Constraining the initial conditions of globular clusters using their radius distribution

Studies of extragalactic globular clusters (GCs) have shown that the peak size of the GC radius distribution (RD) depends only weakly on galactic environment. We model RDs of GC populations using a simple prescription for a Hubble time of relaxation-driven evolution of cluster mass and radius. 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 volume (RV) filling conditions ('filling', meaning that the ratio of halfmass to Jacobi radius is approximately r(h)/r(J) similar or equal to 0.15) or strongly RV under-filling conditions (under-filling', implying that initially rh/r(J) (sic) 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 the galactocentric radius R-G as R-G(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 MF at old ages, but with too many large (massive) clusters at large R-G. An ` under-filling' GC population with a flat CIMF also results in the correct MF, 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 MF. 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.