  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.
