Millimeter-wave polarization of protoplanetary disks due to dust
  scattering
  We present a new method to constrain the grain size in protoplanetary disks
with polarization observations at millimeter wavelengths. If dust grains are
grown to the size comparable to the wavelengths, the dust grains are expected
to have a large scattering opacity and thus the continuum emission is expected
to be polarized due to self-scattering. We perform 3D radiative transfer
calculations to estimate the polarization degree for the protoplanetary disks
having radial Gaussian-like dust surface density distributions, which have been
recently discovered. The maximum grain size is set to be $100 {\rm~\mu m}$ and
the observing wavelength to be 870 ${\rm \mu m}$. We find that the polarization
degree is as high as 2.5 % with a subarcsec spatial resolution, which is likely
to be detected with near-future ALMA observations. The emission is polarized
due to scattering of anisotropic continuum emission. The map of the
polarization degree shows a double peaked distribution and the polarization
vectors are in the radial direction in the inner ring and in the azimuthal
direction in the outer ring. We also find the wavelength dependence of the
polarization degree: the polarization degree is the highest if dust grains have
a maximum size of $a_{\rm max}\sim\lambda/2\pi$, where $\lambda$ is the
observing wavelength. Hence, multi-wave and spatially resolved polarization
observations toward protoplanetary disks enable us to put a constraint on the
grain size. The constraint on the grain size from polarization observations is
independent of or may be even stronger than that from the opacity index.
