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 µm and the observing wavelength to be 870 µm. 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 max ∼ λ/2π, where λ 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.

A giant planet embedded in a protoplanetary disk forms a gap. An analytic relationship among the gap depth, planet mass M p , disk aspect ratio h p , and viscosity α has been found recently, and the gap depth can be written in terms of a single parameter K = (M p /M * ) 2 h −5 p α −1 . We discuss how observed gap features can be used to constrain the disk and/or planet parameters based on the analytic formula for the gap depth. The constraint on the disk aspect ratio is critical in determining the planet mass so the combination of the observations of the temperature and the image can provide a constraint on the planet mass. We apply the formula for the gap depth to observations of HL Tau and HD 169142. In the case of HL Tau, we propose that a planet with ∼ > 0.3 is responsible for the observed gap at 30 AU from the central star based on the estimate that the gap depth is ∼ < 1/3. In the case of HD 169142, the planet mass that causes the gap structure recently found by VLA is ∼ > 0.4M J . We also argue that the spiral structure, if observed, can be used to estimate the lower limit of the disk aspect ratio and the planet mass.

The mechanisms causing millimeter-wave polarization in protoplanetary disks are under debate. To disentangle the polarization mechanisms, we observe the protoplanetary disk around HL Tau at 3.1 mm with the Atacama Large Millimeter/submillimeter Array (ALMA), which had polarization detected with CARMA at 1.3 mm. We successfully detect the ring-like azimuthal polarized emission at 3.1 mm. This indicates that dust grains are aligned with the major axis being in the azimuthal direction, which is consistent with the theory of radiative alignment of elongated dust grains, where the major axis of dust grains is perpendicular to the radiation flux. Furthermore, the morphology of the polarization vectors at 3.1 mm is completely different from those at 1.3 mm. We interpret that the polarization at 3.1 mm to be dominated by the grain alignment with the radiative flux producing azimuthal polarization vectors, while the self-scattering dominates at 1.3 mm and produces the polarization vectors parallel to the minor axis of the disk. By modeling the total polarization fraction with a single grain population model, the maximum grain size is constrained to be 100 µm, which is smaller than the previous predictions based on the spectral index between ALMA at 3 mm and VLA at 7 mm.

A giant planet creates a gap in a protoplanetary disk, which might explain the observed gaps in protoplanetary disks. The width and depth of the gaps depend on the planet mass and disk properties. We have performed two-dimensional hydrodynamic simulations for various planet masses, disk aspect ratios and viscosities, to obtain an empirical formula for the gap width. The gap width is proportional to the square root of the planet mass, −3/4 power of the disk aspect ratio and −1/4 power of the viscosity. This empirical formula enables us to estimate the mass of a planet embedded in the disk from the width of an observed gap. We have applied the empirical formula for the gap width to the disk around HL Tau, assuming that each gap observed by ALMA observations is produced by planets, and discussed the planet masses within the gaps. The estimate of planet masses from the gap widths is less affected by the observational resolution and dust filtration than that from the gap depth.

Here we present high-resolution (15-24 au) observations of CO isotopologue lines from the Molecules with ALMA on Planet-forming Scales (MAPS) ALMA Large Program. Our analysis employs observations of the (J = 2-1) and (1-0) lines of 13 CO and C 18 O and the (J = 1-0) line of C 17 O for five protoplanetary disks. We retrieve CO gas density distributions, using three independent methods: (1) a thermochemical modeling framework based on the CO data, the broadband spectral energy distribution, and the millimeter continuum emission; (2) an empirical temperature distribution based on optically thick CO lines; and (3) a direct fit to the C 17 O hyperfine lines. Results from these methods generally show excellent agreement. The CO gas column density profiles of the five disks show significant variations in the absolute value and the radial shape. Assuming a gas-to-dust mass ratio of 100, all five disks have a global CO-to-H 2 abundance 10-100 times lower than the interstellar medium ratio. The CO gas distributions between 150 and 400 au match well with models of viscous disks, supporting the longstanding theory. CO gas gaps appear to be correlated with continuum gap locations, but some deep continuum gaps do not have corresponding CO gaps. The relative depths of CO and dust gaps are generally consistent with predictions of planet-disk interactions, but some CO gaps are 5-10 times shallower than predictions based on dust gaps. This paper is part of the MAPS special issue of the Astrophysical Journal Supplement.