Seismic modelling of the pulsating mercury-manganese star HD 29589

Photometric and spectroscopic time-series of chemically peculiar mercury-manganese (HgMn) stars show variability, which in most cases is interpreted as the effect of binarity or surface spots. Until recently, pulsations have not been detected in these objects. However, the analysis of the Transiting Exoplanet Survey Satellite (TESS) photometric time-series of a large sample of HgMn stars resulted in a small sample of candidates for pulsating variables. In this paper, we present a thorough analysis of one of them, HD 29589. High-resolution spectroscopic data were used to determine the atmospheric parameters and a detailed chemical composition of the star. The obtained effective temperature, T_eff = 14 400 ± 200 K, places this star among the high-temperature HgMn stars. The determined abundance pattern is typical for this class of objects. Interferometric data were used to discuss the properties of the secondary in the HD 29589 system. More than 20 independent frequencies were identified in the TESS time-series of this star, which we interpret as high-order g-modes. We found regular period spacing, which is due to the presence of consecutive prograde dipole modes. The exact matching of the six frequencies in the series and the position of the star in the Hertzsprung-Russell diagram allowed us to constrain the internal structure of HD 29589. We derived the overshooting parameter, f_ov = 0.028-0.03, mass M = 3.4 M_⊙ and metallicity Z = 0.008. A detailed analysis of pulsating HgMn stars will allow us to understand the influence of the chemical peculiarity on the pulsation characteristics of these objects.