11 The parameters of the CB model

With the current analysis of GRB radio AGs we have completed a first round of the study of GRB properties in the CB model, and it behooves us to look back at the various constraints on the relevant parameters.

In Dar & De Rújula (2000a) we followed Dar (1998) and Dar & Plaga (1999) in suggesting that the large peculiar velocities of neutron stars may be due to a ``natal kick'', induced by a momentum imbalance in the oppositely-directed jets of CBs accompanying their birth. On this basis we chose as a reference value $E_{\rm CB}\sim 10^{52}$ erg, for a jet with a reference number of CBs (or prominent peaks in the GRB light-curve) $n_{_{\rm CB}}\sim 10$. Based on a first analysis of AG properties, and of GRB $\gamma$-ray fluences and individual $\gamma$-ray energies, we set $\gamma_0=10^3$as a reference value.

In Dar & De Rújula (2000b) we investigated two extreme models meant to bracket the behaviour of a CB as it crosses a SN shell or its progenitor's ``wind", is heated by the collision with its constituents, and emits observable $\gamma$-rays as it reaches the shell's transparent outskirts with a radius $R_{_{\rm CB}}^{\rm tr}$, proportional to its early transverse expansion velocity $\beta_{\rm trans}\,c$, which we assumed to be close to the sound speed in a relativistic plasma, $\rm\beta_{trans}=1/\sqrt{3}$. In our ``surface'' model, which is no doubt closer to a realistic description, the energy of the GRB in $\gamma$-rays is proportional to $n_{_{\rm CB}}\, [R_{\rm CB}^{\rm tr}]^2 \,\gamma_0$. (Eq. (45) of Dar & De Rújula 2000b). For the chosen reference parameters, in the surface model, this prediction overestimates the GRB fluences by about one order of magnitude. Since the individual $\gamma$-ray energies corroborate the choice $\gamma_0\sim 10^3$, this means that $[R_{_{\rm CB}}^{\rm tr}]^2$ (and $\rm\beta_{trans}$) are overestimated by roughly one order of magnitude.

In Dar & De Rújula (2001) we analyzed the X-ray ``Fe'' lines observed in the AGs of some GRBs, which we attributed to hydrogen recombination in the CBs, with the corresponding Lyman-$\alpha$ lines boosted by a large factor $\delta/(1+z)\sim 500$. We equated the total number of photons in the lines to the baryon number of the jet of CBs, and found agreement with the baryon number in the jet, $n_{_{\rm CB}}\,N_{\rm CB}$, to within one order of magnitude. But in the current investigation, we have found that the absorption of radio waves keeps the CBs hot and ionized (Appendices C and E). This means that our reference value of  $N_{\rm CB}$ is likely to be an overestimate.

In DDD 2001 we proposed a mechanism that would quench the expansion of a CB in minutes of observer's time, well after it has exited the SN shell. The CBs reach an asymptotic radius (Eq. (16) of DDD 2001):

$$R_{\rm max}^3\simeq {3\,N_{\rm CB}\,\beta_{\rm trans}^2\over 2\,\pi\,n_p\,\gamma_0^2}\cdot$$ (32)

On the basis of this calculated radius, for $\rm\beta_{trans}= 1/(3\sqrt{3})$, we found that the normalization of optical and X-ray AGs agreed with the reference-value expectations. On the same basis, we find now that the normalization is overestimated by an order of magnitude. The reason for the discrepancy is that, in DDD 2001, we effectively placed the spectral discontinuity at a ``cooling break'' frequency corresponding to an electron Lorentz factor $\rm\gamma_e\sim 1$, while we have now argued that the discontinuity should occur at a higher value $\gamma_{\rm e}=\gamma(t)$.

Both the GRB fluence and the AG fluence are, in the CB-model, $F\propto n_{_{\rm CB}}\,R^2$, with $R=R_{_{\rm CB}}^{\rm tr}$ for the $\gamma$ rays and $R=R_{\rm max}$ for the AG. At a value of $x_\infty$ fixed by the fit to the AG's temporal behaviour, the AG fluence is:

$$F_{\rm ~AG}\propto n_{_{\rm CB}}\,n_{e}\,R_{\rm max}^2= n_{_{\rm CB}}\,N_{\rm CB}\,{1\over \pi\, x_\infty}\cdot$$ (33)

All of the above ``problems'' are solved if we reduce our ``typical'' values of $R_{\rm max}^2$ and $N_{\rm CB}$by about one order of magnitude, relative to our reference parameters with, according to Eq. (32), the corresponding reduction of the choice of $\rm\beta_{trans}$ by half an order of magnitude.

The precise location of the injection bend is not predictable and a modification by up to one order of magnitude of its position has a small effect on the quality of the fits to observations. An increase of the cooling break frequency $\rm\nu_b$ implies a corresponding decrease in AG flux, see Eq. (4), adding to the uncertainty in the prediction of the precise overall normalization.

To summarize, the CB model correctly describes, in terms of a very limited set of parameters, the properties of GRBs and their AGs, including their normalizations. This is the case even if we adhere to all of the detailed assumptions we have made, even though they are approximations to a no doubt fairly convoluted physical problem.