Adiabatic and Radiative Cooling of Relativistic Electrons Applied to Synchrotron Spectra and Light-Curves of Gamma-Ray Burst Pulses

TL;DR

This paper models the cooling processes of relativistic electrons in gamma-ray bursts, showing how adiabatic and radiative cooling influence the observed spectra and light-curves, aligning theoretical predictions with observations.

Contribution

It provides analytical and numerical analysis of electron cooling effects on GRB spectra, highlighting the roles of adiabatic and radiative processes in shaping observed pulse features.

Findings

Cooling causes spectral softening and earlier peak times at higher energies.

Power-law injection rates explain observed low-energy spectra.

Cooling tails and spectra depend on magnetic field and injection rate histories.

Abstract

We investigate the adiabatic and radiative (synchrotron and inverse-Compton) cooling of relativistic electrons whose injected/initial distribution with energy is a power-law above a typical energy $\gamma_i$. Analytical and numerical results are presented for the cooling-tail and the cooled-injected distribution that develop below and above the typical energy of injected electrons, for the evolution of the peak-energy $E_p$ of the synchrotron emission spectrum, and for the pulse shape resulting from an episode of electron injection. The synchrotron emission calculated numerically is compared with the spectrum and shape of Gamma-Ray Burst (GRB) pulses. Both adiabatic and radiative cooling processes lead to a softening of the pulse spectrum, and both types of cooling processes lead to pulses peaking earlier and lasting shorter at higher energy, quantitatively consistent with observations. For adiabatic-dominated electron cooling, a power-law injection rate $R_i$ suffices to explain the observed power-law GRB low-energy spectra. Synchrotron-dominated cooling leads to power-law cooling-tails that yield the synchrotron standard slope alpha = -3/2 provided that $R_i \sim B^2$, which is exactly the expectation if the magnetic field is a constant fraction of the post-shock energy density. Increasing (decreasing) $R_i$ and decreasing (increasing) B(t) lead to slopes alpha harder (softer, respectively) than the standard value and to non--power-law (curved) cooling-tails. Inverse-Compton cooling yields four values for the slope alpha but, as for synchrotron, other $R_i$ or B histories yield a wider range of slopes and curved low-energy spectra. Feedback between the power-law segments that develop below and above the typical injected electron leads to a synchrotron spectrum with many breaks above and below the usual 10 keV-1 MeV observing range.