Bulk Lorentz factors of Gamma-Ray Bursts

TL;DR

This paper estimates the distribution of bulk Lorentz factors of Gamma-Ray Bursts using afterglow peak times, revealing their typical values, correlations with burst properties, and implications for jet composition.

Contribution

It introduces a method to derive Lorentz factors from afterglow peak times and provides the first comprehensive distribution analysis considering censored data.

Findings

Median Lorentz factor ~300 for uniform density profile

Confirmed correlations between Lorentz factor and burst energetics

Estimated jet baryon loading around a few 10^{-6} solar masses

Abstract

Knowledge of the bulk Lorentz factor $\Gamma_{0}$ of GRBs allows us to compute their comoving frame properties shedding light on their physics. Upon collisions with the circumburst matter, the fireball of a GRB starts to decelerate, producing a peak or a break (depending on the circumburst density profile) in the light curve of the afterglow. Considering all bursts with known redshift and with an early coverage of their emission, we find 67 GRBs with a peak in their optical or GeV light curves at a time $t_{\rm p}$. For another 106 GRBs we set an upper limit $t_{\rm p}^{\rm UL}$. We show that $t_{\rm p}$ is due to the dynamics of the fireball deceleration and not to the passage of a characteristic frequency of the synchrotron spectrum across the optical band. Considering the $t_{\rm p}$ of 66 long GRBs and the 85 most constraining upper limits, using censored data analysis methods, we reconstruct the most likely distribution of $t_{\rm p}$. All $t_{\rm p}$ are larger than the time $t_{\rm p,g}$ when the prompt emission peaks, and are much larger than the time $t_{\rm ph}$ when the fireball becomes transparent. The reconstructed distribution of $\Gamma_0$ has median value $\sim$300 (150) for a uniform (wind) circumburst density profile. In the comoving frame, long GRBs have typical isotropic energy, luminosity, and peak energy $\langle E_{\rm iso}\rangle=3(8)\times 10^{50}$ erg, $\langle L_{\rm iso}\rangle=3(15) \times 10^{47}$ erg s$^{-1}$ , and $\langle E_{\rm peak}\rangle =1(2)$ keV in the homogeneous (wind) case. We confirm that the significant correlations between $\Gamma$ and the rest frame isotropic energy ($E_{\rm iso}$), luminosity ($L_{\rm iso}$) and peak energy ($E_{\rm peak}$) are not due to selection effects. Assuming a typical opening angle of 5 degrees, we derive the distribution of the jet baryon loading which is centered around a few $10^{-6} {\rm M_{\odot}}$.