Reverse and forward shock afterglow emission from steep jets viewed off axis

We study the morphology of gamma-ray burst (GRB) afterglows viewed off-axis using a simplified analytical model. We consider steep jets, which are expected to be the most common type. These jets, characterized by steep lateral gradients in energy and Lorentz factor, produce highly beamed emission. The observed signal is dominated by their minimum visible angle at any given time. Consequently, the afterglow morphology depends on when this angle begins to decrease, revealing the inner regions of the jet. Depending on whether this decrease occurs before, at, or after the reverse shock crosses the ejecta, three distinct classes of light curves emerge. In the first scenario, the de-beamed emission can produce a rapidly rising signal even prior to the reverse shock crossing. This is expected in GRBs with long duration, low energy, dense circumburst media, or combinations thereof. In some cases, the ejecta shell can be considered as effectively thick in the inner regions and effectively thin in the outer regions. For forward shocks, the temporal slopes in both regimes are identical, which makes it hard to detect the transition. Reverse shocks, however, have distinct temporal slopes, allowing potential detection of the transition in light curves if their emission surpasses that of the forward shock. The characteristic synchrotron frequency of de-beamed emission evolves independently of jet structure for forward shocks but depends on the lateral energy and Lorentz factor gradients for the reverse shock, with slower evolution for steep energy and shallow Lorentz factor gradients.