Probing the wind and evolutionary state of the quiescent luminous blue variable Var$\,$2 in the Galaxy M33: constraints from self-consistent non-LTE modeling
Luminous Blue Variables (LBVs) are massive stars with extreme luminosities, exhibiting significant irregular photometric and spectroscopic variability. This paper is dedicated to the study of dormant LBV Var$\,$2 in the galaxy M33, which has not shown significant brightness changes for almost century and currently observed as Ofpe/WN9 star. In order to obtain reliable estimates of the fundamental parameters of Var$\,$2, we had calculated self-consistent hydrodynamic models of the extended atmosphere, assuming non-local thermodynamic equilibrium (non-LTE) and taking into account the balance of radial forces in the wind of the star. Spectral modeling yields luminosity $L=6.5\times10^{5}\,L_\odot$, current mass $M_* \approx23\,M_\odot$ and hydrogen abundance on the surface ${\rm X}_{\rm H}\approx43\%$. Comparison with evolutionary tracks indicates that these parameters correspond to a star with the initial mass $M_{\rm init}\approx50\,M_\odot$ and age $t_{\rm age}\approx4.4\,$Myr that is evolving from lower temperatures towards the Wolf-Rayet stars. A study of dynamic properties of the wind showed that the shape of the wind velocity profile of Var$\,$2 is close to the one of O- and early B-supergiants rather than Wolf-Rayet stars. In contrast, for the obtained mass-loss rate $\dot{M}=2.1\times10^{-5}\,M_{\odot}\,\text{yr}^{-1}$, the ratio of the wind momentum to the luminosity of Var$\,$2 is in good agreement with the values found for WNL stars. Given the obtained age estimates, Var$\,$2 could potentially have been ejected from the cluster associated with the nearest large star-forming region located at a distance of $\sim100\,$pc. However, statistical analysis of the projected distribution of stars in the vicinity of Var$\,$2 suggests that this LBV could have formed in a local low-populated group.