An excursion-set model for the structure of giant molecular clouds and the interstellar medium

The interstellar medium (ISM) is governed by supersonic turbulence on a range of scales. We use this simple fact to develop a rigorous excursion-set model for the formation, structure and time evolution of dense gas structures [e.g. giant molecular clouds (GMCs), massive clumps and cores]. Supersonic turbulence drives the density distribution in non-self-gravitating regions to a lognormal with dispersion increasing with Mach number. We generalize this to include scales ≳h (the disc scale-height), and use it to construct the statistical properties of the density field smoothed on a scale R. We then compare conditions for self-gravitating collapse including thermal, turbulent and rotational (disc shear) support (reducing to the Jeans/Toomre criterion on small/large scales). We show that this becomes a well-defined barrier crossing problem. As such, an exact 'bound object mass function' can be derived, from scales of the sonic length to well above the disc Jeans mass. This agrees remarkably well with observed GMC mass functions in the Milky Way and other galaxies, with the only inputs being the total mass and size of the galaxies (to normalize the model). This explains the cut-off of the mass function and its power-law slope (close to, but slightly shallower than, -2). The model also predicts the linewidth-size and size-mass relations of clouds and the dependence of residuals from these relations on mean surface density/pressure, in excellent agreement with observations. We use this to predict the spatial correlation function/clustering of clouds and, by extension, star clusters; these also agree well with observations. We predict the size/mass function of 'bubbles' or 'holes' in the ISM, and show that this can account for the observed H I hole distribution without requiring any local feedback/heating sources. We generalize the model to construct time-dependent 'merger/fragmentation trees' which can be used to follow cloud evolution and construct semi-analytic models for the ISM, GMCs and star-forming populations. We provide explicit recipes to construct these trees. We use a simple example to show that if clouds are not destroyed in ∼1-5 crossing times, then all the ISM mass would be trapped in collapsing objects even if the large-scale turbulent cascade were maintained.