Inclination-Induced Polarization of Scattered Millimeter Radiation from Protoplanetary Disks: The Case of HL Tau

Abstract

Spatially resolved polarized millimeter/submillimeter emission has been observed in the disk of HL Tau and two other young stellar objects. It is usually interpreted as coming from magnetically aligned grains, but can also be produced by dust scattering, as demonstrated explicitly by Kataoka et al. for face-on disks. We extend their work by including the polarization induced by disk inclination with respect to the line of sight. Using a physically motivated, semi-analytic model, we show that the polarization fraction of the scattered light increases with the inclination angle $i$, reaching $1/3$ for edge-on disks. The inclination-induced polarization can easily dominate that intrinsic to the disk in the face-on view. It provides a natural explanation for the two main features of the polarization pattern observed in the tilted disk of HL Tau ($i \sim 45^\circ$): the polarized intensity concentrating in a region elongated more or less along the major axis, and polarization in this region roughly parallel to the minor axis. This broad agreement provides support to dust scattering as a viable mechanism for producing, at least in part, polarized millimeter radiation. In order to produce polarization at the observed level ($\sim 1\%$), the scattering grains must have grown to a maximum size of tens of microns. However, such grains may be too small to produce the opacity spectral index of $β\lesssim 1$ observed in HL Tau and other sources; another population of larger, millimeter/centimeter-sized, grains may be needed to explain the bulk of the unpolarized continuum emission.