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

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3. Heating and cooling processes of the model

3.1. Proton illumination of the accretion disk

As mentioned in Sect. 2.2 we equal the energy flux [FORMULA] of the protons into the disk with the viscously liberated energy at the radius r (Eq. 7).

Now consider a fast proton from the corona with temperature [FORMULA] entering the disk and moving through it on a straight trajectory. This is a valid approximation as the protons do not change their path considerably until the very last scattering, where they have already lost almost all of their energy in excess to the electrons.

In a time [FORMULA], such a proton with energy E will lose, through electronic collisions, an amount of energy [FORMULA],

[EQUATION]

where [FORMULA] is the electron density of the plasma and [FORMULA] denotes the velocity of the proton. The stopping power [FORMULA] of a plasma at temperature [FORMULA] is taken from the results discussed by Spitzer (1962) and Ryter et al. (1970).

[EQUATION]

Here [FORMULA] is the Coulomb logarithm, [FORMULA] is the Thomson cross-section, [FORMULA] and [FORMULA] are the usual error function and its derivative, [FORMULA] is the angle between the disk plane and the proton velocity, and [FORMULA] is the proton relative energy. This formula holds for non relativistic electron and proton temperatures.

The energy deposition of a proton moving at an angle [FORMULA] with respect to the vertical, per unit of vertical Thomson depth, is

[EQUATION]

The stopping power of a plasma is plotted in Fig. 2. The stopping power [FORMULA] decreases with increasing electron temperatures. Once a cold layer is heated by the protons the stopping power of this layer decreases and the protons deposit their kinetic energy in greater optical depths, i.e. they penetrate deeper into the accretion disk. This limits the increase of the electron temperature and is one of the factors which regulates it to a fairly narrow range (the other being the temperature dependence of the Comptonization process). In a steady state the heating by the incident protons is balanced at each depth by the cooling through inverse Compton scattering of soft photons. We calculate the approach to equilibrium of the model in a time dependent way. At each time step the penetration of a Maxwellian distribution of protons of temperature [FORMULA] incident on the top of the layer is computed from the temperature [FORMULA] and electron density [FORMULA]. This yields the heating rate [FORMULA] within the layer.

[FIGURE] Fig. 2. Stopping power [FORMULA] vs. proton relative velocity x for different electron temperatures. The electron density is [FORMULA].

3.2. Comptonization in a plane parallel disk

The only radiation process explicitly included in our calculations is electron scattering. This is a good approximation in the proton-heated layer, but in the cooler layers below the production of soft photons by bremsstrahlung and atomic processes would have to be considered. Instead, we replace the gradual soft photon production by a complete thermalization at the nominal base [FORMULA] of the Comptonizing layer. Thus, at this depth, the downward flux [FORMULA] of (Comptonized) photons is assumed to be converted into a black body flux [FORMULA] of upward soft photons.

Since the results depend somewhat on this thermalizing boundary (see Fig. 3), we need a recipe to get a sensible value for its depth. We do this by computing after each time step the integrated free-free emission [FORMULA] over the depth [FORMULA] of the model. Since the thermalizing lower boundary is assumed to represent this gradual photon production, we choose [FORMULA] such that

[EQUATION]

so that the free-free emission, ignored in the Comptonization calculations, matches the soft photon input [FORMULA] assumed at the base. The free-free luminosity [FORMULA] (in erg s-1 cm-2) of the layer [FORMULA] (physically extending from [FORMULA] to [FORMULA]) is calculated from the bremsstrahlung emissivity as given by Rybicki and Lightman (1979)

[EQUATION]

[FORMULA] is the frequency average of the velocity averaged Gaunt factor, which we set to 1.1. We have thus simplified the gradual thermalization with depth through free-free emission by a step at [FORMULA].

[FIGURE] Fig. 3. Spectra for a black hole candidate, as a function of the assumed input depth [FORMULA] of the soft photons. At high [FORMULA] the signature from the blackbody photons completely disappears, whereas at an input depth of [FORMULA] the spectrum is dominated by the soft blackbody bump of unscattered photons.

The electron cooling rates through Comptonization in the heated layer are computed through the Monte Carlo method. This also yields the spectrum emitted at the top of the model and the downward flux [FORMULA] of hard photons incident on the thermalizing boundary, which determines the soft photon input (see Eq. 19).

The method of the Monte Carlo simulation is described in great detail in Pozdnyakov et al. (1983). The algorithms presented there are valid for arbitrary geometries, but rather slow. For speed we have specialized them for use in an inhomogeneous plane parallel medium, i.e. with a one-dimensional stratification of electron densities and temperatures. The density stratification is determined by hydrostatic balance (see Sect. 2.2).

The input photons are created at the lower boundary [FORMULA] with frequency [FORMULA]. The initial photon energy [FORMULA] is selected from a blackbody distribution with a temperature obtained from Eq. 19. To each photon we assign a statistical weight w. The photons are emitted isotropically into the upper half space starting with the weight [FORMULA] (Pozdnyakov et al. 1983). The optical depth [FORMULA] along the trajectory of the photon of frequency [FORMULA] to the upper/lower boundary of the slab is

[EQUATION]

where [FORMULA] is the physical distance to the corresponding boundary, [FORMULA] is the local electron density and [FORMULA] the mean scattering cross section averaged over the electron velocity distribution of temperature [FORMULA] at the position [FORMULA]. To compute the mean cross section we use the Compton cross section given by the Klein-Nishina formula

[EQUATION]

where

[EQUATION]

Here [FORMULA] denotes the direction of the photon, v the velocity of the electron, [FORMULA] is the Lorenz factor and [FORMULA] is the classical electron radius.

Now we follow the photon trajectory form the moment of emission until the photon leaves the slab at the upper/lower boundary. The probability that the photon leaves unscattered is

[EQUATION]

where the index [FORMULA]... denotes succeeding scatterings. The quantity [FORMULA] of the i-th scattering represents the transmitted fraction of the photon and is recorded to calculate the escape spectra. The portion [FORMULA] of the photon remains in the slab and undergoes the i+1-th scattering. The location where the photon scatters is determined by a random number. The velocity of the scattering electron is modeled by a relativistic Maxwell distribution corresponding to the temperature at the scattering position. We follow the photons until w becomes smaller than a certain minimum value [FORMULA]. For the calculations reported here we use 60000 photons and [FORMULA].

The process of scattering is also calculated by the Monte Carlo method described in Podznyakov et al. (1983). The photon energy [FORMULA] after a scattering is given by

[EQUATION]

Thus we obtain the change of energy of an electron in its reference frame through scattering with a photon of weight [FORMULA] by

[EQUATION]

at a position [FORMULA], where the scattering takes place. By recording the positions and energy differences of all scatterings of every photon we can calculate the electron cooling rates [FORMULA] within the slab.

3.3. Energy balance from heating and cooling

We start our calculations with an isothermal slab in hydrostatic equilibrium according to Sect. 2.1 and 2.2. After each Comptonization step we obtain the heating and cooling rates [FORMULA] and [FORMULA] within the slab. The time step in our simulation is adjusted to the shortest energy exchange time scale occurring in the calculation. The energy change (in erg cm-3) after the time step [FORMULA] is then

[EQUATION]

Since heating or cooling of the layer takes place approximately isobarically the temperature change due to [FORMULA] is

[EQUATION]

where [FORMULA] is the enthalpy. With the new temperature profile we can update the hydrostatic disk structure after Eq. 14 and start a new heating/cooling step. After about 200 iterations we obtain an equilibrium state where the Coulomb heating is balanced by the Compton cooling and the temperature and density structure as well as the spectra have reached a stationary state.

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 19100
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