## 7. Relativistic correctionsFrom 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 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 terms, which are of course negligible for our problem), that is where ,
and
are, respectively, the total energy
gain and total energy loss rates per unit volume, and where
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 and its relation to
the thermal pressure 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, , that is the sum of the electron pressure and the proton pressure, where we have defined the non dimensional temperature parameter As for the energy density, we have the usual non relativistic expression for the proton component whereas the relativistic electron ("kinetic") energy density is where is the pressure of the electron component in the ionized gas, and is a function such that (see Björnsson & Svensson 1991, BS91 in the following) where and are modified Bessel functions of order 1 and 2 respectively and of argument , and the function has the following limit values We can now express the relation between the total energy density
and the total gas pressure where we have defined 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: 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 we get the two equations for We note again that, when the heating mechanism does not imply an associated momentum deposition, the factor , multiplying in the definition of (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, , 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 and losses (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 (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: where for () we have taken into account the relativistic bremsstrahlung cooling rate for proton-electron interactions as given by BS91, is the fine-structure constant and the function is given explicitly by BS91 as 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 , so that
the portion of it which is dependent on 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), factors are of order unity or just a little larger than unity. © European Southern Observatory (ESO) 2000 Online publication: December 11, 2000 |