4. Energy equation terms
We now describe the meaning and origin of the energy gain and loss terms appearing on the right side of the energy Eq. (3).
It is interesting to notice that, apart from the radiative cooling rate (where n is the wind number density), all the other energy exchange rates (Compton cooling rate included) have a somehow similar structure, of the type
where is the generic energy exchange rate per unit volume appearing in the hydrodynamic equations, and is a function characteristic of the rate we are referring to, depending on the radial distance and possibly on the temperature of the wind gas.
4.1. Possible interaction of thermal wind plasma with a relativistic electron population
We briefly discuss this possibility, since we had originally considered the plausibility of the contribution to the heating of the thermal wind plasma due to collisional (Coulomb-like or even due to collective effects) interaction with a population of non-thermal relativistic electrons, injected at the base of the wind into the thermal plasma. This non-thermal particle component could be reasonably supposed to be generated in the high energy processes close to the black hole and one possible speculation about their origin in the present context could have been related to magnetic reconnection processes in the magnetically active regions of the AGN coronal structure (see Birk et al. 1999 and references therein).
Indeed, other authors dealing with AGN winds (Weymann et al. 1982) or in any case with AGN hot thermal diffuse medium (David & Durisen 1989) had taken into account the interaction with a relativistic particle component as a source of heating for the thermal plasma. However, we have finally chosen to neglect this specific contribution, although somehow appealing, since it is rather easy to show that, in the presence of a strong radiation field, such as the one we suppose emitted from a central source in the AGN and illuminating the wind, Compton interactions with radiation field photons would much more rapidly and substantially deplete the relativistic electron population of its energy, eventually going into the radiation field itself. Thus, on the one hand Compton losses would be the agent of a rapid evolution of the energy distribution function of a non-thermal relativistic component, and on the other hand they would render the contribution to the heating of thermal wind plasma due to collisional interaction with the injected relativistic electrons essentially negligible, even at the wind base. Comparing the energy loss rate for Compton interactions and that for collisional interactions, this is substantially true over the whole energy range (down to Lorentz factors ) of the non-thermal population energy distribution, for the conditions we would expect in a radio-quiet AGN. Therefore, contrary to the previously cited works, we have chosen to neglect this heating source for the wind plasma as both implausible, in the present context, and, especially, ineffective.
4.2. Radiative cooling rate
As for the radiative cooling rate , we have started using a linear approximation of a numerically evaluated Cooling Function (in erg cm3 s-1) for collisional ionization equilibrium, as described in Landini & Monsignori Fossi (1990); the cooling function can be represented linearly in a number of temperature ranges accurately chosen so as to obtain a good approximation of the numerically obtained functional behaviour.
Indeed, for the high temperature range, bremsstrahlung emission dominates the radiative cooling processes, and it turns out that for our high temperature wind plasma no significant difference in the solution is obtained if we just represent the cooling function with a much simpler approximation, only accounting for bremsstrahlung losses and neglecting line cooling contributions. The approximated expression we have used for the radiative cooling function is the following:
Moreover, although this is clearly a much more schematic representation of the cooling function, it actually seems even more appropriate, with respect to the one accounting for line cooling derived from purely collisional ionization equilibrium, for the present AGN context.
4.3. Compton interaction with the central radiation field
Since the thermal medium of the wind is illuminated by the centrally originated ionizing radiation field, we have to take into account the energy interactions between the electrons of the thermal plasma and the radiation field itself. Since our wind is endowed with sub-relativistic velocity, and we require for our solution to be of physical interest that the thermal wind is substantially optically thin to scattering processes, we can consistently use the well known relations giving energy gain and losses for the plasma due to Compton processes (Levich & Syunyaev 1971, Krolik et al. 1981; Begelman et al. 1983). For a central isotropic source of luminosity , we can write the energy loss rate per unit volume as
where is the electron number density of the thermal wind plasma, i.e. the number density of the wind, since our wind is completely ionized, and the energy deposition rate per unit volume as
where is the Compton temperature, defined as
4.4. Parameterized heating rate contribution
A first analysis of possible complete solutions of our problem has shown that maintaining the wind "alive" on a large range of distances, that is obtaining a physical solution of the wind equations, is not at all easy (actually, it is not feasible), unless some other source of heating, possibly distributed all along the wind way, is taken into account. Also, we recall that in the case for an expanding spherical wind, expansion cooling turns out to be significant as to the temperature evolution, and its role can be important especially in the external region of the wind, where the flow velocity tends to be substantially constant. In spherical symmetry, rewriting the left hand side of Eq. (3) so as to make the temperature T of the plasma appear explicitly, the expansion cooling term in the energy balance equation is essentially expressed as , where , and it is of course characteristic of an expanding flow. For a wind it is therefore impossible to maintain a gas temperature determined by the radiation field Compton temperature, , on a large distance range, since in any case the expansion cooling will act so as to decrease the temperature (Krolik & Begelman 1986). In particular, for the external region this implies that an extra heating term should be considered in order to keep a more or less constant value of the temperature itself.
We have therefore verified the necessity of accounting for some heating source for the wind plasma. One possibility is to try to include one or a combination of other possible heating mechanisms in the problem, implying the definition of the corresponding various physical scenarios. Another way around the problem is to try to define a parameterization, as simple as possible, for an additional contribution to the total heating rate, coming from an unspecified physical mechanism, so as to render the complete integration of the wind problem feasible (i.e. , obtaining non-diverging physical quantities), with a final result solution that effectively meets the physical basic requirements for an AGN wind. We have chosen this second method (see also Sect. 8.1), defining . This type of treatment of the problem, introducing an otherwise non-specified heating rate, is actually quite similar to the one adopted by other authors both in the AGN context (see Raine & O'Reilly 1993) and for the study of solar wind models (see Esser et al. 1997). The functional form of this additional heating rate is essentially the sum of a couple of different (i.e., different spectral index and coefficient) power-laws (with respect to the radial distance r from the central black hole), one of which includes the possibility of a power-law dependence on the temperature as well and it reads:
where we need to specify the values of the various parameters, and is the gravitational radius. Unless otherwise specified, the "physically sensible" solutions we have obtained correspond to an exponent for the temperature power-law.
© European Southern Observatory (ESO) 2000
Online publication: December 11, 2000