Gravitational contraction versus Supernova driving and the origin of the velocity dispersion-size relation in molecular clouds

Molecular cloud observations show that clouds have non-thermal velocity dispersions that scale with the cloud size as $\sigma\propto R^{1/2}$ at constant surface density, and for varying surface density scale with both the cloud`s size and surface density, $\sigma^2 \propto R \Sigma$. The energy source driving these chaotic motions remains poorly understood. We describe the velocity dispersions observed in a cloud population formed in a kiloparsec-scale numerical simulation of a magnetized, supernova-driven, self-gravitating, interstellar medium, including diffuse heating and radiative cooling. We compare the relationships between velocity dispersion, size, and surface density measured in the simulated cloud population to those found in observations of Galactic molecular clouds. We find that external supernova explosions can not drive turbulent motions of the observed magnitudes within dense clouds. On the other hand, self-gravity also induces non-thermal motions as gravitationally bound clouds begin to collapse in our model, and by doing so their internal velocity dispersions recover the observed relations. Energy conservation suggests that the observed behavior is consistent with the kinetic energy being proportional to the gravitational energy. However, the clouds in our model show no sign of reaching a stable equilibrium state at any time, even for strongly magnetized clouds. We conclude that gravitationally bound molecular clouds are always in a state of gravitational collapse and their properties are a natural result of this chaotic collapse. In order to agree with observed star formation efficiencies, this process must be terminated by the early destruction of the clouds, presumably from internal stellar feedback.