Blast waves and reverse shocks: from ultra-relativistic GRBs to moderately relativistic X-ray binaries

Blast wave models are commonly used to model relativistic outflows from ultra-relativistic gamma-ray bursts (GRBs), but are also applied to lower Lorentz factor ejections from X-ray binaries (XRBs). Here we revisit the physics of blast waves and reverse shocks in these systems and explore the similarities and differences between the ultra-relativistic ($\Gamma \gg 1$) and moderately relativistic ($\Gamma \sim$ a few) regimes. We first demonstrate that the evolution of the blast wave radius as a function of the observer frame time is recovered in the on-axis ultra-relativistic limit from a general energy and radius blast wave evolution, emphasizing that XRB ejections are off-axis, moderately relativistic cousins of GRB afterglows. We show that, for fixed blast wave or ejecta energy, reverse shocks cross the ejecta much later (earlier) on in the evolution for less (more) relativistic systems, and find that reverse shocks are much longer-lived in XRBs and off-axis GRBs compared to on-axis GRBs. Reverse shock crossing should thus typically finish after $\sim10-100$ days (in the observer frame) in XRB ejections. This characteristic, together with their moderate Lorentz factors and resolvable core separations, makes XRB ejections unique laboratories for shock and particle acceleration physics. We discuss the impact of geometry and lateral spreading on our results, explore how to distinguish between different shock components, and comment on the implications for GRB and XRB environments. Additionally, we argue that identification of reverse shock signatures in XRBs could provide an independent constraint on the ejecta Lorentz factor.