Star Formation in Massive Clusters via Bondi Accretion

Essentially all stars form in giant molecular clouds (GMCs). However, inside GMCs, most of the gas does not participate in star formation; rather, denser gas accumulates in clumps in the GMC, with the bulk of the stars in a given GMC forming in a few of the most massive clumps. In the Milky Way, these clumps have masses $M_{\rm cl}\lesssim 5\times 10^{-2}$ of the GMC, radii $r_{\rm cl} \sim 1$pc, and free-fall times $\tau_{\rm cl} \sim 2\times 10^5\yr$. We show that clumps inside giant molecular clouds should accrete at a modified Bondi accretion rate, which depends on clump mass as $\dot M_{\rm cl}\sim M_{\rm cl}^{5/4}$. This rate is initially rather slow, usually slower than the initial star formation rate inside the clump (we adopt the common assumption that inside the clump, $\dot M_*=\epsilon_{\rm ff} M_{\rm cl}/\tau_{\rm cl}$, with $\epsilon_{\rm ff} \approx 0.017$). However, after $\sim 2$ GMC free-fall times $\tau_{\rm GMC}$, the clump accretion rate accelerates rapidly; formally, the clump can accrete the entire GMC in $\sim 3\tau_{\rm GMC}$. At the same time, the star formation rate accelerates, tracking the Bondi accretion rate. If the GMC is disrupted by feedback from the largest clump, half the stars in that clump form in the final $\taug$ before the GMC is disrupted. The theory predicts that the distribution of effective star formation rates, measured per GMC free-fall time, is broad, ranging from $\sim 0.001$ up to 0.1 or larger and that the mass spectrum of star clusters is flatter than that of clumps, consistent with observations.