Dark Matter
Virginia Trimble
Physics Department, University of California, Irvine;
Astronomy Department, University of Maryland, College Park
The ancient Greeks supposedly believed that all earthly objects and events result from interactions among four elements -- earth, air, fire, and water -- while heavenly objects consist of a fifth element, or quintessence. This idea went out of fashion during the Renaissance. It became completely untenable when 19th century physicists used the principles of spectroscopy to show that sun, moon, planets, stars, and clouds of gas in space consist of exactly the same chemical elements as found on earth -- hydrogen, helium, carbon, oxygen, sodium, iron, and all the rest.
So far, so good. But spectroscopy (Chapter 38) works only for matter that emits or absorbs light (including radio waves, x-rays, and other forms of electromagnetic radiation; see Chapter 34). Could there be matter out there that does not absorb or emit light? And, if so, could it be something different from ordinary matter, which is made of atoms and molecules? The short answers are almost certainly yes and we cannot be sure. As the 20th century closes, nearly all astronomers agree that 90% or more of the mass in the universe that is capable of exerting gravitational forces does not emit its fair share of light. This is what scientists call dark matter. Its existence is well established. Its nature is not.
How do you look for something you cannot see? By its effects on things you can see, the way a barking dog may alert you to the presence of a silent burglar. When two or more objects ("a system") exert gravitational forces on each other (See Eq.1), they set up a balance between kinetic and potential energy, such that
Eq.1
where M is the mass of the system, v is a characteristic velocity in that system (orbit speed, rotation speed, escape speed, or range of velocities if there are many objects), R is the size scale of the system (radius of orbit, diameter of cluster . . .), and G is the universal constant of gravity. Equations 14.7 and 14.22 , in your textbook, are both special cases of this relationship. (To see this, solve Equation 14.22 for M, and in the second equation preceding Equation 14.7, divide both sides by T2, so that r3/T2 becomes rv2, then solve for M.)
You don't have to be able to see the whole system for this to work. Between 1843 and 1846, John Couch Adams in England and Urbain Jean Joseph Leverrier in France used the velocity and position of Uranus in its orbit to deduce the existence, mass, and location of Neptune before anyone had ever knowingly observed it. Thus, if you can measure the size of a system and a characteristic velocity from a few of the objects in the system, you will know the mass of the entire system.
Many hundreds of astronomers, from Newton's time to the present, have used various versions of Eq.1 and observations of sizes and speeds of astronomical objects to measure their masses. Unfortunately, scientific discoveries almost never happen in the order in which professors would like you to learn about them, so we have to jump somewhat between large and small systems. It may help to glance occasionally at Table 1 to see where we are.
Dutch astronomer Jan H. Oort, born in 1900 and still interested in the subject as I write, studied the velocities and positions of stars relatively close to our own sun, in the disk of the Milky Way galaxy (Fig. 1), between 1932 and 1965. He could measure star speeds relative to the solar system from shifts in the wavelengths of lines in the stellar spectra (the Doppler effect, Chapter 17). The distances to the stars (the "R" of the system) come from comparing how bright they appear with their real luminosities. Oort then used Eq.1 to determine the mass of the system, which he compared with the sum of the masses of the known stars in the solar neighborhood. The value of M came out somewhat larger, by 30 to 50%, than he had expected. Even our own corner of space has some dark matter. Most of it is cool gas and faint stars that emit radio and infrared radiation, which could not be observed when Oort began his work.
Swiss-American astrophysicist Fritz Zwicky (1898 - 1974) was interested in systems having much larger scales. Starting in 1933, he measured velocities and distances in clusters containing hundreds to thousands of galaxies like the Milky Way (Fig. 2). His result was a real surprise -- more than 100 times as much mass as you would have guessed on the basis of the light coming from the galaxies. Zwicky also used a very convenient shorthand for describing the results of his (and other) studies. Measure the brightness in units of the luminosity of the sun (1 L ¤ = 4 
1026 W) and the mass of the system in units of the mass of the sun (1 M¤
= 2 1030 kg). Then a group of stars will have M/L 
1, or maybe as much as 3 - 4 if there are more faint stars than bright ones. Larger values of M/L indicate the presence of dark matter.
Oort and others, studying stars close to the sun, found M/L = 3 - 4, while Zwicky and others, examining large clusters of galaxies, found M/L = 100 - 300. What about single galaxies?
American astronomer Vera C. Rubin is one of the most active investigators of dark matter. Using the world's largest telescopes, she records spectra at many different points within the images of galaxies like the one in Figure 3. Lines in the spectra are shifted by an amount proportional to the velocity of the gas emitting them (Doppler effect, Fig. 4). Applying Eq.1 is particularly easy when the galaxy is edge-on to us. In such cases, R is just the distance from the center to the point measured, and v is the difference in velocity between the center and that point. The light-emitting parts of galaxies have M/L values of about 10, in between the solar neighborhood and clusters of galaxies. Table 1 shows characteristic numbers for clusters of galaxies and some other kinds of astronomical systems.
Most of the velocities used for the systems listed in Table 1 come from Doppler shifts in spectra, but there are other techniques possible. In a hot gas, the speed with which the atoms move is proportional to the square root of the gas temperature
(Eq.2)
as shown in Chapter 21. Thus, if you can measure the temperature of the gas, for instance from the kind of radiation it emits, you can obtain a value for v to use in Eq.1.
A portion of the nonluminous mass implied by the numbers in Table 1 consists of gas and tiny stars. These emit x-rays, infrared radiation, radio waves, and so forth; thus they can be inventoried. After we allow for them, at least 90% of the matter that exerts gravitational forces in clusters of galaxies is still not emitting its fair share of light. This is the minimum amount of dark matter that must be present.
Limits can also be placed on the maximum amount of dark matter possible. The universe as a whole is a (very large!) system in which gravitation is the dominant force. The system is not in equilibrium. Rather, the clusters of galaxies are all moving away from one another, like raisins in a swelling loaf of bread (although this particular bread has no center and no surfaces and doesn't slice very well). In other words, the universe is expanding. It is doing so in a way that says the total mass (luminous + dark matter) is not much larger than the amount implied by Eq.1. The equation cannot be used directly. The total volume (hence mass) of the universe is large compared to the part we can survey and may be infinite. But divide both sides of the equation by R3 (volume), and you get
(Eq.3)
where 
is an average density for our part (or any other part) of the universe.
Essay Problem 4 invites you to derive this equation for yourself in a slightly different way. The measured ratio of velocity to distance (v/R) for clusters of galaxies outside our own is called Hubble's constant, H. Using it, one can set the desired upper limit to density and so to the amount of dark matter in the universe. That upper limit is about 10-26 kg/m3. It is called the "closure density," meaning the amount that will just barely stop the expansion of the universe, if you wait long enough. The closure density corresponds to M/L = 1000, almost 10 times as large as the ratio in clusters of galaxies. Thus, if the universe is closed, there is about 10 times more dark matter between the clusters than in them. Our universe therefore contains at least 90% dark matter, but no more than 99%.
What is the nature of dark matter? Two major candidates are in the running, and there are several ways one might tell the difference between them. First, there could be ordinary dark matter, consisting mostly of hydrogen and helium, as do the stars and galaxies. Second, dark matter could be something else. The standard names for these are baryonic (for ordinary matter) and nonbaryonic dark matter because ordinary atoms have nuclei consisting of protons and neutrons, for which the collective name is baryons. Atoms also have clouds of negatively charged electrons, held to the positively charged nuclei by a force different from gravity. It is called the electromagnetic force. Protons, neutrons, electrons, and all forms of light can exert and be affected by electromagnetic forces.
In addition, protons and neutrons experience a third force, called the strong or nuclear force, which holds them together in nuclei. And finally there is a fourth force, the weak interaction, involved in some kinds of radioactivity and nuclear fusion reactions, like those that keep the sun shining. Exerting and reacting to all four forces is the signature of baryonic material.
Nonbaryonic material, in contrast, can exert and experience only the gravitational and (perhaps) weak forces. This immediately accounts for its darkness -- no electromagnetic force equals no light! Modern theoretical physics predicts that many different kinds of nonbaryonic particles are likely to exist. Most will be unstable, just as there are many kinds of baryons that we normally do not see because they decay into neutrons and protons. The nonbaryonic candidates carry names like WIMP (for Weakly Interacting Massive Particle) and ino (short for photino, gravitino, higgsino, and some of the other names for specific kinds of particles).
Not a single nonbaryonic particle has ever been convincingly detected in a physics laboratory or any other place. This could change soon. Several different kinds of WIMP detectors are under construction or operating (Fig 6). If enough WIMPs or inos pass through such a detector, one will occasionally exert a (weak interaction) force on a baryon inside, thereby depositing some energy and announcing its passage. Meanwhile, perhaps the strongest argument in favor of the dark matter being baryonic is that we know baryons exist!
The strongest argument against large quantities of baryonic dark matter is that it is really quite hard to hide. We can now detect gas at any temperature (as Zwicky could not) and even the faintest stars (as Oort could not), at least in our own galaxy. Only a couple of possibilities remain. One is spheres of gas so small that they never get hot enough for nuclear reactions to occur in them and therefore never shine except by reflected light. Jupiter is an example, and it would be difficult to prove that the dark matter (at least in galaxies) does not consist of billions of Jupiters moving in the space between the stars. Another possibility is remnants from generations of stars that died billions of years ago (although one then has to hide some other products as well as the lingering light).
But there is a generic problem associated with hiding any kind of baryonic dark matter in large quantities. Ten or twenty billion years ago, our expanding universe was so compact and hot that nuclear reactions occurred throughout its constituents. The main products were ordinary hydrogen, heavy hydrogen (deuterium), helium, and a small amount of lithium. (In case you wondered, all the other elements are made by nuclear reactions inside stars.) The relative amounts of each depend on the number of baryons present during the reactions. Qualitatively, it is easy to see which way the relationship must go. With lots of baryons around, they find each other quickly and react, making relatively more helium and lithium. With fewer baryons, they are less likely to interact, and so leave more hydrogen and deuterium behind.
The oldest stars and some gas clouds preserve the relative proportions of H, He, and Li left over from the early universe, so that we can measure them. The proportions correspond to what is produced if the baryons density is 1 to 10% of the closure density. Thus M/L in baryonic material is 10 - 100. This is enough to take care of galaxies. It may not be enough to account for the masses and M/L's of the largest clusters of galaxies. And you really cannot close the universe with baryonic material, unless some important physical process has been left out of the calculations.
We do not know whether the real universe is open or closed. Many tests have been tried, but they invariably get tangled up with changes in the appearance of galaxies as they age or with some other confounding factor. Many theorists, however, strongly suspect that the true density is precisely equal to the critical density, c. Such a universe has the Euclidean geometry of flat space that you learned in high school. But the main reasons for perferring = c have to do with the very early phases of the universal expansion and how they gave rise to what is now here -- for instance, the fact that the universe on large scale looks so very nearly the same in all directions, even though the galaxies we see at large distances in opposite directions in the sky could never have communicated with one another in the past. If one takes these arguments seriously and concludes that the universe must have exactly the critical density, then M/L = 1000. Most of the gravitating mass in the universe in this case is probably not just dark but also nonbaryonic.
The hypothetical nonbaryonic dark matter has some positive virtues of its own, as well as the negative ones of being easy to hide and not messing up nuclear reactions in the early universe. An important one is its potential role in galaxy formation.
Galaxies exist, (after all, we live in one) and have existed for more than 95% of the history of the universe. They had to start forming at a time when everything was still so hot (more than 3000 K) that ordinary matter and light constantly interacted. Thus, growing lumps in the matter would have caused corresponding lumpiness in the radiation left from that period. We see the radiation (as microwaves with a characteristic temperature of 2.7 K), and it is smooth, to better than one part in 100 000, from place to place in the sky. This combination of lumpy matter and smooth radiation is very difficult (perhaps impossible) to achieve if the lumps that become galaxies are made entirely of ordinary matter.
Suppose, however, that we have WIMPs or other nonbaryonic dark matter. Such dark matter will not interact with the light except gravitationally (remember, no electromagnetic forces). Thus lumps can form in it when the universe is young and hot, while still leaving the radiation smooth. Then, later, after everything has cooled below 3000 K, ordinary gas (the baryonic material) will respond to the gravitational forces of those lumps and flow into them. At these lower temperatures, light no longer tags around after ordinary matter, and galaxies like the Milky Way can form without ruffling up the background radiation. The details of the process differ depending on your favorite sort of WIMP or ino, but, quite generally, it is easier to understand galaxy formation if 90% or so of the gravitating mass in the universe is nonbaryonic.
The situation is not entirely a happy one. We know that much of the matter in the universe is nonluminous (not you or me; each of us emits 100 W of infrared radiation), but it could be 99% dark or only 90% dark. And there is no firm way to decide whether the dark stuff is (a) all baryons (hydrogen and helium; carbon and oxygen like us can be ruled out) (b) all nonbaryons (for which there is no independent evidence), or (c) some of each. In the latter two cases, baryonic structures like us, the sun, and stars are not made of the dominant kind of substance in the cosmos. It's a strange sort of minority to be a part of, but perhaps the ancient Greeks would have been pleased.
Suggested Readings
Gribbin, J., and M.J. Rees, Cosmic Coincidences, New York, Bantam Books, 1989.
Krauss, L., The Fifth Essence: Dark Matter in the Universe, New York, Basic Books, 1989.
Parker, B., Invisible Matter and the Fate of the Universe, New York, Plenum, 1989.
Tucker, W., and K. Tucker, The Dark Matter, New York, William Morrow, 1988.
Essay Questions
Problems
1020 m). What is the mass of the galaxy inside our orbit? (Note: 1 parsec = 1pc = 3.261 lightyears = 3.086 x 1010m)
1021 m). If the mass of the galaxy were the value you calculated in Problem 1, what would the velocities of these clusters be? In fact, their velocities are more like 220 km/s. What, therefore, is the total mass of the galaxy out to 50 kpc from its center? As an approximation, treat the galaxy as a sphere.
1023 m), a velocity dispersion of 1400 km/s, and a total luminosity 1000 times that of a single galaxy (e.g., a total of 4 1013 L¤
). What is the ratio of M/L in solar units of such a cluster?
, radius R, and outward velocity at the surface v = RH, where H = Hubble's constant = 100km/s/megaparsec (3.1 10-22 km/s/m) (Fig. 7). Consider the fate of a test particle of mass m on the surface of the sphere. It will fall back in if its potential energy exceeds its kinetic energy; otherwise, it will keep going forever. Calculate (a) its potential energy, (b) its kinetic energy, and (c) the density of the sphere if the two energies are precisely equal. This is the critical density mentioned in Question 1.
|
TABLE 1 Mass to Light Ratios in Astronomical Systems |
||||||||
| System | Methods Used | Scale (parsecs)a | M/L (solar units) | Fraction of Closure Density Seen | ||||
|
Solar neighbor-hood, star clusters |
Velocities of stars |
1 - 3  103 |
1 - 4 | 0.001 - 0.004 | ||||
|
Luminous parts of galaxies |
Rotation speed; velocities of stars |
104 | 3 - 10 | 0.003 - 0.01 | ||||
|
Whole galaxies |
Velocities of globular clusters and companion galaxies Temperature of X-ray gas |
105 | 10 - 30 | 0.01 - 0.03 | ||||
|
Pairs and small groups of galaxies |
Velocities of galaxies |
106 | 30 - 100 | 0.03 - 0.10 | ||||
|
Large clusters of galaxies |
Velocities of galaxies Temperature of X-ray gas |
107 | 100 - 300 | 0.1 - 0.3 | ||||
|
Universe |
Change in expansion rate |
1010 |
 1000 |
 1 |
||||
|
a 1 parsec = 3.086 |
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Figure Captions
Figure 1. Schematic drawing of the Milky Way galaxy. Most of the light comes from the central bulge and thin disk, but globular clusters of stars trace a spheroidal halo (in which most of the mass is dark) that extends five to ten times further out. X marks the solar neighborhood. The actual solar system is so small on the scale of this picture that you would have to expand its diameter by a factor of 5000 to make the orbit of the earth look as big as the period at the end of this sentence.
Figure 2. Messier 84 and 86, bright elliptical galaxies in the nearby Virgo Cluster. (© Royal Observatory Edinburgh/AAT Board 1987)
Figure 3. Spiral galaxy, somewhat like the Milky Way, seen face on. Most of the light comes from this disk, especially the arms, where the young bright stars are concentrated, but the gravitational potential of the galaxy extends much further out. (Courtesy of NASA)
Figure 4. Sketch illustrating the results of placing the slit of a spectrograph along the disk of a spiral galaxy that we happen to see face-on (e.g., see Fig. 5). The lines emitted by gas in the galaxy's disk are moving toward us on one side and away from us on the other side, because the galaxy is rotating. This shifts the lines away from the wavelengths of the same line emitted by gas in the lab ("comparison spectrum") and enables us to measure a v for Equation 14.E1.
Figure 5. Spiral galaxy, somewhat like the Milky Way, seen edge-on. The part you can see extends out about 15,000 parsecs (5 1020 m), but the real galaxy is much larger. (Courtesy of U.S. Naval Observatory)
Figure 6. Interior of the Irvine-Michigan-Brookhaven particle detector. Originally constructed to look for proton decay, its hundreds of thousands of gallons of extremely pure water also function as a WIMP detector. A weakly interacting "ino" can transfer energy to a baryon or an electron in the water. The charged particle then moves away at close to the speed of light, making a shock wave of radiation as it goes. The light so radiated (Cerenkov light) is then seen by phototubes that line the inside of the tank. The diver is not normally present when data are being recorded. No firm detections of WIMPs have been reported from IMB or any other detector operated so far. (Joe Stancampiano and Karl Luttrell, © National Geographic Society)
Figure 7.