Gaia-ESO Survey: massive stars in the Carina Nebula. II. The spectroscopic analysis of the O-star population

The new census of massive stars in the Carina nebula reveals the presence of 54 apparently single O-type stars in the Car OB1 association, an extremely active star-forming region which hosts some of the most luminous stars of the Milky Way. A detailed spectroscopic analysis of the currently most complete sample of O-type stars in the association can be used to inspect the main physical properties of cluster members and test evolutionary and stellar atmospheres models. We perform quantitative spectroscopic analysis for the most complete sample of apparently single O-type stars in Car OB1 with available spectroscopic data. From the high-resolution GES and OWN spectra we obtain a reliable distribution of rotational velocities for a sample of 37 O-type stars. It shows a bimodal structure with a low velocity peak at 60 km s$^{-1}$ and a short tail of fast rotators reaching 320 km s$^{-1}$. We also perform quantitative spectroscopic analysis and derive effective temperature, surface gravity and He abundance for a sample of 47 O-type stars, now including further stars from GOSSS database. Radii, luminosities, and spectroscopic masses were also determined using $Gaia$ astrometry. We create the Hertzsprung-Russell Diagram to inspect the evolutionary status of the region and confirm the lack of stars close to the Zero Age Main Sequence (ZAMS) between $\sim$35 -- 55 M$_\odot$. We confirm a very young population with an age distribution peaking at 1 Myr, some stars close or even on the ZAMS, and a secondary peak at 4 -- 5 Myr in the age distribution. We confirm the youth of Trumpler 14, which is also the only cluster not showing the secondary peak. We also find a clear trend to evolutionary masses higher than derived spectroscopic masses for stars with evolutionary mass below 40 M$_{\odot}$.