## 3. Heating and cooling processes of the model## 3.1. Proton illumination of the accretion diskAs mentioned in Sect. 2.2 we equal the energy flux
of the protons into the disk with
the viscously liberated energy at the radius Now consider a fast proton from the corona with temperature 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 , such a proton with
energy where is the electron density of the plasma and denotes the velocity of the proton. The stopping power of a plasma at temperature is taken from the results discussed by Spitzer (1962) and Ryter et al. (1970). Here is the Coulomb logarithm, is the Thomson cross-section, and are the usual error function and its derivative, is the angle between the disk plane and the proton velocity, and 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 with respect to the vertical, per unit of vertical Thomson depth, is The stopping power of a plasma is plotted in Fig. 2. The stopping power 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 incident on the top of the layer is computed from the temperature and electron density . This yields the heating rate within the layer.
## 3.2. Comptonization in a plane parallel diskThe 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 of the Comptonizing layer. Thus, at this depth, the downward flux of (Comptonized) photons is assumed to be converted into a black body flux 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 over the depth of the model. Since the thermalizing lower boundary is assumed to represent this gradual photon production, we choose such that so that the free-free emission, ignored in the Comptonization
calculations, matches the soft photon input
assumed at the base. The free-free
luminosity (in erg s 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 .
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 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
with frequency
. The initial photon energy
is selected from a blackbody
distribution with a temperature obtained from Eq. 19. To each
photon we assign a statistical weight where is the physical distance to the corresponding boundary, is the local electron density and the mean scattering cross section averaged over the electron velocity distribution of temperature at the position . To compute the mean cross section we use the Compton cross section given by the Klein-Nishina formula where Here denotes the direction of the photon, v the velocity of the electron, is the Lorenz factor and 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 where the index ... denotes
succeeding scatterings. The quantity
of the The process of scattering is also calculated by the Monte Carlo method described in Podznyakov et al. (1983). The photon energy after a scattering is given by Thus we obtain the change of energy of an electron in its reference frame through scattering with a photon of weight by at a position , 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 within the slab. ## 3.3. Energy balance from heating and coolingWe 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
and
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 Since heating or cooling of the layer takes place approximately isobarically the temperature change due to is where 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. © European Southern Observatory (ESO) 2000 Online publication: October 30, 19100 |