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 $E_{iso} \propto \theta^{-3}-\theta^{-4}$.