Robust Cross-correlation-based Measurement of Clump Sizes in Galaxies
  Stars form in molecular complexes that are visible as giant clouds ($\sim
10^{5-6} \mathrm{M}_\odot$) in nearby galaxies and as giant clumps ($\sim
10^{8-9}\mathrm{M}_\odot$) in galaxies at redshifts $z\approx1$$-$$3$.
Theoretical inferences on the origin and evolution of these complexes often
require robust measurements of their characteristic size, which is hard to
measure at limited resolution and often ill-defined due to overlap and
quasi-fractal substructure. We show that maximum and luminosity-weighted sizes
of clumps seen in star formation maps (e.g.\ H$\alpha$) can be recovered
statistically using the two-point correlation function (2PCF), if an
approximate stellar surface density map is taken as the normalizing random
field. After clarifying the link between Gaussian clumps and the 2PCF
analytically, we design a method for measuring the diameters of Gaussian clumps
with realistic quasi-fractal substructure. This method is tested using mock
images of clumpy disk galaxies at different spatial resolutions and perturbed
by Gaussian white noise. We find that the 2PCF can recover the input clump
scale at $\sim20\%$ accuracy, as long as this scale is larger than the spatial
resolution. We apply this method to the local spiral galaxy NGC 5194, as well
as to three clumpy turbulent galaxies from the DYNAMO-HST sample. In both
cases, our statistical H$\alpha$-clump size measurements agree with previous
measurements and with the estimated Jeans lengths. However, the new
measurements are free from subjective choices when fitting individual clumps.
