# The Acceleration Scale in the Data

Sci-Fi By Louis Spencer JR | September 6, 2018

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

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?

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