[Paper Review] A New Derivation of GRB Jet Opening Angles from the Prompt Gamma-Ray Emission

A. Goldstein, R. D. Preece+25 more|arXiv (Cornell University)|Jan 12, 2011

Gamma-ray bursts and supernovae3 citations

TL;DR

This paper presents a novel method to derive gamma-ray burst (GRB) jet opening angles directly from the prompt gamma-ray emission using only $E_{\text{peak}}$ and fluence, bypassing the need for afterglow observations. The approach yields a tighter correlation between rest-frame $E_{\text{peak}}$ and collimation-corrected energy $E_{\gamma}$, placing long and short GRBs on a single power-law relation and ruling out proto-magnetar progenitors for several bursts.

ABSTRACT

The jet opening angle of gamma-ray bursts (GRBs) is an important parameter for determining the characteristics of the progenitor, and the information contained in the opening angle gives insight into the relativistic outflow and the total energy that is contained in the burst. Unfortunately, a confident inference of the jet opening angle usually requires broadband measurement of the afterglow of the GRB, from the X-ray down to the radio and from minutes to days after the prompt gamma-ray emission, which may be difficult to obtain. For this reason, very few of all detected GRBs have constrained jet angles. We present an alternative approach to derive jet opening angles from the prompt emission of the GRB, given that the GRB has a measurable Epeak and fluence, and which does not require any afterglow measurements. We present the distribution of derived jet opening angles for the first two years of the Fermi Gamma-ray Burst Monitor (GBM) operation, and we compare a number of our derived opening angles to the reported opening angles using the traditional afterglow method. We derive the collimation-corrected gamma-ray energy, E_γ, for GRBs with redshift and find that some of the GRBs in our sample are inconsistent with a proto-magnetar progenitor. Finally, we show that the use of the derived jet opening angles results in a tighter correlation between the rest-frame Epeak and E_γ than has previously been presented, which places long GRBs and short GRBs onto one empirical power law.

Motivation & Objective

• To develop a method for estimating GRB jet opening angles without relying on afterglow data, which are often unavailable.
• To enable the calculation of collimation-corrected gamma-ray energy $E_{\gamma}$ for GRBs with redshift using only prompt emission parameters.
• To test whether long and short GRBs can be described by a single empirical power-law relation in $E_{\text{peak}}$–$E_{\gamma}$ space.
• To assess the viability of proto-magnetar progenitors for specific GRBs using derived $E_{\gamma}$ values.

Proposed method

• The method uses the observed $E_{\text{peak}}$ and fluence from Fermi-GBM prompt emission data as input to infer the jet opening angle via a physical scaling relation derived from relativistic beaming and energy conservation.
• It assumes that the prompt emission spectrum follows a Band function, and applies a geometric correction based on the observed peak energy and total fluence to estimate the jet's collimation.
• The derived opening angle is then used to compute the collimation-corrected isotropic-equivalent energy $E_{\gamma}$, assuming a redshift is known.
• The approach is calibrated against a subset of GRBs with independently measured jet angles from afterglow light curves, validating its accuracy.
• The method avoids reliance on pseudo-redshifts or the lag-luminosity relation, unlike previous ensemble methods.
• It leverages the fact that the hard spectral cutoff in the $E_{\text{peak}}$–fluence plane is instrumental-independent, indicating a physical origin tied to jet structure.

Experimental results

Research questions

• RQ1 Can jet opening angles be reliably inferred from prompt emission alone, without afterglow data?
• RQ2 Does the derived $E_{\text{peak}}$–$E_{\gamma}$ relation unify long and short GRBs under a single power law?
• RQ3 Which GRB progenitor models are inconsistent with the derived $E_{\gamma}$ values?
• RQ4 How do geometric viewing-angle effects influence the accuracy of $E_{\gamma}$ estimates?

Key findings

• The derived jet opening angles from prompt emission show good agreement with those obtained via traditional afterglow modeling for a subset of GRBs.
• The method successfully reproduces known jet angles for individual bursts without requiring pseudo-redshifts or additional assumptions.
• A tighter correlation between rest-frame $E_{\text{peak}}$ and $E_{\gamma}$ is observed, with both long and short GRBs lying on a single power-law relation.
• Five long GRBs (080916C, 080810, 090323, 090519, 090902B) and one short GRB (090510) are inconsistent with a proto-magnetar progenitor model based on their derived $E_{\gamma}$ values.
• The method reduces uncertainty in $E_{\gamma}$ estimation for long GRBs with small opening angles due to minimal viewing-angle offset effects.
• The approach enables the estimation of $E_{\gamma}$ for GRBs lacking redshifts, with potential for extrapolating redshift distributions using the $E_{\text{peak}}$–$E_{\gamma}$ relation.