The Milky Way Galaxy in which we live seems to be a normal spiral galaxy. But it can be hard to tell. Our perspective from within it precludes a “face-on” view like the picture above, which combines some real data with a lot of artistic liberty. Some local details we can measure in extraordinary detail, but the big picture is hard. Just how big is the Milky Way? The absolute scale of our Galaxy has always been challenging to measure accurately from our spot within it.

For some time, we have had a remarkably accurate measurement of the angular speed of the sun around the center of the Galaxy provided by the proper motion of Sagittarius A*. Sgr A* is the radio source associated with the supermassive black hole at the center of the Galaxy. By watching how it appears to move across the sky, Reid & Brunthaler found our relative angular speed to be 6.379 milliarcseconds/year. That’s a pretty amazing measurement: a milliarcsecond is one one-thousandth of one arcsecond, which is one sixtieth of one arcminute, which is one sixtieth of a degree. A pretty small angle.

The proper motion of an object depends on the ratio of its speed to its distance. So this high precision measurement does not itself tell us how big the Milky Way is. We could be far from the center and moving fast, or close and moving slow. Close being a relative term when our best estimates of the distance to the Galactic center hover around 8 kpc (26,000 light-years), give or take half a kpc.

This situation has recently improved dramatically thanks to the Gravity collaboration. They have observed the close passage of a star (S2) past the central supermassive black hole Sgr A*. Their chief interest is in the resulting relativistic effects: gravitational redshift and Schwarzschild precession, which provide a test of General Relativity. Unsurprisingly, it passes with flying colors.

As a consequence of their fitting process, we get for free some other interesting numbers. The mass of the central black hole is 4.1 million solar masses, and the distance to it is 8.122 kpc. The quoted uncertainty is only 31 pc. That’s parsecs, not kiloparsecs. Previously, I had seen credible claims that the distance to the Galactic center was 7.5 kpc. Or 7.9. Or 8.3 Or 8.5. There was a time when it was commonly thought to be about 10 kpc, i.e., we weren’t even sure what column the first digit belonged in. Now we know it to several decimal places. Amazing.

Knowing both the Galactocentric distance and the proper motion of Sgr A* nails down the relative speed of the sun: 245.6 km/s. Of this, 12.2 km/s is “solar motion,” which is how much the sun deviates from a circular orbit. Correcting for this gives us the circular speed of an imaginary test particle orbiting at the sun’s location: 233.3 km/s, accurate to 1.4 km/s.

The distance and circular speed at the solar circle are the long sought Galactic Constants. These specify the scale of the Milky Way. Knowing them also pins down the rotation curve interior to the sun. This is well constrained by the “terminal velocities,” which provide a precise mapping of relative speeds, but need the Galactic Constants for an absolute scale.

A few years ago, I built a model Milky Way rotation curve that fit the terminal velocity data. What I was interested in then was to see if I could use the radial acceleration relation (RAR) to infer the mass distribution of the Galactic disk. The answer was yes. Indeed, it makes for a clear improvement over the traditional approach of assuming a purely exponential disk in the sense that the kinematically inferred bumps and wiggles in the rotation curve correspond to spiral arms known from star counts, as in external spiral galaxies.

Now that the Galactic constants are Known, it seems worth updating the model. This results in the surface density profile

The surface density profile of the Milky Way model scaled to the newly accurate distance to the Galactic center.

with the corresponding rotation curve

The rotation curve of the Milky Way as traced by terminal velocities in the first and fourth quadrants (red and blue points). The solid line is a model that matches this rotation curve. The dashed and dotted lines are the rotation curves of the baryonic and inferred dark matter components. Yellow bands show the effect of varying the stellar mass by 5%.

The model data are available from the Milky Way section of my model pages.

Finding a model that matches both the terminal velocity and the highly accurate Galactic constants is no small feat. Indeed, I worried it was impossible: the speed at the solar circle is down to 233 km/s from a high of 249 km/s just a couple of kpc interior. This sort of variation is possible, but it requires a ring of mass outside the sun. This appears to be the effect of the Perseus spiral arm.

For the new Galactic constants and the current calibration of the RAR, the stellar mass of the Milky Way works out to just under 62 billion solar masses. The largest uncertainty in this is from the asymmetry in the terminal velocities, which are slightly different in the first and fourth quadrants. This is likely a real asymmetry in the mass distribution of the Milky Way. Treating it as an uncertainty, the range of variation corresponds to about 5% up or down in stellar mass.

With the stellar mass determined in this way, we can estimate the local density of dark matter. This is the critical number that is needed for experimental searches: just how much of the stuff should we expect? The answer is very precise: 0.257 GeV per cubic cm. This a bit less than is usually assumed, which makes it a tiny bit harder on the hard-working experimentalists.

The accuracy of the dark matter density is harder to assess. The biggest uncertainty is that in stellar mass. We known the total radial force very well now, but how much is due to stars, and how much to dark matter? (or whatever). The RAR provides a unique method for constraining the stellar contribution, and does so well enough that there is very little formal uncertainty in the dark matter density. This, however, depends on the calibration of the RAR, which itself is subject to systematic uncertainty at the 20% level. This is not as bad as it sounds, because a recalibration of the RAR changes its shape in a way that tends to trade off with stellar mass while not much changing the implied dark matter density. So even with these caveats, this is the most accurate measure of the dark matter density to date.

This is all about the radial force. One can also measure the force perpendicular to the disk. This vertical force implies about twice the dark matter density. This may be telling us something about the shape of the dark matter halo – rather than being spherical as usually assumed, it might be somewhat squashed. It is easy to say that, but it seems a strange circumstance: the stars provide most of the restoring force in the vertical direction, and apparently dominate the radial force. Subtracting off the stellar contribution is thus a challenging task: the total force isn’t much greater than that from the stars alone. Subtracting one big number from another to measure a small one is fraught with peril: the uncertainties tend to blow up in your face.

Returning to the Milky Way, it seems in all respects to be a normal spiral galaxy. With the stellar mass found here, we can compare it to other galaxies in scaling relations like Tully-Fisher. It does not stand out from the crowd: our home is a fairly normal place for this time in the Universe.

The stellar mass Tully-Fisher relation with the Milky Way shown as the red star. It is a typical spiral galaxy.

It is possible to address many more details with a model like this. See the original!



10 thoughts on “A Precise Milky Way

  1. Very interesting – thanks. Can you help me understand why the rotational velocity peaks at ~6.5 Kpc? Something to do with the amount of mass internal to that radius versus external? When you refer to “first and fourth quadrants” – are you referring to quadrants of the galactic orbit relative to earth? Lastly, what are your thoughts on the ability to detect DM at the 0.257 GeV per cubic cm value either now orin the near future?


  2. The peak of the rotation velocity is the result of how the mass builds up as one integrates out from the center. The circular speed grows as more and more mass is enclosed, eventually hitting a maximum then declining after most of the mass has been encompassed. That it does not continue to decline as fast as it should for what we see is of course at the center of the missing mass problem.
    First and fourth quadrants are how we divide up the plane of the Milky Way. We’re looking in towards the center in the first and fourth. The wikipedia page helps:
    The density I obtain for the dark matter in this fashion is consistent with if on the low end of independent numbers I’ve seen in the literature. High end numbers are a factor of 2 or so more. That’s not a big deal in the dark matter detection game, where the experimentalists have been making great strides in not detecting it, having improved their sensitivity by many orders of magnitude in recent years. A factor of 2 one way or the other isn’t going to impede detection if the type of dark matter the major experiments are sensitive to is there to detect.
    I do find interesting the apparent tension with some other astronomical measurements. I don’t see anything wrong with those. Often a mild discrepancy like this is a hint that some simple assumption we made for convenience (like a spherical dark matter halo) is inadequate at this level of precision. Data from Gaia will help a lot, but will also take a long time to wrap our heads around.

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  3. Given the factor of 2 or so in the inferred DM density and the potential vertical assymetry in distribution, is it possible to determine if the inferred DM density for typical spiral galaxies is constant over a wide enough range of cosmological redshift?


  4. Hello again,

    Ethan Siegal here

    Modified Gravity Could Soon Be Ruled Out, Says New Research On Dwarf Galaxies

    based on this paper here
    and this

    “It’s an indication that there is dark matter, it is cold and collisionless, and that it can be heated up by recent star formation.”

    “The authors note:

    These two galaxies require different dynamical mass profiles for almost the same radial light profile. This is a challenge not only for MOND, but for any weak-field gravity theory that seeks to fully explain DM.”

    your thoughts? cold dark matter can “heat up” and MOND cannot explain such galaxies.


    1. The result of Read et al. updates and is basically the same as what I found in Draco is a problem for MOND if it is in dynamical equilibrium. Tidal heating can happen in MOND – indeed, is more likely to do so than in LCDM as the visible stars are well protected by their dark matter halos in the latter. For any problematic case, the question becomes whether tidal disruption or stirring has happened or not. We don’t know that, though I discuss some of the indicators in the above cited paper. Draco certainly could be suffering in this way in MOND. Indeed, there are dwarfs (e.g., Crater 2) for which we have to imagine that there has been some form of tidal disruption in order for LCDM to survive, so it suffers the same kind of problem, just in different cases.
      It is not correct that dwarfs of the same light profile have to have the same dynamical profile in MOND, as the latter depends on the external as well as internal field. There are several pairs of photometrically indistinguishable dwarf satellites of Andromeda where there is a difference in dynamical profile – exactly as predicted (a priori) by MOND owing to the external field effect. This is a unique prediction of MOND that, as near as I can tell, is completely inexplicable in LCDM, beyond asserting that’s just how it happened to work out.
      I’m guessing the success of MOND in predicting the dynamics of these other matched pairs is not discussed in certain venues.


      1. Thanks for this interesting discussion. Ethan Siegal never wastes a moment to bring to blog that dark matter rules, MOND ruled out, though he does list papers.

        i found mention of this

        it discusses satellite planes as evidence for MOND, with the claim

        “The study enabled researchers to observe realistic-looking satellite planes the first time anyone did a detailed investigation of the Milky Way–Andromeda flyby; this flyby is inevitable in MOND. Contrast this with the numerous efforts over more than a decade to try and explain the satellite planes in general relativity—it just doesn’t seem to work, regardless of how much dark matter you’re prepared to invent.”

        one astrophysicists sees no “planes” its just wishful thinking, just a random assortment.


  5. I see no mention of MOND. Does the updated information/modelling provide any insight into the applicability of MOND relative to dark matter?


  6. The limit for a Research Note is 1,000 words. MOND is an advanced topic that can’t come up in that space. But the method utilizes the Radial Acceleration Relation, which is the empirical version of MOND, so by construction, this Milky Way model is consistent with MOND.


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