  The discovery of GW170817, the merger of a binary neutron star (NS) triggered
by a gravitational wave detection by LIGO and Virgo, has opened a new window of
exploration in the physics of NSs and their cosmological role. Among the
important quantities to measure are the mass and velocity of the ejecta
produced by the tidally disrupted NSs and the delay - if any - between the
merger and the launching of a relativistic jet. These encode information on the
equation of state of the NS, the nature of the merger remnant, and the jet
launching mechanism, as well as yielding an estimate of the mass available for
r-process nucleosynthesis. Here we derive analytic estimates for the structure
of jets expanding in environments with different density, velocity, and radial
extent. We compute the jet-cocoon structure and the properties of the broadband
afterglow emission as a function of the ejecta mass, velocity, and time delay
between merger and launch of the jet. We show that modeling of the afterglow
light curve can constrain the ejecta properties and, in turn, the physics of
neutron density matter. Our results increase the interpretative power of
electromagnetic observations by allowing for a direct connection with the
merger physics.
