No Universality for Electron's Power-Law Index (p) in Gamma-Ray Bursts and Other Relativistic Sources

The Gamma-Ray Burst (GRB) prompt emission is believed to be from highly relativistic electrons accelerated in relativistic shocks. From the GRB high-energy power-law spectral indices $\beta$ observed by the Burst and Transient Source Experiment (BATSE) Large Area Detectors (LAD), we determine the spectral index, $p$, of the electrons' energy distribution. Both the theoretical calculations and numerical simulations of the particle acceleration in relativistic shocks show that $p$ has a universal value $\approx 2.2-2.3$. We show that the observed distribution of $p$ during GRBs is not consistent with a $\delta$-function distribution or an universal $p$ value, with the width of the distribution $\ge$ 0.54. The distributions of $p$ during X-ray afterglows are also investigated and found to be inconsistent with a $\delta$-function distribution. The $p$-distributions in blazars and pulsar wind nebulae are also broad, inconsistent with a $\delta$-function distribution.