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

First, the reality check. Previously we calibrated the BTFR using galaxies with distances measured by reliable methods like Cepheids and the Tip of the Red Giant Branch (TRGB) method. Application of this calibration obtains the Hubble constant H_{0} = 75.1 +/- 2.3 km/s/Mpc, which is consistent with other local measurements but in tension with the value obtained from fitting the Planck CMB data. All of the calibrator galaxies are nearby (most are within 10 Mpc, which is close by extragalactic standards), but none of them are in the Local Group (galaxies within ~1 Mpc like Andromeda and M33). The distances to Local Group galaxies are pretty well known at this point, so if we got the BTFR calibration right, they had better fall right on it.

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

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

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

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

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

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

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

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

That caveat aside, this works quite well for β_{c} = 2.

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

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

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

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

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

Hi. Is there a connection to what you showed earlier about the rotating galaxies having zero intrinsic scatter around the BTFR? I.e. with this value of beta_c, are the dwarf spheroidals’ deviations from the btfr completely explained by measurement uncertainties? Or am I off on the wrong track?

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Yes, you are on the right track – with the caveat that it only applies to those galaxies to which it applies. It doesn’t work for most of the so-called ultrafaints (the open squares in the first plot), I don’t find these objects to be very informative as they are likely to be out of equilibrium (in any theory). That caveat aside, there is little indication of intrinsic scatter among the remaining dwarfs. Not zero, however: on the bright end, there are some clear deviations. One is Fornax, which has abnormally young stars, so that is probably an offset from the assumed stellar mass-to-light ratio. NGC 147 and NGC 185 are clearly different in velocity dispersion even though they are indistinguishable in luminosity. This might be attributable to differences in anisotropy, which likely happens on occasion. Intriguingly, NGC 147 has distinct tidal tails while NGC 185 does not. So there are some interesting details that lead to real deviations, but in most cases such effects are not perceptible with the current data.

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Tracy, this and other cosmology ‘crisis’ such as Hubble flow, Dark Energy/Matter etc is solved by starting from nonlinear differential rules of Suntola’s Dynamic Universe (DU): dR4/R4=-2 dC4/C4 for differential expansion of 4-radius dR4 step scale dR4/R4 and its deceleration speed rate dC4 of velocity scale dC4/C4..

Relative correction or datum change of z= dR4/R4= 75.1 km/s/Mpc from local group R4 values to DU reference value 71 km/s/Mpc at z=0 converts the ratio 75.1/71 to local velocity ratio Beta_c=2 x 75.1/71=2.1155 = close to MOND prediction 2.12. See DU book to relate heavy mass center values to ‘dips’ -dR4 in direction of R4 of ideal homogeneous mass distribution with corresponding larger dC4 values. This may also be explained as ‘conspiracy of Nature’ fooling GR based cosmologies to see ‘accelerated’ vs decelerating expansion C4 or decelerating dR4/R4 distance scale for expanding R4 – as in attempts of deriving H0 at very small values of R4 near CMB and extrapolating them to present H0 values of large R4 near 13.8 Bly.

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H0 has nothing to do with the value of beta_c as all these Local Group dwarfs are sufficiently nearby that they do not participate in the Hubble flow and have direct distance measurements. There is no scaling by z.

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H0 was replaced in DU by direct predictive power of three different nonlinear scale factors among distances, speed and time – using one single postulate of energy balance. I noticed that your preferred beta_c=2 is related to the linear part of DU nonlinear scale equation among distance (mass) and velocity scales dR4/R4 and dC4/C4. Looks like a similar relation holds also among the velocity scale and its higher order local derivative (dispersion scale = constant EFE term of dR4/R4 distance scale?).

Take a look on DU nested energy frame pyramid and derivation of the basic energy balance equation, evaluated at local distance values of local group members and dwarfs. Take ratio of both sides of these two equations to interconnect the distance/velocity scales. And then interconnect velocity/dispersion = distance scales…Resulting in beta_c=2.1155 vs 2.12 of MOND?!

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Stacy, I noticed from your past blogs that MOND already has discovered DU based energy balance relation among the distance and velocity scales by sc_R4=(sc_C4)^2 so relation among velocity and dispersion scales appears connected to DU sc_T4=(sc_C4)^3 of MOND EFE component.

T4 is the scalar absolute time common for all 4-D metric locations of Riemann 4-sphere vs mistaken GR ‘time’ and simultaneity concepts of GR based cosmologies like FLRW and LCDM. See my past examples of DU emit epoch of R4=13.8/4 Bly, C4=600,000km/s, T4= 2/3 13.8/8 B yrs_today where the slow-down of ticking rate in T4 ‘cosmic clock’ of GPS technology is proven using coral fossil clock from past 1 M years. Also see GDR3 proof of DU and my past PO examples of tying H0 and z measures to these 3 dynamic scale factors of distances R4, c/C4 and T4 based on Suntola’s DU books and papers.

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So the βc=81/4≈2.12 is due to the combination of σ3D=√3σ* and M=9/4 σ3D4/Ga0 only in isotropic isolated equilibrium low-acceleration pressure supported systems (as cited in Milgrom 1995) right? Are there derivations for the right βc in systems in the transition regime and in the Newtonian regime? So suppose I have a system that is mostly pressure supported with a small amount of rotation and with characteristic gbar’s of a0 to 103 a0, what would be the right way of calculating gobs? Because I don’t think gobs=3σ*2/R+V2/R works (unless Hernandez is off in his mass estimates for GCs by like a factor of 2.5).

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Testing subscript and superscript

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Yes, you have the math right, even if html won’t cooperate. One does expect exceptions for the reasons you note – in the Newtonian regime, you ought to get sqrt(3) (and do, in the one case we have here). More generally, objects affected by the external field effect will have lower velocity dispersions than they would as isolated objects. So Crater 2 is predicted to be a 2 km/s object instead of a 4 km/s one that is smack on TF. Hard to tell the difference. As for calculating the more general case, you have the right idea, but it is messy in the tranisiton regime (the math we’ve stated so far only holds formerly in the deep limits for from a0). People have used numerical calculations to work out approximate fitting functions. IO forget the references offhand but can dig them out if you’re keen.

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Idranil beat you to it but thanks for offering! You’re right that the external field is important. There is a fairly strong correlation (R^2=0.6) between the external field and the velocity dispersion when (g_int<g_ext). And that's with a very crude estimation of g_ext which assumes the MW is point like!

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Hi Mark,

I would recommend using the formulae given here:

https://research-repository.st-andrews.ac.uk/handle/10023/18162

They cover two crucial generalisations of the Milgrom (1994) formula, which is actually clearly stated in a paper of his from 1994 (https://ui.adsabs.harvard.edu/abs/1994PhLA..190…17M/abstract) prior to the 1995 ApJ paper but never mind. The generalisations are that the gravity need not be much weaker than a_0, and there could be an external field. This requires a 2D grid of numerical simulations to cover different acceleration strengths internally and externally. The grid was completed and published in 2009. This should address your question of how to handle an arbitrary internal acceleration, and also how to include the external field effect.

Indranil

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Thanks for the reference Idranil! That is exactly what I was looking for. Your work on the WBT is also really picking up steam I see. Can’t wait to read your results for it based on GDR3!

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This was an excellent study that provides an important test of the MOND prediction that the line of sight velocity dispersion should be sqrt(sqrt(4GMa0/81)) for isolated virialised galaxies in the deep-MOND regime. Naturally, objects prone to tidal disruption should be excluded from the analysis, as rightly done here. But testing MOND is not all about simply excluding such objects – it can also be tested based on which ones are affected by tides and which are not. This is a bit hard to do in the Local Group as the whole sky was not surveyed to the same depth or limiting surface brightness. But it could be possible further away for a system that subtends a small angle on the sky and was surveyed homogeneously and very deeply. This is indeed what we did here:

https://darkmattercrisis.wordpress.com/2021/08/

Another suggestion would be to correlate deviations from the isolated MOND prediction with the strength of the external field effect. Perhaps NGC 147 and NGC 185 are feeling the external gravity from M31. The external field effect has been detected in rotationally supported systems (ApJ, 904, 51), so perhaps it also shows up in pressure supported systems.

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