5. Qualitative behaviour of winds
In this section, we want to mention a few general considerations relative to the temperature behaviour we can expect for a thermal AGN wind such as the one we are trying to model, accounting for the energetics we have described above.
If we are supposing a wind originating from a central region in which heating is released to the gas and a kind of corona is generated, of course we have to deal with the specific physical processes that come into play, but there are some very general qualitative lines of reasoning that should apply anyway. Close to a massive object, the physical conditions of the gas must compare with the escape velocity in the gravitational field of the object itself (see Begelman et al. 1983); in fact, a hot, nearly hydrostatic, non-magnetized corona can exist at a certain radius , only if its temperature is such that the following condition is fulfilled
where is a gravitational "escape" temperature and it is defined (dimensionally equating thermal energy density to gravitational binding energy density) as
Therefore, if the gas temperature is higher than the gravitational value defined above, the coronal gas sets up an expansion flow, because its pressure is larger than the gravitational energy density at that radius (). In terms of velocity, this same condition can be expressed as
where is the local sound speed at , defined by , and , is the escape velocity at the same radius ; thus, for an expanding coronal flow to exist, it must be
If we suppose , where , and for example, we obtain
that is K. As a consequence, for a wind to emerge from the very central region of an AGN, its temperature at the base must be pretty high. The only way to circumvent this problem (that is to lower the base temperature of the wind) could be to accelerate the plasma, so that the wind does not set up due to thermal expansion only. To this purpose, momentum deposition at the base of the wind is required: the stronger this deposition is, the lower the temperature value that can be attained at the base of the wind.
A dimensional inspection of momentum and energy equations [Eqs. (2) ad (3)] allows an approximate discussion of the respective relevance of energy and momentum deposition. We shall restrict our attention to the case of constant mass flux, for simplicity.
The case for thermal expansion dominating the starting up of the wind has been discussed above, and corresponds to the situation in which the pressure gradient (, where we have set ) at the wind origin is stronger than the gravitational pull. In our momentum equation we have included radiation force momentum deposition as well as the possible contribution of momentum deposition due to those heating processes that imply the interaction with a relativistic and radially directed population of particles, namely ; this second contribution is not to be taken into account, at least in this simple form, when a different heating mechanism is supposed to be at work in the wind plasma (see Sect. 3).
Analyzing Eq. (2) one has to compare the gravitational term (where ) with additional momentum deposition terms, and ), in case it is present.
Radiation force on the wind plasma of course helps lowering the temperature at the base of the wind, but its effects are not very conspicuous, since in any case the ratio between radiation force density and gravitational pull is in our case just , and this ratio, for the range of parameters here explored (which is typical of Seyfert 1s; see Table 1) is in the approximate interval .
Table 1. Gravitational parameters
Therefore, a possibly relevant contribution to counteract gravitational attraction and lower the requirements on the inner temperature of the wind plasma for acceleration and expansion to take place can only be expected in the case in which the heating process is such to produce an associated momentum deposition (see Sect. 3), when this term, , is sufficiently large. In this case, for a wind to be characterized by significantly smaller than , in principle it should be
when this condition is fulfilled we would expect to have a wind set up without a huge value of the thermal pressure gradient, , required. Moreover, again in the case in which there is contribution to momentum deposition due to the heating process, an even more stringent limit condition that can be required to obtain lower values of the inner temperature of the wind is the following:
Adding this condition to the previous one [relation (10)], a dimensional analysis of the energy equation can give more information. We refer to Appendix A for this analysis, and here we just summarize the indications we obtain. In the light of the considerations in Appendix A, when the two conditions (10) and (11) are simultaneously satisfied,we expect solutions that identify a class, or a "regime", characterized by temperatures in the inner region of the wind that are much lower than , large density gradients with strong acceleration of the wind itself, and high inner region density. However, these properties suggest a closer and more attentive inspection of this class of solutions for what regards their "consistency" with respect to our basic requirements for the wind model (see Sect. 2), especially concerning the total optical depth expected for these solutions, that might well be larger than unity (see Sect. 8.1).
© European Southern Observatory (ESO) 2000
Online publication: December 11, 2000