Study of photometric and spectral variability of the roAp star HD 210684

This paper studies the photometric and spectropolarimetric variability of HD 210684 in order to derive its magnetic properties, rotational period, evolutionary stage, and global stellar parameters. We used the discrete Fourier transform to measure the frequencies and amplitudes of periodic signals present in light curves of HD 210684. We calculated evolution models with MESA, while roAp type pulsations were simulated with GYRE. We derived the values of the T_ eff, log(g), v ̊angle and RV. and radial velocity (RV) from the best fit of the Balmer line profiles using FITSB2. The least square deconvolution (LSD) method was applied to Stokes I & V spectra to measure the mean longitudinal magnetic field łangle B_ ̊m z Detailed analysis of the photometric variability of the light curves provided by for HD 210684 reveals rotational modulation with a period of P=5.02188±0.00005 d and splitting of high-overtone pulsations that corresponds to the same rotational period. Derived łangle B_ TESS ̊m z ̊angle measurements also show periodic variability with P=5.02188 d, but this variability appears to be different from the one predicted by a centred magnetic dipole model, which suggests that the configuration of the surface magnetic field is more complex than the dipolar one. We determined the inclination angle between the line of sight and the rotation axis to be i = 31 ± 2 and the angle between the rotation and magnetic dipole axes as β= 77 ± 3 < 3.85 and show that our best-fitting model depends on whether the observed modes are ell = 1 or ell =2. Simulations of stellar pulsations constrain łog T_ ̊m eff Considering the derived value of β, we prefer the best-fit model with ell =2 modes, which predicts that HD 210684 lies on the main sequence with an age of approximately 1.45 Gyr. Meanwhile, only the most evolved and coolest models in our grid show η > 0 (when frequencies are driven) for the observed triplet. The best fit of the Balmer line profiles resulted in higher values of T_̊m eff and łog(g) for spectra acquired at rotational phases ̌arphi= 0.14 -- 0.39, which suggests visibility of an area with a higher surface temperature. Strong asymmetry of the LSD Stokes I profiles derived at ̌arphi= 0.307 and 0.394 supports this hypothesis. Clear variability of the LSD Stokes I profiles with rotational phase argues for horizontal abundance stratification that may be linked to the inhomogeneity of the surface temperature. Further spectropolarimetric observations of HD 210684 are required to study its unique properties in detail.