We report Submillimeter Array (SMA) observations of CO (J=2--1, 3--2 and 6--5) and its isotopologues (13CO J=2--1, C18O J=2--1 and C17O J=3--2) in the disk around the Herbig Ae star HD 163296 at ~2 (250 AU) resolution, and interpret these data in the framework of a model that constrains the radial and vertical location of the line emission regions. First, we develop a physically self-consistent accretion disk model with an exponentially tapered edge that matches the spectral energy distribution and spatially resolved millimeter dust continuum emission. Then, we refine the vertical structure of the model using wide range of excitation conditions sampled by the CO lines, in particular the rarely observed J=6--5 transition. By fitting 13CO data in this structure, we further constrain the vertical distribution of CO to lie between a lower boundary below which CO freezes out onto dust grains (T ~ 19 K) and an upper boundary above which CO can be photodissociated (the hydrogen column density from the disk surface is ~ 10^{21} cm-2). The freeze-out at 19 K leads to a significant drop in the gas-phase CO column density beyond a radius of ~155 AU, a CO snow line that we directly resolve. By fitting the abundances of all CO isotopologues, we derive isotopic ratios of 12C/13C, 16O/18O and 18O/17O that are consistent with quiescent interstellar gas-phase values. This detailed model of the HD 163296 disk demonstrates the potential of a staged, parametric technique for constructing unified gas and dust structure models and constraining the distribution of molecular abundances using resolved multi-transition, multi-isotope observations.
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