  Empirical correlations between various key parameters have been extensively
explored ever since the discovery of gamma-ray bursts (GRBs) and have been
widely used as standard candles to probe the Universe. The Amati relation and
the Yonetoku relation are two good examples that enjoyed special attention. The
former reflects the connection between the peak photon energy (Ep) and the
isotropic $\gamma$-ray energy release (Eiso), while the latter links Ep with
the isotropic peak luminosity (Lp), both in the form of a power-law function.
Most GRBs are found to follow these correlations well, but a theoretical
interpretation is still lacking. Some obvious outliers may be off-axis GRBs and
may follow correlations that are different from those at the on-axis. Here we
present a simple analytical derivation for the Amati relation and the Yonetoku
relation in the framework of the standard fireball model, the correctness of
which is then confirmed by numerical simulations. The off-axis Amati relation
and Yonetoku relation are also derived. They differ markedly from the
corresponding on-axis relation. Our results reveal the intrinsic physics behind
the radiation processes of GRBs, and they highlight the importance of the
viewing angle in the empirical correlations of GRBs.
