Limitation on the Viewing Angles of Long GRBs Based on Amati Relation

Fermi, Swift, and other observation satellites provided a dataset of 323 Long Gamma-Ray Bursts (LGRBs) with known redshift and well-defined spectral parameters spanning from February 1997 to September 2023. Among these, 123 GRBs (Gamma-Ray Bursts) observed by Fermi-GBM were selected to calibrate the relationship between the isotropic energy E_\rmiso and the peak energy in the cosmological rest frame E_\rmp,i, known as the Amati relation. The analysis revealed that the E_\rmiso of five LGRBs (GRB 980425, GRB 061210, GRB 070714B, GRB 071227, and GRB 080123A) significantly deviated from the Amati relation, suggesting that the jet of them may be off-axis to the line of sight. Utilizing the well-established E_\rmp,i-\varGamma and E_\rmp,i-E_\rmiso correlations, we estimated the viewing angles \theta _\rmobs^' for these five LGRBs. The results show, \theta^'_\rmobs=(0.42^+0.29_-0.14)^\circ for GRB 980425, \theta^'_\rmobs=(0.27^+0.21_-0.10)^\circ for GRB 061210, \theta^'_\rmobs=(0.10^+0.09_-0.02)^\circ for GRB 070714B, \theta^'_\rmobs=(0.13^+0.09_-0.05)^\circ for GRB 071227, and \theta^'_\rmobs=(0.12^+0.14_-0.03)^\circ for GRB 080123A. Notably, all five \theta _\rmobs^' values are small, with the maximum angle being 0.42^\circ for GRB 980425, the nearest long burst observed (z=0.0085). The results suggest that the majority of Long GRBs are likely detected on-axis, with only a small fraction of nearby Long GRBs being observed slightly outside the jet edge.