GRB Afterglow of the Sub-relativistic Materials with Energy Injection

Sub-relativistic materials launched during the merger of binary compact objects and the core-collapse of massive stars acquire velocity structures when expanding in a stratified environment. The remnant (either a spinning magnetized neutron star (NS) or a central black hole) from the compact-object or core-collapse could additionally inject energy into the afterglow via spin-down luminosity or/and by accreting fall-back material, producing a refreshed shock, modifying the dynamics, and leading to rich radiation signatures at distinct timescales and energy bands with contrasting intensities. We derive the synchrotron light curves evolving in a stratified environment when a power-law velocity distribution parametrizes the energy of the shock, and the remnant continuously injects energy into the blastwave. As the most relevant case, we describe the latest multi-wavelength afterglow observations ($\gtrsim 900$ days) of the GW170817/GRB 170817A event via a synchrotron afterglow model with energy injection of a sub-relativistic material. The features of the remnant and the synchrotron emission of the sub-relativistic material are consistent with a spinning magnetized NS and the faster "blue" kilonova afterglow, respectively. Using the multi-band observations of some short-bursts with evidence of kilonova, we provide constraints on the expected afterglow emission.