# Virial coefficient when computing dynamical mass enclosed by a rotating galactic disk of gas

Suppose I have a large thin disk of rotating gas in a galaxy -- the disk has a maximum inclination-corrected $V_{max}$ and a maximum radial extent of $R_{max}$ corresponding to that $V_{max}$ measurement. The virial theorem tells me that the total (aka dynamical) mass enclosed by that rotating disk (including stars, dark matter, etc.) is $V_{max}^2 R_{max} / G$, where $G$ is Newton's gravitational constant. Here I have assumed that the virial coefficient is 1, but the virial coefficient can vary widely based on the geometry of the problem. For example, it can be almost 10 for dwarf galaxies (an order of magnitude!), or a factor of $\sim3-5$ for dispersion-dominated rather than rotation-supported systems like elliptical galaxies where you're using the stars to probe the dynamical mass.

My question is simple: for a simple rotating disk of gas as above (assume it is thin and obviously not dispersion-supported, which I think is one of the simplest possible geometries for a question like this), what is the virial coefficient? I have seen coefficients of both 1 and 2, but I have no idea which derivation is correct. Adding to the problem is that some derivations seem to be using the virial theorem to give an order of magnitude estimate which means they leave out discussion of the virial coefficient altogether.

• Your expression for the virial mass appears to be that of a spherically symmetric distribution of matter, not a disk. The virial coefficient depends on the radial density distribution, which you haven't specified. – Rob Jeffries Jun 22 '16 at 11:34
• I think you're trying to split hairs here. Calculating the Virial coefficient, even for a system with a well defined radial density distribution is never going to be an exact thing. There's really not much you can do to get a specificity better than 1 or 2 and what you end up with depends on the assumptions you make going into the problem. – zephyr Jun 22 '16 at 12:50
• Thanks both! Rob, curious: assuming the radial density distribution for the disk was exponential (typically assumed for astronomical disks), then do you know what the expression for the virial mass would become (unlike the simple $V_{max}^2 R_{max} / G$ for a spherically symmetric distribution I wrote above)? – quantumflash Jun 23 '16 at 15:51
• And what velocity law are you using? – Rob Jeffries Oct 15 '16 at 17:22