The binary neutron star (BNS) merger event GW170817 clearly shows that a BNS merger launches a short Gamma-Ray Burst (sGRB) jet. Unlike collapsars, where the ambient medium is static, in BNS mergers the jet propagates through the merger ejecta that is expanding outward at substantial velocities ($sim 0.2c$). Here, we present semi-analytic and analytic models to solve the propagation of GRB jets through their surrounding media. These models improve our previous model by including the jet collimation by the cocoon self-consistently. We also perform a series of 2D numerical simulations of jet propagation in BNS mergers and in collapsars to test our models. Our models are consistent with numerical simulations in every aspect (the jet head radius, the cocoons lateral width, the jet opening angle including collimation, the cocoon pressure, and the jet-cocoon morphology). The energy composition of the cocoon is found to be different depending on whether the ambient medium is expanding or not; in the case of BNS merger jets, the cocoon energy is dominated by kinetic energy, while it is dominated by internal energy in collapsars. Our model will be useful for estimating electromagnetic counterparts to gravitational waves.
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