Non-equilibrium dynamics in galaxies that appear to have lots of dark matter: ultrafaint dwarfs

This is a long post. It started focused on ultrafaint dwarfs, but can’t avoid more general issues. In order to diagnose non-equilibrium effects, we have to have some expectation for what equilibrium would be. The Tully-Fisher relation is a useful empirical touchstone for that. How the Tully-Fisher relation comes about is itself theory-dependent. These issues are intertwined, so in addition to discussing the ultrafaints, I also review some of the many predictions for Tully-Fisher, and how our theoretical expectation for it has evolved (or not) over time.

In the last post, we discussed how non-equilibrium dynamics might make a galaxy look like it had less dark matter than similar galaxies. That pendulum swings both ways: sometimes non-equilibrium effects might stir up the velocity dispersion above what it would nominally be. Some galaxies where this might be relevant are the so-called ultrafaint dwarfs (not to be confused with ultradiffuse galaxies, which are themselves often dwarfs). I’ve talked about these before, but more keep being discovered, so an update seems timely.

Galaxies and ultrafaint dwarfs

It’s a big universe, so there’s a lot of awkward terminology, and the definition of an ultrafaint dwarf is somewhat debatable. Most often I see them defined as having an absolute magnitude limit M_V > -8, which corresponds to a luminosity less than 100,000 suns. I’ve also seen attempts at something more physical, like being a “fossil” whose star formation was entirely before cosmic reionization, which ended way back at z ~ 6 so all the stars would be at least^{*&^#} 12.5 Gyr old. While such physics-based definitions are appealing, these are often tied up with theoretical projection: the UV photons that reionized the universe should have evaporated the gas in small dark matter halos, so these tiny galaxies can only be fossils from before that time. This thinking pervades much of the literature despite it being obviously wrong, as counterexamples^! exist. For example, Leo P is practically an ultrafaint dwarf by luminosity, but has ample gas (so a larger baryonic mass) and is currently forming stars.

A luminosity-based definition is good enough for us here; I don’t really care exactly where we make the cut. Note that ultrafaint is an appropriate moniker: a luminosity of 10⁵ L_☉ is tiny by galaxy standards. This is a low-grade globular cluster, and some ultrafaints are only a few hundred solar luminosities, which is barely even^# a star cluster. At this level, one has to worry about stochastic effects in stellar evolution. If there are only a handful of stars, the luminosity of the entire system changes markedly as a single star evolves up the red giant branch. Consequently, our mapping from observed quantities to stellar mass is extremely dodgy. For consistency, to compare with brighter dwarfs, I’ve adopted the same boilerplate M_*/L_V = 2 M_☉/L_☉. That makes for a fair comparison luminosity-to-luminosity, but the uncertainty in the actual stellar mass is ginormous.

It gets worse, as the ultrafaints that we know about so far are all very nearby satellites of the Milky Way. They are not discovered in the same way as other galaxies, where one plainly sees a galaxy on survey plates. For example, NGC 7757:

A faint galaxy in the night sky, surrounded by numerous distant star-like points. — *The spiral galaxy NGC 7757 as seen on plates of the Palomar Sky Survey.*

While bright, high surface brightness galaxies like NGC 7757 are easy to see, lower surface brightness galaxies are not. However, they can usually still be seen, if you know where to look:

A faint galaxy amidst numerous distant stars in a dark sky, illustrating the challenges of observing low surface brightness galaxies. — *UGC 1230 as seen on the Palomar Sky Survey. It’s in the middle.*

I like to use this pair as an illustration, as they’re about the same distance from us and about the same angular size on the sky – at least, once you crank up the gain for the low surface brightness UGC 1230:

Comparison of two astronomical images: the left side shows a spiral galaxy with visible structure and brightness, while the right side features a lower surface brightness galaxy, appearing more diffuse and less distinct. — Zoom in on deep CCD images of NGC 7757 (left) and UGC 1230 (right) with the contrast of the latter enhanced. The chief difference between the two is surface brightness – how spread out their stars are. They have a comparable physical diameter, they both have star forming regions that appear as knots in their spiral arms, etc. These galaxies are clearly distinct from the emptiness of the cosmic void around them, being examples of giant stellar systems that gave rise to the term “island universe.”

In contrast to objects that are obvious on the sky as independent island universes, ultrafaint dwarfs are often invisible to the eye. They are recognized as a subset of stars near each other on the sky that also share the same distance and direction of motion in a field that might otherwise be crowded with miscellaneous, unrelated stars. For example, here is Leo IV:

Wide field image of the Ultra-Faint Dwarf Galaxy Leo IV, featuring a zoomed-in view of its faint structure surrounded by numerous background stars and galaxies. — *The ultrafaint dwarf Leo IV as identified by the Sloan Digital Sky Survey and the Hubble Space Telescope.*

See it?

I don’t. I do see a number of background galaxies, including an edge-on spiral near the center of the square. Those are not the ultrafaint dwarf, which is some subset of the stars in this image. To decide which ones are potentially a part of such a dwarf, one examines the color magnitude diagram of all the stars to identify those that are consistent with being at the same distance, and assigns membership in a probabilistic way. It helps if one can also obtain radial velocities and/or proper motions for the stars to see which hang together – more or less – in phase space.

Part of the trick here is deciding what counts as hanging together. A strong argument in favor of these things residing in dark matter halos is that the velocity differences between the apparently-associated stars are too great for them to remain together for any length of time otherwise. This is essentially the same situation that confronted Zwicky in his observations of galaxies in clusters in the 1930s. Here are these objects that appear together in the sky, but they should fly apart unless bound together by some additional, unseen force. But perhaps some of these ultrafaints are not hanging together; they may be in the process of coming apart. Indeed, they may have so few stars because they are well down the path of dissolution.

Since one cannot see an ultrafaint dwarf in the same way as an island universe, I’ve heard people suggest that being bound by a dark matter halo be included in the definition of a galaxy. I see where they’re coming from, but find it unworkable. I know a galaxy when I see one. As did Hubble, as did thousands of other observers since, as can you when you look at the pictures above. It is absurd to make the definition of an object that is readily identifiable by visual inspection be contingent on the inferred presence of invisible stuff.

So are ultrafaints even galaxies? Yes and no. Some of the probabilistic identifications may be mere coincidences, not real objects. However, they can’t all be fakes, and I think that if you put them in the middle of intergalactic space, we would recognize them as galaxies – provided we could detect them at all. At present we can’t, but hopefully that situation will improve with the Rubin Observatory. In the meantime, what we have to work with are these fragmentary systems deep in the potential well of the seventy billion solar mass cosmic gorilla that is the Milky Way. We have to be cognizant that they might have gotten knocked around, as we can see in more massive systems like the Sagittarius dwarf. Of course, if they’ve gotten knocked around too much, then they shouldn’t be there at all. So how do these systems evolve under the influence of a comic gorilla?

Let’s start by looking at the size-mass diagram, as we did before. Ultrafaint dwarfs extend this relation to much lower mass, and also to rather small sizes – some approaching those of star clusters. They approximately follow a line of constant surface density, ~0.1 M_☉ pc^-2 (dotted line)..

A graph illustrating the size-mass relationship of galaxies, plotting effective radius (Re) against stellar mass (M*). Black squares represent data points of larger galaxies, while green squares indicate ultrafaint dwarfs. The dotted line suggests a correlation between size and mass. — *The size and stellar mass of Local Group dwarfs* as discussed previously, with the addition of ultrafaint dwarfs^$ (small gray squares).

This looks weird to me. All other types of galaxies scatter all over the place in this diagram. The ultrafaints are unique in following a tight line in the size-mass plane, and one that follows a line of constant surface brightness. Every element of my observational experience screams that this is likely to be an artifact. Given how these “galaxies” are identified as the loose association of a handful of stars, it is easy to imagine that this trend might be an artifact of how we define the characteristic size of a system that is essentially invisible. It might also arise for physical reasons to do with the cosmic gorilla; i.e., it is a consequence of dynamical evolution. So maybe this correlation is real, but the warning lights that it is not are flashing red.

The Baryonic Tully-Fisher relation as a baseline

Ideally, we would measure accelerations to test theories, particularly MOND. Here, we would need to use the size to estimate the acceleration, but I straight up don’t believe these sizes are physically meaningful. The stellar mass, dodgy as it is, seems robust by comparison. So we’ll proceed as if we know that much – which we don’t, really – but let’s at least try.

With the stellar mass (there is no gas in these things), we are halfway to constructing the baryonic Tully-Fisher relation (BTFR), which is the simplest test of the dynamics that we can make with the available data. The other quantity we need is the characteristic circular speed of the gravitational potential. For rotating galaxies, that is the flat rotation speed, V_f. For pressure supported dwarfs, what is usually measured is the velocity dispersion σ. We’ve previously established that for brighter dwarfs in the Local Group, a decent approximation is V_f = 2σ, so we’ll start by assuming that this should apply to the ultrafaints as well. This allows us to plot the BTFR:

A scatter plot showing the relationship between velocity (Vf in km/s) and baryonic mass (Mb in solar masses), with data points represented by different shapes and colors for various galaxy types. — The baryonic mass and characteristic circular speeds of both rotationally supported galaxies (circles) and pressure supported dwarfs (squares). The colored points follow the same baryonic Tully-Fisher relation (BTFR), but the data for low mass ultrafaint dwarfs (gray squares) *flattens ou*t, *having nearly the same characteristic speed over several decades in mass.*

The BTFR is an emprical relation of the form V_f ~ M_b^1/4 over about six decades in mass. Somewhere around the ultrafaint scale, this no longer appears to hold, with the observed velocity flattening out to become approximately constant for these lowest mass galaxies. I’m not sure this is real, as there many practical caveats to interpreting the observations. Measuring stellar velocities is straightforward but demanding at this level of accuracy. There are many potential systematics, pretty much all of which cause the intrinsic velocity dispersion to be overestimated. For example, observations made with multislit masks tend to return larger dispersions than observations of the same object with fibers. That’s likely because it is hard to build a mask so well that all of the stars perfectly hit the centers of the slitlets assigned to them; offsets within the slit shift the spectrum in a way that artificially adds to the apparent velocity dispersion. Fibers are less efficient in their throughput, but have the virtue of blending the input light in a way that precludes this particular systematic. Another concern is physical – some of the stars that are observed are presumably binaries, and some of the velocity will be due to motion within the binary pair and nothing to do with the gravitational potential of the larger system. This can be addressed with repeated observations to see if some velocities change, but it is hard to do that for each and every system, especially when it is way more fun to discover and explore new systems than follow up on the same one over and over and over again.

There are lots of other things that can go wrong. At some level, some of them probably do – that’s the nature of observational astronomy^&. While it seems likely that some of the velocity dispersions are systematically overestimated, it seems unlikely that all of them are. Let’s proceed as if the bulk of the data is telling us something, even if we treat individual objects with suspicion.

MOND

MOND makes a clear prediction for the BTFR of isolated galaxies: the baryonic mass goes as the fourth power of the flat rotation speed. Contrary to Newtonian expectation, this holds irrespective of surface brightness, which is what attracted my attention to the theory in the first place. So how does it do here?

A graph depicting the relationship between the flat rotation speed (Vf in km/s) and the baryonic mass (Mb in solar masses), showing data points for various galaxies, including ultrafaint dwarfs highlighted with unique markers. — *The same data as above with the addition of the line predicted by MOND (Milgrom 1983).*

Low surface density means low acceleration, so low surface brightness galaxies would make great tests of MOND if they were isolated. Oh, right – they already did. Repeatedly. MOND also correctly predicted the velocities of low mass, gas-rich dwarfs that were unknown when the prediction was made. These are highly nontrivial successes of the theory.

The ultrafaints we’re discussing here are not isolated, so they do not provide the clean tests that isolated galaxies provide. However, galaxies subject to external fields should have low velocities relative to the BTFR, while the ultrafaints have higher velocities. They’re on the wrong side of the relation! Taking this at face value (i.e., assuming equilibrium), MOND fails here.

Whenever MOND has a problem, it is widely seen as a success of dark matter. In my experience, this is rarely true: observations that are problematic for MOND usually don’t make sense in terms of dark matter either. For each observational test we also have to check how LCDM fares.

LCDM

How LCDM fares is often hard to judge because its predictions for the same phenomena are not always clear. Different people predict different things for the same theory. There have been lots of LCDM-based predictions made for both dwarf satellite galaxies and the Tully-Fisher relation. Too many, in fact – it is a practical impossibility to examine them all. Nevertheless, some common themes emerge if we look at enough examples.

The halo mass-velocity relation

The most basic prediction of LCDM is that the mass of a dark matter halo scales with the cube of the circular velocity of a test particle at the virial radius (conventionally taken to be the radius R₂₀₀ that encompasses an average density 200 times the critical density of the universe. If that sounds like gobbledygook to you, just read “halo” for “200”): M₂₀₀ ~ V₂₀₀³. This is a very basic prediction that everyone seems to agree to.

There is a tiny problem with testing this prediction: it refers to the dark matter halo that we cannot see. In order to test it, we have to introduce some scaling factors to relate the dark to the light. Specifically, M_b = f_d M₂₀₀ and V_f = f_v V₂₀₀, where f_d is the observed fraction of mass in baryons and f_v relates the observed flat velocity to the circular speed of our notional test particle at the virial radius. The obvious assumptions to make are that f_d is a constant (perhaps as much as but not more than the cosmic baryon fraction of 16%) and f_v is close to untiy. The latter requirement stems from the need for dark matter to explain the amplitude of the flat rotation speed, but f_v could be slightly different; plausible values range from 0.9 < f_v < 1.4. Values large than one indicate a rotation curve that declines before the virial radius is reached, which is the natural expectation for NFW halos.

Here is a worked example with f_d = 0.025 and f_v = 1:

A graph depicting the relationship between the flat rotation speed (Vf) in kilometers per second and the baryonic mass (Mb) in solar masses. The data points are shown with various markers, including gray squares, green squares, and blue circles, each representing different galaxy types, along with error bars. A solid gray line indicates a trend, while a dotted line marks a theoretical lower bound. — The same data as above with the addition of the nominal prediction of LCDM. The dotted line is the halo mass-circular velocity relation; the gray band is a simple model with f_d = 0.025 and f_v = 1 (e.g., *Mo, Mao, & White 1998)*.

I have illustrated the model with a fat grey line because f_d = 0.025 is an arbitrary choice^* I made to match the data. It could be more, it could be less. The detected baryon fraction can be anythings up to or less than the cosmic value, f_d < fb = 0.16 as not all of the baryons available in a halo cool and condense into cold gas that forms visible stars. That’s fine; there’s no requirement that all of the baryons have to become readily observable, but there is also no reason to expect all halos to cool exactly the same fraction of baryons. Naively one would expect at least some variation in f_d from halo to halo, so there could and probably should be a lot of scatter: the gray line could easily be a much wider band than depicted.

In addition to the rather arbitrary value of f_d, this reasoning also predicts a Tully-Fisher relation with the wrong slope. Picking a favorable value of f_d only matches the data over a narrow range of mass. It was nevertheless embraced for many years by many people. Selection effects bias samples to bright galaxies. Consequently, the literature is rife with TF samples dominated by galaxies with M_b > 10¹⁰ M_☉ (the top right corner of the plot above); with so little dynamic range, a slope of 3 looks fine. Once you look outside that tiny box, it does not look fine.

Personally, I think a slope of 3 is an oversimplification. That is the prediction for dark matter halos; there can be effects that vary systematically with mass. An obvious one is adiabatic compression, the effect by which baryons drag some dark matter along with them as they settle to the center of their halos. This increases f_v by an amount that depends on the baryonic surface density. Surface density correlates with mass, so I would nominally expect higher velocities in brighter galaxies; this drives up the slope. There are various estimates of this effect; typically one gets a slope like 3.3, not the observed 4. Worse, it predicts an additional effect: at a given mass, galaxies of higher surface brightness should also have higher velocity. Surface brightness should be a second parameter in the Tully-Fisher relation, but this is not observed.

The easiest way to reconcile the predicted and observed slopes are to make f_d a function of mass. Since Mb = f_d M₂₀₀ and M₂₀₀ ~ V₂₀₀³, Mb ~ f_d V₂₀₀³. Adopting f_v = 1 for simplicity, Mb ~ V_f⁴ follows if f_d ~ V_f. Problem solved, QED.

There are [at least] two problems with this argument. One is that the scaling f_d ~ V_f must hold perfectly without introducing any scatter. This is a fine-tuning problem: we need one parameter to vary precisely with an another, unrelated parameter. There is no good reason to expect this; we just have to insert the required dependence by hand. This is much worse than choosing an arbitrary value for f_d: now we’re making it a rolling fudge factor to match whatever we need it to. We can make it even more complicated by invoking some additional variation in f_v, but this just makes the fine-tuning worse as the product f_df_v^-3 has to vary just so. Another problem is that what we’re doing all this to adjust the prediction of one theory (LCDM) to match that of a different theory (MOND). It is never a good sign when we have to do that, whether we admit it or not.

Abundance matching

The reasoning leading to a slope 3 Tully-Fisher relation assumes a one-to-one relation between baryonic and halo mass (f_d = constant). This is an eminently reasonable assumption. We spent a couple of decades trying to avoid having to break this assumption. Once we do so and make f_d a freely variable parameter, then it can become a rolling fudge factor that can be adjusted to fit anything. Everyone agrees that is Bad. However, it might be tolerable if there is an independent way of estimating this variation. Rather than make f_d just be what we need it to be as described above, we can instead estimate it with abundance matching.

Abundance matching comes from equating the observed number density of galaxies as a function of mass with the number density of dark matter halos. This process gives f_d, or at least the stellar fraction, f_*, which is close to f_d for bright galaxies. Critically, it provides a way to assign dark matter halo masses to galaxies independently of their kinematics. This replaces an arbitrary, rolling fudge factor with a predictive theory.

Abundance matching models generically introduce curvature into the prediction for the BTFR. This stems from the mismatch in the shape of the galaxy stellar mass function (a Schechter function) and the dark halo mass function (a power law on galaxy scales). This leads to a bend in relations that map between visible and dark mass.

The transition from the M ~ V³ reasoning to abundance matching occurred gradually, but became pronounced circa 2010. There are many abundance matching models; I already faced the problem of the multiplicity of LCDM predictions when I wrote a lengthy article on the BTFR in 2012. To get specific, let’s start with an example from then, the model of Trujillo-Gomez-et al. (2011):

Scatter plot showing the relationship between gravitational potential flat rotation speed (Vf in km/s) and baryonic mass (Mb in solar masses). The plot features varying data points marked with blue circles, green squares, and gray squares, indicating different galaxy types or observational methods. A red curve is drawn, illustrating an empirical relationship fitting the data. — *The same data as above with the addition of the line predicted by LCDM in the model of Trujillo-Gomez-et al. (2011).*

One thing Trujillo-Gomez-et al. (2011) say in their abstract is “The data present a clear monotonic LV relation from ∼50 km s⁻¹ to ∼500 km s⁻¹, with a bend below ∼80 km s⁻¹“. By LV they mean luminosity-velocity, i.e., the regular Tully-Fisher relation. The bend they note is real; that’s what happens when you consider only the starlight and ignore the gas. The bend goes away if you include that gas. This was already known at the time – our original BTFR paper from 2000 has nearly a thousand citations, so it isn’t exactly obscure. Ignoring the gas is a choice that makes no sense empirically but makes a lot of sense from the perspective of LCDM simulations. By 2010, these had become reasonably good at matching the numbers of stars observed in galaxies, but the gas properties of simulated galaxies remained, hmmmmmmm, wanting. It makes sense to utilize the part that works. It makes less sense to pretend that this bend is something physically meaningful rather than an artifact of ignoring the gas. The pressure-supported dwarfs are all star dominated, so this distinction doesn’t matter here, and they follow the BTFR, not the stars-only version.

An old problem in galaxy formation theory is how to calibrate the number density of dark matter halos to that of observed galaxies. For a long time, a choice that people made was to match either the luminosity function or the kinematics. These didn’t really match up, so there was occasional discussion of the virtues and vices of the “luminosity function calibration” vs. the “Tully-Fisher calibration.” These differed by a factor of ~2. This tension between remains with us. Mostly simulations have opted to adopt the luminosity function calibration, updated and rebranded as abundance matching. Again, this makes sense from the perspective of LCDM simulations, because the number density of dark matter halos is something that simulations can readily quantify while the kinematics of individual galaxies are much harder to resolve^**.

The nonlinear relation between stellar mass and halo mass obtained from abundance matching inevitably introduces curvature into the corresponding Tully-Fisher relation predicted by such models. That’s what you see in the curved line of Trujillo-Gomez-et al. (2011) above. They weren’t the first to obtain such a result, and the certainly weren’t the last: this is a feature of LCDM with abundance matching, not a bug.

The line of Trujillo-Gomez-et al. (2011) matches the data pretty well at intermediate masses. It diverges to higher velocities at both small and large galaxy masses. I’ve written about this tension at high masses before; it appears to be real, but let’s concentrate on low masses here. At low masses, the velocity of galaxies with M_b < 10⁸ M_☉ appears to be overestimated. But the divergence between model and reality has just begun, and it is hard to resolve small things in simulations, so this doesn’t seem too bad. Yet.

Moving ahead, there are the “Latte” simulations of Wetzel et al. (2016) that use the well-regarded FIRE code to look specifically at simulated dwarfs, both isolated and satellites – specifically satellites of Milky Way-like systems. (Milky Way. Latte. Get it? Nerd humor.) So what does that find?

A graph displaying the relationship between circular velocity (Vf in km/s) and baryonic mass (Mb in solar masses), featuring various data points distinguished by shape and color, including gray squares, green squares, orange triangles, and blue circles to represent different types of galaxies. — *The same data as above with the addition of* simulated dwarfs (orange triangles) from the Latte LCDM simulation of Wetzel et al. (2016), specifically the simulated satellites in the top panel of their Fig. 3. Note that we plot V_f = 2σ for pressure supported systems, both real and simulated.

The individual simulated dwarf satellites of Wetzel et al. (2016) follow the extrapolation of the line predicted by Trujillo-Gomez-et al. (2011). To first order, it is the same result to higher resolution (i.e., smaller galaxy mass). Most of the simulated objects have velocity dispersions that are higher than observed in real galaxies. Intriguingly, there are a couple of simulated objects with M_* ~ 5 x 10⁶ M_☉ that fall nicely among the data where there are both star-dominated and gas-rich galaxies. However, these two are exceptions; the rule appears to be characteristic speeds that are higher than observed.

The lowest mass simulated satellite objects begin to approach the ultrafaint regime, but resolution continues to be an issue: they’re not really there yet. This hasn’t precluded many people from assuming that dark matter will work where MOND fails, which seems like a heck of a presumption given that MOND has been consistently more successful up until that point. Where MOND underpredicts the characteristic velocity of ultrafaints, LCDM hasn’t yet made a clear prediction, and it overpredicts velocities for objects of slightly larger mass. Ain’t no theory covering itself in glory here, but this is a good example where objects that are a problem for MOND are also a problem for dark matter, and it seems likely that non-equilibrium dynamics play a role in either case.

Comparing apples with apples

A persistent issue with comparing simulations to reality is extracting comparable measures. Where circular velocities are measured from velocity fields in rotating galaxies and estimated from measured velocity dispersions in pressure supported galaxies, the most common approach to deriving rotation curves from simulated objects is to sum up particles in spherical shells and assume V² = GM/R. These are not the same quantities. They should be proxies for one another, but equality holds only in the limit of isotropic orbits in spherical symmetry. Reality is messier than that, and simulations aren’t that simple either^%.

Sales et al. (2017) make the effort to make a better comparison between what is observed given how it is observed, and what the simulations would show for that quantity. Others have made a similar effort; a common finding is that the apparent rotation speeds of simulated gas disks do not trace the gravitational potential as simply as GM/R. That’s no surprise, but most simulated rotation curves do not look like those of real galaxies^{^}, so the comparison is not straightforward. Those caveats aside, Sales et al. (2017) are doing the right thing in trying to make an apples-to-apples comparison between simulated and observed quantities. They extract from simulations a quantity V_out that is appropriate for comparison with what we observe in the outer parts of rotation curves. So here is the resulting prediction for the BTFR:

A graph plotting the baryonic mass (Mb in solar masses) against the characteristic flat rotation speed (Vf in km/s) for various galaxies, showing a curve that describes the baryonic Tully-Fisher relation. The scatter points include different types of galaxies, with green squares indicating specific categories. — *The same data as above with the addition of the line predicted by LCDM in the model of* Sales et al. (2017), specifically the formula for V_out in their Table 2 which is *their proxy for the observable rotation speed.*

That’s pretty good. It still misses at high masses (those two big blue points at the top are Andromeda and the Milky Way) and it still bends away from the data at low masses where there are both star-dominated and gas-rich galaxies. (There are a lot more examples of the latter that I haven’t used here because the plot gets overcrowded.) Despite the overshoot, the use of an observable aspect of the simulations gets closer to the data, and the prediction flattens out in the same qualitative sense. That’s good, so one might see cause for hope that this problem is simply a matter of making a fair comparison between simulations and data. We should also be careful not to over-interpret it: I’ve simply plotted the formula they give; the simulations to which they fit it surely do not resolve ultrafaint dwarfs, so really the line should stop at some appropriate mass scale.

Nevertheless, it makes sense to look more closely at what is observed vs. what is simulated. This has recently been done in greater detail by Ruan et al. (2025). They consider two simulations that implement rather different feedback; both wind up producing rotating, gas rich dwarfs that actually fall on the BTFR.

Scatter plot illustrating the baryonic Tully-Fisher relation, showing the relationship between characteristic circular velocity (Vf) and baryonic mass (Mb) for various galaxy types, including data points for ultrafaint dwarfs. — *The same data as above with the addition of* simulated dwarfs of Ruan et al. (2025), specifically from the top right panel of their Fig. 6. The orange circles are their “massives” and the red triangles the “marvels” (the distinction refers to different feedback models).

Finally some success after all these years! Looking at this, it is tempting to declare victory: problem solved. It was just a matter of doing the right simulation all along, and making an apples-to-apples comparison with the data.

That sounds too goo to be true. Is it repeatable in other simulations? What works now that didn’t before?

These are high resolution simulations, but they still don’t resolve ultrafaints. We’re talking here about gas-rich dwarfs. That’s also an important topic, so let’s look more closely. What works now is in the apples-to-apples assessment: what we would measure for V_out is less than V_max (related to V₂₀₀) of the halo:

A graph displaying two panels: the top panel shows the relation between the ratio of mid-outward velocity to maximum velocity (Vout, mid / Vmax, mid) and the logarithm of baryonic mass (Mbar), with data points represented as circles and triangles. The bottom panel illustrates the relationship between the ratio of outer radius to maximum radius (Rout, mid / Rmax, mid) and the logarithm of baryonic mass, also featuring similar data points. — Two panels from Fig. 7 of *Ruan et al. (2025)* showing the ratio of the velocity we might observe relative to the characteristic circular velocity of the halo (top) and the ratio of the radii where these occur (bottom).

The treatment of cold gas in simulations has improved. In these simulations, V_out(R_out) is measured where the gas surface density falls to 1 M_☉ pc^-2, which is typical of many observations. But the true rotation curve is still rising for objects with M_b < a few x 10⁸ M_☉; it has not yet reached a value that is characteristic of the halo. So the apparent velocity is low, even if the dark matter halos are doing basically the same thing as before:

Graph showing the baryonic Tully-Fisher relation, with velocity Vf (km/s) plotted against baryonic mass Mb (solar masses). Data points include various galaxies and dwarf galaxies, with error bars indicating measurement uncertainties. A red line represents the best-fit relation. — As above, but with the addition of the true V_max *(small black dots*) of the simulated halos discussed by *Ruan et al. (2025)*, which follow the relation of *Sales et al. (2017)* (line for V_max in their Table 2).

I have mixed feelings about this. On the one hand, there are many dwarf galaxies with rising rotation curves that we don’t see flatten out, so it is easy to imagine they might keep going up, and I find it plausible that this is what we would find if we looked harder. So plausible that I’ve spend a fair amount of time doing exactly this. Not all observations terminate at 1 M_☉ pc^-2, and whenever we push further out, we see the same damn thing over and over: the rotation curve flattens out and stays flat^!!. That’s been my anecdotal experience; getting beyond that systematically is the point of the MOHNGOOSE survey. This was constructed to detect much lower atomic gas surface densities, and routinely detects gas at the 0.1 M_☉ pc^-2 level where Ruan et al. suggest we should see something closer to V_max. So far, we don’t.

I don’t want to sound too negative, because how we map what we predict in simulations to what we measure in observations is a serious issue. But it seems a bit of a stretch for a low-scatter power law BTFR to be the happenstance of observational sensitivity that cuts in at a convenient mass scale. So far, we see no indication of that in more sensitive observations. I’ll certainly let you know if that changes.

Survey says…

At this juncture, we’ve examined enough examples that the reader can appreciate my concern that LCDM models can predict rather different things. What does the theory really predict? We can’t really test it until we agree what it should do^!!!.

I thought it might be instructive to combine some of the models discussed above. It is.

Graph illustrating the correlation between the characteristic flat rotation speed (Vf) and baryonic mass (Mb) of galaxies. The plot features data points in different colors representing various galaxy types, with lines indicating theoretical trends and empirical relations. — Some of the LCDM predictions discussed above shown together. The dotted line to the right of the data is the halo mass-velocity relation, which is the one thing we all agree LCDM predicts but which is observationally inaccessible. The grey band is a *Mo, Mao, & White*-type model with f_d = 0.025. The red dotted line is the model of *Trujillo-Gomez-et al. (2011)*; the solid red line that of *Sales et al. (2017)* for V_max.

The models run together, more or less, for high mass galaxies. Thanks to observational selection effects, these are the objects we’ve always known about and matched our theories to. In order to test a theory, one wants to force it to make predictions in new regimes it wasn’t built for. Low mass galaxies do that, as do low surface brightness galaxies, which are often but not always low mass. MOND has done well for both, down to the ultrafaints we’re discussing here. LCDM does not yet explain those, or really any of the intermediate mass dwarfs.

What really disturbs me about LCDM models is their flexibility. It’s not just that they miss, it’s that it is possible to miss the data on either side of the BTFR. The older f_d = constant models predict velocities that are too low for low mass galaxies. The more recent abundance matching models predict velocities that are too high for low mass galaxies. I have no doubt that a model can be constructed that gets it right, because there is obviously enough flexibility to do pretty much anything. Adding new parameters until we get it right is an example of epicyclic thinking, as I’ve been pointing out for thirty years. I don’t know what could be worse for an idea like dark matter that is not falsifiable.

We still haven’t come anywhere close to explaining the ultrafaints in either theory. In LCDM, we don’t even know if we should draw a curved line that catches them as if they’re in equilibrium, or start from a power-law BTFR and look for departures from that due to tidal effects. Both are possible in LCDM, both are plausible, as is some combination of both. I expect theorists will pick an option and argue about it indefinitely.

Tidal effects

The typical velocity dispersion of the ultrafaint dwarfs is too high for them to be in equilibrium in MOND. But there’s also pretty much no way these tiny things could be in equilibrium, being in the rough neighborhood dominated by our home, the cosmic gorilla. That by itself doesn’t make an explanation; we need to work out what happens to such things as they evolve dynamically under the influence of a pronounced external field. To my knowledge, this hasn’t been addressed in detail in MOND any more than in LCDM, though Brada & Milgrom addressed some of the relevant issues.

There is a difference in approach required for the two theories. In LCDM, we need to increase the resolution of simulations to see what happens to the tiniest of dark matter halos and their resident galaxies within the larger dark matter halos of giant galaxies. In MOND we have to simulate the evolution along the orbit of each unique individual. This is challenging on multiple levels, as each possible realization of a MOND theory requires its own code. Writing a simulation code for AQUAL requires a different numerical approach than QUMOND, and those are both modifications of gravity via the Poisson euqation. We don’t know which might be closer to reality; heck, we don’t even know [yet] if MOND is a modification of gravity or intertia, the latter being even harder to code.

Cold dark matter is scale-free, so crudely I expect ultrafaint dwarfs in LCDM to do the same as larger dwarf satellites that have been simulated: their outer dark matter halos are gradually whittled away by tidal stripping for many Gyr. At first the stars are unaffected, but eventually so little dark matter is left that the stars start to be lost impulsively during pericenter passages. Though the dark matter is scale free, the stars and the baryonic physics that made them are not, so that’s where it gets tricky. The apparent dark-to-luminous mass ratio is huge, so one possibility is that the ultrafaints are in equilibrium despite their environment; they just made ridiculously few stars from the amount of mass available. That’s consistent with a wild extrapolation of abundance matching models, but how it comes about physically is less clear. For example, at some low mass, a galaxy would make so few stars that none are massive enough to result in a supernova, so there is no feedback, which is what is preventing too many stars from forming. Awkward. Alternately, the constant exposure to tidal perturbation might stir things up, with the velocity dispersion growing and stars getting stripped to form tidal streams, so they may have started as more massive objects. Or some combination of both, plus the evergreen possibility of things that don’t occur to me offhand.

Equilibrium for ultrafaint satellites is not an option in MOND, but tidal stirring and stripping is. As a thought experiment, let’s imagine what happens to a low mass dwarf typical of the field that falls towards the Milky Way from some large distance. Initially gas-rich, the first environmental effect that it is likely to experience is ram pressure stripping by the hot coronal gas around the Milky Way. That’s a baryonic effect that happens in either theory; it’s nothing to do with the effective law of gravity. A galaxy thus deprived of much of its mass will be out of equilibrium; its internal velocities will be typical of the original mass but the stripped mass is less. Consequently, its structure must adjust to compensate; perhaps dwarf Irregulars puff up and are transformed into dwarf Spheroidals in this way. Our notional infalling dwarf may have time to equilibrate to its new mass before being subject to strong tidal perturbation by the Milky Way, or it may not. If not, it will have characteristic internal velocities that are too high for its new mass, and reside above the BTFR. I doubt this suffices to explain [m]any of the ultrafaints, as their masses are so tiny that some stellar mass loss is also likely to have occurred.

Let’s suppose that our infalling dwarf has time to [approximately] equilibrate, or it simply formed nearby to begin with. Now it is a pressure supported system [more or less] on the BTFR. As it orbits the Milky Way, it feels an extra force from the external field. If it stays far enough out to remain in quasi-equilibrium in the EFE regime, then it will oscillate in size and velocity dispersion in phase with the strength of the external field it feels along its orbit.

If instead a satellite dips too close, it will be tidally disturbed and depart from equilibrium. The extra energy may stir it up, increasing its velocity dispersion. It doesn’t have the mass to sustain that, so stars will start to leak out. Tidal disruption will eventually happen, with the details depending on the initial mass and structure of the dwarf and on the eccentricity of its orbit, the distance of closest approach (pericenter), whether the orbit is prograde or retrograde relative to any angular momentum the dwarf may have… it’s complicated, so it is hard to generalize^##. Nevertheless, we (McGaugh & Wolf 2010) anticipated that “the deviant dwarfs [ultrafaints] should show evidence of tidal disruption while the dwarfs that adhere to the BTFR should not.” Unlike LCDM where most of the damage is done at closest approach, we anticipate for MOND that “stripping of the deviant dwarfs should be ongoing and not restricted to pericenter passage” because tides are stronger and there is no cocoon of dark matter to shelter the stars. The effect is still maximized at pericenter, its just not as impulsive as in the some of the dark matter simulations I’ve seen.

This means that there should be streams of stars all over the sky. As indeed there are. For example:

A color-coded map of the northern sky displaying various stellar streams, indicated by labels such as 'Gaia-1*', 'Gaia-3*', and 'GD-1'. The color gradient represents velocity in kilometers per second, with colors ranging from blue for lower velocities to red for higher velocities. — *Stellar streams in the Milky Way identified using Gaia (Malhan et al. 2018).*

As a tidally influence dwarf dissolves, the stars will leak out and form a trail. This happens in LCDM too, but there are differences in the rate, coherence, and symmetry of the resulting streams. Perhaps ultrafaint dwarfs are just the last dregs of the tidal disruption process. From this perspective, it hardly matters if they originated as external satellites or are internal star clusters: globular clusters native to the Milky Way should undergo a similar evolution.

Evolutionary tracks

Perhaps some of the ultrafaint dwarfs are the nuggets of disturbed systems that have suffered mass loss through tidal stripping. That may be the case in either LCDM or MOND, and has appealing aspects in either case – we went through all the possibilities in McGaugh & Wolf (2010). In MOND, the BTFR provides a reference point for what a stable system in equilibrium should do. That’s the starting point for the evolutionary tracks suggested here:

A graph plotting flat rotation speed (Vf) in km/s against baryonic mass (Mb) in solar masses. The data points include various galaxies represented as blue circles and green squares, with error bars indicating measurement uncertainty. A solid black line demonstrates the overall trend, while red curves suggest alternative theoretical predictions. — *BTFR with conceptual evolutionary tracks (red lines) for tidally-stirred ultrafaint dwarfs.*

Objects start in equilibrium on the BTFR. As they become subject to the external field, their velocity dispersions first decreases as they transition through the quasi-Newtonian regime. As tides kick in, stars are lost and stretched along the satellite’s orbit, so mass is lost but the apparent velocity dispersion increases as stars gradually separate and stretch out along a stream. Their relative velocities no longer represent a measure of the internal gravitational potential; rather than a cohesive dwarf satellite they’re more an association of stars in similar orbits around the Milky Way.

This is crudely what I imagine might be happening in some of the ultrafaint dwarfs that reside above the BTFR. Reality can be more complicated, and probably is. For example, objects that are not yet disrupted may oscillate around and below the BTFR before becoming completely unglued. Moreover, some individual ultrafaints probably are not real, while the data for others may suffer from systematic uncertainties. There’s a lot to sort out, and we’ve reached the point where the possibility of non-equilibrium effects cannot be ignored.

As a test of theories, the better course remains to look for new galaxies free from environmental perturbation. Ultrafaint dwarfs in the field, far from cosmic gorillas like the Milky Way, would be ideal. Hopefully many will be discovered in current and future surveys.

^!Other examples exist and continue to be discovered. More pertinent to my thinking is that the mass threshold at which reionization is supposed to suppress star formation has been a constantly moving goal post. To give an amusing anecdote, while I was junior faculty at the University of Maryland (so at least twenty years ago), Colin Norman called me up out of the blue. Colin is an expert on star formation, and had a burning question he thought I could answer. “Stacy,” he says as soon as I pick up, “what is the lowest mass star forming galaxy?” Uh, Hi, Colin. Off the cuff and totally unprepared for this inquiry, I said “um, a stellar mass of a few times 10⁷ solar masses.” Colin’s immediate response was to laugh long and loud, as if I had made the best nerd joke ever. When he regained his composure, he said “We know that can’t be true as reionization will prevent star formation in potential wells that small.” So, after this abrupt conversation, I did some fact-checking, and indeed, the number I had pulled out of my arse on the spot was basically correct, at that time. I also looked up the predictions, and of course Colin knew his business too; galaxies that small shouldn’t exist. Yet they do, and now the minimum known is two orders of magnitude lower in mass, with still no indication that a lower limit has been reached. So far, the threshold of our knowledge has been imposed by observational selection effects (low luminosity galaxies are hard to see), not by any discernible physics.

More recently, McQuinn et al. (2024) have made a study of the star formation histories of Leo P and a few similar galaxies that are near enough to see individual stars so as to work out the star formation rate over the course of cosmic history. They argue that there seems to be a pause in star formation after reionization, so a more nuanced version of the hypothesis may be that reionization did suppress star forming activity for a while, but these tiny objects were subsequently able to re-accrete cold gas and get started again. I find that appealing as a less simplistic thing that might have happened in the real universe, and not just a simple on/off switch that leaves only a fossil. However, it isn’t immediately clear to me that this more nuanced hypothesis should happen in LCDM. Once those baryons have evaporated, they’re gone, and it is far from obvious that they’ll ever come back to the weak gravity of such a small dark matter halo. It is also not clear to me that this interpretation, appealing as it is, is unique: the reconstructed star formation histories also look consistent with stochastic star formation, with fluctuations in the star formation rate being a matter of happenstance that have nothing to do with the epoch of reionization.

^#So how are ultrafaint dwarfs different from star clusters? Great question! Wish we had a great answer.

Some ultrafaints probably are star clusters rather than independent satellite galaxies. How do we tell the difference? Chiefly, the velocity dispersion: star clusters show no need for dark matter, while ultrafaint dwarfs generally appear to need a lot. This of course assumes that their measured velocity dispersions represent an equilibrium measure of their gravitational potential, which is what we’re questioning here, so the opportunity for circular reasoning is rife.

^$Rather than apply a strict luminosity cut, for convenience I’ve kept the same “not safe from tidal disruption” distinction that we’ve used before. Some of the objects in the 10⁵ – 10⁶ M_☉ range might belong more with the classical dwarfs than with the ultrafaints. This is a reminder that our nomenclature is terrible more than anything physically meaningful.

^&Astronomy is an observational science, not a laboratory science. We can only detect the photons nature sends our way. We cannot control all the potential systematics as can be done in an enclosed, finite, carefully controlled laboratory. That means there is always the potential for systematic uncertainties whose magnitude can be difficult to estimate, or sometimes to even be aware of, like how local variations impact Jeans analyses. This means we have to take our error bars with a grain of salt, often such a big grain as to make statistical tests unreliable: goodness of fit is only as meaningful as the error bars.

I say this because it seems to be the hardest thing for physicists to understand. I also see many younger astronomers turning the crank on fancy statistical machinery as if astronomical error bars can be trusted. Garbage in, garbage out.

^*This is an example of setting a parameter in a model “by hand.”

^**The transition to thinking in terms of the luminosity function rather than Tully-Fisher is so complete that the most recent, super-large, Euclid flagship simulation doesn’t even attempt to address the kinematics of individual galaxies while giving extraordinarily detailed and extensive details about their luminosity distributions. I can see why they’d do that – they want to focus on what the Euclid mission might observe – but it is also symptomatic of the growing tendency to I’ve witnessed to just not talk about those pesky kinematics.

^%Halos in dark matter simulations tend to be rather triaxial, i.e., a 3D bloboid that is neither spherical like a soccer ball nor oblate like a frisbee nor prolate like an American football: each principle axis has a different length. If real halos were triaxial, it would lead to non-circular orbits in dark matter-dominated galaxies that are not observed.

The triaxiality of halos is a result from dark matter-only simulations. Personally, I suspect that the condensation of gas within a dark matter halo (presuming such things exist) during the process of galaxy formation rounds-out the inner halo, making it nearly spherical where we are able to make measurements. So I don’t see this as necessarily a failure of LCDM, but rather an example of how more elaborate simulations that include baryonic physics are sometimes warranted. Sometimes. There’s a big difference between this process, which also compresses the halo (making it more dense when it already starts out too dense), and the various forms of feedback, which may or may not further alter the structure of the halo.

^{^}There are many failure modes in simulated rotation curves, the two most common being the cusp-core problem in dwarfs and sub-maximal disks in giants. It is common for the disks of bright spiral galaxies to be nearly maximal in the sense that the observed stars suffice to explain the inner rotation curve. They may not be completely maximal in this sense, but they come close for normal stellar populations. (Our own Milky Way is a good example.) In contrast, many simulations produce bright galaxies that are absurdly sub-maximal; EAGLE and SIMBA being two examples I remember offhand.

Another common problem is that LCDM simulations often don’t produce rotation curves that are as flat as observed. This was something I also found in my early attempts at model-building with dark matter halos. It is easy to fit a flat rotation curve given the data, but it is hard to predict a priori that rotation curves should be flat.

^!!Gravitational lensing indicates that rotation curves remain flat to even larger radii. However, these observations are only sensitive to galaxies more massive than those under discussion here. So conceivably there could be another coincidence wherein flatness persists for galaxies with M_b > 10¹⁰ M_☉, but not those with M_b < 10⁹ M_☉.

^!!!Many in the community seem to agree that it will surely work out.

^##I’ve tried to estimate dissolution timescales, but find the results wanting. For plausible assumptions, one finds timescales that seem plausible (a few Gyr) but with some minor fiddling one can also find results that are no-way that’s-too-short (a few tens of millions of years), depending on the dwarf and its orbit. These are crude analytic estimates; I’m not satisfied that these numbers were particularly meaningful. Still, this is a worry with the tidal-stirring hypothesis: will perturbed objects persist long enough to be observed as they are? This is another reason we need detailed simulations tailored to each object.

^{*&^#}Note added after initial publication: While I was writing this, a nice paper appeared on exactly this issue of the star formation history of a good number of ultrafaint dwarfs. They find that 80% of the stellar mass formed 12.48 ± 0.18 Gyr ago, so 12.5 was a good guess. Formally, at the one sigma level, this is a little after reionization, but only a tiny bit, so close enough: the bulk of the stars formed long ago, like a classical globular cluster, and these ultrafaints are consistent with being fossils.

Intriguingly, there is a hint of an age difference by kinematic grouping, with things that have been in the Milky Way being the oldest, those on first infall being a little younger (but still very old), and those infalling with the Large Magellanic Cloud a tad younger still. If so, then there is more to the story than quenching by cosmic reionization.

They also show a nice collection of images so you can see more examples. The ellipses trace out the half-light radii, so can see the proclivity for many (not all!) of these objects to be elongated, perhaps as a result of tidal perturbation:

**Figure 2** from Durbin et al. (2025): *Footprints of all HST observations (blue filled patches) overlaid on DSS2 imaging cutouts. Open black ellipses show the galaxy profiles at one half-light radius.*

Non-equilibrium dynamics in galaxies that appear to lack dark matter: ultradiffuse galaxies

Previously, we discussed non-equilibrium dynamics in tidal dwarf galaxies. These are the result of interactions between giant galaxies that are manifestly a departure from equilibrium, a circumstance that makes TDGs potentially a decisive test to distinguish between dark matter and MOND, and simultaneously precludes confident application of that test. There are other galaxies for which I suspect non-equilibrium dynamics may play a role, among them some (not all) of the so-called ultradiffuse galaxies (UDGs).

UDGs

The term UDG has been adopted for galaxies below a certain surface brightness threshold with a size (half-light radius) in excess of 1.5 kpc (van Dokkum et al. 2015). I find the stipulation about the size to be redundant, as surface brightness^* is already a measure of diffuseness. But OK, whatever, these things are really spread out. That means they should be good tests of MOND like low surface brightness galaxies before them: their low stellar surface densities mean^** that they should be in the regime of low acceleration and evince large mass discrepancies when isolated. It also makes them susceptible to the external field effect (EFE) in MOND when they are not isolated, and perhaps also to tidal disruption.

To give some context, here is a plot of the size-mass relation for Local Group dwarf spheroidals. Typically they have masses comparable to globular clusters, but much large sizes – a few hundred parsecs instead of just a few. As with more massive galaxies, these pressure supported dwarfs are all over the place – at a give mass, some are large while others are relatively compact. All but the one most massive galaxy in this plot are in the MOND regime. For convenience, I’ll refer to the black points labelled with names as UDGs⁺.

The size (radius encompassing half of the total light) and stellar mass of Local Group dwarf spheroidals (green points selected by McGaugh et al. 2021 to be relatively safe from external perturbation) along with two more Local Group dwarfs that are subject to the EFE (Crater 2 and Antlia 2) and the two UDGs NGC 1052-DF2 and DF4. Dotted lines show loci of constant surface density. For reference, the solar neighborhood has ~40 M_☉ pc^-2; the centers of high surface brightness galaxies frequently exceed 1,000 M_☉ pc^-2.

The UDGs are big and diffuse. This makes them susceptible to the EFE and tidal effects. The lower the density of a system, the easier it is for external systems to mess with it. The ultimate example is something gets so close to a dominant central mass that it gets tidally disrupted. That can happen conventionally; the stronger effective force of MOND increases tidal effects. Indeed, there is only a fairly narrow regime between the isolated case and tidally-induced disequilibrium where the EFE modifies the internal dynamics in a quasi-static way.

The trouble is the s-word: static. In order to test theories, we assume that the dynamical systems we observe are in equilibrium. Though often a good assumption, it doesn’t always hold. If we forget we made the assumption, we might think we’ve falsified a theory when all we’ve done is discover a system that is out of equilibrium. The universe is a very dynamic place – the whole thing is expanding, after all – so we need to be wary of static thinking.

Equilibrium MOND formulae

That said, let’s indulge in some static thinking. An isolated, pressure supported galaxy in the MOND regime will have an equilibrium velocity dispersion

where M is the mass (the stellar mass in the case of a gas-free dwarf spheroidal), G is Newton’s constant, and a₀ is Milgrom’s acceleration constant. The number 4/81 is a geometrical factor that assumes we’re observing a spherical system with isotropic orbits, neither of which is guaranteed even in the equilibrium case, and deviations from this idealized situation are noticeable. Still, this is as simple as it gets: if you know the mass, you can predict the characteristic speed at which stars move. Mass is all that matters: we don’t care about the radius as we must with Newton (v² = GM/r); the only other quantities are constants of nature.

But what do we mean by isolated? In MOND, it is that the internal acceleration of the system, g_in, exceeds that from external sources, g_ex: g_in ≫ g_ex. For a pressure supported dwarf, g_in ≈ 3σ²/r (so here the size of the dwarf does matter, as does the location of a star within it), while the external field from a giant host galaxy would be g_ex = V_f²/D where V_f is the flat rotation speed stipulated by the baryonic mass of the host and D is the distance from the host to the dwarf satellite. The distance is not a static quantity. As a dwarf orbits its host, D will vary by an amount that depends on the eccentricity of the orbit, and the external field will vary with it, so it is possible to have an orbit in which a dwarf satellite dips in and out of the EFE regime. Many Local Group dwarfs straddle the line g_in ≈ g_ex, and it takes time to equilibrate, so static thinking can go awry.

It is possible to define a sample of Local Group dwarfs that have sufficiently high internal accelerations (but also in the MOND regime with g_ex ≪ g_in ≪ a₀) that we can pretend they are isolated, and the above equation applies. Such dwarfs should^& fall on the BTFR, which they do:

The baryonic Tully-Fisher relation (BTFR) including pressure supported dwarfs (green points) with their measured velocity dispersions matched to the flat rotation speeds of rotationally supported galaxies (blue points) via the prescription of *McGaugh et al. (2021)*. The large blue points are rotators in the Local Group (with Andromeda and the Milky Way up near the top); smaller points are spirals with direct distance measurements (Schombert et al. 2020). The Local Group dwarfs assessed to be safe from external perturbation are on the BTFR (for V_f = 2σ); Crater 2 and the UDGs near NGC 1052 are not.

In contrast, three of the four the UDGs considered here do not fall on the BTFR. Should they?

Conventionally, in terms of dark matter, probably they should. There is no reason for them to deviate from whatever story we make up to explain the BTFR for everything else. That they do means we have to make up a separate story for them. I don’t want to go deeply into this here since the cold dark matter model doesn’t really explain the observed BTFR in the first place. But even accepting that it does so after invoking feedback (or whatever), does it tolerate deviants? In a broad sense, yes: since it doesn’t require the particular form of the BTFR that’s observed, it is no problem to deviate from it. In a more serious sense, no: if one comes up with a model that explains the small scatter of the BTFR, it is hard to make that same model defy said small scatter. I know, I’ve tried. Lots. One winds up with some form of special pleading in pretty much any flavor of dark matter theory on top of whatever special pleading we invoked to explain the BTFR in the first place. This is bad, but perhaps not as bad as it seems once one realizes that not everything has to be in equilibrium all the time.

In MOND, the BTFR is absolute – for isolated systems in equilibrium. In the EFE regime, galaxies can and should deviate from it even if they are in equilibrium. This always goes in the sense of having a lower characteristic velocity for a given mass, so below the line in the plot. To get above the line would require being out of equilibrium through some process that inflates velocities (if systematic errors are not to blame, which also sometimes happens.)

The velocity dispersion in the EFE regime (g_in ≪ g_ex ≪ a₀) is slightly more complicated than this isolated case:

This is just like Newton except the effective value of the gravitational constant is modified. It gets a boost^{^} by how far the system is in the MOND regime: G_eff ≈ G(a₀/g_ex). An easy way to tell which regime an object is in is to calculate both velocity dispersions σ_iso and σ_efe: the smaller one is the one that applies^#. An upshot of this is that systems in the EFE regime should deviate from the BTFR to the low velocity side. The amplitude of the deviation depends on the system and the EFE: both the size and mass matter, as does g_ex. Indeed, if an object is on an eccentric orbit, then the velocity dispersion can vary with the EFE as the distance of the satellite from its host varies, so over time the object would trace out some variable path in the BTFR plane.

Three of the four UDGs fall off the BTFR, so that sounds mostly right, qualitatively. Is it? Yes, for Crater 2, but but not really for the others. Even for Crater 2 it is only a partial answer, as non-equilibrium effects may play a role. This gets involved for Crater 2, then more so for the others, so let’s start with Crater 2.

Crater 2 – the velocity dispersion

The velocity dispersion of Crater 2 was correctly predicted a priori by the formula for σ_efe above. It is a tiny number, 2 km/s, and that’s what was subsequently observed. Crater 2 is very low mass, ~3 x 10⁵ M_☉, which is barely a globular cluster, but it is even more spread out than the typical dwarf spheroidal, having an effective surface density of only ~0.05 M_☉pc^-2. If it were isolated, MOND predicts that it would have a higher velocity dispersion – all of 4 km/s. That’s what it would take to put it on the BTFR above. The seemingly modest difference between 2 and 4 km/s makes for a clear offset. But despite its substantial current distance from the Milky Way (~ 120 kpc), Crater 2 is so low surface density that it is still subject to the external field effect, which lowers its equilibrium velocity dispersion. Unlike isolated galaxies, it should be offset from the BTFR according to MOND.

LCDM struggles to explain the low mass end of the BTFR because it predicts a halo mass-circular speed relation M_halo ~ V_halo³ that differs from the observed M_b ~ V_f⁴. A couple of decades ago, it looked like massive galaxies might be consistent with the lower power-law, but that anticipates higher velocities for small systems. The low velocity dispersion of Crater 2 is thus doubly weird in LCDM. It’s internal velocities are too small not just once – the BTFR is already lower than was expected – but twice, being below even that.

An object with a large radial extent like Crater 2 probes far out into its notional dark matter halo, making the nominal prediction^$ of LCDM around ~17 km/s, albeit with a huge expected scatter. Even if we can explain the low mass end of the BTFR and its unnaturally low scatter in LCDM, we now have to explain this exception to it – an exception that is natural in MOND, but is on the wrong side of the probability distribution for LCDM. That’s one of the troubles with tuning LCDM to mimic MOND: if you succeed in explaining the first thing, you still fail to anticipate the other. There is no EFE^% in LCDM, no reason to anticipate that σ_efe applies rather than σ_iso, and no reason to expect via feedback that this distinction has anything to do with the dynamical accelerations g_in and g_ex.

But wait – this is a post about non-equilibrium dynamics. That can happen in LCDM too. Indeed, one expects that satellite galaxies suffer tidal effects in the field of their giant host. The primary effect is that the dark matter subhalos in which dwarf satellites reside are stripped from the outside in. Their dark matter becomes part of the large halo of the host. But the stars are well-cocooned in the inner cusp of the NFW halo which is more robust than the outskirts of the subhalo, so the observable velocity dispersion barely evolves until most of the dark mass has been stripped away. Eventually, the stars too get stripped, forming tidal streams. Most of the damage occurs during pericenter passage when satellites are closest to their host. What’s left is no longer in equilibrium, with the details depending on the initial conditions of the dwarf on infall, the orbit, the number of pericenter passages, etc., etc.

What does not come out of this process is Crater 2 – at least not naturally. It has stars very far out – these should get stripped outright if the subhalo has been eviscerated to the point where its velocity dispersion is only 2 km/s. This tidal limitation has been noted by Errani et al.: “the large size of kinematically cold ‘feeble giant’ satellites like Crater 2 or Antlia 2 cannot be explained as due to tidal effects alone in the Lambda Cold Dark Matter scenario.” To save LCDM, we need something extra, some additional special pleading on top of non-equilibrium tidal effects, which is why I previously referred to Crater 2 as the Bullet Cluster of LCDM: an observation so problematic that it amounts to a falsification.

Crater 2 – the orbit

We held a workshop on dwarf galaxies on CWRU’s campus in 2017 where issues pertaining to both dark matter and MOND discussed. The case of Crater 2 was one of the things discussed, and it was included in the list of further tests for both theories (see above links). Basically the expectation in LCDM is that most subhalo orbits are radial (highly eccentric), so that is likely to be the case for Crater 2. In contrast, the ultradiffuse blob that is Crater 2 would not survive a close passage by the Milky Way given the strong tidal force exerted by MOND, so the expectation was for a more tangential (quasi-circular) orbit that keeps it at a safe distance.

Subsequently, it became possible to constrain orbits with Gaia data. The exact orbit depends on the gravitational potential of the Milky Way, which isn’t perfectly known. However, several plausible choices of the global potential give an an eccentricity around 0.6. That’s not exactly radial, but it’s pretty far from circular, placing the pericenter around 30 kpc. That’s much closer than its current distance, and well into the regime where it should be tidally disrupted in MOND. No way it survives such a close passage!

So which is it? MOND predicted the correct velocity dispersion, which LCDM struggles to explain. Yet the orbit is reasonable in LCDM, but incompatible with MOND.

Simulations of dwarf satellites

It occurs to me that we might be falling victim to static thinking somewhere. We talked about the impact of tides on dark matter halos a bit above. What should we expect in MOND?

The first numerical simulations of dwarf galaxies orbiting a giant host were conducted by Brada & Milgrom (2000). Their work is specific to the Aquadratic Lagrangian (AQUAL) theory proposed by Bekenstein & Milgrom (1984). This was the first demonstration that it was possible to write a version of MOND that conserved momentum and energy. Since then, a number of different approaches have been demonstrated. These can be subtly different, so it is challenging to know which (if any) is correct. Sorting that out is well beyond the scope of this post, so let’s stick to what we can learn from Brada & Milgrom.

Brada & Milgrom followed the evolution of low surface density dwarfs of a range of masses as they orbited a giant host galaxy. One thing they found was that the behavior of the numerical model could deviate from the analytic expectation of quasi-equilibrium enshrined in the equations above. For an eccentric orbit, the external field varies with distance from the host. If there is enough time to respond to this, the change can be adiabatic (reversible), and the static approximation may be close enough. However, as the external field varies more rapidly and/or the dwarf is more fragile, the numerical solution departs from the simple analytic approximation. For example:

**Fig. 2** of Brada & Milgrom (2000): showing the numerically calculated (dotted line) variation of radius (left) and characteristic velocity (right) for a dwarf on a mildly eccentric orbit (peri- and apocenter of roughly 60 and 90 kpc, respectively, for a Milky Way-like host). Also shown is the variation in the EFE as the dwarf’s distance from the host varies (solid line). Dwarfs go through a breathing mode of increasing/decreasing size and decreasing/increasing velocity dispersion in phase with the orbit. If this process is adiabatic, it tracks the solid line and the static EFE approximation holds. This is not always the case in the simulation, so applying our usual assumption of dynamical equilibrium will result in an error stipulated by the difference between the dotted and solid lines. The amplitude of this error depends on the size, mass, and orbital history of each and every dwarf satellite.

As long as the behavior is adiabatic, the dwarf can be stable indefinitely even as it goes through periodic expansion and contraction in phase with the orbit. Departure from adiabaticity means that every passage will be different. Some damage will be done on the first passage, more on the second, and so on. As a consequence, reality will depart from our simple analytic expectations.

I was aware of this when I made the prediction for the velocity dispersion of Crater 2, and hedged appropriately. Indeed, I worried that Crater 2 should already be out of equilibrium. Nevertheless, I took solace in two things: first, the orbital timescale is long, over a Gyr, so departures from the equilibrium prediction might not have had time to make a dramatic difference. Second, this expectation is consistent with the slow evolution of the characteristic velocity for the most Crater 2-like, m=1 model of Brada & Milgrom (bottom track in the right panel below):

**Fig. 4** of Brada & Milgrom (2000): The variation of the size and characteristic velocity of dwarf models of different mass. The more massive models approximate the adiabatic limit, which gradually breaks down for the lowest mass models. In this example, the m = 1 and 2 models explode, with the scale size growing gradually without recovering.

What about the size? That is not constant except for the most massive (m=16) model. The m=3 and 4 models recover, albeit not adiabatically. The m=4 model almost returns to its original size, but the m=3 model has puffed up after one orbit. The m=1 and 2 models explode.

One can see this by eye. The continuous growth in radii of the lower mass models is obvious. If one looks closely, one can also see the expansion then contraction of the heavier models.

**Fig. 5** of Brada & Milgrom (2000): AQUAL numerical simulations dwarf satellites orbiting a more massive host galaxy. The parameter m describes the mass and effective surface density of the satellite; all the satellites are in the MOND regime and subject to the external field of the host galaxy, which exceeds their internal accelerations. In dimensionless simulation units, m = 5 x 10^-5, which for a satellite of the Milky Way corresponds roughly to a stellar mass of 3 x 10⁶ *M_☉*. For real dwarf satellite galaxies, the scale size is also relevant, but the sequence of m above suffices to illustrate the increasingly severe effects of the external field as m decreases.

The current size of Crater 2 is unusual. It is very extended for its mass. If the current version of Crater 2 has a close passage with the Milky Way, it won’t survive. But we know it already had a close passage, so it should be expanding now as a result. (I did discuss the potential for non-equilibrium effects.) Knowing now that there was a pericenter passage in the (not exactly recent) past, we need to imagine running back the clock on the simulations. It would have been smaller in the past, so maybe it started with a normal size, and now appears so large because of its pericenter passage. The dynamics predict something like that; it is static thinking to assume it was always thus.

The dotted line shows a possible evolutionary track for Crater 2 as it expands after pericenter passage. Its initial condition would have been amongst the other dwarf spheroidals. It could also have lost some mass in the process, so any of the green low-mass dwarfs might be similar to the progenitor.

This is a good example of a phenomena I’ve encountered repeatedly with MOND. It predicts something right, but seems to get something else wrong. If we’re already sure it is wrong, we stop there and never think further. But when one bothers to follow through on what the theory really predicts, more often than not the apparently problematic observation is in fact what we should have expected in the first place.

DF2 and DF4

DF2 and DF4 are two UDGs in the vicinity of the giant galaxy NGC 1052. They have very similar properties, and are practically identical in terms of having the same size and mass within the errors. They are similar to Crater 2 in that they are larger than other galaxies of the same mass.

When it was first discovered, NGC 1052-DF2 was portrayed as a falsification of MOND. On closer examination, had I known about it, I could have used MOND to correctly predict its velocity dispersion, just like the dwarfs of Andromeda. This seemed like yet another case where the initial interpretation contrary to MOND melted away to actually be a confirmation. At this point, I’ve seen literally hundreds^{^^}of cases like that. Indeed, this particular incident made me realize that there would always be new cases like that, so I decided to stop spending my time addressing every single case.

Since then, DF2 has been the target of many intensive observing campaigns. Apparently it is easier to get lots of telescope time to observe a single object that might have the capacity to falsify MOND than it is to get a more modest amount to study everything else in the universe. That speaks volumes about community priorities and the biases that inform them. At any rate, there is now lots more data on this one object. In some sense there is too much – there has been an active debate in the literature over the best distance determination (which affects the mass) and the most accurate velocity dispersion. Some of these combinations are fine with MOND, but others are not. Let’s consider the worst case scenario.

In the worst case scenario, both DF2 and DF4 are too far from NGC 1052 for its current EFE to have much impact, and they have relatively low velocity dispersions for their luminosity, around 8 km/s, so they fall below the BTFR. Worse for MOND is that this is about what one expects from Newton for the stars alone. Consequently, these galaxies are sometimes referred to as being “dark matter free.” That’s a problem for MOND, which predicts a larger velocity dispersion for systems in equilibrium.

Perhaps we are falling prey to static thinking, and these objects are not in equilibrium. While their proximity to neighboring galaxies and the EFE to which they are presently exposed depends on the distance, which is disputed, it is clear that they live in a rough neighborhood with lots of more massive galaxies that could have bullied them in a close passage at some point in the past. Looking at Fig. 4 of Brada & Milgrom above, I see that galaxies whacked out of equilibrium not only expand in radius, potentially explaining the unusually large sizes of these UDGs, but they also experience a period during which their velocity dispersion is below the equilibrium value. The amplitude of the dip in these simulations is about right to explain the appearance of being dark-matter-free.

It is thus conceivable that DF2 and DF4 (the two are nearly identical in the relevant respects) suffered some sort of interaction that perturbed them into their current state. Their apparent absence of a mass discrepancy and the apparent falsification of MOND that follows therefrom might simply be a chimera of static thinking.

Make no mistake: this is a form of special pleading. The period of depressed velocity dispersion does not last indefinitely, so we have to catch them at a somewhat special time. How special depends on the nature of the interaction and its timescale. This can be long in intergalactic space (Gyrs), so it may not be crazy special, but we don’t really know how special. To say more, we would have to do detailed simulations to map out the large parameter space of possibilities for these objects.

I’d be embarrassed for MOND to have to make this kind of special pleading if we didn’t also have to do it for LCDM. A dwarf galaxy being dark matter free in LCDM shouldn’t happen. Galaxies form in dark matter halos; it is very hard to get rid of the dark matter while keeping the galaxy. The most obvious way to do it, in rare cases, is through tidal disruption, though one can come up with other possibilities. These amount to the same sort of special pleading we’re contemplating on behalf of MOND.

Recently, Tang et al. (2024) argue that DF2 and DF4 are “part of a large linear substructure of dwarf galaxies that could have been formed from a high-velocity head-on encounter of two gas-rich galaxies” which might have stripped the dark matter while leaving the galactic material. That sounds… unlikely. Whether it is more or less unlikely than what it would take to preserve MOND is hard to judge. It appears that we have to indulge in some sort of special pleading no matter what: it simply isn’t natural for galaxies to lack dark matter in a universe made of dark matter, just as it is unnatural for low acceleration systems to not manifest a mass discrepancy in MOND. There is no world model in which these objects make sense.

Tang et al. (2024) also consider a number of other possibilities, which they conveniently tabulate:

There are many variations on awkward hypotheses for how these particular UDGs came to be in LCDM. They’re all forms of special pleading. Even putting on my dark matter hat, most sound like crazy talk to me. (Stellar feedback? Really? Is there anything it cannot do?) It feels like special pleading on top of special pleading; it’s special pleading all the way down. All we have left to debate is which form of special pleading seems less unlikely than the others.

I don’t find this debate particularly engaging. Something weird happened here. What that might be is certainly of interest, but I don’t see how we can hope to extract from it a definitive test of world models.

Antlia 2

The last of the UDGs in the first plot above is Antlia 2, which I now regret including – not because it isn’t interesting, but because this post is getting exhausting. Certainly to write, perhaps to read.

Antlia 2 is on the BTFR, which is ordinarily normal. In this case it is weird in MOND, as the EFE should put it off the BTFR. The observed velocity dispersion is 6 km/s, but the static EFE formula predicts it should only be 3 km/s. This case should be like Crater 2.

First, I’d like to point out that, as an observer, it is amazing to me that we can seriously discuss the difference between 3 and 6 km/s. These are tiny numbers by the standard of the field. The more strident advocates of cold dark matter used to routinely assume that our rotation curve observations suffered much larger systematic errors than that in order to (often blithely) assert that everything was OK with cuspy halos so who are you going to believe, our big, beautiful simulations or those lying data?

I’m not like that, so I do take the difference seriously. My next question, whenever MOND is a bit off like this, is what does LCDM predict?

I’ll wait.

Well, no, I won’t, because I’ve been waiting for thirty years, and the answer, when there is one, keeps changing. The nominal answer, as best I can tell, is ~20 km/s. As with Crater 2, the large scale size of this dwarf means it should sample a large portion of its dark matter halo, so the expected characteristic speed is much higher than 6 km/s. So while the static MOND prediction may be somewhat off here, the static LCDM expectation fares even worse.

This happens a lot. Whenever I come across a case that doesn’t make sense in MOND, it usually doesn’t make sense in dark matter either.

In this case, the failure of the static-case prediction is apparently caused by tidal perturbation. Like Crater 2, Antlia 2 may have a large half-light radius because it is expanding in the way seen in the simulations of Brada & Milgrom. But it appears to be a bit further down that path, with member stars stretched out along the orbital path. They start to trace a small portion of a much deeper gravitational potential, so the apparent velocity dispersion goes up in excess of the static prediction.

**Fig. 9** from Ji et al. (2021) showing tidal features in Antlia 2 considering the effects of the Milky Way alone (left panel) and of the Milky Way and the Large Magellanic Cloud together (central panel) along with the position-velocity diagram from individual stars (right panel). The object is clearly not the isotropic, spherical cow presumed by the static equation for the velocity dispersion. Indeed, it is elongated as would be expected from tidal effects, with individual member stars apparently leaking out.

This is essentially what I inferred must be happening in the ultrafaint dwarfs of the Milky Way. There is no way that these tiny objects deep in the potential well of the Milky Way escape tidal perturbation^%% in MOND. They may be stripped of their stars and their velocity dispersions mage get tidally stirred up. Indeed, Antlia 2 looks very much like the MOND prediction for the formation of tidal streams from such dwarfs made by McGaugh & Wolf (2010). Unlike dark matter models in which stars are first protected, then lost in pulses during pericenter passages, the stronger tides of MOND combined with the absence of a protective dark matter cocoon means that stars leak out gradually all along the orbit of the dwarf. The rate is faster when the external field is stronger at pericenter passage, but the mass loss is more continuous. This is a good way to make long stellar streams, which are ubiquitous in the stellar halo of the Milky Way.

So… so what?

It appears that aspects of the observations of the UDGs discussed here that seem problematic for MOND may not be as bad for the theory as they at first seem. Indeed, it appears that the noted problems may instead be a consequence of the static assumptions we usually adopt to do the analysis. The universe is a dynamic place, so we know this assumption does not always hold. One has to judge each case individually to assess whether this is reasonable or not.

In the cases of Crater 2 and Antlia 2, yes, the stranger aspects of the observations fit well with non-equilibrium effects. Indeed, the unusually large half-light radii of these low mass dwarfs may well be a result of expansion after tidal perturbation. That this might happen was specifically anticipated for Crater 2, and Antlia 2 fits the bill described by McGaugh & Wolf (2010) as anticipated by the simulations of Brada & Milgrom (2000) even though it was unknown at the time.

In the cases of DF2 and DF4, it is less clear what is going on. I’m not sure which data to believe, and I want to refrain from cherry-picking, so I’ve discussed the worst-case scenario above. But the data don’t make a heck of a lot of sense in any world view; the many hypotheses made in the dark matter context seem just as contrived and unlikely as a tidally-induced, temporary dip in the velocity dispersion that might happen in MOND. I don’t find any of these scenarios to be satisfactory.

This is a long post, and we have only discussed four galaxies. We should bear in mind that the vast majority of galaxies do as predicted by MOND; a few discrepant cases are always to be expected in astronomy. That MOND works at all is a problem for the dark matter paradigm: that it would do so was not anticipated by any flavor of dark matter theory, and there remains no satisfactory explanation of why MOND appears to happen in a universe made of dark matter. These four galaxies are interesting cases, but they may be an example of missing the forest for the trees.

^*As it happens, the surface brightness threshold adopted in the definition of UDGs is exactly the same as I suggested for VLSBGs (very low surface brightness galaxies: McGaugh 1996), once the filter conversions have been made. At the time, this was the threshold of our knowledge, and I and other early pioneers of LSB galaxies were struggling to convince the community that such things might exist. Up until that time, the balance of opinion was that they did not, so it is gratifying to see that they do.

^**This expectation is specific to MOND; it doesn’t necessarily hold in dark matter where the acceleration in the central regions of diffuse galaxies can be dominated by the cusp of the dark matter halo. These were predicted to exceed what is observed, hence the cusp-core problem.

⁺Measuring by surface brightness, Crater 2 and Antlia 2 are two orders of magnitude more diffuse than the prototypical ultradiffuse galaxies DF2 and DF4. Crater 2 is not quite large enough to count as a UDG by the adopted size definition, but Antlia 2 is. So does that make it super-ultra diffuse? Would it even be astronomy without terrible nomenclature?

^&I didn’t want to use a MOND-specific criterion in McGaugh et al. (2021) because I was making a more general point, so the green points are overly conservative from the perspective of the MOND isolation criterion: there are more dwarfs for which this works. Indeed, we had great success in predicting velocity dispersions in exactly this fashion in McGaugh & Milgrom (2013a, 2013b). And XXVIII was a case not included above that we highlighted as a great test of MOND, being low mass (~4×10⁵ M_☉) but still qualifying as isolated, and its dispersion came in (6.6^+2.9_-2.1 km/s in one measurement, 4.9 ± 1.6 km/s in another) as predicted a priori (4.3^+0.8_-0.7 km/s). Hopefully the Rubin Observatory will discover many more similar objects that are truly isolated; these will be great additional tests, though one wonders how much more piling-on needs to be done.

^{^}This is an approximation that is reasonable for the small accelerations involved. More generally we have G_eff = G/μ(|g_ex+g_in|/a₀) where μ is the MOND interpolation function and one takes the vector sum of all relevant accelerations.

^#This follows because the boost from MOND is limited by how far into the low acceleration regime an object is in. If the EFE is important, the boost will be less than in the isolated case. As we said in 2013, “the case that reports the lower velocity dispersion is always the formally correct one.” I mention it again here because apparently people are good at scraping equations from papers without reading the associated instructions, so one gets statements like “the theory does not specify precisely when the EFE formula should replace the isolated MOND prediction.” Yes it does. We told you precisely when the EFE formula should replace the isolated formula. It is when it reports the lower velocity dispersion. We also noted this as the reason for not giving σ_efe in the tables in cases it didn’t apply, so there were multiple flags. It took half a dozen coauthors to not read that. I’d hate to see how their Ikea furniture turned out.

^$As often happens with LCDM, there are many nominal predictions. One common theme is that “Despite spanning four decades in luminosity, dSphs appear to inhabit halos of comparable peak circular velocity.” So nominally, one would expect a faint galaxy like Crater 2 to have a similar velocity dispersion to a much brighter one like Fornax, and the luminosity would have practically no power to predict the velocity dispersion, contrary to what we observe in the BTFR.

^%There is the 2-halo term – once you get far enough from the center of a dark matter halo (the 1-halo term), there are other halos out there. These provide additional unseen mass, so can boost the velocity. The EFE in MOND has the opposite effect, and occurs for completely different physical reasons, so they’re not at all the same.

^{^^}For arbitrary reasons of human psychology, the threshold many physicists set for “always happens” is around 100 times. That is, if a phenomenon is repeated 100 times, it is widely presumed to be a general rule. That was the threshold Vera Rubin hit when convincing the community that flat rotation curves were the general rule, not just some peculiar cases. That threshold has also been hit and exceeded by detailed MOND fits to rotation curves, and it seems to be widely accepted that this is the general rule even if many people deny the obvious implications. By now, it is also the case for apparent exceptions to MOND ceasing to be exceptions as the data improve. Unfortunately, people tend to stop listening at what they want to hear (in this case, “falsifies MOND”) and fail to pay attention to further developments.

^%%It is conceivable that the ultrafaint dwarfs might elude tidal disruption in dark matter models if they reside in sufficiently dense dark matter halos. This seems unlikely given the obvious tidal effects on much more massive systems like the Sagittarius dwarf and the Magellanic Clouds, but it could in principle happen. Indeed, if one calculates the mass density from the observed velocity dispersion, one infers that they do reside in dense dark matter halos. In order to do this calculation, we are obliged to assume that the objects are in equilibrium. This is, of course, a form of static thinking: the possibility of tidal stirring that enhances the velocity dispersion above the equilibrium value is excluded by assumption. The assumption of equilibrium is so basic that it is easy to unwittingly engage in circular reasoning. I know, as I did exactly that myself to begin with.

Non-equilibrium dynamics in galaxies that appear to lack dark matter: tidal dwarf galaxies

There are a number of galaxies that have been reported to lack dark matter. This is weird in a universe made of dark matter. It is also weird in MOND, which (if true) is what causes the inference of dark matter. So how can this happen?

In most cases, it doesn’t. These claims not only don’t make sense in either context, they are simply wrong. I don’t want to sound too harsh, as I’ve come close to making the same mistake myself. The root cause of this mistake is often a form of static thinking in dynamic situations that the here and now is always a representative test. The basic assumption we have to make to interpret observed velocities in terms of mass is that systems are in (or close to) gravitational equilibrium so that the kinetic energy is a measure of the gravitational potential. In most places, this is a good assumption, so we tend to forget we even made it.

However, no assumption is ever perfect. For example, Gaia has revealed a wealth of subtle non-equilibrium effects in the Milky Way. These are not so large as to invalidate the basic inference of the mass discrepancy, but neither can they be entirely ignored. Even maintaining the assumption of a symmetric but non-smooth mass profile in equilibrium complicates the analysis.

Since the apparent absence of dark matter is unexpected in either theory, one needs to question the assumptions whenever this inference is made. There is one situation in which it is expected, so let’s consider that special case:

Tidal dwarf galaxies

Most dwarf galaxies are primordial – they are the way they are because they formed that way. However, it is conceivable that some dwarfs may form in the tidal debris of collisions between large galaxies. These are tidal dwarf galaxies (TDGs). Here are some examples of interacting systems containing candidate TDGs:

**Fig. 1** from Lelli et al. (2015): *images of interacting systems with TDG candidates noted in yellow.*

I say candidate TDGs because it is hard to be sure a particular object is indeed tidal in origin. A good argument can be made that TDGs require such special conditions to form that perhaps they should not be able to form at all. As debris in tidal arms is being flung about in the (~ 200 km/s) potential well of a larger system, it is rather challenging for material to condense into a knot with a much smaller potential well (< 50 km/s). It can perhaps happen if the material in the tidal stream is both lumpy (to provide a seed to condense on) and sufficiently comoving (i.e., the tidal shear of the larger system isn’t too great), so maybe it happens on rare occasions. One way to distinguish TDGs from primordial dwarfs is metallicity: typical primordial dwarfs have low metallicity while TDGs have the higher metallicity of the giant system that is the source of the parent material.

A clean test of hypotheses

TDGs provide an interesting test of dark matter and MOND. In the vast majority of dark matter models, dark matter halos are dynamically hot, quasi-spherical systems with the particles that compose the dark matter (whatever it is) on eccentric, randomly oriented orbits that sum to a big, messy blob. Arguably it has to be this way in order to stabilize the disks of spiral galaxies. In contrast, the material that composes the tidal tails in which TDGs form originates in the baryonic material of the dynamically cold spiral disks where orbits are nearly circular in the same direction in the same thin plane. The phase space – the combination of position x,y,z and momentum v_x,v_y,v_z – of disk and halo couldn’t be more different. This means that when two big galaxies collide or have a close interaction, everything gets whacked and the two components go their separate ways. Starting in orderly disks, the stars and gas make long, coherent tidal tails. The dark matter does not. The expectation from these basic phase space considerations is consistent with detailed numerical simulations.

We now have a situation in which the dark matter has been neatly segregated from the luminous matter. Consequently, if TDGs are able to form, they must do it only* with baryonic mass. The ironic prediction of a universe dominated by dark matter is that TDGs should be devoid of dark matter.

In contrast, one cannot “turn off” the force law in MOND. MOND can boost the formation of TDGs in the first place, but if said TDGs wind up in the low acceleration regime, they must evince a mass discrepancy. So the ironic prediction here is that, in ignorance of MOND, MOND means that we would infer that TDGs do have dark matter.

Got that? Dark matter predicts TDGs with no dark matter. MOND predicts TDGs that look like they do have dark matter. That’s not confusing at all.

Clean in principle, messy in practice

Tests of these predictions have a colorful history. Bournaud et al. (2007) did a lovely job of combining simulations with observations of the Seashell system (NGC 5291 above) and came to a striking conclusion: the rotation curves of TDGs exceeded that expected for the baryons alone:

**Fig. 2** from Bournaud et al. (2007) *showing the rotation curves for the three TDGs identified in the image above.*

This was a strange, intermediary result. TDGs had more dark matter than the practically zero expected in LCDM, but less than comparable primordial dwarfs as expected in MOND. That didn’t make sense in either theory. They concluded that there must be a component of some other kind of dark matter that was not the traditional dark halo, but rather part of the spiral disk to begin with, perhaps unseen baryons in the form of very cold molecular gas.

Gentile et al. (2007) reexamined the situation, and concluded that the inclinations could be better constrained. When this was done, the result was more consistent with the prediction of MOND and the baryonic Tully-Fisher relation (BTFR. See their Fig. 2).

**Fig. 1** from Gentile et al. (2007): Rotation curve data (full circles) of the 3 tidal dwarf galaxies (Bournaud et al. 2007). The lower (red) curves are the Newtonian contribution V_bar of the baryons (and its uncertainty, indicated as dotted lines). The upper (black) curves are the MOND prediction and its uncertainty (dotted lines). The top panels have as an implicit assumption (following Bournaud et al.) an inclination angle of 45 degrees. In the middle panels the inclination is a free parameter, and the bottom panels show the fits made with the first estimate for the external field effect (EFE).

Clearly there was room for improvement, both in data quality and quantity. We decided to have a go at it ourselves, ultimately leading to Lelli et al. (2015), which is the source of the pretty image above. We reanalyzed the Seashell system, along with some new TDG candidates.

Making sense of these data is not easy. TDG candidates are embedded in tidal features. It is hard to know where the dwarf ends and the tidal stream begins, or even to be sure there is a clear distinction. Here is an example of the northern knot in the Seashell system:

**Fig. 5** from Lelli et al. (2015): *Top panels*: optical image (*left*), total H I map (*middle*), and H I velocity field (*right*). The dashed ellipse corresponds to the disc model described in Sect. 5.1. The cross and dashed line illustrate the kinematical centre and major axis, respectively. In the bottom-left corner, we show the linear scale (optical image) and the H I beam (total H I map and velocity field) as given in Table 6. In the total H I map, contours are at ~4.5, 9, 13.5, 18, and 22.5 M_⊙ pc^-2. *Bottom panels*: position-velocity diagrams obtained from the observed cube (*left*), model cube (*middle*), and residual cube (*right*) along the major and minor axes. Solid contours range from 2σ to 8σ in steps of 1σ. Dashed contours range from −2σ to −4σ in steps of −1σ. The horizontal and vertical lines correspond to the systemic velocity and dynamical centre, respectively.

Both the distribution of gas and the velocities along the tidal tail often blend smoothly across TDG candidates, making it hard to be sure they have formed a separate system. In the case above, I can see what we think is the velocity field of the TDG alone (contained by the ellipse in the upper right panel), but is that really an independent system that has completely decoupled from the tidal material from which it formed? Definite maybe!

Federico Lelli did amazing work to sort through these difficult-to-interpret data. At the end of the day, he found that there was no need for dark matter in any of these TDG candidates. The amplitude of the apparent circular speed was consistent with the enclosed mass of baryons.

**Figs. 11 and 13** from Lelli et al. (2015): the enclosed dynamical-to-baryonic mass ratio (left) and baryonic Tully-Fisher relation (right). TDGs (red points) are consistent with a mass ratio of unity: the observed baryons suffice; no dark matter is inferred. Contrary to Gentile et al., this manifests as a clear offset from the BTFR followed by normal galaxies.

Taken at face value, this absence of dark matter is a win for a universe made of dark matter and a falsification of MOND.

So we were prepared to say that, and did, but as Federico checked the numbers, it occurred to him to check the timescales. Mergers like this happen over the course of a few hundred million years, maybe a billion. The interactions we observe are ongoing; just how far into the process are they? Have the TDGs had time to settle down into dynamical equilibrium? That is the necessary assumption built into the mass ratio plotted above: the dynamical mass assumes the measured speed is that of a test particle in an equilibrium orbit. But these systems are manifestly not in equilibrium, at least on large scales. Maybe the TDGs have had time to settle down?

We can ask how long it takes to make an orbit at the observed speed, which is low by the standards of such systems (hence their offset from Tully-Fisher). To quote from the conclusions of the paper,

These [TDG] discs, however, have orbital times ranging from ~1 to ~3 Gyr, which are significantly longer than the TDG formation timescales (≲1 Gyr). This raises the question as to whether TDGs have had enough time to reach dynamical equilibrium.
Lelli et al. (2015)

So no, not really. We can’t be sure the velocities are measuring the local potential well as we want them to do. A particle should have had time to go around and around a few times to settle down in a new equilibrium configuration; here they’ve made 1/3, maybe 1/2 half of one orbit. Things have not had time to settle down, so there’s not really a good reason to expect that the dynamical mass calculation is reliable.

It would help to study older TDGs, as these would presumably have had time to settle down. We know of a few candidates, but as systems age, it becomes harder to gauge how likely they are to be legitimate TDGs. When you see a knot in a tidal arm, the odds seem good. If there has been time for the tidal stream to dissipate, it becomes less clear. So if such a thing turns out to need dark matter, is that because it is a TDG doing as MOND predicted, or just a primordial dwarf we mistakenly guessed was a TDG?

We gave one of these previously unexplored TDG candidates to a grad student. After much hard work combining observations from both radio and optical telescopes, she has demonstrated that it isn’t a TDG at all, in either paradigm. The metallicity is low, just as it should be for a primordial dwarf. Apparently it just happens to be projected along a tidal tail where it looks like a decent candidate TDG.

This further illustrates the trials and tribulations we encounter in trying to understand our vast universe.

*One expects cold dark matter halos to have subhalos, so it seems wise to suspect that perhaps TDGs condense onto these. Phase space says otherwise. It is not sufficient for tidal debris to intersect the location of a subhalo, the material must also “dock” in velocity space. Since tidal arms are being flung out at the speed that is characteristic of the giant system, the potential wells of the subhalos are barely speed bumps. They might perturb streams, but the probability of them being the seeds onto which TDGs condense is small: the phase space just doesn’t match up for the same reasons the baryonic and dark components get segregated in the first place. TDGs are one galaxy formation scenario the baryons have to pull off unassisted.

The MHONGOOSE survey of atomic gas in and around galaxies

I have been spending a lot of time lately writing up a formal paper on high redshift galaxies, so haven’t had much time to write here. The paper is a lot more involved than I told you so, but yeah, I did. Repeatedly. I do have a start on a post on self-interacting dark matter that I hope eventually to get back to. Today, I want to give a quick note about the MHONGOOSE survey. But first, a non-commercial interruption.

Triton Station joins Rogue Scholar

In internet news, Triton Station has joined Rogue Scholar. The blog itself hasn’t moved; Rogue Scholar is a community of science blogs. It provides some important capabilities, including full-text search, long-term archiving, DOIs, and metadata. The DOIs (Digital Object Identifiers) were of particular interest to me, as they have become the standard for identifying unique articles in regular academic journals now that these have mostly (entirely?) gone on-line. I had not envisioned ever citing this blog in a refereed journal, but a DOI makes it possible to do so. Any scientists who find a post useful are welcome to make use of this feature. I’m inclined to follow the example of JCAP and make the format volume, page be yearmonth, date (YYMM, DD), which comes out to Triton Station (2022), 2201, 03 in the standard astronomy journal format. I do not anticipate continuing to publish in the twenty second century, so no need for YYYYMM, Y2K experience notwithstanding.

For everyone interested in science, Rogue Scholar is a great place to find new blogs.

MHONGOOSE

In science news, the MHONGOOSE collaboration has released its big survey summary paper. Many survey science papers are in the pipeline. Congratulations to all involved, especially PI Erwin de Blok.

Erwin was an early collaborator of mine who played a pivotal role in measuring the atomic gas properties of low surface brightness galaxies, establishing the cusp-core problem, and that low surface brightness galaxies are dark matter dominated (or at least evince large mass discrepancies, as predicted by MOND). He has done a lot more since then, among them playing a leading role in the large VLA survey of nearby galaxies, THINGS. In astronomy we’re always looking forward to the next big survey – its a big universe; there’s always more out there. So after THINGS he conceived and began work on MHONGOOSE. It has been a long road tied to the construction of the MeerKAT array of radio telescopes – a major endeavor on the road to the ambitious Square Kilometer Array.

I was involved in the early phases of the MHONGOOSE project, helping to select the sample of target galaxies (it is really important to cover the full dynamic range of galaxy properties, dwarf to giant) and define the aspirational target sensitivity. HI observations often taper off below a column density of 10²⁰ hydrogen atoms per cm² (about 1 solar mass per square parsec). With work, one can get down to a few times 10¹⁹ cm^-2. We want to go much deeper to see how much farther out the atomic gas extends. It was already known to go further out than the stars, but how far? Is there a hard edge, or just a continuous fall off?

We also hope to detect new dwarf galaxies that are low surface brightness in HI. There could, in theory, be zillions of such things lurking in all the dark matter subhalos that are predicted to exist around big galaxies. Irrespective of theory, are there HI gas-rich galaxies that are entirely devoid of stars? Do such things exist? People have been looking for them a long time, and there are now many examples of galaxies that are well over 95% gas, but there always seem to be at least a few stars associated with them. Is this always true? If we have cases that are 98, 99% gas, why not 100%? Do galaxies with gas always manage to turn at least a little of it into stars? They do have a Hubble time to work on it, so it is also a question why there is so much gas still around in these cases.

And… a lot of other things, but I don’t want to be here all day. So just a few quick highlights from the main survey paper. First, the obligatory sensitivity diagram. This shows how deep the survey reaches (lower column density) as a function of resolution (beam size). You want to see deeply and you want to resolve what you see, so ideally both of these numbers would be small. MHONGOOSE undercuts existing surveys, and is unlikely to be bettered until the full SKA comes on-line, which is still a long way off.

*Sensitivity versus resolution in HI surveys.*

And here are a couple of individual galaxy observations:

Optical images and the HI moment zero, one, and two maps. The moment zero map of the intensity of 21 cm radiation tells us where the atomic gas is, and how much of it there is. The moment one map is the velocity field from which we can construct a rotation curve. The second moment measures the velocity dispersion of the gas.

These are beautiful data. The spiral arms appear in the HI as well as in starlight, and continue in HI to larger radii. The outer edge of the HI disk is pretty hard; there doesn’t seem to be a lot of extra gas at low column densities extending indefinitely into the great beyond. I’m particular struck by the velocity dispersion of NGC 1566 tracking the spiral structure: this means the spiral arms have mass, and any stirring caused by star formation is localized to the spirals where much of the star formation goes on. That’s natural, but the surroundings seem relatively unperturbed: feedback is happening locally, but not globally. The velocity field of NGC 5068 has a big twist in the zero velocity contour (the thick line dividing the red receding side from the blue approaching side); this is a signature of non-circular motion, probably caused in this case by the visible bar. These are two-dimensional examples of Renzo’s rule (Sancisi’s Law), in which features in the visible mass distribution correspond to features in the kinematics.

I’ll end with a quick peak at the environments around some MHONGOOSE target galaxies:

*Fields where additional galaxies (in blue) are present around the central target*.

This is nifty on many levels. First, some (presumptively satellite) dwarf galaxies are detected. That in itself is a treat to me: once upon a time, Renzo Sancisi asked me to smooth the bejeepers out of the LSB galaxy data cubes to look for satellites. After much work, we found nada. Nothing. Zilch. It turns out that LSB galaxies are among the most isolated galaxy types in the universe. So that we detect some things here is gratifying, even in targets that are not LSBs.

Second, there are not a lot of new detections. The halos of big galaxies are not swimming in heretofore unseen swarms of low column density gas clouds. There can always be more at sensitivities yet unreached, but the data sure don’t encourage that perspective. MHONGOOSE is sensitive to very low mass gas clouds. The exact limit is distance-dependent, but a million solar masses of atomic gas should be readily visible. That’s a tiny amount by extragalactic standards, about one globular cluster’s worth of material. There’s just not a lot there.

Disappointing as the absence of zillions of new detections may be discovery-wise, it does teach us some important lessons. Empirically, galaxies look like island universes in gas as well as stars. There may be a few outlying galaxies, but they are not embedded in an obvious cosmic network of ephemeral cold gas. Nor are there thousands of unseen satellites/subhalos suddenly becoming visible – at least not in atomic gas. Theorists can of course imagine other things, but we observers can only measure one thing at a time, as instrumentation and telescope availability allows. This is a big step forward.

Leveling the Playing Field of Dwarf Galaxy Kinematics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Dwarf Satellite Galaxies. III. The dwarfs of Andromeda

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

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

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

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

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

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

Tollerud2011_ml_scatter — Virial mass of the dark matter halo as a function of galaxy luminosity. Dwarfs satellites reside in the wide colored band of low luminosities.

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

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

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

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

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

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

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

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

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

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

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

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

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

AndDwarfswithGoldStars.002 — The velocity dispersions of the dwarf satellites of Andromeda. Each numbered box corresponds to one dwarf (x=1 is for And I and so on). Measured velocity dispersions have a number next to them that is the number of stars on which the measurement is based. MOND predictions are circles: green if isolated, open if the EFE applies. Points appear within each box in the order they appeared in the literature, from left to right. The vast majority of Andromeda’s dwarfs are consistent with MOND (large green circles). Two cases are ambiguous (large yellow circles), having velocity dispersions based only a few stars. Only And V appears to be problematic (large red circle).

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

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

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

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

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

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

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

AndDwarfswithGoldStars.003 — The velocity dispersions of the dwarfs of Andromeda, highlighting photometrically matched pairs – dwarfs that should be indistinguishable, but aren’t because of the EFE.

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

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

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

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

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

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

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

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

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

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

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.

Triton Station

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

Category: Dwarf satellite galaxies