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.
