Exploring the physical properties of the $\gamma$ Dor binary star RX Dra with photometry and asteroseismology

We model the TESS light curve of the binary system RX Dra, and also first calculate a lot of theoretical models to fit the g-mode frequencies previously detected from the TESS series of this system. The mass ratio is determined to be $q$=0.9026${}_{-0.0032}^{+0.0032}$. We newly found that there are 16 frequencies (F1-F7, F11-F20) identified as dipole g-modes, two frequencies (F21, F22) identified as quadrupole g-modes, and another two frequencies (F23, F24) identified as g-sextupole modes, based on these model fits. The primary star is newly determined to be a $\gamma$ Dor pulsator in the main-sequence stage with a rotation period of about 5.7${}_{-0.2}^{+0.7}$ days, rotating slower than the orbital motion. The fundamental parameters of two components are firstly estimated as follows: $M_1$=1.53${}_{-0.17}^{+0.00}$ M ${}_{\odot }$, $M_2$= 1.38${}_{-0.00}^{+0.18}$ M ${}_{\odot }$, $T_1$=7240${}_{-44}^{+490}$ K, $T_2$=6747${}_{-221}^{+201}$ K, $R_1$=1.8288${}_{-0.0959}^{+0.0260}$ R ${}_{\odot }$, $R_2$= 1.3075${}_{-0.2543}^{+0.0450}$ R ${}_{\odot }$, $L_1$=8.2830${}_{-0.6036}^{+1.8015}$ L ${}_{\odot }$ and $L_2$= 3.4145${}_{-0.1843}^{+0.1320}$ L ${}_{\odot }$. Our results show that the secondary star is in the solar-like pulsator region of the H-R diagram, indicating that it could be a pulsating star of this type. Finally, the radius of the convective core of the primary star is estimated to be about 0.1403${}_{-0.0000}^{+0.0206}$ R${}_{\odot }$.