5. Considerations on the GRB structures descending from a constant-thickness approximation
We now proceed to some specific prediction of GRB features computed by using the constant-thickness approximation and the Eqs. (36)-(38), (43)-(44) and (47)-(50) in the range of validation of this approximation just defined in the previous paragraph. As an example for clearly showing the evolution of PEM pulses colliding with baryonic matter, we take the following black hole mass and charge, as well as the mass of baryon remnant as a typical case:
where the total energy of dyadosphere ergs, so the total number of -pairs created in the dyadosphere (given by Eq. (10)) , and baryonic mass . This baryonic mass is close to the limit of validation of the slab model shown in Sect. 4. We have assumed the baryonic matter at a distance of , very close to the transparency condition of the PEM pulse in vacuum (see Ruffini et al. 1999).
In Fig. 8 we represent the Lorentz Factor of the PEM pulse as a function of the radius for collision with different amounts of baryonic matter, corresponding respectively to , and . The diagram extends to values of the radial coordinate at which the transparency condition given by Eq. (34) is reached. Also represented, for each diagram, is the "asymptotic" Lorentz Factor:
The closer the value approaches, at transparency, the "asymptotic" value (52), the smaller the intensity of the radiation emitted in the burst, and the larger the amount of kinetic energy left in the baryonic matter. This point is further clarified in Fig. 9, where are plotted the -factor at transparency and the "asymptotic" one as functions of the baryonic matter. It is interesting that, for a given EMBH, there is a maximum value of the -factor at transparency. After that maximum value, the energy available for the GRB is smaller in intensity, and at decreasing values of the energy, for increasing values of the baryonic mass.
The temperature in the laboratory frame at the transparency point is plotted as a function of the baryonic mass in Fig. 10: it strongly decreases as the baryonic mass increase. The is related to the observed energy-peak of the photon-number spectrum (see e.g., Ruffini et al. 1999).
We plot in Fig. 11 the energy radiated in the burst and the final kinetic energy of baryonic matter. We find that, for small values of B (around ), almost all total energy is radiated as GRB (see also our previous paper Ruffini et al. 1999), and very little energy is left as kinetic energy of baryonic matter as afterglow. While for roughly only of the total initial energy of the dyadosphere is radiated away as GRB, and almost all energy is restored as the kinetic energy of the baryonic matter. It is also clear that for the intensity of the Burst (see also Fig. 8) and the observed radiation frequency drifted to smaller and smaller values and are of little astrophysical interest from the point of view of GRBs. For such values, the energy of the dyadosphere is transferred practically totally to the bulk kinetic energy of the baryonic matter. is also the limit of the validation of our computations based on the analytical slab model, described in Sect. 3. For values of baryonic matter in between these two extreme cases ( and ) the analytical slab model covers the whole range of the observed properties of Gamma Ray Bursts.
We turn now to the thermodynamic parameters relevant in the evolution of the PEM pulse. In Fig. 12 the temperature of PEM pulse, both in the comoving and in the laboratory frame, are given as a function of the radius for a typical case ( and ).
In Fig. 13, we plot the total energy of the non baryonic components of the PEM pulse, which includes both thermal and kinetic energy, and the kinetic energy of the baryonic matter as functions of the radius, for the typical case and . The total energy of the non baryonic components of the PEM pulse is equal to before the collision (see Ruffini et al. 1999) and drops after the collision. While the kinetic energy of baryonic matter
increases as function of radius for . The sum of both them is equal to the total energy ergs during the evolution of the PEM pulse. The value of the total energy of the non baryonic components of the PEM pulse at the transparency point, the ending point of the curve in Fig. 13, is the energy released in the burst. We have discussed this energy as function of baryonic masses in Fig. 11.
where is the thermal energy of baryonic matter, and we can neglect, in Eq. (55), the pressure of the baryonic matter. A sudden increase of the entropy occurs during the collision both for the addition of the baryonic matter, and for the thermal reheating due to the inelastic collapse of the PEM pulse with the baryonic matter at rest. From the energy and momentum conservations, we obtain for the values of the and the temperature during the collision: the proper internal energy, , of the plasma increases as
as baryon mass, , is incrementally gained.
Before the collision the PEM pulse expands keeping its temperature in the laboratory frame constant, while its temperature in the comoving frame falls (see Ruffini et al. 1999). During the collision, a heating of the plasma due to the energy and momentum conservation occurs (see also Fig. 14, where reheating process leads to an increment of the number density of pairs). As the system expands further, both the comoving temperature and the temperature in the laboratory frame decreases, since the total energy of the pairs and the photons before the collision is constant and equal to , while after the collision
where is only the thermal energy of the baryons.
It is also interesting to monitor the change of temperature in the comoving frame before the collision and after the collision (see also Fig. 15). Before the collision, due to the entropy conservation, in the process of annihilation the factor (where the subscript "" means "initial value") increases to a value near to (see Weinberg 1972; Ruffini et al. 1999) since the collision occurs at , near the condition of transparency. The number of pairs has now been reduced drastically. The further jump in the value of the ratio is principally due to the energy and momentum conservation during the inelastic collision. After the collision, there is a small reheating due to the annihilation of the pairs created in the collision (see Fig. 14), which cannot be seen on the scale of our plot. Then, the ratio remains constant.
© European Southern Observatory (ESO) 2000
Online publication: July 13, 2000