On the Effects of Self-Obscuration in the (Sub-)Millimeter Spectral Indices and Appearance of Protostellar Disks

In this paper we explore the effects of self-obscuration in protostellar disks with a radially decreasing temperature gradient and a colder midplane. We are motivated by recent reports of resolved dark lanes (`hamburgers') and (sub)mm spectral indices systematically below the ISM value for optically thin dust $\alpha_{\rm ISM} =3.7$. We explore several model grids, scaling disk mass and varying inclination angle $i$ and observing frequency $\nu$ from the VLA Ka band ($\sim 37$ GHz) to ALMA Band 8 ($\sim 405$ GHz). We also consider the effects of decreasing the index of the (sub-)mm dust opacity power law $\beta$ from 1.7 to 1. We find that a distribution of disk masses in the range $M_{\rm disk} = 0.01-2~M_\odot$ is needed to reproduce the observed distribution of spectral indices, and that assuming a fixed $\beta =1.7$ gives better results than $\beta=1$. A wide distribution of disk masses is also needed to produce some cases with $\alpha \times10$, and are consistent with recent hydrodynamical simulations. Although we do not rule out the possibility of some grain growth occurring within the short protostellar timescales, we conclude that self-obscuration needs to be taken into account.
Comment: Accepted to The Astrophysical Journal Supplement