Hydrodynamic Collimation of Gamma-Ray-Burst Fireballs

Analytic solutions are presented for the hydrodynamic collimation of a relativistic fireball by a surrounding baryonic wind emanating from a torus. The opening angle is shown to be the ratio of the power output of the inner fireball to that of the exterior baryonic wind. The gamma ray burst 990123 might thus be interpreted as a baryon-poor jet (BPJ) with an energy output of order ${10}^{50}\mathrm{erg}$ or less, collimated by a baryonic wind from a torus with an energy output of order ${10}^{52.5}\mathrm{erg}$, roughly the geometric mean of the BPJ and its isotropic equivalent.