  (Shortened) CONTEXT: [...] AIMS: Motivated by the relation proposed by Amati
and collaborators, we look within the ``fireshell'' model for a relation
between the peak energy E_p of the \nu F_\nu total time-integrated spectrum of
the afterglow and the total energy of the afterglow E_{aft}, which in our model
encompasses and extends the prompt emission. METODS: [...] Within the fireshell
model [...] We can then build two sets of ``gedanken'' GRBs varying the total
energy of the electron-positron plasma E^{e^\pm}_{tot} and keeping the same
baryon loading B of GRB050315. The first set assumes for the effective CBM
density the one obtained in the fit of GRB050315. The second set assumes
instead a constant CBM density equal to the average value of the GRB050315
prompt phase. RESULTS: For the first set of ``gedanken'' GRBs we find a
relation E_p\propto (E_{aft})^a, with a = 0.45 \pm 0.01, whose slope strictly
agrees with the Amati one. Such a relation, in the limit B \to 10^{-2},
coincides with the Amati one. Instead, in the second set of ``gedanken'' GRBs
no correlation is found. CONCLUSIONS: Our analysis excludes the Proper-GRB
(P-GRB) from the prompt emission, extends all the way to the latest afterglow
phases and is independent on the assumed cosmological model, since all
``gedanken'' GRBs are at the same redshift. The Amati relation, on the other
hand, includes also the P-GRB, focuses on the prompt emission only, and is
therefore influenced by the instrumental threshold which fixes the end of the
prompt emission, and depends on the assumed cosmology. This may well explain
the intrinsic scatter observed in the Amati relation.
