Exploring the helium core of the delta Scuti star CoRoT 102749568 with asteroseismology

Based on regularities in rotational splittings, we seek possible multiplets for the observed frequencies of CoRoT 102749568. Twenty-one sets of multiplets are identified, including four sets of multiplets with $l=1$, nine sets of multiplets with $l=2$, and eight sets of multiplets with $l=3$. In particular, there are three complete triplets ($f_{10}$, $f_{12}$, $f_{14}$), ($f_{31}$, $f_{34}$, $f_{35}$), and ($f_{41}$, $f_{43}$, $f_{44}$). The rotational period of CoRoT 102749568 is estimated to be $1.34_{-0.05}^{+0.04}$ days. When doing model fittings, three $l=1$ modes ($f_{12}$, $f_{34}$, and $f_{43}$) and the radial first overtone $f_{13}$ are used. Our results shows that the three nonradial modes ($f_{12}$, $f_{34}$, and $f_{43}$) are mixed modes, which mainly provide constraints on the helium core. The radial first overtone $f_{13}$ mainly provides constraint on the stellar envelope. Hence the size of the helium core of CoRoT 102749568 is determined to be $M_{\mathrm{H}\mathrm{e}}$ = 0.148 $\pm$ 0.003 $M_{\odot }$ and $R_{\mathrm{H}\mathrm{e}}$ = 0.0581 $\pm$ 0.0007 $R_{\odot }$. The fundamental parameters of CoRoT 102749568 are determined to be $M$ = 1.54 $\pm$ 0.03 $M_{\odot }$, $Z=$ 0.006, $f_{\mathrm{o}\mathrm{v}}$ = 0.004 $\pm$ 0.002, $\log g$ = 3.696 $\pm$ 0.003, $T_{\mathrm{e}\mathrm{f}\mathrm{f}}$ = 6886 $\pm$ 70 K, $R$ = 2.916 $\pm$ 0.039 $R_{\odot }$, and $L$ = 17.12 $\pm$ 1.13 $L_{\odot }$.