Magnetar as Central Engine of Gamma-Ray Bursts: Quasi-universal Jet, Event Rate, and X-Ray Luminosity Function of Dipole Radiations

Early shallow-decaying X-ray afterglows of gamma-ray bursts (GRBs) may be attributed to the dipole radiations of newly born magnetars. Assuming that the GRB jets powered by magnetars are quasi-universal, we find that the jet structure can be parameterized as a uniform jet with a luminosity of $\mathrm{log}{L}_{{\rm{j}}}/\mathrm{erg}\ {{\rm{s}}}^{-1}={52.68}_{-0.33}^{+0.76}$ (1σ) and an opening angle ${\theta }_{{\rm{j}}}={2.10}_{-1.28}^{+1.90}$ (50% confidence level), surrounding by a power-law decay component with an index of $-{4.00}_{-0.37}^{+0.27}$ (1σ). The inferred local GRB rate is ρ = 9.6 Gpc^-3 yr^-1 by including both the typical GRBs and LL-GRBs as the same population. The typical viewing angle is 3.3°, and may be 20° ∼ 30° for LL-GRBs. The X-ray luminosity function of the dipole radiation wind can be empirically described by a broken power-law function with indices ${\beta }_{1}={0.78}_{-0.15}^{+0.16}$ and ${\beta }_{2}\gt 1.6$ broken at $\mathrm{log}{L}_{{\rm{b}},{\rm{w}}}/\mathrm{erg}\ {{\rm{s}}}^{-1}={48.51}_{-0.65}^{+0.53}$ . In the case where the wind outflow is collimated and coaxial with the GRB jet, we find that the wind structure is similar to the GRB jet, I.e., $\mathrm{log}{L}_{{\rm{c}},{\rm{w}}}/\mathrm{erg}\,{{\rm{s}}}^{-1}={48.38}_{-0.48}^{+0.30}$ , ${\theta }_{{\rm{c}},{\rm{w}}}={{2.65}^{o}}_{-{1.19}^{o}}^{+1.{73}^{o}}$ , and ${k}_{{\rm{w}}}={4.57}_{-0.75}^{+1.21}$ . The observed correlation between the prompt gamma-ray luminosity and X-ray luminosity of the wind may result from the viewing angle effect in such a jet-wind system. A discussion on surveys using the X-ray instruments on board the Einstein Probe mission in the soft X-ray band for the jet and wind emission is also presented.