Title: Opacity limit for supermassive protostars,
Abstract: We present a model for the evolution of supermassive protostars from their
formation at $M_\star \simeq 0.1\,\text{M}_\odot$ until their growth to
$M_\star \simeq 10^5\,\text{M}_\odot$. To calculate the initial properties of
the object in the optically thick regime we follow two approaches: based on
idealized thermodynamic considerations, and on a more detailed one-zone model.
Both methods derive a similar value of $n_{\rm F} \simeq 2 \times 10^{17}
\,\text{cm}^{-3}$ 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 $R_\star \simeq 0.65 \,{\rm AU}$
to $R_\star \simeq 250 \,{\rm AU}$ during its evolution. Finally, we use this
model to construct a sub-grid 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 photo-ionization
feedback can be neglected until very late in the assembly process of the
supermassive object.