Gravity or Turbulence? The velocity dispersion-size relation
  We discuss the nature of the velocity dispersion vs. size relation for
molecular clouds. In particular, we add to previous observational results
showing that the velocity dispersions in molecular clouds and cores are not
purely functions of spatial scale but involve surface gas densities as well. We
emphasize that hydrodynamic turbulence is required to produce the first
condensations in the progenitor medium. However, as the cloud is forming, it
also becomes bound, and gravitational accelerations dominate the motions.
Energy conservation in this case implies $|E_g| \sim E_k$, in agreement with
observational data, and providing an interpretation for two recent
observational results: the scatter in the $\delta v-R$ plane, and the
dependence of the velocity dispersion on the surface density ${\delta v^2/ R}
\propto \Sigma$. We argue that the observational data are consistent with
molecular clouds in a state of hierarchical gravitational collapse, i.e.,
developing local centers of collapse throughout the whole cloud while the cloud
itself is collapsing, and making equilibrium unnecessary at all stages prior to
the formation of actual stars. Finally, we discuss how this mechanism need not
be in conflict with the observed star formation rate.
