2 A cool disk extending inside an ion supported flow

2.1 Optically thick case

Most of the physics of the interaction between the hot supported flow and the underlying cool disk is rather independent of the structure of the cool disk, since it acts mainly as a thermalizing surface. The height above the midplane where the hot ion-supported flow and the cold disk interact is important, however, since this determines the gravitational acceleration, and hence the density within the interaction layer. Processes such as photon production by bremsstrahlung depend on the electron density explicitly. To obtain an estimate for the thickness of the cool disk, the amount of energy released in the cool disk must be specified. For this purpose we adopt an $\alpha$-disk prescription as described in Svensson & Zdziarski (1994). In this model, a fixed fraction f of the gravitational energy release takes place above the disk, and a fraction 1-f in the cool disk. For details we refer also to Paper I.

In order to simplify our treatment we assume that the vertical disk structure can be approximated at a given radius by a plane-parallel, one-dimensional geometry, so that the vertical z-coordinate is the only dimension relevant to our problem. We find the vertical density distribution from the equation of hydrostatic equilibrium,

$$\frac{{\rm d}P}{{\rm d}\tau}=\frac{\Omega^2_{\rm K} z}{\kappa} + \frac{\partial P_{\rm p}}{\partial\tau}(\tau),$$ (1)

where $\Omega_{\rm K}=\left(G M/R^3\right)^{1/2}$ is the local Kepler angular velocity, $\kappa = \kappa_{\rm es}=0.40$ cm² g^-1 is the electron scattering opacity and $\tau$ is the electron scattering optical depth. Note that we use $\tau$ here as a coordinate, replacing the vertical height z. The first term on the right represents the usual weight of the disk; the second term takes into account the momentum exerted by the deceleration of the incident protons as a function of optical depth. The force $\partial P_{\rm p}/ \partial\tau$ is evaluated by recording the change of velocity  $\Delta v_{\rm p}$ of the incoming protons as a function of optical depth (see Sect. 3.1). The contribution of the radiative pressure is neglected as we study cases well below the Eddington luminosity.

Together with the equation for the vertical coordinate z,

$$\frac{{\rm d}z}{{\rm d}\tau} = \frac{-k}{\mu~m_{\rm p}~\kappa}\frac{T(\tau)}{P(\tau)},$$ (2)

we integrate Eq. (1) via a fourth order Runge-Kutta method. We set $\mu=\frac{1}{2}$ for the idealized case of an ionized hydrogen atmosphere.

The initial value for the pressure at the top of the layer is set to a small fraction of the coronal pressure, $P_{\rm top}(\tau=0) = 10^{-2} n_{\rm p}~k~T_{\rm vir}$. The geometrical height $z_{\rm top}$ of this upper boundary layer is not known in advance. We solve the pressure profile by starting with an initial guess for the height of the disk above the mid-plane, $z_{\rm top}$. We integrate Eq. (1) to the maximum optical depth of our model atmosphere $\tau_{\rm bot}$ at the vertical height $z_{\rm bot}$. Usually we set $\tau_{\rm bot}=3$. At this Thomson optical depth the protons have already lost more than 0.99 of their energy. At $\tau_{\rm bot}$ pressure balance between the simulated layer and the underlying cool disk requires $P(\tau_{\rm bot})=P(z_{\rm bot})$. We iterate until we find the right value for which this pressure condition at $z_{\rm bot}$ is fulfilled. Thus we have matched the simulated layer to the underlying cool disk. The temperature of the cool disk and the pressure at the mid-plane are determined as in Paper I.

2.2 Moderate optical depth

Near the inner edge of a cool Shakura-Sunyaev disk at the radial distance $R_{\rm i}$ from the compact object, the surface density drops to low values. In the limit $H/R \ll 1$ (where H is the disk pressure scale height), the surface density varies as $[1-(R_{\rm i}/R)^{1/2}]$. Close to the inner edge of an accretion disk, the assumption of a large optical depth is therefore not valid any more. A certain minimum optical depth is necessary, however, for efficient cooling of the disk by soft photons to be possible. In Sect. 5 we explore cases where the total optical depth of the layer is of order unity or less. The procedure to calculate the hydrostatic balance in such cases is almost the same as in the cases of a surface layer on top of an optically thick disk. But instead of matching the computed layer to an underlying cool disk, we put the bottom of the simulated layer is at $z(\tau_{\rm bot})=0$, the mid-plane of the disk. We are interested in simulating the complete disk, i.e. from the upper surface to the lower surface. We do this by introducing an artificial "mirror'' placed at the mid-plane. All relevant physical quantities of the model are reflected at this mirror. Thus we have to compute one half space only instead of simulating the whole vertical disk structure.