Asteroseismology of the Double-Mode High-Amplitude $\delta$ Scuti Star VX Hydrae

Bi-site time-series photometric and high-resolution spectroscopic observations were made for the double-mode high-amplitude $\delta$ Scuti star VX\ Hya. The fundamental frequency $f_{0}=4.4763\ \rm{c\ days^{-1}}$, the first overtone $f_{1}=5.7897\ \rm{c\ days^{-1}}$ and 23 harmonics and linear combinations of $f_{0}$ and $f_{1}$ are detected by pulsation analysis. From the spectroscopic data, we get $\rm{[Fe/H] = -0.2\pm0.1\ dex}$. The period change rate of the fundamental mode is obtained by using the Fourier-phase diagram method, which gives the value of $(1/P_{0})(dP_{0}/dt)=(1.81\pm0.09) \times 10^{-7}\ \rm{yr}^{-1}$. With these results from the observations, we perform theoretical explorations with the stellar evolution code MESA, and constrain the models by fitting $f_{0}$, $f_{1}$, and $(1/P_{0})(dP_{0}/dt)$ within $3\sigma$ deviations. The results show that the period change of VX\ Hya could be ascribed to the evolutionary effect. The stellar parameters of VX\ Hya could be derived as: the mass $2.385\pm0.025\ M_{\odot}$, the luminosity $\log(L/L_{\odot})=1.93\pm0.02$ and the age $(4.43\pm0.13)\times 10^8$ years. VX\ Hya is found to locate at the post-main-sequence stage with a helium core and a hydrogen-burning shell on the H${-}$R diagram.