An Accretion-Modulated Internal Shock Model for Long GRBs

TL;DR

The study introduces the Accretion-Modulated Internal Shocks (AMIS) framework to explain long GRB prompt emission by tying a time-dependent mass-supply history to the central engine with internal shock dissipation. The approach yields a FRED-like broad envelope driven by fallback- or collapse-regulated accretion and superimposed rapid variability from stochastic shell collisions, captured through the Kobayashi–Piran–Sari (KPS) collision dynamics. Two modulation regimes are explored—mass-driven and rate-driven—demonstrating how pulse widths and amplitudes co-evolve with the engine feeding history, and a Norris FRED fit is used to quantify the envelope. The model also accounts for energy-dependent pulse widths via a phenomenological $E_p$–luminosity relation and a variable low-energy photon index, while acknowledging limitations and outlining paths for future integration with GRMHD jet-launching physics. AMIS thus provides a physically motivated, engine-level interpretation of both global temporal trends and fine-scale GRB variability, with testable predictions for pulse properties and spectral evolution.

Abstract

We introduce the Accretion-Modulated Internal Shock model (AMIS) as a possible framework for explaining the observational properties of long gamma-ray burst (GRB) prompt emission. In this scenario, the envelope of the prompt light curve follows the time-dependent mass-supply history to the central engine, associated with stellar collapse and, where applicable, fallback accretion, whose early-time onset can be approximated by $\dot{M}\propto t^{0-1/2}$ and which subsequently may decay as $\dot{M}\propto t^{-5/3}$, producing a photon count rate with a single fast-rise-exponential-decay (FRED)-like profile. In general, the prompt-emission envelope is regulated by a time-dependent mass supply to the central engine, while internal shocks produce the rapid variability. Since we only aim to introduce this framework here, we focus on the simplest single-FRED shape of the prompt emission profiles, while more complex cases involving multiple episodes and interacting shocks will be explored in forthcoming studies. The model indicates correlations between spectral evolution, FRED-pulse narrowing at high energies, and the mass-supply-controlled envelope. Stochastic Lorentz factor variations of ejected mass- or rate-driven shells, superimposed on the Accretion-Modulated envelope, explain the coexistence of smooth global trends and irregular short-timescale features, such as the widths of individual pulses in long GRB light curves, offering diagnostic tools for probing the inner engine activity.

Figures (3)

• Figure 1: Simulated GRB prompt-emission light curves from an Accretion-Modulated internal-shock (AMIS) model. Mass-driven scenario. The fast-rise, slow-decay shape of the envelope is imposed by a time-dependent, accretion-regulated mass-supply history of the central engine, associated with stellar collapse and accretion-driven mass delivery, which modulates the ejected shell masses ($m_0 = 3\times10^{28}$ g, $t_0=0.1$ s, $t_p=6$ s, $s=5$, and lab-frame widths $l_i \in [10^{9},2\times10^{9}]\,\mathrm{cm}$) at roughly constant ejection intervals. Stochastic variations in shell Lorentz factors ($\Gamma_i \in [400,800]$) generate internal shocks, producing the fine-scale, multi-peaked structure typical of GRB light curves. The simulation yields a total dissipated energy $E_{\rm diss} \simeq 1.036\times10^{52}$ erg, a total initial kinetic energy $E_{\rm kin} \simeq 5.542\times10^{53}$ erg, and a time-averaged radiative efficiency $\epsilon \simeq 0.019$. Pulse FWHMs are measured from the observed peaks of the smoothed light curve, ensuring that closely spaced or merged collisions are treated as single pulses. The resulting FWHMs remain approximately constant, reflecting the nearly uniform hydrodynamic and angular timescales associated with fixed ejection intervals. A locally weighted smoothing (LOWESS) of the individual pulse widths highlights only minor fluctuations around this overall flat trend.
• Figure 2: Simulated GRB prompt-emission light curves from an Accretion-Modulated internal-shock (AMIS) model. Rate-driven scenario. Shell ejection intervals follow the time-dependent, accretion-regulated mass-supply history, while the shell masses remain nearly constant ($m_i \sim 3\times10^{28}$ g). Random fluctuations in the Lorentz factors cause shells to collide at irregular intervals, generating internal shocks that imprint variability on the light curve. The simulation yields a total dissipated energy $E_{\rm diss} \simeq 1.526\times10^{52}$ erg, a total kinetic energy $E_{\rm kin} \simeq 1.062\times10^{54}$ erg, and a time-averaged radiative efficiency $\epsilon \simeq 0.014$. Pulse FWHMs evolve with the mass-supply rate: pulses are narrower and brighter during phases of lower mass supply and broader and dimmer during phases of higher mass supply. The trend is derived using locally weighted smoothing over individual pulse widths to highlight the overall temporal evolution.
• Figure 3: 2005ApJ...627..324N pulse fitting of the mass-driven AMIS luminosity envelope and the corresponding MCMC parameters. The blue points represent the most significant small pulse peaks used for the fit, the dashed curve shows the AMIS simulated luminosity, and the solid black line shows the Norris FRED profile. The full MCMC posterior distributions are shown in the corner plot, while the prior ranges and the posterior median values with 68% credible intervals are summarized in Table \ref{['tab:mcmc_priors_post']}. Best-fit parameters (defined as the posterior median values with 68% credible intervals) are $A = (2.83^{+2.18}_{-1.03})\times10^{51}$ erg s$^{-1}$, $t_{\rm start} = 0.692^{+0.193}_{-0.303}$ s, $\tau_1 = 0.449^{+0.368}_{-0.258}$ s, and $\tau_2 = 13.242^{+3.147}_{-2.478}$ s. The inferred FRED peak time, $t_{\rm pk}=t_{\rm start}+\sqrt{\tau_1\tau_2} = 3.1\pm0.9$ s, coincides with the characteristic peak timescale of the imposed mass-supply envelope ($t_p=3.5$ s).