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Motivated by observational and theoretical work which both suggest very smallscale ($\lesssim 1\,$pc) structure in the circum-galactic medium of galaxiesand in other environments, we study Lyman-$\alpha$ (Ly$\alpha$) radiativetransfer in an extremely clumpy medium with many "clouds" of neutral gas alongthe line of sight. While previous studies have typically considered radiativetransfer through sightlines intercepting $\lesssim 10$ clumps, we explore thelimit 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 somecritical threshold, a multiphase medium behaves similar to a homogeneous mediumin terms of the emergent Ly$\alpha$ spectrum. The value of this thresholddepends on both the clump column density and on the movement of the clumps. Weestimate this threshold analytically and compare our findings to radiativetransfer simulations with a range of covering factors, clump column densities,radii, and motions. Our results suggest that (i) the success in fittingobserved Ly$\alpha$ spectra using homogeneous "shell models" (and thecorresponding failure of multiphase models) hints towards the presence of verysmall-scale structure in neutral gas, in agreement within a number of otherobservations; and (ii) the recurrent problems of reproducing realistic lineprofiles from hydrodynamical simulations may be due to their inability toresolve small-scale structure, which causes simulations to underestimate theeffective covering factor of neutral gas clouds.

Resonant line transfer in a fog: Using Lyman-alpha to probe tiny structures in atomic gas