An analytic derivation of the empirical correlations of gamma-ray bursts

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