On the interpretation of the spectral--energy correlations in long   Gamma--Ray Bursts
  Recently, Liang & Zhang (2005) found a tight correlation involving only observable quantities, namely the isotropic emitted energy $E_{\gamma,iso}$, the energy of the peak of the prompt spectrum $E^\prime_{p}$, and the jet break time $t^\prime_{j}$. This phenomenological correlation can have a first explanation in the framework of jetted fireballs, whose semiaperture angle $\theta_{j}$ is measured by the jet break time $t^\prime_{j}$. By correcting $E_{\gamma, iso}$ for the angle $\theta_{j}$ one obtains the so called Ghirlanda correlation linking the collimation corrected energy $E_\gamma$ and $E^\prime_{p}$. There are two ways to derive $\theta_{j}$ from $t^\prime_{j}$ in the standard scenario, corresponding to an homogeneous or to a wind-like circumburst medium. We show that the Ghirlanda correlation with a wind-like medium is as tight as (if not tighter) than the Ghirlanda correlation found in the case of an homogeneous medium. There are hence two Ghirlanda correlations, both entirely consistent with the phenomenological Liang & Zhang relation. We consider the difference between the observed correlations and the ones one would see in the comoving frame (i.e. moving with the same bulk Lorentz factor of the fireball). Since both $E_{p}$ and $E_\gamma$ transform in the same way, the wind-like Ghirlanda relation, which is linear, remains linear also in the comoving frame, no matter 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 put 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. 
L. Nava (1,2), G. Ghisellini (1), G. Ghirlanda (1), F. Tavecchio (1),   C. Firmani (1,3) ((1) Osservatorio Astronomico Brera...