3 Description of the solutions

Figure 1: Solution plane of the wind equations. The gray region corresponds to the domain where the Bernoulli function is well defined. In the sub-Alfvénic region (y>1), it is limited by the light surface. The thick line (Alfvén mode) separates the sub- and the super-Alfvénic modes. The dashed line indicates the slow (y>1) and fast (y<1) mode Mach curves and the dotted line the gravitational throat curve. The gravitational throat curve is very close to the fast mode Mach curve for this particular case. The slow (S) and fast (F) critical points are the intersections of the Mach and throat curves. The Alfvén point is indicated by A.

Figure 2: A few level contours of the Bernoulli function are shown. The physical solution (thick line) starts in the sub-Alfvénic region, crosses the slow mode Mach curve at the slow critical point (S) then reaches the Alfvén point (A) and enters the super-Alfvénic region where it crosses the fast mode Mach curve at the fast critical point (F). This calculation has been made for a Schwarzschild black hole with m=0.01 and the parameters ( $\gamma =4/3$, $\Theta '=0.04$ and $\omega '=0.5$) have been chosen so that the different points are well separated.

3.1 Properties of the Bernoulli function

As described by Sakurai (1985) in the classical case, the Bernoulli function $\tilde{H}(x,y)$ has the following properties:

1.

At y=1 ( $\rho=\rho_{\rm A}$) $\tilde{H}(x,y)$ diverges if $x \ne 1$ and remains finite if x=1 ( $r=r_{\rm A}$). It means that all solutions going from the sub-Alfvénic region (y>1) to the super-Alfvénic region (y<1) must pass through the Alfvén point x=y=1 which is the only "hole'' in the infinite "wall'' y=1.

2.

Two important curves in the x-y plane are the so called slow/fast mode Mach curve defined by

$$\frac{\partial\tilde{H}}{\partial y}(x,y) = 0$$ (61)

(the slow mode corresponds to the sub-Alfvénic region y>1 and the fast mode to the super-Alfvénic region y<1) and the gravitational throat curve (so called by analogy with the de Laval nozzle) defined by

$$\frac{\partial\tilde{H}}{\partial x}(x,y) = 0.$$ (62)

At the intersections of these two curves, the function $\tilde{H}(x,y)$ is locally flat, corresponding to an X-type critical point (or O-type point).

3.

All level contours of $\tilde{H}(x,y)$ going from $y\to +\infty$ in the sub-Alfvénic region to $y\to 0^+$ in the super-Alfvénic region must cross these critical lines. They have to cross them simultaneously to be not interrupted. This means that the solution must pass through two critical points defined as the slow (respectively fast) critical point $x_{\rm s},y_{\rm s}$ (resp. $x_{\rm f},y_{\rm f}$), intersection of the slow (resp. fast) mode Mach curve and the gravitational throat curve. This imposes two new conditions for the solution:

$$\tilde{H}\left(x_{\rm s},y_{\rm s}\right) = E' \qquad{\rm and}\qquad \tilde{H}\left(x_{\rm f},y_{\rm f}\right) = E'.$$ (63)

Figure 1 shows the x-y plane for a particular choice of the parameters with the slow/fast mode Mach curve, the gravitational throat curve. Different level contours of the function $\tilde{H}(x,y)$ are shown in Fig. 2. The solution is one of the level contours and passes through the slow and fast critical points. These figures are very similar to Fig. 1 of Sakurai (1985) obtained in the classical case. Notice that Begelman & Li (1994) restrict their study to cold flows. A consequence of this assumption is that a purely radial flow is a singular case where the fast critical point is at infinite radius. The fast point moves inward to finite radii only if the flow diverges over-radially, like in the situations we will study in the next section. This is not a necessary condition in our more general model. Because we include gravity and thermal pressure the fast point is always located at finite distances even in a purely radial flow.

3.2 Classification of wind solutions by dimensionless parameters

Figure 3: Parameter space: there are no wind solutions in the lower left part ("Static'': the limits are only approximatively indicated). The upper left part ("Centrifugal'') corresponds to winds where the thermal pressure is negligible whereas in the lower right part ("Thermal'') the centrifugal force plays no essential role. For a Schwarzschild metric with m=0.01 (so $\omega '<0.98$ and $\omega <98$) we have computed several series of solutions with constant terminal Lorentz factor $\Gamma _{\infty }=1.01$, 1.5, 10, 100 and 1000 (solid lines). For $\Gamma _{\infty }=10$ we show also the case where m=0.1 (dashed line: $\omega <8$) and m=0.001 (dotted line: $\omega <998$). The dotted horizontal line corresponds to $\xi =1$ (initially the electromagnetic and the matter energy fluxes are equal) for m=0.01. For m=0.01 and $\Gamma _{\infty }=1.01$ and 1.5 we indicate at different positions the value of the efficiency $\mathit{eff}$ of the magnetic to kinetic energy conversion.

Equation (56) can formally be written as

$$\tilde{H}\left(x,y\,; \beta',\Theta',\omega', m, g, \gamma\right) = E',$$ (64)

where m stands for the parameters defining the metric (see Appendix A for the definition of m in usual cases) and g stands for the parameters fixing the geometry of the flux tubes (i.e. defining $\tilde{s}(x)$). We will for the moment restrict our study to the case where $\tilde{s}={\rm const.}=1$ and we fix the adiabatic index to $\gamma =4/3$ for the rest of this paper. From the two conditions at the slow/fast critical points (63) we can fix two parameters. In practice, following Sakurai (1985), we adjust $\beta'$ so that the value of $\tilde{H}$ is the same at the two critical points and this value gives E'. Therefore all solutions are determined by only three independent parameters: $\Theta '$, $\omega '$ and m. Compared to the classical case studied by Sakurai (1985), there is one supplementary parameter: m. This is due to the presence of a characteristic length scale related to the structure of the space-time (typically the gravitational radius GM/c²) which has no classical counterpart.

The signification of these three parameters is clear: m measures the intensity of the gravitational field, $\Theta '$ gives the strength of the thermal pressure (the pressureless case which is often considered corresponds to $\Theta'=0$) and $\omega '$ measures the strength of the centrifugal force in accelerating the wind, or equivalently the effect of the magnetic field.

In the limit of the flat space-time (Minkowski metric) it is well known that the slow point does not exist anymore and that the solution cannot start at an arbitrary small radius because it cannot cross the slow mode Mach curve without having an infinite derivative . We therefore cannot describe the region near the source but this is clearly because gravity influences the solution significantly at small radii and the Minkowski approximation breaks down. In the case without gravity, as in the classical case, a supplementary parameter ($\beta'$) must be fixed (for instance by fixing the mass flux). In many cases the physical conditions at the basis of the wind are a complex and not well understood question, and some assumptions made everywhere else in the flow are probably not valid here (like the adiabaticity). In this context, just adopting a Minkowski metric and including in the $\beta'$ parameter all the unknown physics fixing the mass flux at the basis of the wind is more elegant than adopting a Schwarzschild (or Kerr) metric and applying the solution up to the source.

Before exploring the three-parameters space we have defined, it is useful to express the relation between m, $\omega '$ and $\Theta '$and more usual physical quantities. If m is the ratio of the gravitational radius of the source over the Alfvén radius, we have

$$m = 0.21\cdot \left(\frac{r_{\rm A}}{10^{6}\,{\rm cm}}\right)^{-1} \left(\frac{M}{1.4~M_\odot}\right).$$ (65)

The angular frequency $\Omega$ can be identified with the rotation rate of the source and fixes the value of $\omega '$:

$$\omega' = 0.11\cdot \left(\frac{\Omega}{10^4\,{\rm Hz}}\right)^2 \left(\frac{r_{\rm A}}{10^6\,{\rm rm}}\right)^2.$$ (66)

The local sound speed is defined by $(c^{\rm s}/c)^2=\gamma P/\rho h$ so that $\Theta '$ can be related to the ratio of the sound speed at the Alfvén point $c^{\rm s}_{\rm A}$ over the speed of light (for $\gamma =4/3$):

$$\Theta' = 1.0\times10^{-2}\cdot \left(\frac{c^{\rm s}_{\rm A... ...ac{1-3\left(c^{\rm s}_{\rm A}/c\right)^2}{0.97} \right)^{-1}.$$ (67)

The three parameters $\Theta '$, $\omega '$ and m being fixed, the solution of the wind equations is simply found by adjusting $\beta'$so that the condition (63) is fulfilled. For each step with $\beta'$ fixed, the slow and fast critical points are determined by a simple Newton-Raphson procedure. The exact expressions of $\tilde{H}$ and its derivatives are used. We have explored in detail the parameter space and the results are presented in Fig. 3. Notice that for a given m, $\omega '$ is limited to the interval

$$\omega'_{\rm min} \le \omega' \le \omega'_{\rm max},$$ (68)

where $\omega'_{\rm max}$ is due to the condition that the Alfvén point lies inside the light surface and $\omega'_{\rm min}$ is non vanishing only in the case of the Kerr metric. In this metric there are no solutions without rotation ($\omega'=0$) because the matter is forced to rotate in the vicinity of the central source. The analytical expressions of $\omega'_{\rm min}$ and $\omega'_{\rm max}$ are given in Appendix A. Here we have considered a Schwarzschild metric with different values of m. We show the results in $\Theta$-$\omega$ coordinates, where $\Theta=\Theta'/m$ and $\omega=\omega'/m$ are the parameters used by Sakurai (1985). Like in the classical case, there are no wind solutions in the lower left part of the plane ("Static'') because neither the centrifugal force (magnetic acceleration) nor the thermal pressure are sufficient to power the wind. The limits of this region are only approximatively indicated. For a pure thermal wind ($\omega=0$) and a Schwarzschild metric, the minimal value of $\Theta$ is given by

$$\Theta_{\rm min} = \left(\gamma-1\right)\frac{\frac{1}{\sqrt{1-2m}}-1}{m},$$ (69)

tending towards $\gamma-1$ for $m\to0$, in agreement with the classical case. In the pressureless case ($\Theta=0$) the minimal value of $\omega$ tends towards (3/2)^3/2 for $m\to0$, which also corresponds to the limit given by Sakurai (1985). The upper left part corresponds to winds where the centrifugal force dominates and the lower right part corresponds to pure thermal winds. In Fig. 3 we have plotted several solutions with constant terminal Lorentz factor $\Gamma _{\infty }=1.01$, 1.5, 10, 100 and 1000 for m=0.01 and in the particular case $\Gamma _{\infty }=10$ we have also plotted the same curves for m=0.1and m=0.01 to show the effect of varying the gravitational field.