The Turbulent Support (TS) and Global Hierarchical Collapse (GHC) models for molecular clouds compared. Differences, convergence, and myths

We provide a detailed comparison between the ``turbulent support'' (TS) and ``global hierarchical collapse'' (GHC) models for molecular clouds and star formation, their respective interpretations of the observational data, the features they share, and suggested tests and observations to discern between them. Also, we clarify common misconceptions in recent literature about the global and hierarchical nature of the GHC scenario, and briefly discuss the evolution of some aspects of both models toward convergence. TS assumes that star-forming molecular clouds and their substructures are either in approximate virial equilibrium between gravity and turbulence or overvirial, so that the cloud is either confined or expanding, and its substructures (clumps, filaments and cores) are produced by turbulent compressions. In this scheme, the star formation rate (SFR) is time-independent and determined by the turbulent and gravitational parameters of the clouds, in particular the virial parameter $\av$. Conversely, GHC assumes that most star-forming molecular clouds and their substructures are part of a continuous gravitationally-driven flow, each accreting from their parent structure. Therefore, GHC is an intrinsically {\it evolutionary} model for the clouds and their star formation rate, determined by the evolution of the collapse flow. It interprets nonthermal motions as a mixture of infall and turbulent components, with the relative importance of the former increasing as the objects become denser and/or more massive, and thus $\av$ is an {\it evolving variable} of the clouds. Tests that may provide clues to distinguishing between TS and GHC must take into account that the innermost parts of globally gravitationally bound structures may not locally appear bound, and thus the binding may have to be searched for at the largest scale of the structure.