The extreme initial kinetic energy allowed by a collapsing turbulent core

We present high-resolution hydrodynamical simulations aimed at following the gravitational collapse of a gas core, in which a turbulent spectrum of velocity is implemented only initially. We determine the maximal value of the ratio of kinetic energy to gravitational energy, denoted here by $\left(\frac{E_{\rm kin} }{E_{\rm grav}}\right)_{\rm max}$, so that the core (i) will collapse around one free-fall time of time evolution or (ii) will expand unboundedly, because it has a value of $\frac{E_{\rm kin}}{E_{\rm grav}}$ larger than $\left( \frac{E_{\rm kin}}{E_{\rm grav}}\right)_{\rm max}$. We consider core models with a uniform or centrally condensed density profile and with velocity spectra composed of a linear combination of one-half divergence-free turbulence type and the other half of a curl-free turbulence type. We show that the outcome of the core collapse are protostars forming either (i) a multiple system obtained from the fragmentation of filaments and (ii) a single primary system within a long filament. In addition, some properties of these protostars are also determined and compared with those obtained elsewhere.