2020ApJ...894...52X

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 logL_j/erg s^–1=52.68_–0.33^+0.76 (1σ) and an opening angle θ_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 β₁=0.78_–0.15^+0.16 and β₂> 1.6 broken at logL_b,w/erg 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., logL_c,w/ergs^–1=48.38_–0.48^+0.30, θ_c,w=2.65^o_-1.19^o+1.73o^, and k_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.