Grain size constraints on HL Tau with polarization signature

The millimeter-wave polarization of the protoplanetary disk around HL Tau has been interpreted as the emission from elongated dust grains aligned with the magnetic field in the disk. However, the self-scattering of thermal dust emission may also explain the observed millimeter-wave polarization. In this paper, we report a modeling of the millimeter-wave polarization of the HL Tau disk with the self-polarization. Dust grains are assumed to be spherical and to have a power-law size distribution. We change the maximum grain size with a fixed dust composition in a fixed disk model to find the grain size to reproduce the observed signature. We find that the direction of the polarization vectors and the polarization degree can be explained with the self-scattering. Moreover, the polarization degree can be explained only if the maximum grain size is $\sim 150 {\rm~\mu m}$. The obtained grain size from the polarization is different from a size of millimeter or larger that has been previously expected from the spectral index of the dust opacity coefficient if the emission is optically thin. We discuss that porous dust aggregates may solve the inconsistency of the maximum grain size between the two constraints.