The Central Density Relation

I promised more results from SPARC. Here is one. The dynamical mass surface density of a disk galaxy scales with its central surface brightness.

This may sound trivial: surface density correlates with surface brightness. The denser the stars, the denser the mass. Makes sense, yes?

Turns out, this situation is neither simple nor obvious when dark matter is involved. The surface brightness traces stellar surface density while the dynamical mass surface density traces stars plus everything else, including dark matter. The latter need not care about the former when dark matter dominates.

Nevertheless, we’ve known there was some connection for some time. This first became clear to me in the mid-90s, when we discovered that low surface brightness galaxies did not shift off of the Tully-Fisher relation as expected. The only way to obtain this situation was to fine-tune the dynamical mass enclosed by the disk with the central surface brightness. Galaxies had to become systematically more dark matter dominated as surface brightness decreased (see Zwaan et al 1995). This was the genesis of the now common statement that low surface brightness galaxies are dark matter dominated (see also de Blok & McGaugh 1996; McGaugh & de Blok 1998).

This is an oversimplification. A more precise statement would be that dark matter dominates to progressively smaller radii in ever lower surface brightness galaxies. Even as dark matter comes to dominate, the dynamics “know” about the stellar distribution.

The rotation curve depends on the enclosed mass, dark as well as luminous. The rate of rise of the rotation curve from the origin correlates with surface brightness. Low surface brightness galaxies have slowly rising rotation curves while high surface brightness galaxies have steeply rising rotation curves. This is very systematic (e.g., Lelli et al 2013).


The rate of rise of rotation curves (dV/dR) as a function of central surface brightness (from Lelli 2014).

Recently, Agris Kalnajs pointed out to us that in a paper written before even I was born, Toomre (1963) had shown how to obtain the central mass surface density of a thin disk from the rotation curve. This has largely been forgotten because dark matter complicates matters. However, we were able to show that Toomre’s formula returns the correct dynamical surface density within a factor of two even in the extreme case of complete domination by a spherical halo component. This breakthrough was enabled by Kalnajs pointing out a straightforward way to include disk thickness (Toomre assumed a razor thin disk) and Lelli pursuing this to the extreme case of a “spherical” disk with a flat rotation curve.

A factor of two is not much to quibble about when one has the large dynamic range of a sample like SPARC. After all, the data cover four orders of magnitude in central surface brightness. Variations in disk thickness and halo domination will only contribute a bit to the scatter.

Without further ado, here is the result:


The central dynamical mass surface density as a function of the central stellar surface density (left) and stellar mass (right). From Lelli et al (2016). Points are color coded by morphological type. The dashed line shows the 1:1 relation expected in the absence of dark matter.

The data show a clear correlation between mass surface density and surface brightness. At high surface brightness, the data have a slope and normalization consistent with stars being the dominant form of mass present. This is the long-known result of “maximum disk” (van Albada & Sancisi 1986). The observed distribution of stars does a good job of matching the inner rotation curve. It is only as you go further out, where rotation curves flatten, or to lower surface brightness that dark matter becomes necessary.

Something interesting happens as surface brightness declines. The data gradually depart from the line of unity. The dynamical surface density begins to exceed the stellar surface density, so we begin to need some dark matter. Stars and Newton alone can no longer explain the data for low surface brightness galaxies.

Note, however, that the data depart from the line of unity with a considerable degree of order. It is not like things go haywire as dark matter comes to dominate, as one might reasonably expect. After all, why should the mass surface density depend on the stellar surface density at all in the limit where the former greatly outweighs the latter? But it does: the correlation persists to the point that the mass surface density is predictable from the surface brightness.

It is not only rotation curves that show this behavior. A similar result was found from the vertical velocity dispersions of disks in the DiskMass survey. Specifically, Swaters et al (2014) show essentially the same plot. However, the DiskMass sample is necessarily restricted to rather high surface brightness galaxies. Consequently, those data show only a hint of the systematic departure from the line of unity, and one could argue that it is linear. A large number of low surface brightness galaxies with good data is key to our result (see the discussion of the sample in SPARC).

This empirical result sheds some light on the debate about dark matter halo profiles. The rotation curves of low surface brightness galaxies rise slowly. This is not consistent with NFW halos, as shown long ago (e.g., de Blok et al. 2001; Kuzio de Naray 2008, 2009, and many others). It is frequently argued that everything is OK with NFW, it is the data that are to blame. The basic idea is that somehow (usually for inadequate resolution, though sometimes other effects are invoked) rotation curves fail to show the steep predicted rise.  It is there! we are assured. We just can’t see it.

Beware of theorists blaming data that doesn’t do what they want. A recent example of this kind of argument is offered by Pineda et al. (2016). Their basic contention is that rotation curves cannot correctly recover the true mass distribution. They invoke a variety of effects to make it sound like one has no chance of recovering the central mass profile from rotation curves.

The figure above shows that this is manifestly nonsense. If the data were incapable of measuring the dynamical surface density, the figure would be a mess. In one galaxy we’d see one wrong thing, in another we’d see a different wrong thing. This would not correlate with surface brightness, or anything else: the y-axis would just be so much garbage. Instead, we see a strong correlation. Indeed, we understand the errors well enough to calculate that most of the scatter is observational. Yes, there are uncertainties. They do add scatter to the plot. A little. That means the true relation is even tighter.

There is a considerable literature in the same vein as Pineda et al. (2016). These concerns can be completely dismissed. Not only are they incorrect, they stem from a form of solution aversion: they don’t like the answer, so deny that it can be true. This attitude has no place in science.