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

Abstract

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 $\log L_{\mathrm{j}}/\mathrm{erg}{\mathrm{s}}^{-1}=52.68_{-0.33}^{+0.76}$ (1$\sigma$) and an opening angle ${\theta }_{\mathrm{j}}=2.10_{-1.28}^{+1.90}$ (50\% confidence level), surrounding by a power-law decay component with an index of $-4.00_{-0.37}^{+0.27}$ (1$\sigma$). The inferred local GRB rate is $\rho =9.6$ Gpc${}^{-3}$ yr${}^{-1}$ by including both the typical GRBs and LL-GRBs as the same population. The typical viewing angle is $3.3^o$, and may be $20^o\sim 30^o$ 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 ${\beta }_2>1.6$ broken at $\log L_{b,w}/\mathrm{erg}{\mathrm{s}}^{-1}=48.51_{-0.65}^{+0.53}$. In case of that the wind outflow is collimated and co-axial with the GRB jet, we find that the wind structure is similar to the GRB jet, i.e., $\log L_{\mathrm{c},\mathrm{w}}/\mathrm{erg}{\mathrm{s}}^{-1}=48.38_{-0.48}^{+0.30}$, ${\theta }_{\mathrm{c},\mathrm{w}}={2.65^o}_{-1.19^o}^{+\mathrm{0.1.7}{3}^o}$, and $k_{\mathrm{w}}=4.57_{-0.75}^{+1.21}$. The observed correlation between the prompt gamma-ray luminosity and X-ray luminosity of the wind may be resulted from the viewing angle effect in such a jet-wind system. Discussion on survey with the X-ray instruments on board the {\em Einstein\ Probe} mission in the soft X-ray band for the jet and wind emission is also presented.

22 pages, 10 figures, and 1 table. Accepted for publication in ApJ