Shell Collisions: Refreshed Shocks¶

This example demonstrates how to model refreshed shocks from delayed collisions between relativistic shells using the trailing_shells parameter.

Background¶

Many GRB afterglow light curves exhibit rebrightenings --- episodes where the flux temporarily increases before resuming its power-law decay. A leading physical explanation is shell collisions: the central engine launches multiple relativistic shells at different times and with different Lorentz factors. Faster (inner) shells catch up to slower (outer, decelerated) shells and collide, depositing energy and momentum into the blast wave.

Key references:

• Sari & Meszaros 2000, ApJ, 535, L33 --- refreshed shocks from multiple shell collisions
• Kumar & Piran 2000, ApJ, 532, 286 --- energy injection from shell interactions
• Akl et al. 2026, A&A (arXiv:2603.08555) --- multi-epoch rebrightenings in GRB 250129A

Physics¶

When a trailing shell (energy \(E_2\), Lorentz factor \(\Gamma_2\), launched at time \(t_\mathrm{launch}\)) catches the decelerating blast wave (Lorentz factor \(\Gamma_1\), total mass \(m_1\)), the collision is resolved via energy-momentum conservation:

The kinetic energy dissipated in the collision thermalizes:

The trailing shell coasts ballistically at \(R = \beta_2 c (t - t_\mathrm{launch})\) until its radius overtakes the blast wave radius. The collision is detected and resolved during the ODE integration in Rust, ensuring proper coupling with the hydrodynamics.

Usage¶

Pass a list of (E_iso, Gamma0, t_launch) tuples to the trailing_shells parameter:

import numpy as np
from blastwave import Jet, TopHat, ForwardJetRes

theta_j = np.radians(6.0)

jet = Jet(
    TopHat(theta_j, 1.7e53, lf0=100.0),  # leading shell
    0.0,                                    # nwind
    1.0,                                    # nism
    tmin=10.0,
    tmax=5e7,
    grid=ForwardJetRes(theta_j, 65),
    tail=True,
    spread=False,
    trailing_shells=[
        (1e54,  40.0,  83.0),         # shell 2: E_iso, Gamma0, t_launch [s]
        (8e53,  25.0,  83.0 + 43200), # shell 3: launched 0.5 days later
    ],
)

Each trailing shell is specified as:

The flux computation is the same as for any Jet:

P = dict(
    Eiso=1.7e53, lf=100.0, theta_c=theta_j, A=0.0, n0=1.0,
    p=2.3, eps_e=0.09, eps_b=2.5e-3,
    theta_v=0.0, z=1.0, d=6700.0,
)

t = np.geomspace(100, 5e7, 400)
flux = jet.FluxDensity(t, 4.68e14, P)  # optical R-band

Multi-band rebrightening¶

The shell collision produces a chromatic rebrightening whose strength depends on frequency:

Radio (1 GHz): The rebrightening is moderate because self-synchrotron absorption limits the peak flux. The post-collision blast wave has more energy, so it peaks later and brighter than the baseline.

Optical (R-band): The rebrightening is strongest in the optical, where the collision boosts both the characteristic frequency and the peak emission. The light curve shows a clear step up at the collision time.

X-ray (2 keV): The collision produces a noticeable bump in the X-ray, though the effect is somewhat weaker since X-ray emission comes from the highest-energy electrons which cool rapidly.

Parameter study¶

The rebrightening amplitude and timing depend on the trailing shell's energy and Lorentz factor:

Left: Varying the shell energy \(E_2\) at fixed \(\Gamma_2 = 40\). More energetic shells produce brighter rebrightenings and inject more thermal energy, delaying the eventual deceleration.

Right: Varying the shell Lorentz factor \(\Gamma_2\) at fixed \(E_2 = 10^{54}\) erg. Faster shells (\(\Gamma_2 = 80\)) catch up later (closer to the leading shell's speed) and deposit less relative kinetic energy. Slower shells (\(\Gamma_2 = 20\)) catch up sooner and produce stronger collisions due to the larger relative Lorentz factor.

Physical parameters¶

Full script¶

The complete analysis script is at examples/shell_collision.py. To regenerate the plots:

python examples/shell_collision.py