On the interpretation of spectral-energy correlations in long gamma-ray bursts

Context. Recently, Liang & Zhang found a tight correlation involving only observable quantities, namely the isotropic emitted energy $E_{\rm \gamma, iso}$, the energy of the peak of the prompt spectrum $E^\prime_{\rm p}$, and the jet break time $t^\prime_{\rm j}$ of Gamma Ray Bursts. This phenomenological correlation can have a first explanation in the framework of jetted fireballs, whose semiaperture angle $\theta_{\rm j}$ is measured by the jet break time $t^\prime_{\rm j}$. By correcting E \gamma, iso for the angle $\theta_{\rm j}$ one obtains the so-called Ghirlanda correlation, linking the collimation-corrected energy $E_\gamma$ and $E^\prime_{\rm p}$. Aims. There are two ways to derive $\theta_{\rm j}$ from $t^\prime_{\rm j}$ in the “standard” scenario, corresponding to a homogeneous or to a wind-like circumburst medium. We compute and compare the $E^\prime_{\rm p}$–$E_\gamma$ correlations derived in these two conditions and study the consistency of these model-dependent correlations with the empirical Liang & Zhang correlation. We consider the difference between the observed correlations and the ones in the comoving frame . Methods. We study 18 GRBs with firmly measured z , E peak and $t_{\rm break}$ and discuss the differences with previously published samples. We compute the correlations accounting for the errors on all the relevant quantities. Results. We show that the Ghirlanda correlation with a wind-like circumburst medium is as tight as (if not tighter) than the Ghirlanda correlation for a homogeneous medium. These two Ghirlanda correlations are both consistent with the phenomenological Liang & Zhang relation. The wind-like Ghirlanda relation, which is linear, remains linear also in the comoving frame, independently of the distribution of bulk Lorentz factors. Instead, in the homogeneous density case, one is forced to assume the existence of a strict relation between the bulk Lorentz factor and the total energy, which in turn places constraints on the radiation mechanisms of the prompt emission. The wind-like Ghirlanda correlation, being linear, corresponds to different bursts having the same number of photons.