4 Efficiency of the magnetic to kinetic energy transfer

4.1 Expressions of the energy fluxes

To study the efficiency of the winds computed in the previous section, we need to express the different components of the energy flux along the flow:

$$\dot{E}_{\rm matter} \dot{m}\left(-h \frac{{u_t}}{{c^2}}\right)$$ =

$$\dot{m}\left( c^2 + E + h \frac{\Omega}{c} \frac{u_\phi}{c} \right)$$ =

$$-\left( \tilde g_{tt} \tilde{u}^t + \tilde g_{t\phi}\frac{\tilde{u}^\phi}{c} \right)\ \tilde{h}(y)\ \dot{m} c^2,$$ = (70)

$$\dot{E}_{\rm em} \dot{m}\frac{\Phi^2\rho}{4\pi\dot{m}^2} \Omega\frac{{g}_{{t}\phi}^2-{g_{tt}g}_{\phi\phi}}{{c^2}} \left(\Omega u^t - u^\phi\right)$$ =

$$\dot{m}\left( \Omega L-h\frac{\Omega}{c}\frac{u_\phi}{c} \right)$$ =

$$\sqrt{\omega'}x\tilde{\varpi}^2 \frac{\tilde{h}(1)y}{M_{\rm A}^2}... ...( \sqrt{\omega'}x\tilde{u}^t-\frac{\tilde{u}^{\rm\phi}}{c} \right)\ \dot{m}c^2,$$ = (71)

$$\dot{E}_{\rm total} \dot{E}_{\rm matter} + \dot{E}_{\rm em}$$ =

$$\dot{m}\left(c^2 + E + \Omega L\right)$$ =

$$-\left( \tilde g_{tt}(1) + \sqrt{\omega}'\tilde g_{t\phi}(1) \right)\ \frac{E'+1}{M_{\rm A}^2}\ \dot{m}c^2 .$$ = (72)

Notice that the matter part is made up of the rest-mass, kinetic and internal energy of the matter. We define two parameters: the initial baryonic load $\eta$

$$\frac{1}{\eta} = \left. \frac{\dot{E}_{\rm matter}}{\dot{m}c^2}\right\vert _{x_0}$$ (73)

and the ratio of the initial power injected in the electromagnetic field over the initial power injected in the matter

$$\xi = \left. \frac{\dot{E}_{\rm em}}{\dot{E}_{\rm matter}} \right\vert _{x_{0}},$$ (74)

where x₀ is the radius where the wind starts. The value of $\xi$depends only weakly on x₀, and we have arbitrary chosen x₀=6m(or r₀=6r_g). One sees that $\eta$ will be fixed by $\Theta '$ [via the initial value of $\tilde{h}(y)$] whereas $\xi$ depends strongly on $\omega '$. Along the flow, the internal energy is converted into kinetic energy which accelerates the wind. Therefore if there were no magnetic field, the Lorentz factor at infinity would be $\Gamma_\infty=1/\eta$. However, the magnetic field can also contribute to the acceleration (when coupled with the rotation) and depending on the efficiency of the conversion of electromagnetic into kinetic energy, the terminal Lorentz factor can be larger than $1/\eta$, with a maximum value (complete conversion) given by

$$\Gamma_\infty^{\rm max} = \frac{1+\xi}{\eta}\cdot$$ (75)

In reality the conversion will never be complete and will be estimated by the following fraction

$$\mathit{eff} = 1 - \frac{\left.\dot{E}_{\rm em}\right\vert _... ...em}\right\vert _{0}} = \frac{\eta\Gamma_{\infty}-1}{\xi}\cdot$$ (76)

4.2 Inefficient conversion

In the pressureless case ($\Theta'=0$) it is well known that the magnetic to kinetic energy transfer is very inefficient for high terminal Lorentz factors (Michel 1969)

$$\frac{\Gamma_\infty \dot{m} c^2}{\dot{E}_{\rm total}} = \frac{1}{\Gamma_\infty^2}\cdot$$ (77)

The curves for constant terminal Lorentz factor in Fig. 3 show clearly that for highly relativistic winds ( $\Gamma _{\infty }=10$, 100 and 1000) the terminal Lorentz factor $\Gamma_\infty$ is independent of $\omega$ (or equivalently of $\xi$) which means that there can only be a tiny magnetic to kinetic energy conversion. When $\omega$ is very close to the maximal allowed value and the outflow is Poynting flux dominated (corresponding to the case where the Alfvén point is at the light surface radius) this tendency is not valid anymore. In this region the terminal Lorentz factor $\Gamma_\infty$ depends strongly on $\omega$ and is almost independent of $\Theta$ (or equivalently of $\eta$). However, even in this case only a tiny fraction of the magnetic energy is converted into kinetic energy. The converted energy amount is great compared to the initial energy in the matter part and therefore leads to a greater increase in $\Gamma$ and $\dot E_{\rm matter}$ throughout the flow. The efficiency $\mathit{eff}$ of the conversion is maximal in the pressureless case ($\Theta=0$) but is still rather small. In this case the parameters $\eta$ and $\xi$ are given by $\eta \simeq 1$ and $\xi \simeq \dot E_{\rm total}/\dot mc^2 - 1 = \Gamma^{3}_{\infty}-1$ which corresponds to

$$\mathit{eff} \simeq \frac{1}{1+\Gamma_\infty+\Gamma_\infty^2}\cdot$$ (78)

These tendencies are still present in mildly relativistic winds, as can be seen in Fig. 3 for $\Gamma_\infty=1.5$ where we have indicated the evolution of $\mathit{eff}$ along the curve. It is only within the classical limit that the conversion becomes important. This is shown in Fig. 3 (The case $\Gamma _{\infty }=1.01$ ( $v_\infty=0.14\,c$)). Here the efficiency of conversion reaches $\sim$60% in the pressureless case, which is in agreement with the classical study of Sakurai (1985).

4.3 Efficient conversion

All wind solutions showing (except in the classical limit) a very bad efficiency of the electromagnetic to kinetic energy conversion have been calculated for a particular geometry corresponding to $\tilde{s}={\rm const.}=1$ in our dimensionless units. This corresponds to magnetic flux tubes of constant opening angle. Under the assumption that the velocity is purely radial and constant at infinity it is possible to predict analytically the asymptotic behavior of the flow for any kind of geometry $\tilde{s}(x)$:

$$\tilde{u}^t \to \Gamma_\infty,$$

$$\tilde{u}^r \to \sqrt{\Gamma_\infty^2-1},$$

$$\tilde{u}^\phi \to 0,$$

$$y \simeq \frac{\sqrt{\beta'}}{\sqrt{\Gamma^2_{\infty}-1}} \frac{1}{\tilde{s}x^2},$$

so that the asymptotic expressions of the energy fluxes are

$$\frac{\dot{E}_{\rm matter}}{\dot{m}c^2} \to \Gamma_\infty,$$ (79)

$$\frac{\dot{E}_{\rm em}}{\dot{m} c^2} \to \frac{\omega'\sqrt... ...(1)}{M_{\rm A}^2} \frac{1}{v_\infty}\frac{1}{\tilde{s}} \cdot$$ (80)

From the last equation, one sees that the magnetic to kinetic energy conversion depends strongly on $\tilde{s}$. At infinity $\tilde{s}\to 0$ is unphysical because it would mean that the energy diverges. The case where the opening angle is constant at infinity corresponds to $\dot{E}_{\rm em}\to{\rm const.}> 0$ at infinity so that the conversion is not complete and the case where the opening angle diverges ( $\tilde{s}\to +\infty$) gives $\dot{E}_{\rm em}\to 0$ so that the conversion is complete. These results indicate that all models considered in the previous section are inefficient due to a particular choice of the geometry: $\tilde{s}={\rm const}$. This assumption is certainly correct at very large distance from the source but the opening angle may have variations at smaller radii. Equation (80) indicates that every region where the opening angle increases is a region of efficient magnetic to kinetic energy transfer. This is in agreement with the results of Begelman & Li (1994).

To check that the geometry is really the key parameter governing the efficiency of such winds we have computed some models using various laws for the evolution of the opening angle $\tilde{s}(x)$. The results are shown in Fig. 4 and confirm the previous analysis. We have considered a Schwarzschild metric with m=0.01 and a wind model characterized by $\omega'=0.97$ and $\Theta '=0.1$ (so the energy flux is initially dominated by the electromagnetic energy flux). We plot the different energy fluxes and the "Lorentz factor'' $\tilde{u}^{\rm t}$ as well as the geometrical function $\tilde{s}(x)$ we have used in each case. Figure 4a corresponds to the inefficient case $\tilde{s}={\rm const.}=1$. Figure 4b corresponds to the case where $\tilde{s}$increases in a region located between x₁=10 and x₂=18: the magnetic to kinetic energy conversion is immediately better. The efficiency $\mathit{eff}$ increases also in geometries with different shapes (Figs. 4d, f, g, h) and different locations of the $\tilde s>1$-region, provided that this region lies beyond the fast point as shown by Begelman & Li (1994). In this case $\tilde{s}_{\infty}$ is the only relevant quantity which governs $\mathit{eff}$. A $\tilde s>1$-region within the fast point like in Fig. 4e does not increase $\mathit{eff}$ and is similar to the purely radial case.

Figure 4: Effect of the geometry on the efficiency of the magnetic to kinetic energy conversion: we consider a Schwarzschild metric with m=0.01 and a wind solution characterized by $\Theta '=0.65$ and $\omega '=0.74$. This corresponds to an initial energy flux which is dominated by the electromagnetic part: $\xi =3.1$ for all models so that initially $75\%$ of the energy is magnetic. All solutions presented here have $\eta =2\times 10^{-2}$ except for cases e) ( $\eta =1.7\times 10^{-2}$) and h) ( $\eta =1.8\times 10^{-2}$). The slow and fast critical points are located at $x_{\rm s}=3.0\times 10^{-2}$ and $x_{\rm f}=2.0$ except for cases d) ( $x_{\rm f}=1.6$) and h) ( $x_{\rm f}=1.6$). On each figure ${\rm d}\log\tilde{s}/{\rm d}\log{x}$, $\tilde{s}(x)$ and the different components of the energy flux (matter/em and total) are presented as functions of the radius x. The "Lorentz factor'' $\tilde{u}^{\rm t}$ is also shown (dotted line). Three vertical dotted lines show the location of the slow (s), the Alfvén (A) and the fast point (f). Case a) $\tilde{s}={\rm const.}=1$. For this particular choice of the geometry, the conversion is extremely inefficient ( $\mathit{eff}=3.8\times 10^{-4}$) and the terminal Lorentz factor equals $\Gamma _{\infty }=50=1/\eta$. Case b) $\tilde{s}$ increases between x1=10 and x2=18 reaching a maximal slope ${\rm d}\log{\tilde{s}}/{\rm d}\log{x}=1$. The efficiency improves a lot: $\mathit{eff}=0.26$ and $\Gamma _{\infty }=90$. Case c) same as  b) but $\tilde{s}$ increases between x1=1000 and x2=1800 ( x2/x1 is the same). It changes neither the efficiency nor the terminal Lorentz factor. Case d) same as b) but $\tilde{s}$ increases between x1=1.5 and x2=2.7 ( x2/x1 is the same), i.e. before the position of the fast point in the reference solution a). Again the efficiency $\mathit{eff}=0.24$ and the terminal Lorentz factor $\Gamma _{\infty }=88$ are almost unchanged. Notice that the fast critical point has moved to be almost at x1. Case e) same as b) but $\tilde{s}$ increases between x1=0.1 and x2=0.18 ( x2/x1 is the same), i.e. before the Alfvén point. The efficiency is again very low: $\mathit{eff}=1.2\times 10^{-4}$ and $\Gamma _{\infty }=58\simeq 1/\eta$. Case f) same as b) but with a maximal slope of ${\rm d}\log{\tilde{s}}/{\rm d}\log{x}=4$. The efficiency is better: $\mathit{eff}=0.69$ and $\Gamma _{\infty }=157$. Case g) same as b) but the region where $\tilde{s}$ increases is larger: x1=10 and x2=100. Again the efficiency is better: $\mathit{eff}=0.68$ and $\Gamma _{\infty }=156$. Case h) we have considered a case where $\tilde{s}$ increases from x1=0.1 to x2=104 with a maximal slope ${\rm d}\log{\tilde{s}}/{\rm d}\log{x}=0.4$. Almost 90% of the magnetic energy is converted into kinetic energy ( $\mathit{eff}=0.88$) so that $\Gamma _{\infty }=206$.

Figure 5: The influence of the geometry, gravitation and thermal energy upon the electromagnetic energy conversion. The solid, dotted and dashed lines correspond to different gravitational field strengths, m=0.001, m=0.01 and m=0.1. a) The case for $\omega '=0.95\,\omega '_{\rm max}$, $\Theta '=0.01$ which is a cold Poynting-flux dominated outflow. b) Thermal energy dominated fireball: $\omega '=0.1\,\omega '_{\rm max}$, $\Theta '=10$. The dotted line lies very close to the solid line and is not visible. c) Non-relativistic case where the rest mass dominates with $\omega '=0.1\,\omega '_{\rm max}$, $\Theta '=0.1$.

Begelman & Li (1994) showed that the electromagnetic energy flux decreases like

$$\frac{\left.\dot{E}_{\rm em}\right\vert _{\infty}} {\left.... ...\vert _{x_0}} = \frac{\tilde{s}(x_{\rm f})}{\tilde{s}_\infty}$$ (81)

for a cold flow in Minkowski metric. If the asymptotic regime is already reached in the region where the opening angle increases, Eq. (80) shows that this relation should still be valid in the most general case, independent of the gravitational field or of the initial amount of thermal energy. To check the validity of this result we consider 9 different $\omega',\theta',m$ combinations, illustrating all possible situations and for each of them we compute the evolution of the efficiency when varying $\tilde{s}_{\infty}$. As Fig. 4 shows, the exact shape of the geometry is not important, so we adopt a particular choice where $\tilde{s}$ rises from $\tilde{s}_{0}=1$ to $\tilde{s}_{\infty}$ between x₁=100 and x₂=200. This region lies always in the super-Alfvénic region, which as discussed above is the condition for magnetic to kinetic energy conversion. Figure 5 shows the quantity

$$\frac{\left.\dot E_{\rm em}\right\vert _{\infty}} {\left.\... ...ht\vert _{x_0}} \cdot\frac{\tilde{s}_{\infty}}{\tilde{s}_{0}}$$ (82)

plotted over $\tilde{s}_{\infty}/\tilde{s}_{0}$ for the 9 different cases. Notice that with our choice of geometry $\tilde{s}(x_{\rm f})=\tilde{s}_{0}$. One sees that gravity and pressure changes the simple picture a bit. In the cold cases (Figs. 5a and c) the converted energy fraction decreases for a stronger gravitational field and high values of m. On the other hand the gravitational field increases the energy conversion by a small amount in the hot thermal dominated case as seen in Fig. 5b. But (81) remains still valid within a factor of 2. For the cases of low gravity and low thermal energy (solid lines in Figs. 5a and c) the quantity (82) approaches 1as expected.

We can therefore conclude that the flow geometry always dominates the energy conversion and all other parameters play an only minor role.