The hydrodynamics of stratified ultra-relativistic outflows and the origin of GRB X-ray plateaus

Abstract

The origin of the X-ray plateau phase observed in a large fraction of gamma-ray burst afterglows remains debated. We present a novel analytic framework for the hydrodynamics of ultra-relativistic, radially stratified outflows interacting with an external medium. By explicitly accounting for a continuous distribution of Lorentz factors within the ejecta, we derive analytic expressions describing the evolution of a long-lived, mildly relativistic reverse shock and determine its crossing time. Then, we compute the resulting synchrotron emission from both the forward and reverse shocks. The forward shock naturally produces a shallow, long-lasting X-ray decay consistent with the observed properties of X-ray plateaus (including the Dainotti relation). We further show that reproducing the observed plateau durations requires the stratified ejecta to extend to Lorentz factors of order $\gtrsim 100$, consistent with the ultra-relativistic outflow that powers the prompt $γ$-ray emission. The reverse shock generates a long-lived millimeter emission component that outshines the forward shock emission at these wavelengths. Both the plateau and reverse shock emission terminate smoothly once the slowest ejecta are processed, marking a transition to the standard Blandford-McKee self-similar evolution without requiring late-time energy injection or an additional emission component. Such stratified outflows are expected on physical grounds, as the ultra-relativistic ejecta responsible for the prompt $γ$-ray emission are unlikely to be launched with a single Lorentz factor. This model provides a unified picture in which the same outflow powers the prompt emission, the X-ray plateau, and the subsequent afterglow evolution.

Figures (7)

• Figure 1: A schematic illustration of the forward–reverse shock structure, showing the four dynamical regions: (1) unshocked external medium, (2) shocked external medium, (3) shocked ejecta, and (4) unshocked ejecta. The pressure in the unshocked regions is negligible compared to that in the shocked regions, $p_1,p_4\ll p_2,p_3$. In our analytic treatment, the pressure and velocity are approximated as uniform throughout the shocked region, while the density is taken to be uniform within each of the regions (2) and (3), separated by the contact discontinuity.
• Figure 2: Exact analytic solutions for the LF ratio, $\gamma_4/\gamma_2$, and for the shocked-ejecta LF, $\bar{\gamma}_3$ (measured in the unshocked ejecta frame), shown as functions of the single dimensionless parameter $\gamma_4/\sqrt{f}$ (with $f\equiv\rho_4/\rho_1$). Also shown are the asymptotic limits: the UR RS regime ($\gamma_4^2\gg f$, Eq. (\ref{['eq:LorUR']})) and the Newtonian RS regime ($\gamma_4^2\ll f$, Eq. (\ref{['eq:LorNew']})).
• Figure 3: Schematic illustration of the ejecta LF structure. The cumulative mass profile $M(>\gamma)$ is shown as a function of LF, featuring a steep high-$\gamma$ tail above a characteristic LF $\gamma_\text{max}$ and a shallow, approximately constant profile at lower $\gamma_\text{min}$. The RS initially propagates through the steep part of the ejecta and subsequently enters the shallow region, as indicated by the arrows. The figure is qualitative and not drawn to scale.
• Figure 4: In Blue: numerical solution of Eq. (\ref{['eq:numeric']}) showing the dependence of the LF ratio $x\equiv\gamma_4/\gamma_2$ on the ejecta power-law index $s$. In red: the reverse shock LF in the frame of the unshocked ejecta, $\bar{\gamma}_3=\frac{1}{2}\left(x+\frac{1}{x}\right)$. In yellow: the ratio of the swept-up external medium mass, $M_{\text{ext}}$, to the canonical deceleration mass, $M_\text{iso}(>\gamma_4)/\gamma_4$rees_relativistic_1992. In green: The ratio between the shocked external medium energy, $E_2$, and the shocked ejecta energy, $E_3$. All are shown as a function of the ejecta power-law index $s$. For $s\gg1$, corresponding to a Newtonian RS, the outflow experiences negligible deceleration due to the RS, consequently, most of the energy remains in the ejecta ($E_3$).
• Figure 5: The corresponding correction factors for the RS crossing time, $h_\gamma$ (uniform ISM; $k=0$) and $h_a$ (wind environment; $k=2$), are shown. For moderate values $2\lesssim s\lesssim 4$, relevant when the RS propagates through the shallow part of the ejecta, the RS is mildly relativistic and the crossing time is significantly extended relative to the standard thin-shell estimates.