Leveling the Playing Field of Dwarf Galaxy Kinematics

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 Vc of a test particle by Vc = √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 βc that makes Vo = βc σ* an equality?

One notices that the data for pressure supported dwarfs nicely parallels that for rotating galaxies. We estimate βc 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 βc = 2.

The BTFR plane with the outer velocity of dwarf spheroidals taken to be Vo = 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 βc is that the outer velocities of rotation curves are measured at a larger radii than the velocity dispersions of dwarf spheroidals. The value of βc 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 Vc = √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 Vo is marked for both. Note that Vo > √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; βc = 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 βc 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 βc = 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 Vo = 2σ*.

Dwarf Satellite Galaxies. III. The dwarfs of Andromeda

Dwarf Satellite Galaxies. III. The dwarfs of Andromeda

Like the Milky Way, our nearest giant neighbor, Andromeda (aka M31), has several dozen dwarf satellite galaxies. A few of these were known and had measured velocity dispersions at the time of my work with Joe Wolf, as discussed previously. Also like the Milky Way, the number of known objects has grown rapidly in recent years – thanks in this case largely to the PAndAS survey.

PAndAS imaged the area around M31 and M33, finding many individual red giant stars. These trace out the debris from interactions and mergers as small dwarfs are disrupted and consumed by their giant host. They also pointed up the existence of previously unknown dwarf satellites.

M31fromPANDAS_ McC2012_EPJ_19_01003
The PAndAS survey field. Dwarf satellites are circled.

As the PAndAS survey started reporting the discovery of new dwarf satellites around Andromeda, it occurred to me that this provided the opportunity to make genuine a priori predictions. These are the gold standard of the scientific method. We could use the observed luminosity and size of the newly discovered dwarfs to predict their velocity dispersions.

I tried to do this for both ΛCDM and MOND. I will not discuss the ΛCDM case much, because it can’t really be done. But it is worth understanding why this is.

In ΛCDM, the velocity dispersion is determined by the dark matter halo. This has only a tenuous connection to the observed stars, so just knowing how big and bright a dwarf is doesn’t provide much predictive power about the halo. This can be seen from this figure by Tollerud et al (2011):

Virial mass of the dark matter halo as a function of galaxy luminosity. Dwarfs satellites reside in the wide colored band of low luminosities.

This graph is obtained by relating the number density of galaxies (an observed quantity) to that of the dark matter halos in which they reside (a theoretical construct). It is highly non-linear, deviating strongly from the one-to-one line we expected early on. There is no reason to expect this particular relation; it is imposed on us by the fact that the observed luminosity function of galaxies is rather flat while the predicted halo mass function is steep. Nowadays, this is usually called the missing satellite problem, but this is a misnomer as it pervades the field.

Addressing the missing satellites problem would be another long post, so lets just accept that the relation between mass and light has to follow something like that illustrated above. If a dwarf galaxy has a luminosity of a million suns, one can read off the graph that it should live in a dark halo with a mass of about 1010 M. One could use this to predict the velocity dispersion, but not very precisely, because there’s a big range corresponding to that luminosity (the bands in the figure). It could be as much as 1011 M or as little as 109 M. This corresponds to a wide range of velocity dispersions. This wide range is unavoidable because of the difference in the luminosity function and halo mass function. Small variations in one lead to big variations in the other, and some scatter in dark halo properties is unavoidable.

Consequently, we only have a vague range of expected velocity dispersions in ΛCDM. In practice, we never make this prediction. Instead, we compare the observed velocity dispersion to the luminosity and say “gee, this galaxy has a lot of dark matter” or “hey, this one doesn’t have much dark matter.” There’s no rigorously testable prior.

In MOND, what you see is what you get. The velocity dispersion has to follow from the observed stellar mass. This is straightforward for isolated galaxies: M* ∝ σ4 – this is essentially the equivalent of the Tully-Fisher relation for pressure supported systems. If we can estimate the stellar mass from the observed luminosity, the predicted velocity dispersion follows.

Many dwarf satellites are not isolated in the MONDian sense: they are subject to the external field effect (EFE) from their giant hosts. The over-under for whether the EFE applies is the point when the internal acceleration from all the stars of the dwarf on each other is equal to the external acceleration from orbiting the giant host. The amplitude of the discrepancy in MOND depends on how low the total acceleration is relative to the critical scale a0. The external field in effect adds some acceleration that wouldn’t otherwise be there, making the discrepancy less than it would be for an isolated object. This means that two otherwise identical dwarfs may be predicted to have different velocity dispersions is they are or are not subject to the EFE. This is a unique prediction of MOND that has no analog in ΛCDM.

It is straightforward to derive the equation to predict velocity dispersions in the extreme limits of isolated (aex ≪ ain < a0) or EFE dominated (ain ≪ aex < a0) objects. In reality, there are many objects for which ain ≈ aex, and no simple formula applies. In practice, we apply the formula that more nearly applies, and pray that this approximation is good enough.

There are many other assumptions and approximations that must be made in any theory: that an object is spherical, isotropic, and in dynamical equilibrium. All of these must fail at some level, but it is the last one that is the most serious concern. In the case of the EFE, one must also make the approximation that the object is in equilibrium at the current level of the external field. That is never true, as both the amplitude and the vector of the external field vary as a dwarf orbits its host. But it might be an adequate approximation if this variation is slow. In the case of a circular orbit, only the vector varies. In general the orbits are not known, so we make the instantaneous approximation and once again pray that it is good enough. There is a fairly narrow window between where the EFE becomes important and where we slip into the regime of tidal disruption, but lets plow ahead and see how far we can get, bearing in mind that the EFE is a dynamical variable of which we only have a snapshot.

To predict the velocity dispersion in the isolated case, all we need to know is the luminosity and a stellar mass-to-light ratio. Assuming the dwarfs of Andromeda to be old stellar populations, I adopted a V-band mass-to-light ratio of 2 give or take a factor of 2. That usually dominates the uncertainty, though the error in the distance can sometimes impact the luminosity at a level that impacts the prediction.

To predict the velocity dispersion in the EFE case, we again need the stellar mass, but now also need to know the size of the stellar system and the intensity of the external field to which it is subject. The latter depends on the mass of the host galaxy and the distance from it to the dwarf. This latter quantity is somewhat fraught: it is straightforward to measure the projected distance on the sky, but we need the 3D distance – how far in front or behind each dwarf is as well as its projected distance from the host. This is often a considerable contributor to the error budget. Indeed, some dwarfs may be inferred to be in the EFE regime for the low end of the range of adopted stellar mass-to-light ratio, and the isolated regime for the high end.

In this fashion, we predicted velocity dispersions for the dwarfs of Andromeda. We in this case were Milgrom and myself. I had never collaborated with him before, and prefer to remain independent. But I also wanted to be sure I got the details described above right. Though it wasn’t much work to make the predictions once the preliminaries were established, it was time consuming to collect and vet the data. As we were writing the paper, velocity dispersion measurements started to appear. People like Michelle Collins, Erik Tollerud, and Nicolas Martin were making follow-up observations, and publishing velocity dispersion for the objects we were making predictions for. That was great, but they were too good – they were observing and publishing faster than we could write!

Nevertheless, we managed to make and publish a priori predictions for 10 dwarfs before any observational measurements were published. We also made blind predictions for the other known dwarfs of Andromeda, and checked the predicted velocity dispersions against all measurements that we could find in the literature. Many of these predictions were quickly tested by on-going programs (i.e., people were out to measure velocity dispersions, whether we predicted them or not). Enough data rolled in that we were soon able to write a follow-up paper testing our predictions.

Nailed it. Good data were soon available to test the predictions for 8 of the 10* a priori cases. All 8 were consistent with our predictions. I was particularly struck by the case of And XXVIII, which I had called out as perhaps the best test. It was isolated, so the messiness of the EFE didn’t apply, and the uncertainties were low. Moreover, the predicted velocity dispersion was low – a good deal lower than broadly expected in ΛCDM: 4.3 km/s, with an uncertainty just under 1 km/s. Two independent observations were subsequently reported. One found 4.9 ± 1.6 km/s, the other 6.6 ± 2.1 km/s, both in good agreement within the uncertainties.

We made further predictions in the second paper as people had continued to discover new dwarfs. These also came true. Here is a summary plot for all of the dwarfs of Andromeda:

The velocity dispersions of the dwarf satellites of Andromeda. Each numbered box corresponds to one dwarf (x=1 is for And I and so on). Measured velocity dispersions have a number next to them that is the number of stars on which the measurement is based. MOND predictions are circles: green if isolated, open if the EFE applies. Points appear within each box in the order they appeared in the literature, from left to right. The vast majority of Andromeda’s dwarfs are consistent with MOND (large green circles). Two cases are ambiguous (large yellow circles), having velocity dispersions based only a few stars. Only And V appears to be problematic (large red circle).

MOND works well for And I, And II, And III, And VI, And VII, And IX, And X, And XI, And XII, And XIII, And XIV, And XV, And XVI, And XVII, And XVIII, And XIX, And XX, And XXI, And XXII, And XXIII, And XXIV, And XXV, And XXVIII, And XXIX, And XXXI, And XXXII, and And XXXIII. There is one problematic case: And V. I don’t know what is going on there, but note that systematic errors frequently happen in astronomy. It’d be strange if there weren’t at least one goofy case.

Nevertheless, the failure of And V could be construed as a falsification of MOND. It ought to work in every single case. But recall the discussion of assumptions and uncertainties above. Is falsification really the story these data tell?

We do have experience with various systematic errors. For example, we predicted that the isolated dwarfs spheroidal Cetus should have a velocity dispersion in MOND of 8.2 km/s. There was already a published measurement of 17 ± 2 km/s, so we reported that MOND was wrong in this case by over 3σ. Or at least we started to do so. Right before we submitted that paper, a new measurement appeared: 8.3 ± 1 km/s. This is an example of how the data can sometimes change by rather more than the formal error bars suggest is possible. In this case, I suspect the original observations lacked the spectral resolution to resolve the velocity dispersion. At any rate, the new measurement (8.3 km/s) was somewhat more consistent with our prediction (8.2 km/s).

The same predictions cannot even be made in ΛCDM. The velocity data can always be fit once they are in hand. But there is no agreed method to predict the velocity dispersion of a dwarf from its observed luminosity. As discussed above, this should not even be possible: there is too much scatter in the halo mass-stellar mass relation at these low masses.

An unsung predictive success of MOND absent from the graph above is And IV. When And IV was discovered in the general direction of Andromeda, it was assumed to be a new dwarf satellite – hence the name. Milgrom looked at the velocities reported for this object, and said it had to be a background galaxy. No way it could be a dwarf satellite – at least not in MOND. I see no reason why it couldn’t have been in ΛCDM. It is absent from the graph above, because it was subsequently confirmed to be much farther away (7.2 Mpc vs. 750 kpc for Andromeda).

The box for And XVII is empty because this system is manifestly out of equilibrium. It is more of a stellar stream than a dwarf, appearing as a smear in the PAndAS image rather than as a self-contained dwarf. I do not recall what the story with the other missing object (And VIII) is.

While writing the follow-up paper, I also noticed that there were a number of Andromeda dwarfs that were photometrically indistinguishable: basically the same in terms of size and stellar mass. But some were isolated while others were subject to the EFE. MOND predicts that the EFE cases should have lower velocity dispersion than the isolated equivalents.

The velocity dispersions of the dwarfs of Andromeda, highlighting photometrically matched pairs – dwarfs that should be indistinguishable, but aren’t because of the EFE.

And XXVIII (isolated) has a higher velocity dispersion than its near-twin And XVII (EFE). The same effect might be acting in And XVIII (isolated) and And XXV (EFE). This is clear if we accept the higher velocity dispersion measurement for And XVIII, but an independent measurements begs to differ. The former has more stars, so is probably more reliable, but we should be cautious. The effect is not clear in And XVI (isolated) and And XXI (EFE), but the difference in the prediction is small and the uncertainties are large.

An aggressive person might argue that the pairs of dwarfs is a positive detection of the EFE. I don’t think the data for the matched pairs warrant that, at least not yet. On the other hand, the appropriate use of the EFE was essential to all the predictions, not just the matched pairs.

The positive detection of the EFE is important, as it is a unique prediction of MOND. I see no way to tune ΛCDM galaxy simulations to mimic this effect. Of course, there was a  very recent time when it seemed impossible for them to mimic the isolated predictions of MOND. They claim to have come a long way in that regard.

But that’s what we’re stuck with: tuning ΛCDM to make it look like MOND. This is why a priori predictions are important. There is ample flexibility to explain just about anything with dark matter. What we can’t seem to do is predict the same things that MOND successfully predicts… predictions that are both quantitative and very specific. We’re not arguing that dwarfs in general live in ~15 or 30 km/s halos, as we must in ΛCDM. In MOND we can say this dwarf will have this velocity dispersion and that dwarf will have that velocity dispersion. We can distinguish between 4.9 and 7.3 km/s. And we can do it over and over and over. I see no way to do the equivalent in ΛCDM, just as I see no way to explain the acoustic power spectrum of the CMB in MOND.

This is not to say there are no problematic cases for MOND. Read, Walker, & Steger have recently highlighted the matched pair of Draco and Carina as an issue. And they are – though here I already have reason to suspect Draco is out of equilibrium, which makes it challenging to analyze. Whether it is actually out of equilibrium or not is a separate question.

I am not thrilled that we are obliged to invoke non-equilibrium effects in both theories. But there is a difference. Brada & Milgrom provided a quantitative criterion to indicate when this was an issue before I ran into the problem. In ΛCDM, the low velocity dispersions of objects like And XIX, XXI, XXV and Crater 2 came as a complete surprise despite having been predicted by MOND. Tidal disruption was only invoked after the fact – and in an ad hoc fashion. There is no way to know in advance which dwarfs are affected, as there is no criterion equivalent to that of Brada. We just say “gee, that’s a low velocity dispersion. Must have been disrupted.” That might be true, but it gives no explanation for why MOND predicted it in the first place – which is to say, it isn’t really an explanation at all.

I still do not understand is why MOND gets any predictions right if ΛCDM is the universe we live in, let alone so many. Shouldn’t happen. Makes no sense.

If this doesn’t confuse you, you are not thinking clearly.

*The other two dwarfs were also measured, but with only 4 stars in one and 6 in the other. These are too few for a meaningful velocity dispersion measurement.

Dwarf Satellite Galaxies. II. Non-equilibrium effects in ultrafaint dwarfs

Dwarf Satellite Galaxies. II. Non-equilibrium effects in ultrafaint dwarfs

I have been wanting to write about dwarf satellites for a while, but there is so much to tell that I didn’t think it would fit in one post. I was correct. Indeed, it was worse than I thought, because my own experience with low surface brightness (LSB) galaxies in the field is a necessary part of the context for my perspective on the dwarf satellites of the Local Group. These are very different beasts – satellites are pressure supported, gas poor objects in orbit around giant hosts, while field LSB galaxies are rotating, gas rich galaxies that are among the most isolated known. However, so far as their dynamics are concerned, they are linked by their low surface density.

Where we left off with the dwarf satellites, circa 2000, Ursa Minor and Draco remained problematic for MOND, but the formal significance of these problems was not great. Fornax, which had seemed more problematic, was actually a predictive success: MOND returned a low mass-to-light ratio for Fornax because it was full of young stars. The other known satellites, Carina, Leo I, Leo II, Sculptor, and Sextans, were all consistent with MOND.

The Sloan Digital Sky Survey resulted in an explosion in the number of satellites galaxies discovered around the Milky Way. These were both fainter and lower surface brightness than the classical dwarfs named above. Indeed, they were often invisible as objects in their own right, being recognized instead as groupings of individual stars that shared the same position in space and – critically – velocity. They weren’t just in the same place, they were orbiting the Milky Way together. To give short shrift to a long story, these came to be known as ultrafaint dwarfs.

Ultrafaint dwarf satellites have fewer than 100,000 stars. That’s tiny for a stellar system. Sometimes they had only a few hundred. Most of those stars are too faint to see directly. Their existence is inferred from a handful of red giants that are actually observed. Where there are a few red giants orbiting together, there must be a source population of fainter stars. This is a good argument, and it is likely true in most cases. But the statistics we usually rely on become dodgy for such small numbers of stars: some of the ultrafaints that have been reported in the literature are probably false positives. I have no strong opinion on how many that might be, but I’d be really surprised if it were zero.

Nevertheless, assuming the ultrafaints dwarfs are self-bound galaxies, we can ask the same questions as before. I was encouraged to do this by Joe Wolf, a clever grad student at UC Irvine. He had a new mass estimator for pressure supported dwarfs that we decided to apply to this problem. We used the Baryonic Tully-Fisher Relation (BTFR) as a reference, and looked at it every which-way. Most of the text is about conventional effects in the dark matter picture, and I encourage everyone to read the full paper. Here I’m gonna skip to the part about MOND, because that part seems to have been overlooked in more recent commentary on the subject.

For starters, we found that the classical dwarfs fall along the extrapolation of the BTFR, but the ultrafaint dwarfs deviate from it.

Fig. 1 from McGaugh & Wolf (2010, annotated). The BTFR defined by rotating galaxies (gray points) extrapolates well to the scale of the dwarf satellites of the Local Group (blue points are the classical dwarf satellites of the Milky Way; red points are satellites of Andromeda) but not to the ultrafaint dwarfs (green points). Two of the classical dwarfs also fall off of the BTFR: Draco and Ursa Minor.

The deviation is not subtle, at least not in terms of mass. The ultrataints had characteristic circular velocities typical of systems 100 times their mass! But the BTFR is steep. In terms of velocity, the deviation is the difference between the 8 km/s typically observed, and the ~3 km/s needed to put them on the line. There are a large number of systematic effects errors that might arise, and all act to inflate the characteristic velocity. See the discussion in the paper if you’re curious about such effects; for our purposes here we will assume that the data cannot simply be dismissed as the result of systematic errors, though one should bear in mind that they probably play a role at some level.

Taken at face value, the ultrafaint dwarfs are a huge problem for MOND. An isolated system should fall exactly on the BTFR. These are not isolated systems, being very close to the Milky Way, so the external field effect (EFE) can cause deviations from the BTFR. However, these are predicted to make the characteristic internal velocities lower than the isolated case. This may in fact be relevant for the red points that deviate a bit in the plot above, but we’ll return to that at some future point. The ultrafaints all deviate to velocities that are too high, the opposite of what the EFE predicts.

The ultrafaints falsify MOND! When I saw this, all my original confirmation bias came flooding back. I had pursued this stupid theory to ever lower surface brightness and luminosity. Finally, I had found where it broke. I felt like Darth Vader in the original Star Wars:

I have you now!

The first draft of my paper with Joe included a resounding renunciation of MOND. No way could it escape this!


I had this nagging feeling I was missing something. Darth should have looked over his shoulder. Should I?

Surely I had missed nothing. Many people are unaware of the EFE, just as we had been unaware that Fornax contained young stars. But not me! I knew all that. Surely this was it.

Nevertheless, the nagging feeling persisted. One part of it was sociological: if I said MOND was dead, it would be well and truly buried. But did it deserve to be? The scientific part of the nagging feeling was that maybe there had been some paper that addressed this, maybe a decade before… perhaps I’d better double check.

Indeed, Brada & Milgrom (2000) had run numerical simulations of dwarf satellites orbiting around giant hosts. MOND is a nonlinear dynamical theory; not everything can be approximated analytically. When a dwarf satellite is close to its giant host, the external acceleration of the dwarf falling towards its host can exceed the internal acceleration of the stars in the dwarf orbiting each other – hence the EFE. But the EFE is not a static thing; it varies as the dwarf orbits about, becoming stronger on closer approach. At some point, this variation becomes to fast for the dwarf to remain in equilibrium. This is important, because the assumption of dynamical equilibrium underpins all these arguments. Without it, it is hard to know what to expect short of numerically simulating each individual dwarf. There is no reason to expect them to remain on the equilibrium BTFR.

Brada & Milgrom suggested a measure to gauge the extent to which a dwarf might be out of equilibrium. It boils down to a matter of timescales. If the stars inside the dwarf have time to adjust to the changing external field, a quasi-static EFE approximation might suffice. So the figure of merit becomes the ratio of internal orbits per external orbit. If the stars inside a dwarf are swarming around many times for every time it completes an orbit around the host, then they have time to adjust. If the orbit of the dwarf around the host is as quick as the internal motions of the stars within the dwarf, not so much. At some point, a satellite becomes a collection of associated stars orbiting the host rather than a self-bound object in its own right.

Deviations from the BTFR (left) and the isophotal shape of dwarfs (right) as a function of the number of internal orbits a star at the half-light radius makes for every orbit a dwarf makes around its giant host (Fig. 7 of McGaugh & Wolf 2010).

Brada & Milgrom provide the formula to compute the ratio of orbits, shown in the figure above. The smaller the ratio, the less chance an object has to adjust, and the more subject it is to departures from equilibrium. Remarkably, the amplitude of deviation from the BTFR – the problem I could not understand initially – correlates with the ratio of orbits. The more susceptible a dwarf is to disequilibrium effects, the farther it deviated from the BTFR.

This completely inverted the MOND interpretation. Instead of falsifying MOND, the data now appeared to corroborate the non-equilibrium prediction of Brada & Milgrom. The stronger the external influence, the more a dwarf deviated from the equilibrium expectation. In conventional terms, it appeared that the ultrafaints were subject to tidal stirring: their internal velocities were being pumped up by external influences. Indeed, the originally problematic cases, Draco and Ursa Minor, fall among the ultrafaint dwarfs in these terms. They can’t be in equilibrium in MOND.

If the ultrafaints are out of equilibrium, the might show some independent evidence of this. Stars should leak out, distorting the shape of the dwarf and forming tidal streams. Can we see this?

A definite maybe:

The shapes of some ultrafaint dwarfs. These objects are so diffuse that they are invisible on the sky; their shape is illustrated by contours or heavily smoothed grayscale pseudo-images.

The dwarfs that are more subject to external influence tend to be more elliptical in shape. A pressure supported system in equilibrium need not be perfectly round, but one departing from equilibrium will tend to get stretched out. And indeed, many of the ultrafaints look Messed Up.

I am not convinced that all this requires MOND. But it certainly doesn’t falsify it. Tidal disruption can happen in the dark matter context, but it happens differently. The stars are buried deep inside protective cocoons of dark matter, and do not feel tidal effects much until most of the dark matter is stripped away. There is no reason to expect the MOND measure of external influence to apply (indeed, it should not), much less that it would correlate with indications of tidal disruption as seen above.

This seems to have been missed by more recent papers on the subject. Indeed, Fattahi et al. (2018) have reconstructed very much the chain of thought I describe above. The last sentence of their abstract states “In many cases, the resulting velocity dispersions are inconsistent with the predictions from Modified Newtonian Dynamics, a result that poses a possibly insurmountable challenge to that scenario.” This is exactly what I thought. (I have you now.) I was wrong.

Fattahi et al. are wrong for the same reasons I was wrong. They are applying equilibrium reasoning to a non-equilibrium situation. Ironically, the main point of the their paper is that many systems can’t be explained with dark matter, unless they are tidally stripped – i.e., the result of a non-equilibrium process. Oh, come on. If you invoke it in one dynamical theory, you might want to consider it in the other.

To quote the last sentence of our abstract from 2010, “We identify a test to distinguish between the ΛCDM and MOND based on the orbits of the dwarf satellites of the Milky Way and how stars are lost from them.” In ΛCDM, the sub-halos that contain dwarf satellites are expected to be on very eccentric orbits, with all the damage from tidal interactions with the host accruing during pericenter passage. In MOND, substantial damage may accrue along lower eccentricity orbits, leading to the expectation of more continuous disruption.

Gaia is measuring proper motions for stars all over the sky. Some of these stars are in the dwarf satellites. This has made it possible to estimate orbits for the dwarfs, e.g., work by Amina Helmi (et al!) and Josh Simon. So far, the results are definitely mixed. There are more dwarfs on low eccentricity orbits than I had expected in ΛCDM, but there are still plenty that are on high eccentricity orbits, especially among the ultrafaints. Which dwarfs have been tidally affected by interactions with their hosts is far from clear.

In short, reality is messy. It is going to take a long time to sort these matters out. These are early days.