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Astron. Astrophys. 362, 1-8 (2000)

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2. A cool disk inside an ion supported accretion flow

The 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 f of the energy release is transported to the accretion disk corona (ADC) above the cold disk. All the angular momentum transport and the accretion take place in the cool disk. This allows us to use a standard thin disk model for the cool disk. In the calculations reported, 95% of the accretion energy is released in the ADC.

2.1. The radial structure of the cold disk

We 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 f the transition between the gas pressure supported solutions and the radiation pressure supported solutions (corresponding to the break in the curves in Fig. 1) moves to higher [FORMULA]. The case [FORMULA] represents the standard [FORMULA] disk.

[FIGURE] Fig. 1. Disc solutions at a fixed radius [FORMULA] from the compact object for various values [FORMULA], the viscosity parameter [FORMULA], and [FORMULA]. The accretion rate is super-Eddington ([FORMULA] for an accretion efficiency [FORMULA]) above the horizontal line. The transition from the radiation pressure dominated solutions to the gas pressure dominated solutions (break in the curve) moves to higher [FORMULA] with increasing f. For [FORMULA] only a gas pressure dominated solution exists.

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 [FORMULA].

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Here [FORMULA] 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 [FORMULA], where [FORMULA] is the Schwarzschild radius of a black hole of mass M, accretion rate [FORMULA], where [FORMULA] is the Eddington luminosity and [FORMULA] denotes the accretion efficiency. [FORMULA] is the standard viscosity parameter according to Shakura & Sunyaev (1973). We have abbreviated [FORMULA] for the inner boundary condition at the inner edge of the disk, [FORMULA] is the fine-structure constant, [FORMULA] is the classical electron radius, [FORMULA] is the Boltzmann constant, [FORMULA] and [FORMULA] are the masses of the proton and the electron, respectively. The numerical value of the coefficient [FORMULA] is given by

[EQUATION]

2.2. Hydrostatic balance of cool disk and Comptonizing layer

For 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 [FORMULA] as function of optical depth [FORMULA] is consistently updated throughout our calculation according to the temperature profile [FORMULA].

In hydrostatic equilibrium the coronal pressure, [FORMULA], at the coronal base is in equilibrium with the disk pressure at disk surface, with [FORMULA], the pressure in the mid-plane of the disk. Above this slab we locate the hot protons. A fraction f of the gravitational power [FORMULA] is assumed to be directly dissipated to the protons. To calculate the pressure that the corona exerts on the top of the disk we have to figure out some numbers first.

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, [FORMULA], 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.

[EQUATION]

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

[EQUATION]

where [FORMULA] 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).

[EQUATION]

At the distance dominating the energy release, [FORMULA], the protons have a temperatures around 20 MeV.

Knowing the proton temperature [FORMULA] and the energy flux [FORMULA] at a certain radius we can calculate the number density [FORMULA], and with the equation of state the coronal pressure [FORMULA] of the proton gas at the coronal base, i.e. the surface of the disk:

[EQUATION]

As we have assumed a two temperature plasma in the corona with the electron temperature [FORMULA], the contribution to the pressure by the coronal electron gas can be neglected.

[FORMULA] provides a boundary condition for the pressure at the top of our Comptonizing layer. The height [FORMULA] 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 [FORMULA]. The choice of [FORMULA] is discussed in Sect. 3.2.

If [FORMULA] is the geometric height at [FORMULA], pressure balance between the Comptonizing layer and the underlying cool disk requires that [FORMULA]=[FORMULA]. 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 [FORMULA]. 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, [FORMULA], and iterating until we get the right value for which the pressure condition condition at [FORMULA] 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 [FORMULA], the pressure at mid-plane is given by [FORMULA] and the scale height is [FORMULA].

The vertical hydrostatic equilibrium calculated from the top of the disk to the mid-plane yields

[EQUATION]

where

[EQUATION]

is the local Kepler angular velocity.

Together with the equation for the scattering optical depth

[EQUATION]

one obtains a differential equation describing the pressure profile as a function of optical depth for any temperature profile.

[EQUATION]

Eq. 14 is solved via a forth order Runge-Kutta method. We have abbreviated

[EQUATION]

where [FORMULA] denotes the electron scattering opacity, which for ionized hydrogen is [FORMULA]. This calculation is done for each time step to account for the new temperature profile after each cooling/heating step.

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 19100
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