tokens: the result by about a factor of 2. Therefore, the shock width in terms of the pair unloaded Thomson optical depth is Fig. 6 shows , obtained from integration of the full equations (without assuming x l > 1), as a function μΓ u / f for several values of Γ u (i.e. on each curve, Γ u remains constant while f varies). It shows that equation ( 13 ) provides an order of magnitude estimate of the shock width as a function of the photon escape fraction. Fig. 7 shows the Lorentz factor profile as a function of . It shows how the shock becomes narrower when the fraction of escaping photons increases. Once the optical depth to infinity becomes too small and photon escape starts affecting the shock structure, the shock adjusts itself and becomes narrower to match the finite optical depth. This is in contrast to a Newtonian shock that cannot adjust itself, and therefore full shock breakout follows within a dynamical time after photon escape starts becoming non-negligible. Figs 8 – 9 show the mechanism that allows the shock to adjust its width – a rapid buildup of x l within a unit interval of the optical depth τ, which compensates for the lack of backscattered photons at τ > τ ∞ . As can be seen from Figs 7 – 9 , the structures of all shocks with different values of f converge while the flow is still relativistic, before it approaches the downstream (τ ≲ 1), even when photon escape strongly affects the shock width. This result supports our assumption that the conditions in the immediate downstream are similar in all shocks, even in the presence of a significant photon escape. Figure 6. The shock width as a function of μΓ u / f . On each of the four curves, f varies while Γ u = 5, 15, 50 and 150 remains constant. is measured using the full analytic solution with η = 0.5 and a = 2, and is defined as the pair unloaded Thompson optical depth, where Γ = 0.9Γ u . The circles mark the location where , at which equation ( 13 ) predicts a transition in the shock width dependence on f . The solid black line is a plot of , and is given for guidance. This figure indicates that the width is given by equation ( 13 ) to within an order of magnitude. Figure 7. Lorentz factor profiles as a function of for shocks with Γ u = 10 and different f values. Figure 8. Profiles of the loading parameter as a function of for shocks with Γ u = 10 and different f values. Figure 9. Profiles of the loading parameter as a function of τ for shocks with Γ u = 10 and different f values. 4 SHOCK BREAKOUT In the previous section, we derived the steady-state structure of an RRMS with photon escape. In astrophysical settings, the shock propagates in a medium having a finite Thomson depth to infinity . Note that we use to describe the optical depth of the circumstellar medium since no pairs are created before it is encountered by the shock. For typical settings, we expect both Γ sh = Γ u and to vary with time. Then, once , photons starts leaking out. However, since the shock width continuously
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