Transitional disks around young stars are promising candidates to look for recently formed, embedded planets. Planet-disk interaction models predict that planets clear a gap in the gas while trapping dust at larger radii. Other physical mechanisms could be responsible for cavities as well. Previous observations have revealed that gas is still present inside these cavities, but the spatial distribution of this gas remains uncertain. We present high spatial resolution observations with the Atacama Large Millimeter/submillimeter Array (ALMA) of 13CO and C18O lines of four well-studied transitional disks. The observations are used to set constraints on the gas surface density, specifically cavity size and density drop inside the cavity. The physical-chemical model DALI is used to analyze the gas images of SR21, HD135344B, DoAr44 and IRS48. The main parameters of interest are the size, depth and shape of the gas cavity. CO isotope-selective photodissociation is included to properly constrain the surface density in the outer disk from C18O emission. The gas cavities are up to 3 times smaller than those of the dust in all four disks. Model fits indicate that the surface density inside the gas cavities decreases by a factor of 100-10000 compared with the surface density profile derived from the outer disk. A comparison with an analytical model of gap depths by planet-disk interaction shows that the disk viscosities are likely low, with a<1E-3 for planet masses <10 MJup. The resolved measurements of the gas and dust in transition disk cavities support the predictions of models that describe how planet-disk interactions sculpt gas disk structures and influence the evolution of dust grains. These observed structures strongly suggest the presence of giant planetary companions in transition disk cavities, although at smaller orbital radii than is typically indicated from the dust cavity radii alone.
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