3 Effect of a medium of uniform density

We now consider the effect of the ISM, whose density n is supposed to be uniform. The external shock (forward shock) produces the afterglow. Simultaneously a reverse shock propagates into the wind. Its strength is comparable to that of the internal shocks, which are midly relativistic while the external shock is initially very strong and relativistic. We therefore adopt the same assumptions to compute the emission of the shocked material behind the reverse and the internal shocks.

It is possible to derive a critical ratio of the total energy injected over the density (see Sari & Piran 1995):

$$\left.\frac{E^{\rm inj}}{n}\right\vert _{\rm crit} \sim \frac{4 \pi}{3} m_{\rm p} c^{5} \bar{\Gamma}^{8} T^{3}$$ (2)

for which the reverse shock will interfere with the internal shocks. Injecting the typical values used in our example (Lorentz factor $\bar{\Gamma} \sim 300$ and duration T=4 s) gives

$$\left.\frac{E^{\rm inj}_{52}}{n}\right\vert _{\rm crit} \sim 0.071 \bar{\Gamma}_{300}^{8} T_{4}^{3},$$ (3)

where $E^{\rm inj}_{52} = \frac{E^{\rm inj}}{10^{52}/ 4 \pi\ {\rm erg/sr}}$. Assuming an efficiency $f_{\gamma}$ for the conversion of wind energy into gamma-rays by internal shocks allows to obtain $E^{\rm inj}=\frac{E_{\gamma}}{f_{\gamma}}$ from the observed burst energy $E_{\gamma}$.

The total energy being fixed, if the density is smaller than the critical value, the reverse shock does not contribute in gamma-rays and only produces a delayed X-ray emission with an intensity which increases with n. If the density reaches or exceeds the critical value, the X-ray and the gamma-ray profiles become very affected.

We have represented in Fig. 2 the profiles for a ratio $E^{\rm inj}_{52} / n = 0.5$ ($n \sim\ 0.15 n_{\rm crit}$). The gamma-ray profile is unchanged and the X-ray profiles are improved. The corresponding E-W(E) diagram in Fig. 2 shows a well reproduced power-law over the complete energy range with an index -0.45 in agreement with the observations of GRB 960720. In the spectrum the contribution of the reverse shock to the late emission appears like a X-ray plateau, which is observed (and is even more extended) in GRB 960720.