2025ApJ...987...76X

The properties of X-ray flashes (XRFs) and X-ray-rich gamma-ray bursts (XRRs) as compared with classical gamma-ray bursts (C-GRBs) have been widely discussed during the SWIFT era. It has been proposed that XRFs and XRRs are low-energy extensions of the gamma-ray burst (GRB) population, and therefore, they should follow similar correlations. To further examine this idea, we collect a sample of 303 GRBs detected by SWIFT over the past two decades, all of which have reliable redshifts and spectral parameters. The bursts are classified into XRFs, XRRs, and C-GRBs based on their softness ratio (SR), which is calculated by dividing the 25-50 keV fluence with the 50-100 keV fluence. A strong correlation is found among the isotropic energy E_iso, peak luminosity L_p, and rest-frame burst duration T_90,rest, i.e., ${E}_{{\rm{iso}}}\propto {L}_{{\rm{p}}}^{0.88\pm 0.02}{T}_{90,{\rm{rest}}}^{0.58\pm 0.02}$. Additionally, two tight three-parameter correlations involving SR and the rest-frame peak energy E_p are also derived, i.e., ${E}_{{\rm{p}}}\propto {E}_{{\rm{iso}}}^{0.20\pm 0.02}{{\rm{SR}}}^{-2.27\pm 0.15}$ and ${E}_{{\rm{p}}}\propto {L}_{{\rm{p}}}^{0.17\pm 0.02}{{\rm{SR}}}^{-2.33\pm 0.14}$. It is interesting to note that XRFs, XRRs, and C-GRBs all follow the above correlations. The implications of these correlations and their potential application in cosmology are discussed.