Transition from Clumpy to Smooth Angular Diameter Distances

Distance relations in a locally inhomogeneous universe are expected to behave like the Dyer-Roeder solution on small angular scales and the Friedmann-Robertson-Walker solution on large angular scales. Within a simple compact clump model the transition between these asymptotic behaviors is demonstrated and quantified. The redshift dependent transition scale is of order a few arcseconds; this implies it should have little influence on large angular scale cosmological tests such as the volume-redshift relation but possibly significant effects on arcsecond angular diameter measurements of radio galaxies and AGNs. For example, at $z=2$ on arcsecond scales a clumpy flat universe mimics the angular diameter distance of a smooth $\Omega=0.27$ model.