Apparent positive correlation of the low-energy spectral index α and the polarization degree (left) and the possible photosphere explanation (right, polarization degree Π = |Q|/I). (a) Notably, this positive correlation can be obtained from both the POLAR sample (blue circles, with a Pearson correlation coefficient of r = 0.69) and the AstroSAT bursts (green triangles, r = 0.85). The P values standing for the significance level are both ∼0.06, and thus the correlation is almost significant (combining POLAR and AstroSAT, P ∼ 0.03 can be achieved). (b) For the photosphere polarization, the polarization degree is shown to be positively correlated with the p value (the power-law decreasing index of Γ, for the jet angular distribution), for the smaller θ_c, Γ case available for bursts with higher energy. Also, for Γ ⋅ θ_c, Γ ≃ 1, α is positively correlated with the p value, α ≃ ( − 1/4)(1 + 3/p). The predicted positive correlation of α and the polarization degree for θ_v = 0.015 (red pluses in the left panel) can match the observations of POLAR and the slope of the AstroSAT sample, approximately. Notice that GRB 170127C in POLAR may have larger θ_c, Γ, thus possessing a larger α of 0.25 and a smaller polarization degree (see Figure 5a, maybe two to three times smaller).

Fig. 5.

Correlations of photosphere polarization degree (Π = |Q|/I), θ_c, Γ (or θ_jet), and E_iso. (a) Comparison of the photosphere polarization degree for smaller θ_c, Γ (Γ ⋅ θ_c, Γ = 1, solid lines; predicting a much larger polarization degree) and larger θ_c, Γ (Γ ⋅ θ_c, Γ = 2, dashed lines; predicting a much smaller polarization degree). (b) The smaller θ_jet (likely θ_c, Γ also) is found for the high-energy sample (E_iso ≳ 10⁵³ erg), indicating the polarization detection.