How fast is the fastest-rotating O-type star in the Galaxy?

Context . Rapid rotation distorts massive stars and produces latitude-dependent effective temperature and gravity. Stellar parameters inferred from a single spherical atmosphere convolved with a rotational profile can therefore be apparent inclination-dependent quantities. LAMOST J040643.69+542347.8 is an extreme Galactic O-type runaway star for which these effects should be appreciable. Aims . We re-determined the stellar and rotational parameters of LAMOST J040643.69+542347.8 with an oblate gravity-darkened photospheric model and used selected He II transitions treated with tabulated Stark profiles. Methods . We fitted continuum-normalised spectra over 4000–4600 Å with ZPEKTR . The local TLUSTY/SYNSPEC inputs are emergent flux spectra, and we introduced their angular contribution to the visible Roche surface through the limb-darkening treatment implemented in ZPEKTR . The gravity darkening we employed follows the Espinosa-Lara & Rieutord model, and for the selected He II transitions, we used the Schöning–Butler Stark profiles implemented in SYNSPEC. We masked interstellar and wind- or circumstellar-dominated features. Results . The model reproduces the fitted He II λ 4200, He I λ 4471, and He II λ 4542 profiles. The parent non-rotating counterpart (PNRC) parameters are T eff PNRC = 38 716 ± 1041 K and log g PNRC = 4.12 ± 0.12, and we obtained v e sin i = 587 ± 73 km s −1 . The central grid solution has Ω/Ω c = 0.93, v e / v c = 0.77, i = 65°, and R e / R p = 1.25. The corresponding ratio of centrifugal to gravitational acceleration at the equator is η e = 0.50, whereas η e = 1 and R e / R p = 1.5 at Roche critical rotation. Thus, the angular rotation rate is high, but the central solution is not critical in the equatorial force-balance sense. The projected rotational velocity is higher than but statistically consistent with the value previously obtained from a spherical rotational-convolution analysis. Conclusions . LAMOST J040643.69+542347.8 is an exceptionally rapid O-type rotator with a markedly oblate gravity-darkened photosphere. The ratios Ω/Ω c , v e / v c , and η e should be quoted together when discussing its proximity to critical rotation. A controlled spherical fit with the same atmosphere grid would be required to quantitatively isolate the bias introduced by neglecting rotational deformation and gravity darkening.