Wide Binary Weirdness

My last post about the Milky Way was intended to be a brief introduction to our home galaxy in order to motivate the topic of binary stars. There’s too much interesting to say about the Milky Way as a galaxy, so I never got past that. Even now I feel the urge to say more, like with this extended rotation curve that I included in my contribution to the proceedings of IAU 379.

The RAR-based model rotation curve of the Milky Way extrapolated to large radii (note the switch to a logarithmic scale at 20 kpc!) for comparison to the halo stars of Bird et al (2022) and the globular clusters of Watkins et al (2019). The location of the solar system is noted by the red circle.

But instead I want to talk about data for binary stars from the Gaia mission. Gaia has been mapping the positions and proper motions of stars in the local neighborhood with unprecedented accuracy. These can be used to measure distances via trigonometric parallax, and speeds along the sky. The latter once seemed impossible to obtain in numbers with much precision; thanks to Gaia such data now outnumber radial (line of sight) velocities of comparable accuracy from spectra. That is a mind-boggling statement to anyone who has worked in the field; for all of my career (and that of any living astronomer), radial velocities have vastly outnumbered comparably well-measured proper motions. Gaia has flipped that forever-reality upside down in a few short years. It’s third data release was in June of 2022; this provides enough information to identify binary stars, and we’ve had enough time to start (and I do mean start) sorting through the data.

OK, why are binary stars interesting to the missing mass (really the acceleration discrepancy) problem? In principle, they allow us to distinguish between dark matter and modified gravity theories like MOND. If galactic mass discrepancies are caused by a diffuse distribution of dark matter, gravity is normal, and binary stars should orbit each other as Newton predicts, no matter their separation: the dark matter is too diffuse to have an impact on such comparatively tiny scales. If instead the force law changes at some critical scale, then the orbital speeds of widely separated binary pairs that exceed this scale should get a boost relative to the Newtonian case.

The test is easy to visualize for a single binary system. Imagine two stars orbiting one another. When they’re close, they orbits as Newton predicts. This is, after all, how we got Newtonian gravity – as an explanation for Kepler’s Laws or planetary motion. Ours is a lonely star, not a binary, but that makes no difference to gravity: Jupiter (or any other planet) is an adequate stand-in. Newton’s universal law of gravity (with tiny tweaks from Einstein) is valid as far out in the solar system as we’ve been able to probe. For scale, Pluto is about 40 AU out (where Earth, by definition, is 1 AU from the sun).

Let’s start with a pair of stars orbiting at a distance that is comfortably in the Newtonian regime, say with a separation of 40 AU. If we know the mass of the stars, we can calculate what their orbital speed will be. Now imagine gradually separating the stars so they are farther and farther apart. For any new separation s, we can predict what the new orbital speed will be. According to Newton, this will decline in a Keplerian fashion, v ~ 1/√s. This will continue indefinitely if Newton remains forever the law of the land. If instead the force law changes at some critical scale sc, then we would expect to see a change when the separation exceeds that scale. Same binary pair, same mass, but relatively faster speed – a faster speed that on galaxy scales leads to the inference of dark matter. In essence, we want to check if binary stars also have flat rotation curves if we look far enough out.

We have long known that simply changing the force law at some length scale sc does not work. In MOND, the critical scale is an acceleration, a0. This will map to a different sc for binary stars of different masses. For the sun, the critical acceleration scale is reached at sc ≈ 7000 AU ≈ 0.034 parsecs (pc), about a tenth of a light-year. That’s a lot bigger than the solar system (40 AU) but rather smaller than the distance to the next star (1.3 pc = 4.25 light-years). So it is conceivable that there are wide binaries in the solar neighborhood for which this test can be made – pairs of stars with separations large enough to probe the MOND regime without being so far apart that they inevitably get broken up by random interactions with unrelated stars.

Gaia is great for identifying binaries, and space is big. There are thousands of wide binaries within 200 pc of the sun where Gaia can obtain excellent measurements. That’s not a big piece of the galaxy – it is a patch roughly the size of the red circle in the rotation curve plot above – but it is still a heck of a lot of stars. A signal should emerge, and a number of papers have now appeared that attempt this exercise. And ooooo-buddy, am I confused. Frequent readers will have noticed that it has been a long time between posts. There are lots of reasons for this, but a big one is that every time I think I understand what is going on here, another paper appears with a different result.

OK, first, what do we expect? Conventionally, binaries should show Keplerian behavior whatever their separation. Dark matter is not dense enough locally to have any perceptible impact. In MOND, one might expect an effect analogous to the flattening of rotation curves, hence higher velocities than predicted by Newton. And that’s correct, but it isn’t quite that simple.

In MOND, there is the External Field Effect (EFE) in which the acceleration from distant sources can matter to the motion of a local system. This violates the strong but not the weak Equivalence Principle. In MOND, all accelerative tugs matter, whereas conventionally only local effects matter.

This is important here, as we live in a relatively high acceleration neighborhood that is close to a0. The acceleration the sun feels towards the Galactic center is about 1.8 a0. This applies to all the stars in the solar neighborhood, so even if one finds a binary pair that is widely separated enough for the force of one star on another to be less than a0, they both feel the 1.8 a0 of the greater Galaxy. A lot of math intervenes, with the net effect being that the predicted boost over Newton is less than it would have been in the absence of this effect. There is still a boost, but its predicted amplitude is less than one might naively hope.

The location of the solar system along the radial acceleration relation is roughly (gbar, gobs) = (1.2, 1.8) a0. At this acceleration, the effects of MOND are just beginning to appear, and the external field of the Galaxy can affect local binary stars.

One of the first papers to address this is Hernandez et al (2022). They found a boost in speed that looks like MOND but is not MOND. Rather, it is consistent with the larger speed that is predicted by MOND in the absence of the EFE. This implies that the radial acceleration relation depicted above is absolute, and somehow more fundamental than MOND. This would require a new theory that is very similar to MOND but lacks the EFE, which seems necessary in other situations. Weird.

A thorough study has independently been made by Pittordis & Sutherland (2023). I heard a talk by them over Zoom that motivated the previous post to set the stage for this one. They identify a huge sample of over 73,000 wide binaries within 300 pc of the sun. Contrary to Hernandez et al., they find no boost at all. The motions of binaries appear to remain perfectly Keplerian. There is no hint of MOND-like effects. Different.

OK, so that is pretty strong evidence against MOND, as Indranil Banik was describing to me at the IAU meeting in Potsdam, which is why I knew to tune in for the talk by Pittordis. But before I could write this post, yet another paper appeared. This preprint by Kyu-Hyun Chae splits the difference. It finds a clear excess over the Newtonian expectation that is formally highly significant. It is also about right for what is expected in MOND with the EFE, in particular with the AQUAL flavor of MOND developed by Bekenstein & Milgrom (1984).

So we have one estimate that is MOND-like but too much for MOND, one estimate that is straight-laced Newton, and one estimate that is so MOND that it can start to discern flavors of MOND.

I really don’t know what to make of all this. The test is clearly a lot more complicated than I made it sound. One does not get to play God with a single binary pair; one instead has to infer from populations of binaries of different mass stars whether a statistical excess in orbital velocity occurs at wide separations. This is challenging for lots of reasons.

For example, we need to know the mass of each star in each binary. This can be gauged by the mass-luminosity relation – how bright a main sequence star is depends on its mass – but this must be calibrated by binary stars. OK, so, it should be safe to use close binaries that are nowhere near the MOND limit, but it can still be challenging to get this right for completely mundane, traditional astronomical reasons. It remains challenging to confidently infer the properties of impossibly distant physical objects that we can never hope to visit, much less subject to laboratory scrutiny.

Another complication is the orientation and eccentricity of orbits. The plane of the orbit of each binary pair will be inclined to our line of sight so that the velocity we measure is only a portion of the full velocity. We do not have any way to know what the inclination of any one wide binary is; it is hard enough to identify them and get a relative velocity on the plane of the sky. So we have to resort to statistical estimates. The same goes for the eccentricities of the orbits: not all orbits are circles; indeed, most are not. The orbital speed depends on where an object is along its elliptical orbit, as Kepler taught us. So yet again we must make some statistical inference about the distribution of eccentricities. These kinds of estimates are both doable and subject to going badly wrong.

The net effect is that we wind up looking at distributions of relative velocities, and trying to perceive whether there is an excess high-velocity tail over and above the Newtonian expectation. This is far enough from my expertise that I do not feel qualified to judge between the works cited above. It takes time to sort these things out, and hopefully we can all come to agreement on what it is that we’re seeing. Right now, we’re not all seeing eye-to-eye.

There is a whole session devoted to this topic at the upcoming meeting on MOND. The primary protagonists will be there, so hopefully some progress can be made. At least it should be entertaining.

Take it where?

Take it where?

I had written most of the post below the line before an exchange with a senior colleague who accused me of asking us to abandon General Relativity (GR). Anyone who read the last post knows that this is the opposite of true. So how does this happen?

Much of the field is mired in bad ideas that seemed like good ideas in the 1980s. There has been some progress, but the idea that MOND is an abandonment of GR I recognize as a misconception from that time. It arose because the initial MOND hypothesis suggested modifying the law of inertia without showing a clear path to how this might be consistent with GR. GR was built on the Equivalence Principle (EP), the equivalence1 of gravitational charge with inertial mass. The original MOND hypothesis directly contradicted that, so it was a fair concern in 1983. It was not by 19842. I was still an undergraduate then, so I don’t know the sociology, but I get the impression that most of the community wrote MOND off at this point and never gave it further thought.

I guess this is why I still encounter people with this attitude, that someone is trying to rob them of GR. It’s feels like we’re always starting at square one, like there has been zero progress in forty years. I hope it isn’t that bad, but I admit my patience is wearing thin.

I’m trying to help you. Don’t waste you’re entire career chasing phantoms.

What MOND does ask us to abandon is the Strong Equivalence Principle. Not the Weak EP, nor even the Einstein EP. Just the Strong EP. That’s a much more limited ask that abandoning all of GR. Indeed, all flavors of EP are subject to experimental test. The Weak EP has been repeatedly validated, but there is nothing about MOND that implies platinum would fall differently from titanium. Experimental tests of the Strong EP are less favorable.

I understand that MOND seems impossible. It also keeps having its predictions come true. This combination is what makes it important. The history of science is chock full of ideas that were initially rejected as impossible or absurd, going all the way back to heliocentrism. The greater the cognitive dissonance, the more important the result.

Continuing the previous discussion of UT, where do we go from here? If we accept that maybe we have all these problems in cosmology because we’re piling on auxiliary hypotheses to continue to be able to approximate UT with FLRW, what now?

I don’t know.

It’s hard to accept that we don’t understand something we thought we understood. Scientists hate revisiting issues that seem settled. Feels like a waste of time. It also feels like a waste of time continuing to add epicycles to a zombie theory, be it LCDM or MOND or the phoenix universe or tired light or whatever fantasy reality you favor. So, painful as it may be, one has find a little humility to step back and take account of what we know empirically independent of the interpretive veneer of theory.

As I’ve said before, I think we do know that the universe is expanding and passed through an early hot phase that bequeathed us the primordial abundances of the light elements (BBN) and the relic radiation field that we observe as the cosmic microwave background (CMB). There’s a lot more to it than that, and I’m not going to attempt to recite it all here.

Still, to give one pertinent example, BBN only works if the expansion rate is as expected during the epoch of radiation domination. So whatever is going on has to converge to that early on. This is hardly surprising for UT since it was stipulated to contain GR in the relevant limit, but we don’t actually know how it does so until we work out what UT is – a tall order that we can’t expect to accomplish overnight, or even over the course of many decades without a critical mass of scientists thinking about it (and not being vilified by other scientists for doing so).

Another example is that the cosmological principle – that the universe is homogeneous and isotropic – is observed to be true in the CMB. The temperature is the same all over the sky to one part in 100,000. That’s isotropy. The temperature is tightly coupled to the density, so if the temperature is the same everywhere, so is the density. That’s homogeneity. So both of the assumptions made by the cosmological principle are corroborated by observations of the CMB.

The cosmological principle is extremely useful for solving the equations of GR as applied to the whole universe. If the universe has a uniform density on average, then the solution is straightforward (though it is rather tedious to work through to the Friedmann equation). If the universe is not homogeneous and isotropic, then it becomes a nightmare to solve the equations. One needs to know where everything was for all of time.

Starting from the uniform condition of the CMB, it is straightforward to show that the assumption of homogeneity and isotropy should persist on large scales up to the present day. “Small” things like galaxies go nonlinear and collapse, but huge volumes containing billions of galaxies should remain in the linear regime and these small-scale variations average out. One cubic Gigaparsec will have the same average density as the next as the next, so the cosmological principle continues to hold today.

Anyone spot the rub? I said homogeneity and isotropy should persist. This statement assumes GR. Perhaps it doesn’t hold in UT?

This aspect of cosmology is so deeply embedded in everything that we do in the field that it was only recently that I realized it might not hold absolutely – and I’ve been actively contemplating such a possibility for a long time. Shouldn’t have taken me so long. Felten (1984) realized right away that a MONDian universe would depart from isotropy by late times. I read that paper long ago but didn’t grasp the significance of that statement. I did absorb that in the absence of a cosmological constant (which no one believed in at the time), the universe would inevitably recollapse, regardless of what the density was. This seems like an elegant solution to the flatness/coincidence problem that obsessed cosmologists at the time. There is no special value of the mass density that provides an over/under line demarcating eternal expansion from eventual recollapse, so there is no coincidence problem. All naive MOND cosmologies share the same ultimate fate, so it doesn’t matter what we observe for the mass density.

MOND departs from isotropy for the same reason it forms structure fast: it is inherently non-linear. As well as predicting that big galaxies would form by z=10, Sanders (1998) correctly anticipated the size of the largest structures collapsing today (things like the local supercluster Laniakea) and the scale of homogeneity (a few hundred Mpc if there is a cosmological constant). Pretty much everyone who looked into it came to similar conclusions.

But MOND and cosmology, as we know it in the absence of UT, are incompatible. Where LCDM encompasses both cosmology and the dynamics of bound systems (dark matter halos3), MOND addresses the dynamics of low acceleration systems (the most common examples being individual galaxies) but says nothing about cosmology. So how do we proceed?

For starters, we have to admit our ignorance. From there, one has to assume some expanding background – that much is well established – and ask what happens to particles responding to a MONDian force-law in this background, starting from the very nearly uniform initial condition indicated by the CMB. From that simple starting point, it turns out one can get a long way without knowing the details of the cosmic expansion history or the metric that so obsess cosmologists. These are interesting things, to be sure, but they are aspects of UT we don’t know and can manage without to some finite extent.

For one, the thermal history of the universe is pretty much the same with or without dark matter, with or without a cosmological constant. Without dark matter, structure can’t get going until after thermal decoupling (when the matter is free to diverge thermally from the temperature of the background radiation). After that happens, around z = 200, the baryons suddenly find themselves in the low acceleration regime, newly free to respond to the nonlinear force of MOND, and structure starts forming fast, with the consequences previously elaborated.

But what about the expansion history? The geometry? The big questions of cosmology?

Again, I don’t know. MOND is a dynamical theory that extends Newton. It doesn’t address these questions. Hence the need for UT.

I’ve encountered people who refuse to acknowledge4 that MOND gets predictions like z=10 galaxies right without a proper theory for cosmology. That attitude puts the cart before the horse. One doesn’t look for UT unless well motivated. That one is able to correctly predict 25 years in advance something that comes as a huge surprise to cosmologists today is the motivation. Indeed, the degree of surprise and the longevity of the prediction amplify the motivation: if this doesn’t get your attention, what possibly could?

There is no guarantee that our first attempt at UT (or our second or third or fourth) will work out. It is possible that in the search for UT, one comes up with a theory that fails to do what was successfully predicted by the more primitive theory. That just lets you know you’ve taken a wrong turn. It does not mean that a correct UT doesn’t exist, or that the initial prediction was some impossible fluke.

One candidate theory for UT is bimetric MOND. This appears to justify the assumptions made by Sanders’s early work, and provide a basis for a relativistic theory that leads to rapid structure formation. Whether it can also fit the acoustic power spectrum of the CMB as well as LCDM and AeST has yet to be seen. These things take time and effort. What they really need is a critical mass of people working on the problem – a community that enjoys the support of other scientists and funding institutions like NSF. Until we have that5, progress will remain grudgingly slow.

1The equivalence of gravitational charge and inertial mass means that the m in F=GMm/d2 is identically the same as the m in F=ma. Modified gravity changes the former; modified inertia the latter.

2Bekenstein & Milgrom (1984) showed how a modification of Newtonian gravity could avoid the non-conservation issues suffered by the original hypothesis of modified inertia. They also outlined a path towards a generally covariant theory that Bekenstein pursued for the rest of his life. That he never managed to obtain a completely satisfactory version is often cited as evidence that it can’t be done, since he was widely acknowledged as one of the smartest people in the field. One wonders why he persisted if, as these detractors would have us believe, the smart thing to do was not even try.

3The data for galaxies do not look like the dark matter halos predicted by LCDM.

4I have entirely lost patience with this attitude. If a phenomena is correctly predicted in advance in the literature, we are obliged as scientists to take it seriously+. Pretending that it is not meaningful in the absence of UT is just an avoidance strategy: an excuse to ignore inconvenient facts.

+I’ve heard eminent scientists describe MOND’s predictive ability as “magic.” This also seems like an avoidance strategy. I, for one, do not believe in magic. That it works as well as it doesthat it works at all – must be telling us something about the natural world, not the supernatural.

5There does exist a large and active community of astroparticle physicists trying to come up with theories for what the dark matter could be. That’s good: that’s what needs to happen, and we should exhaust all possibilities. We should do the same for new dynamical theories.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

You have zero data.

Other measurements often provide a proxy measurement that seems like it ought to be close enough to use. If not the flat rotation speed, maybe you have a line width or a maximum speed or V2.2 or the hybrid S0.5 or some other metric. That’s fine, so long as you recognize you’re plotting something different so should expect to get something different – not the BTFR. Again, we’ve shown that the flat rotation speed is the measure that minimizes the scatter; if you utilize some other measure you’re gonna get more scatter. That may be useful for some purposes, but it only tells you about what you measured. It doesn’t tell you anything about the scatter in the BTFR constructed with the flat rotation speed if you didn’t measure the flat rotation speed.

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

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

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

Astronomical Acceleration Scales

Astronomical Acceleration Scales

A quick note to put the acceleration discrepancy in perspective.

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

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

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

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

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

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

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

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

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

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

We seem to understand gravity well in this regime.

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

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

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

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

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

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

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

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

RAR fits to individual galaxies

RAR fits to individual galaxies

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The radial acceleration relation with optimized parameters.

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

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

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

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

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

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

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

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

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

One Law to Rule Them All

One Law to Rule Them All

One Law to rule them all, One Law to guide them,
One Law to form them all and in the dark halo bind them.


Galaxies appear to obey a single universal effective force law.


Early indications of this have been around for some time. It has become particularly clear in our work using near-infrared surface photometry to trace the stellar mass distribution of late type galaxies (SPARC). It takes a while to wrap our heads around the implications.


The observed phenomenology constitutes a new law of nature. One Law to rule all galaxies.


The Astrophysical Journal just published our long and thorough investigation of this issue eponymously titled One Law to Rule Them All: The Radial Acceleration Relation of GalaxiesIt includes this movie showing the build-up of the radial acceleration relation in the data.

So far, the ubiquitous effective force law had only been clearly demonstrated in rotating galaxies. Federico Lelli and Marcel Pawlowski went to great lengths to also include pressure supported galaxies, from giant ellipticals to dwarf spheroidals. They appear to follow the same effective force law as rotating galaxies.

The Radial Acceleration Relation defined by rotating late type galaxies (blue points) is also obeyed by early type galaxies, regardless of whether they be fast rotators (orange points) or pressure supported slow rotators (red point) or dark matter dominated dwarf spheroidal satellite galaxies (grey and green points).

This is not a fluke of a few special galaxies. It involves galaxies of all known morphological types spanning an enormous range in mass, size, and surface brightness. I have spent the last twenty years adding new data for all varieties of galaxy types to this relation in the expectation that it would break. Instead it has become stronger and clearer.

Understanding the observed relation is one of the pre-eminent challenges in modern physics. Once we exclude metaphysical nonsense like multiverses, it is arguably the most important unsolved problem. Why does this happen?

The usual ad hoc interpretation of rotation curves in terms of dark matter does nothing to anticipate the observed phenomenology, which is in fact quite troubling from this perspective as it requires excessive fine-tuning. This has been known (if widely ignored) for a while, but doesn’t preclude the more rabid advocates of dark matter from asserting that it all comes about naturally. Lets not mince words here: claims that the radial acceleration relation occurs naturally with dark matter are pure, unadulterated bullshit fueled by confirmation bias and cognitive dissonance. Perhaps dark matter is the root cause, but there is nothing natural about it.

The natural explanation of a single effective force law is that it is caused by a truly universal force law.

So far, the theory that comes closest to explaining these data is MOND. Milgrom, understandably enough, argues that these data require MOND. He has a valid point. It is a good argument, but does it suffice to overcome the other problems MOND faces? These are not as great as widely portrayed, but they aren’t exactly negligible, either. I tried to look at the problem from both perspectives in this review for the Canadian Journal of Physics. [Being able to see things from both sides is an essential skill if one is to be objective, an important value in science that seems disturbingly absent in its modern practice.]

MOND anticipates an asymptotic slope of 1/2 at low acceleration (gobs ~ gbar1/2). In the figure above, the data for the faintest (“ultrafaint”) dwarf spheroidals show a flattening in the empirical law at low accelerations that is not predicted by MOND. Perhaps the underlying force law is subtly different from pure MOND? On the other hand, weak lensing observations show that the MOND slope extrapolates well to much lower accelerations.

It is possible that the data for ultrafaint dwarfs are in some cases misleading. Are these objects in dynamical equilibrium (a prerequisite for analysis)? Are they even dwarf galaxies? Some of the ultrafaints are not clearly distinct objects in the sense of dSph satellites like Crater 2: it is not clear that all of them deserve the status of “dwarf galaxy.” Some are little more than a handful of stars that occupy a similar cell in phase space – perhaps they are fragmentary structures in the Galactic stellar halo? Or the rump end of dissolving satellites? This is anticipated to occur in both ΛCDM and MOND. If so, their velocity dispersions probably tell us more about their disruption history than their gravitational potential, in which case their location in the plot is misleading.

Detailed questions like these are the subject of much current research. For now, lets take a step back and appreciate the data for what they say, irrespective of the underlying theoretical reason for it. We’re looking at a new law of nature! How cool is that?

Ash nazg durbatulûk, ash nazg gimbatul, ash nazg thrakatulûk, agh burzum-ishi krimpatul.

Ode to Vera

Ode to Vera

Vera Rubin passed away a few weeks ago. This was not surprising: she had lived a long, positive, and fruitful life, but had faced the usual health problems of those of us who make it to the upper 80s. Though news of her death was not surprising, it was deeply saddening. It affected me more than I had anticipated, even armed with the intellectual awareness that the inevitable must be approaching. It saddens me again now trying to write this, which must inevitably be an inadequate tribute.

In the days after Vera Rubin passed away, I received a number of inquiries from the press asking me to comment on her life and work for their various programs. I did not respond. I guess I understand the need to recognize and remark on the passing of a great scientist and human being, and I’m glad the press did in fact acknowledge her many accomplishments. But I wondered if, by responding, I would be providing a tribute to Vera, or merely feeding the needs of the never-ending hyperactive news cycle. Both, I guess. At any rate, I did not feel it was my place to comment. It did not seem right to air my voice where hers would never be heard again.

I knew Vera reasonably well, but there are plenty who knew her better and were her colleagues over a longer period of time. Also, at the back of my mind, I was a tiny bit afraid that no matter what I said, someone would read into it some sort of personal scientific agenda. My reticence did not preclude other scientists who knew her considerably less well from doing exactly that. Perhaps it is unavoidable: to speak of others, one must still use one’s own voice, and that inevitably is colored by our own perspective. I mention this because many of the things recently written about Vera do not do justice to her scientific opinions as I know them from conversations with her. This is important, because Vera was all about the science.

One thing I distinctly remembering her saying to me, and I’m sure she repeated this advice to many other junior scientists, was that you had to do science because you had a need to Know. It was not something to be done for awards or professional advancement; you could not expect any sort of acknowledgement and would likely be disappointed if you did. You had to do it because you wanted to find out how things work, to have even a brief moment when you felt like you understood some tiny fraction of the wonders of the universe.

Despite this attitude, Vera was very well rewarded for her science. It came late in her career – she did devote a lot of energy to raising a large family; she and her husband Bob Rubin were true life partners in the ideal sense of the term: family came first, and they always supported each other. It was deeply saddening when Bob passed, and another blow to science when their daughter Judy passed away all too early. We all die, sometimes sooner rather than later, but few of us take it well.

Professionally, Vera was all about the science. Work was like breathing. Something you just did; doing it was its own reward. Vera always seemed to take great joy in it. Success, in terms of awards, came late, but it did come, and in many prestigious forms – membership in the National Academy of Sciences, the Gold Medal of the Royal Astronomical Society, and the National Medal of Science, to name a few of her well-deserved honors. Much has been made of the fact that this list does not include a Nobel Prize, but I never heard Vera express disappointment about that, or even aspiration to it. Quite the contrary, she, like most modest people, didn’t seem to consider it to be appropriate. I think  part of the reason for this was that she self-identified as an astronomer, not as a physicist (as some publications mis-report). That distinction is worthy of an entire post so I’ll leave it for now.

Astronomer though she was, her work certainly had an outsized impact on physics. I have written before as to why she was deserving of a Nobel Prize, if for slightly different reasons than others give. But I do not dread that she died in any way disappointed by the lack of a Nobel Prize. It was not her nature to fret about such things.

Nevertheless, Vera was an obvious scientist to recognize with a Nobel Prize. No knowledgeable scientist would have disputed her as a choice. And yet the history of the physics Nobel prize is incredibly lacking in female laureates (see definition 4). Only two women have been recognized in the entire history of the award: Marie Curie (1903) and Maria Goeppert-Mayer (1963). She was an obvious woman to have honored in this way. It is hard to avoid the conclusion that the awarding of the prize is inherently sexist. Based on two data points, it has become more sexist over time, as there is a longer gap between now and the last award to a woman (63 years) than between the two awards (60 years).

Why should gender play any role in the search for knowledge? Or the recognition of discoveries made in that search? And yet women scientists face antiquated attitudes and absurd barriers all the time. Not just in the past. Now.

Vera was always a strong advocate of women in science. She has been an inspiration to many. A Nobel prize awarded to Vera Rubin would have been great for her, yes, but the greater tragedy of this missed opportunity is what it would have meant to all the women who are scientists now and who will be in the future.

Well, those are meta-issues raised by Vera’s passing. I don’t think it is inappropriate, because these were issues dear to her heart. I know the world is a better place for her efforts. But I hadn’t intended to go off on meta-tangents. Vera was a very real, warm, positive human being. So I what I had meant to do was recollect a few personal anecdotes. These seem so inadequate: brief snippets in a long and expansive life. Worse, they are my memories, so I can’t see how to avoid making it at least somewhat about me when it should be entirely about her. Still. Here are a few of the memories I have of her.

I first met Vera in 1985 on Kitt Peak. In retrospect I can’t imagine a more appropriate setting. But at the time it was only my second observing run, and I had no clue as to what was normal or particularly who Vera Rubin was. She was just another astronomer at the dinner table before a night of observing.

A very curious astronomer. She kindly asked what I was working on, and followed up with a series of perceptive questions. She really wanted to know. Others have remarked on her ability to make junior people feel important, and she could indeed do that. But I don’t think she tried, in particular. She was just genuinely curious.

At the time, I was a senior about to graduate from MIT. I had to beg permission to take some finals late so I could attend this observing run. My advisor, X-ray astronomer George Whipple Clark, kindly bragged about how I had actually got my thesis in on time (most students took advantage of a default one-week grace period) in order to travel to Kitt Peak. Vera, ever curious, asked about my thesis, what galaxies were involved, how the data were obtained… all had been from a run the semester before. As this became clear, Vera got this bemused look and asked “What kind of thesis can be written from a single observing run?” “A senior thesis!” I volunteered: undergraduate observers were rare on the mountain in those days; up till that point I think she had assumed I was a grad student.

I encountered Vera occasionally over the following years, but only in passing. In 1995, she offered me a Carnegie fellowship at DTM. This was a reprieve in a tight job market. As it happened, we were both visiting the Kapteyn Institute, and Renzo Sancisi had invited us both to dinner, so she took the opportunity to explain that their initial hire had moved on to a faculty position so the fellowship was open again. She managed to do this without making me feel like an also-ran. I had recently become interested in MOND, and here was the queen of dark matter offering me a job I desperately needed. It seemed right to warn her, so I did: would she have a problem with a postdoc who worked on MOND? She was visibly shocked, but only for an instant. “Of course not,” she said. “As a Carnegie Fellow, you can work on whatever you want.”

Vera was very supportive throughout my time at DTM, and afterwards. We had many positive scientific interactions, but we didn’t really work together then. I tried to get her interested in the rotation curves of low surface brightness galaxies, but she had a full plate. It wasn’t until a couple of years after I left DTM that we started collaborating.

Figure made by Vera Rubin from her measurements of the rotation curves of low surface brightness galaxies. Published in McGaugh, Rubin, & de Blok (2001).

Vera loved to measure. The reason I chose the picture featured at top is that it shows her doing what she loved. By the time we collaborated, she had moved on to using a computer to measure line positions for velocities. But that is what she loved to do. She did all the measurements for the rotation curves we measured, like the ones shown above. As the junior person, I had expected to do all that work, but she wanted to do it. Then she handed it on to me to write up, with no expectation of credit. It was like she was working for me as a postdoc. Vera Rubin was an awesome postdoc!

She also loved to observe. Mostly that was a typically positive, fruitful experience. But she did have an intense edge that rarely peaked out. One night on Las Campanas, the telescope broke. This is not unusual, and we took it in stride. For a half hour or so. Then Vera started calmly but assertively asking the staff why we were not yet back up and working. Something was very wrong, and it involved calling in extra technicians who led us into the mechanical bowels of the du Pont telescope, replete with steel cables and unidentifiable steam-punk looking artifacts. Vera watched them like a hawk. She never said a negative word. But she silently, intently watched them. Tension mounted; time slowed to a crawl till it seemed that I could feel like a hard rain the impact of every photon that we weren’t collecting. She wanted those photons. Never said a negative word, but I’m sure the staff felt a wall of pressure that I was keenly aware of merely standing in its proximity. Perhaps like a field mouse under a raptor’s scrutiny.

Vera was not normally like that, but every good observer has in her that urgency to get on sky. This was the only time I saw it come out. Other typical instrumental guffaws she bore in stride. This one took too long. But it did get fixed, and we were back on sky, and it was as if there had never been a problem in the world.

Ultimately, Vera loved the science. She was one of the most intrinsically curious souls I ever met. She wanted to know, to find out what was going on up there. But she was also content with what the universe chose to share, reveling in the little discoveries as much as the big ones. Why does the Hα emission extend so far out in UGC 2885? What is the kinematic major axis of DDO 154, anyway? Let’s put the slit in a few different positions and work it out. She kept a cheat sheet taped on her desk for how the rotation curve changed if the position angle were missed – which never happened, because she prepared so carefully for observing runs. She was both thorough and extremely good at what she did.

Vera was very positive about the discoveries of others. Like all good astronomers, she had a good BS detector. But she very rarely said a negative word. Rarely, not never. She was not a fan of Chandrasekhar, who was the editor of the ApJ when she submitted her dissertation paper there. Her advisor, Gamow, had posed the question to her, is there a length scale in the sky? Her answer would, in the modern parlance, be called the correlation length of galaxies. Chandrasekhar declined to consider publishing this work, explaining in a letter that he had a student working on the topic, and she should wait for the right answer. The clear implication was that this was a man’s job, and the work of a woman was not to be trusted. Ultimately her work was published in the proceedings of the National Academy, of which Gamow was a member. He had predicted that this is how Chandrasekhar would behave, afterwards sending her a postcard saying only “Told you so.”

On another occasion, in the mid-90s when “standard” CDM meant SCDM with Ωm = 1, not ΛCDM, she confided to me in hushed tones that the dark matter had to be baryonic. Other eminent dynamicists have said the same thing to me at times, always in the same hushed tones, lest the cosmologists overhear. As well they might. To my ears this was an absurdity, and I know well the derision it would bring. What about Big Bang Nucleosynthesis? This was the only time I recall hearing Vera scoff. “If I told the theorists today that I could prove Ωm = 1, tomorrow they would explain that away.”

I was unconvinced. But it made clear to me that I put a lot of faith in Big Bang Nucleosynthesis, and this need not be true for all intelligent scientists. Vera – and the others I allude to, who still live so I won’t name – had good reasons for her assertion. She had already recognized that there was a connection between the baryon distribution and the dynamics of galaxies, and that this made a lot more sense if the dark and luminous component were closely related – for example, if the dark matter – or at least some important fraction of it in galaxies – were itself baryonic. Even if we believe in Big Bang Nucleosynthesis, we’re still missing a lot of baryons.

The proper interpretation of this evidence is still debated today. What I learned from this was to be more open to the possibility that things I thought I knew for sure might turn out to be wrong. After all, that pretty much sums up the history of cosmology.

It was widely reported that Vera discovered dark matter or “proved” or “confirmed” its existence. I don’t think Vera would agree with this assessment, nor would many of her colleagues at DTM. I know this because we talked about it. A lot.

To my mind, what Vera discovered is both more specific and more profound than the dark matter paradigm it helped to create. What she discovered observationally is that rotation curves are very nearly flat, and continue to be so to indefinitely large radius. Over and over again, for every galaxy in the sky. It is a law of nature for galaxies, akin to Kepler’s laws for planets. Dark matter is an inference, a subsidiary result. It is just one possible interpretation, a subset of amazing and seemingly unlikely possibilities opened up by her discovery.

The discovery itself is amazing enough without conflating it with dark matter or MOND or any other flavor of interpretation of which the reader might be fond. Like many great discoveries, it has many parents. I would give a lot of credit to Albert Bosma, but there are also others who had early results, like Mort Roberts and Seth Shostak. But it was Vera whose persistence overcame the knee-jerk conservatism of cosmologists like Sandage, who she said dismissed her early flat rotation curve of M31 (obtained in collaboration with Roberts) as “the effect of looking at a bright galaxy.” “What does that even mean?” she asked me rhetorically. She also recalled Jim Gunn gasping “But… that would mean most of the mass is dark!” Indeed. It takes time to wrap our heads around these things. She obtained rotation curve after rotation curve in excess of a hundred to ensure we realized we had to do so.

Vera realized the interpretation was never as settled as the data. Her attitude (and that of many of us, including myself) is nicely summarized by her exchange with Tohline at the end of her 1982 talk at IAU 100. One starts with the most conservative – or at least, least outrageous – possibility, which at that time was a mere factor of two in hidden mass, which could easily have been baryonic. Yet much more more recently, at the last conference I attended with her (in 2009), she reminded the audience (to some visible consternation) that it was still “early days” for dark matter, and we should not be surprised to be surprised – up to, and including, how gravity works.

At this juncture, I expect some readers will accuse me of what I warned about above: using this for my own agenda. I have found it is impossible to avoid having an agenda imputed to me by people who don’t like what they imagine my agenda to be, whether they imagine right or not – usually not. But I can’t not say these things if I want to set the record straight – these were Vera’s words. She remained concerned all along that it might be gravity to blame rather than dark matter. Not convinced, nor even giving either the benefit of the doubt. There was, and remains, so much to figure out.

“Early days.”

I suppose, in the telling, it is often more interesting to relate matters of conflict and disagreement than feelings of goodwill. In that regards, some of the above anecdotes are atypical: Vera was a very positive person. It just isn’t compelling to relate episodes like her gushing praise for Rodrigo Ibata’s discovery of the Sagittarius dwarf satellite galaxy. I probably only remember that myself because I had, like Rodrigo, encountered considerable difficulty in convincing some at Cambridge that there could be lots of undiscovered low surface brightness galaxies out there, even in the Local Group. Some of these same people now seem to take for granted that there are a lot more in the Local Group than I find plausible.

I have been fortunate in my life to have known many talented scientists. I have met many people from many nations, most of them warm, wonderful human beings. Vera was the best of the best, both as a scientist and as a human being. The world is a better place for having had her in it, for a time.

What is Natural?

I have been musing for a while on the idea of writing about Naturalness in science, particularly as it applies to the radial acceleration relation. As a scientist, the concept of Naturalness is very important to me, especially when it comes to the interpretation of data. When I sat down to write, I made the mistake of first Googling the term.

The top Google hits bear little resemblance to what I mean by Naturalness. The closest match is specific to a particular, rather narrow concept in theoretical particle physics. I mean something much more general. I know many scientific colleagues who share this ideal. I also get the impression that this ideal is being eroded and cheapened, even among scientists, in our post-factual society.

I suspect the reason a better hit for Naturalness doesn’t come up more naturally in a Google search is, at least in part, an age effect. As wonderful a search engine as Google may be, it is lousy at identifying things B.G. (Before Google).  The concept of Naturalness has been embedded in the foundations of science for centuries, to the point where it is absorbed by osmosis by students of any discipline: it doesn’t need to be formally taught; there probably is no appropriate website.

In many sciences, we are often faced with messy and incomplete data. In Astronomy in particular, there are often complicated astrophysical processes well beyond our terrestrial experience that allow a broad range of interpretations. Some of these are natural while others are contrived. Usually, the most natural interpretation is the correct one. In this regard, what I mean by Naturalness is closely related to Occam’s Razor, but it is something more as well. It is that which follows – naturally – from a specific hypothesis.

An obvious astronomical example: Kepler’s Laws follow naturally from Newton’s Universal Law of Gravity. It is a trivial amount of algebra to show that Kepler’s third Law, P2 = a3, follows as a direct consequence of Newton’s inverse square law. The first law, that orbits are ellipses, follows with somewhat more math. The second law follows with the conservation of angular momentum.

It isn’t just that Newtonian gravity is the simplest explanation for planetary orbits. It is that all the phenomena identified by Kepler follow naturally from Newton’s insight. This isn’t obvious just by positing an inverse square law. But in exploring the consequences of such a hypothesis, one finds that one clue after another falls into place like the pieces of a jigsaw puzzle. This is what I mean by Naturalness.

I expect that this sense of Naturalness – the fitting together of the pieces of the puzzle – is what gave Newton encouragement that he was on the right path with the inverse square law. Let’s not forget that both Newton and his inverse square law came in for a lot of criticism at the time. Both Leibniz and Huygens objected to action at a distance, for good reason. I suspect this is why Newton prefaced his phrasing of the inverse square law with the modifier as if: “Everything happens… as if the force between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.” He is not claiming that this is right, that it has to be so. Just that it sure looks that way.

The situation with the radial acceleration relation in galaxies today is the same. Everything happens as if there is a single effective force law in galaxies. This is true regardless of what the ultimate reason proves to be.

The natural explanation for the single effective force law indicated by the radial acceleration relation is that there is indeed a unique force law at work. In this case, such a force law has already been hypothesized: MOND. Often MOND is dismissed for other reasons, though reports of its demise have repeatedly been exaggerated. Perhaps MOND is just the first approximation of some deeper theory. Perhaps, like action at a distance, we simply don’t yet understand the underlying reasons for it.

Four Strikes

Four Strikes

So the radial acceleration relation is a new law of nature. What does it mean?

One reason we have posed it as a law of nature is that it is interpretation-free. It is a description of how nature works – in this case, a rule for how galaxies rotate. Why nature behaves thus is another matter.

Some people have been saying the RAR (I tire of typing out “radial acceleration relation”) is a problem for dark matter, while others seem to think otherwise. Lets examine this.

The RAR has a critical scale g = 1.2 · 10-10 m s-2. At high acceleration, above this scale, we don’t need dark matter: systems like the solar system or the centers of high surface brightness galaxies are WYSIWYG. At low accelerations, below this scale, we begin to need dark matter. The lower the acceleration, the more dark matter we need.

OK, so this means there is little to no dark matter when the baryons are dense (high gbar), but progressively more as gbar becomes smaller than the critical scale g. Low gbar happens when the surface density of baryons is low. So the amount of dark matter scales inversely with baryonic surface density.

That’s weird.

This is weird for a number of reasons. First, there is no reason for the dark matter to care what the baryons are doing when dark matter dominates. When gobs ≫ gbar the dark matter greatly outweighs the baryons, which simply become tracer particles in the gravitational potential of the dark matter halo. There is no reason for the dark matter to know or care about what the baryonic tracer particles are doing. And yet the RAR persists as a tight correlation well into this regime. It is as if the baryonic tail wags the dark matter dog.

Second, there should be more dark matter where there are more baryons. Galaxies form by baryons falling into dark matter halos. As they do so, they dissipate energy and sink to the center of the halo. In this process, the drag some of the dark matter along with them in a process commonly referred to as “adiabatic compression.” In practice, the process need not be adiabatic, but the dark matter must respond to the rearrangement of the gravitational potential caused by the dissipative infall of the baryons.

These topics have been discussed at great length in the galaxy formation literature. Great arguments have erupted time and again about how best to implement the compression in models, and how big the effect is in practice. These details need not concern us here. What matters is that they are non-negotiable fundamentals of the dark matter paradigm.

Galaxies form by baryonic infall within dark matter halos. The halos form first while the baryons are still coupled to the photons prior to last scattering. This is one of the fundamental reasons we need non-baryonic cold dark matter that does not interact with photons: to get a jump on structure formation. Without it, we cannot get from the smooth initial condition observed in the cosmic microwave background to the rich amount of structure we see today.

As the baryons fall into halos, they must sink to the center to form galaxies. Why? Dark matter halos are much bigger than the galaxies that reside within them. All tracers of the gravitational potential say so. Initially, this might seem odd, as the baryons might to just track the dominant dark matter. But baryons are different: they can dissipate energy. By so doing, they can sink to the center – not all baryons need to sink to the centers of their dark matter halos, but enough to make a galaxy. This they must do in order to form the galaxies that we observe – galaxies that are more centrally condensed than their dark matter halos.

That’s enough, in return, to affect the dark matter. As the baryons dissipate, the gravitational potential is non-stationary. The dark matter distribution must respond to this change in the total gravitational potential. The net result is a further concentration of the dark matter towards the center of the halo: in effect, the baryons drag some dark matter along with them.

I have worked on adiabatic compression myself, but a nice illustration is given by this figure from Elbert et al. (2016):

Dark matter halos formed in numerical simulations illustrating the effect of adiabatic compression. One the left is a pristine halo without baryons. In the middle is a halo after formation of a disk galaxy. On right is a halo after formation of a more compact disk.

One can see by eye the compression caused by the baryons. The more dense the baryons become, the more dark matter they drag towards the center with them.

The fundamental elements of the dark matter paradigm, galaxy formation by baryonic infall and dissipation accompanied by the inevitable compression of the dark matter halo, inevitably lead us to expect that more baryons in the center means more dark matter as well. We observe the exact opposite in the RAR. As baryons become denser, they become the dominant component, to the point where they are the only component. Rather than more dark matter as we expect, more baryons means less dark matter in reality.

Third, the RAR correlation is continuous and apparently scatter-free over all accelerations. The data map from the regime of no dark matter at high accelerations to lots of dark matter at low accelerations in perfect 1:1 harmony with the distribution of the baryons. If we observe the distribution of baryons, we know the corresponding distribution of dark matter. The tail doesn’t just wag the dog. It tells it to sit, beg, and roll over.

Fourth, there is a critical scale in the data, g. That’s the scale where the mass discrepancy sets in. This is a purely empirical statement.

Cold dark matter is scale free. Being scale free is fundamental to its nature. It is essential to fitting the large scale structure, which it does quite well.

So why is there this ridiculous acceleration scale in the data?!? Who ordered this?! It should not be there.

So yes, the radial acceleration relation is a problem for the cold dark matter paradigm.

The Third Law of Galactic Rotation

Flat rotation curves were the first clear evidence that the dynamics of galaxies do not follow the same rules as planetary systems. But they do follow rules. These include asymptotic flatness, Tully-Fisher, the luminosity-size-rotation curve shape relation (aka the `universal‘ rotation curve), Renzo’s rule, and the central density relation.

Rotation curves color coded by the characteristic surface density of stars and gas, ranging from low surface brightness galaxies (blue) to those  of high surface brightness (red).

These various relations sound like a hodge-podge of random astronomical effects. This is misleading. There is a great deal of organization in the data. The surface density of stars and gas, and the acceleration (gbar) determined by their gravitational potential, plays a defining role. Indeed, the known relations are all manifestations of a single, more fundamental relation, the radial acceleration relation.

The radial acceleration relation. The centripetal acceleration measured by the rotation curve (gobs) correlates with that predicted by the observed distribution of stars and gas (gbar). The data consist of 2,693 resolved points along the rotation curves of 153 rotating galaxies from the SPARC database. All galaxies fall along the same relation, within the uncertainties. Red squares are binned data. The lower panel shows residuals from a fit to the data. The dashed lines are the scatter in the data; the red lines are the amount of scatter expected from measurement uncertainties.

The radial acceleration relation connects what you see in galaxies with what you get for the gravitational force. This would be a trivial statement if galaxies behave as planetary systems do. They would follow the 1:1 dotted line in the figure. Instead, they bend away from that line.

Indeed, the data are consistent with a single effective force law, which can be written


Other functional forms could also work. But they would all necessarily have a critical acceleration scale g ≈ 10-10 m s-2. This is an important scale that is ubiquitous in extragalactic astronomy. It seems to be a new fundamental scale in physics.

The critical acceleration marks the onset of the missing mass problem. Above this scale, there is no need for dark matter. Below it, the difference between the 1:1 line and the data is what we attribute to dark matter. The more the observed acceleration exceeds that which can be explained by the stars and gas, the larger the mass discrepancy.

Irrespective of interpretation, the data establish the radial acceleration relation in a purely empirical way. There is nothing but data here. The axes are independent: one is measured from rotation curves, the other from photometry. These need not be well connected – the dark matter could cause any sort of acceleration independently of the stars and gas. But they are intimately coupled.

There are no deviations from the radial acceleration relation beyond those attributable to experimental error. The residuals do not correlate with mass, size, surface brightness, color, environment, how many intelligent civilizations a galaxy hosts, or anything else. The scale that matters is not luminosity or halo mass or size. It is the acceleration determined from the surface density of stars and gas.

The radial acceleration relation is a fundamental relation. In effect, it is a law of nature. Third in our counting, but first in importance, as both flat rotation curves and the Tully-Fisher relation follow from it. It must be explained by any theory that claims to provide a satisfactory description of galaxy dynamics.