An Accretion-Modulated Internal Shock Model for Long GRBs

This paper introduces the Accretion-Modulated Internal Shock (AMIS) model, which proposes that the smooth envelope of long gamma-ray burst prompt emission is governed by time-dependent mass accretion from stellar collapse and fallback, while rapid variability arises from internal shocks within the ejected shells.

The Gist

Imagine a Gamma-Ray Burst (GRB) as the most powerful fireworks display in the universe. For decades, astronomers have been trying to figure out exactly how the "fireworks factory" (the central engine) works.

This paper introduces a new theory called AMIS (Accretion-Modulated Internal Shock model). Think of it as a new recipe for how these cosmic explosions are cooked.

Here is the breakdown using simple analogies:

1. The Engine: A Star Eating Itself

Imagine a massive star that runs out of fuel and collapses. Instead of just disappearing, it leaves behind a dense core (like a black hole) that starts "eating" the leftover debris from the star.

• The Feeding Rate: The star doesn't eat at a steady pace. First, it grabs a huge mouthful quickly (the "fast rise"), and then it slowly chews and swallows the rest over time (the "slow decay").
• The Analogy: Think of a person eating a giant bowl of spaghetti. They take a huge bite at the start, then spend the next few minutes slowly slurping up the rest. The paper calls this the "Mass-Supply History."

2. The Light Curve: The Shape of the Meal

The paper argues that the overall shape of the GRB's light curve (the graph of brightness over time) is just a direct reflection of this eating schedule.

• The Envelope: If the engine is fed a lot of material quickly and then less and less, the light curve looks like a FRED (Fast-Rise, Exponential-Decay). It shoots up bright and then fades away slowly.
• The Analogy: The "envelope" of the light curve is like the silhouette of the person eating. If they eat fast and then slow down, the silhouette of their hand moving to their mouth goes up fast and comes down slow.

3. The Sparkles: Internal Shocks

But wait! GRB light curves aren't just smooth hills; they are jagged, spiky, and chaotic. They look like a mountain range with many peaks.

• The Cause: This is where the "Internal Shock" part comes in. As the engine spits out material to make the jet, it doesn't shoot it out perfectly evenly. Sometimes it shoots a fast shell of gas, and later, a slower shell.
• The Collision: The fast shell catches up to the slow shell and crashes into it. This crash creates a shockwave, which flashes a burst of light.
• The Analogy: Imagine a highway where cars are leaving a garage.
  • The Envelope: The total number of cars leaving the garage follows the "eating schedule" (lots at first, then fewer).
  • The Spikes: But, some cars are speeding (fast shells) and some are driving slowly (slow shells). When a speeder catches a slowpoke, they crash (Internal Shock). That crash is the little spike of light you see on the graph.

4. Two Ways the Factory Can Run

The authors tested two different ways this "factory" could operate to create the same jagged light curve:

• Scenario A: The "Mass-Driven" Factory

  • How it works: The factory spits out shells at a steady rhythm (like a clock ticking), but the size of the shells changes.
  • The Result: Early on, when the engine is "hungry," it spits out giant, heavy shells. Later, it spits out tiny pebbles.
  • Visual: You get big, loud crashes early on, and smaller, quieter crashes later. The width of the spikes stays roughly the same, but their height changes.

• Scenario B: The "Rate-Driven" Factory

  • How it works: The factory spits out shells of the same size, but the timing changes.
  • The Result: When the engine is "hungry," it spits shells out very frequently (thick shells). When it's "full," it spits them out slowly (thin shells).
  • Visual: You get wide, dim spikes when the engine is busy, and narrow, bright spikes when it slows down.

5. Why Does This Matter?

For a long time, scientists thought the jagged spikes were just random chaos. This paper suggests that the chaos is actually organized.

• The Big Picture: The smooth, slow fade of the burst tells us about the star's collapse and how the black hole is feeding.
• The Small Details: The jagged spikes tell us about the speed and timing of the shells crashing into each other.

The Takeaway

The AMIS model is like a conductor for an orchestra.

• The Conductor (the mass-supply from the collapsing star) sets the overall tempo and volume (the smooth rise and fall).
• The Musicians (the shells of gas) are playing their own notes. Sometimes they play loud, sometimes soft, sometimes they clash (shocks).
• The result is a symphony that has a clear, predictable structure (the envelope) but is full of exciting, unpredictable improvisation (the spikes).

By understanding this, astronomers can look at a GRB light curve and say, "Ah, that shape tells me exactly how the star collapsed and how the black hole is feeding itself."

Technical Summary

Here is a detailed technical summary of the paper "An Accretion-Modulated Internal Shock Model for Long GRBs" by Moradi et al.

1. Problem Statement

Long Gamma-Ray Bursts (GRBs) exhibit prompt emission light curves characterized by two distinct features:

1. Global Temporal Trends: A broad, smooth envelope often resembling a Fast-Rise–Exponential-Decay (FRED) profile.
2. Fine-Scale Variability: Rapid, multi-peaked stochastic fluctuations superimposed on the global trend.

While the Internal Shock (IS) model successfully explains the non-thermal spectra and rapid variability (via collisions of relativistic shells), it struggles to account for the systematic, global temporal trends (e.g., pulse waiting times, FWHM evolution, and peak flux correlations) observed in many bursts. Conversely, fallback accretion models explain the long-duration energy injection but have not been fully integrated into the prompt emission phase to explain the interplay between the smooth envelope and the fine-scale spikes. There is a need for a unified framework that links the central engine's mass-supply history to both the macroscopic light-curve envelope and the microscopic pulse structure.

2. Methodology

The authors propose the Accretion-Modulated Internal Shock (AMIS) model, which couples a time-dependent mass-supply history from the central engine with the standard Kobayashi-Piran-Sari (KPS) internal shock dynamics.

• Mass-Supply Framework:

  • The model assumes the central engine (a black hole or neutron star formed from a collapsar) is fed by a time-dependent mass supply rate, $\dot{M}(t)$.
  • This rate is phenomenologically parameterized to mimic stellar collapse and fallback accretion:
    • Rising Phase: $\dot{M} \propto t^{\beta_{onset}}$ (approx. $t^{-1/2}$ or $t^0$).
    • Decaying Phase: $\dot{M} \propto t^{-5/3}$ (characteristic of self-similar fallback).
  • The jet luminosity is assumed to scale linearly with this mass supply: $L_{\gamma}(t) = \epsilon_{rad} \eta \dot{M}(t) c^2$.

• Internal Shock Dynamics:

  • The engine ejects a sequence of relativistic shells.
  • Collisions occur when faster shells catch up with slower ones, dissipating kinetic energy into radiation.
  • The authors utilize the KPS analytic pulse shape and collision dynamics (conservation of energy-momentum) to simulate the light curve.

• Simulation Scenarios:
  The authors explore two distinct modulation mechanisms to link $\dot{M}(t)$ to shell properties:

  1. Mass-Driven Scenario: Shells are ejected at roughly constant time intervals ($\Delta t_{ej} \approx \text{const}$), but the mass of each shell ($m_i$) scales with the instantaneous mass supply rate ($m_i \propto \dot{M}(t_i)$).
  2. Rate-Driven Scenario: Shell masses remain nearly constant, but the ejection interval ($\Delta t_i$) and shell thickness are modulated by the mass supply rate ($\Delta t_i \propto 1/\dot{M}(t_i)$).

• Spectral Evolution:

  • The model incorporates a time-resolved spectral evolution where the peak energy $E_p(t)$ tracks the luminosity ($E_p \propto L^{\gamma_{ep}}$).
  • It accounts for energy-dependent Band function indices to explain why pulse widths vary with energy.

3. Key Contributions

• Unified Framework: The AMIS model provides a physical mechanism where the global envelope is dictated by the fallback/accretion history of the progenitor, while the fine-scale variability arises from stochastic fluctuations in shell Lorentz factors and ejection parameters.
• Dual Modulation Scenarios: The paper distinguishes between mass-driven and rate-driven modulation, offering testable predictions for how pulse widths and amplitudes evolve differently under each regime.
• Analytic Derivation of Pulse Widths: The authors derive analytic estimates showing how energy-dependent spectral indices lead to the observed scaling of pulse width with energy ($w(E) \propto E^{-0.4}$).
• FRED Envelope Reproduction: The model demonstrates that a smooth FRED-like envelope naturally emerges from a mass-supply history that rises and then decays as $t^{-5/3}$, without requiring ad-hoc smoothing of the light curve.

4. Results

• Light Curve Morphology:
  • Simulations of both scenarios successfully reproduce the observed multi-peaked structure superimposed on a smooth FRED envelope.
  • Mass-Driven: Pulse amplitudes follow the mass-supply envelope (growing then decaying), while pulse widths (FWHM) remain approximately constant. This aligns with observations that pulse widths in long GRBs show little systematic evolution.
  • Rate-Driven: Pulse widths evolve inversely with the mass-supply rate (narrower/brighter during low supply, broader/dimmer during high supply). This scenario produces a width-luminosity evolution.
• Spectral-Temporal Correlations:
  • The model successfully reproduces the "hard-to-soft" evolution and the narrowing of pulses at higher energies. This is achieved because the effective decay index of the photon flux depends on the energy-dependent spectral index $\alpha(E)$.
• Energetics and Efficiency:
  • Simulations yield radiative efficiencies ($\epsilon_{rad}$) of $\sim 1.4\% - 1.9\%$, consistent with standard internal shock expectations.
  • The model accommodates both Type I collapsars (high accretion, prompt collapse) and Type II collapsars (lower accretion, fallback-dominated), explaining both bright and underluminous GRBs.
• Norris FRED Fitting:
  • The simulated mass-driven light curve was fitted with the Norris et al. (1996) FRED function. The derived peak time ($t_{pk} \approx 3.1$ s) matched the characteristic peak of the imposed mass-supply envelope ($t_p = 3.5$ s), validating the mapping between engine activity and observed pulses.

5. Significance

• Progenitor Diagnostics: The AMIS model offers a diagnostic tool to probe the central engine's activity and the progenitor's structure. By analyzing the global envelope shape, one can infer the mass-supply history (e.g., distinguishing between prompt collapse and fallback-dominated scenarios).
• Bridging Scales: It resolves the disconnect between macroscopic engine evolution (accretion physics) and microscopic emission mechanisms (internal shocks), suggesting that the "driver" of GRB variability is the accretion rate itself.
• Future Directions: The paper highlights that while the current mapping is phenomenological, future work integrating General Relativistic Magnetohydrodynamic (GRMHD) simulations will be crucial to derive the precise relationship between accretion rate and shell properties. Additionally, testing the model against large GRB samples (e.g., from Fermi/GBM) will determine the prevalence of mass-supply modulation versus other variability drivers.

In conclusion, the AMIS model posits that the complex light curves of long GRBs are not purely stochastic but are fundamentally organized by the time-dependent mass supply from the collapsing star, with internal shocks acting as the mechanism that translates this smooth engine evolution into the observed rapid, multi-peaked emission.