a factor 15$\pm$5$\times$ larger than those locally (see also @Swinbank09 [@Swinbank10Nature; @Jones10; @Wisnioski11b]). We note that high luminosities at fixed size have been found in local starbursts, such as in the Antennae [@Bastian06], whilst offsets of factors $\sim $50$\times$ have been inferred for star-forming regions in high-redshift galaxies [e.g.  @Swinbank09; @Jones10; @Wisnioski11b].

The Relation between the Disk and Clump Properties {#sec:Sigma_disk_Sigma_clump}
--------------------------------------------------

It is possible to relate the properties of the clumps to the overall properties of the disk [e.g.  @Hopkins12a]. For example, the velocity dispersion of the fastest growing Jeans unstable mode which can not be stabilised by rotation in a gas disk is given by $$\sigma_t(R)^2\,=\,\pi\,G\,\Sigma_{\rm disk}\,R
  \label{eqn:sigma_Sigma}$$ [e.g.  @Escala08; @Elmegreen09b; @Dekel09; @Genzel11; @Livermore12a]. The critical density for collapse ($\rho_c$), on scale $R$ from a turbulent ISM is given by $$\rho_c\,=\,\frac{3}{4\,\pi\,R^{3}}\,M_{\rm J}\,\simeq\,\frac{9}{8\,\pi\,R^2\,G}\,\sigma_t(R)^2 
\label{eqn:MJeans}$$ where $\sigma_t(R)$ is the line of sight velocity turbulent velocity dispersion and $M_{\rm J}$ is the Jeans mass. The critical density for collapse therefore scales as $$\rho_c(R)\,=\,\frac{9}{8R}\,\Sigma_{\rm disk}
  \label{eqn:R_SigmaDisk}$$ Assuming that the cloud contracts by a factor $\simeq $2.5 as it collapses, the post-collapse surface density of the cloud is $$\Sigma_{\rm cloud}\simeq\,10\,\rho_c R\simeq \,10\,\Sigma_{\rm disk}
  \label{eqn:Sigmacloud_Sigmadisk}$$ [see also @Livermore12a]. Thus, the surface density of the 