## 4. Energy equation termsWe 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 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 populationWe 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 rateAs for the radiative cooling rate
, we have started using a linear
approximation of a numerically evaluated Cooling Function
(in
erg cm 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 fieldSince 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, where is the Compton temperature, defined as , with , and spectral luminosity of the central source (see Krolik et al. 1981, Begelman et al. 1983). ## 4.4. Parameterized heating rate contributionA 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 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 ( 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 |