A measurement of the turbulence-driven density distribution in a
  non-star-forming molecular cloud
  Molecular clouds are supersonically turbulent. This turbulence governs the
initial mass function and the star formation rate. In order to understand the
details of star formation, it is therefore essential to understand the
properties of turbulence, in particular the probability distribution of density
in turbulent clouds. We present formaldehyde volume density measurements of a
non-star-forming cloud along the line of sight towards W49A. We use these
measurements in conjunction with total mass estimates from 13CO to infer the
shape of the density probability distribution function. This method is
complementary to measurements of turbulence via the column density distribution
and should be applicable to any molecular cloud with detected CO. We show that
turbulence in this cloud is probably compressively driven, with a
compressive-to-total Mach number ratio $b = \mathcal{M}_C/\mathcal{M}>0.4$. We
measure the standard deviation of the density distribution, constraining it to
the range $1.5 < \sigma_s < 1.9$ assuming that the density is lognormally
distributed. This measurement represents an essential input into star formation
laws. The method of averaging over different excitation conditions to produce a
model of emission from a turbulent cloud is generally applicable to optically
thin line observations.
