# Do galaxy clusters follow De Vaucouleurs' law?

Do galaxy clusters follow De Vaucouleurs' law? If objects in the universe are built hierarchically, wouldn't one expect a galaxy cluster that is roughly spherical to follow the same profile as an elliptical galaxy?

They do indeed.

This page is a good starting point for further reading; it also leads to de Vaucouleurs (1948), in which de Vaucouleurs applies his eponymous relation to galaxy clusters. He uses a density profile $$\sigma_b=\sigma_0\exp\left[-7.67\left(b/r_e\right)^{1/4}\right]$$ where $r_e$ is the radius such that one half of all galaxies lie at radii $b\leq r$. The similarity to the common logarithmic form of the law for galaxies should be apparent; re-arranging, it comes to $$\log \sigma_b=\log \sigma_b-7.67\left(b/r_e\right)^{1/4}$$ De Vaucouleurs' application is the only one I can find at present. Mellier & Mathez (1986) applied the cluster version of de Vaucouleurs' Law to the Coma cluster

Related models, such as the fundamental plane, have also been fitted to galaxy clusters; see Marmo et al. (2004) and Schaeffer et al. (1993) for some interesting investigations into this.

• A. Thank you for the background on de Vaucouleurs, but let's start with the assumption that we're pretty familiar with the law. B. The Donzelli article is about Bright Galaxy Clusters which, while very massive, are not the same as galaxy clusters. C. Do you have any similar papers for Galaxy Clusters (e.g. where the radius of objects is on the order of Mpc)? – Quarkly Oct 1 '15 at 16:41
• @DonaldRoyAirey I stated the law as it applies to galaxy clusters, not individual galaxies, to show the difference; my answer assumes that you are already familiar with it. I can remove the Donzelli paper, if you feel it's not relevant. I have been able to find another paper for the Coma cluster. – HDE 226868 Oct 1 '15 at 18:27
• Mellier & Mathez paper is about deprojection. I've actually been over it several times and I can't get their analytical formula to match up with Young's (1961) tables. While I agree that this document appears to show a de Vaucouleurs profile of the Coma Cluster, the formula that is at the heart of their analysis doesn't produce anything that looks remotely like a Sersic or de Vaucouleurs profile. Don't take my word for it, please just try using their Formula (2), plug in a β and a scaled radius and see if you get anything that looks like table (2). – Quarkly Oct 1 '15 at 19:44
• @DonaldRoyAirey I haven't crunched the numbers yet, but are you positive you scaled the radius correctly? – HDE 226868 Oct 1 '15 at 23:00
• You don't need to get that far. The left column is the already scaled number. You just take any ratio, say, 0.01, and look up the Young number and the Mellier & Mathez equivalent using their analytical formula. The math couldn't be simpler. Since the numbers don't add up, the formula and just about everything else in this paper is suspect. – Quarkly Oct 2 '15 at 10:05

I'm really hoping someone else will chime in, but here's what I've found on my own. These fellows tried to fit the Coma Cluster into a NFW profile. They referenced an earlier, apparently seminal work by Kent and Gunn. Kent and Gunn attempted to fit the cluster into a inverse r-squared law (probably due to the fact that they used the inverse r-squared to select the members of the cluster). The general form of their law is:

$$μ = \frac{μ_0}{ 1 + \left(\frac{r}{r_e}\right)^2}$$

Again, I'd love to see a respectable article where someone demonstrates a fit of a galaxy cluster to either a de Vaucouleurs or Sersic profile.