2 The internal shocks

GRB 960720 has been observed both by BATSE and Beppo-SAX. It is a single-pulse burst, with a "FRED'' profile. Its duration in the 50-700 keV band is around 2-3 s but the X-ray emission lasts longer: Piro et al. (1998) show that the power-law between the width of the pulse and the energy (already known in the gamma-ray range) is observed down to 2 keV. They find $W(E) \propto E^{-0.46}$.

We use a simple model to simulate internal shocks and build synthetic bursts: all pressure waves are neglected so that we consider only direct collisions between solid layers. In the shocked material, the magnetic field reaches equipartition values (10-1000 G) and the Lorentz factor of the electrons is obtained from the dissipated energy per proton $\epsilon$ using the formula given by Bykov & Mészáros (1996) who suppose that only a fraction $\zeta$ of the electrons is accelerated:

$$\Gamma_{\rm e} \sim \left[ \frac{\alpha_{\rm M}}{\zeta} \... ...{\rm M} \sim 0.1 \to 1\ {\rm and}\ 1 \le \alpha \le 1.5 {\rm)}.$$ (1)

  

Figure 2: Results obtained for a ratio $E^{\rm inj}_{52} / n = 0.5$. For $E_{\gamma}=10^{51}/4\pi\ {\rm erg/sr}$ and $f_{\gamma} = 0.05$ it corresponds to $n=4\ {\rm cm}^{-3}$. Left panel: Profiles of the synthetic burst in the X- and gamma-ray bands. The full line takes into account the presence of the ISM whereas the dotted line only shows the contribution of the internal shocks. Middle panel: Pulse width as a function of energy. The dotted line corresponds to $W(E) \propto E^{-0.45}\!$. Right pannel: Burst spectrum. The full line takes into account the ISM and the dashed line does not

For $\zeta \sim 1$ the usual equipartition assumption yields values of $\Gamma_{\rm e}$of a few hundreds: the gamma-ray emission is due to inverse Compton scattering on the synchrotron photons. Smaller values for the fraction of accelerated electrons ($\zeta < 10^{-2}$) lead to larger Lorenz factors ($\Gamma_{\rm e}$ of a few thousands) so that the gamma-rays are directly produced by the synchrotron process, which is the assumption made here. Internal shocks have been shown to successfully reproduce the main temporal and spectral properties of GRBs (Daigne & Mochkovitch 1998).

We model GRB 960720 with a wind emitted during 4 s and consisting in a slow and a rapid part of equal mass (see Fig. 1). Two internal shocks are generated and we sum both contributions to the emission to construct the synthetic burst. The profile in the SAX 50-700 keV band looks very similar to GRB 960720 as can be seen in Fig. 2. However the X-ray emission does not last long enough so that the power-law relating W(E) and E is not reproduced in this spectral range.