3 Evaporation vs. condensation rates

In Sect. 2 we have shown that the internal viscous heating of the disk protons leads to a mass evaporation rate according to Eq. (16).

At this point we do not yet know whether the mass loss  $\dot\mathcal{M}$from a warm disk region is high enough to completely evaporate the disk. At the same time as the upper atmosphere of the warm disk evaporates, the hot protons from ISAF condense into it and increase the surface density. For an effective evaporation of the disk the mass loss rate must be higher than the condensation rate. To compute the condensation rate we need an estimate of the density in the ISAF.

In our previous numerical simulations of warm disks (Paper II) we parameterized proton mass flux from the ISAF by scaling the energy flux of the incident protons with the local energy dissipation rate in the ISAF. Here we adopt a slightly more realistic mass flux rate in an ISAF.

In a thin disk approximation the surface density $\Sigma_{\rm I}$ of the ISAF, with accretion rate $M_{\rm I}$, is

$$\Sigma_{\rm I}=\frac{\Omega_{\rm K}}{3\pi\alpha c_{\rm s}^2}~ M_{\rm I} ~\left[~1-(3R_{\rm S}/R)^{1/2}\right].$$ (24)

We assume that the protons in the ISAF have a Maxwellian velocity distribution according to their local virial temperature. The mass flux rate $\phi$ [in g cm^-2 s^-1] from a Maxwellian through a surface is given by

$$\phi=\rho_{\rm p}~\sqrt{\frac{k~T_{\rm p}}{2\pi~m_{\rm p}}}\cdot$$ (25)

From Eq. (24) we can calculate the proton density $\rho_{\rm I}=\Sigma_{\rm I}/(2 H_{\rm I})$ in the ISAF. This yields the local mass flux rate onto the cool disk for protons at their virial temperature, i.e. the condensation rate:

$$\phi_{\rm I}=\frac{1}{\alpha}\frac{\dot M_{\rm I}}{8\pi^{3/2} R^2}~\left[~1-(3R_{\rm S}/R)^{1/2}\right]\;.$$ (26)

We can now compare the mass condensation rate with the evaporation rate according to Eq. (16). The evaporation rate is $\propto$ $(\alpha T)^2$. Therefore high disk temperatures and high $\alpha$'s favor mass evaporation. In Paper II we have calculated these temperatures with a radiative transfer calculation that includes brems photon production, $\gamma\gamma$ pair production and Comptonization. We found there that the temperature at the inner edge of a cool accretion disk reaches several 100 keV ("warm state''), while its optical depth $\tau_{\rm 1/2}$ (at the border with the cool part of the disk) is of the order unity for accretion rates around $0.1~\dot M_{\rm Edd}$. We assume here that at a certain radius the warm disk state exists and, ignoring its radial structure, compare the local evaporation and condensation rates.

Figure 4: Decrease of the proton mass flux with depth in a thin, hydrostatic, isothermal disk with total optical depth $\tau _{\rm tot}= 2\cdot \tau _{\rm 1/2}=3$, at disk temperatures of 1, 50 and 200 keV, for $R=8~R_{\rm S}$ and $\alpha =0.1$. At temperatures above 100 keV, only a fraction of the incident protons is captured by the disk.

Figure 3 shows this comparison for different values for the viscosity parameter $\alpha$, different temperatures and accretion rates. For values of the viscosity parameter $\alpha\ga 0.3$ and $\dot M=0.1~\dot M_{\rm Edd}$ evaporation dominates when the temperature of the warm state exceeds $\sim$300 keV. As the ISAF density (i.e. the accretion rate) decreases, the condensation rate of the protons into the the disk decreases and evaporation dominates over the condensation rates over a wider range of radii.

The condensation rate given by Eq. (26) is actually an overestimate, since it assumes that all incident protons are stopped in the disk. While this is correct for cool disks, for a high temperature plasma the rate of the electron-proton energy exchange is small and a disk with low surface density gets optically thin for the penetrating hot protons. This is demonstrated by Fig. (5) where we show how the incident proton flux changes with depth into a warm disk. The Thomson optical depth in this example is ( $\tau_{\rm tot} = 2\cdot\tau_{1/2}=3$). At the temperatures of a cool standard disk ($\la$1 keV) almost all protons are absorbed in this layer. But at the high temperatures of the warm state the disk is optically thin and practically all ISAF protons fly through the disk without being absorbed. The penetrating protons do not add to the surface density in this case, and evaporation should therefore be possible even at lower values of $\alpha$.

At temperatures below 100 keV, the evaporation rate from the warm state into the ISAF is quite small and can not balance the loss from it by condensation. This is roughly the temperature of the warm surface layer on a cool optically thick disk heated by proton illumination (the situation sketched in Fig. 1). Thus the relative importance of evaporation and condensation reverses just at the point where the cool component disappears. As long as a cool disk is present, the thermal instability in its warm surface layer is relatively weak, while it effectively absorbs all incoming protons. Once the cool component is gone, the temperature and evaporation rate goes up, while at the same time mass condensation by stopping of protons in the disk becomes ineffective.

3.1 Dependence on the accretion rate

Figures 3 and 4 show that net evaporation takes place close to the hole preferentially at low accretion rates in the ISAF, which may sound counterintuitive. It is a consequence of the fact that the evaporation rate does not depend on the flux of incident protons (cf. Eq. (16)). This again is a result of the fact that the field protons in the warm disk lie "outside the main energy channel'': the incident proton energy flux sets the electron temperature (through the Comptonization balance) but does not affect the proton temperature directly. The temperature of the field protons is determined by the secondary balance between viscous heating of the field protons and their energy loss to the electrons.

At low accretion rates, however, the hot proton flux eventually becomes insufficient to keep the layer "warm'': it will cool down to low ($\sim$1 keV) temperatures by bremsstrahlung losses. We have studied this transition in Paper II, where we found that a layer of optical depth $\tau _{1/2}=1.5$ can just be kept in the warm state for an accretion rate in the ISAF of about $0.1~\dot{M}_{\rm E}$.

At lower accretion rates, a warm disk is possible only when the bremsstrahlung losses are lower, at lower electron densities. Since the electron density is proportional to the optical depth $\tau$ of the disk, and the brems losses proportional to $n_{\rm e}^2$, the minimum ISAF accretion rate needed to maintain a warm disk state by proton illumination scales as $\tau^2$. The optical depth of the disk vanishes towards its inner edge, so we expect that there is always a region close to the inner edge of the disk where evaporation takes place, even at very low accretion rates.

Summarizing this argument, the optical depth $\tau_{\rm m}$ of a warm disk at the point where it matches onto the cool disk, depends on the accretion rate in the ISAF as

$$\tau_{\rm m}= 2~\cdot \tau_{{1/2},{\rm m}}\approx 3 \left({\dot M_{\rm I}\over 0.1~\dot M_{\rm E}}\right)^{1/2}\cdot$$ (27)