TIC 322208686: An Eclipsing System with $\gamma$ Doradus Pulsations and a Third Component on a Wider Orbit

TIC 322208686 is known to be a detached binary that exhibits two types of variability: pulsation and eclipse. We present the physical properties of the target star using the short-cadence TESS data from sectors 24, 57, and 58, and our echelle spectra that show the presence of a tertiary companion. The spectral analysis led to the triple-lined radial velocities and the atmospheric parameters of the eclipsing components. Joint modeling of these observations reveals that the eclipsing pair contains two F-type stars with masses $1.564\pm0.012$ $M_\odot$ and $1.483\pm0.012$ $M_\odot$, radii $1.588\pm0.011$ $R_\odot$ and $1.500\pm0.012$ $R_\odot$, effective temperatures $7028\pm100$ K and $7020\pm110$ K, and luminosities $5.51\pm0.32$ $L_\odot$ and $4.90\pm0.32$ $L_\odot$. The light contributions of the three stars obtained from this modeling match well with those calculated from the observed spectra. The binary star parameters are in satisfactory agreement with evolutionary model predictions for age $t$ = 0.4 Gyr and metallicity $Z$ = 0.03. We extracted 11 significant frequencies from the TESS light residuals with the binary effects removed. Of these, five signals between 0.65 day$^{-1}$ and 1.89 day$^{-1}$ can be considered as $\gamma$ Dor pulsations originating mainly from the primary component, while the other frequencies are likely instrumental artifacts or combination terms. These results suggest that TIC 322208686 is a hierarchical triple, containing a pulsating eclipsing pair and a tertiary companion.