The odd primordial halo of the Milky Way

The mass distribution of dark matter halos that we infer from observations tells us where the dark matter needs to be now. This differs form the mass distribution it had to start, as it gets altered by the process of galaxy formation. It is the primordial distribution that dark matter-only simulations predict most robustly. We* reverse-engineer the collapse of the baryons that make up the visible Galaxy to infer the primordial distribution, which turns out to be… odd.

The Gaia rotation curve and the mass of the Milky Way

As we discussed a couple of years ago, Gaia DR3 data indicate a declining rotation curve for the Milky Way. This decline becomes more steep, nearly Keplerian, in the outskirts of the Milky Way (17 < R < 30 kpc). This is may or may not be consistent with data further out, which gets hard to interpret as the LMC (at 50 kpc) perturbs orbits and the observed motions may not correspond to orbits in dynamical equilibrium. So how much do the data inform us about the gravitational potential?

Milky Way rotation curve (various data) including Gaia DR3 (multiple analyses). Also shown is the RAR model (blue line) that was fit to the terminal velocities from 3 < R < 8.2 kpc (gray points) and predates other data illustrated here.

I am skeptical of the Keplerian portion of this result (as discussed at length at the time) because other galaxies don’t do that. However, I am a big fan of listening to the data, and the people actually doing the work. Taken at face value, the Gaia data show a Keplerian decline with a total mass around 2 x 10¹¹ M_☉. If correct, this falsifies MOND.

How does dark matter fare? There is an implicit assumption made by many in the community that any failing of MOND is an automatic win for dark matter. However, it has been my experience that observations that are problematic for MOND are also problematic for dark matter. So let’s check.

Short answer: this is really weird in terms of dark matter. How weird? For starters, most recent non-Gaia dynamical analyses suggest a total mass closer to 10¹² M_☉, a factor of five higher than the Gaia value. I’m old enough to remember when the accepted mass was 2 x 10¹² M_☉, an order of magnitude higher. Yet even this larger mass is smaller than suggested by abundance matching recipes, which give more like 4 x 10¹² M_☉. So somewhere in the range 2 – 40 x 10¹¹ M_☉.

The Milky Mass has been adjusted so often, have we finally hit it?

The guy was all over the road. I had to swerve a number of times before I hit him.
Boston Driver’s Handbook (1982 edition)^&

If it sounds like we’re all over the map, that’s because we are. It is very hard to constrain the total mass of a dark matter halo. We can’t see it, nor tell where it ends. We infer, indirectly, that the edge is way out beyond the tracers we can see. Heck, even speaking of an “edge” is ill-defined. Theoretically, we expect it to taper off with the density of dark matter falling as ρ ~ r^-3, so there is no definitive edge. Somewhat arbitrarily,** we adopt the radius that encloses a density 200 times the average density of the universe as the “virial” radius. This is all completely notional, and it gets worse, as the process of forming a galaxy changes the initial mass distribution. What we observe today is the changed form, not the primordial initial condition for which the notional mass is defined.

Adiabatic compression during galaxy formation

To form a visible galaxy, baryons must dissipate and sink to the center of their parent dark matter halo. This process changes the mass distribution and alters the halo from its primordial state. In effect, the gravity of the sinking baryons drags some dark matter along^# with them.

The change to the dark matter halo is often called adiabatic compression. The actual process need not be adiabatic, but that’s how we approximate it. We’ve tested this approximation with detailed numerical simulations, and it works pretty well, at least if you do it right (there are boring debates about technique). What happens makes sense intuitively: the response of the primordial halo to the infall of baryons is to become more dense at the center. While this makes sense physically, it is problematic for LCDM as it takes an NFW halo that is already too dense at the center to be consistent with data and makes it more dense. This has been known forever, so opposing this is one thing feedback is invoked to do, which it may or may not do, depending on how it really works. Even if feedback can really turn a compressed cusp into a core, it is widely to expected to be important only in low mass galaxies where the gravitational potential well isn’t too deep. It isn’t supposed to be all that important in galaxies as massive as the Milky Way, though I’m sure that can change as needed.

There are a variety of challenges to implementing an accurate compression computation, so we usually don’t bother: the standard practice is to assume a halo model and fit it to the data. That will, at best, given a description of the current dark matter halo, not what it started as, which is our closest point of comparison with theory. To give an example of the effect, here is a Milky Way model I built a decade ago:

**Figure 13** from McGaugh (2016): Milky Way rotation curve from the data of Luna et al. (2006, red points) and McClure-Griffiths & Dickey (2007, gray points) together with a bulgeless baryonic mass model (black line). The total rotation is approximately fit (blue line) with an adiabatically compressed NFW halo (solid green line) using the procedure implemented by Sellwood & McGaugh (2005). The primordial halo before compression is shown as the dashed line. The parameters of the primordial halo are a concentration c = 7 and a mass *M₂₀₀ = 6 x 10¹¹ M_☉*. *Fitting NFW to the present halo instead gives c = 14, M₂₀₀ = 4 x 10¹¹ M_☉, so the difference is appreciable and depend on the quality and radial extent of the available data.*

The change from the green dashed line to the solid green line is the difference compression makes. That’s what happens if a baryon distribution like that of the Milky Way settles in an NFW halo. The inferred mass M₂₀₀ is lower and the concentration c higher than it originally was – and it is the original version that we should compare to the expectations of LCDM.

When I built this model, I considered several choices for the bulge/bar fraction: something reasonable, something probably too large, and something definitely too small (zero). The model above is the last case of zero bulge/bar. I show it because it is the only case for which the compression procedure worked. If there is a larger central concentration of baryons – i.e., a bulge and/or a bar – then the compression is greater. Too great, in fact: I could not obtain a fit (see also Binney & Piffl and this related discussion).

The calculation of the compression requires knowledge of the primordial halo parameters, which is what one is trying to obtain. So one has to guess an initial state, run the code, check how close it came, then iterate the initial guess. This is computationally expensive, so I was just eyeballing the fit above. Pengfei has done a lot of work to implement a method that iteratively computes the compression and rigorously fits it to data. So we decided to apply it to the newer Gaia DR3 data.

Fitting the Gaia rotation curve with adiabatically compressed halos

We need two inputs here: one, the rotation curve to fit, and two, the baryonic distribution of the Milky Way. The latter is hard to specify given our location within the Milky Way, so there are many different estimates. We tried a dozen.

Another challenge of doing this is deciding which data rotation curve data to fit. We chose to focus on the rotation curve of Jiao et al. (2023) because they made estimates of the systematic as well as random errors. The statistics of Gaia are so good it is practically impossible to fit any equilibrium model to them. There are aspects of the data for which we have to consider non-equilibrium effects (spiral arms, the bar, “snails” from external perturbations) so the usual assumptions are at best an approximation, plus there can always be systematic errors. So the approach is to believe the data, but with the uncertainty estimate of Jiao et al. (2023) that includes systematics.

For a halo model, we started with the boilerplate LCDM NFW halo^$. This doesn’t fit the data. Indeed, all attempts to fit NFW halos fail in similar ways for all of the different baryonic mass models we tried. The quasi-Keplerian part of the Gaia rotation curve simply cannot be fit: the NFW halo inevitably requires more mass further out.

Here are a few examples of the NFW fits:

**Fig. A.3** from Li et al. (2025). Fits of Galactic circular velocities using the NFW model implementing adiabatic halo contraction using 3 baryonic models. [Another 9 appear in the paper.] Data points with errors are the rotation velocities from Jiao et al. (2023), while open triangles show the data from Eilers et al. (2019), which are not fitted. [The radius ranges from 5 to 30 kpc.] Blue, purple, green and black solid lines correspond to the contributions by the stellar disk, central bar, gas (and dust if any), and compressed dark matter halo, respectively. The total contributions are shown using red solid lines. Black dashed lines are the inferred primordial halos.

LCDM as represented by NFW suffers the same failure mode as seen in MOND (plot at top): both theories overshoot the Gaia rotation curve at R > 17 kpc. This is an example of how data that are problematic for MOND are also problematic for dark matter.

We do have more freedom in the case of dark matter. So we tried a different halo model, Einasto. (For this and many other halo models, see Pengfei’s epic compendium of dark matter halo fits.) Where NFW has two parameters, a concentration c and mass M₂₀₀, Einasto has a third parameter that modulates the shape of the density profile^%. For a very specific choice of this third parameter (α = 0.17), it looks basically the same as NFW. But if we let α be free, then we can obtain a fit. Of all the baryonic models, the RAR model+compressed Einasto fits best:

**Fig. 1** from Li et al. (2025). Example of a circular velocity fit using the McGaugh19^$$ model for baryonic mass distributions. The purple, blue, and green lines represent the contributions of the bar, disk, and gas components, respectively. The solid and dashed black lines show the current and primordial dark matter halos, respectively. The solid red line indicates the total velocity profile. The black points show the latest Gaia measurements (Jiao et al. 2023), and the gray upward triangles and squares show the terminal velocities from (McClure-Griffiths & Dickey 2007, 2016), and Portail et al. (2017), respectively. The data marked with open symbols were not fit because they do not consider the systematic uncertainties.

So it is possible to obtain a fit considering adiabatic compression. But at what price? The parameters of the best-fit primordial Einasto halo shown above are c = 5.1, M₂₀₀ = 1.2 x 10¹¹ M_☉, and α = 2.75. That’s pretty far from the α = 0.17 expected in LCDM. The mass is lower than low. The concentration is also low. There are expectation values for all these quantities in LCDM, and all of them miss the mark.

**Fig. 2** from Li et al. (2025). Halo masses and concentrations of the primordial Galactic halos derived from the Gaia circular velocity fits using 12 baryonic models. The red and blue stars with errors represent the halos with and without adiabatic contraction, respectively. The predicted halo mass-concentration relation within 1 σ from simulations (Dutton & Macciò 2014) is shown as the declining band. The vertical band shows the expected range of the MW halo mass according to the abundance-
matching relation (Moster et al. 2013). The upper and lower limits are set by the highest stellar mass model plus 1 σ and the lowest stellar mass model minus 1 σ, respectively.

The expectation for mass and concentration is shown as the bands above. If the primordial halo were anything like what it should be in LCDM, the halo parameters represented by the red stars should be where the bands intersect. They’re nowhere close. The same goes for the shape parameter. The halo should have a density profile like the blue band in the plot below; instead it is more like the red band.

**Fig. 3** from Li et al. (2025). Structure of the inferred primordial and current Galactic halos, along with predictions for the cold and warm dark matter. The density profiles are scaled so that there is no need to assume or consider the masses or concentrations for these halos. The gray band indicates the range of the current halos derived from the Gaia velocity fits using the 12 baryonic models, and the red band shows their corresponding primordial halos within 1σ. The blue band presents the simulated halos with cold dark matter only (Dutton & Macciò 2014). The purple band shows the warm dark matter halos (normalized to match the primordial Galactic halo) with a core size spanning from 4.56 kpc (WDM5 in Macciò et al. 2012) to 7.0 kpc, corresponding to a particle mass of 0.05 keV and lower.

So the primordial halo of the Milky Way is pretty odd. From the perspective of LCDM, the mass is too low and the concentration is too low. The inner profile is too flat (a core rather than a cusp) and the outer profile is too steep. This outer steepness is a large part of why the mass comes out so low; there just isn’t a lot of halo out there. The characteristic density ρ_s is at least in the right ballpark, so aside from the inner slope, the outer slope, the mass, and the concentration, LCDM is doing great.

What if we ignore the naughty bits?

It is really hard for any halo model to fit the steep decline of the Gaia rotation curve at R > 17 kpc. Doing so is what makes the halo mass so small. I’m skeptical about this part of the data, so do things improve if we don’t sweat that part?

Ignoring the data at R > 17 kpc allows the mass to be larger, consistent with other dynamical determinations if not quite with abundance matching. However, the inner parts of the rotation curve still prefer a low density core. That is, something like the warm dark matter halo depicted as the purple band above rather than NFW with its dense central cusp. Or self-interacting dark matter. Or cold dark matter with just-so feedback. Or really anything that obfuscates the need to confront the dangerous question: why does MOND perform better?

*This post is based on the recently published paper by my former student Pengfei Li, who is now faculty at Nanjing University. They have a press release about it.

^&A few months after reading this in the Boston Driver’s Handbook, this exact thing happened to me.

**This goes back to BBKS in 1986 when the bedrock assumption was that the universe had Ω_m = 1, for which the virial radius was 188 times the critical density. 200 was close enough, and stuck, even though for LCDM the virial radius is more like an overdensity close to 100, which is even further out.

^#This is one of many processes that occur in simulations, which are great for examining the statistics of simulated galaxy-like objects but completely useless for modeling individual galaxies in the real universe. There may be similar objects, but one can never say “this galaxy is represented by that simulated thing.” To model a real galaxy requires a customized approach.

^$NFW halos consistently perform worse in fitting data than any other halo model, of which there are many. It has been falsified as a viable representation of reality so many times that I can’t recall them all, and yet they remain the go-to model. I think that’s partly thanks to their simplicity – it is mathematically straightforward to implement – and to the fact that is what simulations predict: LCDM halos should look like NFW. People, including scientists, often struggle to differentiate simulation from reality, so we keep flogging the dead horse.

^%The density profile of the NFW halo model asymptotes to power laws at both small and large radii: ρ → r^-1 as r → 0 and ρ → r^-3 as r → ∞. The third parameter of Einasto allows a much wider ranges of shapes.

^$$The McGaugh19 model user here is the one with a reasonable bulge/bar. This dense component can be fit in this case because we start with a halo model with a core rather than a cusp (closer to α = 1 than to the α = 0.17 of NFW/LCDM).

Progressive Approximations in Mass Modeling

I have said I wasn’t going to attempt to teach an entire graduate course on galaxy dynamics in this forum, and I’m not. But I can give some pointers for those who want to try it for themselves. It also provides some useful context for fans of Deur’s approach.

The go-to textbook for this topic is Galactic Dynamics by Binney & Tremaine. The first edition was published in 1987, conveniently when I switched to grad school in astronomy. It was already a deep and well-developed field at that time; this is a compendium of considerable scientific knowledge.

Fun story: a colleague in a joint physics & astronomy department once complained to me that she wanted to develop a course in galaxy dynamics, which is a staple of graduate programs in astronomy & astrophysics. However, there was a certain senior colleague who objected, saying that since it was astronomy, it couldn’t possibly be a rigorous course worthy of a full semester graduate course. This is a casual bias that astronomers often encounter when talking to physicists, many of whom have attitudes about the subject that were trapped in amber sometime in the Jurassic. I suggested that she walk into his office and drop a copy of Galactic Dynamics on his desk from on high, as (1) it would make a hefty impact, and (2) no one who so much as skims this book could persist in this toxic attitude.

She later reported that she had done this, and it had worked.

Galactic Dynamics is not a starter book. It is the textbook we use when teaching the graduate course that this is not. A useful how-to guide for the specific material I’ll discuss here is provided by Federico Lelli. In brief, to model the gravitational potential of an observed distribution of matter, we can make one of the following series of approximations:

*This is a slide I sometimes use to introduce mass modeling in science talks as a reminder for expert audiences.*

All science is an approximation at some level. The most crude approximation we can employ here is to imagine that all of the mass resides at a central point. In this limit, the potential is simply

V² = GM/R

where V is the orbital speed of a test particle on a circular orbit, G is Newton’s constant, M is the mass, and R is the distance from the point mass. Galaxies are not point masses, so this is a terrible approximation, as can be seen by the divergent V ~ R^-1/2 behavior as R → 0 (the dotted line above).

The next bad approximation one can make is a spherical cow: assume the mass is distributed in a sphere that is projected as the image we see on the sky. This at least incorporates the fact that the mass is not all concentrated at a point, so

V² = GM(R)/R

acknowledges that the mass M is spread out as a function of radius. This is a spherical cow. Since we cannot see dark matter, we almost always assume it to be a spherical cow.

For the luminous disk of a spiral galaxy, a common approximation is the so-called exponential disk:

Σ(R) = Σ₀ e^-R/R_d

where Σ₀ is the central surface density of stars and R_d is the scale length of the disk – the characteristic size over which the surface brightness declines exponentially. This can be integrated by parts to obtain an expression for the enclosed mass M(R) which I leave as an exercise for the eager reader. This provides a handy analytic formula, the rotation curve of which is illustrated above by the dashed line.

Spiral galaxies are fairly thin when seen edge-on, so the spherical cow is not a great approximation. In a classic paper, Freeman (1970) solved the Poisson equation for the case of a razor-thin exponential disk, where one meets modified Bessel functions of the first and second kind (denoted “ikik” above). These must be solved numerically, but one can make a tabulation for use with any choice of disk mass and scale length. Such a thin disk is illustrated by the grey line above for a choice of stellar mass and scale length appropriate to NGC 6946.

*The spiral galaxy NGC 6946, aka the fireworks galaxy.*

Spiral galaxies are not razor thin of course. We only see a projected image on the sky, so for a galaxy like NGC 6946, we may have a good measurement of its azimuthally averaged light (and presumable stellar mass) distribution Σ(R) but we have no idea how thick it is. Here, we have to make an educated guess based on observations of edge-on galaxies. A ballpark average is R:z = 8:1, but some galaxies are thicker and others thinner, so this becomes an approximation with an associated uncertainty. This uncertainty cannot be unambiguously eliminated; it is one of the known unknowns that comprise the inevitable systematic errors in astronomy. Fortunately, allowing for a finite thickness only takes the harsh edge off of the thin disk case, and the assumption one chooses makes little difference to the result (compare the lines labeled thick and thin above).

The exponential disk formula Σ(R) is an azimuthal average over an image like that of NGC 6946. This approximation captures none of the spiral structure: it only tells us about the average rate at which the surface brightness falls off. It also imposes a smooth shape on that fall off that our eyes can see is not necessarily a great approximation. So the next level of approximation is to solve the Poisson equation numerically for the observed surface brightness profile, Σ(R), not just the exponential approximation thereto. This is the blue line in the bottom right graph above.

There are important differences between using the numerical solution for the observed light distribution and the exponential disk approximation. This has been known since the 1980s, but the analytic expression is so convenient that people need an occasional reminder not to trust it too much. Jerry Sellwood felt the need to provide this reminder in 1999:

Small apparent differences in the shape of the mass profile (left) correspond to pronounced differences in the rotation curve (right). I chose the example of NGC 6946 in part because the exponential approximation for it is pretty good. Nevertheless, the details matter, so the best practice is to build numerical mass models, as we did for SPARC.

Building numerical mass models is tractable for external galaxies, where we can see the entire light distribution. It is not possible for our own Milky Way, since we are located within it and cannot see it as a whole. Consequently, the vast majority of Milky Way models rely on the exponential approximation; so far as I’m aware, I’m the only one who has built a model that attempts to get beyond this.

Numerical mass models are still an approximation. We’re assuming that the gravitational potential is static and azimuthally symmetric. Taking the next step would require abandoning these assumptions to model the spiral arms. The Poisson equation can handle that, but it becomes dicey because the arms rotate with some pattern speed (generally unknown) and may grow or dissolve or reform on some unknown timescale. The potential at any given point is time variable even in equilibrium, so we need not just a numerical solution but a live numerical simulation to keep track of it. That can be done, but it has to be done on a case by case basis, and the answer will depend somewhat on additional assumptions that have to be introduced to run the simulation, like specifying a dark matter halo.

One can generalize further to consider the full 3D potential, e.g., to allow for asymmetry in the z-direction as well as in azimuth. One can further imagine non-equilibrium processes, such as an external perturbations. There is good evidence that the Milky Way suffers both of these effects, the passage of the Large Magellanic Cloud being one obvious and apparently large perturbation. So we are in the awkward position that the Gaia data now oblige us to consider the entire run of possible effects through non-equilibrium processes in a mass distribution that is not completely symmetric in any of the three spatial dimensions, but for the main mass component we are stuck with the inadequate approximation of an exponential disk.

Geometry appears to play a crucial role in the approach of Deur to the acceleration discrepancy problem. The essential claim is that the discrepancy correlates with flattening, with highly flattened systems like spirals evincing the classic discrepancy while spherical systems like E0 galaxies showing none. Big if true!

A useful plot appears on slide 44:

*Some measure of the discrepancy as a function of apparent ellipticity.*

This is the one example shown that goes into the plot of many determinations of the slope a on the following slide. It being the only one, it is the only thing I have to evaluate without chasing down every other case. Looking at this, I am not inclined to do so.

At first it looks persuasive: the best fit slope is clear. There is no reason why the discrepancy should depend on the projected ellipticity of a triaxial 3D blob of stars, so this must be telling us something important. I’d be on board with that if it were true, but I’ve seen too many non-correlations masquerading as correlations to believe this one. The fitted slope is strongly influenced by the one point at large ellipticity; absent that, a slope of zero works fine. Mostly what I see here is a lot of scatter, which is normal in extragalactic astronomy. Since there are only a few points at high and low ellipticity, we don’t know what would happen if we went out and got more data. But I bet that what would happen is that the high ellipticity points would wind up looking like those in the middle: a big blob of scatter, with no significant correlation.

I’d kinda like to be wrong about this one, so I won’t even get into the theory side, which I find sorta compelling but ultimately unpersuasive. Why are gravitons confined to a disk? What happens way far out? Surely the flatness of the disk at tens of kpc is not dictating the flatness at 1000 kpc.

Surely.

Tully-Fisher from gravitational lensing

Last time, we discussed the remarkable result that gravitational lensing extends the original remarkable result of flat rotation curves much farther out, as far as the data credibly probe. This corroborates and extends the result of Brouwer et al. They did a thorough job, but one thing they did not consider was Tully-Fisher. If the circular speed inferred from gravitational lensing remains constant, does this flat velocity fall on the same Tully-Fisher relation that is seen in kinematic data?

We set out to answer this question. Along the way, we did three new things: 1. Dr. Mistele derived an improved method for doing the lensing analysis, extending the radial range over which the data were credible. 2. He explored the criteria by which galaxies were judged to be isolated, finding a morphological type dependence on how far out one had to exclude. 3. We reanalyzed the stellar masses of the KiDS sample to be consistent with those we used when analyzing the kinematic data of SPARC galaxies. The first two are connected, as how far out we can trust the data depends on how well we can define a clean sample of isolated galaxies. The third resolved an apparent offset between early type galaxies (ETGs, aka ellipticals) and late type galaxies (LTGs, aka spirals) seen by Brouwer et al. That appears to be an artifact of stellar population modeling, as I suspected when I first discussed their result. We don’t need to do any fitting of the mass-to-light ratio; the the apparent offset between types disappears when we use use the same population models for both kinematic and lensing data.

I could write a lot about each of these, but most of it is the stuff of technical details that would be dull to many people. If you’re into that sort of thing, go and read the long science paper which is where such details reside. Here I just want to describe the Tully-Fisher result. Spoiler alert: it is the same as that from kinematics.

First off, I’m talking strictly about the Baryonic Tully-Fisher relation: the scaling between baryonic mass and the flat rotation speed. To address this, we bin the lensing data by mass. The mass of each bin is well defined by the average of the many thousands of galaxies within the bin. By far the dominant uncertainty is the systematic in stellar mass caused by stellar population modeling. We went through this with a fine tooth comb, and I’m confident we have an internally self-consistent result. That doesn’t preclude it being wrong in an absolute sense – such is the nature of astronomy – but we can at least make a straight comparison between kinematic and lensing data using the same best-effort stellar mass estimates.

For the velocity, we estimate the average effective rotation curve for each mass bin from the lensing data. We also split the data into morphological types to look for differences. The statistics go down when one divvies up the data like this, so the uncertainties go up, but there are enough KiDS galaxies to define four mass bins. Here are their inferred rotation curves:

**Figure 1** from Mistele et al. (2024). Circular velocities implied by weak lensing for four baryonic mass bins (most to least massive from the top row to the bottom) for the whole sample (left column), for LTGs (middle column), and for ETGs (right column). The lowest ETG mass bin is not shown because it contains too few lenses. Instead we show results for lenses with spectroscopic redshifts from GAMA, without splitting by mass or type due to the small sample size (gray and white symbols). For comparison, we also show results for KiDS without splitting by mass or type (small yellow symbols). Open symbols at small radii indicate where lenses are not yet effective point masses. Light-colored symbols (not-outlined) at large radii indicate data points that may still be reliable but where the isolation criterion is less certain. The error bars show the statistical errors. Horizontal lines and the corresponding shaded regions indicate the inferred V_flat values and uncertainties that we use for the BTFR. The extent of the horizontal lines indicates the radial range we consider when calculating V_flat.

Note that the average over all KiDS data shown in the lower right bin is the data shown in the press release image in the previous post, but the x-axis is logarithmic here. The GAMA data in that bin provide an important cross-check, as these galaxies have spectroscopic redshifts. They give the same answer as the larger KiDS sample, which relies on photometric redshifts. We need the larger sample to consider finer bins in mass, which is the rest of the plot.

Another thing to note here is that all the data in all the bins are consistent with remaining flat. There are some hints of a turn down at very large radii, particularly for LTGs in the second and third row, but these are not statistically significant, and only happen where the data start to become untrustworthy. Where exactly that happens is a judgement call.

Let’s take a closer look, with a comparison to radio data:

**Figure 2** from Mistele et al. (2024). The circular velocities from weak lensing (circles) compared with those from gas kinematics (diamonds). The individual galaxies illustrated here have among the most extended 21 cm rotation curves in their mass bins; the lensing data continue to much larger radii still. The error bars show the statistical error, while the gray band indicates the systematic uncertainty in the radial accelerations. Symbol colors are as in Figure 1. Open symbols at large radii indicate where lenses are not sufficiently isolated. The solid green lines indicate the circular velocities of NFW halos and baryonic point masses appropriate for each mass bin. Green crosses indicate each NFW halo’s virial radius. The light green band adds a qualitative estimate of a two-halo term contribution to the NFW halo, which may become important at large radii in case our isolation criterion is imperfect there.

Again we see that the lensing data, averaged over many galaxies, extend much further out than the rotation curve of any one individual. The x-axis is again logarithmic, so the lensing data go way further out. They trace to 1 Mpc, which is crazy far beyond the observed ends of the most extended individual galaxies. A more conservative limit is the 300 kpc estimated by Brouwer et al. Surely we can go further than that, but how much further remains a judgement call.

What should we expect? The green lines show the rotation curve we’d expect for galaxy in an NFW halo with parameters specified by the stellar mass-halo mass relation of Kravtsov et al. (2018). Not all such relations agree well with kinematic data; this is the case that agrees most closely. We have intentionally cherry-picked the relation that makes LCDM look best. And it does look good up to a point, for example in the top two mass bins out to the virial radius of the halo (tick marks). Beyond that, not so much, and not at all for the two lower mass bins. The data extend far enough out that we should see the predicted decline. We do not.

The green line only represents the expected halo of the primary galaxy. When one gets so far out, one has to worry about all the other stuff out there. We’ve selected galaxies to be isolated, so there isn’t much that is luminous. But we can only exclude down to some sensitivity limit, there might be lots of tiny dwarf galaxies whose mass adds up and starts to affect the result. And of course there can be completely invisible dark matter. The green band attempts to account for this extra stuff in the so-called 2-halo term. This is hard to do, but we’ve made our best estimate based on the LCDM power spectrum. I’m sure the 2-halo term can be adjusted, but the shape is wrong. It will take some fine-tuning to get an effectively flat rotation curve out of the 1-halo+2 halo terms. They don’t naturally do that.

Something that is easy to do is define a flat value of the rotation speed. That’s just the average over the lensing data. We exclude the points at R < 50 kpc, as the assumption of a spherical mass that we make in the lensing analysis isn’t really valid at those comparatively small scales. We tried averaging over a bunch of different ranges, all of which gave pretty much the same answer. For illustration, we show two cases: a conservative one that only uses the data at R < 300 kpc, and another that goes out to 1 Mpc. Having measured V_flat over these ranges, we can plot Tully-Fisher:

**Figure 3** from Mistele et al. (2024). The baryonic Tully–Fisher relation implied by weak lensing for the entire sample (yellow symbols, left column) and for ETGs and LTGs separately (red and blue symbols, right column). The V_flat values are weighted averages of the V_c values shown in Figure 1 for 50 kpc < R < 300 kpc (first row) and 50 kpc < R < 1000 kpc (second row). Vertical error bars represent a 0.1 dex systematic uncertainty on M_*/L. For comparison, we also show the best fit to the kinematic data from Lelli et al. (2019; solid gray line) and the corresponding binned kinematic data (white diamonds).

Lo and behold, we find the same Baryonic Tully-Fisher relation from lensing data as we find with kinematics. This does not surprise me, but it didn’t have to be true. It shouldn’t be true in LCDM: if we can measure out to the virial radius, we should see some indication of a decline in velocity. We have and we don’t.

We also see no statistically significant separation between ETGs and LTGs. This is important, as a theory like MOND predicts that there should be no morphology dependence: only the baryonic mass matters. Brouwer et al. did see an indication of such a split, but it was small compared to the uncertainty in stellar population models. We don’t see it when we use our own stellar mass estimates. This is particularly true in the more conservative (300 kpc) case. There is a hint of a segregation when we average out to 1000 kpc, but the statistics say this isn’t significant. Since the lowest mass bin is most affected, I suspect this is a hint that the isolation criterion is failing first for the smallest galaxies. That makes sense, as the sensitivity limit on interlopers makes the lowest mass bin most susceptible to having its signal inappropriately boosted. It also makes sense that ETGs would be affected first, as ETGs are known to be more clustered than LTGs. It is really hard to define an isolated sample of ETGs, as discussed at length by Mistele et al.

The lensing data corroborate previous kinematic work. Rotation curves are flat. The amplitude of the flat rotation speed correlates with baryonic mass as M_b ∝ V_f⁴. The radial acceleration relation extends to very low accelerations. These are all predictions of MOND. Moreover they are unique predictions: predictions made a priori by MOND and only by MOND. Dark matter models so far provide no satisfactory explanation*.

That hasn’t prevented people from overlooking these basic facts in order to get to the apparent if statistically meaningless difference between ETGs and LTGs. Nevermind the successes! The slight offset between ETGs and LTGs falsify MOND! Seriously: other scientists have already made this argument to me while completely eliding the successes of MOND. It’s a case of refusing to see the forest for a tree that’s a little away from the others.

I think I said something about how this would happen when I first wrote about Brouwer et al‘s lensing result. Ah yes, here it is:

MOND predicted this behavior well in advance of the observation, so one would have to bend over backwards, rub one’s belly, and simultaneously punch oneself in the face to portray this as anything short of a fantastic success of MOND.

I say that because I’m sure people will line up to punch themselves in the face in exactly this fashion.

And so it has come to pass. Sometimes human behavior is as predictable as galaxy dynamics.

*There are many claims to explain limited portions of these results, but none are satisfactory. There is no LCDM model that matches the entire dynamic range of the radial acceleration relation. See, for example, Fig. 5 of Brouwer et al. (reproduced below), which shows the MICE and BAHAMAS simulations. Neither extend into the regime that is well-constrained by kinematic data; there is no reason to think they would successfully do so and good reason to think otherwise. MICE comes nowhere close to this regime and has no baryonic physics that would allow it do even address this question. BAHAMAS comes close but appears to turn away from the kinematic data before it gets there. We’ve built our own LCDM models; they don’t work either. We can make them come close, but only over a limited dynamic range, not over the full span of the data. It isn’t good enough to only explain a limited range of the data. One has to explain the full range, and the only theory that does that so far is MOND.

Fig. 5 from Brouwer et al. showing the radial acceleration relation inferred from the MICE (red band) and BAHAMAS (orange band) simulations. Not also that in our assessment of stellar masses, the lower acceleration points translate a bit to the right.

Rotation curves: still flat after a million light-years

That rotation curves become flat at large radii is one of the most famous results in extragalactic astronomy. This had been established by Vera Rubin and her collaborators by the late 1970s. There were a few earlier anecdotal cases to this effect, but these seemed like mild curiosities until Rubin showed that the same thing was true over and over again for a hundred spiral galaxies. Flat rotation curves took on the air of a de facto natural law and precipitated the modern dark matter paradigm.

Optical and radio data

Rotation curves shouldn’t be flat. If what we saw was what we got, the rotation curve would reach a peak within the light distribution and decline further out. Perhaps an illustration is in order:

The rotation curve (data points, left) of NGC 6946 (right). The red line shows the expected rotation curve for the detected normal matter, which includes both the stars (yellow, from 2MASS) and atomic gas (blue, from THINGS). This provides a good description of the inner rotation curve but falls short further out. The excess observed rotation leads to the need for dark matter or MOND. Also noted is the extent of the rotation curve measured optically to the effective edge of the stars (Daigle et al. 2006; Epinat et al. 2008) and that measured with radio interferometric observations of the gas (Boomsma et al. 2008).

An obvious question is how far out rotation curves remain flat. In the rotation curves traced with optical observations by Rubin et al., the discrepancy was clear but modest – typically a factor of two in mass. It was possible to imagine that the mass-to-light ratios of stars increased with radius in a systematic way, bending the red line above to match the data out to the edge of the stars. This seemed unlikely, but neither did it seem like a huge ask.

Once one gets to the edge of the stellar distribution, most of the mass has been encompassed, and the rotation curve really should start to decline. Increasing the mass-to-light ratio of the stars ceases to be an option once we run out of stars*. Fortunately, the atomic gas typically extends to larger radii, so provides a tracer further out. Albert Bosma pursued this until there were again enough examples to establish that yes, flat rotation curves were the rule. They extended much further out, well beyond where the mass of the observed stars and gas could explain the data.

How much further out? It depends on the galaxy. A convenient metric is the scale length of the disk, which is a measure of the extent of the light distribution. Some galaxies are bigger than others. The peak of the contribution of the stars to the rotation curve occurs around 2.2 scale lengths. The rotation curve of NGC 6946 extends to about 7 scale lengths, far enough to make the discrepancy clear. For a long time, the record holder was NGC 2403, with a rotation curve that remains flat for 20 scale lengths.

Twenty scale lengths is a long way out. It is observations like this that demanded dark matter halos that are much larger than the galaxies they contain. They also posed a puzzle, since we were still nowhere near finding the edge of the mass distribution. Rotation curves seemed to persist in being flat indefinitely.

Results from gravitational lensing

Weak gravitational lensing provides a statistical technique to probe the gravitational potential of galaxies. Brouwer et al. did pioneering work with data from the KiDS survey, and found that the radial acceleration relation extended to much lower accelerations than probed by the types of kinematic data discussed above. That implies that rotation curves remain flat way far out. How far?

Postdoc Tobias Mistele worked out an elegant technique to improve the analysis of lensing data. His analysis corroborates the findings of Brouwer et al. It also provides the opportunity to push further out.

Weak gravitational lensing is a subtle effect – so subtle that one must coadd thousands of galaxies to get a signal. Beyond that, the limiting effect on the result is how isolated the galaxies are. Lensing is sensitive to all mass; if you go far enough out you start to run into other galaxies whose mass contributes to the signal. So one key is to identify isolated galaxies, and restrict the sample to them. KiDS is large enough to do this. Indeed, Mistele was able to show that while neighbors⁺ were a definite concern for elliptical galaxies, they were much less of a problem for spirals. Consequently, we can trace the implied rotation curve way far out.

How far out? In a new paper, Mistele shows that rotation curves continue way far out. Way way way far out. I mean, damn.

The average rotation curve of isolated galaxies (blue points) inferred from KiDS gravitational lensing data. This remains flat well beyond a million light-years with no end in sight. The width of the figure is the distance between the Milky Way and Andromeda. For comparison, the rotation curve of a single galaxy, UGC 6614, is shown in red. An image of the galaxy is shown to scale centered at the origin. UGC 6614 was selected for this illustration because it has a comparable rotation speed to the KiDS average and because it is one of the largest galaxies known: the red points are already a very extended rotation curve. Image credit: Mistele, Lelli, & McGaugh 2024.

Optical rotation curves typically extend to the edge of the stellar disk. That’s about 8 kpc in the example of NGC 6946 given above. Radio observations of the atomic gas of that galaxy extend to 17 kpc. That fits within the first two tick marks on the graph with the lensing rotation curve.

UGC 6614 is a massive galaxy with a very extended low surface brightness disk. Its rotation curve is traced by radio data to over 60 kpc. It is one of the most extended individual rotation curves known. The statistical lensing data push this out by a factor of ten, and more, with no end in sight. The flat rotation curves found by Rubin and Bosma and everyone else appear to persist indefinitely.

So what does it mean? First, flat rotation curves really are a law of nature, in the same sense of Kepler’s laws of planetary motion. Galaxies don’t obey those planetary rules, they have their own set of rules. This is what nature does.

In terms of dark matter halos, the extent of isolated galaxy rotation curves is surprisingly large. Just as we come to the edge of the stellar disk, then the gas disk, we should eventually hit the edge of the dark matter halo. In principle we can imagine this to be arbitrarily large, but in practice there are other galaxies in the universe so this cannot go one forever.

In the context of LCDM, we now have a pretty good idea of how extended halos should be from abundance matching. A galaxy of the mass of UGC 6614 should live in a halo with a virial radius of about 300 kpc or less. There is some uncertainty in this, of course, but we really should have hit the edge with the lensing data. There should be some sign of it, but we see none.

One complication is the so-called 2-halo term. In addition to the primary dark matter halo that hosts a galaxy, when you get very far out, you run into other halos. Isolated galaxies are selected to avoid this to the extent possible, but eventually there will be some extra mass that causes extra lensing signal that would cause an overestimate of the rotation speed. I’ll forgo a detailed discussion of this for now (see Mistele et al. if you’re eager), but the bottom line is that it would require some unnatural fine-tuning for the 1+2 halo terms to add up to such flat rotation curves. There ought to be a perceptible feature in the transition from the primary halo to the surrounding environment. We don’t see that.

In the context of MOND, a flat rotation curve that persists indefinitely is completely natural. That’s what an isolated galaxy should do. Even in MOND there should be an environmental effect: the mass of everything else in the universe should impose an external field effect that eventually limits the extent of the rotation curve. How this transition happens depends on the density of other galaxies; by selecting isolated galaxies this effect is put off as much as possible. Hopefully it will be detected as the data improve from projects like Euclid.

The primary prediction of MOND is an indefinitely extended rotation curve; the external field effect is a subtle detail. Yet again, that is what we see: MOND gets it right without really trying, and in a way that makes little sense in terms of dark matter. Sometimes I wish MOND had never been invented so we could claim to have discovered something profoundly new, or at least discuss the empirical result without concern that the data would get confused with the theory. MOND predictions keep being corroborated, yet the community persists in ignoring its implications, even in terms of dark matter. It’s gotta be telling us something.

We have a press release about this result, so perhaps you will see it kicking around your news feed.

*We could, of course, invoke dark stars, but that’s just an invisible horse of a different color.

⁺There is a well known correlation between morphology and density such that elliptical galaxies tend to live in the densest environments. This means that they are more likely to have neighbors that interfere with the lensing measurement, so finding that identifying isolated ellipticals with a clean lensing signal is more challenging that finding isolated spirals comes as no surprise. Isolated ellipticals do exist so it is possible, but one has to be very restrictive with the sample.

Is the Milky Way’s rotation curve declining?

Yes, some. That much is a step forward from a decade ago, when a common assumption was that the Milky Way’s rotation curve remained flat at the speed at which the sun orbited. This was a good guess based on empirical experience with other galaxies, but not all galaxies have rotation curves that are completely flat, nor can we be sure the sun is located where that is the case.

A bigger question whether the Milky Way’s rotation curve is declining in a Keplerian fashion. This would indicate that the total mass has been enclosed. That would be a remarkable result. If true, it would be the first time that the total mass of an individual galaxy has been measured. There have been claims to this effect before that have not panned out when the data have been extended to larger radii, so one might be inclined to be skeptical.

There are several claims now to see a distinctly declining rotation curve based on the third data release (DR3) from Gaia. The most recent, Jiao et al., has gained some note by virtue of putting “Keplerian decline” in the title, but very similar results have also been reported by Ou et al., Wang et al. and Sylos Labini et al. They all obtain basically the same answer using the same data, with minor differences in the error assessment and other details. There are also differences in interpretation*, which is always possible even when everyone agrees about what the data say.

Jiao et al. measure a total mass for the Milky Way of about 2 x 10¹¹ M_☉. Before looking at the data, let’s take a moment to think about that number. Most mass determinations – and there are lots, see Fig. 2 of Wang et al. – for the Milky Way have been in the neighborhood of 10¹² M_☉. Indeed, for most of my career, it was traditionally Known to be 2 x 10¹² M_☉. The new measurement is an order of magnitude smaller. That’s a lot to be off by, even in extragalactic astronomy. The difference, as we’ll see, has to do with what data we use.

The mass of stars and gas in the Milky Way is about 6 x 10¹⁰ M_☉, give or take ten billion. That means that nearly a third of the total mass is normal baryonic matter that we can readily see. So the ratio of dark-to-baryonic mass is only 2.3:1, well short of the cosmic ratio of about 6:1. That’s embarrassing – especially since much of the effort in galaxy formation theory has been to explain why the baryon fraction is much less than the cosmic fraction, not much more. And here our Galaxy is an outlier, having much less dark matter for its stellar mass than everything else. It is always a bad sign when the Galaxy appears to violate the Copernican Principle.

Nonetheless, this is what we find if we look at the Gaia DR3 data. Here is a model I’ve shown before, extrapolated to larger radii with some new data added. The orange circles are the Gaia DR3 rotation curve as given by Jiao et al. For radii greater than 18 kpc, they show a clear decline consistent with a Keplerian curve for a 1.95 x 10¹¹ M_☉ point mass (dotted line), as per Fig. 9 of Jiao et al.

*Milky Way model (blue line) compared with various data.*

This is the first time we’ve been able to trace the rotation curve so far out with stars in the disk of the Milky Way, and the Keplerian line is a good match. If that’s all we know, then a total mass of only 2 x 10¹¹ M_☉ is a reasonable inference. That’s not all we know.

As I alluded above, a halo mass this small makes no sense in the context of cosmology. Not only is 2 x 10¹¹ M_☉ too small, the more commonly inferred dynamical mass of 10¹² M_☉ is also too small. According to abundance matching, which has become an important aspect of LCDM, the Milky Way should reside in a 3 or 4 x 10¹² M_☉ halo. So the new mass makes a factor of 2 or 3 problem into a factor a ten problem. That is too large to attribute to scatter in the stellar mass-halo mass relation. Worse, there is no evidence that the Milky Way is an outlier from scaling relations like Tully-Fisher. We can’t have it one way and not the other.

The traditional mass estimates that obtain ~10¹² M_☉ rely on dwarf satellite galaxies as tracers of the gravitational potential of the Milky Way. Maybe they’re not fair tracers? We have to make assumptions about their orbits to use them to infer a mass; perhaps these assumptions are wrong? It is conceivable that many of our satellites are on first infall rather than in well-established orbits. Indeed, the consensus is that our largest satellites, the Magellanic Clouds, are on first infall, and that they cause a substantial perturbation to the halo of the Milky Way. This was an absurd thought 15 years ago – the Magellanic clouds must have been here forever, and were far too small to do damage – but now this is standard lore.

There are tracers at large radii besides dwarf satellite galaxies. The figure above shows three: globular clusters (pink triangles) and two types of stars in the halo: blue horizontal branch stars (green squares) and K giants (red squares). These are well-known parts of the Milky Way that have been with us for many billions of years, so they’ve had plenty of time to become equilibrium tracers of the gravitational potential. They clearly indicate a larger enclosed mass than predicted by the Keplerian decline traced by the Gaia rotation curve, and are consistent with traditional satellite analyses. Perhaps these data are somehow misleading, but it is hard to see how.

Gaia is great, but has its limits. It is really optimized for nearby stars (within a few kpc). Outside of that, the statistics… leave something to be desired. Is it safe to push out beyond 20 kpc? I don’t know, but I did notice this panel from Fig. 8 of Wang et al.:

*Radial velocities of stars at different heights above the Galactic plane.*

The radial velocity is a minor component of disk motion, where azimuthal motion dominates. However, one does need to know it to solve the Jeans equation. Having it wrong will cause a perceptible systematic error. You notice the bifurcation in the data for R > 22 kpc? That, in technical terms, is Messed Up. I don’t know what goes awry there, but I’ve done this exercise enough times for the sight of this to scare the bejeepers out of me. No way I trust any of these data at R > 22 kpc, and I hope having seen this doesn’t give me nightmares tonight.

Perhaps the uncertainty caused by this is adequately reflected in the large error bars on the orange points above. Those with R > 22 kpc are nicely Keplerian, but also consistent with a lot of things, including the blue line that successfully predicts the halo stars and globular clusters. That’s not true for the data around R = 20 kpc where the error bars are much smaller: there the discrepancy with the blue line I take seriously. But that is a much more limited affair that might indicate the presence of a ring of mass – that’s what gives the bumps and wiggles at smaller radii – and certainly isn’t enough to imply the entire mass of the Milky Way has been enclosed.

But who knows? Perhaps fifteen years hence it will be the standard lore that all galaxies reside in dark matter halos that are only twice the mass of their luminous disks. At that mass ratio, all the galactic dark matter could be baryonic. I wouldn’t bet on it, but stranger things have happened before, and will happen again.

*A difference in interpretation is largely what the debate about dark matter and MOND boils down to. There is no doubt that there are acceleration discrepancies in extragalactic objects that require something beyond what you see is what you get with normal gravity. Whether we should blame what we can’t see or the assumption of normal gravity is open to interpretation. I would hope this is obvious, but this elementary point seems to be lost on many.

A few words about the Milky Way

I recently traveled to my first international meeting since the Covid pandemic began. It was good to be out in the world again. It also served as an excellent reminder of the importance of in-person interactions. On-line interactions are not an adequate substitute. I’d like to be able to recount all that I learned there, but it is too much. This post will touch on one of the much-discussed topics, our own Milky Way Galaxy.

When I put on a MOND hat, there are a few observations that puzzle me. The most persistent of these include the residual mass discrepancy in clusters, the cosmic microwave background, and the vertical motions of stars in the Milky Way disk. Though much hyped, the case for galaxies lacking dark matter does not concern me much: the examples I’ve seen so far appear to be part of the normal churn of early results that are likely to regress toward the norm as the data improve. I’ve seen this movie literally hundreds of times. I’m more interested in understanding the forest than a few outlying trees.

The Milky Way is a normal galaxy – it is part of the forest. It is easy to get lost in the leaves when one has access to data for millions going on billions of individual stars. These add up to a normal spiral galaxy, and we know a lot about external spirals that can help inform our picture of our own home.

For example, by assuming that the Milky Way falls along the radial acceleration relation defined by other spiral galaxies, I was able to build a mass model of its surface density profile. The resulting mass distribution is considerably more detailed than the usual approach of assuming a smooth exponential disk, which would be a straight line in the right-hand plot below. With the level of detail becoming available from missions like the Gaia satellite, it is necessary to move beyond such approximations.

Left: Spiral structure in the Milky Way traced by regions of gas ionized by young stars (HII regions, in red) and by the birthplaces of giant molecular clouds (GMCs, in blue). Right: the azimuthally-averaged surface density profile of stars inferred from the rotation curve of the Milky Way using the Radial Acceleration Relation. The features inferred kinematically correspond to the spiral arms known from star counts, providing a local example of Renzo’s Rule.

This model was built before Gaia data became available, and is not informed by it. Rather, I took the terminal velocities measured by McClure-Griffiths & Dickey, which provide the estimate of the Milky Way rotation curve that is most directly comparable to what we measure in external spirals, and worked out the surface density profile using the radial acceleration relation. The resulting model possesses bumps and wiggles like those we see corresponding to spiral arms in external galaxies. And indeed, it turns out that the locations of these features correspond with known spiral arms. Those are independent observations: one is from the kinematics of interstellar gas, the other from traditional star counts.

The model turns out to have a few further virtues. It matches the enclosed mass profile of the inner bulge/bar region of the Galaxy without any attempt at a specific fit. It reconciles the rotation curve measured with stars using Gaia data with that measured using gas in the interstellar medium – a subtle difference that was nevertheless highly significant. It successfully predicts that the rotation curve beyond the solar radius would not be perfectly flat, but rather decline at a specific rate – and exactly that rate was subsequently measured using Gaia. These are the sort of results that inclines one to believe that the underlying physics has to be MOND. Inferring maps of the mass distribution with this level of detail is simply not possible using a dark matter model.

The rotation curve of the Milky as observed in interstellar gas (light grey) and as fit to the radial acceleration relation (blue line). Only the region from 3 to 8 kpc has been fit; the rest follows. This matches well stellar observations from the inner, barred region of the Milky Way (dark grey squares: Portail et al. 2017) and the gradual decline of the outer rotation curve (black squares: Eilers et al. 2019) once corrected for the presence of bumps and wiggles due to spiral arms. These require taking numerical derivatives for use in the Jeans equation; the red squares show the conventional result obtained when neglecting this effect by assuming a smooth exponential surface density profile. See McGaugh (2008 [when the method was introduced and the bulge/bar model for the inner region was built], 2016 [the main fitting paper], 2018 [an update to the distance to the Galactic center], 2019 [including bumps & wiggles in the Gaia analysis]).

Great, right? It is. It also makes a further prediction: we can use the mass model to predict the vertical motions of stars perpendicular to the Milky Way’s disk.

Most of the kinetic energy of stars orbiting in the solar neighborhood is invested in circular motion: the vast majority of stars are orbiting in the same direction in the same plane at nearly the same speed. There is some scatter, of course, but radial motions due to orbital eccentricities represent a small portion of the kinetic energy budget. As stars go round and round, the also bob up and down, oscillating perpendicular to the plane of the disk. The energy invested in these vertical motions is also small, which is why the disk of the Milky Way is thin.

View of the Milky Way in the infrared provided by the COBE satellite. The dust lanes that afflict optical light are less severe at these wavelengths, revealing that the stellar disk of the Milky Way is thin but for the peanut-shaped bulge/bar at the center.

Knowing the surface density profile of the Milky Way disk, we can predict the vertical motions. In the context of dark matter, most of the restoring force that keeps stars near the central plane is provided by the stars themselves – the dark matter halo is quasi-spherical, and doesn’t contribute much to the restoring force of the disk. In MOND, the stars and gas are all there is. So the prediction is straightforward (if technically fraught) in both paradigms. Here is a comparison of both predictions with data from Bovy & Rix (2013).

The dynamical surface density implied by vertical motions (data from Bovy & Rix 2013). The dark blue line is the prediction of the model surface density described above – assuming Newtonian gravity. The light blue line is the naive prediction of MOND.

Looks great again, right? The dark blue line goes right through the data with zero fitting. The only exception is in the radial range 5.5 to 6.4 kpc, which turns out to be where the stars probing the vertical motion are maximally different from the gas informing the prediction: we’re looking at different Galactic longitudes, right where there is or is not a spiral arm, so perhaps we should get a different answer in this range. Theory gives us the right answer, no muss, no fuss.

Except, hang on – the line that fits is the Newtonian prediction. The prediction of MOND overshoots the data. It gets the shape right, but the naive MOND prediction is for more vertical motion than we see.

By the “naive” MOND prediction, I mean that we assume that MOND gives the same boost in the vertical direction as it does in the radial direction. This is the obvious first thing to try, but it is not necessarily what happens in all possible MOND theories. Indeed, there are some flavors of modified inertia in which it should not. However, one would expect some boost, and in these data there appears to be none. We get the right answer with just Newton and stars. There’s not even room for much dark matter.

I hope Gaia helps us sort this out. I worry that it will provide so much information that we risk missing the big picture for all the leaves.

This leaves us in a weird predicament. The radial force is extraordinarily well-described by MOND, which reveals details that we could never hope to access if all we know about is dark matter. But if we spot Newtonian gravity this non-Newtonian information from the radial motion, it predicts the correct vertical motion. It’s like we have MOND in one direction and Newton in another.

This makes no sense, so is one of the things that worries me most about MOND. It is not encouraging for dark matter either – we don’t get to spot ourselves MOND in the radial direction then pretend that dark matter did it. At present, it feels like we are up the proverbial creek without a paddle.

Are there credible deviations from the baryonic Tully-Fisher relation?

There is a rule of thumb in scientific publication that if a title is posed a question, the answer is no.

It sucks being so far ahead of the field that I get to watch people repeat the mistakes I made (or almost made) and warned against long ago. There have been persistent claims of deviations of one sort or another from the Baryonic Tully-Fisher relation (BTFR). So far, these have all been obviously wrong, for reasons we’ve discussed before. It all boils down to data quality. The credibility of data is important, especially in astronomy.

Here is a plot of the BTFR for all the data I have ready at hand, both for gas rich galaxies and the SPARC sample:

*Baryonic mass (stars plus gas) as a function of the rotation speed measured at the outermost detected radius.*

A relation is clear in the plot above, but it’s a mess. There’s lots of scatter, especially at low mass. There is also a systematic tendency for low mass galaxies to fall to the left of the main relation, appearing to rotate too slowly for their mass.

There is no quality control in the plot above. I have thrown all the mud at the wall. Let’s now do some quality control. The plotted quantities are the baryonic mass and the flat rotation speed. We haven’t actually measured the flat rotation speed in all these cases. For some, we’ve simply taken the last measured point. This was an issue we explicitly pointed out in Stark et al (2009):

Fig. 1 from Stark et al (2009): Examples of rotation curves (Swaters et al. 2009) that do and do not satisfy the flatness criterion. The rotation curve of UGC 4173 (top) rises continuously and does not meet the flatness criterion. UGC 5721 (center) is an ideal case with clear flattening of the rotational velocity. UGC 4499 marginally satisfies the flatness criterion.

If we include a galaxy like UGC 4173, we expect it will be offset to the low velocity side because we haven’t measured the flat rotation speed. We’ve merely taken that last point and hoped it is close enough. Sometimes it is, depending on your tolerance for systematic errors. But the plain fact is that we haven’t measured the flat rotation speed in this case. We don’t even know if it has one; it is only empirical experience with other examples that lead us to expect it to flatten if we manage to observe further out.

For our purpose here, it is as if we hadn’t measured this galaxy at all. So let’s not pretend like we have, and restrict the plot to galaxies for which the flat velocity is measured:

*The same as the first plot, restricted to galaxies for which the flat rotation speed has been measured.*

The scatter in the BTFR decreases dramatically when we exclude the galaxies for which we haven’t measured the relevant quantities. This is a simple matter of data quality. We’re no longer pretending to have measured a quantity that we haven’t measured.

There are still some outliers as there are still things that can go wrong. Inclinations are a challenge for some galaxies, as are distances determinations. Remember that Tully-Fisher was first employed as a distance indicator. If we look at the plot above from that perspective, the outliers have obviously been assigned the wrong distance, and we would assign a new one by putting them on the relation. That, in a nutshell, is how astronomical distance indicators work.

If we restrict the data to those with accurate measurements, we get

*Same as the plot above, restricted to galaxies for which the quantities measured on both axes have been measured to an accuracy of 20% or better.*

Now the outliers are gone. They were outliers because they had crappy data. This is completely unsurprising. Some astronomical data are always crappy. You plot crap against crap, you get crap. If, on the other hand, you look at the high quality data, you get a high quality correlation. Even then, you can never be sure that you’ve excluded all the crap, as there are often unknown unknowns – systematic errors you don’t know about and can’t control for.

We have done the exercise of varying the tolerance limits on data quality many times. We have shown that the scatter varies as expected with data quality. If we consider high quality data, we find a small scatter in the BTFR. If we consider low quality data, we get to plot more points, but the scatter goes up. You can see this by eye above. We can quantify this, and have. The amount of scatter varies as expected with the size of the uncertainties. Bigger errors, bigger scatter. Smaller errors, smaller scatter. This shouldn’t be hard to understand.

So why do people – many of them good scientists – keep screwing this up?

There are several answers. One is that measuring the flat rotation speed is hard. We have only done it for a couple hundred galaxies. This seems like a tiny number in the era of the Sloan Digitial Sky Survey, which enables any newbie to assemble a sample of tens of thousands of galaxies… with photometric data. It doesn’t provide any kinematic data. Measuring the stellar mass with the photometric data doesn’t do one bit of good for this problem if you don’t have the kinematic axis to plot against. Consequently, it doesn’t matter how big such a sample is.

Other measurements often provide a proxy measurement that seems like it ought to be close enough to use. If not the flat rotation speed, maybe you have a line width or a maximum speed or V_2.2 or the hybrid S_0.5 or some other metric. That’s fine, so long as you recognize you’re plotting something different so should expect to get something different – not the BTFR. Again, we’ve shown that the flat rotation speed is the measure that minimizes the scatter; if you utilize some other measure you’re gonna get more scatter. That may be useful for some purposes, but it only tells you about what you measured. It doesn’t tell you anything about the scatter in the BTFR constructed with the flat rotation speed if you didn’t measure the flat rotation speed.

Another possibility is that there exist galaxies that fall off the BTFR that we haven’t observed yet. It is a big universe, after all. This is a known unknown unknown: we know that we don’t know if there are non-conforming galaxies. If the relation is indeed absolute, then we never can find any, but never can we know that they don’t exist, only that we haven’t yet found any credible examples.

I’ve addressed the possibility of nonconforming galaxies elsewhere, so all I’ll say here is that I have spent my entire career seeking out the extremes in galaxy properties. Many times I have specifically sought out galaxies that should deviate from the BTFR for some clear reason, only to be surprised when they fall bang on the BTFR. Over and over and over again. It makes me wonder how Vera Rubin felt when her observations kept turning up flat rotation curves. Shouldn’t happen, but it does – over and over and over again. So far, I haven’t found any credible deviations from the BTFR, nor have I seen credible cases provided by others – just repeated failures of quality control.

Finally, an underlying issue is often – not always, but often – an obsession with salvaging the dark matter paradigm. That’s hard to do if you acknowledge that the observed BTFR – its slope, normalization, lack of scale length residuals, negligible intrinsic scatter; indeed, the very quantities that define it, were anticipated and explicitly predicted by MOND and only MOND. It is easy to confirm the dark matter paradigm if you never acknowledge this to be a problem. Often, people redefine the terms of the issue in some manner that is more tractable from the perspective of dark matter. From that perspective, neither the “cold” baryonic mass nor the flat rotation speed have any special meaning, so why even consider them? That is the road to MONDness.

A brief history of the Radial Acceleration Relation

In science, all new and startling facts must encounter in sequence the responses
1. It is not true!
2. It is contrary to orthodoxy.
3. We knew it all along.
Louis Agassiz (circa 1861)

This expression exactly depicts the progression of the radial acceleration relation. Some people were ahead of this curve, others are still behind it, but it quite accurately depicts the mass sociology. This is how we react to startling new facts.

For quotation purists, I’m not sure exactly what the original phrasing was. I have paraphrased it to be succinct and have substituted orthodoxy for religion, because even scientists can have orthodoxies: holy cows that must not be slaughtered.

I might even add a precursor stage zero to the list above:

0. It goes unrecognized.

This is to say, that if a new fact is sufficiently startling, we don’t just disbelieve it (stage 1); at first we fail to see it at all. We lack the cognitive framework to even recognize how important it is. An example is provided by the 1941 detection of the microwave background by Andrew McKellar. In retrospect, this is as persuasive as the 1964 detection of Penzias and Wilson to which we usually ascribe the discovery. At the earlier time, there was simply no framework for recognizing what it was that was being detected. It appears to me that P&Z didn’t know what they were looking at either until Peebles explained it to them.

The radial acceleration relation was first posed as the mass discrepancy-acceleration relation. They’re fundamentally the same thing, just plotted in a slightly different way. The mass discrepancy-acceleration relation shows the ratio of total mass to that which is visible. This is basically the ratio of the observed acceleration to that predicted by the observed baryons. This is useful to see how much dark matter is needed, but by construction the axes are not independent, as both measured quantities are used in forming the ratio.

The radial acceleration relation shows independent observations along each axis: observed vs. predicted acceleration. Though measured independently, they are not physically independent, as the baryons contribute some to the total observed acceleration – they do have mass, after all. One can construct a halo acceleration relation by subtracting the baryonic contribution away from the total; in principle the remainders are physically independent. Unfortunately, the axes again become observationally codependent, and the uncertainties blow up, especially in the baryon dominated regime. Which of these depictions is preferable depends a bit on what you’re looking to see; here I just want to note that they are the same information packaged somewhat differently.

To the best of my knowledge, the first mention of the mass discrepancy-acceleration relation in the scientific literature is by Sanders (1990). Its existence is explicit in MOND (Milgrom 1983), but here it is possible to draw a clear line between theory and data. I am only speaking of the empirical relation as it appears in the data, irrespective of anything specific to MOND.

I met Bob Sanders, along with many other talented scientists, in a series of visits to the University of Groningen in the early 1990s. Despite knowing him and having talked to him about rotation curves, I was unaware that he had done this.

Stage 0: It goes unrecognized.

For me, stage one came later in the decade at the culmination of a several years’ campaign to examine the viability of the dark matter paradigm from every available perspective. That’s a long paper, which nevertheless drew considerable praise from many people who actually read it. If you go to the bother of reading it today, you will see the outlines of many issues that are still debated and others that have been forgotten (e.g., the fine-tuning issues).

Around this time (1998), the dynamicists at Rutgers were organizing a meeting on galaxy dynamics, and asked me to be one of the speakers. I couldn’t possibly discuss everything in the paper in the time allotted, so was looking for a way to show the essence of the challenge the data posed. Consequently, I reinvented the wheel, coming up with the mass discrepancy-acceleration relation. Here I show the same data that I had then in the form of the radial acceleration relation:

*The Radial Acceleration Relation from the data in McGaugh (1999). Plot credit: Federico Lelli. (There is a time delay in publication: the 1998 meeting’s proceedings appeared in 1999.)*

I recognize this version of the plot as having been made by Federico Lelli. I’ve made this plot many times, but this is version I came across first, and it is better than mine in that the opacity of the points illustrates where the data are concentrated. I had been working on low surface brightness galaxies; these have low accelerations, so that part of the plot is well populated.

The data show a clear correlation. By today’s standards, it looks crude. Going on what we had then, it was fantastic. Correlations practically never look this good in extragalactic astronomy, and they certainly don’t happen by accident. Low quality data can hide a correlation – uncertainties cause scatter – but they can’t create a correlation where one doesn’t exist.

This result was certainly startling if not as new as I then thought. That’s why I used the title How Galaxies Don’t Form. This was contrary to our expectations, as I had explained in exhaustive detail in the long paper and revisit in a recent review for philosophers and historians of science.

I showed the same result later that year (1998) at a meeting on the campus of the University of Maryland where I was a brand new faculty member. It was a much shorter presentation, so I didn’t have time to justify the context or explain much about the data. Contrary to the reception at Rutgers where I had adequate time to speak, the hostility of the audience to the result was palpable, their stony silence eloquent. They didn’t want to believe it, and plenty of people got busy questioning the data.

Stage 1: It is not true.

I spent the next five years expanding and improving the data. More rotation curves became available thanks to the work of many, particularly Erwin de Blok, Marc Verheijen, and Rob Swaters. That was great, but the more serious limitation was how well we could measure the stellar mass distribution needed to predict the baryonic acceleration.

The mass models we could build at the time were based on optical images. A mass model takes the observed light distribution, assigns a mass-to-light ratio, and makes a numerical solution of the Poisson equation to obtain the the gravitational force corresponding to the observed stellar mass distribution. This is how we obtain the stellar contribution to the predicted baryonic force; the same procedure is applied to the observed gas distribution. The blue part of the spectrum is the best place in which to observe low contrast, low surface brightness galaxies as the night sky is darkest there, at least during new moon. That’s great for measuring the light distribution, but what we want is the stellar mass distribution. The mass-to-light ratio is expected to have a lot of scatter in the blue band simply from the happenstance of recent star formation, which makes bright blue stars that are short-lived. If there is a stochastic uptick in the star formation rate, then the mass-to-light ratio goes down because there are lots of bright stars. Wait a few hundred million years and these die off, so the mass-to-light ratio gets bigger (in the absence of further new star formation). The time-integrated stellar mass may not change much, but the amount of blue light it produces does. Consequently, we expect to see well-observed galaxies trace distinct lines in the radial acceleration plane, even if there is a single universal relation underlying the phenomenon. This happens simply because we expect to get M*/L wrong from one galaxy to the next: in 1998, I had simply assumed all galaxies had the same M*/L for lack of any better prescription. Clearly, a better prescription was warranted.

In those days, I traveled through Tucson to observe at Kitt Peak with some frequency. On one occasion, I found myself with a few hours to kill between coming down from the mountain and heading to the airport. I wandered over to the Steward Observatory at the University of Arizona to see who I might see. A chance meeting in the wild west: I encountered Eric Bell and Roelof de Jong, who were postdocs there at the time. I knew Eric from his work on the stellar populations of low surface brightness galaxies, an interest closely aligned with my own, and Roelof from my visits to Groningen.

As we got to talking, Eric described to me work they were doing on stellar populations, and how they thought it would be possible to break the age-metallicity degeneracy using near-IR colors in addition to optical colors. They were mostly focused on improving the age constraints on stars in LSB galaxies, but as I listened, I realized they had constructed a more general, more powerful tool. At my encouragement (read their acknowledgements), they took on this more general task, ultimately publishing the classic Bell & de Jong (2001). In it, they built a table that enabled one to look up the expected mass-to-light ratio of a complex stellar population – one actively forming stars – as a function of color. This was a big step forward over my educated guess of a constant mass-to-light ratio: there was now a way to use a readily observed property, color, to improve the estimated M*/L of each galaxy in a well-calibrated way.

Combining the new stellar population models with all the rotation curves then available, I obtained an improved mass discrepancy-acceleration relation:

The Radial Acceleration Relation from the data in McGaugh (2004); version using Bell’s stellar population synthesis models to estimate M*/L (see Fig. 5 for other versions). Plot credit: Federico Lelli.

Again, the relation is clear, but with scatter. Even with the improved models of Bell & de Jong, some individual galaxies have M*/L that are wrong – that’s inevitable in this game. What you cannot know is which ones! Note, however, that there are now 74 galaxies in this plot, and almost all of them fall on top of each other where the point density is large. There are some obvious outliers; those are presumably just that: the trees that fall outside the forest because of the expected scatter in M*/L estimates.

I tried a variety of prescriptions for M*/L in addition to that of Bell & de Jong. Though they differed in texture, they all told a consistent story. A relation was clearly present; only its detailed form varied with the adopted prescription.

The prescription that minimized the scatter in the relation was the M*/L obtained in MOND fits. That’s a tautology: by construction, a MOND fit finds the M*/L that puts a galaxy on this relation. However, we can generalize the result. Maybe MOND is just a weird, unexpected way of picking a number that has this property; it doesn’t have to be the true mass-to-light ratio in nature. But one can then define a ratio Q

that relates the “true” mass-to-light ratio to the number that gives a MOND fit. They don’t have to be identical, but MOND does return M*/L that are reasonable in terms of stellar populations, so Q ~ 1. Individual values could vary, and the mean could be a bit more or less than unity, but not radically different. One thing that impressed me at the time about the MOND fits (most of which were made by Bob Sanders) was how well they agreed with the stellar population models, recovering the correct amplitude, the correct dependence on color in different bandpasses, and also giving the expected amount of scatter (more in the blue than in the near-IR).

Fig. 7 *of McGaugh (2004).* Stellar mass-to-light ratios of galaxies in the blue B-band (*top*) and near-IR K-band (*bottom*) as a function of B–V color for the prescription of maximum disk (*left*) and MOND (*right*). Each point represents one galaxy for which the requisite data were available at the time. The line represents the mean expectation of stellar population synthesis models from Bell et al. (2003). These lines are completely independent of the data: neither the normalization nor the slope has been fit to the dynamical data. The red points are due to Sanders & Verheijen (1998); note the weak dependence of M*/L on color in the near-IR.

The obvious interpretation is that we should take seriously a theory that obtains good fits with a single free parameter that checks out admirably well with independent astrophysical constraints, in this case the M*/L expected for stellar populations. But I knew many people would not want to do that, so I defined Q to generalize to any M*/L in any (dark matter) context one might want to consider.

Indeed, Q allows us to write a general expression for the rotation curve of the dark matter halo (essentially the HAR alluded to above) in terms of that of the stars and gas:

Equation 22 of McGaugh (2004).

The stars and the gas are observed, and μ is the MOND interpolation function assumed in the fit that leads to Q. Except now the interpolation function isn’t part of some funny new theory; it is just the shape of the radial acceleration relation – a relation that is there empirically. The only fit factor between these data and any given model is Q – a single number of order unity. This does leave some wiggle room, but not much.

I went off to a conference to describe this result. At the 2006 meeting Galaxies in the Cosmic Web in New Mexico, I went out of my way at the beginning of the talk to show that even if we ignore MOND, this relation is present in the data, and it provides a strong constraint on the required distribution of dark matter. We may not know why this relation happens, but we can use it, modulo only the modest uncertainty in Q.

Having bent over backwards to distinguish the data from the theory, I was disappointed when, immediately at the end of my talk, prominent galaxy formation theorist Anatoly Klypin loudly shouted

“We don’t have to explain MOND!”

But you do have to explain the data. The problem was and is that the data look like MOND. It is easy to conflate one with the other; I have noticed that a lot of people have trouble keeping the two separate. Just because you don’t like the theory doesn’t mean that the data are wrong. What Anatoly was saying was that

2. It is contrary to orthodoxy.

Despite phrasing the result in a way that would be useful to galaxy formation theorists, they did not, by and large, claim to explain it at the time – it was contrary to orthodoxy so didn’t need to be explained. Looking at the list of papers that cite this result, the early adopters were not the target audience of galaxy formation theorists, but rather others citing it to say variations of “no way dark matter explains this.”

At this point, it was clear to me that further progress required a better way to measure the stellar mass distribution. Looking at the stellar population models, the best hope was to build mass models from near-infrared rather than optical data. The near-IR is dominated by old stars, especially red giants. Galaxies that have been forming stars actively for a Hubble time tend towards a quasi-equilibrium in which red giants are replenished by stellar evolution at about the same rate they move on to the next phase. One therefore expects the mass-to-light ratio to be more nearly constant in the near-IR. Not perfectly so, of course, but a 2 or 3 micron image is as close to a map of the stellar mass of a galaxy as we’re likely to get.

Around this time, the University of Maryland had begun a collaboration with Kitt Peak to build a big infrared camera, NEWFIRM, for the 4m telescope. Rob Swaters was hired to help write software to cope with the massive data flow it would produce. The instrument was divided into quadrants, each of which had a field of view sufficient to hold a typical galaxy. When it went on the telescope, we developed an efficient observing method that I called “four-shooter”, shuffling the target galaxy from quadrant to quadrant so that in processing we could remove the numerous instrumental artifacts intrinsic to its InSb detectors. This eventually became one of the standard observing modes in which the instrument was operated.

NEWFIRM in the lab in Tucson. Most of the volume is for cryogenics: the IR detectors are *helium*–*cooled to 30 K. Partial student for scale.*

I was optimistic that we could make rapid progress, and at first we did. But despite all the work, despite all the active cooling involved, we were still on the ground. The night sky was painfully bright in the IR. Indeed, the thermal component dominated, so we could observe during full moon. To an observer of low surface brightness galaxies attuned to any hint of scattered light from so much as a crescent moon, I cannot describe how discombobulating it was to walk outside the dome and see the full fricking moon. So bright. So wrong. And that wasn’t even the limiting factor: the thermal background was.

We had hit a surface brightness wall, again. We could do the bright galaxies this way, but the LSBs that sample the low acceleration end of the radial acceleration relation were rather less accessible. Not inaccessible, but there was a better way.

The Spitzer Space Telescope was active at this time. Jim Schombert and I started winning time to observe LSB galaxies with it. We discovered that space is dark. There was no atmosphere to contend with. No scattered light from the clouds or the moon or the OH lines that afflict that part of the sky spectrum. No ground-level warmth. The data were fantastic. In some sense, they were too good: the biggest headache we faced was blotting out all the background galaxies that shown right through the optically thin LSB galaxies.

Still, it took a long time to collect and analyze the data. We were starting to get results by the early-teens, but it seemed like it would take forever to get through everything I hoped to accomplish. Fortunately, when I moved to Case Western, I was able to hire Federico Lelli as a postdoc. Federico’s involvement made all the difference. After many months of hard, diligent, and exacting work, he constructed what is now the SPARC database. Finally all the elements were in place to construct an empirical radial acceleration relation with absolutely minimal assumptions about the stellar mass-to-light ratio.

In parallel with the observational work, Jim Schombert had been working hard to build realistic stellar population models that extended to the 3.6 micron band of Spitzer. Spitzer had been built to look redwards of this, further into the IR. 3.6 microns was its shortest wavelength passband. But most models at the time stopped at the K-band, the 2.2 micron band that is the reddest passband that is practically accessible from the ground. They contain pretty much the same information, but we still need to calculate the band-specific value of M*/L.

Being a thorough and careful person, Jim considered not just the star formation history of a model stellar population as a variable, and not just its average metallicity, but also the metallicity distribution of its stars, making sure that these were self-consistent with the star formation history. Realistic metallicity distributions are skewed; it turn out that this subtle effect tends to counterbalance the color dependence of the age effect on M*/L in the near-IR part of the spectrum. The net results is that we expect M*/L to be very nearly constant for all late type galaxies.

This is the best possible result. To a good approximation, we expected all of the galaxies in the SPARC sample to have the same mass-to-light ratio. What you see is what you get. No variable M*/L, no equivocation, just data in, result out.

We did still expect some scatter, as that is an irreducible fact of life in this business. But even that we expected to be small, between 0.1 and 0.15 dex (roughly 25 – 40%). Still, we expected the occasional outlier, galaxies that sit well off the main relation just because our nominal M*/L didn’t happen to apply in that case.

One day as I walked past Federico’s office, he called for me to come look at something. He had plotted all the data together assuming a single M*/L. There… were no outliers. The assumption of a constant M*/L in the near-IR didn’t just work, it worked far better than we had dared to hope. The relation leapt straight out of the data:

*The Radial Acceleration Relation from the data in McGaugh et al. (2016)*. Plot credit: Federico Lelli.

Over 150 galaxies, with nearly 2700 resolved measurements within each galaxy, each with their own distinctive mass distribution, all pile on top of each other without effort. There was plenty of effort in building the database, but once it was there, the result appeared, no muss, no fuss. No fitting or fiddling. Just the measurements and our best estimate of the mean M*/L, applied uniformly to every individual galaxy in the sample. The scatter was only 0.12 dex, within the range expected from the population models.

No MOND was involved in the construction of this relation. It may look like MOND, but we neither use MOND nor need it in any way to see the relation. It is in the data. Perhaps this is the sort of result for which we would have to invent MOND if it did not already exist. But the dark matter paradigm is very flexible, and many papers have since appeared that claim to explain the radial acceleration relation. We have reached

3. We knew it all along.

On the one hand, this is good: the community is finally engaging with a startling fact that has been pointedly ignored for decades. On the other hand, many of the claims to explain the radial acceleration relation are transparently incorrect on their face, being nothing more than elaborations of models I considered and discarded as obviously unworkable long ago. They do not provide a satisfactory explanation of the predictive power of MOND, and inevitably fail to address important aspects of the problem, like disk stability. Rather than grapple with the deep issues the new and startling fact poses, it has become fashionable to simply assert that one’s favorite model explains the radial acceleration relation, and does so naturally.

There is nothing natural about the radial acceleration relation in the context of dark matter. Indeed, it is difficult to imagine a less natural result – hence stages one and two. So on the one hand, I welcome the belated engagement, and am willing to consider serious models. On the other hand, if someone asserts that this is natural and that we expected it all along, then the engagement isn’t genuine: they’re just fooling themselves.

Early Days. This was one of Vera Rubin’s favorite expressions. I always had a hard time with it, as many things are very well established. Yet it seems that we have yet to wrap our heads around the problem. Vera’s daughter, Judy Young, once likened the situation to the parable of the blind men and the elephant. Much is known, yes, but the problem is so vast that each of us can perceive only a part of the whole, and the whole may be quite different from the part that is right before us.

So I guess Vera is right as always: these remain Early Days.

The curious case of AGC 114905: an isolated galaxy devoid of dark matter?

It’s early in the new year, so what better time to violate my own resolutions? I prefer to be forward-looking and not argue over petty details, or chase wayward butterflies. But sometimes the devil is in the details, and the occasional butterfly can be entertaining if distracting. Today’s butterfly is the galaxy AGC 114905, which has recently been in the news.

There are a couple of bandwagons here: one to rebrand very low surface brightness galaxies as ultradiffuse, and another to get overly excited when these types of galaxies appear to lack dark matter. The nomenclature is terrible, but that’s normal for astronomy so I would overlook it, except that in this case it gives the impression that there is some new population of galaxies behaving in an unexpected fashion, when instead it looks to me like the opposite is the case. The extent to which there are galaxies lacking dark matter is fundamental to our interpretation of the acceleration discrepancy (aka the missing mass problem), so bears closer scrutiny. The evidence for galaxies devoid of dark matter is considerably weaker than the current bandwagon portrays.

If it were just one butterfly (e.g., NGC 1052-DF2), I wouldn’t bother. Indeed, it was that specific case that made me resolve to ignore such distractions as a waste of time. I’ve seen this movie literally hundreds of times, I know how it goes:

Observations of this one galaxy falsify MOND!
Hmm, doing the calculation right, that’s what MOND predicts.
OK, but better data shrink the error bars and now MOND falsified.
Are you sure about…?
Yes. We like this answer, let’s stop thinking about it now.
As the data continue to improve, it approaches what MOND predicts.
<crickets>

Over and over again. DF44 is another example that has followed this trajectory, and there are many others. This common story is not widely known – people lose interest once they get the answer they want. Irrespective of whether we can explain this weird case or that, there is a deeper story here about data analysis and interpretation that seems not to be widely appreciated.

My own experience inevitably colors my attitude about this, as it does for us all, so let’s start thirty years ago when I was writing a dissertation on low surface brightness (LSB) galaxies. I did many things in my thesis, most of them well. One of the things I tried to do then was derive rotation curves for some LSB galaxies. This was not the main point of the thesis, and arose almost as an afterthought. It was also not successful, and I did not publish the results because I didn’t believe them. It wasn’t until a few years later, with improved data, analysis software, and the concerted efforts of Erwin de Blok, that we started to get a handle on things.

The thing that really bugged me at the time was not the Doppler measurements, but the inclinations. One has to correct the observed velocities by the inclination of the disk, 1/sin(i). The inclination can be constrained by the shape of the image and by the variation of velocities across the face of the disk. LSB galaxies presented raggedy images and messy velocity fields. I found it nigh on impossible to constrain their inclinations at the time, and it remains a frequent struggle to this day.

Here is an example of the LSB galaxy F577-V1 that I find lurking around on disk from all those years ago:

*The LSB galaxy F577-V1 (B-band image, left) and the run of the eccentricity of ellipses fit to the atomic gas data (right).*

A uniform disk projected on the sky at some inclination will have a fixed corresponding eccentricity, with zero being the limit of a circular disk seen perfectly face-on (i = 0). Do you see a constant value of the eccentricity in the graph above? If you say yes, go get your eyes checked.

What we see in this case is a big transition from a fairly eccentric disk to one that is more nearly face on. The inclination doesn’t have a sudden warp; the problem is that the assumption of a uniform disk is invalid. This galaxy has a bar – a quasi-linear feature that is common in many spiral galaxies that is supported by non-circular orbits. Even face-on, the bar will look elongated simply because it is. Indeed, the sudden change in eccentricity is one way to define the end of the bar, which the human eye-brain can do easily by looking at the image. So in a case like this, one might adopt the inclination from the outer points, and that might even be correct. But note that there are spiral arms along the outer edge that is visible to the eye, so it isn’t clear that even these isophotes are representative of the shape of the underlying disk. Worse, we don’t know what happens beyond the edge of the data; the shape might settle down at some other level that we can’t see.

This was so frustrating, I swore never to have anything to do with galaxy kinematics ever again. Over 50 papers on the subject later, all I can say is D’oh! Repeatedly.

Bars are rare in LSB galaxies, but it struck me as odd that we saw any at all. We discovered unexpectedly that they were dark matter dominated – the inferred dark halo outweighs the disk, even within the edge defined by the stars – but that meant that the disks should be stable against the formation of bars. My colleague Chris Mihos agreed, and decided to look into it. The answer was yes, LSB galaxies should be stable against bar formation, at least internally generated bars. Sometimes bars are driven by external perturbations, so we decided to simulate the close passage of a galaxy of similar mass – basically, whack it real hard and see what happens:

Simulation of an LSB galaxy during a strong tidal encounter with another galaxy. Closest approach is at t=24 in simulation units (between the first and second box). A linear bar does not form, but the model galaxy does suffer a strong and persistent oval distortion: all these images are shown face-on (i=0). From Mihos et al (1997).

This was a conventional simulation, with a dark matter halo constructed to be consistent with the observed properties of the LSB galaxy UGC 128. The results are not specific to this case; it merely provides numerical corroboration of the more general case that we showed analytically.

Consider the image above in the context of determining galaxy inclinations from isophotal shapes. We know this object is face-on because we can control our viewing angle in the simulation. However, we would not infer i=0 from this image. If we didn’t know it had been perturbed, we would happily infer a substantial inclination – in this case, easily as much as 60 degrees! This is an intentionally extreme case, but it illustrates how a small departure from a purely circular shape can be misinterpreted as an inclination. This is a systematic error, and one that usually makes the inclination larger than it is: it is possible to appear oval when face-on, but it is not possible to appear more face-on than perfectly circular.

Around the same time, Erwin and I were making fits to the LSB galaxy data – with both dark matter halos and MOND. By this point in my career, I had deeply internalized that the data for LSB galaxies were never perfect. So we sweated every detail, and worked through every “what if?” This was a particularly onerous task for the dark matter fits, which could do many different things if this or that were assumed – we discussed all the plausible possibilities at the time. (Subsequently, a rich literature sprang up discussing many unreasonable possibilities.) By comparison, the MOND fits were easy. They had fewer knobs, and in 2/3 of the cases they simply worked, no muss, no fuss.

For the other 1/3 of the cases, we noticed that the shape of the MOND-predicted rotation curves was usually right, but the amplitude was off. How could it work so often, and yet miss in this weird way? That sounded like a systematic error, and the inclination was the most obvious culprit, with 1/sin(i) making a big difference for small inclinations. So we decided to allow this as a fit parameter, to see whether a fit could be obtained, and judge how [un]reasonable this was. Here is an example for two galaxies:

UGC 1230 (left) and UGC 5005 (right). Ovals show the nominally measured inclination (i=22^o for UGC 1230 and 41^o for UGC 5005, respectively) and the MOND best-fit value (i=17^o and 30^o). From de Blok & McGaugh (1998).

The case of UGC 1230 is memorable to me because it had a good rotation curve, despite being more face-on than widely considered acceptable for analysis. And for good reason: the difference between 22 and 17 degrees make a huge difference to the fit, changing it from way off to picture perfect.

*Rotation curve fits for UGC 1230 (top) and UGC 5005 (bottom) with the inclination fixed (left) and fit (right). From de Blok & McGaugh (1998).*

What I took away from this exercise is how hard it is to tell the difference between inclination values for relatively face-on galaxies. UGC 1230 is obvious: the ovals for the two inclinations are practically on top of each other. The difference in the case of UGC 5005 is more pronounced, but look at the galaxy. The shape of the outer isophote where we’re trying to measure this is raggedy as all get out; this is par for the course for LSB galaxies. Worse, look further in – this galaxy has a bar! The central bar is almost orthogonal to the kinematic major axis. If we hadn’t observed as deeply as we had, we’d think the minor axis was the major axis, and the inclination was something even higher.

I remember Erwin quipping that he should write a paper on how to use MOND to determine inclinations. This was a joke between us, but only half so: using the procedure in this way would be analogous to using Tully-Fisher to measure distances. We would simply be applying an empirically established procedure to constrain a property of a galaxy – luminosity from line-width in that case of Tully-Fisher; inclination from rotation curve shape here. That we don’t understand why this works has never stopped astronomers before.

Systematic errors in inclination happen all the time. Big surveys don’t have time to image deeply – they have too much sky area to cover – and if there is follow-up about the gas content, it inevitably comes in the form of a single dish HI measurement. This is fine; it is what we can do en masse. But an unresolved single dish measurement provides no information about the inclination, only a pre-inclination line-width (which itself is a crude proxy for the flat rotation speed). The inclination we have to take from the optical image, which would key on the easily detected, high surface brightness central region of the image. That’s the part that is most likely to show a bar-like distortion, so one can expect lots of systematic errors in the inclinations determined in this way. I provided a long yet still incomplete discussion of these issues in McGaugh (2012). This is both technical and intensely boring, so not even the pros read it.

This brings us to the case of AGC 114905, which is part of a sample of ultradiffuse galaxies discussed previously by some of the same authors. On that occasion, I kept to the code, and refrained from discussion. But for context, here are those data on a recent Baryonic Tully-Fisher plot. Spoiler alert: that post was about a different sample of galaxies that seemed to be off the relation but weren’t.

Baryonic Tully-Fisher relation showing the ultradiffuse galaxies discussed by Mancera Piña et al. (2019) as gray circles. These are all outliers from the relation; AGC 114905 is highlighted in orange. Placing much meaning in the outliers is a classic case of missing the forest for the trees. The outliers are trees. The Tully-Fisher relation is the forest.

On the face of it, these ultradiffuse galaxies (UDGs) are all very serious outliers. This is weird – they’re not some scatter off to one side, they’re just way off on their own island, with no apparent connection to the rest of established reality. By calling them a new name, UDG, it makes it sound plausible that these are some entirely novel population of galaxies that behave in a new way. But they’re not. They are exactly the same kinds of galaxies I’ve been talking about. They’re all blue, gas rich, low surface brightness, fairly isolated galaxies – all words that I’ve frequently used to describe my thesis sample. These UDGs are all a few billion solar mass is baryonic mass, very similar to F577-V1 above. You could give F577-V1 a different name, slip into the sample, and nobody would notice that it wasn’t like one of the others.

The one slight difference is implied by the name: UDGs are a little lower in surface brightness. Indeed, once filter transformations are taken into account, the definition of ultradiffuse is equal to what I arbitrarily called very low surface brightness in 1996. Most of my old LSB sample galaxies have central stellar surface brightnesses at or a bit above 10 solar masses per square parsec while the UDGs here are a bit under this threshold. For comparison, in typical high surface brightness galaxies this quantity is many hundreds, often around a thousand. Nothing magic happens at the threshold of 10 solar masses per square parsec, so this line of definition between LSB and UDG is an observational distinction without a physical difference. So what are the odds of a different result for the same kind of galaxies?

Indeed, what really matters is the baryonic surface density, not just the stellar surface brightness. A galaxy made purely of gas but no stars would have zero optical surface brightness. I don’t know of any examples of that extreme, but we came close to it with the gas rich sample of Trachternach et al. (2009) when we tried this exact same exercise a decade ago. Despite selecting that sample to maximize the chance of deviations from the Baryonic Tully-Fisher relation, we found none – at least none that were credible: there were deviant cases, but their data were terrible. There were no deviants among the better data. This sample is comparable or even extreme than the UDGs in terms of baryonic surface density, so the UDGs can’t be exception because they’re a genuinely new population, whatever name we call them by.

The key thing is the credibility of the data, so let’s consider the data for AGC 114905. The kinematics are pretty well ordered; the velocity field is well observed for this kind of beast. It ought to be; they invested over 40 hours of JVLA time into this one galaxy. That’s more than went into my entire LSB thesis sample. The authors are all capable, competent people. I don’t think they’ve done anything wrong, per se. But they do seem to have climbed aboard the bandwagon of dark matter-free UDGs, and have talked themselves into believing smaller error bars on the inclination than I am persuaded is warranted.

Here is the picture of AGC 114905 from Mancera Piña et al. (2021):

AGC 114905 in stars (left) and gas (right). The contours of the gas distribution are shown on top of the stars in white. Figure 1 from Mancera Piña et al. (2021).

This messy morphology is typical of very low surface brightness galaxies – hence their frequent classification as Irregular galaxies. Though messier, it shares some morphological traits with the LSB galaxies shown above. The central light distribution is elongated with a major axis that is not aligned with that of the gas. The gas is raggedy as all get out. The contours are somewhat boxy; this is a hint that something hinky is going on beyond circular motion in a tilted axisymmetric disk.

The authors do the right thing and worry about the inclination, checking to see what it would take to be consistent with either LCDM or MOND, which is about i=11^o in stead of the 30^o indicated by the shape of the outer isophote. They even build a model to check the plausibility of the smaller inclination:

*Contours of models of disks with different inclinations (lines, as labeled) compared to the outer contour of the gas distribution of AGC 114905. Figure 7 from Mancera Piña et al. (2021).*

Clearly the black line (i=30^o) is a better fit to the shape of the gas distribution than the blue dashed line (i=11^o). Consequently, they “find it unlikely that we are severely overestimating the inclination of our UDG, although this remains the largest source of uncertainty in our analysis.” I certainly agree with the latter phrase, but not the former. I think it is quite likely that they are overestimating the inclination. I wouldn’t even call it a severe overestimation; more like par for the course with this kind of object.

As I have emphasized above and elsewhere, there are many things that can go wrong in this sort of analysis. But if I were to try to put my finger on the most important thing, here it would be the inclination. The modeling exercise is good, but it assumes “razor-thin axisymmetric discs.” That’s a reasonable thing to do when building such a model, but we have to bear in mind that real disks are neither. The thickness of the disk probably doesn’t matter too much for a nearly face-on case like this, but the assumption of axisymmetry is extraordinarily dubious for an Irregular galaxy. That’s how they got the name.

It is hard to build models that are not axisymmetric. Once you drop this simplifying assumption, where do you even start? So I don’t fault them for stopping at this juncture, but I can also imagine doing as de Blok suggested, using MOND to set the inclination. Then one could build models with asymmetric features by trial and error until a match is obtained. Would we know that such a model would be a better representation of reality? No. Could we exclude such a model? Also no. So the bottom line is that I am not convinced that the uncertainty in the inclination is anywhere near as small as the adopted ±3^o.

That’s very deep in the devilish details. If one is worried about a particular result, one can back off and ask if it makes sense in the context of what we already know. I’ve illustrated this process previously. First, check the empirical facts. Every other galaxy in the universe with credible data falls on the Baryonic Tully-Fisher relation, including very similar galaxies that go by a slightly different name. Hmm, strike one. Second, check what we expect from theory. I’m not a fan of theory-informed data interpretation, but we know that LCDM, unlike SCDM before it, at least gets the amplitude of the rotation speed in the right ballpark (V_flat ~ V₂₀₀). Except here. Strike two. As much as we might favor LCDM as the standard cosmology, it has now been extraordinarily well established that MOND has considerable success in not just explaining but predicting these kind of data, with literally hundreds of examples. One hundred was the threshold Vera Rubin obtained to refute excuses made to explain away the first few flat rotation curves. We’ve crossed that threshold: MOND phenomenology is as well established now as flat rotation curves were at the inception of the dark matter paradigm. So while I’m open to alternative explanations for the MOND phenomenology, seeing that a few trees stand out from the forest is never going to be as important as the forest itself.

The Baryonic Tully-Fisher relation exists empirically; we have to explain it in any theory. Either we explain it, or we don’t. We can’t have it both ways, just conveniently throwing away our explanation to accommodate any discrepant observation that comes along. That’s what we’d have to do here: if we can explain the relation, we can’t very well explain the outliers. If we explain the outliers, it trashes our explanation for the relation. If some galaxies are genuine exceptions, then there are probably exceptional reasons for them to be exceptions, like a departure from equilibrium. That can happen in any theory, rendering such a test moot: a basic tenet of objectivity is that we don’t get to blame a missed prediction of LCDM on departures from equilibrium without considering the same possibility for MOND.

This brings us to a physical effect that people should be aware of. We touched on the bar stability above, and how a galaxy might look oval even when seen face on. This happens fairly naturally in MOND simulations of isolated disk galaxies. They form bars and spirals and their outer parts wobble about. See, for example, this simulation by Nils Wittenburg. This particular example is a relatively massive galaxy; the lopsidedness reminds me of M101 (Watkins et al. 2017). Lower mass galaxies deeper in the MOND regime are likely even more wobbly. This happens because disks are only marginally stable in MOND, not the over-stabilized entities that have to be hammered to show a response as in our early simulation of UGC 128 above. The point is that there is good reason to expect even isolated face-on dwarf Irregulars to look, well, irregular, leading to exactly the issues with inclination determinations discussed above. Rather than being a contradiction to MOND, AGC 114905 may illustrate one of its inevitable consequences.

I don’t like to bicker at this level of detail, but it makes a profound difference to the interpretation. I do think we should be skeptical of results that contradict well established observational reality – especially when over-hyped. God knows I was skeptical of our own results, which initially surprised the bejeepers out of me, but have been repeatedly corroborated by subsequent observations.

I guess I’m old now, so I wonder how I come across to younger practitioners; perhaps as some scary undead monster. But mates, these claims about UDGs deviating from established scaling relations are off the edge of the map.

Leveling the Playing Field of Dwarf Galaxy Kinematics

We have a new paper on the arXiv. This is a straightforward empiricist’s paper that provides a reality check on the calibration of the Baryonic Tully-Fisher relation (BTFR) and the distance scale using well-known Local Group galaxies. It also connects observable velocity measures in rotating and pressure supported dwarf galaxies: the flat rotation speed of disks is basically twice the line-of-sight velocity dispersion of dwarf spheroidals.

First, the reality check. Previously we calibrated the BTFR using galaxies with distances measured by reliable methods like Cepheids and the Tip of the Red Giant Branch (TRGB) method. Application of this calibration obtains the Hubble constant H₀ = 75.1 +/- 2.3 km/s/Mpc, which is consistent with other local measurements but in tension with the value obtained from fitting the Planck CMB data. All of the calibrator galaxies are nearby (most are within 10 Mpc, which is close by extragalactic standards), but none of them are in the Local Group (galaxies within ~1 Mpc like Andromeda and M33). The distances to Local Group galaxies are pretty well known at this point, so if we got the BTFR calibration right, they had better fall right on it.

They do. From high to low mass, the circles in the plot below are Andromeda, the Milky Way, M33, the LMC, SMC, and NGC 6822. All fall on the externally calibrated BTFR, which extrapolates well to still lower mass dwarf galaxies like WLM, DDO 210, and DDO 216 (and even Leo P, the smallest rotating galaxy known).

The BTFR for Local Group galaxies. Rotationally supported galaxies with measured flat rotation velocities (circles) are in good agreement with the BTFR calibrated independently with fifty galaxies external to the Local Group (solid line; the dashed line is the extrapolation below the lowest mass calibrator). Pressure supported dwarfs (squares) are plotted with their observed velocity dispersions in lieu of a flat rotation speed. Filled squares are color coded by their proximity to M31 (red) or the Milky Way (orange) or neither (green). Open squares are dwarfs whose velocity dispersions may not be reliable tracers of their equilibrium gravitational potential (see McGaugh & Wolf).

The agreement of the BTFR with Local Group rotators is so good that it is tempting to say that there is no way to reconcile this with a low Hubble constant of 67 km/s/kpc. Doing so would require all of these galaxies to be more distant by the factor 75/67 = 1.11. That doesn’t sound too bad, but applying it means that Andromeda would have to be 875 kpc distant rather than the 785 ± 25 adopted by the source of our M31 data, Chemin et al. There is a long history of distance measurements to M31 so many opinions can be found, but it isn’t just M31 – all of the Local Group galaxy distances would have to be off by this factor. This seems unlikely to the point of absurdity, but as colleague and collaborator Jim Schombert reminds me, we’ve seen such things before with the distance scale.

So that’s the reality check: the BTFR works as it should in the Local Group – at least for the rotating galaxies (circles in the plot above). What about the pressure supported galaxies (the squares)?

Galaxies come in two basic kinematic types: rotating disks or pressure supported ellipticals. Disks are generally thin, with most of the stars orbiting in the same direction in the same plane on nearly circular orbits. Ellipticals are quasi-spherical blobs of stars on rather eccentric orbits oriented all over the place. This is an oversimplification, of course; real galaxies have a mix of orbits, but usually most of the kinetic energy is invested in one or the other, rotation or random motions. We can measure the speeds of stars and gas in these configurations, which provides information about the kinetic energy and corresponding gravitational binding energy. That’s how we get at the gravitational potential and infer the need for dark matter – or at least, the existence of acceleration discrepancies.

The elliptical galaxy M105 (left) and the spiral galaxy NGC 628 (right). Typical orbits are illustrated by the colored lines: predominantly radial (highly eccentric in & out) orbits in the pressure supported elliptical; more nearly circular (low eccentricity, round & round) orbits in rotationally supported disks. (Galaxy images are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain as part of the Palomar Observatory Sky Survey-II. Digital versions of the scanned photographic plates were obtained for reproduction from the Digitized Sky Survey.)

We would like to have full 6D phase space information for all stars – their location in 3D configuration space and their momentum in each direction. In practice, usually all we can measure is the Doppler line-of-sight speed. For rotating galaxies, we can [attempt to] correct the observed velocity for the inclination of the disk, and get an idea or the in-plane rotation speed. For ellipticals, we get the velocity dispersion along the line of sight in whatever orientation we happen to get. If the orbits are isotropic, then one direction of view is as good as any other. In general that need not be the case, but it is hard to constrain the anisotropy of orbits, so usually we assume isotropy and call it Close Enough for Astronomy.

For isotropic orbits, the velocity dispersion σ_* is related to the circular velocity V_c of a test particle by V_c = √3 σ_*. The square root of three appears because the kinetic energy of isotropic orbits is evenly divided among the three cardinal directions. These quantities depend in a straightforward way on the gravitational potential, which can be computed for the stuff we can see but not for that which we can’t. The stars tend to dominate the potential at small radii in bright galaxies. This is a complication we’ll ignore here by focusing on the outskirts of rotating galaxies where rotation curves are flat and dwarf spheroidals where stars never dominate. In both cases, we are in a limit where we can neglect the details of the stellar distribution: only the dark mass matters, or, in the case of MOND, only the total normal mass but not its detailed distribution (which does matter for the shape of a rotation curve, but not its flat amplitude).

Rather than worry about theory or the gory details of phase space, let’s just ask the data. How do we compare apples with apples? What is the factor β_c that makes V_o = β_c σ_* an equality?

One notices that the data for pressure supported dwarfs nicely parallels that for rotating galaxies. We estimate β_c by finding the shift that puts the dwarf spheroidals on the BTFR (on average). We only do this for the dwarfs that are not obviously affected by tidal effects whose velocity dispersions may not reflect the equilibrium gravitational potential. I have discussed this at great length in McGaugh & Wolf, so I refer the reader eager for more details there. Here I merely note that the exercise is meaningful only for those dwarfs that parallel the BTFR; it can’t apply to those that don’t regardless of the reason.

That caveat aside, this works quite well for β_c = 2.

*The BTFR plane with the outer velocity of dwarf spheroidals taken to be V_o = 2σ*.

The numerically inclined reader will note that 2 > √3. One would expect the latter for isotropic orbits, which we implicitly average over by using the data for all these dwarfs together. So the likely explanation for the larger values of β_c is that the outer velocities of rotation curves are measured at a larger radii than the velocity dispersions of dwarf spheroidals. The value of β_c is accounts for the different effective radii of measurement as illustrated by the rotation curves below.

The rotation curve of the gas rich Local Group dIrr WLM (left, Iorio et al.) and the equivalent circular velocity curve of the pressure supported dSph Leo I (right). The filled point represents the luminosity weighted circular speed V_c = √3 σ_* at the 3D half light radius where variation due to anisotropy is minimized (Wolf et al). The dotted lines illustrate how the uncertainty grows away from this point due to the compounding effects of anisotropy. The outer circular speed *V_o* is marked for both. Note that V_o > √3 σ_* simply because of the shape of the circular velocity curve, which has not yet reached the flat plateau where the velocity dispersion is measured.

Once said, this seems obvious. The velocity dispersions of dwarf spheroidals are measured by observing the Doppler shifts of individual member stars. This measurement is necessarily made where the stars are. In contrast, the flat portions of rotation curves are traced by atomic gas at radii that typically extend beyond the edge of the optical disk. So we should expect a difference; β_c = 2 quantifies it.

One small caveat is that in order to compare apples with apples, we have to adopt a mass-to-light ratio for the stars in dwarfs spheroidals in order to compare them with the combined mass of stars and gas in rotating galaxies. Indeed, the dwarf irregulars that overlap with the dwarf spheroidals in mass are made more of gas than stars, so there is always the risk of some systematic difference between the two mass scales. In the paper, we quantify the variation of β_c with the choice of M*/L. If you’re interested in that level of detail, you should read the paper.

I should also note that MOND predicts β_c = 2.12. Taken at face value, this implies that MOND prefers an average mass-to-light ratio slightly higher than what we assumed. This is well within the uncertainties, and we already know that MOND is the only theory capable of predicting the velocity dispersions of dwarf spheroidals in advance. We can always explain this after the fact with dark matter, which is what people generally do, often in apparent ignorance that MOND also correctly predicts which dwarfs they’ll have to invoke tidal disruption for. How such models can be considered satisfactory is quite beyond my capacity, but it does save one from the pain of having to critically reassess one’s belief system.

That’s all beyond the scope of the current paper. Here we just provide a nifty empirical result. If you want to make an apples-to-apples comparison of dwarf spheroidals with rotating dwarf irregulars, you will do well to assume V_o = 2σ_*.

Triton Station

A Blog About the Science and Sociology of Cosmology and Dark Matter

Category: Rotation curves