The radial acceleration relation connects what we see in visible mass with what we get in galaxy dynamics. This is true in a statistical sense, with remarkably little scatter. The SPARC data are consistent with a single, universal force law in galaxies. One that appears to be sourced by the baryons alone.

This was not expected with dark matter. Indeed, it would be hard to imagine a less natural result. We can only salvage the dark matter picture by tweaking it to make it mimic its chief rival. This is not a healthy situation for a theory.

On the other hand, if these results really do indicate the action of a single universal force law, then it should be possible to fit each individual galaxy. This has been done many times before, with surprisingly positive results. Does it work for the entirety of SPARC?

For the impatient, the answer is yes. Graduate student Pengfei Li has addressed this issue in a paper in press at A&A. There are some inevitable goofballs; this is astronomy after all. But by and large, it works much better than I expected – the goof rate is only about 10%, and the worst goofs are for the worst data.

Fig. 1 from the paper gives the example of NGC 2841. This case has been historically problematic for MOND, but a good fit falls out of the Bayesian MCMC procedure employed.  We marginalize over the nuisance parameters (distance and inclination) in addition to the stellar mass-to-light ratio of disk and bulge. These come out a tad high in this case, but everything is within the uncertainties. A long standing historical problem is easily solved by application of Bayesian statistics.

RAR fit (equivalent to a MOND fit) to NGC 2841. The rotation curve and components of the mass model are shown at top left, with the fit parameters at top right. The fit is also shown in terms of acceleration (bottom left) and where the galaxy falls on the RAR (bottom right).

Another example is provided by the low surface brightness (LSB) dwarf galaxy IC 2574. Note that like all LSB galaxies, it lies at the low acceleration end of the RAR. This is what attracted my attention to the problem a long time ago: the mass discrepancy is large everywhere, so conventionally dark matter dominates. And yet, the luminous matter tells you everything you need to know to predict the rotation curve. This makes no physical sense whatsoever: it is as if the baryonic tail wags the dark matter dog.

RAR fit for IC 2574, with panels as in the figure above.

In this case, the mass-to-light ratio of the stars comes out a bit low. LSB galaxies like IC 2574 are gas rich; the stellar mass is pretty much an afterthought to the fitting process. That’s good: there is very little freedom; the rotation curve has to follow almost directly from the observed gas distribution. If it doesn’t, there’s nothing to be done to fix it. But it is also bad: since the stars contribute little to the total mass budget, their mass-to-light ratio is not well constrained by the fit – changing it a lot makes little overall difference. This renders the formal uncertainty on the mass-to-light ratio highly dubious. The quoted number is correct for the data as presented, but it does not reflect the inevitable systematic errors that afflict astronomical observations in a variety of subtle ways. In this case, a small change in the innermost velocity measurements (as happens in the THINGS data) could change the mass-to-light ratio by a huge factor (and well outside the stated error) without doing squat to the overall fit.

We can address statistically how [un]reasonable the required fit parameters are. Short answer: they’re pretty darn reasonable. Here is the distribution of 3.6 micron band mass-to-light ratios.

Histogram of best-fit stellar mass-to-light ratios for the disk components of SPARC galaxies. The red dashed line illustrates the typical value expected from stellar population models.

From a stellar population perspective, we expect roughly constant mass-to-light ratios in the near-infrared, with some scatter. The fits to the rotation curves give just that. There is no guarantee that this should work out. It could be a meaningless fit parameter with no connection to stellar astrophysics. Instead, it reproduces the normalization, color dependence, and scatter expected from completely independent stellar population models.

The stellar mass-to-light ratio is practically inaccessible in the context of dark matter fits to rotation curves, as it is horribly degenerate with the parameters of the dark matter halo. That MOND returns reasonable mass-to-light ratios is one of those important details that keeps me wondering. It seems like there must be something to it.

Unsurprisingly, once we fit the mass-to-light ratio and the nuisance parameters, the scatter in the RAR itself practically vanishes. It does not entirely go away, as we fit only one mass-to-light ratio per galaxy (two in the handful of cases with a bulge). The scatter in the individual velocity measurements has been minimized, but some remains. The amount that remains is tiny (0.06 dex) and consistent with what we’d expect from measurement errors and mild asymmetries (non-circular motions).

The radial acceleration relation with optimized parameters.

For those unfamiliar with extragalactic astronomy, it is common for “correlations” to be weak and have enormous intrinsic scatter. Early versions of the Tully-Fisher relation were considered spooky-tight with a mere 0.4 mag. of scatter. In the RAR we have a relation as near to perfect as we’re likely to get. The data are consistent with a single, universal force law – at least in the radial direction in rotating galaxies.

That’s a strong statement. It is hard to understand in the context of dark matter. If you think you do, you are not thinking clearly.

So how strong is this statement? Very. We tried fits allowing additional freedom. None is necessary. One can of course introduce more parameters, but we find that no more are needed. The bare minimum is the mass-to-light ratio (plus the nuisance parameters of distance and inclination); these entirely suffice to describe the data. Allowing more freedom does not meaningfully improve the fits.

For example, I have often seen it asserted that MOND fits require variation in the acceleration constant of the theory. If this were true, I would have zero interest in the theory. So we checked.

Here we learn something important about the role of priors in Bayesian fits. If we allow the critical acceleration g to vary from galaxy to galaxy with a flat prior, it does indeed do so: it flops around all over the place. Aha! So g is not constant! MOND is falsified!

Best fit values of the critical acceleration in each galaxy for a flat prior (light blue) and a Gaussian prior (dark blue). The best-fit value is so consistent in the latter case that the inset is necessary to see the distribution at all. Note the switch to a linear scale and the very narrow window.

Well, no. Flat priors are often problematic, as they have no physical motivation. By allowing for a wide variation in g, one is inviting covariance with other parameters. As g goes wild, so too does the mass-to-light ratio. This wrecks the stellar mass Tully-Fisher relation by introducing a lot of unnecessary variation in the mass-to-light ratio: luminosity correlates nicely with rotation speed, but stellar mass picks up a lot of extraneous scatter. Worse, all this variation in both g and the mass-to-light ratio does very little to improve the fits. It does a tiny bit – χ2 gets infinitesimally better, so the fitting program takes it. But the improvement is not statistically meaningful.

In contrast, with a Gaussian prior, we get essentially the same fits, but with practically zero variation in g. wee The reduced χ2 actually gets a bit worse thanks to the extra, unnecessary, degree of freedom. This demonstrates that for these data, g is consistent with a single, universal value. For whatever reason it may occur physically, this number is in the data.

We have made the SPARC data public, so anyone who wants to reproduce these results may easily do so. Just mind your priors, and don’t take every individual error bar too seriously. There is a long tail to high χ2 that persists for any type of model. If you get a bad fit with the RAR, you will almost certainly get a bad fit with your favorite dark matter halo model as well. This is astronomy, fergodssake.

21 thoughts on “RAR fits to individual galaxies

  1. Hello
    In the news, there is this

    A new experiment to understand dark matter
    Space Daily-9 hours ago
    Is dark matter a source of a yet unknown force in addition to gravity? The mysterious dark matter is little understood and trying to understand its …

    would a new long range “fifth force” that is attractive between dark matter and baryonic matter explain these relations you describe in the paper such as baryonic tully fishcer and RAR?


    1. We need some sort of connection between the two, so in principle yes. In practice, it has to be a very specific connection – one that “looks like” MOND. So whether this works to do the right thing is the first question.


      1. There seems to be a fundamental “looks like”-MOND issue.
        Most of the Mass in the Universe is Invisible (Dark Matter), or
        Dynamical Laws must be Modified (MOND).”
        I think the basic problem is:
        Most of the Mass in the Universe is Invisible (Dark Matter) that somehow looks like MOND at galactic scales, or
        Dynamical Laws must be Modified in some way that approximates MOND on galactic scales.


      2. wouldn’t such a dark matter model, that reproduces MOND, also explain weak gravitational lensing and CMB acoustic peaks large scale structure?


  2. “… it is as if if the baryonic tail wags the dark matter dog.” If the dog never barks, then is the dog really there? Google “sanders milgrom” and “scarpa milgrom”. Does dark energy have negative gravitational mass-energy and zero inertial mass-energy? Does dark matter have positive gravitational mass-energy and zero inertial mass-energy?


      1. The CMB requires a form of dark matter that does not interact via the electromagnetic force and exists with the cosmic density obtained from fits to its acoustic power spectrum.


    1. Sure. It is wrong. Simple as that.

      They perform an analysis that is a subset of the analysis I describe here. They discover nothing new. Indeed, they only demonstrate the dangers of flat priors that I alluded to.


  3. Consider 4 hypotheses:
    (1) There exist 2 basic types of dark matter particles: MONDian and non-MONDian.
    (2) The MONDian dark matter particles have variable effective mass depending upon nearby gravitational acceleration. The non-MONDian dark matter particles have ordinary mass-energy, which does not depend upon nearby gravitational acceleration.
    (3) In the standard form of Einstein’s field equations, replace the -1/2 by -1/2 + fake-function, where the hypothetical fake-function = sqrt((60±10)/4) * 10^-5 in the range of gravitational accelerations where MOND is valid and = approximately 0 in the range of gravitational accelerations where MOND is invalid. The fake-function arises from a false belief that MONDian dark matter particles do not exist.
    (4) All of the non-MOND dark matter problems (i.e. the dark matter problems that MOND cannot successfully model) can be explained by non-MONDian dark matter particles.


      1. If dark matter does not consist of dark particles, then there must be some form of relativistic MOND that passes all empirical tests.
        Consider Sean Carroll’s analysis of the CMB:
        “Sean, March 6:
        Hi Stacy–I’m not sure what you are saying about the third peak in the CMB. We agree that “pure baryons shouldn’t do that.” I can only think of three possibilities.
        (1) There is some sort of source for gravity other than baryons.
        (2) There is a modification of gravity that doesn’t include new sources, but also doesn’t respond directly to where the sources actually are.
        (3) The data aren’t good enough to say that the odd-numbered peaks are boosted relative to what we would expect from damped oscillations of baryons alone.”
        Possibility (2) might be wrong if Einsteinian gravitational lensing is slightly wrong. However, someone needs to precisely define gravitational lensing in relativistic MOND and then satisfactorily answer Sean Carroll’s CMB analysis. (He seems to be arguing that any MOND-like theory without dark matter particles must be wrong. It is not clear that his argument attempts to refute MOND with some dark matter particles.)


  4. Unfortunately this article is chock full of log-log and semi-log plots. Such presentations are used nowadays by astronomers to hide poor data with large errors…and because everybody else does it. Common decades ago because it was impossible to use anything but linear regression. Why not present a plot of “number of points” vs. g (m/s^2) and find out what the data is really telling you? And then there are the several log vs log diagrams to properly treat…


  5. Have you seen “A precise extragalactic test of General Relativity” by Collett et all, published in Science:

    They find a consistency between the bending of light from a more distant galaxy by the galaxy ESO 325-G004 and the rotational velocity of stars around the centre of the galaxy.


  6. Yes, I have read this paper. It is interesting as a test of GR, confirming in this case that the mass implied by dynamics is the same as the mass implied by gravitational lensing. It would be very strange if it were otherwise, even in most modified gravity theories that I’m aware of. Viable theories contain GR in the appropriate limit. Strong lensing occurs in the regime of high surface density where one does not expect to see any MOND-like effect. The paper mentions f(R) gravity, which attempts to explain dark energy rather than dark matter, but it is unclear to me that it is tested in any serious way either. So this is a nice test that things work as they should, but it is neither a surprise nor a problem for viable modified gravity theories.


    1. “Strong lensing occurs in the regime of high surface density where one does not expect to see any MOND-like effect.” Without relativistic MOND, what can the pro-MOND astrophysicists predict for strong lensing? Consider the alleged Fernández-Rañada-Milgrom effect: Replace the -1/2 in the standard form of Einstein’s field equations by -1/2 + dark-matter-compensation-constant, where the constant is approximately (3.9 ± .8 ) * 10^-5 — the Gravity Probe B science team says that they have proved this is wrong. I conjecture that the 4 ultra-precise gyroscopes worked correctly.
      Sean Carroll has claimed that “… MOND is not true …”
      I conjecture that my version of gravitational lensing disconfirms what Carroll claims about the Bullet Cluster — the Bullet Cluster data actually confirm Milgromian gravitational lensing.


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