4 Stability

In the above section we have derived some geometrical constraints on the emitting plasma in order to reproduce the observed features and continua. Since the results envisage particular geometric conditions, we here analyze their stability. First, since the heating energy is supposed to come from a central point (the GRB progenitor), one has to make sure that the emitting material is not accelerated to high velocities.

Consider a shell of mass M absorbing energy from a relativistic outflow. If the material is radiative, it acquires a bulk velocity v=E/(M c). A bulk velocity $v\simeq0.1~c$ was measured in GRB 011211 (R02) and GRB 991216 (Piro et al. 2000). Requiring that our optically thin shell is accelerated to a comparable or smaller speed implies a radius larger than:

$$R\ge{{L~\sigma_T}\over{4\pi~m_{\rm p}~c^2~\eta_{\rm {line}}~\... ...line},45}}\over{\eta_{\rm {line}}~\tau_T~v_9}} \qquad{\rm cm}.$$ (10)

This is not a compelling limit, given the radii discussed above.

Provided that the emitting medium is not accelerated to relativistic speeds by the energy input, we also want that the thin emitting shell (or blobs) do not expand in a time scale smaller than the emission one (which can be either the heating or cooling time scale). The shell (or blobs) was in fact in equilibrium with the ambient medium when it was cold. Now that its temperature is increased it will tend to expand under the effect of the increased internal pressure. If it expands at the speed of sound $c_{\rm s}$, its density will be sizably modified in a time scale $t_{\exp}=R~\eta_R/c_{\rm s}$. It is therefore required that the expansion time is longer than the emission time scale.

In the case of steady heating we obtain:

$$R>c_{\rm s}~{{t_{\rm {heat}}\over{\eta_R}}} = 10^{8}~T_8^{1/2}~ {{t_{\rm {heat}}\over{\eta_R}}} \ga10^{17}~T_8^{1/2}.$$ (11)

Also in the steady heating case, therefore, the radius has to be large in order to allow for the production of thermal components in the early GRB X-ray afterglows. For such large radii, however, the density required to fulfill the steady heating conditions is $n\ga10^{15}/t_{\rm {heat}}\ga10^{12}$, several orders of magnitudes larger than what is inferred by Piro et al. (2002). It seems therefore that the steady heating case requires more extreme conditions than the flash heating one (the same distance from the explosion site and limits on the total mass involved, but larger densities and smaller filling factors). Applying the same stability condition to a flash heating case, we are left with the more relaxed constraint

$$\tau_T>0.1$$ (12)

which is therefore not difficult to fulfill. These stability considerations suggest therefore that the only viable way to produce a sizable thermal afterglow component is by heating a small portion of a massive $\tau_T\sim1$ (either very thin or clumpy) shell of material located at a relatively large distance from the burst explosion site (Fig. 3). It should however be emphasized that these stability considerations cannot be applied if the heating is provided hydrodynamically, in such a way that the source of heating is providing also the confining pressure. In this case also a steady heating scenario, with a much smaller shell, may be viable.