## 3. Hydrodynamical equations for a wind-type flowWe are dealing with a wind-type outflow, implying that we are
assuming stationarity of the flow as a physically reasonable
approximation; this may be effectively realized since the dynamical
crossing time of the wind region we are interested in (that can be
estimated as , being The description we choose for the problem is a purely
hydrodynamical one, thus totally neglecting any magnetic field
dynamical influence. Also, as mentioned in Sect. 2, we assume
that the wind is basically radial, and it can be described in
spherical symmetry, that is, if The starting equations are thus the usual stationary hydrodynamical equations: together with In the following we define explicitly the notations used and we describe the meaning of the various terms appearing in the equations. Eq. (1) is the steady state continuity equation, generically
written so as to allow for the possibility of inclusion of mass
sources along the wind; is a mass
flux per steradiant, and can be expressed as
, with
a dimensional constant and the
non-dimensional function that takes
into account the dependence of the mass flux on radial distance from
the central black hole, thus allowing for a mass source along the
wind. In the simple case of no mass sources along the wind,
so that Of course, when the mass flux is constant, and there are thus no mass sources in the wind, we have and all the terms multiplied by in momentum equation and energy equation disappear. On the contrary, those terms are intrinsic to the derivation of hydrodynamic equations in case there is mass input in the wind (Bittencourt 1988). For an analogous formulation see also David et al. 1987; notice, however, that, differently from this last work, in our treatment we do not account explicitly for energy gain-loss to the wind due to the input of mass in the wind itself, since at this level we have no specific information about the thermal properties of the input material; instead, we suppose that possible energy exchanges due to mass loading of the wind can be implicitly accounted for by a parameterized heating rate that is discussed in the following, in Sect. 4.4. In momentum equation (2), is the mass of the central black hole, and represents the total momentum deposition rate, that we have written in a general form as the sum of two distinct contributing terms. The first term on the right hand side represents radiation pressure effects, that is radiation force density (force per unit volume) , where in our problem is the radial and only non-vanishing component of the radiation flux vector . In our approximation, it is simply ; in the following, we write with a notation similar to the other possible momentum deposition term, defining which has the same physical dimensions as an energy exchange rate per unit volume, so that in general we can write We allow for a second momentum deposition term () representing the possible contribution due to the fraction of energy gain that goes directly into bulk kinetic energy, when the energy deposition rate (dimensionally energy per unit time per unit volume and explicitly described in the following section) comes from a heating mechanism due to the interaction of the wind thermal plasma with a population of radially directed relativistic particles injected in the wind at its base, in principle similarly to the case examined in Weymann et al. 1982(see also Begelman et al. 1991). We have formally taken into account this contribution to allow for the possibility of such an energy deposition mechanism and in that case, the actual energy deposition rate is , as it appears in our energy equation [Eq. (3)]. Indeed, this can be considered as a limit situation, but other physical mechanisms depositing heat are to some degree likely to transfer some momentum to the flow (see Esser at al. 1997), although possibly less efficiently and with a more complex relation between the energy deposition rate and the corresponding momentum deposition rate. The other limit condition is that in which no momentum deposition is associated with the heating process, and in that case the second term in momentum deposition rate [Eq. (5) or (6)] vanishes and the factor , multiplying in Eq. (3), reduces to unity. In the following, we build up models corresponding to the two limit conditions identified above. Eq. (4) is the equation of state for the wind gas, that we suppose a pure hydrogen gas; in general, for the typical solutions we obtain, the temperature turns out to be quite high, so that the gas is essentially completely ionized, and we can safely assume . © European Southern Observatory (ESO) 2000 Online publication: December 11, 2000 |