Long-term optical variability of the Be/X-ray binary GRO J2058+42

We investigate the long-term optical variability of the Be/X-ray binary GRO J2058+42 and the possible connection with periods of enhanced X-ray activity. We performed an optical spectroscopic and photometric analysis on data collected during about 18 years. We also present the first optical polarimetric observations of this source. The long-term optical light curves in the $BVRI$ bands and the evolution of the H$\alpha$ equivalent width display a sinusoidal pattern with maxima and minima that repeat every $\sim$9.5 years. The amplitude of this variability increases as the wavelength increases. The H$\alpha$ equivalent width varied from about $-0.3$ to $-15$ \AA. We found a significant decrease in the polarization degree during the low optical state. The optical maxima occur near periods of enhanced X-ray activity and are followed by a drop in the optical emission. Unlike many other Be/X-ray binaries, GRO 2058+42 does not display $V/R$ variability. The long-term optical variability agrees with the standard model of a Be/X-ray binary, where the circumstellar disk of the Be star grows and dissipates on timescales of 9--10 years. We find that the dissipation of the disk started after a major X-ray outburst. However, the stability of the H$\alpha$ line shape as a double-peak profile and the lack of asymmetries suggest the absence of a warped disk and argue against the presence of a highly distorted disk during major X-ray outbursts.