5.2. Slim Disks
The usual -viscosity prescription, in which the viscous stress is proportional to the total (gas plus radiation) pressure, leads to disks that are thermally and viscously unstable in cases in which the accretion rate is high enough that the inner regions of the disk are supported by radiation pressure (see, e.g., Shakura & Sunyaev 1976). Whether such instabilities are real is not at all clear because of the strong dependence on the ad hoc viscosity prescription (see, e.g., Piran 1978). However, if they are real, it may be that the disk jumps to a higher accretion rate, optically thick, geometrically "slim" (not thin) disk, in which the flow is stabilized by advection of heat into the black hole (Abramowicz et al. 1988). Even ignoring such instabilities, thin disk models are inconsistent at accretion rates above 20%-30% of the Eddington rate, where the flow probably adopts a slim disk geometry and advection must be included. Slim disks are essentially the same as the geometrically thick disks discussed above, except that the radial and vertical structure can be modeled separately. In particular, one can still use vertically integrated equations to describe the radial variation of flow variables. At high, super-Eddington accretion rates, this approximation must eventually break down, and the full two-dimensional thick disk equations must be solved (see, e.g., Papaloizou & Szuszkiewicz 1994).
Slim disk models are probably more appropriate for bright AGNs than standard thin disks, but once again, very little work has been done on their predicted SED. Szuszkiewicz et al. (1996) have calculated emergent spectra of slim disks by assuming that each annulus emits as a modified blackbody. Such models therefore make no predictions of discrete features such as the Lyman edge. Within this approximation, slim disk spectra are identical to thin disk spectra at low accretion rates, but at high (super-Eddington) accretion rates, the SED of a slim disk becomes redder in the extreme UV than that of a thin disk model with the same mass and accretion rate. The differences in the optical/UV SED predictions between the thin and slim disk models remain small, and slim disks still do not provide a satisfactory fit to the observations if the infrared bump is indeed due to thermal dust emission. Szuszkiewicz et al. (1996) propose that the optical/UV continuum may arise from reprocessing of radiation from the inner disk, which as noted above is one way out of this difficulty (see Section 4). Slim disks do have an advantage over thin disks in that they can produce thermal soft X-rays without being inconsistent, because Eddington or super-Eddington accretion rates are required. Even here though the modified blackbody approximation produces a steeply falling soft X-ray excess, which does not seem to explain the comparatively flat spectra of QSOs. Better atmosphere modeling may provide better agreement. Slim disks might be able to explain the steep soft X-ray excesses observed in narrow line Seyfert 1 galaxies (Szuszkiewicz 1997).