Experimental foundations of the "otherwise galaxies would not hold together" argument for dark matter

Question

On the question of motivation and evidence for dark matter, there are some illuminating (so to speak) answers on this site. I'd like to understand in detail the foundations of one of the lines of reasoning, as stated succinctly in this answer:

...stars within galaxies are moving faster than the escape velocity of the gravitational pull of the center of mass of the galaxy...

If I were to attempt to unpack this, I might say something like the following:

1. There a formula for escape velocity, based on on the classical laws of physics, which should be valid for measuring the rotation of a galaxy (ie, general relativity correction would not change the conclusion we are after?). $$ v = \sqrt(2GM/r) $$

2. However, there are other techniques for measuring M, v, and r.

   • M can be measured indirectly via the mass-luminosity relationship
   • v (here the speed of the outermost stars in the galaxy) and r (the radius of the galaxy) can be inferred, via the Doppler Effect, by measuring the galaxy's red-shift (and some other measurement, I suppose?).

3. The velocity formula does not add up when the measurements are plugged in. Therefore, we can assume that either one of the measurements is incorrect, or the escape velocity formula is incorrect.

4. The most likely candidate is that the measurement for mass is incorrect, specifically because the mass-luminosity relationship is invalid. The theory being that some mass doesn't interact with light.

Am I roughly correct that the above is the argument for the existence of dark matter? I'm looking for the building blocks of the argument, its foundation in terms of measurements and theories. (And I do understand that the case for dark matter lies in a convergence of several lines of evidence, but I'm attempting to wrap my arms around just this one for now.)

Answer 1

The outline is pretty good, but there are a few points I should mention.

First, you're right: GR corrections don't solve the discrepancy.

Second, the mass-luminosity description you linked is just for main-sequence stars. The main sequence is defined as the phase of a star's life when nuclear fusion of hydrogen powers the star. Before and after this stage things are very different. Moreover, there is plenty of ordinary gas and dust floating between the stars - this is known as the interstellar medium. So converting the luminosity of a galaxy into a mass of known "stuff" is a bit trickier. You have to be sure there aren't plausible models of stars or dust that could account for the discrepancy.

As for measuring radius and velocity: The radius is obtained from the angular separation together with a distance to the galaxy (obtained with particularly bright and well-understood stars, or perhaps supernovae, or some other technique). And yes, the velocity is simply obtained from the Doppler shift. Of course, this is a velocity along your line of sight - you may need to throw in a factor accounting for the inclination of the galaxy's rotation (this will only ever make the actual velocity larger than the line-of-sight velocity, so it increases the discrepancy).

[Minor detail: astronomers don't generally measure a single velocity and compare it to the escape velocity. Instead, they use the assumed mass distribution to derive the circular velocity - the velocity of a circular orbit - as a function of radius. They then plot this theoretical curve along with the data points over a range of radii. A schematic of what this tends to look like is shown in this article. This tells us not only that there is missing matter, but also something about its distribution. In particular, it doesn't seem to clump up near the center of the galaxy, as one might expect for "normal" matter that can experience drag and other electromagnetic-mediated forces.]

Third, it could be that galaxies are not in equilibrium, and we happen to be observing them just as a bunch of random stars coalesced into deceptively organized structures that will fly apart in the near future. Not very likely, but it's important to understand what all your assumptions are.

Finally, all you can really conclude, as you noted, is that your luminosity $\to$ mass conversion didn't work. This does not mean there absolutely has to be some exotic new form of matter. The most convincing evidence for that comes from cosmology, as noted in the accepted answer to the question you linked at the beginning. The thing is, it's all too easy to come up with ways normal matter is "hiding" today, like being obscured by dust or locked up in cold planetary bodies not bound to any stars. If our missing matter interacted with light at all, though, it would certainly have done so in the early universe, and in fact the existence of such dark matter is necessary for our standard cosmological models to agree with reality.