The First Law of Galactic Rotation

Nota Bene:  One of the authors, Stacy McGaugh, pointed out to me that he considers the principle described below as the Third Law of Galactic Rotation.  If you would like to know why, please go read his fascinating article on the law, as well as his articles on the other two laws.

The online magazine Quanta recently published a hot button article called “The Case Against Dark Matter”. The gist of the article is that a flurry of papers and lectures were presented over the last months of 2016 that call into question the existence of dark matter. One of the papers stood out to me, so I’ll review it here. It is called Radial Acceleration Relation in Rotationally Supported Galaxies, composed by Stacy McGaugh, Federico Lelli, and James Schombert. The abstract is reprinted below, after I summarize what I think is understated by the word extraordinary in the paper.  To be clear, for me, this appears to be the kind of discovery that will usher in a new view of the universe, as the Michelson-Moreley results motivated a similar paradigm shift in the early 20th Century.

The relationship

The authors compiled two sets of observations into one database called SPARC – Spitzer Photometry and Accurate Rotation Curves. One set, at wavelength 3.6 μm, was obtained from observations with the Spitzer Space Telescope. The other set, at wavelength 21 cm, was compiled from decades of observations with arrays of radio telescopes like New Mexico’s Very Large Array. The 3.6 μm observations see stars, and are used to measure the amount of stellar mass in galaxies. The 21 cm observations see dark dust and gas in galaxies due to a spectral line in neutral hydrogen, and is used to create rotation curves for the galaxies.

The rotation curves represent the original problem Dark Matter was invented to solve. According to Kepler’s laws of planetary motion, as modified by Newton’s law of gravitational force, the speed of an orbiting object depends on how much mass is within the orbit. More mass = faster object. For rough point masses, like the objects in our Solar System, the orbits are determined by how far they are from the central star. In a galaxy, things get more complicated because the mass is distributed. As you go further away from the center, more and more stars and other matter end up within your orbit, thus increasing your orbital speed. This increase of mass with distance is tempered by the fact that the amount of stars and gas gets less as you approach the edge of the galaxy. Therefore, the speed of orbiting objects should drop off as you leave the galaxy.

Galactic rotation curves from McGough et al. (2016)
Rotation curves of two types of galaxy from McGaugh et al. (2016: Fig. 2). The black dots with error bars represent the observed orbital velocity at increased radial distance from the galactic center. The other curves show what the orbital velocity should be, due to various masses in the galaxy: dotted = gas, dashed = stars, dot-dash = galactic bulge. The solid blue line is the total orbital velocity expected due to all observed masses put together. In other words, far away from the galactic center, objects are moving much faster than they should be due to observed matter.

Exactly the opposite is found. The 21 cm observations have shown that the orbital velocity of gas usually speeds up as you leave the galaxy, thus implying that the amount of matter, in fact, goes up as you leave the galaxy! But, to date, nobody has actually observed the mass. This missing mass got the apt name Dark Matter in the 1930s by astronomer Fritz Zwicky. Over 80% of matter, based on these galactic rotation curves as well as other observational evidence, is this dark matter. Other theories that don’t invoke dark matter have been invented – such as the Modified Newtonian Dynamics of Mordehai Milgrom – but none has had the success enjoyed by dark matter.

In the future, the work by McGaugh et al. might be seen as the silver bullet that killed dark matter. Using both sets of observations, they calculate the centripetal acceleration exerted on the objects in over 150 galaxies. Centripetal acceleration can be calculated in two ways: one way requires knowledge of orbital velocity, the other way requires knowledge of mass. McGaugh et al. get the mass with the 3.6 μm observations, which can be converted from luminosity to mass quite directly. This gives them g_{bar}, the centripetal acceleration due to observed mass. Then they get the orbital velocity from the 21 cm observations, via the rotation curves. This gives them g_{obs}, the centripetal acceleration observed to exist. Then, these are plotted against each other:

Centripetal acceleration due to observed matter versus centripetal acceleration observed. Plots from McGough et al. (2016)
Centripetal acceleration due to observed matter versus centripetal acceleration observed. Plots from McGough et al. (2016)

The relationship is absolutely extraordinary! It’s not directly proportional, but follows a slight curve toward the center of the galaxy. What is remarkable is that the exact same relation holds for all galaxies in the study, and is not dependent on galactic type. They map the relationship to a relatively simple equation

g_{obs} = \mathcal{F}(g_{bar}) = \frac{g_{bar}}{1-e^{\sqrt{g_{bar}/{g_{\dagger}}}}}

which only requires one parameter for the fit, g_{\dagger}.

Why is this extraordinary? It does NOT mean that observed mass causes what we see in the rotation curve – there is still something missing. However, it shows that there is a simple, absolute, relationship between the rotation curve and the observed mass. In other words, in a given galaxy, using this new law, if you are given the distribution of observed, not dark, matter, you can produce the galaxy’s rotation curve. There is absolutely no need to invoke invisible mass.

The First Law

Kepler developed his laws after hundreds of hours of observational, computational, and hypothetical work. With these laws, the orbits of any object in the solar system can be completely determined (disregarding important small deviations). There is no concept of gravitational force in Kepler’s laws, only geometry, time, and motion. The laws hold for every stellar system.

McGaugh et al. have found a new law that applies to galaxies. And, they are not blind to this law’s significance. In a more extensive paper published in the Astrophysical Journal, Lelli et al. state

The radial acceleration relation describes the local link between baryons and dynamics in galaxies, encompassing and generalizing several well-known galaxy scaling laws. This is tantamount to a Natural Law: a sort of Kepler law for galactic systems. A tight coupling between baryons and DM [dark matter] is difficult to understand within the standard ΛCDM cosmology. Our results may point to the need for a revision of the current DM paradigm.

One difference between Kepler’s laws and this new galactic law, is that Kepler believed he had found the causes of his laws. Later on, his causes were tossed out by the new theory of Newtonian gravitation. McGaugh et al. have not yet settled on a cause for their newfound relationship. However, they do hint at some possibilities:

Possible interpretations for the radial acceleration relation fall into three broad categories: (1) it represents the end product of galaxy formation; (2) it represents new dark sector physics that leads to the observed coupling; (3) it is the result of new dynamical laws rather than dark matter. None of these options are entirely satisfactory.

I’ll conclude by drawing another parallel. Case Western University, home of two of this paper’s authors, was host to another critical observational result. In 1887, Albert Michelson and Edward Morely performed an interferometry experiment there to measure the Earth’s velocity through the ether. They famously returned a “null” result. Almost 30 years later, Albert Einstein shocked the world by declaring, based on Michelson and Morely’s result, that the ether – a fundamental substance of physics since the Enlightenment – does not exist.  Never again could theory or experiment rely on belief in the existence of the ether.  Case Western may, again, be the source of a critical result that will provide reason to both discard a fictional substance and provoke a new paradigm shift in our view of the cosmos.

But, maybe not.  If this discovery doesn’t kill dark matter, then dark matter’s place in the universe is about to become much stronger.  Either way, we now have a new, first law of galactic rotation.


We report a correlation between the radial acceleration traced by rotation curves and that predicted by the observed distribution of baryons. The same relation is followed by 2693 points in 153 galaxies with very different morphologies, masses, sizes, and gas fractions. The correlation persists even when dark matter dominates. Consequently, the dark matter contribution is fully specified by that of the baryons. The observed scatter is small and largely dominated by observational uncertainties. This radial acceleration relation is tantamount to a natural law for rotating galaxies.

S. S. McGaugh, F. Lelli, and J. M. Schombert, Physical Review Letters 117, 201101 (2016).

2 thoughts on “The First Law of Galactic Rotation”

    1. @Eric DeBlackmere, yes, it usually takes a while for an important paradigm changing discovery sinks in. Thank you for reading and reposting!


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