Resonant line transfer in a fog: Using Lyman-alpha to probe tiny
  structures in atomic gas
  Motivated by observational and theoretical work which both suggest very small
scale ($\lesssim 1\,$pc) structure in the circum-galactic medium of galaxies
and in other environments, we study Lyman-$\alpha$ (Ly$\alpha$) radiative
transfer in an extremely clumpy medium with many "clouds" of neutral gas along
the line of sight. While previous studies have typically considered radiative
transfer through sightlines intercepting $\lesssim 10$ clumps, we explore the
limit of a very large number of clumps per sightline (up to $f_{\mathrm{c}}
\sim 1000$). Our main finding is that, for covering factors greater than some
critical threshold, a multiphase medium behaves similar to a homogeneous medium
in terms of the emergent Ly$\alpha$ spectrum. The value of this threshold
depends on both the clump column density and on the movement of the clumps. We
estimate this threshold analytically and compare our findings to radiative
transfer simulations with a range of covering factors, clump column densities,
radii, and motions. Our results suggest that (i) the success in fitting
observed Ly$\alpha$ spectra using homogeneous "shell models" (and the
corresponding failure of multiphase models) hints towards the presence of very
small-scale structure in neutral gas, in agreement within a number of other
observations; and (ii) the recurrent problems of reproducing realistic line
profiles from hydrodynamical simulations may be due to their inability to
resolve small-scale structure, which causes simulations to underestimate the
effective covering factor of neutral gas clouds.
