Escaping the maze: a statistical sub-grid model for cloud-scale density
  structures in the interstellar medium
  The interstellar medium (ISM) is a turbulent, highly structured multi-phase
medium. State-of-the-art cosmological simulations of the formation of galactic
discs usually lack the resolution to accurately resolve those multi-phase
structures. However, small-scale density structures play an important role in
the life cycle of the ISM, and determine the fraction of cold, dense gas, the
amount of star formation and the amount of radiation and momentum leakage from
cloud-embedded sources. Here, we derive a $statistical\, model$ to calculate
the unresolved small-scale ISM density structure from coarse-grained,
volume-averaged quantities such as the $gas\, clumping\, factor$,
$\mathcal{C}$, and mean density $\left<\rho\right>_V$. Assuming that the
large-scale ISM density is statistically isotropic, we derive a relation
between the three-dimensional clumping factor, $\mathcal{C}_\rho$, and the
clumping factor of the $4\pi$ column density distribution on the cloud surface,
$\mathcal{C}_\Sigma$, and find $\mathcal{C}_\Sigma=\mathcal{C}_\rho^{2/3}$.
Applying our model to calculate the covering fraction, i.e., the $4\pi$ sky
distribution of optically thick sight-lines around sources inside interstellar
gas clouds, we demonstrate that small-scale density structures lead to
significant differences at fixed physical ISM density. Our model predicts that
gas clumping increases the covering fraction by up to 30 per cent at low ISM
densities compared to a uniform medium. On the other hand, at larger ISM
densities, gas clumping suppresses the covering fraction and leads to increased
scatter such that covering fractions can span a range from 20 to 100 per cent
at fixed ISM density. All data and example code is publicly available at
GitHub.
