The Sizes and Luminosities of Massive Star Clusters

The masses of star clusters range over seven decades, from ten up to one hundred million solar masses. Remarkably, clusters with masses in the range 10^4 to 10^6 solar mases show no systematic variation of radius with mass. However, recent observations have shown that clusters with masses greater than 3x10^6 solar masses do show an increase in size with increasing mass. We point out that clusters with m>10^6 solar masses were optically thick to far infrared radiation when they formed, and explore the hypothesis that the size of clusters with m> 3x10^6 solar masses is set by a balance between accretion powered radiation pressure and gravity when the clusters formed, yielding a mass-radius relation r~0.3(m/10^6M_odot)^{3/5} pc. We show that the Jeans mass in optically thick objects increases systematically with cluster mass. We argue, by assuming that the break in the stellar initial mass function is set by the Jeans mass, that optically thick clusters are born with top heavy initial mass functions; it follows that they are over-luminous compared to optically thin clusters when young, and have a higher mass to light ratio Upsilon_V=m/L_V when older than ~1 Gyr. Old, optically thick clusters have Upsilon_V~ mcl^{0.1-0.3}. It follows that L_V~sigma^{beta}, where sigma is the cluster velocity dispersion, and beta~4. It appears that Upsilon_V is an increasing function of cluster mass for compact clusters and ultra-compact dwarf galaxies. We show that this is unlikely to be due to the presence of non-baryonic dark matter, by comparing clusters to Milky Way satellite galaxies, which are dark matter dominated. The satellite galaxies appear to have a fixed mass inside a fiducial radius, M(r=r_0)=const.