5 Discussion and conclusions

Our model describes a situation which naturally develops if a cold standard Shakura-Sunyaev disk is truncated within a hot, optically thin flow (ISAF) in the inner regions of the accreting system. We have proposed a SSD-ISAF transition based on few, well-known physical processes: Spitzer's theory of the energy exchange in a fully ionized plasma, and standard viscous heating due to friction in the accretion disk. We have shown that the transition of the cool disk material into an ISAF is the logical and inevitable consequence of these basic interactions. The process involves two steps:

(i) At the inner edge of the disk the surface density of the cool disk gets low. Virial protons penetrating from the ISAF heat the cool disk electrons. The electron temperature is limited ( $T_{\rm e}\sim 80$ keV) since they can radiate their energy efficiently via bremsstrahlung and Comptonization. Once the disk is too thin, proton heating overcomes the radiative losses everywhere in the disk. The disk heats up, expands and radiative losses become even more inefficient. With increasing temperature pair production sets in, but also the proton heating gets less efficient at higher temperature. Finally a new equilibrium for the thin disk is found at several 100 keV. We label this the "warm state'', since its temperature is intermediate between that of the cool disk and the virial temperature. An important aspect in this process is that protons in the disk are outside of the main energy channel. The main energy is transferred from the external ISAF protons to the disk electrons, which loose this energy via radiation. For the formation of the warm disk state the internal viscous heating of the protons is completely unimportant, but not so in the second step of the process:

(ii) At the temperatures of the warm state the disk protons and electrons are not coupled very tightly any more, as the timescale for establishing thermal equilibrium is not short compared to the thermal timescale. Now the minor energy channel due to viscous heating in the warm disk becomes important for the energy budget of the protons, because the viscously released energy can not be exchanged very efficiently with the ambient electrons. The upper part of the warm disk, where the densities are lowest and the Coulomb exchange time scales longest, is subject to a thermal instability. The size of the unstable region depends mainly on the viscosity parameter and the temperature of the warm disk. The higher the temperatures and $\alpha$, the deeper is the unstable region. The protons there are heated to their local virial temperature, and become part of the ISAF: the warm state evaporates. In the warm state no effective mass condensation takes place, since it at the same time becomes optically thin for penetrating hot protons. The mass evaporation due to the viscous instability of the warm state is therefore the main process which determines the mass budget in this region.

On the basis of this picture, we have also looked at the radial structure of an accretion disk in which mass exchange with an ISAF takes place. Using the evaporation and condensation rates derived, we have investigated the conditions under which accretion is possible in such a way that the entire accretion flow eventually evaporates. In this case a steady state is possible with an inner edge to the disk at some finite radius $R_{\rm i}$ outside the last stable orbit. We find that such a steady state is indeed possible for plausible values for the accretion rate and viscosity parameter. The steady state condition determines a relation between the accretion rate and the value of $R_{\rm i}$. Also, it determines the width $\delta=\Delta R/R_{\rm i}$of the warm, evaporating disk region; we find $\delta\approx 0.3$ for $\alpha$ of order 0.3 and accretion at a tenth of Eddington.

A potentially important factor which we have not been able to include in our picture of the warm disk region is the cooling effect of soft photons from the cool disk extending just outside the warm disk. If such photons can enter the warm region, they will cause a cooling by inverse Compton scattering on the warm electrons. Since both the cool and warm disk are quite thin ($H/R\ll 1$), the radial optical depth of the warm disk is large, and the angle subtended by the cool disk as seen from the interior of the warm disk is small (to visualize this, see Fig. 5). It thus seems likely that the effect of such cooling on the energy balance in the warm disk can be small, but this point requires closer scrutiny.

The extent of the region (in distance from the hole) where the physical conditions assumed here apply is limited. At large radii the proton temperature in the optically thin, hot region decreases ($\propto$R^-1). At lower proton temperatures the proton penetration depth into the cool disk also gets smaller. This limits the region where a warm state can be produced. Without the warm state our mechanism will probably not work efficiently enough to transfer all material from a cool SSD into material from an ISAF. Therefore we do not expect our mechanism to work at large radii from the black hole. But a combination of the coronal evaporation flow model as suggested by Meyer et al. (2000) and the two stage model proposed here could in principle cover a large range in the radial direction for the SSD-ISAF transition to occur.

Acknowledgements

This work was done in the research network "Accretion onto black holes, compact stars and proto stars'' funded by the European Commission under contract number ERBFMRX-CT98-0195.