Size-Luminosity Scaling Relations of Local and Distant Star Forming
  Regions
  We investigate star forming scaling relations using Bayesian inference on a
comprehensive data sample of low- (z<0.1) and high-redshift (1<z<5) star
forming regions. This full data set spans a wide range of host galaxy stellar
mass ($M_{*} \sim10^6-10^{11} M_{\odot}$) and clump star formation rates (SFR $
\sim10^{-5}-10^2 M_\odot yr^{-1}$). We fit the power-law relationship between
the size (r$_{H\alpha}$) and luminosity (L$_{H\alpha}$) of the star forming
clumps using the Bayesian statistical modeling tool Stan that makes use of
Markov Chain Monte Carlo (MCMC) sampling techniques. Trends in the scaling
relationship are explored for the full sample and subsets based on redshift and
selection effects between samples. In our investigation we find no evidence of
redshift evolution of the size-luminosity scaling relationship, nor a
difference in slope between lensed and unlensed data. There is evidence of a
break in the scaling relationship between high and low star formation rate
surface density ($\Sigma_{SFR}$) clumps. The size-luminosity power law fit
results are L$_{H\alpha}\sim$ r$_{H\alpha}^{2.8}$ and L$_{H\alpha}\sim$
r$_{H\alpha}^{1.7}$ for low and high $\Sigma_{SFR}$ clumps, respectively. We
present a model where star forming clumps form at locations of gravitational
instability and produce an ionized region represented by the Str\"{o}mgren
radius. A radius smaller than the scale height of the disk results in a scaling
relationship of $L \propto r^3$ (high $\Sigma_{SFR}$ clumps), and a scaling of
$L \propto r^2$ (low $\Sigma_{SFR}$ clumps) if the radius is larger than the
disk scale height.
