Probing the Temperature Structure of Optically Thick Disks Using
  Polarized Emission of Aligned Grains
  Polarized continuum emission from aligned grains in disks around young
stellar objects can be used to probe the magnetic field, radiation anisotropy,
or drift between dust and gas, depending on whether the non-spherical grains
are aligned magnetically, radiatively or mechanically. We show that it can also
be used to probe another key disk property -- the temperature gradient -- along
sight lines that are optically thick, independent of the grain alignment
mechanism. We first illustrate the technique analytically using a simple 1D
slab model, which yields an approximate formula that relates the polarization
fraction to the temperature gradient with respect to the optical depth tau at
the tau=1 surface. The formula is then validated using models of stellar
irradiated disks with and without accretion heating. The promises and
challenges of the technique are illustrated with a number of Class 0 and I
disks with ALMA dust polarization data, including NGC 1333 IRAS4A1, IRAS
16293B, BHB 07-11, L1527, HH 212 and HH 111. We find, in particular, that the
sight lines passing through the near-side of a highly inclined disk trace
different temperature gradient directions than those through the far-side,
which can lead to a polarization orientation on the near-side that is
orthogonal to that on the far-side, and that the HH 111 disk may be such a
case. Our technique for probing the disk temperature gradient through dust
polarization can complement other methods, particularly those using molecular
lines.
