Bulk Lorentz factors of gamma-ray bursts

¹ INAF–Osservatorio Astronomico di Brera, via E. Bianchi 46, 23807 Merate, Italy
e-mail: giancarlo.ghirlanda@brera.inaf.it
² Dipartimento di Fisica G. Occhialini, Università di Milano Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
³ Università degli Studi dell’Insubria, via Valleggio 11, 22100 Como, Italy
⁴ INAF–IASF Milano, via E. Bassini 15, 20133 Milano, Italy
⁵ INAF–Osservatorio Astronomico di Trieste, via G.B. Tiepolo, 11, 34143 Trieste, Italy

Received: 19 July 2017
Accepted: 11 October 2017

Abstract

Knowledge of the bulk Lorentz factor Γ₀ of gamma-ray bursts (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 (including one short event) with a peak in their optical or GeV light curves at a time t_p. For another 106 GRBs we set an upper limit t_p^UL. The measure of t_p provides the bulk Lorentz factor Γ₀ of the fireball before deceleration. We show that t_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_p of 66 long GRBs and the 85 most constraining upper limits, we estimate Γ₀ or a lower limit Γ₀^LL. Using censored data analysis methods, we reconstruct the most likely distribution of t_p. All t_p are larger than the time T_p,γ when the prompt γ-ray emission peaks, and are much larger than the time T_ph when the fireball becomes transparent, that is, t_p>T_p,γ>T_ph. The reconstructed distribution of Γ₀ has median value ~300 (150) for a uniform (wind) circumburst density profile. In the comoving frame, long GRBs have typical isotropic energy, luminosity, and peak energy ⟨ E_iso ⟩ = 3(8) × 10⁵⁰ erg, ⟨ L_iso ⟩ = 3(15) × 10⁴⁷ erg s^-1, and ⟨ E_peak ⟩ = 1(2) keV in the homogeneous (wind) case. We confirm that the significant correlations between Γ₀ and the rest frame isotropic energy (E_iso), luminosity (L_iso), and peak energy (E_p) are not due to selection effects. When combined, they lead to the observed E_p−E_iso and E_p−L_iso correlations. Finally, assuming a typical opening angle of 5 degrees, we derive the distribution of the jet baryon loading which is centered around a few 10^-6M_⊙.