These are all fitting formulae (not physically motivated, for the most part), so their form can vary from paper to paper. However, there are some standards. The
singular isothermal sphere should always take the simple form:
where there will be some set of normalizing constants in the above equation. Sometimes it's expressed in terms of the associated rotation velocity and other times just some scaled density, but as long as they're both 1/r
2, it's really the same thing.
When one moves to the non-singular isothermal spheres (meaning the density doesn't approach infinity as r -> 0), the standards are less straightforward. The available formulae can be characterized (and are chosen) based on their limiting behavior. The formula the OP gives approaches the singular isothermal sphere at large radii and a constant density at small radii. This is a decent fit to some galaxies, but a poor fit to most clusters (for which similar formulae are often used). Another example of a non-singular isothermal sphere is:
There are several normalizing constants and scale radii, but the important thing is that it has roughly constant density at the center, approaches 1/r
2 at moderate radii, and has an
exponential truncation at the outskirts.
The point behind the isothermal sphere is that it produces flat rotation curves at moderate radii. It's meant to fit to the observational data that suggest a roughly flat rotation curve for spiral galaxies. We can only measure rotation curves out to moderate distance from the galaxy's center (the galaxy becomes too dim at the outskirts), so we don't know how far out the 1/r
2 dependence goes. It must change form at some point because the total integrated mass diverges for a halo that is 1/r
2 all the way to infinity. The curve the OP gives ignores this problem, presumably because the outer profile doesn't matter for the purposes of the reference paper.
The profile Garth gives is quite different, both qualitatively and quantitatively. It is the standard NFW profile. This profile was made to fit
simulations rather than observations, so if its parameters can be chosen to successfully fit real galaxy data, then it means theory matches observation. Again, look at the limiting behavior. At large radii, it approaches a 1/r
3 profile, while at small radii it goes as 1/r. At first glance, it would seem to never have a 1/r
2 dependence and thus be inconsistent with observations of a flat rotation curve. However, there must be some transition zone between the two dependences, so it turns out that at the radii to which we can measure rotation curves, the NFW profile is approximately 1/r
2.
There has been a lot of research on how well the NFW profile fits real data and the general consensus seems to be that it's better than isothermal models, but not dramatically so. The biggest discrepancy is near the centers of the halos, where the simulations appear to have a sharper cusp than the data. This was already discussed in another thread:
Cuspy Halo Problems
As simulation and observations improve, we'll be able to get a better handle on just how well they match. The fits do seem to be getting better with time, but I would still be cautious to say anything definitive.
