[Paper Review] Gamma-ray burst afterglows and evolution of postburst fireballs with energy injection from strongly magnetic millisecond pulsars

Z. G. Dai, T. Lu|arXiv (Cornell University)|Oct 25, 1998

Gamma-ray bursts and supernovae74 citations

TL;DR

This paper proposes that gamma-ray burst (GRB) afterglows, particularly GRB970228's optical lightcurve, result from a postburst relativistic fireball energized by magnetic dipole radiation from a strongly magnetized millisecond pulsar. The model explains the observed lightcurve—initial decay, flattening, and final decline—through energy injection that alters the fireball's deceleration, with constraints indicating a pulsar period of 1.4–1.7 ms and surface magnetic field of 20–30×10¹² G.

ABSTRACT

Millisecond pulsars with strong magnetic fields may be formed through several processes, e.g. accretion-induced collapse of magnetized white dwarfs, merger of two neutron stars. During the birth of such a pulsar, an initial fireball available for a gamma-ray burst (GRB) may occur. We here study evolution of a postburst relativistic fireball with energy injection from the pulsar through magnetic dipole radiation, and find that the magnitude of the optical afterglow from this fireball first decreases with time, subsequently flattens, and finally declines again. This may provide a natural explanation for the behavior of the lightcurve of the afterglow of GRB970228 if this burst resulted from the birth of a strongly magnetic millisecond pulsar.

Motivation & Objective

• To explain the complex optical afterglow lightcurve of GRB970228, which shows an initial decay, a flattening phase, and a final decline.
• To investigate whether energy injection from a newborn strongly magnetized millisecond pulsar via magnetic dipole radiation can account for the observed afterglow evolution.
• To constrain the physical parameters of the central engine—specifically pulsar period, magnetic field strength, and moment of inertia—based on observational afterglow data.
• To assess the viability of pulsar-powered fireballs as a central engine model for cosmological gamma-ray bursts.

Proposed method

• Models the evolution of a relativistic postburst fireball decelerated by the swept-up interstellar medium, with continuous energy injection from a pulsar via magnetic dipole radiation.
• Uses the standard fireball model with forward and reverse shocks, incorporating energy injection that modifies the deceleration timescale and flux evolution.
• Applies the magnetic dipole radiation luminosity formula $ L = \frac{2}{3c^3} \left(\frac{2\pi}{P}\right)^4 R^6 B_s^2 \sin^2\theta $, parameterized as $ L \approx 4 \times 10^{43} \, B_{\bot,12}^2 P_{\rm ms}^{-4} R_6^6 \, \text{erg s}^{-1} $.
• Derives the timescale $ \tau \approx 5 \times 10^7 \, \text{s} \, E_{51} B_{\bot,12}^{-2} P_{\rm ms}^4 R_6^{-6} $ for the fireball to absorb half its initial energy from the pulsar.
• Compares the model's predicted flux decay laws with observed afterglow magnitudes at early and late times to constrain pulsar parameters.
• Uses observational constraints: flat afterglow magnitude at ~6 days (R ~ 27), decline to V=28.0 at 4 September (~188 days), and flattening ending by ~60 days.

Experimental results

Research questions

• RQ1 Can energy injection from a strongly magnetized millisecond pulsar explain the multi-segmented lightcurve of GRB970228’s optical afterglow?
• RQ2 What are the required pulsar parameters (period, magnetic field, moment of inertia) to reproduce the observed afterglow evolution?
• RQ3 Under what conditions does the pulsar's energy injection lead to a flattening phase in the afterglow lightcurve?
• RQ4 How does the timescale of pulsar energy injection ($\tau$) compare to the fireball deceleration timescale ($T$) in determining the lightcurve shape?
• RQ5 Is the observed afterglow behavior consistent with a pulsar-powered fireball model, or does it require alternative engines?

Key findings

• The model successfully reproduces the three-phase lightcurve of GRB970228: initial decay, flattening phase, and final decline, due to energy injection from a pulsar.
• The flattening phase occurs only when the pulsar's energy injection timescale $\tau$ is shorter than the fireball deceleration timescale $T$, i.e., $\tau < T$.
• From observational constraints, the pulsar period is constrained to $1.4 \leq P_{\rm ms} \leq 1.7$ ms, assuming $E_{51} \sim 1$, $R_6 \sim 1$, and $I_{45} \sim 2$.
• The effective surface magnetic field is estimated at $20 \leq B_{\bot,12} \leq 30$, corresponding to $B_{\bot} \sim 2 \times 10^{13}$ to $3 \times 10^{13}$ G.
• The model predicts that for extremely strong magnetic fields ($B_{\bot} \sim 10^{15}$ G), the energy injection occurs too quickly ($T < 100$ s), making the flattening phase unobservable in optical afterglows.
• The analysis shows that the observed afterglow behavior is inconsistent with pulsar energy injection if $\tau > T$, i.e., if the pulsar period is too long or the magnetic field too weak.