Numerical Simulations of an Initially Top-Hat Jet and the Afterglow of GW170817\,/\,GRB170817A

The afterglow of GRB$\,$170817A/GW$\,$170817 was very unusual, slowly rising as $F_\nu\propto{}t_{\rm{}obs}^{0.8}\nu^{-0.6}$, peaking at $t_{\rm{obs,pk}}\sim\,150\;$days, and sharply decaying as $\sim{}t_{\rm{}obs}^{-2.2}$. VLBI observations revealed an unresolved radio afterglow image whose flux centroid moved superluminally with $v_{\rm{app}}\approx4c$, clearly indicating that the afterglow was dominated by a relativistic jet's compact core. Different jet angular structures explained the afterglow lightcurves: Gaussian and steep power-law profiles with narrow core angles $\theta_c\lesssim5^\circ$ and larger viewing angles $\theta_{\rm{}obs}/\theta_c\sim3-5$. However, a top-hat jet (with sharp edges at $\theta=\theta_0$) was ruled out since it appeared to produce an early flux rise much steeper than observed. Using 2D relativistic hydrodynamic simulations we show that the initial steep flux rise is an artifact caused by the simulation's finite start time, $t_0$, missing its flux contributions from $t<t_0$ and sometimes "compensated" using an analytic top-hat jet. While an initially top-hat jet is not very physical, such simulations are particularly useful at $t_{\rm{}obs}\gtrsim{}t_{\rm{obs,pk}}$ when the afterglow emission is dominated by the jet's core and becomes insensitive to its exact initial angular profile if it drops off sharply outside of the core. We demonstrate that an initially top-hat jet fits GW$\,$170817/GRB$\,$170817A's afterglow lightcurves and flux centroid motion at $t_{\rm{}obs}\gtrsim{}t_{\rm{obs,pk}}$, for $\theta_{\rm{}obs}/\theta_0\approx3$ and may also fit the earlier lightcurves for $\Gamma_0=\Gamma(t_0)\gtrsim10^{2.5}$. We analytically express the degeneracies between the model parameters, and find a minimal jet energy of $E_{\rm{}min}\approx5.3\times10^{48}\;$erg and circum-burst medium density of $n_{\min}\approx5.3\times10^{-6}~{\rm cm}^{-3}$.