Constraining Circum-burst Environments of GRBs with Jet Break Features in X-ray Afterglows
The nature of the circum-burst medium serves as a key diagnostic for probing the progenitor systems and the physics of relativistic jet propagation in gamma-ray bursts (GRBs). In this work, we systematically infer the density profile index $k$ (where $n \propto r^{-k}$) from the change in the temporal decay index at the jet break ($\Delta\alpha$). Within the framework of the uniform jet model, the two quantities are linked by the relation $\Delta\alpha = (3 - k)/(4 - k)$. We apply this diagnostic to a substantial and uniformly selected sample of 170 GRBs with clear jet breaks, identified from over 1,400 Swift/XRT X-ray afterglows observed from 2004 to 2024. By fitting the light curves with a broken power-law model, we obtain $\Delta\alpha$ for each burst and subsequently derive the corresponding $k$ value. We then use the derived $k$ values to classify the circum-burst environment of each GRB. Our results reveal a near-even split: 82 bursts ($\sim48\%$) are consistent with a constant-density interstellar medium (ISM, $k \approx 0$), while 88 bursts ($\sim52\%$) favor a wind environment ($k \approx 2$). For the 35 bursts with optical data, our X-ray-based classifications are generally consistent with independent multi-band analyses. Additionally, we derive jet opening angles and true beaming-corrected energies for bursts with known redshifts.