5 Application to gamma-ray bursts

Since the discovery of their optical afterglows gamma-ray bursts (hereafter GRBs) have been known to be located at cosmological distance. More than ten redshifts have been measured from z=0.43(GRB990712) to z=4.5 (GRB000131). The corresponding radiated energy in the gamma-ray domain (20-20000keV) ranges from $5\times10^{51}\,$erg (GRB970228) to $2\times 10^{54}\,$erg (GRB990123) assuming isotropic emission. Most sources that have been proposed to explain such a huge release of energy in a few seconds involve a rapidly rotating compact stellar-mass core. Among them the two most popular are mergers of compact objects (neutron stars binary or neutron star - black hole systems) or collapses of very massive stars to a black hole (collapsars) (Mészáros & Rees 1992; Narayan et al. 1992; Mochkovitch et al. 1993; Woosley 1993; Paczynski 1998). In both cases, the resulting system is a stellar mass black hole surrounded by a thick torus made of stellar debris or of infalling stellar material partially supported by centrifugal forces. An other interesting proposition (Usov 1992; Kluzniak & Ruderman 1998; Spruit 1999) associates GRBs with highly magnetized millisecond pulsars. The location of the detected optical counterparts, well inside their host galaxy and possibly associated with star-forming regions, seems to favor the collapsar scenario. However the other propositions cannot be ruled out, at least for short bursts, for which no optical counterpart has been detected yet.

Whatever the source is, the released energy must initially be injected in a wind which eventually becomes relativistic. The existence of such a relativistic wind has been directly inferred from the observations of radio scintillation in GRB970508 (Frail et al. 1997) and is also needed to avoid photon-photon annihilation. The absence of signature of this last process in the BATSE spectra of GRBs implies very high Lorentz factor for the wind: $\Gamma \sim 100$-1000(Goodman 1986; Baring 1995). The second step consists in the conversion of a fraction of the wind kinetic energy into gamma-rays, probably via the formation of shocks within the wind itself (Rees & Mészáros 1994; Daigne & Mochkovitch 1998). Such internal shocks are expected if the wind is generated with a highly non uniform distribution of the Lorentz factor so that rapid layers catch up with slower ones. In the last step, the wind is decelerated when it interacts with the environment of the source and the resulting external shock is responsible for the afterglow observed in X-ray, optical and radio bands.

The origin of the relativistic wind is the most complex of the three steps in this scenario. Several proposals have been made but only few calculations have been performed so that none appears to be fully conclusive. However it is suspected that large magnetic fields play an important role. In a previous paper (Spruit et al. 2001) we have considered different possible geometries of magnetic fields in GRB outflows and we have proposed that in many cases, dissipation of magnetic energy by reconnection should occur. The model we have presented in this paper allows us to investigate these questions in more details. In particular we focus on the case where the outflow generated by the central engine is initially Poynting flux dominated (in the following, we assume that only $10\%$ of the energy is initially injected in the matter). To be consistent with the observations showing that at the beginning of the afterglow emission, the matter flow is highly relativistic, we also impose that the terminal Lorentz factor has a large value (in the following, we will adopt $\Gamma_\infty=100$). This implies a reasonable efficiency of the magnetic to kinetic energy conversion. The goal of the study presented in this section is to illustrate that there are geometries allowing such an efficiency and to discuss the possibility of magnetic reconnection in this scenario.

Spruit et al. (2001) have shown that for typical GRB outflows the MHD approximation is valid to very large distance ( $10^{19}\,$cm) which is the main assumption of our calculations. The second main assumption - the stationarity of the flow - is of course less justified in the case of GRBs. However we can estimate the time scale to reach the stationary regime in our wind solutions as the time needed by a particle starting from the basis of a flow line to reach the Alfvén point:

$$t_{\rm stat} = \frac{r_{\rm a}}{c}\int_{x_{0}}^{1} \frac{\tilde{u}^{t}}{\tilde{u}^{r}}{\rm d}x$$ (83)

(in the source frame). Let us estimate this time scale in a particular case. We consider a Poynting flux dominated wind (we adopt $\xi=9.0$so that only 10% of the energy flux is initially injected in the matter) with a moderately low initial baryonic load (we take $\eta=1/50$). We impose that the terminal Lorentz factor is $\Gamma_\infty=100$. If there were no magnetic to kinetic energy conversion, the Lorentz factor at infinity would only be $1/\eta=50$. In order to get a final Lorentz factor of 100, we need to assume that the geometry allows an efficiency $\mathit{eff} = \left(\eta\Gamma_{\infty}-1\right)/\xi = 1/9$. We have shown in Sect. 4 that this implies $\tilde{s}_{\infty}/\tilde{s}_{0} \simeq \xi / \left(1+\xi-\eta\Gamma_{\infty}\right)=1.125$. For a given m, the two other parameters $\Theta '$ and $\omega '$ are fixed by the values of $\xi$ and $\eta$. We find that the following set of parameters: m=0.069, $\Theta'=3.8$ and $\omega'=0.78$ fulfill the requirements and corresponds to a reasonable value of the Alfvén radius $r_{\rm A}$ and the angular frequency $\Omega$ in the case of a millisecond pulsar-like source ( $M=1.4~M_\odot$) which is most likely leading to an equatorial flow as we are considering here: $r_{\rm A}=3.0\times 10^{6}\,$cm and $\Omega = 8.8\times 10^{3}\,$Hz. Figure 6 shows the evolution of the "Lorentz factor'' and the electromagnetic and matter energy fluxes in this case. We have assumed a simple geometry like those in Sect. 4 where $\tilde{s}$ increases in a region located between x₁=300 and x₂=900, well outside the fast critical point radius. The corresponding time scale to reach the stationary regime $t_{\rm stat}$ is between $\sim$ $r_{\rm A}/c$ and $\sim$ $2 r_{\rm A} /c$, depending on the adopted value of the initial radius x₀. As $r_{\rm A}/c = 10^{-4}\,$cms^-1 here, this is compatible with the timescale of the variability observed in GRBs profiles. This means that when the physical conditions at the basis of the flow vary on a time scale ms the flow reacts instantaneously to reach a new stationary state corresponding to the new boundary conditions. Thus our calculation is a good approximation for the relativistic wind of GRBs. If the wind produced by the source lasts for a duration $t_{\rm w}$, our solution is appropriate for the physical quantities within the corresponding shell when it is located at radius r.

On the solution we present in Fig. 6, the acceleration occurs in two phases. First the initial thermal energy is converted into kinetic energy, the magnetic energy remaining unchanged. This phase ends at $r\simeq 10^{9}\,$cm where $\Gamma_{\infty}\simeq 1/\eta \simeq 50$. The second phase occurs in the region where the opening angle increases. Here a magnetic to kinetic energy conversion takes place. We define the acceleration radius $r_{\rm acc}$ as the radius where the flow reaches a Lorentz factor of $\Gamma=0.95\Gamma_\infty$ and the acceleration can be considered as finished. The value of this radius is completely dominated by the unknown flow geometry and equals $r_{\rm acc} \simeq 2.7\times 10^{9}\,$cm in this case. Even if the location of the region where the opening angle diverges would extend to higher radii up to 10¹⁰-$10^{11}\,$cm, this radius is well below two other important radii: the photosphere radius $r_{\rm ph}$ where the wind becomes transparent and the reconnection radius $r_{\rm rec}$ where the reconnection of the magnetic field should occur. These two radii have been estimated in Spruit et al. (2001). The photosphere radius is the solution of

$$1 = \int_{r}^{r+2\Gamma^2 c t_{\rm w}} \frac{\kappa \rho}{2\Gamma^2}{\rm d}r$$ (84)

and is independent of the duration of the burst $t_{\rm w}$ as long as $r_{\rm ph} \ll 2\Gamma^2 c t_{\rm w} = 6\times 10^{14}\,{\rm cm}\cdot\left(\Gamma/100\right)^2 \left(t_{\rm w} / 1\,{\rm s}\right)$. Here we have $r_{\rm ph}=6.2\times 10^{10}\,$cm for $\dot{E}_{\rm total}=10^{51}\,$erg/s and $r_{\rm ph}=5.9\times 10^{11}\,$cm for $\dot{E}_{\rm total}=10^{52}\,$erg/s. This interval is marked by a thick line in Fig. 6. As the remaining thermal energy in the wind at such a large radius is very small, our adiabatic wind solution applies up to the reconnection radius, where magnetic dissipation starts. This radius is given by

$$r_{\rm rec} \simeq \frac{\pi c}{\epsilon \Omega}\Gamma^2 \left(1+\frac{1}{\xi}\right)^{1/2}$$ (85)

where $\epsilon<1$ is a numerical factor of order unity measuring the reconnection speed in unit of the Alfvén speed. In our case we have $r_{\rm rec}\simeq 1.1\times 10^{11}\,{\rm cm}\cdot\epsilon$. As the magnetic energy flux is still 80% of the total energy flux at $r_{\rm rec}$, a very large amount of energy can possibly be dissipated at this large distance. Depending on the value of $\dot{E}_{\rm total}$ and $\epsilon$, such reconnection events may start when the wind is still optically thick (low $\epsilon$, high $\dot{E}_{\rm total}$) or when the wind is already transparent (high $\epsilon$, low $\dot{E}_{\rm total}$). As the dissipated magnetic energy is probably first converted into thermal energy, the consequences for the wind may be very different in these two cases. (i) if the wind is optically thick, this injection of thermal energy should be converted, at least partially (up to the photosphere radius) into kinetic energy, leading to a third phase of acceleration; (ii) on the other hand, if the wind is transparent, reconnection events could directly contribute to the observed emission. Notice that all the radii we have computed are usually small compared to the typical radius where internal shocks occur (with $\Gamma_\infty=100$)

$$r_{\rm IS}\simeq 3\times 10^{14}\,{\rm cm}\cdot \left(\frac{t_{\rm var}}{1\,{\rm s}}\right),$$ (86)

where $t_{\rm var}$ is the typical time scale of the variability in the initial distribution of the Lorentz factor and also small compared to the deceleration radius where the external shock becomes efficient (with $\Gamma_\infty=100$)

$$r_{\rm dec}\simeq 5\times 10^{16}\,{\rm cm}\cdot \left(\fr... ...\right)^{1/3} \left(\frac{n}{1\,{\rm cm}^{-3}}\right)^{-1/3},$$ (87)

where n is the density of the external medium and E the total energy of the wind at this radius. So these two "standard'' mechanisms are not affected by the reconnection events. However the relevant energy flux will be the kinetic energy flux at $r_{\rm acc}$, possibly increased to a larger value if the reconnection starts in the optically thick regime.

Figure 6: Geometry, "Lorentz factor'' and energy fluxes for our example. The vertical dotted lines mark the radii of the slow-, the Alfvén-, fast point and the acceleration radius.