Afterglow Lightcurves from Misaligned Structured Jets

Abstract: GRB 170817A/GW 170817 is the first GRB clearly viewed far from the GRB jet's symmetry axis. Its afterglow was densely monitored over a wide range of frequencies and times. It has been modeled extensively, primarily numerically, and although this endeavour was very fruitful, many of the underlying model parameters remain undetermined. We provide analytic modelling of GRB afterglows observed off-axis, considering jets with a narrow core (of half-opening angle $\theta_{\rm c}$) and power-law wings in energy per unit solid angle ($\epsilon=\epsilon_c\Theta^{-a}$ where $\Theta=[1+(\theta/\theta_{\rm c})^2]^{1/2}$) and initial specific kinetic energy ($\Gamma_0-1=[\Gamma_{\rm c,0}-1]\Theta^{-b}$), as well as briefly discuss Gaussian jets. Our study reveals qualitatively different types of lightcurves that can be viewed in future off-axis GRBs, with either single or double peaks, depending on the jet structure and the viewing angle. Considering the lightcurve shape rather than the absolute normalizations of times and / or fluxes, removes the dependence of the lightcurve on many of the highly degenerate burst parameters. This study can be easily used to determine the underlying jet structure, significantly reduce the effective parameter space for numerical fitting attempts and provide physical insights. As an illustration, we show that for GRB 170817A, there is a strong correlation between the allowed values of $\Gamma_{\rm c,0}$ and $b$, leading to a narrow strip of allowed solutions in the $\Gamma_{\rm c,0}$-$b$ plane above some minimal values $\Gamma_{\rm c,0}\gtrsim 40, b\gtrsim1.2$. Furthermore, the Lorentz factor of the material dominating the early lightcurve can be constrained by three independent techniques to be $\Gamma_{0}(\theta_{\rm min,0})\approx5-7$.