The environment of the fast rotating star Achernar III. Photospheric parameters revealed by the VLTI

Context. Rotation significantly impacts on the structure and life of stars. In phases of high rotation velocity (close to critical), the photospheric structure can be highly modified, and present in particular geometrical deformation (rotation flattening) and latitudinal-dependent flux (gravity darkening). The fastest known rotators among the nondegenerate stars close to the main sequence, Be stars, are key targets for studying the effects of fast rotation on stellar photospheres. Aims. We seek to determine the purely photospheric parameters of Achernar based on observations recorded during an emission-free phase (normal B phase). Methods. Several recent works proved that optical/IR long-baseline interferometry is the only technique able to sufficiently spatially resolve and measure photospheric parameters of fast rotating stars. We thus analyzed ESO-VLTI (PIONIER and AMBER) interferometric observations of Achernar to measure its photospheric parameters by fitting our physical model CHARRON using a Markov chain Monte Carlo method. This analysis was also complemented by spectroscopic, polarimetric, and photometric observations to investigate the status of the circumstellar environment of Achernar during the VLTI observations and to cross-check our model-fitting results. Results. Based on VLTI observations that partially resolve Achernar, we simultaneously measured five photospheric parameters of a Be star for the first time: equatorial radius (equatorial angular diameter), equatorial rotation velocity, polar inclination, position angle of the rotation axis projected on the sky, and the gravity darkening beta coefficient (effective temperature distribution). The close circumstellar environment of Achernar was also investigated based on contemporaneous polarimetry, spectroscopy, and interferometry, including image reconstruction. This analysis did not reveal any important circumstellar contribution, so that Achernar was essentially in a normal B phase at least from mid-2009 to end-2012, and the model parameters derived in this work provide a fair description of its photosphere. Finally, because Achernar is the flattest interferometrically resolved fast rotator to-date, the measured beta and flattening, combined with values from previous works, provide a crucial test for a recently proposed gravity darkening model. This model offers a promising explanation to the fact that the measured beta parameter decreases with flattening and shows significantly lower values than the classical prediction of von Zeipel.