Planetesimal Initial Mass Functions following Diffusion Regulated Gravitational Collapse

The initial mass function (IMF) of planetesimals is of key importance for understanding the initial stages of planet formation, yet theoretical predictions so far have been insufficient in explaining the variety of IMFs found in simulations. Here, we connect diffusion-tidal-shear limited planetesimal formation within the framework of a Toomre-like instability in the particle mid-plane of a protoplanetary disk to an analytic prediction for the planetesimal IMF. The shape of the IMF is set by the stability parameter $Q_\mathrm{p}$, which in turn depends on the particle Stokes number, the Toomre $Q$ value of the gas, the local dust concentration and the local diffusivity. We compare our prediction to high-resolution numerical simulations of the streaming instability and planetesimal formation via gravitational collapse. We find that our IMF prediction agrees with numerical results, and is consistent with both the `planetesimals are born big' paradigm and the power law description commonly found in simulations.