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

Abstract

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 $E_{\mathrm{iso}}$ and the peak energy in the cosmological rest frame $E_{p,i}$, known as the Amati relation. The analysis revealed that the $E_{\mathrm{iso}}$ 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 $E_{\mathrm{iso}}-E_{p,i}$ and Lorentz factor $\Gamma -E_{\mathrm{iso}}$ correlations, we estimated the viewing angles ${\theta }_{\mathrm{obs}}^{\prime }$ for these five LGRBs. The results show, ${\theta }_{\mathrm{obs}}^{\prime }={(0.{42}_{-0.14}^{+0.29})}^{\circ }$ for GRB 980425, ${\theta }_{\mathrm{obs}}^{\prime }={(0.{27}_{-0.10}^{+0.21})}^{\circ }$ for GRB 061210, ${\theta }_{\mathrm{obs}}^{\prime }={(0.{10}_{-0.02}^{+0.09})}^{\circ }$ for GRB 070714B, ${\theta }_{\mathrm{obs}}^{\prime }={(0.{13}_{-0.05}^{+0.09})}^{\circ }$ for GRB 071227, and ${\theta }_{\mathrm{obs}}^{\prime }={(0.{12}_{-0.03}^{+0.14})}^{\circ }$ for GRB 080123A. Notably, all five ${\theta }_{\mathrm{obs}}^{\prime }$ 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.