People seem to like to do retrospectives at year’s end. I take a longer view, but the end of 2020 seems like a fitting time to do that. Below is the text of a paper I wrote in 1995 with collaborators at the Kapteyn Institute of the University of Groningen. The last edit date is from December of that year, so this text (in plain TeX, not LaTeX!) is now a quarter century old. I am just going to cut & paste it as-was; I even managed to recover the original figures and translate them into something web-friendly (postscript to jpeg). This is exactly how it was.
This was my first attempt to express in the scientific literature my concerns for the viability of the dark matter paradigm, and my puzzlement that the only theory to get any genuine predictions right was MOND. It was the hardest admission in my career that this could be even a remote possibility. Nevertheless, intellectual honesty demanded that I report it. To fail to do so would be an act of reality denial antithetical to the foundational principles of science.
It was never published. There were three referees. Initially, one was positive, one was negative, and one insisted that rotation curves weren’t flat. There was one iteration; this is the resubmitted version in which the concerns of the second referee were addressed to his apparent satisfaction by making the third figure a lot more complicated. The third referee persisted that none of this was valid because rotation curves weren’t flat. Seems like he had a problem with something beyond the scope of this paper, but the net result was rejection.
One valid concern that ran through the refereeing process from all sides was “what about everything else?” This is a good question that couldn’t fit into a short letter like this. Thanks to the support of Vera Rubin and a Carnegie Fellowship, I spent the next couple of years looking into everything else. The results were published in 1998 in a series of three long papers: one on dark matter, one on MOND, and one making detailed fits.
This had started from a very different place intellectually with my efforts to write a paper on galaxy formation that would have been similar to contemporaneous papers like Dalcanton, Spergel, & Summers and Mo, Mao, & White. This would have followed from my thesis and from work with Houjun Mo, who was an office mate when we were postdocs at the IoA in Cambridge. (The ideas discussed in Mo, McGaugh, & Bothun have been reborn recently in the galaxy formation literature under the moniker of “assembly bias.”) But I had realized by then that my ideas – and those in the papers cited – were wrong. So I didn’t write a paper that I knew to be wrong. I wrote this one instead.
Nothing substantive has changed since. Reading it afresh, I’m amazed how many of the arguments over the past quarter century were anticipated here. As a scientific community, we are stuck in a rut, and seem to prefer to spin the wheels to dig ourselves in deeper than consider the plain if difficult path out.
Testing hypotheses of dark matter and alternative gravity with low surface density galaxies
The missing mass problem remains one of the most vexing in astrophysics. Observations clearly indicate either the presence of a tremendous amount of as yet unidentified dark matter1,2, or the need to modify the law of gravity3-7. These hypotheses make vastly different predictions as a function of density. Observations of the rotation curves of galaxies of much lower surface brightness than previously studied therefore provide a powerful test for discriminating between them. The dark matter hypothesis requires a surprisingly strong relation between the surface brightness and mass to light ratio8, placing stringent constraints on theories of galaxy formation and evolution. Alternatively, the observed behaviour is predicted4 by one of the hypothesised alterations of gravity known as modified Newtonian dynamics3,5 (MOND).
Spiral galaxies are observed to have asymptotically flat [i.e., V(R) ~ constant for large R] rotation curves that extend well beyond their optical edges. This trend continues for as far (many, sometimes > 10 galaxy scale lengths) as can be probed by gaseous tracers1,2 or by the orbits of satellite galaxies9. Outside a galaxy’s optical radius, the gravitational acceleration is aN = GM/R2 = V2/R so one expects V(R) ~ R-1/2. This Keplerian behaviour is not observed in galaxies.
One approach to this problem is to increase M in the outer parts of galaxies in order to provide the extra gravitational acceleration necessary to keep the rotation curves flat. Indeed, this is the only option within the framework of Newtonian gravity since both V and R are directly measured. The additional mass must be invisible, dominant, and extend well beyond the optical edge of the galaxies.
Postulating the existence of this large amount of dark matter which reveals itself only by its gravitational effects is a radical hypothesis. Yet the kinematic data force it upon us, so much so that the existence of dark matter is generally accepted. Enormous effort has gone into attempting to theoretically predict its nature and experimentally verify its existence, but to date there exists no convincing detection of any hypothesised dark matter candidate, and many plausible candidates have been ruled out10.
Another possible solution is to alter the fundamental equation aN = GM/R2. Our faith in this simple equation is very well founded on extensive experimental tests of Newtonian gravity. Since it is so fundamental, altering it is an even more radical hypothesis than invoking the existence of large amounts of dark matter of completely unknown constituent components. However, a radical solution is required either way, so both possibilities must be considered and tested.
A phenomenological theory specifically introduced to address the problem of the flat rotation curves is MOND3. It has no other motivation and so far there is no firm physical basis for the theory. It provides no satisfactory cosmology, having yet to be reconciled with General Relativity. However, with the introduction of one new fundamental constant (an acceleration a0), it is empirically quite successful in fitting galaxy rotation curves11-14. It hypothesises that for accelerations a < a0 = 1.2 x 10-10 m s-2, the effective acceleration is given by aeff = (aN a0)1/2. This simple prescription works well with essentially only one free parameter per galaxy, the stellar mass to light ratio, which is subject to independent constraint by stellar evolution theory. More importantly, MOND makes predictions which are distinct and testable. One specific prediction4 is that the asymptotic (flat) value of the rotation velocity, Va, is Va = (GMa0)1/4. Note that Va does not depend on R, but only on M in the regime of small accelerations (a < a0).
In contrast, Newtonian gravity depends on both M and R. Replacing R with a mass surface density variable S = M(R)/R2, the Newtonian prediction becomes M S ~ Va4 which contrasts with the MOND prediction M ~ Va4. These relations are the theoretical basis in each case for the observed luminosity-linewidth relation L ~ Va4 (better known as the Tully-Fisher15 relation. Note that the observed value of the exponent is bandpass dependent, but does obtain the theoretical value of 4 in the near infrared16 which is considered the best indicator of the stellar mass. The systematic variation with bandpass is a very small effect compared to the difference between the two gravitational theories, and must be attributed to dust or stars under either theory.) To transform from theory to observation one requires the mass to light ratio Y: Y = M/L = S/s, where s is the surface brightness. Note that in the purely Newtonian case, M and L are very different functions of R, so Y is itself a strong function of R. We define Y to be the mass to light ratio within the optical radius R*, as this is the only radius which can be measured by observation. The global mass to light ratio would be very different (since M ~ R for R > R*, the total masses of dark haloes are not measurable), but the particular choice of definition does not affect the relevant functional dependences is all that matters. The predictions become Y2sL ~ Va4 for Newtonian gravity8,16 and YL ~ Va4 for MOND4.
The only sensible17 null hypothesis that can be constructed is that the mass to light ratio be roughly constant from galaxy to galaxy. Clearly distinct predictions thus emerge if galaxies of different surface brightnesses s are examined. In the Newtonian case there should be a family of parallel Tully-Fisher relations for each surface brightness. In the case of MOND, all galaxies should follow the same Tully-Fisher relation irrespective of surface brightness.
Recently it has been shown that extreme objects such as low surface brightness galaxies8,18 (those with central surface brightnesses fainter than s0 = 23 B mag./[] corresponding 40 L☉ pc-2) obey the same Tully-Fisher relation as do the high surface brightness galaxies (typically with s0 = 21.65 B mag./[] or 140 L☉ pc-2) which originally15 defined it. Fig. 1 shows the luminosity-linewidth plane for galaxies ranging over a factor of 40 in surface brightness. Regardless of surface brightness, galaxies fall on the same Tully-Fisher relation.

MOND predicts this behaviour in spite of the very different surface densities of low surface brightness galaxies. In order to understand this observational fact in the framework of standard Newtonian gravity requires a subtle relation8 between surface brightness and the mass to light ratio to keep the product sY2 constant. If we retain normal gravity and the dark matter hypothesis, this result is unavoidable, and the null hypothesis of similar mass to light ratios (which, together with an assumed constancy of surface brightness, is usually invoked to explain the Tully-Fisher relation) is strongly rejected. Instead, the current epoch surface brightness is tightly correlated with the properties of the dark matter halo, placing strict constraints on models of galaxy formation and evolution.
The mass to light ratios computed for both cases are shown as a function of surface brightness in Fig. 2. Fig. 2 is based solely on galaxies with full rotation curves19,20 and surface photometry, so Va and R* are directly measured. The correlation in the Newtonian case is very clear (Fig. 2a), confirming our inference8 from the Tully-Fisher relation. Such tight correlations are very rare in extragalactic astronomy, and the Y-s relation is probably the real cause of an inferred Y-L relation. The latter is much weaker because surface brightness and luminosity are only weakly correlated21-24.

The Y-s relation is not predicted by any dark matter theory25,26. It can not be purely an effect of the stellar mass to light ratio, since no other stellar population indicator such as color21-24 or metallicity27,28 is so tightly correlated with surface brightness. In principle it could be an effect of the stellar mass fraction, as the gas mass to light ratio follows a relation very similar to that of total mass to light ratio20. We correct for this in Fig. 2 by subtracting the known atomic gas mass so that Y refers only to the stars and any dark matter. We do not correct for molecular gas, as this has never been detected in low surface brightness galaxies to rather sensitive limits30 so the total mass of such gas is unimportant if current estimates31 of the variation of the CO to H2 conversion factor with metallicity are correct. These corrections have no discernible effect at all in Fig. 2 because the dark mass is totally dominant. It is thus very hard to see how any evolutionary effect in the luminous matter can be relevant.
In the case of MOND, the mass to light ratio directly reflects that of the stellar population once the correction for gas mass fraction is made. There is no trend of Y* with surface brightness (Fig. 2b), a more natural result and one which is consistent with our studies of the stellar populations of low surface brightness galaxies21-23. These suggest that Y* should be roughly constant or slightly declining as surface brightness decreases, with much scatter. The mean value Y* = 2 is also expected from stellar evolutionary theory17, which always gives a number 0 < Y* < 10 and usually gives 0.5 < Y* < 3 for disk galaxies. This is particularly striking since Y* is the only free parameter allowed to MOND, and the observed mean is very close to that directly observed29 in the Milky Way (1.7 ± 0.5 M☉/L☉).
The essence of the problem is illustrated by Fig. 3, which shows the rotation curves of two galaxies of essentially the same luminosity but vastly different surface brightnesses. Though the asymptotic velocities are the same (as required by the Tully-Fisher relation), the rotation curve of the low surface brightness galaxy rises less quickly than that of the high surface brightness galaxy as expected if the mass is distributed like the light. Indeed, the ratio of surface brightnesses is correct to explain the ratio of velocities at small radii if both galaxies have similar mass to light ratios. However, if this continues to be the case as R increases, the low surface brightness galaxy should reach a lower asymptotic velocity simply because R* must be larger for the same L. That this does not occur is the problem, and poses very significant systematic constraints on the dark matter distribution.

Satisfying the Tully-Fisher relation has led to some expectation that haloes all have the same density structure. This simplest possibility is immediately ruled out. In order to obtain L ~ Va4 ~ MS, one might suppose that the mass surface density S is constant from galaxy to galaxy, irrespective of the luminous surface density s. This achieves the correct asymptotic velocity Va, but requires that the mass distribution, and hence the complete rotation curve, be essentially identical for all galaxies of the same luminosity. This is obviously not the case (Fig. 3), as the rotation curves of lower surface brightness galaxies rise much more gradually than those of higher surface brightness galaxies (also a prediction4 of MOND). It might be possible to have approximately constant density haloes if the highest surface brightness disks are maximal and the lowest minimal in their contribution to the inner parts of the rotation curves, but this then requires fine tuning of Y* with this systematically decreasing with surface brightness.
The expected form of the halo mass distribution depends on the dominant form of dark matter. This could exist in three general categories: baryonic (e.g., MACHOs), hot (e.g., neutrinos), and cold exotic particles (e.g., WIMPs). The first two make no specific predictions. Baryonic dark matter candidates are most subject to direct detection, and most plausible candidates have been ruled out10 with remaining suggestions of necessity sounding increasingly contrived32. Hot dark matter is not relevant to the present problem. Even if neutrinos have a small mass, their velocities considerably exceed the escape velocities of the haloes of low mass galaxies where the problem is most severe. Cosmological simulations involving exotic cold dark matter33,34 have advanced to the point where predictions are being made about the density structure of haloes. These take the form33,34 p(R) = pH/[R(R+RH)b] where pH characterises the halo density and RH its radius, with b ~ 2 to 3. The characteristic density depends on the mean density of the universe at the collapse epoch, and is generally expected to be greater for lower mass galaxies since these collapse first in such scenarios. This goes in the opposite sense of the observations, which show that low mass and low surface brightness galaxies are less, not more, dense. The observed behaviour is actually expected in scenarios which do not smooth on a particular mass scale and hence allow galaxies of the same mass to collapse at a variety of epochs25, but in this case the Tully-Fisher relation should not be universal. Worse, note that at small R < RH, p(R) ~ R-1. It has already been noted32,35 that such a steep interior density distribution is completely inconsistent with the few (4) analysed observations of dwarf galaxies. Our data19,20 confirm and considerably extend this conclusion for 24 low surface brightness galaxies over a wide range in luminosity.
The failure of the predicted exotic cold dark matter density distribution either rules out this form of dark matter, indicates some failing in the simulations (in spite of wide-spread consensus), or requires some mechanism to redistribute the mass. Feedback from star formation is usually invoked for the last of these, but this can not work for two reasons. First, an objection in principle: a small mass of stars and gas must have a dramatic impact on the distribution of the dominant dark mass, with which they can only interact gravitationally. More mass redistribution is required in less luminous galaxies since they start out denser but end up more diffuse; of course progressively less baryonic material is available to bring this about as luminosity declines. Second, an empirical objection: in this scenario, galaxies explode and gas is lost. However, progressively fainter and lower surface brightness galaxies, which need to suffer more severe explosions, are actually very gas rich.
Observationally, dark matter haloes are inferred to have density distributions1,2,11 with constant density cores, p(R) = p0/[1 + (R/R0)g]. Here, p0 is the core density and R0 is the core size with g ~ 2 being required to produce flat rotation curves. For g = 2, the rotation curve resulting from this mass distribution is V(R) = Va [1-(R0/R) tan-1({R/R0)]1/2 where the asymptotic velocity is Va = (4πG p0 R02)1/2. To satisfy the Tully-Fisher relation, Va, and hence the product p0 R02, must be the same for all galaxies of the same luminosity. To decrease the rate of rise of the rotation curves as surface brightness decreases, R0 must increase. Together, these two require a fine tuning conspiracy to keep the product p0 R02 constant while R0 must vary with the surface brightness at a given luminosity. Luminosity and surface brightness themselves are only weakly correlated, so there exists a wide range in one parameter at any fixed value of the other. Thus the structural properties of the invisible dark matter halo dictate those of the luminous disk, or vice versa. So, s and L give the essential information about the mass distribution without recourse to kinematic information.
A strict s-p0-R0 relation is rigorously obeyed only if the haloes are spherical and dominate throughout. This is probably a good approximation for low surface brightness galaxies but may not be for the those of the highest surface brightness. However, a significant non-halo contribution can at best replace one fine tuning problem with another (e.g., surface brightness being strongly correlated with the stellar population mass to light ratio instead of halo core density) and generally causes additional conspiracies.
There are two perspectives for interpreting these relations, with the preferred perspective depending strongly on the philosophical attitude one has towards empirical and theoretical knowledge. One view is that these are real relations which galaxies and their haloes obey. As such, they provide a positive link between models of galaxy formation and evolution and reality.
The other view is that this list of fine tuning requirements makes it rather unattractive to maintain the dark matter hypothesis. MOND provides an empirically more natural explanation for these observations. In addition to the Tully-Fisher relation, MOND correctly predicts the systematics of the shapes of the rotation curves of low surface brightness galaxies19,20 and fits the specific case of UGC 128 (Fig. 3). Low surface brightness galaxies were stipulated4 to be a stringent test of the theory because they should be well into the regime a < a0. This is now observed to be true, and to the limit of observational accuracy the predictions of MOND are confirmed. The critical acceleration scale a0 is apparently universal, so there is a single force law acting in galactic disks for which MOND provides the correct description. The cause of this could be either a particular dark matter distribution36 or a real modification of gravity. The former is difficult to arrange, and a single force law strongly supports the latter hypothesis since in principle the dark matter could have any number of distributions which would give rise to a variety of effective force laws. Even if MOND is not correct, it is essential to understand why it so closely describe the observations. Though the data can not exclude Newtonian dynamics, with a working empirical alternative (really an extension) at hand, we would not hesitate to reject as incomplete any less venerable hypothesis.
Nevertheless, MOND itself remains incomplete as a theory, being more of a Kepler’s Law for galaxies. It provides only an empirical description of kinematic data. While successful for disk galaxies, it was thought to fail in clusters of galaxies37. Recently it has been recognized that there exist two missing mass problems in galaxy clusters, one of which is now solved38: most of the luminous matter is in X-ray gas, not galaxies. This vastly improves the consistency of MOND with with cluster dynamics39. The problem with the theory remains a reconciliation with Relativity and thereby standard cosmology (which is itself in considerable difficulty38,40), and a lack of any prediction about gravitational lensing41. These are theoretical problems which need to be more widely addressed in light of MOND’s empirical success.
ACKNOWLEDGEMENTS. We thank R. Sanders and M. Milgrom for clarifying aspects of a theory with which we were previously unfamiliar. SSM is grateful to the Kapteyn Astronomical Institute for enormous hospitality during visits when much of this work was done. [Note added in 2020: this work was supported by a cooperative grant funded by the EU and would no longer be possible thanks to Brexit.]
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