The efficiency per free-fall time as a ratio of the Star Formation Rate to the gas-infall rate in collapsing cores: dependence on the core definition, accretion, and radial structure

A parameter used to characterise star formation activity in MCs is the efficiency per free-fall time, $\epsilon_{\rm ff}$, although commonly referred to as an efficiency, it is formally the ratio between the star formation rate (SFR) and the gas-infall rate. Here we numerically study the collapse of cores and define $\epsilon_{\rm ff}\equiv\langle\dot{M}_\star\rangle/(M_{\rm core}/\tau_{\rm ff})$, where $\langle\dot{M}_\star\rangle$ is the average SFR, $M_{\rm core}$ is the gas mass within the core (as the gas cells above a density threshold), and $\tau_{\rm ff}$ is the free-fall time of the core gas. We perform simplified numerical experiments of the gravitational collapse of an isolated core, varying the initial mean number density ($n_0=100$ and $1000~\rm cm^{-3}$) and adopting closed/open BCs to (dis)allow fresh gas accretion into the domain. The simulations start with a slight central Gaussian overdensity that evolved into a power-law profile, $n\propto r^{-p}$ with $p\to2$. As the collapse proceeds, a sink particle forms in the center of the core. We find that both the BCs and the adopted core definition modify the measured core properties and, consequently, the inferred $\epsilon_{\rm ff}$. Low-density models have less mass available, and their accretion histories are therefore much more sensitive to the choice of BCs, while high-density runs, with their larger mass reservoirs, maintain similar accretion histories regardless of the BCs. In all models, after sink formation, $\epsilon_{\rm ff}$ rises and then remains relatively stable while accretion continues to replenish the core's mass, but increases once the gas reservoir is exhausted. Somewhat counterintuitively, $\epsilon_{\rm ff}$ is higher in the low-mass cores, since the larger gas infall rates onto the high-mass cores compensate for their higher SFR. We conclude that the inferred $\epsilon_{\rm ff}$ depends sensitively on both the adopted core definition and external mass supply