tokens: of periapsis ω , and time of periastron t peri ). These can be used to calculate the projected position of both components on the sky at any moment in time. In all previous cases of binary events with significant binary motion, Keplerian orbital motion provided only a small improvement relative to the linear approximation ( Skowron et al. 2011 ; Shin et al. 2012 ). This is not the case here, because, as we show below, the orbital period of the lens is similar to the duration of the event (e.g. Penny et al. 2011 ). Modelling of this event is an iterative process: for given microlensing parameters, we estimated the angular radius and distance to the source, we calculated best-fit microlensing parameters, and we repeated the procedure until all parameters converged. The best-fit microlensing parameters are presented in Table 5 . Uncertainties were calculated using the Markov chain Monte Carlo approach (MCMC; Foreman-Mackey et al. 2013 ) and represent 68% confidence intervals of marginalized posterior distributions. We note that another degenerate solution exists for the microlensing model that differs only by the signs of s z and γ z (( s z , γ z ) → − ( s z , γ z )). The second solution has the same physical parameters (except for Ω → π − Ω and ω → ω − π ) and differs by the sign of the radial velocity. Thus, the degeneracy can be broken with additional radial velocity measurements of the lens ( Skowron et al. 2011 ). Table 5. Best-fit microlensing model parameters of the Gaia16aye binary event. Notes. HJD′=HJD-2450000. We adopt t 0, par = t 0, kep = 7675. 4.3. Source star Spectroscopic observations of the event indicate that the source is a K5-type giant or a super-giant. If the effective temperature of the source were higher than 4250 K, TiO absorption features would be invisible. If the temperature were lower than 3800 K, these features would be stronger than those in the observed spectra. Spectral modelling indicates that the effective temperature of the source is 3933 ± 135 K. According to Houdashelt et al. (2000) , the intrinsic Johnson–Cousins colours of a star of this spectral type and solar metallicity should be , and (error bars correspond to the source of K4- and M0-type, respectively). We used a model-independent regression to calculate the observed colours of the source (we used observations collected in the Bialkow Observatory, which were calibrated to the standard system): V − R = 0.99 ± 0.01 and V − I = 1.91 ± 0.01. Thus, the colour excess is E ( V − I ) = 0.31 and E ( V − R ) = 0.16, consistent with the standard reddening law ( Cardelli et al. 1989 ) and A V = 0.62. According to the best-fitting microlensing model, the amount of light coming from the magnified source is V s = 16.61 ± 0.02 and I s = 14.70 ± 0.02. The V -band brightness of the source after correcting for extinction is therefore V 0 = 15.99 mag. Subsequently, we used the colour–surface brightness relations for giants from Adams et al. (2018) to estimate the
ner_tags: 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-Model I-Model I-Model O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation 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 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 B-Model I-Model I-Model I-Model I-Model B-Model 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 B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula 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 B-CelestialObject O O O B-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula 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-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 B-Observatory I-Observatory O O O O O O O B-Wavelength O B-Wavelength O O O O O B-Wavelength O B-Wavelength O O O O O O O O O O O B-Wavelength O B-Wavelength O O O O O O B-Wavelength O B-Wavelength O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Wavelength O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O B-Wavelength 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