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

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6. Wind equations and critical point definition

From the system of hydrodynamical equations written in Sect. 3, we can derive a system of two first order differential equations in two unknowns that we define as combinations of the original physical quantities, that is a) the sound speed [FORMULA], such that

[EQUATION]

where [FORMULA], and b) the Mach number M,

[EQUATION]

The wind equations we obtain are the following:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where we have set [FORMULA], and we remind that [FORMULA], so that [FORMULA]. Similarly to what noticed at the end of Sect. 3 regarding the energy equation (3), the factor [FORMULA] multiplying [FORMULA] in both Eqs. (14) and (15) reduces to unity when the heating mechanism does not imply an associated momentum deposition [FORMULA].

Since we are interested in wind-type flows, we are looking for transonic solutions, and specifically for those solutions starting from sub-sonic flow speed and getting to super-sonic; this of course means that the Mach number of the flow has to pass through the critical value [FORMULA], where the flow speed equals the local sound speed. To have a physically meaningful solution, we have to ask that, in the radial position at which it is [FORMULA], the radial derivatives of the physical unknowns [FORMULA] and [FORMULA] maintain finite values; from the structure of the equations above, it is then clear that this condition is only attained if the right hand side of both the equations is set equal to zero at that same location; this in turn defines an implicit relation for the radial position at which the transonic transition takes place in the flow, that is the sonic (or critical) point, [FORMULA],

The sonic point position [FORMULA] is therefore obtained by the following implicit equation:

[EQUATION]

where the subscript "c" indicates that the quantity is evaluated at the sonic point [FORMULA], [FORMULA], and [FORMULA]. From mass conservation equation (Eq. 1), one gets [FORMULA]; defining the dimensional constant as [FORMULA], we have [FORMULA], and if there are no mass sources along the wind way, so that [FORMULA], the constant [FORMULA] effectively represents the density of the wind at the sonic point. Again, in Eq. (16) the factor [FORMULA] reduces to unity for the cases in which there is no momentum deposition contribution associated with the heating process represented by the parameterized heating rate [FORMULA].

The critical point position is a function of general physical parameters of the problem (such as the mass of the central black hole [FORMULA], the luminosity of the central source [FORMULA] and the Compton temperature [FORMULA]) of other parameters more specific to the chosen representation of the physical mechanisms determining heating and cooling rates, of the form of the non-dimensional mass deposition function [FORMULA], and of the temperature at the sonic point itself, [FORMULA]. The specification of these quantities defines the sonic point position. Notice that [FORMULA] does not depend on the value of [FORMULA], as it is clear from the non-dimensional version of the sonic point equation (Eq. (B.16)).

To build up the transonic solution we need then to integrate the wind equations starting from the sonic point, both inwards (towards the base of the wind at [FORMULA]) and outwards, up to the region where the wind has reached an asymptotic flow speed. This is actually possible by first expanding the equations around the sonic point to evaluate the derivatives of M and [FORMULA] at [FORMULA], as "initial" conditions for the integration. Appendix B shows the procedure and parameters for deriving non-dimensional equations to be numerically integrated.

To a first approximation, for the sub-relativistic wind solutions we are interested in and in the case in which the radiative cooling rate (in the region of the sonic point at least) is much smaller than the other energy exchange rates, i.e. ,
[FORMULA] where the subscript "c" implies the quantities are evaluated at the critical point [FORMULA] and we refer to Eq. (7) for the generic expression of energy exchange rates per unit volume, we obtain

[EQUATION]

In order to obtain a wind solution, typically the dominant term is the first one, that is

[EQUATION]

where again [FORMULA], showing the dependence on the temperature value, while the energy rate balance term in Eq. (17) can be significant to move the sonic point farther away or closer to the central black hole, depending on whether, respectively, energy gain is larger than losses or viceversa.

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© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
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