tokens: . 4.6. Perspectives with Gaia At the considered range of distances, Gaia will deliver exquisite parallax accuracy (%), especially because the binary effect can be adequately subtracted thanks to the knowledge of the astrometric orbital parameters presented in this study. Combined with the size of the apparent orbits and the periods, we expect the total masses to be measured with an uncertainty of 5%, following Eq. ( 1 ). For most systems, the error budget will be equally shared between the interferometric resolved astrometry and the Gaia distance. These measurements of the total masses will be completely independent from radial velocities. Gaia will also constrain the orbital photocentre displacement . This quantity can be extracted even if the orbital period is poorly sampled by Gaia because of the knowledge of all orbital parameters. It permits calculating the mass ratio : where is the (known) size of the apparent orbit and the flux ratio in the Gaia band. Interestingly, is well approximated by the (known) flux ratio in the band because hot stars have their optical and near-infrared spectra in the Rayleigh-Jeans regime. In order to test feasibility, we computed the expected photocentre displacement from the orbits derived in this study. This was done by inverting Eq. ( 5 ), guessing the expected mass ratio. We used two approaches, both relying on the calibrations from Martins et al. (2005) . We first converted the observed flux ratio into mass ratio. For a pair of main-sequence stars, the calibration is well approximated by . The exponent is even lowered for a massive, evolved primary with a less-massive main sequence secondary. In a second approach, we simply use the expected mass ratio from the individual spectral types when available (see Table 1 ). The computed are reported in Fig. 2 . The predicted displacements range from 0.1 mas to 1 mas. This is a factor ten to a hundred larger than the Gaia performances (assuming the orbital photocentre shift is measured with a similar accuracy than the parallaxes). Accounting for 5% uncertainty on the total mass (shared between the interferometric resolved astrometry and the Gaia distance, see Eq. ( 1 )), and for 1 to 10% uncertainty on the mass-ratio (shared between the interferometric flux ratio and the photocentre shift amplitude from Gaia , see Eq. ( 5 )), we expect to measure the individual masses with an uncertainty of %, independently of radial velocities. Fig. 2 Predicted photocentre displacement versus the flux ratio for the ten binaries of the sample. The mass ratios were estimated either from the flux ratio (assuming main sequence stars, filled symbols), or derived from the individual spectral types when available (open symbols). The dashed red line is for the expected Gaia performances. 5. Conclusion In this paper we report observations of spatially resolved astrometry on ten massive O-type binaries, using OLBI. This dataset is used to derive the elements of the apparent orbits. Combined
ner_tags: O O O O B-Telescope O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O