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

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Appendix A: qualitative issues on wind energetics

A.1. Energetics and momentum deposition

The following analysis refers to the case in which the energy deposition process implies a corresponding momentum deposition contribution (see Sects. 3 and 5). Since we are interested in analyzing winds with internal temperatures much lower than the huge value of [FORMULA] we have estimated above, we look for the consequences on the energy equation of the limit conditions (10) and (11) of Sect. 5. In the constant mass flux limit, and supposing that, given the first of the conditions above, we can approximate with [FORMULA] the whole of the energy balance (gain minus loss rates) appearing on the right hand side of the energy equation (Eq. (3)), this same equation, divided by c, can be rewritten, close to the wind origin, as

[EQUATION]

Since the second of conditions (10) must apply (and the more so [FORMULA], because the flow is sub-relativistic), this equation leads to the condition

[EQUATION]

or, neglecting the factor of order unity [FORMULA],

[EQUATION]

From this relation we can conclude that, allowing for small [FORMULA] and large momentum deposition [FORMULA], we can expect very large gradients in density in the inner wind region, that, no matter what is the value of the density itself at the distance at which the wind becomes supersonic, lead to a very large density in the internal region. This in turn can easily imply a total optical depth to scattering that is by far exceeding unity (by orders of magnitude) and resulting total power exchanges in the energy deposition and loss processes that are very large with respect to the central source luminosity; as a consequence, in this case the corresponding solution would turn out to be unacceptable with respect to our criteria, and inconsistent as for our treatment.

A.2. Energy balance

In Sect. 8.1 we have discussed the importance of an additional heating source distributed in distance along the wind way for a wind model, mentioning the well known case of simple polytropic models, corresponding to a situation in which the energetic balance ([FORMULA]) is in favour of energy gain. Nevertheless, we do expect our wind model energetics to be more complex than that of a wind resembling a simple polytropic model. In fact, exploring the parameter space to understand and identify the conditions to effectively get well-behaved solutions, we have found that the parameters chosen to define [FORMULA] are pretty critical and determine a rather narrow range, if we want to keep our solution in the physically reasonable regime (see Sect. 5). Moreover, an inspection of the energetics shows that for our model the balance [FORMULA] can locally be [FORMULA], although, in general, when this is verified, the relative value of the difference is quite small, i.e. [FORMULA].

Our model is closer to a polytropic-mimicking wind in its supersonic, external region, where [FORMULA] and, in the asymptotic limit, [FORMULA]; in this case, in fact, it is always [FORMULA], and also, due to the very small densities reached in this region, it can be [FORMULA]. Indeed, for a model with constant mass flux, combining Eqs. (2) and (3) (non-relativistic energy equation is of course appropriate for the description of this supersonic region) in the asymptotic limit, in which [FORMULA] and for [FORMULA], we get

[EQUATION]

In this region, even for negligible [FORMULA], it is [FORMULA], since at these distances, being [FORMULA] and the density of the wind very low, [FORMULA]; however, in this case, the wind temperature tends to decrease, since the expansion cooling term ([FORMULA]) is decreasing more slowly than [FORMULA]. Thus, to avoid a too strong decrease of the temperature in this external region or to maintain a sort of asymptotic, more or less constant (or slowly decreasing) temperature, the contribution of [FORMULA] is again necessary, and it must be [FORMULA] or a close power of r.

Appendix B: non-dimensional parameters and equations

In this Appendix, we explicitly describe our choice for normalization of the physical quantities appearing in the wind equations derived in Sect. 3.

Apart from the Mach number [FORMULA], which is non-dimensional due to its own definition, all the physical quantities appearing in the wind equations are dimensional, and we have to reduce them to the corresponding non-dimensional ones with a normalization appropriate for our problem. A natural choice for a wind problem, in which the sonic point is a critical starting point for the resolution of the problem itself, is to build up the normalization of the problem making use of the chosen physical parameters at the sonic point itself, namely [FORMULA], and [FORMULA], which is directly the value of the density at the critical point only if [FORMULA], what is for sure verified when no mass sources along the wind are taken into account. From the temperature at the critical point, [FORMULA], we obtain the sound speed at the critical point itself, since

[EQUATION]

where [FORMULA]. We choose however to normalize distances with the gravitational radius [FORMULA], to account for a basic ingredient in the problem such as the central black hole gravitational pull. We thus define the following non-dimensional quantities:

[EQUATION]

from the definitions given in Sect. 3, the non-dimensional mass flux per steradiant can be written as

[EQUATION]

so that the non-dimensional wind density can be recovered from the results of integration for [FORMULA] and [FORMULA] as

[EQUATION]

where, of course, [FORMULA].

As for the energy exchange rates per unit volume, for those that can be written in the form defined by Eq. 7 in the paper ([FORMULA], [FORMULA], [FORMULA], and the quantity [FORMULA], defined by Eq. 5 and related to radiation pressure momentum deposition in the wind), we can isolate a dimensional factor [FORMULA] so that the non-dimensional corresponding quantity can be expressed as

[EQUATION]

that is

[EQUATION]

where [FORMULA] is the non-dimensional version of the characteristic rate function [FORMULA]. To be more specific, we have

[EQUATION]

where [FORMULA] and [FORMULA] are two non-dimensional parameters,

[EQUATION]

and other quantities appearing in the definitions above have been introduced in the sections referring to the heating and cooling rates in the text.

The same dimensional factor can be of course used to obtain the non-dimensional radiative cooling rate per unit volume from [FORMULA], for which we have in fact

[EQUATION]

where the quantity in square parentheses is the non-dimensional rate per unit volume.

B.1. Normalized form of non-relativistic wind equations

Making use of the definitions above, we obtain the non-dimensional form of Eqs. (13) and (14):

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where the non-dimensional functions [FORMULA], [FORMULA], [FORMULA], and [FORMULA] represent the various rate functions [FORMULA] defined generically by Eq. (B8), corresponding to the energy exchange rates per unit volume labeled with the same subscript; [FORMULA] is the normalized cooling function as defined by Eq. (B.13)

Also, the normalized sonic point ([FORMULA]) implicit equation is corresponding to Eq. (15)

[EQUATION]

B.2. Normalized form of relativistically correct wind equations

Relativistically corrected equations of Sect. 7 have to be reduced to their non-dimensional form. Our choices for normalization of the various dimensional quantities are the same defined at the beginning of the present Appendix B, except for velocity normalization; here we define a non-dimensional velocity by making use of the sound speed at the critical point, [FORMULA], which is still given by Eq. (B.1), so that

[EQUATION]

notice that, since,as we have mentioned in Sect. 7, the temperature value at the critical point for the solutions we are interested in is always chosen in the non-relativistic range, the definition of the sound speed at the critical point is always the one given in Eq. (B.1) and it keeps it physical meaning in any case. With this notation, the expression for the normalized gas density now turns out to be

[EQUATION]

As for the heating and cooling rates, normalizations factors are the same of course, and the only difference we have here is in the definition of the normalized rate functions for [FORMULA] and [FORMULA] (see Sect. 7), so that in the relativistic temperature regime, we define

[EQUATION]

where [FORMULA]. With the definitions above, and setting also

[EQUATION]

normalized equations for relativistic electron temperature regime are the following:

[EQUATION]

[EQUATION]

where [FORMULA] and [FORMULA] are non-dimensional and, in terms of normalized quantities can be written explicitly as follows

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
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