3 Thermal models for GRB afterglows

Consider a cloud of plasma of electron density n at a temperature T which covers a solid angle $\Omega=2\pi(1-\cos\theta)$ along the line of sight to the observer at a distance R from the burst explosion site (see Fig. 3). The luminosity that is inferred from infinity, taking into account the light travel time effects, is given by:

$$L\approx1.7\times10^{-23}~T_8^{1/2}~EI~{{t_{\rm {heat}}+t_{\r... ...ool}}} \over{R/c(1-\cos\theta)+t_{\rm {heat}}+t_{\rm {cool}}}}$$ (1)

where EI is the emission integral ( $EI=\int_V~n_{\rm e}n_{\rm i}~$dV), $t_{\rm {cool}}\sim1.2\times10^{15}~T_8^{1/2}~n^{-1}$ s is the plasma cooling time and $t_{\rm {heat}}$ is the heating time, i.e. the time during which heat (or energy) is supplied to the emitting plasma. It is assumed that during this heating time the temperature is held constant, i.e. the plasma is in equilibrium. In the following we will consider a uniform density plasma and approximate $n_{\rm e}\approx{}n_{\rm i}\approx{}n$. In this case EI=n² V, where V is the volume of the emitting cloud.

Figure 2: Photoionization optical depth of a $\tau _T=1$ plasma in thermal equilibrium as a function of the temperature and frequency. The range of frequencies of the considered K$_\alpha$ lines is shown by gray shading.

In many astrophysically relevant situations (e.g. the IGM in galaxy clusters), $R/c\ll{}t_{\rm {cool}}$ and both the heating and cooling time scales are irrelevant and Eq. (1) simplifies to the usual free-free equation:

$$L=1.7\times10^{-23}~T_8^{1/2}~EI.$$ (2)

In GRBs this may not be the case. In order to reproduce afterglow observations in the non-cooling regime, the plasma cloud must satisfy simultaneously three conditions. First, it must produce the observed luminosity in the non-cooling regime

$$L=1.7\times10^{-23} ~ T_8^{1/2} ~ n^2 ~ {{4\pi}\over{3}} R^3 ... ...ver{4\pi}} \sim 10^{46} ~ L_{46} \quad {{\rm erg}\over{\rm s}}$$ (3)

where the numeric value is typical for the early X-ray afterglow (see e.g. GRB 011211, R02) and $\eta_R$ is the volume filling factor of a possible shell-like or clumpy cloud. Second, it must fulfill the non-cooling condition:

$$R~(1-\cos\theta) \le c~t_{\rm {cool}} = 3.6\times10^{25}~T_8^{1/2}~n^{-1}$$ (4)

finally, it must require a total amount of energy smaller than the total energy of a GRB:

$$E=8.7\times10^{-8} ~ T_8 ~ n ~ R^3 ~ \eta_R ~ {{\Omega}\over{4\pi}} \le 10^{52} ~ E_{52} \quad {\rm erg}.$$ (5)

Conditions (3) and (4) yield the density constraint:

$$n \le 3.7\times10^8 ~ T_8^2 ~ \eta_R ~ L_{46}^{-1} ~ {{\Omega}\over{4\pi}}$$ (6)

while conditions (3) and (5) give:

$$n \ge 1.2\times10^9 ~ T_8^{1/2} ~ L_{46} ~ E_{52}^{-1}.$$ (7)

The above conditions cannot clearly be satisfied simultaneously if all the parameters are taken equal to their fiducial values. Since $\eta_R$ and $\Omega/4\pi$ are both numbers smaller than unity, a change in the geometry does not help, making the constraint of Eq. (6) more stringent. Also the temperature cannot be changed significantly, especially as long as X-ray lines must be taken into account.

In conclusion, a thermal component that contributes to the early afterglow of a typical GRB cannot be emitted in the non-cooling regime. In fact, in order to reach a luminosity large enough without involving a too large thermal energy, the plasma density must be so large as to make the cooling time very short. As an example, R02 derived a cooling time of $\sim$2 s for the afterglow of GRB 011211. We cannot therefore adopt the simple Eq. (2) but we must use the more complete Eq. (1).

Figure 3: Cartoon of the geometrical set-up for the GRB and thermal reprocessing material. The GRB is surrounded by a thin shell (or clumped thicker shell) of material, which is heated by GRB photons within the fireball solid angle.

We therefore consider in the following two limiting cases: $R/c(1-\cos\theta)\gg{}t_{\rm {cool}}\gg{}t_{\rm {heat}}$ (flash heating) and $R/c(1-\cos\theta)\gg{}t_{\rm {heat}}\gg{}t_{\rm {cool}}$(steady heating). In principle also the case $t_{\rm {heat}}\gg{}R/c(1-\cos\theta)\gg{}t_{\rm {cool}}$ may be interesting. It is however difficult to identify a heating source that can be active for a time comparable to the line emission time scale. Even if the emitting plasma would be heated by afterglow photons, the heating time could be at most a few per cent of the line emission time scale (Lazzati et al. 2002b).

Figure 4: The luminosity coefficient $c_{\rm h}$ (see text) as a function of the plasma temperature and of metallicity for the considered elements. As in Fig. 1, the dotted, dashed, solid, dot-dash and dot-dot-dot-dash lines are relative to 0.1, 0.3, 1, 3 and 10 time solar metallicity, respectively.

3.1 Flash heating

Consider a portion of a spherical shell of plasma of radius R, temperature T and uniform density n covering a solid angle $\Omega=1-\cos\theta$ (see, again, Fig. 3). Its emission integral can be written as $EI=n^2~R^3~\eta_R\Omega$ where $\eta_R\equiv\Delta{R}/R$ is the ratio of the thickness of the shell to its radius. Considering the values $\eta _{\rm {line}}$ defined above (and shown in Fig. 1), the cloud will produce a line with luminosity (see Eq. (1)):

$$L_{\rm {line}}=1.9\times10^{27} \eta_{\rm {line}}~R~T_8~\tau_T$$ (8)

where we have used the Thomson opacity $\tau_T=n~R~\eta_R~\sigma_T$. Equation (8) coupled with the efficiencies shown in Fig. 1 implies that when we observe a line with luminosity $L_{\rm {line}}\ga5\times10^{44}$ erg s^-1(Ghisellini et al. 2002) a thermal model in the flash heating regime requires a radius larger than $R\ga2\times10^{17}$ cm. Analogous results are obtained by imposing a ten times larger continuum luminosity. Such a large radius implies that the line luminosity should remain constant for a time $t=R(1-\cos\theta)/c$, to be compared with the observed variability time scales $t_{\rm {var}}\sim2\times10^4$ s observed in e.g. GRB 011211 (R02). This constraint can be satisfied if the heated plasma covers a very small solid angle, of opening angle $\theta\sim4.5^\circ$, at the limit but consistent with the smallest opening angles inferred for GRB jets from afterglow modelling (Frail et al. 2001; Panaitescu & Kumar 2002). In order to satisfy the cooling condition, however, the plasma must be dense ( $n\ga10^{11}$ cm^-3) and therefore confined in a very thin shell (or small clumps) with $\eta_R\la5\times10^{-5}$. In addition, if we assume that the plasma is due to a spherical shell surrounding the GRB explosion site which is heated only in the small polar cap by afterglow photons, the shell mass would turn out to be $M=4\pi~R^2~m_{\rm p}~\tau_T/\sigma_T \sim 600~M_\odot$ for the case explored above. Both the geometry factor $\eta_R$ and the total mass required seem to be rather extreme, even though the whole picture is, in this case, completely self consistent.

3.2 Steady heating

Let us now consider a plasma in which a source of heating is active for a time scale $t_{\rm {heat}}>1.2\times10^{15}~T_8/n=t_{\rm {cool}}$. In the case of radiative heating, this time scale may be for example the GRB duration or even longer, if the afterglow photons can contribute to the heating. In steady heating, the line luminosity (continuum luminosity for $\eta_{\rm {line}}=1$) can be written as:

$$L_{\rm {line}}=7.2\times10^{36}~\tau_T^2~T_8^{1/2}~ \eta_{\rm... ...over{\eta_R}} \equiv c_{\rm h} {{t_{\rm {heat}}}\over{\eta_R}}$$ (9)

where we have defined the parameter $c_{\rm h}$ in order to emphasize the ratio of the heating time scale over the geometry parameter. In Fig. 4 we show the highest possible values of the coefficient $c_{\rm h}$ as a function of temperature for five different values of metallicity (as in Fig. 1). The lines are plotted only when the equivalent width of the line is predicted to be larger than 100 eV, corresponding to the unshaded areas in Fig. 1.

To compare these results with afterglow data, consider a Fe line with L=10⁴⁵ erg s^-1 (GRB 991216; Piro et al. 2000) or the S line in GRB 011211 (R02) with luminosity $L=4\times10^{44}$ erg s^-1. In both cases a ratio $t_{\rm {heat}}/\eta_R\ga10^9$ is required, by comparison with the appropriate panel in Fig. 4. A typical heating time can be considered to be the fireball transit time, which is equivalent to the total duration of the prompt GRB emission. In both cases such duration is of the order of $\sim$100 s, leaving us with an extreme requirement of anisotropy $\eta_R\la10^{-7}$. Alternatively, the heating may be provided radiatively by the absorption of GRB and early afterglow photons. In that case, the duration of the heat supply may be of several per cent of the line emission time scale (Lazzati et al. 2002b). In that case the constraint would be relaxed to the (yet still challenging) value $\eta_R\la10^{-6}$. The most convenient solution is represented by an (unknown) heating mechanism acting for a time comparable to the line emission time scale itself. In this case, the constraint on the geometric factor $\eta_R$ would be similar to the one derived above for the flash heating case. Such a heating mechanism is however presently obscure and will have to face the problem of stability discussed below (Sect. 4).

In a steady heating scenario, therefore, thermal lines and continua can dominate the early afterglow emission, only in case of extreme geometric conditions, in which the emission is produced either in a sheet like shell or an extremely clumpy medium. Interestingly, analogous extreme conditions were independently inferred for the environment of GRB 000210 (Piro et al. 2002) in order to account for the lack of ionization features in the soft X-ray afterglow spectrum. Piro et al. (2002, and references therein) argue also that such conditions may be realized in giant molecular clouds. Stability considerations (see Sect. 4), however, show that more extreme conditions are required in this case. Unlike what is derived in the flash heating condition, in this case the radius of the shell does not have to be very large, and therefore the total mass required is not huge.