. Martin Rees suggests that “we keep our minds open (or at least ajar) to the possibility that our ideas on gravity need reappraisal.”
Dark Matter in Our Own Galaxy
We can apply the method used to discover Neptune to search for dark matter in our own galaxy. It can be shown mathematically that the average speed of a star in a galaxy is related to the strength of the gravitational force that the star is experiencing. The gravitational force is caused by all matter, not just visible matter. We can thus use the strength of the gravitational force inferred from the motion of visible objects to calculate the total mass that is present. This is the theme of this chapter. The whole issue can be formulated as a simple question: Does there exist in the universe a vast amount of dark matter, which remains undetected except for its gravitational influence?
The first evidence for dark matter outside the solar system came from the study of the motion of stars in our galaxy in the early twentieth century. Let me begin by giving a quick sketch of the structure of our galaxy, which is believed to be an average spiral galaxy. The Milky Way consists of three major structures; a disk, a bulge, and a halo shown in Fig. 4.1. The disk is a gigantic rotating structure about 100,000 light years in diameter. The disk is a very flattened structure with a thickness of 2,000 light years. Our Sun, which is located in the disk, is located about 30,000 light years from the center of our galaxy. At this distance from the center, the disk rotates at about 200 km s − 1. It thus takes about 250 million years for the Sun (and of course the planets) to rotate once about the center of our galaxy. The halo of our galaxy is a spherical structure consisting of about 200 globular clusters. The structures are located within this sphere. Shapley used the globular clusters to estimate the size of our galaxy. Globular clusters are star clusters that contain as many as a million stars in a region 60 light years in diameter. The third component of the Milky Way is the nuclear bulge. The stars in the bulge are similar to halo stars (old stars) whereas the disk is the site of ongoing star formation and contains very young stars as well as older stars like our Sun. As they rotate about the center of the galaxy, disk stars such as our Sun also move vertically, oscillating about the plane of the disk. If we think of the disk as lying in a horizontal plane, the disk stars have a vertical component to their motion. By measuring the vertical speed of these stars and the thickness of the disk, we can estimate the strength of the vertical gravitational field due to the disk near the Sun. The vertical velocities are about 20 km s − 1, 10 times less than the rotational velocities. It turns out that the method is difficult to implement in practice, but some studies suggest there is between two to five times as much dark matter as visible matter in the solar neighborhood.
Fig. 4.1Overview of the structure of the Milky Way galaxy. There are three major components: a disk of stars, gas and dust; a central bulge; and a halo of old stars. The galaxy shown here is embedded in a dark matter halo several times its own size (Fig. 4.4)
Since the Sun rotates about the center of the galaxy, we can use the rotation speed and distance from the galactic center of the Sun to estimate the mass of the Milky Way. This method is a simple application of Newton’s laws. We use the same method to estimate the mass of the Sun from the period and size of Earth’s orbit, or the mass of Jupiter from the period and size of the orbits of its moons. In this manner, we calculate the mass of our galaxy to be about 100 billion (1011) solar masses.
We can actually do better than this and learn about the distribution of mass in our galaxy, not just the total mass. Hydrogen
Fig. 4.2Rotation speed of the planets around the Sun plotted against their distance from the Sun. Sincet almost all the mass in the solar system is in the Sun the more distant planets rotate slower
atoms give off radio waves having a wavelength of 21 cm. The disk of our galaxy contains substantial amounts of hydrogen gas. With radio telescopes, we can measure the rotation speed of this gas at various distances from the galactic center. We can use this information to plot what we refer to as the rotation curve of our galaxy. We can do the same for our solar system. Instead of using gas we use the planets as tracers. We know the orbital speed of the planets, and we know their distances from the Sun. The orbital speed decreases as we go farther and farther from the Sun, as shown in Fig. 4.2. The Earth orbits the Sun at about 30 km s − 1. The planet Saturn, 10 times farther from the Sun than the Earth, has a speed of 10 km s − 1, Pluto, which lies 40 times further from the Sun than the Earth, orbits at about 5 km s − 1. The reason for the decrease in speed as we move out from the Sun is that the Sun’s gravitational force is weaker at the distance of Saturn than at the Earth’s distance. When we plot a rotation curve for the Milky Way, we find a surprising result. As we see from Fig. 4.3, the rotation speed does not decrease with increasing distance from the galactic center. If anything, it increases slightly. The Sun revolves at about 200 km s − 1 around the center of our galaxy. The rotation speed twice as far out is about 250 km s − 1. This immediately tells us that the mass in our galaxy is not distributed like the mass in our solar system. In the solar system, the mass is nearly all concentrated in one blob at the center, the Sun. The mass in our galaxy is
Fig. 4.3Rotation speed of gas clouds around the Milky Way versus distance from the center of the galaxy. The fact that the shape of the curve is so different to the shape of the curve in Fig. 4.2 tells us that the mass in the Milky Way is much more spread out than the mass in the solar system
much more spread out. A detailed analysis suggests that most of the mass in our galaxy is distributed in the halo of our galaxy. We infer that the halo contains about 10 times as much mass as the disk. However, there is relatively very little light coming from the halo. The globular clusters cannot account for the halo mass. The conclusion is, therefore, that most of the mass of our galaxy is not in the form of visible matter (Fig. 4.4).
Gravitational Lensing and the Search for Dark Matter in the Milky Way
So far, the only two things we know about the dark matter is that it has mass and that it does not emit radiation. We would like to know more. An ingenious experiment has been devised to study this issue. This involves a process known as gravitational lensing, which we encountered in Chap. 1. Gravitational lensing is the bending of light by matter. The idea was proposed in 1916. In 1919, a British expedition observed an eclipse from Brazil and confirmed Einstein’s General Theory of Relativity. It has been suggested that the dark matter in our galaxy consists of objects of comparable mass to the planet Jupiter, which never got hot enough in their centers to become stars and start glowing brightly. These objects,
Fig. 4.4We conclude from using Newton’s laws together with the measurements in Fig. 4.3 that our Milky Way galaxy is embedded in a halo of dark matter which contains about 20 times as much mass as that which is visible in the form of gas and stars
brown dwarves, aret also known as massive compact halo objects, which gives the acronym MACHO. If a MACHO crosses the line of sight between us and a background star, it will act as a gravitational lens and cause the apparent brightness of the star to increase. These chancet alignments are quite rare and should affect about one star in a million each year. Large-scale monitoring projects are under way to detect such lensing events and are finding several such events per year. The lensing objects seem to have a mass between a tenth and half that of the Sun. These mass limits suggest that the MACHOs are white dwarves, dead remnants of stars like the Sun and much less
luminous. We therefore have direct evidence for MACHOS, but the current belief is that they are not present in sufficient numbers to account for all the dark matter in the halo of our galaxy. The latest studies reveal that the proportion of the dark halo mass contributed by the MACHOS could be as high as 20 %.
The sheer numbers involved in the MACHO project are impressive. The data are obtained using a 50 in. telescope at the Mt. Stromlo and Siding Spring Observatories in Australia. Since 1992, 27,000 images have been obtained, and the variability of almost 20 million stars has been determined. From this database 50 lensing events have been extracted. Analyzing these huge amounts of data is done automatically using computers. Another project, OGLE, the Optical Gravitational Lensing Experiment, got under way in 1992 as well. More recently, a 1.3 m telescope at the Las Campanas Observatory in Chile has obtained similar data to the MACHO project. These projects involve collaborations between U.S., Polish, and Australian scientists, a good example of successful international collaborations in scientific research. The results of these projects are not yet conclusive as to the origin of the halo dark matter, but they suggest that at least some of the dark halo may be comprised of stellar remnants such as white dwarfs.
Dark Matter in Other Galaxies
Of course, our galaxy is but one of many. It is, in fact, easier to study dark matter in galaxies other than our own. For spiral galaxies whose internal motions are dominated by rotation we have to determine rotation curves. This is done at optical and radio wavelengths. At radio wavelengths, we use radio telescopes to measure the redshifts of hydrogen atoms in the galaxy by observing the 21-cm line emitted by neutral hydrogen. For most galaxies, this gas can be detected farther out than the stars. This is because the gas can be detected more easily than any stars that may be present.
The rotation curves of spiral galaxies resemble that of our own. The example shown in Fig. 4.5 illustrates this, they rise from the center of the galaxy, and reach a maximum value, and remain flat as far out as we can detect them. By a flat rotation curve, I mean that the gas rotates at constant speed around the center of the galaxy, and this speed does not change as one moves away from the galactic center. The implication is that the mass enclosed within a given radius increases as the radius increases. We can, however, see the gas, as we have mentioned, out to considerably larger radii than the stars. This immediately tells us that the stars cannot be responsible for the missing matter. The missing mass must be dark.
Fig. 4.5Rotation speed of gas clouds in the spiral galaxy NGC 7332 versus distance from the center of the galaxy. The distance is measured in units of thousands of light years. The measurements thus extend out to 120,000 light years from the center of the galaxy (about twice as far as the Milky Way measurements shown in Fig. 4.3). The plot illustrates that the rotation pattern of our own galaxy is not unique but is common to many other spiral galaxies
The way these observations are usually explained is to assume three components for spiral galaxies: the bulge, the disk, and the halo. The assumption is that the halo contains most of the dark matter. It is also agreed that the density of the halo decreases as one moves away from the center of a galaxy. There are other reasons for wanting spiral galaxies to be embedded in dark halos. Computer models of isolated disks cannot produce stable disks. The disks distort into oval shapes and form large bars. We do not see this in nature. When the models are modified to include the presence of a dark halo, the disks can be shown to be stable. This is of course circumstantial evidence, for no dark halo has been directly detected. We can, however, clearly state that if our theories of gravity are correct, spiral galaxies should contain large amounts of dark matter.
There exists another class of galaxies, ellipticals, that are quite different from spiral galaxies. Elliptical galaxies contain little gas and dust, they do not have disks, and they rotate much slower than spirals. It is much less straightforward to search for evidence of dark matter in ellipticals than in spirals. However, the evidence, once again, seems to point to the existence of dark halos. Elliptical galaxies have been found to contain hot gas at a temperature of about 10 million degrees Kelvin. This temperature is so high that the gas emits large amounts of X-ray radiation, which our telescopes can detect. To confine this gas in a galaxy requires large amounts of dark matter. Because the atoms in the hot gas are moving very quickly, the gravitational field of the galaxy must be strong in order to prevent the atoms from escaping. This in turn requires large amounts of dark matter.
The conclusion is that galaxies seem to be embedded in massive dark halos containing several times the mass of the visible matter and extending to several times the visible radius of the galaxy. This raises the question of the nature of the dark matter. We would also like to know how the dark matter got there. These are questions that theories of the formation of galaxies must try to answer.
Professor Zwicky Weighs the Coma Cluster
Let us now turn our attention to structures larger than galaxies: the great galaxy clusters (Fig. 4.6). Clusters of galaxies can contain many hundreds of galaxies in a region a few million light years in diameter. The Swiss astronomer Fritz Zwicky (1898–1974) was the first to suggest that clusters of galaxies contain dark matter. The method he used is based on a theorem of physics called the “virial theorem”. The idea is that for systems in equilibrium, the kinetic energy is comparable to the potential energy. What does all this jargon mean? The kinetic energy of a system is the energy of motion. It is the energy that the system has due to the motion, that is to say, the speed of its constituent parts. For a cluster, the kinetic energy arises due to the speed of the galaxies relative to the cluster center. The potential energy is the energy due to the gravitational force acting between the particles (or galaxies). When we say the system is in equilibrium, we mean that its size does not change with time. Of course, the system is changing because the galaxies are moving, but we mean that, on average, the size of the system does not change.
Fig. 4.6The Coma cluster of galaxies. The Coma cluster consists mostly of elliptical galaxies each housing billions of stars (Credit: Dean Rowe)
Clusters of galaxies can be thought of as a swarm of galaxies held together by gravity. We can thus think of each galaxy as a point mass flying back and forth around the cluster. We assume in this discussion that the clusters are in equilibrium. There is evidence to the contrary, but let us proceed with this assumption. We can measure the motion of the galaxies in the cluster by measuring their redshifts. We will notice that not all the galaxies have exactly the same redshift. They have roughly the same redshift, corresponding to the cosmological distance of the cluster, but there is a spread about this average. Some galaxies have a slightly larger redshift than the average, and some have a slightly lower redshift. The spread of redshifts about the mean is a measure of the speed of a galaxy through the cluster. The galaxies in clusters move through the cluster with speeds of about 1,000 km s − 1. We can use this number to calculate the total energy of motion of the cluster galaxies, then using this number we can infer the potential energy using the virial theorem. The potential energy is also a measure of the total mass of the cluster.
To summarize the previous two paragraphs: we can measure the average speed of galaxies in a given cluster and use that to infer the cluster’s total mass. We can then add up the masses of all the stars in the galaxies that are cluster members and see if the mass is accounted for. Zwicky, who was ahead of his time in many fields of astronomy did this in the 1930s and found that the sums did not add up. There is a lot more matter in clusters than can be accounted for by the stars in the member galaxies. Clusters contain many galaxies, by the way, up to several thousand for the most massive ones. The conclusion that Zwicky reached in 1933 is that if the galaxies contain stars having a mass like the Sun, then, for every sun, there must be 300 solar masses of dark matter.
Let us take as an example the Coma cluster of galaxies, one of the best studied clusters. The mass contained within 3 million l
ight years of the cluster center is about 6 ×1014 solar masses. Clusters of galaxies are known to emit large amounts of X-rays. This was shown to be the case in the 1960s when the first X-ray telescopes were launched into Earth orbit. The X-rays are believed to be emitted by gas that is extremely hot, as much or more than 100 million degrees Kelvin. We can estimate the mass of the X-ray gas to be about 20 % of the total cluster mass and about 10 times that of the stars in the galaxies. We can also use the temperature of the X-ray gas to estimate the total masses the clustering the same way that we used galaxy motions to estimate the total cluster mass. Let us review the mass budget. Twenty percent of the total cluster mass consists of X-ray emitting hot gas. An additional 2 % of the total mass comes in the form of stars in galaxies. We have thus accounted for 22 % of the total cluster mass, which leaves the vast majority, 78 % of the mass in the form of some undetected dark matter, whose nature is unknown to us. The total cluster masses calculated by the X-ray method and the galaxy motion method agree to a few percent.
There is yet another fascinating way to measure cluster masses: the gravitational lensing method. Clusters are so massive that they can bend the light coming from even more distant galaxies behind the cluster. These background galaxies thus appear distorted in the form of arcs. They also appear bluer and fainter than the cluster galaxies, and they can therefore be distinguished from true cluster members. The shape and extent of the lensing distortions can be used to estimate the cluster mass. The gravitational lensing mass estimates agree with the two previous mass estimates for individual clusters. Since three independent measures provide the same answer, the suggestion is that they are measuring the same thing and that there is indeed a vast amount of dark matter in clusters.