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 $f_0=4.4763\mathrm{c}\mathrm{d}\mathrm{a}\mathrm{y}{\mathrm{s}}^{-1}$, the first overtone $f_1=5.7897\mathrm{c}\mathrm{d}\mathrm{a}\mathrm{y}{\mathrm{s}}^{-1}$ and 23 harmonics and linear combinations of $f_0$ and $f_1$ are detected by pulsation analysis. From the spectroscopic data, we get $[\mathrm{F}\mathrm{e}\mathrm{/}\mathrm{H}]=-0.2\pm 0.1\mathrm{d}\mathrm{e}\mathrm{x}$. 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 $(1\mathrm{/}{P}_0)(d{P}_0\mathrm{/}dt)$ 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 $2.385\pm 0.025M_{\odot }$, the luminosity $\log (L\mathrm{/}{L}_{\odot })=1.93\pm 0.02$ and the age $(4.43\pm 0.13)\times {10}^8$ 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.