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 $\alpha <2$, as reported for some sources in the literature. Such extremely low spectral indices arise naturally when the selected observing frequencies sample the appropriate change in the temperature structure of the optically thick model disk. Our results show that protostellar disk masses could often be underestimated by $> \times10$, 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.