Are passive protostellar disks stable to self-shadowing?

The uniqueness and stability of irradiated flaring passive protostellar disks is investigated in the context of a simplified set of equations for the vertical height H as a function of radius R. It is found that the well-known flaring disk solution with H ~ R^{9/7} is not unique. Diverging solutions and asymptotically conical (H ~ R) solutions are also found. Moreover, using time-dependent linear perturbation analysis, it is found that the flaring disk solution may become unstable to self-shadowing. A local enhancement in the vertical height alters the functional form of irradiation grazing angle, and causes the 'sunny side' of the enhancement to grow and the 'shadow side' to collapse in a run-away fashion. This instability operates in regions of the disk in which the cooling time is much shorter than the vertical sound crossing time, which may occur in the outer regions of the passive irradiated disk if dust and gas are sufficiently strongly thermally coupled. Processes that may stabilize the disk, which include active accretion, irradiation from above (e.g. a scattering corona) and low disk optical depth, are likely to operate only at small or at large radius. The simple analysis of this Letter therefore suggests that the instability may alter the flaring disk structure at intermediate radii (between the actively accreting and fast rotating inner regions and the optically thin outer regions).