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Astron. Astrophys. 363, 455-475 (2000)

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7. Relativistic corrections

From the qualitative considerations on the temperature of the plasma of a nuclear AGN wind mentioned in Sect. 5, we do expect very high temperature values in the inner region of the wind. Indeed, the condition for the temperature T to reach values that are to be considered relativistic for the electron component of the plasma is that [FORMULA], i.e. [FORMULA] K; therefore, in the light of the discussion in Sect. 5, the electron component of the wind plasma can easily be characterized by relativistic temperatures, and such values are not much consistent with the use of totally non-relativistic equations, although the wind flow speeds are indeed sub-relativistic. As a consequence, we have appropriately modified the wind equations as it is described in the following of this section, and all the solutions that are explicitly described and shown in the present paper are those pertaining to the relativistically corrected problem.

In fact, although non-relativistic fluid motion is expected, relativistic temperatures require a relativistic treatment of energy density and energy exchanges, that is the use of a relativistically correct energy equation (see Landau & Lifshitz 1959).

In case of sub-relativistic fluid motion, the energy conservation equation can be formally written in the same way as for the non-relativistic case, as it is the case for the momentum equation (to [FORMULA] terms, which are of course negligible for our problem), that is

[EQUATION]

where [FORMULA], [FORMULA] and [FORMULA] are, respectively, the total energy gain and total energy loss rates per unit volume, and where [FORMULA] is the total particle energy density; Eq. (18) is another form of the energy equation, and, in the totally sub-relativistic case, Eq. (3) in Sect. 3 can be easily derived from it. Here the difference with the non-relativistic case lies in the different expression for the relativistic electron energy density [FORMULA] and its relation to the thermal pressure p; also, it is important to stress that here [FORMULA] represents only the kinetic energy density of the relativistic electron component, due to thermal motions, that is it does not include the rest energy density.

Notice that here we only have to deal with relativistic electrons, since at the relevant temperature values the proton component of the plasma is still non-relativistic, and here we suppose that the electron and proton components have the same temperature. The total gas pressure expression for a completely ionized hydrogen gas is thus the same as in the non-relativistic case, namely, [FORMULA], that is the sum of the electron pressure and the proton pressure, where we have defined the non dimensional temperature parameter

[EQUATION]

As for the energy density, we have the usual non relativistic expression for the proton component

[EQUATION]

whereas the relativistic electron ("kinetic") energy density is

[EQUATION]

where [FORMULA] is the pressure of the electron component in the ionized gas, and [FORMULA] is a function such that (see Björnsson & Svensson 1991, BS91 in the following)

[EQUATION]

where [FORMULA] and [FORMULA] are modified Bessel functions of order 1 and 2 respectively and of argument [FORMULA], and the function [FORMULA] has the following limit values

[EQUATION]

We can now express the relation between the total energy density [FORMULA] and the total gas pressure p as follows:

[EQUATION]

where we have defined

[EQUATION]

With the definitions above, and using the continuity equation, which is still Eq. (1) since the fluid motion is sub-relativistic, one gets to a form of the energy equation perfectly analogous to Eq. (3), which is correct for the totally non-relativistic case:

[EQUATION]

this equation, together with Eqs. (1),(2) and (4), provides the system of hydrodynamic equations correct for relativistic electron temperature regime.

These equations can be combined to obtain a system of two equations in two unknowns, which are essentially the flow velocity and the temperature, representing a wind that is flowing at sub-relativistic speed, but whose plasma may be characterized by relativistic temperatures for the electronic component. Defining the following quantities

[EQUATION]

we get the two equations for v and T

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

We note again that, when the heating mechanism does not imply an associated momentum deposition, the factor [FORMULA], multiplying [FORMULA] in the definition of [FORMULA] (Eq. (31)), reduces to unity.

The above representation of the problem is sufficient to our purposes. In fact, we are interested in wind solutions whose critical point temperatures, [FORMULA], are still in the non-relativistic range for the electrons and therefore we proceed by looking for the critical point of the wind in the non-relativistic regime, that is still using Eq. (16); then we obtain the wind solution, first, that is close to the sonic point, by integrating equations (14) and (15), and then, going inwards to a region in which the temperature increases to finally reach the relativistic range for the electrons, by solving the system of relativistically corrected Eqs. (33) and (34), with the solution of the first step integration (non-relativistic equations) used as "initial" condition.

As a matter of fact, in writing down explicitly the expressions for energy gain [FORMULA] and losses [FORMULA] (energy density /time, dimensionally) we have to account for the fact that the electrons are in a relativistic regime; this turns out to change the rate at which energy is lost by the hot gas to the radiation field through Inverse Compton process, which is now [FORMULA] (see Krolik et al. 1981), as well as the appropriate expression for the bremsstrahlung loss rate (see BS91), that turns out to be higher than what it would be if the non-relativistic, high temperature loss rate would be extrapolated up to relativistic temperatures. Thus, in the relativistic regime for electron temperatures, these rates turn out to have the following expressions:

[EQUATION]

where for ([FORMULA]) we have taken into account the relativistic bremsstrahlung cooling rate for proton-electron interactions as given by BS91, [FORMULA] is the fine-structure constant and the function [FORMULA] is given explicitly by BS91 as

[EQUATION]

We can now solve the wind problem consistently. However, to successfully integrate the relativistically correct system of equations and solve for the wind in its internal region, closer to its origin, we find that it is also necessary to modify the additional parameterized heating rate [FORMULA], so that the portion of it which is dependent on T (see Eq. (8)) now is no longer simply linear with respect to T itself, but is instead [FORMULA] (similarly to the modification for "relativistically" correct Compton losses of the wind plasma). With this last change and choosing the values of the parameters appropriately, we can solve the equations in a consistent way and obtain physically sensible behaviour of the wind we are modeling.

We would like to stress at this point that all the considerations and dimensional analysis of Sect. 5 still apply for the relativistically correct case as well; in fact, in the energy equation (25), substituting non-relativistic Eq. (3), [FORMULA] factors are of order unity or just a little larger than unity.

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© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
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