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.

Tully-Fisher: the Second Law

Tully-Fisher: the Second Law

Previously I noted how we teach about Natural Law, but we no longer speak in those terms. All the Great Laws are already know, right? Surely there can’t be such things left to discover!

That rotation curves tend towards asymptotic flatness is, for all practical purposes, a law of nature. It is tempting to leap straight to the interpretation (dark matter!), but it is worth appreciating the discovery for itself. It isn’t like rotation curves merely exceed what can be explained by the stars and gas, nor that they rise and fall willy-nilly. The striking, ever-repeated observation is an indefinitely extended radial range with near-constant rotation velocity.

The rotation curves of galaxies over a large dynamic range in mass, from the most massive spiral with a well measured rotation curve (UGC 2885) to tiny, low mass, low surface brightness, gas rich dwarfs.

New Laws of Nature aren’t discovered every day. This discovery should have warranted a Nobel prize for Vera Rubin and Albert Bosma. If only we were able to see it in those terms three decades ago. Instead, we phrased it in terms of dark matter, and that was a radical enough idea it has to await verification in the laboratory. Now the prize will go to some experimental group (should there ever be a successful detection) while the new law of nature goes unrecognized. That’s OK – there should be a Nobel prize for a verified laboratory detection of non-baryonic dark matter, should that ever occur – but there should also be a Nobel prize for flat rotation curves, and it should have been awarded a long time ago.

It takes a while to appreciate these things. Another well known yet unrecognized Law of Nature is the Tully-Fisher relation. First discovered as a relation between luminosity and line-width (figure from Tully & Fisher 1977), this relation is most widely known for its utility in measuring the cosmic distance scale.

The original Tully-Fisher relation.

At the time, it gave the “wrong” answer (H0 ≠ 50), and Sandage is reputed to have suppressed its publication for a couple of years. This is one reason astronomy journals have, and should have, a high acceptance rate – too many historical examples of bad behavior to protect sacred cows.

Besides its utility as a distance indicator, the Tully-Fisher relation has profound implications for physical theory. It is not merely a relation between two observables of which only one is distance-dependent. It is a link between the observed mass and the physics that sets the flat velocity.

The stellar mass Tully-Fisher relation (left) and the baryonic Tully-Fisher relation (right). In both cases, the x-axis is the flat rotation velocity measured from resolved rotation curves. In the right panel, the y-axis is the baryonic mass – the sum of observed stars and gas. The latter appears to be a law of nature from which galaxies never stray.

The original y-axis of the Tully-Fisher relation, luminosity, was a proxy for stellar mass. The line-width was a proxy for rotation velocity, of which there are many variants. At this point it is clear that the more fundamental variables are baryonic mass – the sum of observed stars and gas – and the flat rotation velocity.

I had an argument – of the best scientific sort – with Renzo Sancisi in 1995. I was disturbed that our then-new low surface brightness galaxies were falling on the same Tully-Fisher relation as previously known high surface brightness galaxies of comparable luminosity. The conventional explanation for the Tully-Fisher relation up to that point invoked Freeman’s Law – the notion (now deprecated) that all spirals had the same central surface brightness. This had the effect of suppressing the radius term in Newton’s

V2 = GM/R.

Galaxies followed a scaling between luminosity (mass) and velocity because they all had the same R at a given M.

By construction, this was not true for low surface brightness galaxies. They have larger radii at fixed luminosity (representing the mass M). That’s what makes them low surface brightness – their stars are more spread out. Yet they fall smack on the same Tully-Fisher relation!

Renzo and I looked at the result and argued up and down, this way and that about the data, the relation, everything. We were getting no closer to understanding it, or agreeing on what it meant. Finally he shouted “TULLY-FISHER IS GOD!” to which I retorted “NEWTON IS GOD!”

It was a healthy exchange of viewpoints.

Renzo made his assertion because, in his vast experience as an observer, galaxies always fell on the Tully-Fisher relation. I made mine, because, well, duh. The problem is that the observed Tully-Fisher relation does not follow from Newton.

But Renzo was right. Galaxies do always fall on the Tully-Fisher relation. There are no residuals from the baryonic Tully-Fisher relation. Neither size nor surface brightness are second parameters. The relation cares not whether a galaxy disk has a bar or not. It does not matter whether a galaxy is made of stars or gas. It does not depend on environment or pretty much anything else one can imagine. Indeed, there is no intrinsic scatter to the relation, as best we can tell. If a galaxy rotates, it follows the baryonic Tully-Fisher relation.

The baryonic Tully-Fisher relation is a law of nature. If you measure the baryonic mass, you know what the flat rotation speed will be, and vice-versa. The baryonic Tully-Fisher relation is the second law of rotating galaxies.