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.

Fig1_annotated — 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:

darth-vader-i-have-you-now_1 — I have you now!

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

But…

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.

Fig7_annotated — 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:

Ell_D_wImages — 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.

Astronomical Acceleration Scales

A quick note to put the acceleration discrepancy in perspective.

The acceleration discrepancy, as Bekenstein called it, more commonly called the missing mass or dark matter problem, is the deviation of dynamics from those of Newton and Einstein. The quantity D is the amplitude of the discrepancy, basically the ratio of total mass to that which is visible. The need for dark matter – the discrepancy – only manifests at very low accelerations, of order 10^-10 m/s/s. That’s one part in 10¹¹ of what you feel standing on the Earth.

MDacc_wclusters_uptomergingBH — The mass discrepancy as a function of acceleration. There is no discrepancy (D=1) at high acceleration: everything is normal in the solar system and at the highest accelerations probed. The discrepancy only manifests at very low accelerations.

Astronomical data span enormous, indeed, astronomical, ranges. This is why astronomers so frequently use logarithmic plots. The abscissa in the plot above spans 25 orders of magnitude, from the lowest accelerations measured in the outskirts of galaxies to the highest conceivable on the surface of a neutron star on the brink of collapse into a black hole. If we put this on a linear scale, you’d see one point (the highest) and all the rest would be crammed into x=0.

Galileo established that the we live in a regime where the acceleration due to gravity is effectively constant; g = 9.8 m/s/s. This suffices to describe the trajectories of projectiles (like baseballs) familiar to everyday experience. At least is suffices to describe the gravity; air resistance plays a non-negligible role as well. But you don’t need Newton’s Universal Law of Gravity; you just need to know everything experiences a downward acceleration of one gee.

As we move to higher altitude and on into space, this ceases to suffice. As Newton taught us, the strength of the gravitational attraction between two bodies decreases as the distance between them increases. The constant acceleration recognized by Galileo was a special case of a more general phenomenon. The surface of the Earth is a [very nearly] constant distance from its center, so gee is [very nearly] constant. Get off the Earth, and that changes.

In the plot above, the acceleration we experience here on the surface of the Earth lands pretty much in the middle of the range known to astronomical observation. This is normal to us. The orbits of the planets in the solar system stretch to lower accelerations: the surface gravity of the Earth exceeds the centripetal force it takes to keep Earth in its orbit around the sun. This decreases outward in the solar system, with Neptune experiencing less than 10^-5 m/s/s in its orbit.

We understand the gravity in the solar system extraordinarily well. We’ve been watching the planets orbit for ages. The inner planets, in particular, are so well known that subtle effects have been known for ages. Most famous is the tiny excess precession of the perihelion of the orbit of Mercury, first noted by Le Verrier in 1859 but not satisfactorily* explained until Einstein applied General Relativity to the problem in 1916.

The solar system probes many decades of acceleration accurately, but there are many decades of phenomena beyond the reach of the solar system, both to higher and lower accelerations. Two objects orbiting one another intensely enough for the energy loss due to the emission of gravitational waves to have a measurable effect on their orbit are the two neutron stars that compose the binary pulsar of Hulse & Taylor. Their orbit is highly eccentric, pulling an acceleration of about 270 m/s/s at periastron (closest passage). The gravitational dynamics of the system are extraordinarily well understood, and Hulse & Taylor were awarded the 1993 Nobel prize in physics for this observation that indirectly corroborated the existence of gravitational waves.

ghostbusters-20090702101358857 — The mass-energy tensor was dancing a monster jig as the fabric of space-time was rent asunder, I can tell you!

Direct detection of gravitational waves was first achieved by LIGO in 2015 (the 2017 Nobel prize). The source of these waves was the merger of a binary pair of black holes, a calamity so intense that it converted the equivalent of 3 solar masses into the energy carried away as gravitational waves. Imagine two 30 solar mass black holes orbiting each other a few hundred km apart 75 times per second just before merging – that equates to a centripetal acceleration of nearly 10¹¹ m/s/s.

We seem to understand gravity well in this regime.

The highest acceleration illustrated in the figure above is the maximum surface gravity of a neutron star, which is just a hair under 10¹³ m/s/s. Anything more than this collapses to a black hole. The surface of a neutron star is not a place that suffers large mountains to exist, even if by “large” you mean “ant sized.” Good luck walking around in an exoskeleton there! Micron scale crustal adjustments correspond to monster starquakes.

High-end gravitational accelerations are 20 orders of magnitude removed from where the acceleration discrepancy appears. Dark matter is a problem restricted to the regime of tiny accelerations, of order 1 Angstrom/s/s. That isn’t much, but it is roughly what holds a star in its orbit within a galaxy. Sometimes less.

Galaxies show a large and clear acceleration discrepancy. The mob of black points is the radial acceleration relation, compressed to fit on the same graph with the high acceleration phenomena. Whatever happens, happens suddenly at this specific scale.

I also show clusters of galaxies, which follow a similar but offset acceleration relation. The discrepancy sets in a littler earlier for them (and with more scatter, but that may simply be a matter of lower precision). This offset from galaxies is a small matter on the scale considered here, but it is a serious one if we seek to modify dynamics at a universal acceleration scale. Depending on how one chooses to look at this aspect of the problem, the data for clusters are either tantalizingly close to the [far superior] data for galaxies, or they are impossibly far removed. Regardless of which attitude proves to be less incorrect, it is clear that the missing mass phenomena is restricted to low accelerations. Everything is normal until we reach the lowest decade or two of accelerations probed by current astronomical data – and extragalactic data are the only data that test gravity in this regime.

We have no other data that probe the very low acceleration regime. The lowest acceleration probe we have with solar system accuracy is from the Pioneer spacecraft. These suffer an anomalous acceleration whose source was debated for many years. Was it some subtle asymmetry in the photon pressure due thermal radiation from the spacecraft? Or new physics?

Though the effect is tiny (it is shown in the graph above, but can you see it?), it would be enormous for a MOND effect. MOND asymptotes to Newton at high accelerations. Despite the many AU Pioneer has put between itself and home, it is still in a regime 4 orders of magnitude above where MOND effects kick in. This would only be perceptible if the asymptotic approach to the Newtonian regime were incredibly slow. So slow, in fact, that it should be perceptible in the highly accurate data for the inner planets. Nowadays, the hypothesis of asymmetric photon pressure is widely accepted, which just goes to show how hard it is to construct experiments to test MOND. Not only do you have to get far enough away from the sun to probe the MOND regime (about a tenth of a light-year), but you have to control for how hard itty-bitty photons push on your projectile.

That said, it’d still be great experiment. Send a bunch of test particles out of the solar system at high speed on a variety of ballistic trajectories. They needn’t be much more than bullets with beacons to track them by. It would take a heck of a rocket to get them going fast enough to return an answer within a lifetime, but rocket scientists love a challenge to go real fast.

*Le Verrier suggested that the effect could be due to a new planet, dubbed Vulcan, that orbited the sun interior to the orbit of Mercury. In the half century prior to Einstein settling the issue, there were many claims to detect this Victorian form of dark matter.

Dwarf Satellite Galaxies and Low Surface Brightness Galaxies in the Field. I.

The Milky Way and its nearest giant neighbor Andromeda (M31) are surrounded by a swarm of dwarf satellite galaxies. Aside from relatively large beasties like the Large Magellanic Cloud or M32, the majority of these are the so-called dwarf spheroidals. There are several dozen examples known around each giant host, like the Fornax dwarf pictured above.

Dwarf Spheroidal (dSph) galaxies are ellipsoidal blobs devoid of gas that typically contain a million stars, give or take an order of magnitude. Unlike globular clusters, that may have a similar star count, dSphs are diffuse, with characteristic sizes of hundreds of parsecs (vs. a few pc for globulars). This makes them among the lowest surface brightness systems known.

This subject has a long history, and has become a major industry in recent years. In addition to the “classical” dwarfs that have been known for decades, there have also been many comparatively recent discoveries, often of what have come to be called “ultrafaint” dwarfs. These are basically dSphs with luminosities less than 100,000 suns, sometimes being comprised of as little as a few hundred stars. New discoveries are being made still, and there is reason to hope that the LSST will discover many more. Summed up, the known dwarf satellites are proverbial drops in the bucket compared to their giant hosts, which contain hundreds of billions of stars. Dwarfs could rain in for a Hubble time and not perturb the mass budget of the Milky Way.

Nevertheless, tiny dwarf Spheroidals are excellent tests of theories like CDM and MOND. Going back to the beginning, in the early ’80s, Milgrom was already engaged in a discussion about the predictions of his then-new theory (before it was even published) with colleagues at the IAS, where he had developed the idea during a sabbatical visit. They were understandably skeptical, preferring – as many still do – to believe that some unseen mass was the more conservative hypothesis. Dwarf spheroidals came up even then, as their very low surface brightness meant low acceleration in MOND. This in turn meant large mass discrepancies. If you could measure their dynamics, they would have large mass-to-light ratios. Larger than could be explained by stars conventionally, and larger than the discrepancies already observed in bright galaxies like Andromeda.

This prediction of Milgrom’s – there from the very beginning – is important because of how things change (or don’t). At that time, Scott Tremaine summed up the contrasting expectation of the conventional dark matter picture:

“There is no reason to expect that dwarfs will have more dark matter than bright galaxies.” *

This was certainly the picture I had in my head when I first became interested in low surface brightness (LSB) galaxies in the mid-80s. At that time I was ignorant of MOND; my interest was piqued by the argument of Disney that there could be a lot of as-yet undiscovered LSB galaxies out there, combined with my first observing experiences with the then-newfangled CCD cameras which seemed to have a proclivity for making clear otherwise hard-to-see LSB features. At the time, I was interested in finding LSB galaxies. My interest in what made them rotate came later.

The first indication, to my knowledge, that dSph galaxies might have large mass discrepancies was provided by Marc Aaronson in 1983. This tentative discovery was hugely important, but the velocity dispersion of Draco (one of the “classical” dwarfs) was based on only 3 stars, so was hardly definitive. Nevertheless, by the end of the ’90s, it was clear that large mass discrepancies were a defining characteristic of dSphs. Their conventionally computed M/L went up systematically as their luminosity declined. This was not what we had expected in the dark matter picture, but was, at least qualitatively, in agreement with MOND.

My own interests had focused more on LSB galaxies in the field than on dwarf satellites like Draco. Greg Bothun and Jim Schombert had identified enough of these to construct a long list of LSB galaxies that served as targets my for Ph.D. thesis. Unlike the pressure-supported ellipsoidal blobs of stars that are the dSphs, the field LSBs we studied were gas rich, rotationally supported disks – mostly late type galaxies (Sd, Sm, & Irregulars). Regardless of composition, gas or stars, low surface density means that MOND predicts low acceleration. This need not be true conventionally, as the dark matter can do whatever the heck it wants. Though I was blissfully unaware of it at the time, we had constructed the perfect sample for testing MOND.

Having studied the properties of our sample of LSB galaxies, I developed strong ideas about their formation and evolution. Everything we had learned – their blue colors, large gas fractions, and low star formation rates – suggested that they evolved slowly compared to higher surface brightness galaxies. Star formation gradually sputtered along, having a hard time gathering enough material to make stars in their low density interstellar media. Perhaps they even formed late, an idea I took a shining to in the early ’90s. This made two predictions: field LSB galaxies should be less strongly clustered than bright galaxies, and should spin slower at a given mass.

The first prediction follows because the collapse time of dark matter halos correlates with their larger scale environment. Dense things collapse first and tend to live in dense environments. If LSBs were low surface density because they collapsed late, it followed that they should live in less dense environments.

I didn’t know how to test this prediction. Fortunately, fellow postdoc and office mate in Cambridge at the time, Houjun Mo, did. It came true. The LSB galaxies I had been studying were clustered like other galaxies, but not as strongly. This was exactly what I expected, and I thought sure we were on to something. All that remained was to confirm the second prediction.

At the time, we did not have a clear idea of what dark matter halos should be like. NFW halos were still in the future. So it seemed reasonable that late forming halos should have lower densities (lower concentrations in the modern terminology). More importantly, the sum of dark and luminous density was certainly less. Dynamics follow from the distribution of mass as Velocity² ∝ Mass/Radius. For a given mass, low surface brightness galaxies had a larger radius, by construction. Even if the dark matter didn’t play along, the reduction in the concentration of the luminous mass should lower the rotation velocity.

Indeed, the standard explanation of the Tully-Fisher relation was just this. Aaronson, Huchra, & Mould had argued that galaxies obeyed the Tully-Fisher relation because they all had essentially the same surface brightness (Freeman’s law) thereby taking variation in the radius out of the equation: galaxies of the same mass all had the same radius. (If you are a young astronomer who has never heard of Freeman’s law, you’re welcome.) With our LSB galaxies, we had a sample that, by definition, violated Freeman’s law. They had large radii for a given mass. Consequently, they should have lower rotation velocities.

Up to that point, I had not taken much interest in rotation curves. In contrast, colleagues at the University of Groningen were all about rotation curves. Working with Thijs van der Hulst, Erwin de Blok, and Martin Zwaan, we set out to quantify where LSB galaxies fell in relation to the Tully-Fisher relation. I confidently predicted that they would shift off of it – an expectation shared by many at the time. They did not.

BTFSBallwlinessmall — The Tully-Fisher relation: disk mass vs. flat rotation speed (circa 1996). Galaxies are binned by surface brightness with the highest surface brightness galaxies marked red and the lowest blue. The lines show the expected shift following the argument of Aaronson et al. Contrary to this expectation, galaxies of all surface brightnesses follow the same Tully-Fisher relation.

I was flummoxed. My prediction was wrong. That of Aaronson et al. was wrong. Poking about the literature, everyone who had made a clear prediction in the conventional context was wrong. It made no sense.

I spent months banging my head against the wall. One quick and easy solution was to blame the dark matter. Maybe the rotation velocity was set entirely by the dark matter, and the distribution of luminous mass didn’t come into it. Surely that’s what the flat rotation velocity was telling us? All about the dark matter halo?

Problem is, we measure the velocity where the luminous mass still matters. In galaxies like the Milky Way, it matters quite a lot. It does not work to imagine that the flat rotation velocity is set by some property of the dark matter halo alone. What matters to what we measure is the combination of luminous and dark mass. The luminous mass is important in high surface brightness galaxies, and progressively less so in lower surface brightness galaxies. That should leave some kind of mark on the Tully-Fisher relation, but it doesn’t.

CRVfresid — Residuals from the Tully-Fisher relation as a function of size at a given mass. Compact galaxies are to the left, diffuse ones to the right. The red dashed line is what Newton predicts: more compact galaxies should rotate faster at a given mass. Fundamental physics? Tully-Fisher don’t care. Tully-Fisher don’t give a sh*t.

I worked long and hard to understand this in terms of dark matter. Every time I thought I had found the solution, I realized that it was a tautology. Somewhere along the line, I had made an assumption that guaranteed that I got the answer I wanted. It was a hopeless fine-tuning problem. The only way to satisfy the data was to have the dark matter contribution scale up as that of the luminous mass scaled down. The more stretched out the light, the more compact the dark – in exact balance to maintain zero shift in Tully-Fisher.

This made no sense at all. Over twenty years on, I have yet to hear a satisfactory conventional explanation. Most workers seem to assert, in effect, that “dark matter does it” and move along. Perhaps they are wise to do so.

repomanfoxharris — Working on the thing can drive you mad.

As I was struggling with this issue, I happened to hear a talk by Milgrom. I almost didn’t go. “Modified gravity” was in the title, and I remember thinking, “why waste my time listening to that nonsense?” Nevertheless, against my better judgement, I went. Not knowing that anyone in the audience worked on either LSB galaxies or Tully-Fisher, Milgrom proceeded to derive the MOND prediction:

“The asymptotic circular velocity is determined only by the total mass of the galaxy: V_f⁴ = a₀GM.”

In a few lines, he derived rather trivially what I had been struggling to understand for months. The lack of surface brightness dependence in Tully-Fisher was entirely natural in MOND. It falls right out of the modified force law, and had been explicitly predicted over a decade before I struggled with the problem.

I scraped my jaw off the floor, determined to examine this crazy theory more closely. By the time I got back to my office, cognitive dissonance had already started to set it. Couldn’t be true. I had more pressing projects to complete, so I didn’t think about it again for many moons.

When I did, I decided I should start by reading the original MOND papers. I was delighted to find a long list of predictions, many of them specifically to do with surface brightness. We had just collected fresh data on LSB galaxies, which provided a new window on the low acceleration regime. I had the data to finally falsify this stupid theory.

Or so I thought. As I went through the list of predictions, my assumption that MOND had to be wrong was challenged by each item. It was barely an afternoon’s work: check, check, check. Everything I had struggled for months to understand in terms of dark matter tumbled straight out of MOND.

I was faced with a choice. I knew this would be an unpopular result. I could walk away and simply pretend I had never run across it. That’s certainly how it had been up until then: I had been blissfully unaware of MOND and its perniciously successful predictions. No need to admit otherwise.

Had I realized just how unpopular it would prove to be, maybe that would have been the wiser course. But even contemplating such a course felt criminal. I was put in mind of Paul Gerhardt’s admonition for intellectual honesty:

“When a man lies, he murders some part of the world.”

Ignoring what I had learned seemed tantamount to just that. So many predictions coming true couldn’t be an accident. There was a deep clue here; ignoring it wasn’t going to bring us closer to the truth. Actively denying it would be an act of wanton vandalism against the scientific method.

Still, I tried. I looked long and hard for reasons not to report what I had found. Surely there must be some reason this could not be so?

Indeed, the literature provided many papers that claimed to falsify MOND. To my shock, few withstood critical examination. Commonly a straw man representing MOND was falsified, not MOND itself. At a deeper level, it was implicitly assumed that any problem for MOND was an automatic victory for dark matter. This did not obviously follow, so I started re-doing the analyses for both dark matter and MOND. More often than not, I found either that the problems for MOND were greatly exaggerated, or that the genuinely problematic cases were a problem for both theories. Dark matter has more flexibility to explain outliers, but outliers happen in astronomy. All too often the temptation was to refuse to see the forest for a few trees.

The first MOND analysis of the classical dwarf spheroidals provides a good example. Completed only a few years before I encountered the problem, these were low surface brightness systems that were deep in the MOND regime. These were gas poor, pressure supported dSph galaxies, unlike my gas rich, rotating LSB galaxies, but the critical feature was low surface brightness. This was the most directly comparable result. Better yet, the study had been made by two brilliant scientists (Ortwin Gerhard & David Spergel) whom I admire enormously. Surely this work would explain how my result was a mere curiosity.

Indeed, reading their abstract, it was clear that MOND did not work for the dwarf spheroidals. Whew: LSB systems where it doesn’t work. All I had to do was figure out why, so I read the paper.

As I read beyond the abstract, the answer became less and less clear. The results were all over the map. Two dwarfs (Sculptor and Carina) seemed unobjectionable in MOND. Two dwarfs (Draco and Ursa Minor) had mass-to-light ratios that were too high for stars, even in MOND. That is, there still appeared to be a need for dark matter even after MOND had been applied. One the flip side, Fornax had a mass-to-light ratio that was too low for the old stellar populations assumed to dominate dwarf spheroidals. Results all over the map are par for the course in astronomy, especially for a pioneering attempt like this. What were the uncertainties?

Milgrom wrote a rebuttal. By then, there were measured velocity dispersions for two more dwarfs. Of these seven dwarfs, he found that

“within just the quoted errors on the velocity dispersions and the luminosities, the MOND M/L values for all seven dwarfs are perfectly consistent with stellar values, with no need for dark matter.”

Well, he would say that, wouldn’t he? I determined to repeat the analysis and error propagation.

MdB98bFig8_dSph — Mass-to-light ratios determined with MOND for eight dwarf spheroidals (named, as published in McGaugh & de Blok 1998). The various symbols refer to different determinations. Mine are the solid circles. The dashed lines show the plausible range for stellar populations.

The net result: they were both right. M/L was still too high for Draco and Ursa Minor, and still too low for Fornax. But this was only significant at the 2σ level, if that – hardly enough to condemn a theory. Carina, Leo I, Leo II, Sculptor, and Sextans all had fairly reasonable mass-to-light ratios. The voting is different now. Instead of going 2 for 5 as Gerhard & Spergel found, MOND was now 5 for 8. One could choose to obsess about the outliers, or one could choose to see a more positive pattern. Either a positive or a negative spin could be put on this result. But it was clearly more positive than the first attempt had indicated.

The mass estimator in MOND scales as the fourth power of velocity (or velocity dispersion in the case of isolated dSphs), so the too-high M*/L of Draco and Ursa Minor didn’t disturb me too much. A small overestimation of the velocity dispersion would lead to a large overestimation of the mass-to-light ratio. Just about every systematic uncertainty one can think of pushes in this direction, so it would be surprising if such an overestimate didn’t happen once in a while.

Given this, I was more concerned about the low M*/L of Fornax. That was weird.

Up until that point (1998), we had been assuming that the stars in dSphs were all old, like those in globular clusters. That corresponds to a high M*/L, maybe 3 in solar units in the V-band. Shortly after this time, people started to look closely at the stars in the classical dwarfs with the Hubble. Low and behold, the stars in Fornax were surprisingly young. That means a low M*/L, 1 or less. In retrospect, MOND was trying to tell us that: it returned a low M*/L for Fornax because the stars there are young. So what was taken to be a failing of the theory was actually a predictive success.

Hmm.

And Gee. This is a long post. There is a lot more to tell, but enough for now.

*I have a long memory, but it is not perfect. I doubt I have the exact wording right, but this does accurately capture the sentiment from the early ’80s when I was an undergraduate at MIT and Scott Tremaine was on the faculty there.

A Precise Milky Way

The Milky Way Galaxy in which we live seems to be a normal spiral galaxy. But it can be hard to tell. Our perspective from within it precludes a “face-on” view like the picture above, which combines some real data with a lot of artistic liberty. Some local details we can measure in extraordinary detail, but the big picture is hard. Just how big is the Milky Way? The absolute scale of our Galaxy has always been challenging to measure accurately from our spot within it.

For some time, we have had a remarkably accurate measurement of the angular speed of the sun around the center of the Galaxy provided by the proper motion of Sagittarius A*. Sgr A* is the radio source associated with the supermassive black hole at the center of the Galaxy. By watching how it appears to move across the sky, Reid & Brunthaler found our relative angular speed to be 6.379 milliarcseconds/year. That’s a pretty amazing measurement: a milliarcsecond is one one-thousandth of one arcsecond, which is one sixtieth of one arcminute, which is one sixtieth of a degree. A pretty small angle.

The proper motion of an object depends on the ratio of its speed to its distance. So this high precision measurement does not itself tell us how big the Milky Way is. We could be far from the center and moving fast, or close and moving slow. Close being a relative term when our best estimates of the distance to the Galactic center hover around 8 kpc (26,000 light-years), give or take half a kpc.

This situation has recently improved dramatically thanks to the Gravity collaboration. They have observed the close passage of a star (S2) past the central supermassive black hole Sgr A*. Their chief interest is in the resulting relativistic effects: gravitational redshift and Schwarzschild precession, which provide a test of General Relativity. Unsurprisingly, it passes with flying colors.

As a consequence of their fitting process, we get for free some other interesting numbers. The mass of the central black hole is 4.1 million solar masses, and the distance to it is 8.122 kpc. The quoted uncertainty is only 31 pc. That’s parsecs, not kiloparsecs. Previously, I had seen credible claims that the distance to the Galactic center was 7.5 kpc. Or 7.9. Or 8.3 Or 8.5. There was a time when it was commonly thought to be about 10 kpc, i.e., we weren’t even sure what column the first digit belonged in. Now we know it to several decimal places. Amazing.

Knowing both the Galactocentric distance and the proper motion of Sgr A* nails down the relative speed of the sun: 245.6 km/s. Of this, 12.2 km/s is “solar motion,” which is how much the sun deviates from a circular orbit. Correcting for this gives us the circular speed of an imaginary test particle orbiting at the sun’s location: 233.3 km/s, accurate to 1.4 km/s.

The distance and circular speed at the solar circle are the long sought Galactic Constants. These specify the scale of the Milky Way. Knowing them also pins down the rotation curve interior to the sun. This is well constrained by the “terminal velocities,” which provide a precise mapping of relative speeds, but need the Galactic Constants for an absolute scale.

A few years ago, I built a model Milky Way rotation curve that fit the terminal velocity data. What I was interested in then was to see if I could use the radial acceleration relation (RAR) to infer the mass distribution of the Galactic disk. The answer was yes. Indeed, it makes for a clear improvement over the traditional approach of assuming a purely exponential disk in the sense that the kinematically inferred bumps and wiggles in the rotation curve correspond to spiral arms known from star counts, as in external spiral galaxies.

Now that the Galactic constants are Known, it seems worth updating the model. This results in the surface density profile

with the corresponding rotation curve

MW_2018_VR — The rotation curve of the Milky Way as traced by terminal velocities in the first and fourth quadrants (red and blue points). The solid line is a model that matches this rotation curve. The dashed and dotted lines are the rotation curves of the baryonic and inferred dark matter components. Yellow bands show the effect of varying the stellar mass by 5%.

The model data are available from the Milky Way section of my model pages.

Finding a model that matches both the terminal velocity and the highly accurate Galactic constants is no small feat. Indeed, I worried it was impossible: the speed at the solar circle is down to 233 km/s from a high of 249 km/s just a couple of kpc interior. This sort of variation is possible, but it requires a ring of mass outside the sun. This appears to be the effect of the Perseus spiral arm.

For the new Galactic constants and the current calibration of the RAR, the stellar mass of the Milky Way works out to just under 62 billion solar masses. The largest uncertainty in this is from the asymmetry in the terminal velocities, which are slightly different in the first and fourth quadrants. This is likely a real asymmetry in the mass distribution of the Milky Way. Treating it as an uncertainty, the range of variation corresponds to about 5% up or down in stellar mass.

With the stellar mass determined in this way, we can estimate the local density of dark matter. This is the critical number that is needed for experimental searches: just how much of the stuff should we expect? The answer is very precise: 0.257 GeV per cubic cm. This a bit less than is usually assumed, which makes it a tiny bit harder on the hard-working experimentalists.

The accuracy of the dark matter density is harder to assess. The biggest uncertainty is that in stellar mass. We known the total radial force very well now, but how much is due to stars, and how much to dark matter? (or whatever). The RAR provides a unique method for constraining the stellar contribution, and does so well enough that there is very little formal uncertainty in the dark matter density. This, however, depends on the calibration of the RAR, which itself is subject to systematic uncertainty at the 20% level. This is not as bad as it sounds, because a recalibration of the RAR changes its shape in a way that tends to trade off with stellar mass while not much changing the implied dark matter density. So even with these caveats, this is the most accurate measure of the dark matter density to date.

This is all about the radial force. One can also measure the force perpendicular to the disk. This vertical force implies about twice the dark matter density. This may be telling us something about the shape of the dark matter halo – rather than being spherical as usually assumed, it might be somewhat squashed. It is easy to say that, but it seems a strange circumstance: the stars provide most of the restoring force in the vertical direction, and apparently dominate the radial force. Subtracting off the stellar contribution is thus a challenging task: the total force isn’t much greater than that from the stars alone. Subtracting one big number from another to measure a small one is fraught with peril: the uncertainties tend to blow up in your face.

Returning to the Milky Way, it seems in all respects to be a normal spiral galaxy. With the stellar mass found here, we can compare it to other galaxies in scaling relations like Tully-Fisher. It does not stand out from the crowd: our home is a fairly normal place for this time in the Universe.

TFMW — The stellar mass Tully-Fisher relation with the Milky Way shown as the red star. It is a typical spiral galaxy.

It is possible to address many more details with a model like this. See the original!

A brief history of the acceleration discrepancy

As soon as I wrote it, I realized that the title is much more general than anything that can be fit in a blog post. Bekenstein argued long ago that the missing mass problem should instead be called the acceleration discrepancy, because that’s what it is – a discrepancy that occurs in conventional dynamics at a particular acceleration scale. So in that sense, it is the entire history of dark matter. For that, I recommend the excellent book The Dark Matter Problem: A Historical Perspective by Bob Sanders.

Here I mean more specifically my own attempts to empirically constrain the relation between the mass discrepancy and acceleration. Milgrom introduced MOND in 1983, no doubt after a long period of development and refereeing. He anticipated essentially all of what I’m going to describe. But not everyone is eager to accept MOND as a new fundamental theory, and often suffer from a very human tendency to confuse fact and theory. So I have gone out of my way to demonstrate what is empirically true in the data – facts – irrespective of theoretical interpretation (MOND or otherwise).

What is empirically true, and now observationally established beyond a reasonable doubt, is that the mass discrepancy in rotating galaxies correlates with centripetal acceleration. The lower the acceleration, the more dark matter one appears to need. Or, as Bekenstein might have put it, the amplitude of the acceleration discrepancy grows as the acceleration itself declines.

Bob Sanders made the first empirical demonstration that I am aware of that the mass discrepancy correlates with acceleration. In a wide ranging and still relevant 1990 review, he showed that the amplitude of the mass discrepancy correlated with the acceleration at the last measured point of a rotation curve. It did not correlate with radius.

AccDisc_Sanders1990 — The acceleration discrepancy from Sanders (1990).

I was completely unaware of this when I became interested in the problem a few years later. I wound up reinventing the very same term – the mass discrepancy, which I defined as the ratio of dynamically measured mass to that visible in baryons: D = M_tot/M_bar. When there is no dark matter, M_tot = M_bar and D = 1.

My first demonstration of this effect was presented at a conference at Rutgers in 1998. This considered the mass discrepancy at every radius and every acceleration within all the galaxies that were available to me at that time. Though messy, as is often the case in extragalactic astronomy, the correlation was clear. Indeed, this was part of a broader review of galaxy formation; the title, abstract, and much of the substance remains relevant today.

MD1998_constantML — The mass discrepancy – the ratio of dynamically measured mass to that visible in luminous stars and gas – as a function of centripetal acceleration. Each point is a measurement along a rotation curve; two dozen galaxies are plotted together. A constant mass-to-light ratio is assumed for all galaxies.

I spent much of the following five years collecting more data, refining the analysis, and sweating the details of uncertainties and systematic instrumental effects. In 2004, I published an extended and improved version, now with over 5 dozen galaxies.

MDaccpoponly — One panel from Fig. 5 of McGaugh (2004). The mass discrepancy is plotted against the acceleration predicted by the baryons (in units of km² s² kpc^-1).

Here I’ve used a population synthesis model to estimate the mass-to-light ratio of the stars. This is the only unknown; everything else is measured. Note that the vast majority galaxies land on top of each other. There are a few that do not, as you can perceive in the parallel sets of points offset from the main body. But that happens in only a few cases, as expected – no population model is perfect. Indeed, this one was surprisingly good, as the vast majority of the individual galaxies are indistinguishable in the pile that defines the main relation.

I explored the how the estimation of the stellar mass-to-light ratio affected this mass discrepancy-acceleration relation in great detail in the 2004 paper. The details differ with the choice of estimator, but the bottom line was that the relation persisted for any plausible choice. The relation exists. It is an empirical fact.

At this juncture, further improvement was no longer limited by rotation curve data, which is what we had been working to expand through the early ’00s. Now it was the stellar mass. The measurement of stellar mass was based on optical measurements of the luminosity distribution of stars in galaxies. These are perfectly fine data, but it is hard to map the starlight that we measured to the stellar mass that we need for this relation. The population synthesis models were good, but they weren’t good enough to avoid the occasional outlier, as can be seen in the figure above.

One thing the models all agreed on (before they didn’t, then they did again) was that the near-infrared would provide a more robust way of mapping stellar mass than the optical bands we had been using up till then. This was the clear way forward, and perhaps the only hope for improving the data further. Fortunately, technology was keeping pace. Around this time, I became involved in helping the effort to develop the NEWFIRM near-infrared camera for the national observatories, and NASA had just launched the Spitzer space telescope. These were the right tools in the right place at the right time. Ultimately, the high accuracy of the deep images obtained from the dark of space by Spitzer at 3.6 microns were to prove most valuable.

Jim Schombert and I spent much of the following decade observing in the near-infrared. Many other observers were doing this as well, filling the Spitzer archive with useful data while we concentrated on our own list of low surface brightness galaxies. This paragraph cannot suffice to convey the long term effort and enormity of this program. But by the mid-teens, we had accumulated data for hundreds of galaxies, including all those for which we also had rotation curves and HI observations. The latter had been obtained over the course of decades by an entire independent community of radio observers, and represent an integrated effort that dwarfs our own.

On top of the observational effort, Jim had been busy building updated stellar population models. We have a sophisticated understanding of how stars work, but things can get complicated when you put billions of them together. Nevertheless, Jim’s work – and that of a number of independent workers – indicated that the relation between Spitzer’s 3.6 micron luminosity measurements and stellar mass should be remarkably simple – basically just a constant conversion factor for nearly all star forming galaxies like those in our sample.

Things came together when Federico Lelli joined Case Western as a postdoc in 2014. He had completed his Ph.D. in the rich tradition of radio astronomy, and was the perfect person to move the project forward. After a couple more years of effort, curating the rotation curve data and building mass models from the Spitzer data, we were in the position to build the relation for over a dozen dozen galaxies. With all the hard work done, making the plot was a matter of running a pre-prepared computer script.

Federico ran his script. The plot appeared on his screen. In a stunned voice, he called me into his office. We had expected an improvement with the Spitzer data – hence the decade of work – but we had also expected there to be a few outliers. There weren’t. Any.

All. the. galaxies. fell. right. on. top. of. each. other.

rar — The radial acceleration relation. The centripetal acceleration measured from rotation curves is plotted against that predicted by the observed baryons. 2693 points from 153 distinct galaxies are plotted together (bluescale); individual galaxies do not distinguish themselves in this plot. Indeed, the width of the scatter (inset) is entirely explicable by observational uncertainties and the expected scatter in stellar mass-to-light ratios. From McGaugh et al. (2016).

This plot differs from those above because we had decided to plot the measured acceleration against that predicted by the observed baryons so that the two axes would be independent. The discrepancy, defined as the ratio, depended on both. D is essentially the ratio of the y-axis to the x-axis of this last plot, dividing out the unity slope where D = 1.

This was one of the most satisfactory moments of my long career, in which I have been fortunate to have had many satisfactory moments. It is right up there with the eureka moment I had that finally broke the long-standing loggerhead about the role of selection effects in Freeman’s Law. (Young astronomers – never heard of Freeman’s Law? You’re welcome.) Or the epiphany that, gee, maybe what we’re calling dark matter could be a proxy for something deeper. It was also gratifying that it was quickly recognized as such, with many of the colleagues I first presented it to saying it was the highlight of the conference where it was first unveiled.

Regardless of the ultimate interpretation of the radial acceleration relation, it clearly exists in the data for rotating galaxies. The discrepancy appears at a characteristic acceleration scale, g_† = 1.2 x 10^-10 m/s/s. That number is in the data. Why? is a deeply profound question.

It isn’t just that the acceleration scale is somehow fundamental. The amplitude of the discrepancy depends systematically on the acceleration. Above the critical scale, all is well: no need for dark matter. Below it, the amplitude of the discrepancy – the amount of dark matter we infer – increases systematically. The lower the acceleration, the more dark matter one infers.

The relation for rotating galaxies has no detectable scatter – it is a near-perfect relation. Whether this persists, and holds for other systems, is the interesting outstanding question. It appears, for example, that dwarf spheroidal galaxies may follow a slightly different relation. However, the emphasis here is on slighlty. Very few of these data pass the same quality criteria that the SPARC data plotted above do. It’s like comparing mud pies with diamonds.

Whether the scatter in the radial acceleration relation is zero or merely very tiny is important. That’s the difference between a new fundamental force law (like MOND) and a merely spectacular galaxy scaling relation. For this reason, it seems to be controversial. It shouldn’t be: I was surprised at how tight the relation was myself. But I don’t get to report that there is lots of scatter when there isn’t. To do so would be profoundly unscientific, regardless of the wants of the crowd.

Of course, science is hard. If you don’t do everything right, from the measurements to the mass models to the stellar populations, you’ll find some scatter where perhaps there isn’t any. There are so many creative ways to screw up that I’m sure people will continue to find them. Myself, I prefer to look forward: I see no need to continuously re-establish what has been repeatedly demonstrated in the history briefly outlined above.

The Acceleration Scale in the Data

One experience I’ve frequently had in Astronomy is that there is no result so obvious that someone won’t claim the exact opposite. Indeed, the more obvious the result, the louder the claim to contradict it.

This happened today with a new article in Nature Astronomy by Rodrigues, Marra, del Popolo, & Davari titled Absence of a fundamental acceleration scale in galaxies. This title is the opposite of true. Indeed, they make exactly the mistake in assigning priors that I warned about in the previous post.

There is a very obvious acceleration scale in galaxies. It can be seen in several ways. Here I describe a nice way that is completely independent of any statistics or model fitting: no need to argue over how to set priors.

Simple dimensional analysis shows that a galaxy with a flat rotation curve has a characteristic acceleration

g_† = 0.8 V_f⁴/(G M_b)

where V_f is the flat rotation speed, M_b is the baryonic mass, and G is Newton’s constant. The factor 0.8 arises from the disk geometry of rotating galaxies, which are not spherical cows. This is first year grad school material: see Binney & Tremaine. I include it here merely to place the characteristic acceleration g_† on the same scale as Milgrom’s acceleration constant a₀.

These are all known numbers or measurable quantities. There are no free parameters: nothing to fiddle; nothing to fit. The only slightly tricky quantity is the baryonic mass, which is the sum of stars and gas. For the stars, we measure the light but need the mass, so we must adopt a mass-to-light ratio, M*/L. Here I adopt the simple model used to construct the radial acceleration relation: a constant 0.5 M_⊙/L_⊙ at 3.6 microns for galaxy disks, and 0.7 M_⊙/L_⊙ for bulges. This is the best present choice from stellar population models; the basic story does not change with plausible variations.

This is all it takes to compute the characteristic acceleration of galaxies. Here is the resulting histogram for SPARC galaxies:

ascale_hist — Characteristic accelerations for SPARC galaxies. The gray histogram includes all galaxies; the blue includes only higher quality data (quality flag 1 or 2 in SPARC and distance accuracy better than 20%). The range of the x-axis is chosen to match the range shown in Fig. 1 of Rodrigues et al.

Do you see the acceleration scale? It’s right there in the data.

I first employed this method in 2011, where I found <g_†> = 1.24 ± 0.14 Å s^-2 for a sample of gas rich galaxies that predates and is largely independent of the SPARC data. This is consistent with the SPARC result <g_†> = 1.20 ± 0.02 Å s^-2. This consistency provides some reassurance that the mass-to-light scale is near to correct since the gas rich galaxies are not sensitive to the choice of M*/L. Indeed, the value of Milgrom’s constant has not changed meaningfully since Begeman, Broeils, & Sanders (1991).

The width of the acceleration histogram is dominated by measurement uncertainties and scatter in M*/L. We have assumed that M*/L is constant here, but this cannot be exactly true. It is a good approximation in the near-infrared, but there must be some variation from galaxy to galaxy, as each galaxy has its own unique star formation history. Intrinsic scatter in M*/L due to population difference broadens the distribution. The intrinsic distribution of characteristic accelerations must be smaller.

I have computed the scatter budget many times. It always comes up the same: known uncertainties and scatter in M*/L gobble up the entire budget. There is very little room left for intrinsic variation in <g_†>. The upper limit is < 0.06 dex, an absurdly tiny number by the standards of extragalactic astronomy. The data are consistent with negligible intrinsic scatter, i.e., a universal acceleration scale. Apparently a fundamental acceleration scale is present in galaxies.

maxresdefault — Do you see the acceleration scale?

RAR fits to individual galaxies

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

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

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

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

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

NGC2841_RAR_MCMC — RAR fit (equivalent to a MOND fit) to NGC 2841. The rotation curve and components of the mass model are shown at top left, with the fit parameters at top right. The fit is also shown in terms of acceleration (bottom left) and where the galaxy falls on the RAR (bottom right).

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

IC2574_RAR_MCMC — RAR fit for IC 2574, with panels as in the figure above.

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

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

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

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

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

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

RAR_MCMC — The radial acceleration relation with optimized parameters.

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

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

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

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

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

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

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

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

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

The dwarf galaxy NGC1052-DF2

A recently discovered dwarf galaxy designated NGC1052-DF2 has been in the news lately. Apparently a satellite of the giant elliptical NGC 1052, DF2 (as I’ll call it from here on out) is remarkable for having a surprisingly low velocity dispersion for a galaxy of its type. These results were reported in Nature last week by van Dokkum et al., and have caused a bit of a stir.

It is common for giant galaxies to have some dwarf satellite galaxies. As can be seen from the image published by van Dokkum et al., there are a number of galaxies in the neighborhood of NGC 1052. Whether these are associated physically into a group of galaxies or are chance projections on the sky depends on the distance to each galaxy.

NGC 1052 is listed by the NASA Extragalactic Database (NED) as having a recession velocity of 1510 km/s and a distance of 20.6 Mpc. The next nearest big beastie is NGC 1042, at 1371 km/s. The difference of 139 km/s is not much different from 115 km/s, which is the velocity that Andromeda is heading towards the Milky Way, so one could imagine that this is a group similar to the Local Group. Except that NED says the distance to NGC 1042 is 7.8 Mpc, so apparently it is a foreground object seen in projection.

Van Dokkum et al. assume DF2 and NGC 1052 are both about 20 Mpc distant. They offer two independent estimates of the distance, one consistent with the distance to NGC 1052 and the other more consistent with the distance to NGC 1042. Rather than wring our hands over this, I will trust their judgement and simply note, as they do, that the nearer distance would change many of their conclusions. The redshift is 1803 km/s, larger than either of the giants. It could still be a satellite of NGC 1052, as ~300 km/s is not unreasonable for an orbital velocity.

So why the big fuss? Unlike most galaxies in the universe, DF2 appears not to require dark matter. This is inferred from the measured velocity dispersion of ten globular clusters, which is 8.4 km/s. That’s fast to you and me, but rather sluggish on the scale of galaxies. Spread over a few kiloparsecs, that adds up to a dynamical mass about equal to what we expect for the stars, leaving little room for the otherwise ubiquitous dark matter.

This is important. If the universe is composed of dark matter, it should on occasion be possible to segregate the dark from the light. Tidal interactions between galaxies can in principle do this, so a galaxy devoid of dark matter would be good evidence that this happened. It would also be evidence against a modified gravity interpretation of the missing mass problem, because the force law is always on: you can’t strip it from the luminous matter the way you can dark matter. So ironically, the occasional galaxy lacking dark matter would constitute evidence that dark matter does indeed exist!

DF2 appears to be such a case. But how weird is it? Morphologically, it resembles the dwarf spheroidal satellite galaxies of the Local Group. I have a handy compilation of those (from Lelli et al.), so we can compute the mass-to-light ratio for all of these beasties in the same fashion, shown in the figure below. It is customary to refer quantities to the radius that contains half of the total light, which is 2.2 kpc for DF2.

dwarfMLdyn — The dynamical mass-to-light ratio for Local Group dwarf Spheroidal galaxies measured within their half-light radii, as a function of luminosity (left) and average surface brightness within the half-light radius (right). DF2 is the blue cross with low M/L. The other blue cross is Crater 2, a satellite of the Milky Way discovered after the compilation of Local Group dwarfs was made. The dotted line shows M/L = 2, which is a good guess for the stellar mass-to-light ratio. That DF2 sits on this line implies that stars are the only mass that’s there.

Perhaps the most obvious respect in which DF2 is a bit unusual relative to the dwarfs of the Local Group is that it is big and bright. Most nearby dwarfs have half light radii well below 1 kpc. After DF2, the next most luminous dwarfs is Fornax, which is a factor of 5 lower in luminosity.

DF2 is called an ultradiffuse galaxy (UDG), which is apparently newspeak for low surface brightness (LSB) galaxy. I’ve been working on LSB galaxies my entire career. While DF2 is indeed low surface brightness – the stars are spread thin – I wouldn’t call it ultra diffuse. It is actually one of the higher surface brightness objects of this type. Crater 2 and And XIX (the leftmost points in the right panel) are ultradiffuse.

Astronomers love vague terminology, and as a result often reinvent terms that already exist. Dwarf, LSB, UDG, have all been used interchangeably and with considerable slop. I was sufficiently put out by this that I tried to define some categories is the mid-90s. This didn’t catch on, but by my definition, DF2 is VLSB – very LSB, but only by a little – it is much closer to regular LSB than to extremely (ELSB). Crater 2 and And XIX, now they’re ELSB, being more diffuse than DF2 by 2 orders of magnitude.

SBdefinitiontable — Surface brightness categories from McGaugh (1996).

Whatever you call it, DF2 is low surface brightness, and LSB galaxies are always dark matter dominated. Always, at least among disk galaxies: here is the analogous figure for galaxies that rotate:

MLdynDisk — Dynamical mass-to-light ratios for rotationally supported disk galaxies, analogous to the plot above for pressure supported disks. The lower the surface brightness, the higher the mass discrepancy. The correlation with luminosity is secondary, as a result of the correlation between luminosity and surface brightness. From McGaugh (2014).

Pressure supported dwarfs generally evince large mass discrepancies as well. So in this regard, DF2 is indeed very unusual. So what gives?

Perhaps DF2 formed that way, without dark matter. This is anathema to everything we know about galaxy formation in ΛCDM cosmology. Dark halos have to form first, with baryons following.

Perhaps DF2 suffered one or more tidal interactions with NGC 1052. Sub-halos in simulations are often seen to be on highly radial orbits; perhaps DF2 has had its dark matter halo stripped away by repeated close passages. Since the stars reside deep in the center of the subhalo, they’re the last thing to be stripped away. So perhaps we’ve caught this one at that special time when the dark matter has been removed but the stars still remain.

This is improbable, but ought to happen once in a while. The bigger problem I see is that one cannot simply remove the dark matter halo like yanking a tablecloth and leaving the plates. The stars must respond to the change in the gravitational potential; they too must diffuse away. That might be a good way to make the galaxy diffuse, ultimately perhaps even ultradiffuse, but the observed motions are then not representative of an equilibrium situation. This is critical to the mass estimate, which must perforce assume an equilibrium in which the gravitational potential well of the galaxy is balanced against the kinetic motion of its contents. Yank away the dark matter halo, and the assumption underlying the mass estimate gets yanked with it. While such a situation may arise, it makes it very difficult to interpret the velocities: all tests are off. This is doubly true in MOND, in which dwarfs are even more susceptible to disruption.

Then there are the data themselves. Blaming the data should be avoided, but it does happen once in a while that some observation is misleading. In this case, I am made queasy by the fact that the velocity dispersion is estimated from only ten tracers. I’ve seen plenty of cases where the velocity dispersion changes in important ways when more data are obtained, even starting from more than 10 tracers. Andromeda II comes to mind as an example. Indeed, several people have pointed out that if we did the same exercise with Fornax, using its globular clusters as the velocity tracers, we’d get a similar answer to what we find in DF2. But we also have measurements of many hundreds of stars in Fornax, so we know that answer is wrong. Perhaps the same thing is happening with DF2? The fact that DF2 is an outlier from everything else we know empirically suggests caution.

Throwing caution and fact-checking to the wind, many people have been predictably eager to cite DF2 as a falsification of MOND. Van Dokkum et al. point out the the velocity dispersion predicted for this object by MOND is 20 km/s, more than a factor of two above their measured value. They make the MOND prediction for the case of an isolated object. DF2 is not isolated, so one must consider the external field effect (EFE).

The criterion by which to judge isolation in MOND is whether the acceleration due to the mutual self-gravity of the stars is less than the acceleration from an external source, in this case the host NGC 1052. Following the method outlined by McGaugh & Milgrom, and based on the stellar mass (adopting M/L=2 as both we and van Dokkum assume), I estimate an internal acceleration of DF2 to be g_in = 0.15 a₀. Here a₀ is the critical acceleration scale in MOND, 1.2 x 10^-10 m/s/s. Using this number and treating DF2 as isolated, I get the same 20 km/s van Dokkum et al. estimate.

Estimating the external field is more challenging. It depends on the mass of NGC 1052, and the separation between it and DF2. The projected separation at the assumed distance is 80 kpc. That is well within the range that the EFE is commonly observed to matter in the Local Group. It could be a bit further granted some distance along the line of sight, but if this becomes too large then the distance by association with NGC 1052 has to be questioned, and all bets are off. The mass of NGC 1052 is also rather uncertain, or at least I have heard wildly different values quoted in discussions about this object. Here I adopt 10¹¹ M_☉ as estimated by SLUGGS. To get the acceleration, I estimate the asymptotic rotation velocity we’d expect in MOND, V⁴ = a₀GM. This gives 200 km/s, which is conservative relative to the ~300 km/s quoted by van Dokkum et al. At a distance of 80 kpc, the corresponding external acceleration g_ex = 0.14 a₀. This is very uncertain, but taken at face value is indistinguishable from the internal acceleration. Consequently, it cannot be ignored: the calculation published by van Dokkum et al. is not the correct prediction for MOND.

The velocity dispersion estimator in MOND differs when g_ex < g_in and g_ex > g_in (see equations 2 and 3 of McGaugh & Milgrom). Strictly speaking, these apply in the limits where one or the other field dominates. When they are comparable, the math gets more involved (see equation 59 of Famaey & McGaugh). The input data are too uncertain to warrant an elaborate calculation for a blog, so I note simply that the amplitude of the mass discrepancy in MOND depends on how deep in the MOND regime a system is. That is, how far below the critical acceleration scale it is. The lower the acceleration, the larger the discrepancy. This is why LSB galaxies appear to be dark matter dominated; their low surface densities result in low accelerations.

For DF2, the absolute magnitude of the acceleration is approximately doubled by the presence of the external field. It is not as deep in the MOND regime as assumed in the isolated case, so the mass discrepancy is smaller, decreasing the MOND-predicted velocity dispersion by roughly the square root of 2. For a factor of 2 range in the stellar mass-to-light ratio (as in McGaugh & Milgrom), this crude MOND prediction becomes

σ = 14 ± 4 km/s.

Like any erstwhile theorist, I reserve the right to modify this prediction granted more elaborate calculations, or new input data, especially given the uncertainties in the distance and mass of the host. Indeed, we should consider the possibility of tidal disruption, which can happen in MOND more readily than with dark matter. Indeed, at one point I came very close to declaring MOND dead because the velocity dispersions of the ultrafaint dwarf galaxies were off, only realizing late in the day that MOND actually predicts that these things should be getting tidally disrupted (as is also expected, albeit somewhat differently, in ΛCDM), so that the velocity dispersions might not reflect the equilibrium expectation.

In DF2, the external field almost certainly matters. Barring wild errors of the sort discussed or unforeseen, I find it hard to envision the MONDian velocity dispersion falling outside the range 10 – 18 km/s. This is not as high as the 20 km/s predicted by van Dokkum et al. for an isolated object, nor as small as they measure for DF2 (8.4 km/s). They quote a 90% confidence upper limit of 10 km/s, which is marginally consistent with the lower end of the prediction (corresponding to M/L = 1). So we cannot exclude MOND based on these data.

That said, the agreement is marginal. Still, 90% is not very high confidence by scientific standards. Based on experience with such data, this likely overstates how well we know the velocity dispersion of DF2. Put another way, I am 90% confident that when better data are obtained, the measured velocity dispersion will increase above the 10 km/s threshold.

More generally, experience has taught me three things:

In matters of particle physics, do not bet against the Standard Model.
In matters cosmological, do not bet against ΛCDM.
In matters of galaxy dynamics, do not bet against MOND.

The astute reader will realize that these three assertions are mutually exclusive. The dark matter of ΛCDM is a bet that there are new particles beyond the Standard Model. MOND is a bet that what we call dark matter is really the manifestation of physics beyond General Relativity, on which cosmology is based. Which is all to say, there is still some interesting physics to be discovered.

Dwarf Galaxies on the Shoulders of Giants

The week of June 5, 2017, we held a workshop on dwarf galaxies and the dark matter problem. The workshop was attended by many leaders in the field – giants of dwarf galaxy research. It was held on the campus of Case Western Reserve University and supported by the John Templeton Foundation. It resulted in many fascinating discussions which I can’t possibly begin to share in full here, but I’ll say a few words.

Dwarf galaxies are among the most dark matter dominated objects in the universe. Or, stated more properly, they exhibit the largest mass discrepancies. This makes them great places to test theories of dark matter and modified gravity. By the end, we had come up with a few important tests for both ΛCDM and MOND. A few of these we managed to put on a white board. These are hardly a complete list, but provide a basis for discussion.

First, ΛCDM.

LCDM_whiteboard — A few issues for ΛCDM identified during the workshop.

UFDs in field: Over the past few years, a number of extremely tiny dwarf galaxies have been identified as satellites of the Milky Way galaxy. These “ultrafaint dwarfs” are vaguely defined as being fainter than 100,000 solar luminosities, with the smallest examples having only a few hundred stars. This is absurdly small by galactic standards, having the stellar content of individual star clusters within the Milky Way. Indeed, it is not obvious to me that all of the ultrafaint dwarfs deserve to be recognized as dwarf galaxies, as some may merely be fragmentary portions of the Galactic stellar halo composed of stars coincident in phase space. Nevertheless, many may well be stellar systems external to the Milky Way that orbit it as dwarf satellites.

That multitudes of minuscule dark matter halos exist is a fundamental prediction of the ΛCDM cosmogony. These should often contain ultrafaint dwarf galaxies, and not only as satellites of giant galaxies like the Milky Way. Indeed, one expects to see many ultrafaints in the “field” beyond the orbital vicinity of the Milky Way where we have found them so far. These are predicted to exist in great numbers, and contain uniformly old stars. The “old stars” portion of the prediction stems from the reionization of the universe impeding star formation in the smallest dark matter halos. Upcoming surveys like LSST should provide a test of this prediction.

From an empirical perspective, I do expect that we will continue to discover galaxies of ever lower luminosity and surface brightness. In the field, I expect that these will be predominantly gas rich dwarfs like Leo P rather than gas-free, old stellar systems like the satellite ultrafaints. My expectation is an extrapolation of past experience, not a theory-specific prediction.

No Large Cores: Many of the simulators present at the workshop showed that if the energy released by supernovae was well directed, it could reshape the steep (‘cuspy’) interior density profiles of dark matter halos into something more like the shallow (‘cored’) interiors that are favored by data. I highlight the if because I remain skeptical that supernova energy couples as strongly as required and assumed (basically 100%). Even assuming favorable feedback, there seemed to be broad (in not unanimous) consensus among the simulators present that at sufficiently low masses, not enough stars would form to produce the requisite energy. Consequently, low mass halos should not have shallow cores, but instead retain their primordial density cusps. Hence clear measurement of a large core in a low mass dwarf galaxy (stellar mass < 1 million solar masses) would be a serious problem. Unfortunately, I’m not clear that we quantified “large,” but something more than a few hundred parsecs should qualify.

Radial Orbit for Crater 2: Several speakers highlighted the importance of the recently discovered dwarf satellite Crater 2. This object has a velocity dispersion that is unexpectedly low in ΛCDM, but was predicted by MOND. The “fix” in ΛCDM is to imagine that Crater 2 has suffered a large amount of tidal stripping by a close passage of the Milky Way. Hence it is predicted to be on a radial orbit (one that basically just plunges in and out). This can be tested by measuring the proper motion of its stars with Hubble Space Telescope, for which there exists a recently approved program.

DM Substructures: As noted above, there must exist numerous low mass dark matter halos in the cold dark matter cosmogony. These may be detected as substructure in the halos of larger galaxies by means of their gravitational lensing even if they do not contain dwarf galaxies. Basically, a lumpy dark matter halo bends light in subtly but detectably different ways from a smooth halo.

No Wide Binaries in UFDs: As a consequence of dynamical friction against the background dark matter, binary stars cannot remain at large separations over a Hubble time: their orbits should decay. In the absence of dark matter, this should not happen (it cannot if there is nowhere for the orbital energy to go, like into dark matter particles). Thus the detection of a population of widely separated binary stars would be problematic. Indeed, Pavel Kroupa argued that the apparent absence of strong dynamical friction already excludes particle dark matter as it is usually imagined.

Short dynamical times/common mergers: This is related to dynamical friction. In the hierarchical cosmogony of cold dark matter, mergers of halos (and the galaxies they contain) must be frequent and rapid. Dark matter halos are dynamically sticky, soaking up the orbital energy and angular momentum between colliding galaxies to allow them to stick and merge. Such mergers should go to completion on fairly short timescales (a mere few hundred million years).

MOND

A few distinctive predictions for MOND were also identified.

Tangential Orbit for Crater 2: In contrast to ΛCDM, we expect that the `feeble giant’ Crater 2 could not survive a close encounter with the Milky Way. Even at its rather large distance of 120 kpc from the Milky Way, it is so feeble that it is not immune from the external field of its giant host. Consequently, we expect that Crater 2 must be on a more nearly circular orbit, and not on a radial orbit as suggested in ΛCDM. The orbit does not need to be perfectly circular of course, but is should be more tangential than radial.

This provides a nice test that distinguishes between the two theories. Either the orbit of Crater 2 is more radial or more tangential. Bear in mind that Crater 2 already constitutes a problem for ΛCDM. What we’re discussing here is how to close what is basically a loophole whereby we can excuse an otherwise unanticipated result in ΛCDM.

EFE: The External Field Effect is a unique prediction of MOND that breaks the strong equivalence principle. There is already clear if tentative evidence for the EFE in the dwarf satellite galaxies around Andromeda. There is no equivalent to the EFE in ΛCDM.

I believe the question mark was added on the white board to permit the logical if unlikely possibility that one could write a MOND theory with an undetectably small EFE.

Position of UFDs on RAR: We chose to avoid making the radial acceleration relation (RAR) a focus of the meeting – there was quite enough to talk about as it was – but it certainly came up. The ultrafaint dwarfs sit “too high” on the RAR, an apparent problem for MOND. Indeed, when I first worked on this subject with Joe Wolf, I initially thought this was a fatal problem for MOND.

My initial thought was wrong. This is not a problem for MOND. The RAR applies to systems in dynamical equilibrium. There is a criterion in MOND to check whether this essential condition may be satisfied. Basically all of the ultrafaints flunk this test. There is no reason to think they are in dynamical equilibrium, so no reason to expect that they should be exactly on the RAR.

Some advocates of ΛCDM seemed to think this was a fudge, a lame excuse morally equivalent to the fudges made in ΛCDM that its critics complain about. This is a false equivalency that reminds me of this cartoon:

hqdefault — I dare ya to step over this line!

The ultrafaints are a handful of the least-well measured galaxies on the RAR. Before we obsess about these, it is necessary to provide a satisfactory explanation for the more numerous, much better measured galaxies that establish the RAR in the first place. MOND does this. ΛCDM does not. Holding one theory to account for the least reliable of measurements before holding another to account for everything up to that point is like, well, like the cartoon… I could put an NGC number to each of the lines Bugs draws in the sand.

Long dynamical times/less common mergers: Unlike ΛCDM, dynamical friction should be relatively ineffective in MOND. It lacks the large halos of dark matter that act as invisible catchers’ mitts to make galaxies stick and merge. Personally, I do not think this is a great test, because we are a long way from understanding dynamical friction in MOND.

Non-evolution with redshift: If the Baryonic Tully-Fisher relation and the RAR are indeed the consequence of MOND, then their form is fixed by the theory. Consequently, their slope shouldn’t evolve with time. Conceivably their normalization might (e.g., the value of a₀ could in principle evolve). Some recent data for high redshift galaxies place constraints on such evolution, but reports on these data are greatly exaggerated.

These are just a few of the topics discussed at the workshop, and all of those are only a few of the issues that matter to the bigger picture. While the workshop was great in every respect, perhaps the best thing was that it got people from different fields/camps/perspectives talking. That is progress.

I am grateful for progress, but I must confess that to me it feels excruciatingly slow. Models of galaxy formation in the context of ΛCDM have made credible steps forward in addressing some of the phenomenological issues that concern me. Yet they still seem to me to be very far from where they need to be. In particular, there seems to be no engagement with the fundamental question I have posed here before, and that I posed at the beginning of the workshop: Why does MOND get any predictions right?

Degenerating problemshift: a wedged paradigm in great tightness

Reading Merritt’s paper on the philosophy of cosmology, I was struck by a particular quote from Lakatos:

A research programme is said to be progressing as long as its theoretical growth anticipates its empirical growth, that is as long as it keeps predicting novel facts with some success (“progressive problemshift”); it is stagnating if its theoretical growth lags behind its empirical growth, that is as long as it gives only post-hoc explanations either of chance discoveries or of facts anticipated by, and discovered in, a rival programme (“degenerating problemshift”) (Lakatos, 1971, pp. 104–105).

The recent history of modern cosmology is rife with post-hoc explanations of unanticipated facts. The cusp-core problem and the missing satellites problem are prominent examples. These are explained after the fact by invoking feedback, a vague catch-all that many people agree solves these problems even though none of them agree on how it actually works.

FeedbackCartoonSilkMamon — Cartoon of the feedback explanation for the difference between the galaxy luminosity function (blue line) and the halo mass function (red line). From Silk & Mamon (2012).

There are plenty of other problems. To name just a few: satellite planes (unanticipated correlations in phase space), the emptiness of voids, and the early formation of structure (see section 4 of Famaey & McGaugh for a longer list and section 6 of Silk & Mamon for a positive spin on our list). Each problem is dealt with in a piecemeal fashion, often by invoking solutions that contradict each other while buggering the principle of parsimony.

It goes like this. A new observation is made that does not align with the concordance cosmology. Hands are wrung. Debate is had. Serious concern is expressed. A solution is put forward. Sometimes it is reasonable, sometimes it is not. In either case it is rapidly accepted so long as it saves the paradigm and prevents the need for serious thought. (“Oh, feedback does that.”) The observation is no longer considered a problem through familiarity and exhaustion of patience with the debate, regardless of how [un]satisfactory the proffered solution is. The details of the solution are generally forgotten (if ever learned). When the next problem appears the process repeats, with the new solution often contradicting the now-forgotten solution to the previous problem.

This has been going on for so long that many junior scientists now seem to think this is how science is suppose to work. It is all they’ve experienced. And despite our claims to be interested in fundamental issues, most of us are impatient with re-examining issues that were thought to be settled. All it takes is one bold assertion that everything is OK, and the problem is perceived to be solved whether it actually is or not.

8631e895433bc3d1fa87e3d857fc7500 — *“Is there any more?”*

That is the process we apply to little problems. The Big Problems remain the post hoc elements of dark matter and dark energy. These are things we made up to explain unanticipated phenomena. That we need to invoke them immediately casts the paradigm into what Lakatos called degenerating problemshift. Once we’re there, it is hard to see how to get out, given our propensity to overindulge in the honey that is the infinity of free parameters in dark matter models.

Note that there is another aspect to what Lakatos said about facts anticipated by, and discovered in, a rival programme. Two examples spring immediately to mind: the Baryonic Tully-Fisher Relation and the Radial Acceleration Relation. These are predictions of MOND that were unanticipated in the conventional dark matter picture. Perhaps we can come up with post hoc explanations for them, but that is exactly what Lakatos would describe as degenerating problemshift. The rival programme beat us to it.

In my experience, this is a good description of what is going on. The field of dark matter has stagnated. Experimenters look harder and harder for the same thing, repeating the same experiments in hope of a different result. Theorists turn knobs on elaborate models, gifting themselves new free parameters every time they get stuck.

On the flip side, MOND keeps predicting novel facts with some success, so it remains in the stage of progressive problemshift. Unfortunately, MOND remains incomplete as a theory, and doesn’t address many basic issues in cosmology. This is a different kind of unsatisfactory.

In the mean time, I’m still waiting to hear a satisfactory answer to the question I’ve been posing for over two decades now. Why does MOND get any predictions right? It has had many a priori predictions come true. Why does this happen? It shouldn’t. Ever.

Triton Station

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

Category: MOND