Dark Matter Don Lincoln Citation: Phys. Teach. 51, 134 (2013); doi: 10.1119/1.4792003 View online: http://dx.doi.org/10.1119/1.4792003 View Table of Contents: http://tpt.aapt.org/resource/1/PHTEAH/v51/i3 Published by the American Association of Physics Teachers Related Articles The Einstein Theory of Relativity: A Trip to the Fourth Dimension: Lillian R. Lieber, The Einstein Theory of Relativity: Lillian R. Lieber Phys. Teach. 51, 191 (2013) THE 3 Rs Phys. Teach. 51, 132 (2013) Fermi Questions Phys. Teach. 51, 187 (2013) Tutorials in Physics Science Making by Andy Elby, University of Maryland, tinyurl.com/WSEIbyMakeSense Phys. Teach. 51, 190 (2013) Washing the Plates Phys. Teach. 51, 184 (2013) Additional information on Phys. Teach. Journal Homepage: http://tpt.aapt.org/ Journal Information: http://tpt.aapt.org/about/about_the_journal Top downloads: http://tpt.aapt.org/most_downloaded Information for Authors: http://www.aapt.org/publications/tptauthors.cfm Downloaded 04 Mar 2013 to 24.27.111.209. Redistribution subject to AAPT license or copyright; see http://tpt.aapt.org/authors/copyright_permission Dark Matter Don Lincoln, Fermi National Accelerator Laboratory, Batavia, IL t’s a dark, dark universe out there, and I don’t mean where, from the mentioned motion of clusters of galaxies and because the night sky is black. After all, once you leave gravitational lensing, to objects like NGC 4555, an elliptical the shadow of the Earth and get out into space, you’re galaxy surrounded by hydrogen gas heated to 10,000,000 K. Isurrounded by countless lights glittering everywhere you The temperature of this gas is so high that it should have dis- look. But for all of Sagan’s billions and billions of stars and persed, were it not for something holding it in. However, the galaxies, it’s a jaw-dropping fact that the ordinary kind of evidence for dark matter that is simplest for an introductory matter like that which makes up you and me is but 5% of the physics student to understand is the rotation curves of galax- energy budget of the universe. The glittering spectacle of the ies. Rotation curves are simply plots of the orbital velocity of heavens is a rather thin icing on a very large and dark cake. stars as a function of their distance from the galactic center. According to the most current estimates, ordinary matter Astronomers can exploit the Doppler shift to measure makes up merely 4.6% of the universe, with a form of mat- the velocity of stars in galaxies. Further, they can use known ter called “dark matter” being 22.7%. An even more esoteric relationships between the brightness and color of stars to the component of the cosmos is called “dark energy” and it com- star’s mass to work out the distribution of the light-emitting prises a whopping 72.8% of the energy and matter budget of (i.e., visible) mass in the galaxy. By combining simple Newto- the universe. This article describes our current understand- nian principles, it is easy to work out the dominant features ing of dark matter and why so many astronomers are confi- expected to be present in the rotation curve. The calculation dent that it exists. One of the various strands of evidence for begins by identifying gravity as the centripetal force. We then the existence of dark matter is also of pedagogical interest, insert the standard formulae for these terms: as it is perhaps unique in being a conundrum on the cosmic frontier that is easily understood using only algebra-based (1) introductory physics. The first inkling astronomers had that perhaps their and with some manipulation, we get telescopes were not telling the entire story came not long after a publication in 1925 by Edwin Hubble. Combining the (2) observations of others along with his own, Hubble made the scientific community aware of the existence of other galaxies, Mattractive is the amount of the galaxy’s mass that attracts with the implicit consequence that we lived in a galaxy of our the star to move in its orbit. Outside the galaxy, this is trivial; own—the Milky Way galaxy. The realization that the Milky Mattractive= Mgalaxy. However, inside the galaxy, not all of Way was a compact, gravitationally bound conglomeration the galaxy’s mass plays a role in determining the motion of of stars led theorist Bertil Lindblad and observer Jan Oort (of the stars. The distribution of mass within a typical galaxy is Oort cloud fame) to compare Newtonian predictions of the complex, necessitating that we turn to numerical techniques. rotation of the Milky Way with observations. The implications However, as an illustration we can treat the galaxy as a sphere were clear. The Milky Way was rotating faster than predicted of uniform density. By employing Newton’s shell theorem, using Newtonian principles and the observed amount of mat- which is the same logic familiar to introductory students in ter. Linblad and Oort’s work led Oort to state in 1932 that their exploration of Gauss’ law, one sees that the mass that at- there seemed to be two to three times more mass in the Milky tracts a star is the mass inside a sphere of radius equal to the Way than could be observed. Of course, this was the 1920s distance between the center of the galaxy and the star. Thus, 3 and 1930s, and a data/theory agreement within a factor of two inside the galaxy, Mattractive= Mgalaxy × (r/Rgalaxy) . By insert- was actually pretty good. It was quite possible that the sim- ing this term into Eq. (2), we are able to predict the rotation plest explanation (observational error) was the correct one. curve of stars inside this simplified galaxy. In 1933, astronomer Fritz Zwicky studied the Coma cluster of galaxies and ascertained that the galaxies in the outskirts of the cluster were moving far too fast to remain . (3) gravitationally bound to the cluster core. This discrepancy was much larger than Oort’s, with the luminous mass able to account for only 10% of the gravity necessary to describe Thus we see that the orbital velocity of a star rises linearly the motion of these galaxies. The plot thickened. Subsequent as a function of radius within the body of the galaxy and then attempts to measure the mass of galaxies using gravitational falls as the inverse of the square root of the radius outside lensing added to the tension. Zwicky invented the term “dark of the mass distribution of the galaxy. While physical gal- matter” to describe this invisible component of the cosmos. axies have a more complex mass distribution than the one In the intervening decades, there have been many observa- used here, the actual rotation curves have similar features, tions1 supporting the contention that dark matter is every- as shown in Fig. 1. Near the center of the galaxy, the veloc- 134 The Physics Teacher ◆ Vol. 51, March 2013 DOI: 10.1119/1.4792003 Downloaded 04 Mar 2013 to 24.27.111.209. Redistribution subject to AAPT license or copyright; see http://tpt.aapt.org/authors/copyright_permission By invoking the standard relationship between centripetal acceleration and velocity, a = v2/r, and making the appropri- ate substitutions into Eq. (1), we find that MOND predicts that at large orbital radii, which is independent of the radius. At smaller radii, MOND makes the same pre- dictions as traditional Newtonian theory. This behavior is consistent with observations. Of course, this agreement is by construction. There are many valid criticisms of the MOND theory. First, the form of the function m(a/a0) is known only in its limiting cases. Another criticism of the simplified version Fig. 1. Rotation curves of thousands of galaxies tell the same tale. given in Eq. (4) is that it conserves neither energy nor mo- Inside the galaxy, data and theory are in agreement. However, in the outskirts of the galaxy, stars are observed to orbit with nearly mentum. This serious deficiency was overcome in a 1984 constant velocity. This is in striking contrast with predictions. paper by Milgrom and Jakob Bekenstein, in which a Lagrang- (Figure adapted from Ref. 2.) ian formulation was employed. Another criticism of the early versions of MOND is that it is not a relativistic theory. Subse- ity is roughly proportional to the orbital radius, and outside quent work, including some by Bekenstein, has found various the galaxy the velocity decreases, since the increased radius ways to marry MOND with relativity. incorporates no new mass but does decrease the force due The proof of a theory is in how well it works. So how to gravity. In the outskirts of the galaxy, the rotation curve is well does MOND work? For the question of galaxy rotation predicted to smoothly bridge these two behaviors, reflecting curves, it works extremely well. It also has some successes for the fact that real galaxies aren’t uniform spheres but instead the myriad other bits of evidence that has led to the dark mat- have a gradient in the distribution of mass. ter conundrum. We will return to the strengths of the various Figure 1 shows a prediction and observation for a typical proposed solutions to this problem after other solutions have galaxy. Over the decades, thousands of galaxies have been been discussed. investigated. Time and time again, astronomers found that at large radii, stars all tend to orbit with the same velocity, in Dark matter: Baryonic clear disagreement with the predictions. The general idea of dark matter is that there exists mat- The pedagogical beauty of the dark matter conundrum is ter in the universe that does not emit or absorb light.