An increasing number of young circumstellar disks show strikingly ordered ( sub ) millimeter polarization orientations along the minor axis , which is strong evidence for polarization due to scattering by \sim 0.1 mm-sized grains .
To test this mechanism further , we model the dust continuum and polarization data of HD 163296 , one of the best observed disks with prominent rings and gaps , using the RADMC-3D radiative transfer code .
We find that scattering by grains with a maximum size of 90 \mu m can simultaneously reproduce the polarization observed at 0.87 mm ( ALMA Band 7 ) and the unusually low spectral index of \alpha \sim 1.5 between 0.87 and 1.25 mm ( ALMA Band 6 ) in the optically thick inner disk ( \lesssim 30 AU ) as a result of more efficient scattering at a shorter wavelength .
The relatively low spectral index of \alpha \sim 2.5 inferred for the optically thin gaps is also reproduced by the same ( relatively small ) grains , as a result of telescope beam averaging of the gaps ( with an intrinsic \alpha \sim 4 ) and their adjacent optically thick rings ( where \alpha \lesssim 2 ) .
In this case , the long-standing tension between the grain sizes inferred from polarization and spectral index disappears because the relatively low \alpha values are illusory and do not require large mm-sized grains .
In addition , the polarization fraction has a unique pattern of azimuthal variation : higher along the major axis than the minor axis in the gaps but higher along the minor axis than the major axis in the rings .
We find a rapidly declining polarization spectrum ( with the fraction p \propto \lambda ^ { -3 } approximately ) in the gaps , which becomes flattened or even inverted towards short wavelengths in the optically thick rings .
These contrasting behaviors in the rings and gaps provide further tests of scattering-induced polarization that can be tested via multi-wavelength observations that resolve the disk substructure .