[Paper Review] GRBs have preferred jet opening angles and bulk Lorentz factors

G. Ghisellini, G. Ghirlanda+8 more|arXiv (Cornell University)|Nov 9, 2012

Gamma-ray bursts and supernovae21 references3 citations

TL;DR

This paper proposes that all gamma-ray bursts (GRBs) have identical intrinsic peak energy $E'_{\rm p} \sim 1.5$ keV and isotropic energy $E'_{\gamma} \sim 2 \times 10^{48}$ erg in their comoving frame, with observed diversity arising from varying bulk Lorentz factors ($\Gamma_0$) and jet opening angles ($\theta_{\rm jet}$). Using population synthesis simulations, it finds log-normal distributions for $\Gamma_0$ and $\theta_{\rm jet}$ with preferred values $\langle\Gamma_0\rangle \sim 275$ and $\langle\theta_{\rm jet}\rangle \sim 8.7^\circ$, linked by $\theta_{\rm jet}^{2.5}\Gamma_0 = \text{const}$, and predicts that ~6% of on-axis GRBs lack a jet break due to $\sin\theta_{\rm jet} < 1/\Gamma_0$. The model explains the $E_{\rm p}-E_{\rm iso}$ correlation as a result of these distributions and their geometric constraints.

ABSTRACT

We recently found that Gamma Ray Burst energies and luminosities, in their comoving frame, are remarkably similar. This, coupled with the clustering of energetics once corrected for the collimation factor, suggests the possibility that all bursts, in their comoving frame, have the same peak energy E'peak (of the order of a few keV) and the same energetics of the prompt emission E'gamma (of the order of 2e48 erg). The large diversity of bursts energies is then due to the different bulk Lorentz factor Gamma and jet aperture angle theta_jet. We investigated, through a population synthesis code, what are the distributions of Gamma and theta_jet compatible with the observations. Both quantities must have preferred values, with log-normal best fitting distributions and ~ 275 and ~ 8.7 degree. Moreover, the peak values of the Gamma and theta_jet distributions must be related - theta_jet^2.5 Gamma =const: the narrower the jet angle, the larger the bulk Lorentz factor. We predict that ~6% of the bursts that point to us should not show any jet break in their afterglow light curve since they have sin(theta_jet)&lt;1/Gamma. Finally, we estimate that the local rate of GRBs is ~0.3% of all local SNIb/c and ~2.5% of local hypernovae, i.e. SNIb/c with broad absorption lines.

Motivation & Objective

• To investigate whether GRBs have preferred values of bulk Lorentz factor ($\Gamma_0$) and jet opening angle ($\theta_{\rm jet}$), rather than being randomly distributed.
• To determine if the observed $E_{\rm p}-E_{\rm iso}$ correlation in GRBs arises from intrinsic similarity in comoving-frame energetics and luminosities, with diversity due to $\Gamma_0$ and $\theta_{\rm jet}$ variations.
• To test whether the $E_{\rm p}-E_{\rm iso}$ correlation is a selection effect or reflects a physical distribution of $\Gamma_0$ and $\theta_{\rm jet}$, using population synthesis simulations.
• To quantify the fraction of on-axis GRBs that may lack a detectable jet break due to $\sin\theta_{\rm jet} < 1/\Gamma_0$, and assess their observational impact.

Proposed method

• Assumes all GRBs have identical comoving-frame peak energy $E'_{\rm p} \sim 1.5$ keV and isotropic energy $E'_{\gamma} \sim 2 \times 10^{48}$ erg, independent of $\Gamma_0$.
• Uses population synthesis simulations to generate synthetic GRB populations with varying $\Gamma_0$ and $\theta_{\rm jet}$, applying the observed $E_{\rm p}-E_{\rm iso}$ correlation as a constraint.
• Imposes geometric constraints: for $\Gamma_0 \lesssim 1/\theta_{\rm jet}$, the effective collimation angle is $1/\Gamma_0$, leading to $E_{\rm p} \propto E_{\rm iso}^{1/3}$, which defines forbidden regions in the $E_{\rm p}-E_{\rm iso}$ plane.
• Fits log-normal distributions to the simulated $\Gamma_0$ and $\theta_{\rm jet}$ distributions, optimizing for agreement with the Swift complete sample in the $E_{\rm p}-E_{\rm iso}$ plane.
• Imposes a physical relation $\theta_{\rm jet}^{2.5}\Gamma_0 = \text{const}$ between the peak values of the distributions to match the observed correlation slope.
• Compares simulated distributions with observed $\Gamma_0$ and $\theta_{\rm jet}$ values from 30 bursts with measured $\Gamma_0$ and 27 with measured $\theta_{\rm jet}$.

Experimental results

Research questions

• RQ1 Do GRBs have preferred values of bulk Lorentz factor $\Gamma_0$ and jet opening angle $\theta_{\rm jet}$, or are they randomly distributed?
• RQ2 Is the observed $E_{\rm p}-E_{\rm iso}$ correlation in GRBs a result of intrinsic similarity in comoving-frame properties combined with varying $\Gamma_0$ and $\theta_{\rm jet}$?
• RQ3 What is the functional form of the $\Gamma_0$ and $\theta_{\rm jet}$ distributions that best reproduce the observed $E_{\rm p}-E_{\rm iso}$ correlation?
• RQ4 What fraction of on-axis GRBs should lack a detectable jet break due to $\sin\theta_{\rm jet} < 1/\Gamma_0$, and how does this affect sample completeness?
• RQ5 What is the local rate of GRBs relative to core-collapse supernovae (SNIbc) and hypernovae, based on the derived $\Gamma_0$ and $\theta_{\rm jet}$ distributions?

Key findings

• The best-fitting log-normal distribution for $\Gamma_0$ has a mean of $\sim 275$, a mode of $\sim 274$, and a median of $\sim 221$, with $\sigma \sim 0.06$ in log space.
• The best-fitting log-normal distribution for $\theta_{\rm jet}$ has a mean of $\sim 8.7^\circ$, a mode of $\sim 8.7^\circ$, and a median of $\sim 11^\circ$, with $\sigma \sim 0.04$ in log space.
• The peak values of the $\Gamma_0$ and $\theta_{\rm jet}$ distributions are related by $\theta_{\rm jet}^{2.5}\Gamma_0 = \text{const}$, indicating that narrower jets are associated with higher Lorentz factors.
• Approximately 6% of GRBs that are pointed toward Earth should not exhibit a jet break in their afterglow light curves, due to $\sin\theta_{\rm jet} < 1/\Gamma_0$, potentially leading to misclassification as outliers.
• The local GRB rate is estimated to be $\sim 0.3\%$ of all local SNIbc and $\sim 2.5\%$ of local hypernovae (SNIbc with broad lines), based on the derived $\Gamma_0$ and $\theta_{\rm jet}$ distributions.
• The model successfully reproduces the observed $E_{\rm p}-E_{\rm iso}$ correlation with a slope of $\sim 0.6$, and explains the harder correlation slope in bright bursts as a selection effect favoring wider jets in less sensitive surveys.