Evolution of the density PDF in star forming clouds: the role of gravity
  We derive an analytical theory of the PDF of density fluctuations in
supersonic turbulence in the presence of gravity in star-forming clouds. The
theory is based on a rigorous derivation of a combination of the Navier-Stokes
continuity equations for the fluid motions and the Poisson equation for the
gravity. It extends upon previous approaches first by including gravity, second
by considering the PDF as a dynamical system, not a stationary one. We derive
the transport equations of the density PDF, characterize its evolution and
determine the density threshold above which gravity strongly affects and
eventually dominates the dynamics of turbulence. We demonstrate the occurence
of {\it two} power law tails in the PDF, with two characteristic exponents,
corresponding to two different stages in the balance between turbulence and
gravity. Another important result of this study is to provide a procedure to
relate the observed {\it column density} PDFs to the corresponding {\it volume
density} PDFs. This allows to infer, from the observation of column-densities,
various physical parameters characterizing molecular clouds, notably the virial
parameter. Furthermore, the theory offers the possibility to date the clouds in
units of ${t}_{\rm coll}$, the time since a statistically significant fraction
of the cloud started to collapse. The theoretical results and diagnostics
reproduce very well numerical simulations and observations of star-forming
clouds. The theory provides a sound theoretical foundation and quantitative
diagnostics to analyze observations or numerical simulations of star-forming
regions and to characterize the evolution of the density PDF in various regions
of molecular clouds.
