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

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

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

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

g = 0.8 Vf4/(G Mb)

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

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

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

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

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

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

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

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

Do you see the acceleration scale?

39 thoughts on “The Acceleration Scale in the Data

  1. Is calibration of “unresolved feedback” to debunk MOND a prime example of data dredging or p-hacking?
    Read the 1st page of the following publication:
    Ludlow, Aaron D., Alejandro Benitez-Llambay, Matthieu Schaller, Tom Theuns, Carlos S. Frenk, Richard Bower, Joop Schaye et al. “The Mass-Discrepancy Acceleration Relation: A Natural Outcome of Galaxy Formation in CDM Halos.” arXiv preprint arXiv:1610.07663 (2016).


  2. It is basically scientific gaslighting. These people denied that the mass discrepancy-acceleration relation was a valid entity the needed to explain before they claimed to explain it.


      1. I have asked them that. No answer yet. I get asked to referee lots of papers, more than I can actually do. It would seem natural to at least ask, since they use multiple referees and this analysis is based on data we provided. It isn’t like these authors have obtained new data or even worked in the field. They simply grabbed the data from the internet and turned a crank on a black box statistical package that they only dimly comprehend. I speculate that the authors specifically requested that we not be consulted, but I have no way to know that. Nature is NOT a prestigious journal in Astronomy: it is, in fact, the butt of many well-deserved jokes.


    1. Why should scientists try to gaslight the reality of MOND? In Newtonian terms, MOND implies that gravitational energy is not conserved, linear momentum is not conserved, and angular momentum is not conserved. In the non-relativistic approximation, Milgrom thinks Newton’s 2nd law of motion is slightly wrong — but I think that the 2nd law is correct and the 3rd law is slightly wrong. I conjecture that string theory with the finite nature hypothesis implies MOND and no supersymmetry (because string vibrations are confined to 3 copies of the Leech lattice), while string theory with the infinite nature hypothesis implies supersymmetry and no MOND. Is my conjecture foolish? Think about Woit versus Witten —Woit says that string theory is a failed research program — what does Witten say?
      “String theory is the only known generalization of quantum field theory that makes sense.” “Unravelling string theory”, Edward Witten, 2005
      How can Newton’s 3rd law of nature be wrong? Gravitons might be escaping from the boundary of the multiverse into the interior of the universe. This escape process would not be DIRECTLY detectable but might be indicated by dark matter, dark energy, and inflation. The escape process might bend gravitational lines of force (dark matter) and create gravitational energy loss (dark energy) and create inflation (excess flattening of spacetime).
      In the standard form of Einstein’s field equations, replace the -1/2 by -1/2 + dark-matter-compensation-constant. I conjecture that the Gravity Probe B science team found the confirmation that dark-matter-compensation is roughly equal to 3.9 ± .4 * 10^-5 BUT they thought that the weird anomaly is due to “very small misalignments” in their 4 ultra-precise gyroscopes due to “random patches of electrostatic potential”. I think that the 4 gyroscopes worked correctly. “Viewpoint: Finally, results from Gravity Probe B” by Clifford M. Will, 31 May 2011


      1. Readers of this blog might take a look at this recent article
        Dark Energy After GW170817: Dead Ends and the Road Ahead
        Jose María Ezquiaga and Miguel Zumalacárregui
        Phys. Rev. Lett. 119, 251304 – Published 18 December 2017

        To me, there are many faults in the comments by David Brown, including his appeal to authority and acceptance of gravitons as real. The strength of the Einstein field equation, without the cosmological constant, is being consistent with all of Newton’s laws – and vice versa.


  3. thanks for creating a new post, on this issue,

    “The only slightly tricky quantity is the baryonic mass, which is the sum of stars and gas.”

    does that goes into calculation also include mass of black holes and neutrinos and non-luminous baryonic matter also enter into the equation ?


    1. I compute the baryonic mass as Mb=M*+Mg. M* = (M*/L)*L (here L is the luminosity in the Spitzer 3.6 micron band) and Mg = 1.33*M(HI) is the atomic gas mass corrected to include Helium. Other components like molecular gas, ionized gas, black holes, etc. can be added in, but these are the proverbial drops in the bucket. The molecular gas is comparable to the atomic gas in a handful of the brightest galaxies, but is much less in the vast majority of the SPARC sample. There is extremely little ionized gas in the disk (millions of solar masses vs. billions in atomic gas) and all other components are negligible. There may be more ionized gas in extended corona around galaxies, but this is an order of magnitude farther out than any of the quantities we are concerned with here.

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      1. What about the mass-equivalence of spatial energy? In the Foundation of the General Theory of Relativity* Einstein said the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy. It isn’t made of WIMPs. Then in his 1920 Leyden Address he described a gravitational field as a place where space is neither homogeneous nor isotropic. Space expands between the galaxies but not within, as per the raisin cake analogy. Conservation of energy means the spatial energy density within a galaxy is greater than that outside a galaxy. The non-uniform expansion means every galaxy is sitting in a region of inhomogeneous space – which is what a gravitational field is. All the more so for older galaxies. See for a hint of this.


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  4. Very, very interesting results compatible with recent underground “lab” observations of dark matter have been total busts. The histogram presented here as N vs. g+/a_0 is much more believable than a log(data) presentation. One suspects the data are Gaussian, or nearly so, in distribution. Will be interesting to see the same type of graph with more data in the near future.


      1. If “the data are consistent with filling out a Gaussian distribution” for the Milgromian acceleration scale, then are the data also consistent with negligible dark matter halos around galaxies (how negligible in quantitative terms)?
        What might be wrong with the following?
        Kroupa, Pavel. “The observed spatial distribution of matter on scales ranging from 100kpc to 1Gpc is inconsistent with the standard dark-matter-based cosmological models.” arXiv preprint arXiv:1610.03854 (2016).


  5. btw have you heard of this paper?

    Radial Acceleration Relation of
    Satellite Galaxies
    Enrico Garaldi, Emilio Romano-Díaz, Cristiano Porciani, and Marcel S. Pawlowski
    Phys. Rev. Lett. 120, 261301 – Published 25 June 2018

    ” Moreover, the acceleration due to the gravitational field of the host has no effect on the RAR. This is in contrast with the external field effect in modified Newtonian dynamics (MOND) ”

    sounds like there’s a lot of effort to show dark matter reproduces RAR and MOND does not.


    1. This is in principle a good test: the external field effect (EFE) is unique to MOND, and indeed predicts that some dwarfs should fall off the RAR. Strictly speaking, the RAR is a prediction of MOND for isolated galaxies; the EFE applies only when a giant host galaxy ups the acceleration felt by stars in a dwarf satellite. The RAR itself was uniquely predicted by MOND; it is not an inevitable outcome of galaxy formation in LCDM. Certainly many groups are trying to make it so, but whether this is feasible remains an open question: at best LCDM can be made to “look like” MOND after the fact. To explain the EFE on top of the RAR would require extra epicycles on top of the epicycles we’ve already added. The go-to epicycle for this purpose has already become tidal disruption. The EFE successfully predicted the velocity dispersions in some Local Group dwarfs like Andromeda XIX and Crater 2; the LCDM explanation is simply that these objects were disrupted by a bigger halo so their velocity dispersions are lower than expected. This provides no explanation for why MOND predicted which objects would have low velocity dispersions in the first place.


      1. “… the external field effect (EFE) is unique to MOND .. ” This might apply to all theories published in refereed journals but consider (2) basic theories (or suggestions of theories):
        (1) In the standard form of Einstein’s field equations, replace the -1/2 by -1/2 + dark-matter-compensation-function, where this is a function (of unknown parameters) that might mimic MOND and might indicate that Newtonian-Einsteinian gravitational theory is slightly wrong.
        (2) In the standard form of Einstein’s field equations, replace the -1/2 by -1/2 + fake-function, where this fake-function indicates a weird unexpected property of dark matter particles that somehow simulate MOND but the fake-function does not indicate a problem with Newtonian-Einsteinian gravitational theory.
        It seems to me that (alternative 1 ) implies a real EFE, which might not entirely agree with MOND; (alternative 2) indicates that MOND simulates dark matter particles with a fake EFE.


  6. @Michael L. Smith
    Concerning the following:
    Ezquiaga, Jose María, and Miguel Zumalacárregui. “Dark Energy after GW170817.” arXiv preprint arXiv:1710.05901 (2017).
    Newton’s law of gravity and non-relativistic MOND predict that the speed of gravitational propagation is infinite. Some MOND-like relativistic theory should be correct because (A) MOND has many empirical successes and (B) relativity theory has many empirical successes. I conjecture the following:
    MILGROM DENIAL HYPOTHESIS: The main problem with string theory is that string theorists fail to realized that Milgrom is the Kepler of contemporary cosmology.
    If MOND were wrong, then McGaugh, Kroupa, Sanders, and Scarpa would not agree that the MOND acceleration constant is approximately 1.2±.2 * 10^-10 meter/sec^2 — am I wrong on this?


  7. Stacy have you read Bee’s latest blog post

    Monday, July 23, 2018
    Evidence for modified gravity is now evidence against it.

    her claim is that the paper doesn’t test against dark matter.

    do you plan a follow up paper to Rodrigues, have you spoken to them?

    also i look forward to the august issue of SciAm i understand you and Bee have an article there.

    have you and Bee thought about writing down a relativistic completion of MOND?


  8. No, they don’t test against dark matter. If they did, they would get the same result as they claim against MOND. There is a tail of galaxies with high (formally unacceptable) chi^2 in any model of any flavor. There are a number of reasons for this, one being that the uncertainties on the measured Doppler velocity are not equivalent to what we want them to be: Gaussian uncertainties on the circular velocity of the gravitational potential. People who work in the field understand this. That’s why we did not claim that all flavors of dark matter are excluded at 10 sigma in even though the chi^2 distribution is similarly bad.

    We have submitted to Nature Astronomy a correspondence pointing out some of our concerns. Frankly, there are too many to fit into the word limits of such a contribution. But I have no interest in spending my time writing papers pointing out that other people made glaringly obvious mistakes. I don’t even like writing blog posts about it, which is why I rarely post here – there could be so much more along these lines. It has to be especially misleading to earn mention here.


  9. ok thanks.

    are there any tests to distinguish dark matter vs MOND?

    and do you favor any relativistic extension of MOND that hasn’t been ruled out like TEVS ?


  10. There are many tests that should in principle distinguish MOND from DM. Rotation curves, for starters. The presence or absence of the external field effect. Etc. – I have written many original papers about this and at least a half a dozen review articles about this, e.g., . The problem is that DM is a concept, not a theory. As such, it is not subject to falsification, and can always pretend to “predict” whatever MOND predicts. So it depends on what you consider adequate to “distinguish” between DM and MOND.

    I am not aware of a satisfactory relativistic version of MOND. I’m not even sure that is the right tree to bark at. MONDian phenomenology may point to something more fundamental in the nature of inertia rather than a “simple” modification of gravity. That is to say, not only do we not have a satisfactory theory, most people are not even working on the right problem.


    1. “not only do we not have a satisfactory [MONDian] theory, most people are not even working on the right problem.”

      I am curious: what exactly is the “right problem” to work on re. MOND? I ask from genuine ignorance.

      sean s.


      1. That’s my point. I don’t think we know what the right problem is.

        Most people are working to find dark matter. That is the wrong problem to spend time on IF the stuff doesn’t exist.

        Some people are working on modified gravity theories. Not nearly enough. The number of person-years of effort that have gone into the fruitless DM search exceed those that have looked into alternatives by many orders of magnitude.

        Very few people have even considered modified inertia theories. They’re inherently non-local and unpleasant to work with. They also appear to be the most likely basis for the MOND phenomenology. But one’s judgement of “most likely” is necessarily biased.

        It remains possible that we haven’t even conceived of the right way to approach the problem.


  11. What about the observation that the MOND scale of acceleration ao is on the same order as the cosmological constant as suggesting quantum gravity? specifically, below ao, set by the cosmological constant the gravitational physics is dominated as quantum gravity and above ao, physics becomes classical

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    1. what sort of theoretical legwork needs to be done to make this a concrete and serious proposal, namely that objects with acceleration near the cosmological constant behave as MOND due to quantum effects (either gravity or inertia) and above this it becomes classical as described by Newton and GR?

      when acceleration approaches the scale set by cosmological constant, the quantum mechanical effects of dark energy interacts with objects to change its inertia, or gravity?

      what physicists are best able to model such an effect, and give it details?


    2. It does indeed. To quote @skdh ‘s comment in – space expands “if the average matter density is so small that the gravitational self-atttraction becomes smaller than the outward pull from the expansion. That’s a rough estimate. To be precise you’d have to do a numerical calculation with the exact matter distribution of some region of space-time and there’s no general statement that can be made. But since galaxies are stable, we know they’re gravitationally bound and don’t expand. It’s somewhere beyond the scale of galaxy clusters that expansion takes over.”

      Could it be that the effects of expansion start to occur around mass densities that support a0? Or is that a dumb question. If not dumb, can you point me at additional reading?


      1. That is a good question.

        The transition from stable, gravitationally bound objects to the metric of the expanding space in which they’re embedded is a neat problem we usually ignore. We tend to work in one limit or the other: treating galaxies as we do the solar system, or as mere motes of dust in an expanding universe.

        So it is intriguing that a0 is the scale in the outskirts of galaxies and also about the scale of the accelerated expansion. There are attempts to connect the two, but I’m not aware of any convincing ones. But then, there hasn’t been enough effort in this direction to say much.

        Liked by 1 person

  12. “I don’t think we know what the right problem is. … It remains possible that we haven’t even conceived of the right way to approach the problem.”

    Okay, so … what comes next?

    As I understand it [merely a layman’s understanding] we know there’s a discrepancy between “observable” galactic masses and those required to make sense of galactic dynamics. Either the laws governing dynamics are misunderstood, or there are unknown effects/fields, or there is unobserved matter. Any defects in my “understanding” are my fault.

    GR seems unbroken, so a “fix” there seems unreasonable, no one has found dark matter even though others think its effects can easily observed; and MONDish explanations are incomplete, immature, uncertain, and/or bogus.

    So what now?

    Is it that the problem is unknown? or that everything tried has come up empty and now frustration and a sense of futility are becoming inexorable? Is the problem a lack of data? or a mass failure of imagination?

    “… when you have eliminated the impossible, whatever remains, however improbable, must be the truth …” — Sherlock Holmes, “The Sign of Four”

    sean s.

    BTW, I am no physicist; it’s easy for me to sit on the sidelines and wonder “WTF?”. I do try to appreciate the dilemma for those of you on (or in) the field. There must be something so crazy it’s been overlooked.



    1. No, the problem is not unknown. Nor is it a lack of data. It is something akin to a “mass failure of imagination” because you don’t know how gravity works. Even through Einstein made it clear. He didn’t say light curves because spacetime is curved. He said light curves because the speed of light is spatially variable. Because a concentration of energy “conditions” the surrounding space, such that it is “neither homogeneous nor isotropic”. He also said “the energy of the gravitational field shall act gravitatively like any other kind of energy”. This energy is spatial energy, it has a mass-equivalence, and it isn’t made of particles. And space, in case you hadn’t noticed, is dark.

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  13. I think you sum up the situation well.

    The biggest problem is a failure of imagination. Most of the community is stuck in the dark matter rut. I fear a thousand years of dark epicycles, since we seem to be unwilling to think outside that box.

    But once you open the box, the universe becomes a big, scary place in which you have to be comfortable admitting that we don’t know many things we thought we did. Most scientists are not comfortable with that, and indeed, many seem unwilling or unable to even contemplate such a thing. I struggled mightily with it myself, as I kept thinking “but there’s so much evidence for dark matter.” Took me to realize this was a linguistic problem: something is messed up; dark matter is just on possible solution. Most of the scientific community seems to remain deeply trapped in this mindset.

    So I don’t know what’s next. All we really know so far is what doesn’t work. Pretty sure digging deeper into an invisible rabbit hole is a waste of time, talent, and resources.

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  14. “But once you open the box, the universe becomes a big, scary place in which you have to be comfortable admitting that we don’t know many things we thought we did. Most scientists are not comfortable with that, and indeed, many seem unwilling or unable to even contemplate such a thing.”

    I think this is a human condition; perhaps many scientists think of their discipline as giving them an exception to the human condition. That would be silly.

    You are so right; if DM is a dead end, there’s no point continuing. But then even scientists fall prey to the “sunk costs” fallacy. They are human …

    “I struggled mightily with it myself, …”

    As do I; as everyone should. “I don’t know” is powerfully liberating, but it really annoys those who cannot handle uncertainty or are dogmatic.

    Well, carry on. I enjoy your posts; to the extent I can comprehend them.

    sean s.


  15. In the early ‘80’s H. Sato and K. Maeda published a series of papers concerning the newly discovered Voids and their likely impacts on an early matter filled Universe. Using israel’s Formalism they predicted that expanding Voids would exert a pressure on Matter, forcing it into filamentary structures along their perimeters. Instead of imbedding bound structures in an expanding metric, why not postulate bound metrics and expanding metrics as two interwoven dynamical aspects for our Universe. In essence the developing Voids and ensuing gravitational collapse along their perimeters have divided the Universe into two separate metric configurations. Matter structures aren’t embedded, they’re constrained! And those constraints produce real physical effects….(?)


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