The evolution of beamed GRB afterglow: non-relativistic case

There has been increasing evidence that at least some GRBs are emission beamed. The beamed GRB afterglow evolution has been discussed by several authors in the ultra-relativistic case. It has been shown that the dynamics of the blast wave will be significantly modified by the sideways expansion, and there may be a sharp break in the afterglow light curves under certain circumstances. However, this is true only when the fireball is still relativistic. Here we present an analytical approach to the evolution of the beamed GRB blast wave expanding in the surrounding medium (density $n\propto r^{-s}$) in the non-relativistic case, our purpose is to explore whether the sideways expansion will strongly affect the blast wave evolution as in the relativistic case. We find that the blast wave evolution is strongly dependent on the speed of the sideways expansion. If it expands with the sound speed, then the jet angle $\theta $ increases with time as $\theta \propto ln t$, which means that the sideways expansion has little effect on the afterglow light curves, the flux $F\propto t^{-\frac{3(5\alpha -1)}{5}}$ for $s=0$ and $F\propto t^{-\frac{7\alpha +1}{3}}$ for $s=2$.It is clear that the light curve of $s=2$ is not always steeper than that of $s=0$, as in the relativistic case.We also show that if the expansion speed is a constant, then the jet angle $\theta \propto t$, and the radius $r \propto t^{0}$, in this case the sideways expansion has the most significant effect on the blast wave evolution, the flux $F\propto t^{-(5\alpha -1)}$ independent of $s$, and we expect that there should be a smooth and gradual break in the light curve.