4 An evaporating cool disk

4.1 Thin disks with evaporation/condensation

In the above we have started with the assumption that an ISAF and a cool disk coexist, and found the conditions under which the disk can feed mass into the ISAF. We have done this by considering the conditions at each distance from the hole separately. To turn the ingredients into a consistent picture, we have to consider the mass flux through the system, so that conditions as a function of distances from the hole are connected, in the way sketched in Fig. 5. At large distance, we have a standard cool disk. Closer in, an ISAF surrounds it and condenses onto it, producing what we have called here the warm, proton-illuminated layer. Even further in, the vertical optical depth of the disk becomes too low to sustain cooling by brems losses, and the whole disk transforms to a "warm disk'' state (Paper II) of nearly uniform temperature. The upper layers of this warm disk evaporate to feed the ISAF assumed at the outset, and at the inner edge of the disk, at some radius $R_{\rm i}$, the entire mass flux through the disk has evaporated into the ISAF. We now investigate what the conditions are for such a radial structure to be possible in a steady state.

The disk, whether in a cool ($\la$1 keV) or warm ($\sim$300 keV) state, is still very cold compared to the local virial temperature, so that the standard thin disk approximation is valid. The difference with a standard steady disk is that the mass flux is now a function of distance, due to the condensation and evaporation from/to the ISAF. We first consider the modifications to the $\alpha$-disk diffusion equation that this causes.

Figure 5: Evaporation from the warm disk: a cool disk (black, $\sim$1 keV) partly extends into an ISAF and is exposed to the hot virialized protons from the hot torus (open arrows, light grey). Above and below the cool disk a heated surface layer is produced ($\sim$80 keV, dark grey) due to proton illumination. At the inner edge at R0 the surface density of the cool disk gets small and a warm disk develops ($\sim$300-500 keV, grey). Net mass loss into the ISAF evaporates the warm region between $R_{\rm i}<R<R_0$ (filled arrows). Due to the high temperatures in the warm disk this region is a source for hard X-rays.

The surface mass density of the disk material is

$$\Sigma = \int_{-\infty}^{\infty} \rho ~{\rm d}z \approx 2\rho_0~H\; ,$$ (28)

where $\rho_0$, H denote the density at the midplane and the scale height of the cool disk, respectively.

The change of the mass accretion rate with radius through the cool disk due to evaporation can be expressed by

$$\frac{\partial\dot M }{\partial R}= 4 \pi R ~\dot\mathcal{M}\; ,$$ (29)

where we include an additional factor 2 to account for the mass loss on both sides of the disk. Integrating this equation yields the mass loss (in g sec^-1) due to evaporation in a ring between $R_{\rm i}$and R,

$$\dot M_{\rm ev} = 4\pi \int_{R_{\rm i}}^{R} R \dot \mathcal{M} {\rm d}R\;.$$ (30)

The conservation of mass in a cool disk including the evaporation term (Eq. (29)) can be expressed by

$$\frac{\partial}{\partial t}(R\Sigma)+\frac{\partial}{\partial R} (R\Sigma v_{\rm R}) + R \dot\mathcal{M} = 0 \; ,$$ (31)

where $v_{\rm R}$ is the radial drift velocity of the disk material.

As in the standard derivation, the thin disk diffusion equation follows from the angular momentum equation, which now includes a term for the angular momentum carried with the evaporating/condensing material. For the present purpose, it is sufficient to assume that condensing and evaporating material just has the same specific angular momentum, $\Omega_{\rm K}R^2=$ as the disk.

The equation for the angular momentum balance in an evaporating disk then is

 

$$\frac{\partial}{\partial t}(R \Sigma \Omega R^2)+ \frac{\partial}... ...R^2 = \frac{\partial}{\partial R}(S R^3 \frac{\partial \Omega}{\partial R})\; ,$$ (32)

where

$$S=\int_{-\infty}^{\infty} \rho~\nu~{\rm d}z \approx \Sigma\nu\;.$$ (33)

We use the usual $\alpha$ prescription for the viscosity (Shakura & Sunyaev 1973),

$$\nu = \alpha \frac{c_s^2}{\Omega}\cdot$$ (34)

The approximate equality in Eq. (33) holds if $\nu$ can be considered independent of the geometrical height z.

4.2 Steady state

Multiplying the continuity equation (Eq. (31)) with $R^2 \Omega$ yields after subtraction from Eq. (32) an expression for the mass flux $\dot M$ in the cool disk in the stationary case ( $\partial/\partial t = 0$):

$$\dot M = 6~\pi R^{1/2} \frac{\partial}{\partial R}\left(\nu \Sigma R^{1/2}\right)\;.$$ (35)

The thin disk equations, Eqs. (31), (32) can be used in their time dependent form. For example, the evaporating inner regions could expand inward towards the last stable orbit, or outwards. As a first application, however, we are interested in steady-state conditions. In the stationary case we can use Eq. (35) to replace $\dot M$ in Eq. (29). Integrating this expression yields

$$R^{1/2} \frac{\partial}{\partial R}(\nu \Sigma R^{1/2})= \frac{2}{3}\int_{R_{\rm i}}^{R} R \dot \mathcal{M} {\rm d}R + C_1\;.$$ (36)

We integrate one more time and obtain the surface density distribution:

$$\nu \Sigma = \frac{2}{3 R^{1/2}} \int_{R_{\rm i}}^R \frac{{\r... ...R} R~\dot\mathcal{M} {\rm d}R + 2 C_1 + \frac{C_2}{R^{1/2}}\;,$$ (37)

where C₁,C₂ are integration constants. Let the total mass loss due to evaporation from $R_{\rm i}$ to infinity be $\dot M_{\rm ev,\infty}$ (cf. Eq. (30)). With Eqs. (35), (36) we find the integration constant C₁,

$$C_1 = \frac{1}{6 \pi}(\dot M - \dot M_{{\rm ev},\infty})\;.$$ (38)

As in the standard treatment, the second integration constant is fixed by considering the conditions at the inner edge of the disk. This could be the last stable orbit if the disk extends all the way down to the hole, but more interesting is the case when the entire mass flux has evaporated into an ISAF before reaching the hole. Thus we now assume that a steady state exists with the inner edge at some as yet unspecified distance $R_{\rm i}$. The requirement of a steady state will then impose a condition on the parameters of the system that has to be satisfied. This will turn out to be a condition on the accretion rate. At the end of the calculation, we thus obtain a relation between the mass accretion rate and the position of the inner edge.

Thus we set $\nu \Sigma_i=0$ at the inner edge of the disk, as in standard accretion theory. With Eqs. (30), (37), (38) and $R=R_{\rm i}$ we get an expression for the second integration constant C₂,

$$C_2 = -\frac{R_{\rm i}^{1/2}}{3\pi}(\dot M -\dot M_{{\rm ev},\infty})\;.$$ (39)

With the previous results the surface density distribution as a function of radius in a disk with evaporation losses is now

 

$$\nu\Sigma = \frac{1}{6\pi~R^{1/2}}\int_{R_{\rm i}}^{R}\frac{{\rm ... ...M - \dot M_{{\rm ev},\infty}\right]~\left[~ 1 - (R_{\rm i}/R)^{1/2}\right] \; .$$ (40)

From this expression the classical formula for the surface density in a cool disk can be recovered if one sets $\dot M_{\rm ev}=0$, i.e. when no evaporation takes place.

This expression for the surface density, though strictly derived for steady conditions, is still approximately valid if the position of the inner edge changes slowly. We are interested in a true stationary case, however. In this case the accretion rate in the disk will be exactly equal to the total mass loss due to evaporation in the disk, $\dot M = \dot M_{{\rm ev},\infty}$, or in other words all matter drifting inward through the cool disk has evaporated when $R_{\rm i}$ is reached. Eq. (36) then simplifies to

$$\nu \Sigma = \frac{1}{6\pi~R^{1/2}}\int_{R_{\rm i}}^{R} \frac{{\rm d} R}{R^{1/2}} \dot M_{\rm ev}(R).$$ (41)

4.3 Radial extent of the warm disk

We can now estimate the distance over which the process of evaporation into the ISAF takes place. Let R₀ be the innermost radius where the cool disk component exists. Inside this, there is only a warm disk (see Fig. 5). Evaporation takes place both from the warm layer on top of the cool disk at R>R₀ and from the warm disk region $R_{\rm i}<R<R_0$, but the warm disk region is expected to contribute most, since its temperature is significantly higher. Thus we equate, for the present approximate purpose, the evaporating region with the warm disk region. Assume that the relative extent $\delta_0=R_0/R_{\rm i}-1$ of the warm disk is small. The evaporation rate Eq. (16), with $\Omega\approx \Omega_{\rm i}$, depends only on temperature. The temperature of the warm disk is relatively fixed (Paper II), so we can set $\dot\mathcal{M}$ constant as well. Equation (40) for the surface density as a function of distance from the inner edge is then

$$\nu \Sigma = \frac{2}{3}\int_{R_{\rm i}}^{R}{\rm d} R\int_{R... ...\rm d}R \approx \frac{1}{3} \dot\mathcal{M}~(R-R_{\rm i})^2 ,$$ (42)

or

$$\Sigma = \frac{1}{3} \mathcal{A}\;\delta_0^2\;,$$ (43)

where

$$\mathcal{A} = \frac{\Omega}{\alpha~c_s^2}~\dot\mathcal{M}~R_{\rm i}^2 \; .$$ (44)

In the warm state, little net condensation takes place (as we have seen in Sect. 3) and $\dot\mathcal{M}$ is dominated by the evaporation losses. Using Eq. (16) for $\dot\mathcal{M}$we then find

$$\mathcal{A} = \alpha~\theta~f~\frac{3}{16\sqrt{2\pi}~\ln\Lamb... ... p}}{m_{\rm e}}\right)^{3/2} \frac{R_{\rm S}}{R_{\rm i}} \; ,$$ (45)

where r₀ is the classical electron radius and $\theta=k T_{\rm e}/m_{\rm e}c^2$. This result depends on the temperature of the warm disk, the viscosity parameter and the distance from the hole.

We can now make an estimate of the relative extent $\delta_0$ of the warm disk. R₀ is the maximum radius where the warm disk can exist (at larger surface density, it develops a cool disk component). The optical depth at R₀ is thus given by Eq. (27). Computing the optical depth from Eq. (43) and equating this to  $\tau_{\rm m}$ we have

$$\kappa ~\frac{\Omega \dot \mathcal{M} R_{\rm i}^2}{3\alpha~c_... ...ac{\dot M_{\rm I}}{0.1 \cdot \dot M_{\rm Edd}}\right)^{1/2}\;,$$ (46)

which shows that $\delta_0$ is a function of the warm disk temperature, the viscosity, the relative distance from the hole and the accretion rate. Numerically, we have

 

$$\delta_0 \approx 0.1 \left(\frac{1}{\alpha~\theta ~f}\right)^\fra... ...frac{1}{2} ~\left(\frac{\dot M_{\rm I}}{\dot M_{\rm Edd}}\right)^\frac{1}{4}\;,$$ (47)

where we have set $\ln \Lambda \approx 20$. With typical values, $\theta\sim O(1)$, $R_{\rm i}/R_{\rm S}\sim 8$, $\dot M_{\rm I}/\dot M_{\rm Edd}=0.1$, f=0.88 and $\alpha=0.3$ this yields $\delta_0 \approx 0.3$. We conclude that the extent of the warm disk is in an `interesting' range. It is smaller than the distance to the hole, but still large enough that the width of the warm disk is large compared to its thickness, so our thin-disk treatment of the evaporation region is justified.

4.4 Conditions for a steady mass flux to exist

We now estimate the radial distance of the transition radius from the hole as a function of the mass accretion rate $\dot M$ in the system. At the distance $R_{\rm i}$ all mass accreted in the cool disk is evaporated, $\dot M_{\rm ev} = \dot M$. The accretion rate $\dot M$ of the system is now also equal to the accretion rate  $\dot M_{\rm I}$ in the ISAF and we can write

$$\dot M_{\rm ev} \approx 2 \cdot 2 \pi ~R_{\rm i} \Delta R_{\r... ...= 4\pi~R_{\rm i}^2~\delta_0~\dot\mathcal{M} = \dot M_{\rm I}.$$ (48)

We use Eqs. (16, 47) and find for $R_{\rm i}$ (in units of the Schwarzschild radius)

$$\frac{R_{\rm i}}{R_{\rm S}} \approx \;7 \times \frac{1}{\alph... ...frac{\dot M_{\rm I}}{\dot M_{\rm Edd}}\right)^\frac{3}{4}\cdot$$ (49)

Again with the same values as above, $\theta\sim O(1)$, $\dot M_{\rm I}/\dot M_{\rm Edd}=0.1$, f=0.88 and $\alpha=0.3$ this yields $R_{\rm i}/R_{\rm S}\approx 8$. We conclude that, using plausible values, the estimated transition radius is in the region where we expect our model to work.