What is the dimensionless central potential in a King Model? King Models are commonly used to model stellar clusters. I understand that they are described by a surface brightness profile $$ \Sigma (r) = \frac{\Sigma (0)}{(1+r/r_0)^2} $$ as described here.
In the literature, a 'central dimensionless potential' $W$ is quoted, which appears to describe the King model. How is $W$ related to the surface brightness profile? Also is this the correct description of a King Model - I seem to get conflicting reports on the web?
 A: I'll give this a try, though I don't have the time or inclination to work through the gory details myself. I'll start by defining the notation I'll use:
Poisson's equation:
$$\nabla^2\Phi({\bf x}) = 4\pi G \rho({\bf x})$$
Define a new potential by a constant offset, where $\Phi_0$ is the potential at infinity:
$$\Psi=-\Phi+\Phi_0$$
And define the velocity dispersion (which is a constant in the King model):
$$\sigma^2 = \langle v^2\rangle/3$$
The radial density profile (radial in the spherical sense) is:
$$\rho(r) = \rho_0\left[e^{\Psi/\sigma^2}{\rm erf\left(\frac{\sqrt{\Psi}}{\sigma}\right)}-\sqrt{\frac{4\Psi}{\pi\sigma^2}}\left(1+\frac{2\Psi}{3\sigma^2}\right)\right]$$
This needs some work before being useful. Apparently the trick is to insert it into Poisson's equation and solve for $\Psi$ using the boundary conditions $\Psi(r=0)=\Psi_0$ and $\left.\frac{{\rm d}\Psi}{{\rm d}r}\right|_{r=0}=0$.
Then I think the dimensionless central potential is:
$$W=\Psi_0/\sigma^2$$
You could then get the surface density profile (which in this case is the same as the surface brightness profile) by doing something like this:
$$\Sigma(R) = \lim_{\Delta R \rightarrow 0}\frac{2\pi\int_{-\infty}^\infty\int_R^{R+\Delta R}\rho(r)\,R\,{\rm d}z\,{\rm d }R}{2\pi \left[\left(R+\Delta R\right)^2-R^2\right]}$$
I think there's a way to write that integral without the limit, but this is what I managed to scribble down off hand. From there you should be able to see how $W$ comes in.
Hopefully that helps, and please let me know if/when you figure out definitively what $W$ is.
My reference for all this (other than the parts that came out of my head, e.g. that last integral) is Mo, van den Bosch & White.
