Radiatively damped density waves in optically thick protostellar disks

The redistribution of angular momentum and mass in circumstellar disks by spiral density waves is governed by processes that dissipate the waves. Dissipation is usually attributed to unspecified nonlinear processes or an ad hoc viscosity. In this paper, we examine dissipation by radiative losses associated with the cyclic compression and expansion of disk material by the density waves. We consider linear, discrete modes in optically thick disks and use a simplified treatment of the radiative losses in order to assess the parameter dependences and importance of radiative damping. Wave action conservation principles are generalized to describe the effect of dissipation (by any means) and the consequent coupling of wave and disk energy and momenta. A relation is derived (in the WKBJ limit) between radial mass transport of disk: material and the azimuthally averaged dissipation rate. A wave amplitude equation is similarly derived, which describes the radial evolution of wave angular momentum, and therefore the radial deposition of angular momentum in the disk. Associated with the equation is a characteristic damping length, proportional to the cooling time from the base state temperature times the radial wave speed, and a factor that depends weakly on optical depth. The damping length is used to define, in terms of the disk base state temperature and surface density, regimes of strong and weak radiative damping. Waves in warm disks may be damped close to the resonances at which they are excited, while waves in cold disks can propagate to distances at which other dissipative mechanisms may dominate. It is found that the boundary between these regimes cuts across values that are thought to have characterized the primitive solar nebula. Thus both strong and weak radiative damping may be encountered in circumstellar disks in general.