How the Cookie Crumbles: A Model for Star-forming Clumps in High-redshift Disk Galaxies

We present a simple model for the number distribution of maximally star-forming clumps in rotating disk galaxies, at high-$z$ with high gas surface densities. By combining assumptions surrounding marginal stability of disks against gravitational fragmentation and collapse (i.e., Toomre's $Q\approx 1$), star cluster formation efficiency scaling with local gas surface density, and star formation rates being tied to the relevant local dynamical/free-fall times, we find a star-forming clump distribution of $N_c(> \dot M_\star) \propto \dot M_\star^{-4/3}$ when assuming a power-law form for the gas surface density profile, and a numerically integrable relation for arbitrary gas disk profiles. We compare this model with recent high-redshift observations of lensed clumpy star-forming rotation-dominated galaxies, and find good agreement with the distribution of clump star formation rates and number of clumps. Moreover, we argue that any rotation-supported galaxy should have a significantly higher number of identifiable star-forming clumps relative to dispersion supported objects at a similar mass as $N_c \sim (V_c/\sigma)^2$.