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I am trying to understand a typical explanation for the need of a dark matter halo in galaxies. The rotational velocity of stars around the centre of a galaxy seems to be constant past a certain radius. Assuming we can use

$v(r)=\sqrt\frac{GM(r)}{r}$,

where $M(r)$ is the mass of the galaxy enclosed within radius $r$, we seem to need $M(r)\propto r$ for the velocity to be constant. Assuming we can use $M(r)=\frac{4}{3}\pi r^3\rho(r)$, this requires $\rho(r) \propto 1/r^2$. Below is an image of the surface density profile of galaxy NGC 3198. Let's say the volume density profile looks similar and it can be modelled as $1/r^2$. This would make the constant velocity behaviour plausible without the need for dark matter.

enter image description here

The problem arises from the fact that the visible matter (stars, gas) in galaxies have a cut-off radius $R_{\textrm{visible}}$ past which there is virtually none. We do not receive any luminosity from regions located at $r>R_{\textrm{visible}}$. The claim is then that the velocity curve is still constant past $R_{\textrm{visible}}$, and hence the mass of the galaxy has to keep increasing linearly with radius. This extra invisible mass leads to postulate that galaxies are embedded in a dark matter halo of radius $R_{\textrm{dark}}>R_{\textrm{visible}}$.

My question is, how can we know that the velocity curve is still constant for $r>R_{\textrm{visible}}$ if we cannot detect anything coming from such outer region?

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Indeed this can be confusing at first. The velocity curve is measured using various tracer objects, stars and hydrogen clouds being the most obvious ones. Hydrogen clouds can be found way beyond where stars shine. As a concrete (and pretty) example I slightly modified Figure 3 from this paper:

enter image description here

This is the galaxy Messier 74. On the left you see the integrated hydrogen (HI) map from the THINGS survey. On the right is, at the same scale, the optical image from the DSS survey. Clearly, what we see with stars is just a small part of the galaxy.

Your calculation gives a good first estimate what to expect, but is only correct for a spherically symmetric distribution. In contrast, we observe both the stars as well as the hydrogen clouds to be limited to a relatively thin disk. Thus neither stars nor hydrogen clouds can explain the observed flat rotation curve.

BTW, indeed a simple $M(r)\propto r$ gives the right constant rotation speed, as is seen from your first equation. This is the mass distribution of an isothermal halo. Such a distribution is consistent with what you would expect e.g. from simulations of structure formation. Taken together, it is yet another consistent piece in the puzzle of dark matter evidence.

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Clearly a rotation curve cannot be measured beyond the point at which no visible matter can be seen.

Rotation velocity measurements can come from clouds of hydrogen gas, distant planetary nebulae or globular clusters and satellite galaxies. These objects are scarce enough that they make a negligible contribution to the Galactic mass.

The dark matter doesn't begin at some radius, and the normal matter doesn't abruptly end at some radius.

Surface density isn't volume density and has different units.

The velocity in a Keplerian disc, which might be the appropriate model for a disc galaxy, does not depend on radius as $v \propto r^{-1/2}$, since that assumes a spherically symmetric mass distribution. Thus if you want to avoid dark matter (you can't, with Newtonian gravity) then you should begin by comparing with a self-consistent model of the velocity field according to the visible mass distribution.

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  • $\begingroup$ Yea that all makes sense. As you can see my Q has a lot of "let's assume" just for sake of argument. And I realise myself that it leads down the wrong path in many respects. I was just intrigued by the following paragraph here: w.astro.berkeley.edu/~mwhite/darkmatter/rotcurve.html $\endgroup$ – Luismi98 Nov 29 '20 at 19:38
  • $\begingroup$ The first real surprise in the study of dark matter lay in the outermost parts of galaxies, known as galaxy halos. Here there is negligible luminosity, yet there are occasional orbiting gas clouds which allow one to measure rotation speeds and distances. The rotation speed is found not to decrease with increasing distance from the galactic center, implying that the mass distribution of the galaxy cannot be concentrated, like the light distribution. $\endgroup$ – Luismi98 Nov 29 '20 at 19:39
  • $\begingroup$ The mass must continue to increase: since the rotation speed satisfies v^2=GM/r, where M is the mass within radius r, we infer that M increases proportionally to r. This rise appears to stop at about 50kpc, where halos appear to be truncated. $\endgroup$ – Luismi98 Nov 29 '20 at 19:39
  • $\begingroup$ I was intrigued about the fact that we might be relying on velocity measurements of occasional orbiting gas clouds on outer regions of the galaxy disk, which seemed a bit dodgy to me $\endgroup$ – Luismi98 Nov 29 '20 at 19:41
  • $\begingroup$ @Luismi98 Why do you think occasional orbiting gas clouds would obey different physical laws? As I said - there are planetary nebulae (which are particularly accurate velocity indicators), globular clusters and satellite galaxies in halos. $\endgroup$ – ProfRob Nov 30 '20 at 8:19

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