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Validating Scattering-Induced (Sub)millimeter Disk Polarization through the Spectral Index, Wavelength-Dependent Polarization Pattern, and Polarization Spectrum: The Case of HD 163296

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 ~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$μ$m can simultaneously reproduce the polarization observed at 0.87 mm (ALMA Band 7) and the unusually low spectral index of $α$ ~ 1.5 between 0.87 and 1.25 mm (ALMA Band 6) in the optically thick inner disk as a result of more efficient scattering at a shorter wavelength. The relatively low spectral index of $α$ ~ 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 $α$ ~ 4) and their adjacent optically thick rings (where $α$ << 2). In this case, the long-standing tension between the grain sizes inferred from polarization and spectral index disappears because the relatively low $α$ 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 $p \propto λ^{-3}$ 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.