4 Analysis of the results

   
4.1 The photospheric radius

The photosphere is located where the optical depth reaches a value of 1. The optical depth depends on the density and the the radial velocity u and must be integrated from a finite radius to infinity. Because we only want to know the photospheric radius within a factor of, say 2, we define it to be where the mean free path of a photon equals the distance from the source r. In the comoving frame a photon sees the mass density $\rho$ and the mean free path for Thompson scattering is $1/(\kappa\rho)$. The source distance in this frame is $r/\Gamma$ so that the photosphere is located at

$$r_{\rm ph} = \frac{u_{\rm ph}}{\kappa\rho_{\rm ph}}$$ (51)

which yields

$$\frac{\kappa L}{c^3 \sigma_0^{3/2}} = u_{\rm ph}^2 r_{\rm ph}.$$ (52)

If we neglect not only the initial velocity but also the initial radius r₀ compared to the photospheric radius, (46) and (47) simplify to

$$u_{\rm ph}^3 = \frac{3}{\pi c} \epsilon\Omega (1-\mu^2)\sigma_0^{3/2} r_{\rm ph} \hfill\mbox{(longitudinal case),}\quad$$ (53)

$$u_{\rm ph}^2 = \frac{4\epsilon (1-\mu^2)\sigma_0^{3/2}}{\v... ...r_{\rm ph}}{r_0}\right) \hfill\mbox{(transversal case).}\quad$$ (54)

Together with condition (52) at the photosphere we arrive at the equations for the photospheric radius and the 4-velocity in the longitudinal case

  

$$u_{\rm ph} \left[ \frac{3\kappa}{\pi c^4} \epsilon\Omega (1-\mu^2)L \right]^{1/5}$$ =

$$119\cdot \left[ \epsilon_{-1} \Omega_4 \left(\frac{1-\mu^2}{0.5}\right) L_{50} \right]^{1/5} \ ,$$ = (55)

$$r_{\rm ph} \left[ \frac{\pi^2 \kappa^3}{9 c^7} \frac{L^3}{\left(\epsilon\Omega(1-\mu^2)\right)^2} \right]^{1/5} \sigma_0^{-3/2}$$ =

$$1.05\times 10^{11}~{\rm cm}$$ =

$$\times \left[ \epsilon_{-1} \Omega_4 \left(\frac{1-\mu^2}{0.5}\right) \right]^{-2/5} L_{50}^{3/5} \sigma_{0,2}^{-3/2}.$$ (56)

Note that the 4-velocity at the photosphere $u_{\rm ph}$ does not depend on the initial Poynting flux ratio $\sigma_0$ and only weakly on L.

For the transversal case the flow velocity always depends greatly on the initial radius r₀. The dissipation time scale is $\tau_{\rm tr}\sim ru$ and most energy is released at small rnear the source. The acceleration depends crucially on the onset of the dissipation and therefore on r₀. In our simple model r₀ and $\vartheta$ are not well determined by physical arguments so that the transversal case is rather uncertain and highly speculative. One cannot write down robust equations for the photosphere like in the longitudinal case without many degrees of freedom.

4.2 Energy available for prompt radiation

The energy dissipated beyond the photospheric radius is

$$L_{\rm D} = \left(u_\infty - u_{\rm ph}\right) \dot M c^2.$$ (57)

Using (20), (43) and (55) this yields

$$L_{\rm D} = e \left(1-\mu^2\right) L$$ (58)

with

$$e = 1 - \left( \frac{3\kappa}{\pi c^4} \frac{\epsilon\Omega L}{\left(1-\mu^2\right)^4} \right)^{1/5} \sigma_0^{-3/2}.$$ (59)

For a Poynting flux dominated flow, the magnetic energy flux equals the total energy flux L. Of this, a fraction $1-\mu^2$ is dissipated internally while a fraction $e(1-\mu^2)$ can be converted to radiation beyond the photosphere. Thus e is an efficiency factor, which gives the ratio between the energy dissipated in the optically thin domain and the total dissipated energy. Efficient conversion of free magnetic energy into non-thermal radiation can happen if e is of the order unity which requires that the second term in (59) is small:

$$0.24\times \left( \epsilon_{-1} \Omega_4 L_{50} \right)^... ...ft(\frac{1-\mu^2}{0.5}\right)^{-4/5} \sigma_{0,2}^{-3/2} < 1$$ (60)

or written differently

$$\sigma_0 > 39\times \left( \epsilon_{-1} \Omega_4 L_{50} \right)^{2/15} \left(\frac{1-\mu^2}{0.5}\right)^{-8/15}$$ (61)

where $\epsilon_{-1} = \epsilon/0.1$, $\Omega_4 = \Omega/\left(10^4~{\rm s^{-1}}\right)$, $L_{50} = L/\left(10^{50}~{\rm erg~s^{-1}~sterad^{-1}}\right)$ and $\sigma_{0,2} = \sigma_0/100$ are parameters scaled to fiducial GRB values.

When (61) is satisfied, part of the magnetic energy is released beyond the photosphere, and powers the prompt radiation. If it is not satisfied, the energy is released inside the photosphere and is converted, instead, into bulk kinetic energy. Some other means of conversation into radiation is then needed, such as internal shocks. Since the dependence on parameters other than the initial Poynting flux ratio $\sigma_0$ is small in (61), we conclude that efficient powering of prompt radiation by magnetic dissipation in GRB is possible for $\sigma_0\ga100$.

4.3 Longitudinal and transversal cases in comparison

The major difference between the longitudinal and the transversal case is the different dissipation time scale. While the decay time scale for the longitudinal case (44) is $\tau_{\rm lo}\sim u^2$ and therefore limited by $u\le u_\infty$, the time scale for the transversal case $\tau_{\rm tr}\sim ru$ is not limited. At small radii it starts at low values but grows then to infinity. This major difference is visualised in Fig. 1 where the flow Lorentz factor is plotted depending on the radius.

Figure 2: The influence of r0 on the Lorentz factor: the solid lines correspond to longitudinal case solutions and the dashed one to transversal case solutions with $1.5\times 10^{7}~{\rm cm}\le r_0\le 1.2\times 10^9~{\rm cm}$. The constants used where $\sigma _0=100$, $\mu ^2=0.5$, $\epsilon =0.1$, $\Omega=10^4~{\rm s}^{-1}$. For the transversal cases $\vartheta =2\pi c\sqrt {\sigma _0}/(\Omega r_0)$ is chosen so that the initial acceleration (slope at r0) is the same as in the corresponding longitudinal cases.

As mentioned in Sect. 4.1 the transversal case depends strongly on the initial radius r₀. This is seen in Fig. 2 where the numerical solutions of (39) are shown for various initial radii. While all longitudinal case solutions merge toward the $u\sim r^{1/3}$ power-law there is a large spread in the transversal case solutions.

The longitudinal and transversal cases are the two limits for a general case where both kinds of scaling of the decay time scale occur. One can model the mixing of both cases by writing the dissipation time scale as

$$\tau = k \left(\frac{r}{r_0}\right)^\alpha \left(\frac{u}{u_0}\right)^{2-\alpha}$$ (62)

where $0<\alpha<1$ is a dimensionless parameter which determines the mixing. $\alpha=0$ corresponds to the pure longitudinal case and $\alpha=1$ to the transversal case. The constant k can be written depending on the corresponding model parameters as

$$k = \frac{2\pi\sigma_0}{\epsilon\Omega}$$ (63)

or as

$$k = \frac{\vartheta r_0 \sqrt{\sigma_0}}{\epsilon c}\cdot$$ (64)

Figure 3: The Lorentz factor $\Gamma$ for different parameters $\alpha \in \{0,0.2,0.4,0.6,0.8,1\}$. The other wind parameters were set to $\sigma _0=100$, $\mu ^2=0.5$, $\epsilon =0.1$, $r_0=10^8~{\rm cm}$, $\Omega=10^4~{\rm s}^{-1}$ (corresponding to $\vartheta =10.8^\circ$ in the transversal description). The different winds all start with the same dissipation rate so that they show the same initial acceleration. $\Gamma _\infty$ is marked by a horizontally dotted line.

Figure 3 shows the velocity profiles for various $\alpha$ values. All graphs result from a numerical integration of (39). Beyond the photosphere, assuming it is around $10^{11}~{\rm cm}$, the dissipation is only efficient if the field variation does not point in transversal direction. The efficiency estimation in (59) is therefore an upper limit for the general case with $\alpha\not=0$.

   
4.4 The validity of the MHD condition

Because we work with the ideal MHD approximation we have to make sure that there are enough charges in the flow to make up the required electric current density. Because the reconnection processes will destroy the ordered initial field configuration quickly it does not make much sense to consider this configuration throughout the flow. But one can at least estimate needed currents by looking at a sinusoidal wave in the equatorial plane. In Paper I we derived the limiting radius where the MHD condition breaks down by using a constant flow speed and assumed $\mu=0$. The condition that enough charges are available to carry the current is

$$\frac{\Omega B}{4\pi\Gamma^2} = \frac{2\rho ec}{m_{\rm p}}\cdot$$ (65)

This yields the radius up to which the MHD approximation holds:

  

$$r_{\rm MHD} \frac{4e \sqrt{\pi L}}{m_{\rm p} c^{3/2} \Omega} \frac{u/\sigma_0^{3/2}}{\sqrt{1-u/\sigma_0^{3/2}}}$$ (66)

$$4\times10^{18}~{\rm cm}\cdot L_{50}^{1/2} \Omega_4^{-1} \sigma_{0,2}^{-1}.$$ (67)

Here, we have used the dependence (21) of the magnetic field strength on velocity. In (66) $r_{\rm MHD}$ is written as a function of u and depends implicitly on r. At r₀where $u=u_0=\sqrt{\sigma_0}$ it starts at the value (67) and rises strongly until the final velocity $u = u_\infty = \sigma_0^{3/2}$ is reached where $r_{\rm MHD}$ diverge to $\infty$. For the GRB parameters assumed here, we find $r_{\rm MHD}\gg r_{\rm sr}$ and the MHD approximation is always fulfilled, as in Paper I.

   
4.5 Comparison with the striped pulsar wind

Dissipation of magnetic energy was applied to the Crab pulsar wind by Lyubarsky & Kirk (2001). Their model setup included a striped pulsar wind (Coroniti 1990) that is the equatorial wind of on inclined rotator. This is quite similar to our longitudinal case where all Poynting flux can decay so that $\mu=0$. The wind starts with $\Gamma=\sqrt{\sigma_0}$ and reaches $\Gamma_\infty = \sigma_0^{3/2}$as in our model. Due to a difference approach to model the reconnection rate they obtain a flow acceleration of $\Gamma \sim r^{1/2}$ (Eq. (30) Lyubarsky & Kirk 2001) which is faster than $\Gamma\approx u \sim r^{1/3}$ form (50).

The findings of Lyubarsky & Kirk (2001) that the reconnection is inefficient for the Crab wind seems to contradict our result, that it efficiently accelerates the GRB outflow. The reason for that is the different initial Poynting flux values used for the Crab pulsar and in our study. $\sigma_0$ is the critical parameter controlling the final Lorentz factor and the spatial size of the accelerating wind.

Discussing the Crab pulsar wind in detail and speculating why reconnection fails is beyond the scope of the present paper. Instead, we simply take the flow parameter values $\sigma_{0,2}=400$, $\Omega_4=0.02$ from Lyubarsky & Kirk (2001) and show that our model gives basically the same result as the striped wind model. However, see Yubarsky & Eichler (2001) for a critical revision of the Crab pulsar wind parameters. Equation (50) yields for the 4-velocity at the observed termination shock at $r=3\times 10^{17}~{\rm cm}$

$$\frac{u}{u_\infty} = 0.023~\epsilon_{-1}^{-1/3},\qquad \frac{u}{u_0} = 920~\epsilon_{-1}^{-1/3}.$$ (68)

When the wind reaches its termination shock only a small fraction of the Poynting flux was converted. Though, due to the small amount of mass in the flow large acceleration occurs and the Lorentz factor increases by almost 4 orders of magnitude. This is the same result as obtained by Lyubarsky & Kirk (2001). The different acceleration laws of the two models does not change the picture. The observed pulsar wind bubble is to small to allow for efficient reconnection.

The radius $r_{\rm sr}$ from (49) denotes the radius where the Poynting flux conversion ends. Its value scales with the third power of $\sigma_0$. Plausible Lorentz factors for GRB winds of around 10²-10⁴ imply $\sigma_{0,2} \approx 0.2$-5 (or larger for $\mu>0$). This lowers $r_{\rm sr}$ by 6 orders of magnitudes compared to the Crab wind. Thus $r_{\rm sr} \la 3\times 10^{15}~{\rm cm}$ which is smaller than the radius $r_{\rm a} \approx 10^{16}~{\rm cm}$ where the flow runs into the ambient medium (Piran 1999). The requirement on $\sigma_0$ for the dissipation to take place inside a radius $r_{\rm a}$ can be expressed by (49) which yields an upper limit for the initial Poynting flux ratio:

$$\sigma_0 \la 1100 \cdot \left( r_{{\rm a},16} \epsilon_{-1... ...a_4\right)^{1/3} \left(\frac{1-\mu^2}{0.5}\right)^{-2/3}\cdot$$ (69)

For GRBs there is no size problem as for the Crab wind and Poynting flux can be efficiently converted.