The anomalously low (sub)millimeter spectral indices of some
  protoplanetary disks may be explained by dust self-scattering
  Previous (sub)millimeter observations have found that the spectral indices of
dust emission from some young stellar objects are lower than that of the black
body emission in the Rayleigh-Jeans limit (i.e., 2.0). In particular, the
recent Atacama Large Millimeter Array observations have spatially resolved that
the innermost regions of the protoplanetary disks TW\,Hya and HD\,163296
present anomalously low (i.e., $<$2.0) millimeter spectral indices. In some
previous works, such anomalously low millimeter spectral indices were
considered unphysical and were attributed to measurement errors. The present
work clarifies that if the albedo is high and is increasing with frequency, it
is possible to reproduce such anomalously low spectral indices when the
emission source is optically thick. In addition, to yield lower than 2.0
spectral index at (sub)millimeter bands, the required dust maximum grain size
\amax is on the order of 10-100 \micron, which is very well consistent with the
previously derived \amax values based on multi-wavelength dust polarimetric
observations. In light of this, measuring Stokes I spectral index may also
serve as an auxiliary approach for assessing whether the observed dust
polarization is mainly due to dust scattering or is due to the aligned dust
grains.
