The Structure and Evolution of Relativistic Jetted Blast Waves

Abstract: We study, analytically and numerically, the structure and evolution of relativistic jetted blast waves that propagate in uniform media, such as those that generate afterglows of gamma-ray bursts. Similar to previous studies, we find that the evolution can be divided into two parts: (i) a pre-spreading phase, in which the jet core angle is roughly constant, $\theta_{c,0}$, and the shock Lorentz factor along the axis, $\Gamma_a$, evolves as a part of the Blandford-Mckee solution, and (ii) a spreading phase, in which $\Gamma_a$ drops exponentially with the radius and the core angle, $\theta_c$, grows rapidly. Nevertheless, the jet remains collimated during the relativistic phase, where $\theta_c(\Gamma_a\beta_a=1)\simeq 0.4\theta_{c,0}^{1/3}$. The transition between the phases takes place when $\Gamma_a\simeq 0.2\theta_{c,0}^{-1}$. We find that the "wings" of jets with initial "narrow" structure ($\frac{d \log\,E_{iso}}{d\log\,\theta}<-3$ outside of the core, where $E_{iso}$ is isotropic equivalent energy), start evolving during the pre-spreading phase. By the spreading phase these jets evolve to a self-similar profile, which is independent of the initial structure, where in the wings $\Gamma(\theta)\propto\theta^{-1.5}$ and $E_{iso}(\theta)\propto \theta^{-2.6}$. Jets with initial "wide" structure roughly keep their initial profile during their entire evolution. We provide analytic description of the jet lateral profile evolution for a range of initial structures, as well as the evolution of $\Gamma_a$ and $\theta_c$. For off-axis GRBs, we present a relation between the initial jet structure and the light curve rising phase. Applying our model to GW170817, we find that initially the jet had $\theta_{c,0}=0.4-4.5~\deg$ and wings which are consistent with $E_{iso} \propto \theta^{-3}-\theta^{-4}$.