Profiles of the Toomre Q parameter as a function of radius. The left panel shows the case where we extrapolate the optical depth in order to remain constant inward of the radius for which α ≈ 2, while the right panel assumes the optical depth increases as r^−1.5 therein, thus providing an upper and lower limit for Q. The curves are colored by the resultant dust optical depth at 3 mm. The horizontal dotted line corresponds to Q = 1.7, a reference value below which the disk can develop gravitational instabilities. (See Section 6.2.1 for more details.