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 $E_{\rm{iso}}$ and the peak energy in the cosmological rest frame $E_{\rm{p,i}}$, known as the Amati relation. The analysis revealed that the $E_{\rm{iso}}$ 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 $E_{\rm{p,i}}-\varGamma$ and $E_{\rm{p,i}}-E_{\rm{iso}}$ correlations, we estimated the viewing angles $\theta _{\rm{obs}}^{'}$ for these five LGRBs. The results show, $\theta^{'}_{\rm{obs}}=(0.42^{+0.29}_{-0.14})^\circ$ for GRB 980425, $\theta^{'}_{\rm{obs}}=(0.27^{+0.21}_{-0.10})^\circ$ for GRB 061210, $\theta^{'}_{\rm{obs}}=(0.10^{+0.09}_{-0.02})^\circ$ for GRB 070714B, $\theta^{'}_{\rm{obs}}=(0.13^{+0.09}_{-0.05})^\circ$ for GRB 071227, and $\theta^{'}_{\rm{obs}}=(0.12^{+0.14}_{-0.03})^\circ$ for GRB 080123A. Notably, all five $\theta _{\rm{obs}}^{'}$ 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.