Probability distribution functions of gas in M31 and M51
  We present probability distribution functions (PDFs) of the surface densities
of ionized and neutral gas in the nearby spiral galaxies M31 and M51, as well
as of dust emission and extinction Av in M31. The PDFs are close to lognormal
and those for HI and Av in M31 are nearly identical. However, the PDFs for H2
are wider than the HI PDFs and the M51 PDFs have larger dispersions than those
for M31. We use a simple model to determine how the PDFs are changed by
variations in the line-of-sight (LOS) pathlength L through the gas, telescope
resolution and the volume filling factor of the gas, f_v. In each of these
cases the dispersion sigma of the lognormal PDF depends on the variable with a
negative power law. We also derive PDFs of mean LOS volume densities of gas
components in M31 and M51. Combining these with the volume density PDFs for
different components of the ISM in the Milky Way (MW), we find that sigma
decreases with increasing length L with an exponent of -0.76 +/- 0.06, which is
steeper than expected. We show that the difference is due to variations in f_v.
As f_v is similar in M31, M51 and the MW, the density structure in the gas in
these galaxies must be similar. Finally, we demonstrate that an increase in f_v
with increasing distance to the Galactic plane explains the decrease in sigma
with latitude of the PDFs of emission measure and FUV emission observed for the
MW.
