Rethinking "Star Soup"

By Dana Mackenzie

The Andromeda Galaxy, the first of many galaxies analyzed by Vera Rubin. Image courtesy of NASA/JPL-Caltech/UCLA.

To date, nobody has ever actually seen a “lump” of dark matter. All the evidence for its existence depends on mathematical calculations—something that ought to make mathematicians happy. But Don Saari, a longtime specialist in celestial mechanics at the University of California at Irvine, is far from convinced.

At a standing-room-only lecture at this year’s Joint Mathematics Meetings, Saari called into question a classical—in fact the earliest—method for estimating the amount of dark matter, which uses galactic rotation curves. “It’s a mathematical computation, and that computation is not correct,” Saari says.

The method, which was used by Vera Rubin in the 1970s to convince her fellow astrophysicists that dark matter was real, assumes that a galaxy can be approximated as what Saari calls a “star soup”—a uniform, homogeneous distribution of matter. Using this assumption and Newton’s law of gravitation, it is easy to deduce a relation between the velocity of rotation \(v(r)\) of a star at radius \(r\) from the center of a galaxy, and the total mass of the galaxy \(M(r)\) that lies inside its orbit. The relation looks like this:

\[\begin{equation}
M(r)= \frac{rv(r)^2}{G},
\end{equation}\]

where \(G\) is Newton’s gravitational constant. In our solar system, for example, the masses of the planets are negligible compared to the sun. Thus, \(M(r)\) is effectively constant beyond the sun’s radius, and the rotation speeds of the planets drop off proportionally to \(1\sqrt{r}\). For a galaxy, which has a fuzzier edge than the solar system, we would not expect quite such a rapid decrease, but we would still expect \(v(r)\) to decrease near the visible edge of the galaxy.

But that’s not the way galaxies behave. Rubin showed that the velocity of orbiting stars (and gas clouds) in a spiral galaxy remains constant out to the edge of the galaxy. If the velocity is constant, then \(M(r)\) must increase proportionally to \(r\). And this means there must be a vast amount of matter in the galaxy that we cannot see.

It’s a simple argument that, according to Saari, has one big flaw: A galaxy is not a “star soup.” To put it mathematically, the limit of billions of point masses will not necessarily behave like a continuum. In a galaxy of point masses, Saari points out, the masses are attracted much more strongly to nearby masses than to distant ones. This tugging can exaggerate local variations in velocity, and can lead to a “crack-the-whip” effect, with stars pulling their neighbors along, which is not permitted by the star soup approximation. This instability may be how galaxies get their spiral arms in the first place—a fascinating issue, Saari believes, for mathematicians to examine. 

Central Configurations

What Saari has proved to date is that the standard method for estimating \(M(r)\) produces extravagantly wrong answers for the total mass of special arrangements of stars called “central configurations.” A symmetric configuration of point masses of this type rotates essentially as a rigid body under Newton’s law of gravitation. 

Certain central configurations are quite common. The most widely known is an equilateral triangle formed by three bodies: typically a large mass and two smaller masses that revolve around it in the same orbit, but 60 degrees apart. Such configurations were predicted mathematically by Joseph Louis Lagrange in 1772. One example consists of the sun, Jupiter, and the Trojan asteroids. Another consists of Earth, the moon, and a spacecraft, which could hypothetically be placed at either of two Lagrange points, one in front of and the other trailing the moon; these orbits are stable and would make excellent spots for fuel depots for interplanetary missions.

Another central configuration that can actually be found in the solar system is a ringed planet. In 1859, James Clerk Maxwell showed that a symmetric arrangement of masses in a circle around a central mass can rotate as if the masses were a rigid body. His intended application, the rings of Saturn, is a bit problematic because the rings are so wide that they cannot be considered particles in a single circle. A better example might be the very tenuous ring surrounding Uranus, which was unknown in Maxwell’s day.

Saari has introduced a new central configuration, which is in fact a step toward a multi-ringed planet. What he envisions is a cosmic spiderweb, with equal spacing of the spokes, but not necessarily of the rings. A mass is placed at each intersection point of a ring and a spoke; all the masses on a given ring are equal, but the masses can differ from ring to ring. He has proved that if the spacing of the rings is chosen just right, the masses will orbit as a rigid body, which means that \(v(r)\) grows linearly as a function of \(r\). The usual argument would suggest, then, that the matter distribution \(M(r)\) grows proportionally to \(r^3\), a growth rate even more dramatic than that found in spiral galaxies. And yet the masses can actually be chosen in such a way that \(M(r)\) is proportional to ln\((r)\). This is a vast discrepancy: An astrophysicist would conclude that 99.999% (or more) of the spiderweb galaxy is dark matter—and yet there is none at all.

Saari argues that the reason for the discrepancy is the tugging from nearest neighbors that the “star soup” model ignores. In fact, the choice of spacing for the rings is very sensitive to this effect: Too far apart and there is not enough tugging, too close and there is too much.

Galaxy as Guinea Pig

The reaction to Saari’s paper in the astrophysics community has so far been frosty: While referees have agreed that the mathematical argument is correct, the paper has been rejected by two astrophysical journals. Saari believes that the editors and referees missed the point. He views the spiderweb galaxy not as a configuration that is actually found in the universe, but as a test case to learn whether the method for computing \(M(r)\) works, much like a guinea pig in a biological experiment. “If the guinea pig gets sick, then we would say there is a concern about the treatment,” Saari says. 

One reason that Saari may be fighting an uphill battle is that astrophysicists have by now gathered multiple independent lines of evidence that converge on roughly the same estimates for the cosmic ratio of ordinary matter (baryons) to dark matter. To begin with, the amount of matter in a galaxy can be estimated not only by rotation curves, but also by such methods as gravitational lensing. In this relativistic effect, two galaxies line up with the observer, and the light from the more distant galaxy bends around the closer one. This produces multiple images of the more distant galaxy, or sometimes even a complete ring. While such arrangements are rare, a few dozen of the billions of galaxies in the observable universe have an orientation that would produce lensing.

Another dark matter test is weak lensing, which occurs when the intervening matter is insufficient to produce multiple images but is abundant enough to distort the image of the distant galaxy in predictable ways—by shearing or bending, for example. Weak lensing has the advantage of being more or less ubiquitous, and it allows for the calculation of a distribution of matter in the nearer galaxy. These calculations consistently show baryonic matter concentrated in the center of a galaxy, with dark matter more abundant in the fringes. “There is now better agreement about the profile of the dark matter halo than the distribution of the central baryons!” wrote Richard Massey, Thomas Kitching, and Johan Richard in a 2010 survey article.

Finally, a different kind of evidence is provided by surveys of the cosmic microwave background, such as the one by the Planck telescope. These measurements allow astrophysicists to study the interaction of normal and dark matter before the formation of galaxies, in the era before space became transparent. At that time the universe was filled by a plasma of matter and photons that vibrated acoustically. The variations in temperature produced by these sound waves were “frozen in” when the universe became transparent, and the size of the variations provides a good clue to the amount of dark matter in the universe—just as the sounds of bells made of dense and light materials will be different.

Tremendous Opportunity for Mathematicians

Saari’s argument does not address any of these other pieces of evidence for dark matter. In fact, he believes that dark matter most likely exists, although perhaps not in the amounts proposed by cosmologists. “I don’t want to criticize astrophysicists,” he says. “They can’t wait for mathematicians to come up with a perfect model of a galaxy; they have to make simplifying assumptions. But there is a tremendous opportunity for mathematicians to look at those assumptions and see whether we believe them.”

One of the first pieces of evidence for dark matter came from observations of the rotation speeds of stars in spiral galaxies, such as NGC3198 (shown here). The observed rotation speeds are nearly constant out to the very edge of the galaxy (data points). The rotation speeds predicted by the “star soup” model, assuming the galaxy has only visible matter, are shown by the dashed line. Astrophysicists interpreted the discrepancy to mean that there must be unseen dark matter. In fact, a model that combines the visible bulge and disk with a dark matter halo (solid line) matches the data very well. From Particles and Fundamental Interactions, Chapter 13: Microcosm and Macrocosm, Sylvie Braibant, Giorgio Giacomelli, and Maurizio Spurio, Springer Netherlands, 2012; reprinted with permission from Springer Science+Business Media BV.

Among mathematicians, the reception to Saari’s work has been warmer, although until the paper is subjected to peer review it can’t be said definitively to be correct. “I do like the argument, and I think it needs to be countered,” says Gareth Roberts, a celestial mechanics theorist at the College of the Holy Cross. “If I was interested in dark matter, I would want to get to the bottom of this.”

Cédric Villani, a Fields medalist from the University of Lyon, who attended Saari’s lecture, commented in an e-mail, “It may be a discrete-to-continuous issue, it may also have something to do with the inhomogeneity [of the mass distribution], or both. The only thing I can say is that one of the main arguments used for dark matter, viewed from an outsider’s perspective, looks quite fragile, and Saari’s counterexample showed this very well.”

Freelance writer Dana Mackenzie writes from Santa Cruz, California. He is the author of the 2012 book The Universe in Zero Words: The Story of Mathematics as Told Through Equations.