Thermal Emission and Scattering by Aligned Grains: Plane-Parallel Model and Application to Multiwavelength Polarization of the HL Tau Disk

Telescopes are now able to resolve dust polarization across circumstellar disks at multiple wavelengths, allowing the study of the polarization spectrum. Most disks show clear evidence of dust scattering through their unidirectional polarization pattern typically at the shorter wavelength of $\sim 870 \mu$m. However, certain disks show an elliptical pattern at $\sim 3$mm, which is likely due to aligned grains. With HL Tau, its polarization pattern at $\sim 1.3$mm shows a transition between the two patterns making it the first example to reveal such transition. We use the T-matrix method to model elongated dust grains and properly treat scattering of aligned non-spherical grains with a plane-parallel slab model. We demonstrate that a change in optical depth can naturally explain the polarization transition of HL Tau. At low optical depths, the thermal polarization dominates, while at high optical depths, dichroic extinction effectively takes out the thermal polarization and scattering polarization dominates. Motivated by results from the plane-parallel slab, we develop a simple technique to disentangle thermal polarization of the aligned grains $T_{0}$ and polarization due to scattering $S$ using the azimuthal variation of the polarization fraction. We find that, with increasing wavelength, the fractional polarization spectrum of the scattering component $S$ decreases, while the thermal component $T_{0}$ increases, which is expected since the optical depth decreases. We find several other sources similar to HL Tau that can be explained by azimuthally aligned scattering prolate grains when including optical depth effects. In addition, we explore how spirally aligned grains with scattering can appear in polarization images.