## 2. A cool disk inside an ion supported accretion flowThe geometry for the accretion flow we consider here consists of an ion supported advection torus or ADAF (Rees et al. 1982; Narayan & Yi 1994, 1995). This flow is assumed to coexist with an optically thick accretion disk, such that the cool disk extends partly into the hot flow. We do not address the question here how much of an overlap between the two is physically realistic. Since the incident ions loose essentially all their energy once they have entered the cool disk, the overlap region is a significant sink of energy and mass from the ADAF. If it is too wide, these losses might be too high for an ADAF to be sustainable. The distance of the overlap region from the hole is treated as a free parameter of the problem. The properties of the cool disk depend on its accretion rate. We
assume here that a fixed fraction ## 2.1. The radial structure of the cold diskWe set up our cool disk model according to Svensson & Zdziarski
(1994) (SZ94). They have shown that if a sufficiently large fraction
of the accretion power is dissipated in the accretion disk corona
(ADC), a cold, optically thick and pressure supported disk can exist
down to small radii very close to the black hole horizon. With
increasing
Our solutions of the equations of SZ94 are slightly different, for the gas pressure supported case, because of minor algebraic inaccuracies we detected in SZ94. We present our version of the solutions in Eqs. 1-5. In Fig. 1 we plot the numerical solution for Eq. 28 of SZ94 substituting our result for . Here are the pressure scale
height, scattering optical depth, mass density, pressure and
temperature of the cold disk (subscript d), respectively. We have used
the following dimensionless quantities: radius
, where
is the Schwarzschild radius of a
black hole of mass ## 2.2. Hydrostatic balance of cool disk and Comptonizing layerFor our calculations we consider the simple idealized case of an accretion disk in a plan-parallel geometry. We assume that the disk and the corona are in hydrostatic equilibrium. The pressure profile as function of optical depth is consistently updated throughout our calculation according to the temperature profile . In hydrostatic equilibrium the coronal
pressure, , at the coronal base is in
equilibrium with the disk pressure at disk surface, with
, the pressure in the mid-plane of
the disk. Above this slab we locate the hot protons. A fraction
The energy exchange between the corona and the disk in our model is mediated by protons only [a model which accounts for the interactions by radiation only was obtained by Haardt & Maraschi (1991; 1993)]. We equal the energy flux by the protons, , into the cool disk with the viscous energy dissipation in the corona. That is, we neglect both the radiation loss from the corona and energy loss by advection in the corona. These assumptions are for definiteness of the model only, and can easily be generalized. We assume that the protons above the cool disk have a Maxwellian velocity distribution. This is probably not the case but as the velocity distribution in the corona is not known we consider this assumption to be adequate for our calculations. The energy flux from a Maxwellian proton distribution is given by where denotes the number density of the protons. For the temperature of the protons in our model we take the local virial temperature (Rees et al. 1982). At the distance dominating the energy release, , the protons have a temperatures around 20 MeV. Knowing the proton temperature and the energy flux at a certain radius we can calculate the number density , and with the equation of state the coronal pressure of the proton gas at the coronal base, i.e. the surface of the disk: As we have assumed a two temperature plasma in the corona with the electron temperature , the contribution to the pressure by the coronal electron gas can be neglected. provides a boundary condition for the pressure at the top of our Comptonizing layer. The height of this upper boundary above the mid-plane of the disk is not known in advance, but must be found by matching of the Comptonizing layer to the underlying cool disk. The transition between the Comptonizing layer and the cool disk is gradual, and determined by the processes which reprocess the downward flux of hard photons into soft photons. In our calculations this reprocessing is not treated in detail, but replaced by reprocessing into a black body spectrum at an assumed base of the Comptonizing layer, at Thomson depth . The choice of is discussed in Sect. 3.2. If is the geometric height at , pressure balance between the Comptonizing layer and the underlying cool disk requires that =. If the Comptonizing layer is thin compared with its height above the mid-plane, the acceleration of gravity would be constant and the pressure the layer exerts on the cool disk would be proportional to . In most of our results, the Comptonizing layer is indeed thin, but to be sure we have allowed for arbitrary thickness. The pressure exerted on the cool disk then depends on this thickness since the acceleration of gravity increases with height. The layer thickness in turn depends on its temperature distribution, which varies with time as the cooling and heating processes settle towards equilibrium. Thus we solve the pressure profile by starting with an initial guess for the height of the disk surface above the mid-plane, , and iterating until we get the right value for which the pressure condition condition at is fulfilled. The underlying part of the disk is given by the solutions presented in Sect. 2.1, i.e. it is isothermal with temperature , the pressure at mid-plane is given by and the scale height is . The vertical hydrostatic equilibrium calculated from the top of the disk to the mid-plane yields where is the local Kepler angular velocity. Together with the equation for the scattering optical depth one obtains a differential equation describing the pressure profile as a function of optical depth for any temperature profile. Eq. 14 is solved via a forth order Runge-Kutta method. We have abbreviated where denotes the electron scattering opacity, which for ionized hydrogen is . This calculation is done for each time step to account for the new temperature profile after each cooling/heating step. © European Southern Observatory (ESO) 2000 Online publication: October 30, 19100 |