Opacity Limit for Supermassive Protostars

We present a model for the evolution of supermassive protostars from their formation at ${M}_{\star }\simeq 0.1\,{M}_{\odot }$ until their growth to ${M}_{\star }\simeq {10}^{5}\,{M}_{\odot }$. To calculate the initial properties of the object in the optically thick regime, we follow two approaches: one based on idealized thermodynamic considerations, and another based on a more detailed one-zone model. Both methods derive a similar value of ${n}_{{\rm{F}}}\simeq 2\times {10}^{17}\,{\mathrm{cm}}^{-3}$ for the density of the object when opacity becomes important, i.e., the opacity limit. The subsequent evolution of the growing protostar is determined by the accretion of gas onto the object and can be described by a mass–radius relation of the form ${R}_{\star }\propto {M}_{\star }^{1/3}$ during the early stages, and of the form ${R}_{\star }\propto {M}_{\star }^{1/2}$ when internal luminosity becomes important. For the case of a supermassive protostar, this implies that the radius of the star grows from ${R}_{\star }\simeq 0.65\,\mathrm{au}$ to ${R}_{\star }\simeq 250\,\mathrm{au}$ during its evolution. Finally, we use this model to construct a subgrid recipe for accreting sink particles in numerical simulations. A prime ingredient thereof is a physically motivated prescription for the accretion radius and the effective temperature of the growing protostar embedded inside it. From the latter, we can conclude that photoionization feedback can be neglected until very late in the assembly process of the supermassive object.