Abstract
Studies of extra-galactic globular clusters have shown that the peak size of the globular cluster (GC) radius distribution (RD) depends only weakly on galactic environment, and can be used as a standard ruler. We model RDs of GC populations using a simple prescription for a Hubble time of relaxation driven evolution of cluster mass and radius, and explore the conditions under which the RD can be used as a standard ruler. We consider a power-law cluster initial mass function (CIMF) with and without an exponential truncation, and focus in particular on a flat and a steep CIMF (power-law indices of 0 and -2, respectively). For the initial half-mass radii at birth we adopt either Roche-lobe filling conditions ('filling',meaning that the ratio of half-mass to Jacobi radius is approximately rh/rJ ~ 0.15) or strongly Roche-lobe under-filling conditions ('under-filling', implying that initially rh/rJ << 0.15). Assuming a constant orbital velocity about the galaxy centre we find for a steep CIMF that the typical half-light radius scales with galactocentric radius RG as RG^1/3. This weak scaling is consistent with observations, but this scenario has the (well known) problem that too many low-mass clusters survive. A flat CIMF with 'filling' initial conditions results in the correct mass function at old ages, but with too many large (massive) clusters at large RG. An 'underfilling' GC population with a flat CIMF also results in the correct mass function, and can also successfully reproduce the shape of the RD, with a peak size that is (almost) independent of RG. In this case, the peak size depends (almost) only on the peak mass of the GC mass function. The (near) universality of the GC RD is therefore because of the (near) universality of the CIMF. There are some extended GCs in the outer halo of the Milky Way that cannot be explained by this model.