The propagation of relativistic jets in expanding media

Abstract: We present a comprehensive analytic model of relativistic jet propagation in expanding homologous media (ejecta). This model covers the entire jet evolution as well as a range of configurations that are relevant to binary neutron star mergers. These include low and high luminosity jets, unmagnetized and mildly magnetized jets, time-dependent luminosity jets, and Newtonian and relativistic head velocities. We also extend the existing solution of jets in a static medium to power-law density media with index $\alpha<5$. Our model provides simple analytic formulae (calibrated by 3D simulations) for the jet head propagation and breakout times. We find that the system evolution has two main regimes: strong and weak jets. Strong jets start their propagation immediately within the ejecta. Weak jets are unable to penetrate the ejecta at first, and breach it only after the ejecta expands significantly, thus their evolution is independent of the delay between the onset of the ejecta and the jet launching. After enough time, both strong and weak jets approach a common asymptotic phase. We find that a necessary, but insufficient, criterion for the breakout of unmagnetized [weakly magnetized] jets is $E_{j,{\rm iso,tot}} \gtrsim 3[0.4]E_{ej,{\rm tot}}\left({\theta_j}/{0.1{\rm~rad}}\right)^2$, where $E_{j,{\rm iso,tot}}$ is the jet total isotropic equivalent energy, $\theta_j$ is its opening angle, and $E_{ej,{\rm tot}}$ is the ejecta energy. Applying our model to short GRBs, we find that there is most likely a large diversity of ejecta mass, where mass $ \lesssim 10^{-3}~{\rm M}_{\odot} $ (at least along the poles) is common.