OO Dra: An Algol-type Binary Formed through an Extremely Helium-poor Mass Accretion Revealed by Asteroseismology

Eclipsing binaries with pulsating components are very important objects in understanding stellar structure and evolution. The nature of the eclipse allows us to derive accurate physical parameters of the components, and the oscillation frequencies provide significant insight into their interiors as well as opportunities to identify physical processes behind the pulsating nature of the components (Aerts & Harmanec 2004; Mkrtichian et al. 2005, 2007). Various types of pulsating stars have been found in eclipsing binaries, such as red giants (Gaulme et al. 2013, 2014, 2016; Beck et al. 2014), γ Dor stars (Maceroni et al. 2013; Guo et al. 2017; Ibanoglu et al. 2018; Guo & Li 2019), and δ Scuti stars (Kahraman et al. 2017; Liakos & Niarchos 2017; Gaulme & Guzik 2019). Many detailed studies of pulsating stars in eclipsing binaries have been carried out (Schmid & Aerts 2016; Kahraman et al. 2017; Beck et al. 2018; Zhang et al. 2018; Bowman et al. 2019; Chen et al. 2020). Schmid & Aerts (2016) presented an asteroseismic study of the δ Scuti-γ Doradus pulsating binary KIC 10080943 and showed that the size of core overshooting and diffusive mixing can be well constrained with the hypothesis of the same age of the two components. Beck et al. (2018) analyzed the eccentric binary KIC 9163796 and reported that the two pulsating red giant components are in early and late phases of the first dredge-up event on the red giant branch. Most eclipsing binaries with δ Scuti-type pulsations are found to be Algol-type systems. The pulsating components exhibit similar pulsation behaviors to those of single δ Scuti pulsators, but they have a distinct evolutionary history due to mass transfer between the two components. Kahraman et al. (2017) presented a comparison between pulsating eclipsing binaries and single δ Scuti pulsators and found that the δ Scuti components in binaries pulsate at lower amplitudes and shorter periods than those of single pulsators. Guo et al. (2017) analyzed the post mass-transfer binary KIC 9592855 and found that the core and envelope of the pulsator rotate nearly uniformly and both of their rotation rates are similar to the orbital motion. Bowman et al. (2019) investigated the Transiting Exoplanet Survey Satellite (TESS) light curve of the oEA system U Gru and showed that tidally perturbed oscillations can occur in the p-mode region. Chen et al. (2020) carried out an analysis of the Algol system KIC 10736223 through binary properties and asteroseismology and reported that KIC 10736223 had just undergone the rapid mass-transfer stage. Post mass-transfer binaries carry evidence of binary interaction, thus the study of them offers new and strict constraints to refine stellar evolution theories, which can help us further understand the influences of tidal forces and mass-transfer processes among interacting binaries. However, mode identification is more difficult for δ Scuti stars due to the low radial orders. So far, comprehensive asteroseismic modeling of δ Sct-type pulsators in post mass-transfer binaries is still lacking.

OO Dra was first discovered by Biyalieva & Khruslov (2007) to be an eclipsing binary with an orbital period of 1.23838 days. The pulsating nature of OO Dra was later found by Dimitrov et al. (2008) through their BVR passbands light curves. They made a preliminary frequency analysis of the out-of eclipse photometric data, and detected a main frequency about 37 cycle days⁻¹. Besides, Dimitrov et al. (2008) contributed spectroscopy of OO Dra, from which the effective temperature, surface gravity, and projected rotational velocity of the primary star were determined to be T_eff,1 = 8500 K, log g₁ = 4.0 dex, and ${\upsilon }_{1}\sin i$ = ∼60 km s⁻¹, respectively. Zhang et al. (2014) presented more comprehensive BV band photometric observations of OO Dra, and found that the binary may be a detached system with the less-massive secondary nearly filling its Roche lobe. They detected two frequencies of 41.87 and 34.75 cycle days⁻¹ in both bands of the light residuals, and confirmed the primary star to be a pulsator with δ Scuti pulsations. Lee et al. (2018) presented time series spectroscopy of OO Dra, and derived the effective temperature and the projected rotational velocity of the primary component to be 8260 ± 210 K and 72 ± 5 km s⁻¹, respectively. Furthermore, Lee et al. (2018) modeled their new radial velocity curves and the BV photometric data of Zhang et al. (2014), and obtained the physical parameters of the primary star to be M₁ = 2.031 ± 0.058 M_⊙, R₁ = 2.078 ± 0.026 R_⊙, log g₁ = 4.110 ± 0.017, and L₁ = 18.0 ± 1.9 L_⊙, and those of the secondary star to be M₂ = 0.187 ± 0.009 M_⊙, R₂ = 1.199 ± 0.017 R_⊙, log g₂ = 3.552 ± 0.026, and L₂ = 1.99 ± 0.20 L_⊙, respectively.

In this work, we extend the work of Zhang et al. (2014) and Lee et al. (2018), and perform a more comprehensive asteroseismic analysis for the eclipsing binary OO Dra. In Section 2, we present the pulsational features of the primary component. We introduce the details of the input physics in Section 3, and elaborate our asteroseismic modeling in Section 4. Finally, we conclude and discuss our results in Section 5.

OO Dra was observed in 2 minute cadence mode by the TESS satellite during Sectors 20 and 21 from 2019 December 24 to 2020 February 18 with a time span of ΔT = 58.28 days. We downloaded the data from the Mikulski Archive for Space Telescopes (MAST; https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html), and all TESS data used in this paper can be found in MAST: 10.17909/t9-kfsv-6g77. From the data file, we extracted the BJD times and the "PDCSAP_FLUX" fluxes, which were processed with the Pre-search Data Conditioning Pipeline (Jenkins et al. 2016) to eliminate instrumental trends. Afterward, we removed outliers and normalized the fluxes with the method described in Slawson et al. (2011).

Lee et al. (2018) determined the physical parameters of OO Dra by using the B- and V-band photometry light curves, while observations by TESS are in white light and therefore do not include color information. In this work, we adopt the physical parameters obtained by Lee et al. (2018). Using the ephemerides given by Lee et al. (2018), we computed phases and folded the light curve as illustrated in Figure 1.

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In order to remove the light changes due to eclipses, and obtain the pulsational light variations, we adopt a simple approach, i.e., computing a mean curve for the folded the light curve and then subtracting it from the 2 minute cadence data. When computing the mean curve, the number of bins in phase should not be too large so as to ensure that each bin can contain enough data points. Given that the 58.28 day's TESS data of OO Dra include 35,927 data points, we use 1500 bins in phase in this work. Moreover, we also use 500 bins and 1000 bins in phase for the folded light curve, which results in slightly different wiggles. However, we find those differences and interpolation algorithms do not impact the extracted frequencies. In Figure 1, we show the mean curve with the red line in the upper panel, and show the light residuals in plots of magnitude versus phase in the lower panel. Afterward, we perform a multiple frequency analysis of the light residuals with the software Period04 (Lenz & Breger 2005) to investigate the nature of the pulsation in detail. No peaks higher than 100 day⁻¹ are found, thus, we perform further frequency extractions in the range of 0−100 day⁻¹.

At each step of the iteration, we choose the frequency with the highest amplitude and carry out a multi-period least-squares fit to the data using all frequencies that have been already detected. The data were then pre-whitened and the residuals were used for further analysis. Following Breger et al. (1993), we adopt the empirical threshold of the signal-to-noise ratio, S/N = 4, for a frequency to be accepted as significant. Finally, we extracted a total of 45 frequencies with S/N > 4, and listed them in Table 1. The uncertainties of frequencies and amplitudes are calculated using Monte Carlo simulations as described in Fu et al. (2013), and noises are computed in the range of 2 day⁻¹ around each frequency. Figure 2 depicts the Fourier amplitude spectra of the light residuals, with the original spectrum illustrated in the upper panel and the residual spectrum of the 45 frequencies after pre-whitening illustrated in the lower panel.

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Following Pápics (2012) and Kurtz et al. (2015), we examine the extracted frequencies to search for possible orbital harmonics and linear combinations in the form of f_i = f_j ± mf_orb or f_i = mf_j + nf_k , where f_j and f_k are the parent frequencies, f_i is the combination term, m and n are integers, and f_orb = 0.8075 day⁻¹. We accepted a peak as a combination if the amplitudes of both parent frequencies were larger than that of the presumed combination term, and the difference between the observed frequency and the predicted frequency is smaller than the frequency resolution 1.5/ΔT = 0.027 day⁻¹ (Loumos & Deeming 1978). We identify a total of 33 such frequencies, and mark them in Table 1. Besides, we accepted the lower amplitude frequency as an unresolved peak if two frequencies were closer than 1.5/ΔT = 0.027 day⁻¹. Finally, seven confident independent frequencies f₁, f₂, f₃, f₄, f₇, f₁₁, and f₁₂ are retained as shown in Table 1.

Our models are computed with the one-dimensional stellar evolution code Modules for Experiments in Stellar Astrophysics version 10398 (Paxton et al. 2011, 2013, 2015, 2018). In particular, we use its submodule "pulse_adipls" to generate evolutionary models of stars and calculate the corresponding adiabatic frequencies (Christensen-Dalsgaard 2008).

In the calculations, we choose the 2005 update of the OPAL equation of state tables (Rogers & Nayfonov 2002). We adopt the OPAL opacity tables of Iglesias & Rogers (1996) in the high temperature region, and use the tables of Ferguson et al. (2005) in the low temperature region. The solar metal composition AGSS09 (Asplund et al. 2009) is used as the initial ingredient in metal, and a simple photosphere is adopted for the atmospheric surface boundary conditions. The classical mixing length theory of Böhm-Vitense (1958) with the solar value of α = 1.9 (Paxton et al. 2013) is used in the convective region. In addition, we do not consider the effects of the stellar rotation, the element diffusion, the convective overshooting, nor the magnetic fields on the structure and evolution of the star in our models (for more details, refer to the Appendix).

4.1. Single-star Evolutionary Models

Our single-star evolutionary grid considers variations of the stellar mass M, the helium abundance Y, and the metallicity Z, which dictate the evolutionary track of a star. Therein, the stellar mass M varies from 1.80–2.20 M_⊙ at intervals of 0.02 M_⊙, and metallicities Z vary from 0.005–0.030 at intervals of 0.001. For the helium abundance Y, we adopt three different values: the extremely helium-poor abundance 0.220, the moderate helium abundance 0.249 + 1.33Z (Li et al. 2018) as a function of Z, and the helium-rich abundance 0.300.

Each star in the grid is computed evolving from the pre-main-sequence stage to the evolutionary status where the central hydrogen of the star exhausts (X_c < 1 × 10⁻⁵). Based on the results of binary model given by Lee et al. (2018), we use 8000 K < T_eff < 8600 K and 1.95 R_⊙ < R < 2.15 R_⊙ as the observational constraints. When a star evolves along its evolutionary track, we calculate the adiabatic frequencies of the radial oscillations (ℓ = 0) and nonradial oscillations with ℓ = 1 and 2 for all stellar models that meet the observational constraint. Because of the cancellation effects of the surface geometry, oscillation modes with a higher degree are hardly visible, thus, oscillation modes with ℓ ≥ 3 are not included in this work.

In general, the component stars in binaries rotate along with the orbital motion. Following Chen et al. (2020), we consider the rotation period P_rot of the star as the fourth adjustable parameter, varying from 1.0–1.50 days with a step of 0.01 day. According to the theory of stellar oscillations, rotation will result in each nonradial oscillation mode having ℓsplits into 2ℓ + 1 different frequencies. The general expression of the first-order effect of rotation on pulsation was deduced to be

$$\begin{eqnarray}&&{\nu }_{{\ell },n,m}={\nu }_{{\ell },n}+m\delta {\nu }_{{\ell },n}={\nu }_{{\ell },n}+{\beta }_{{\ell },n}\displaystyle \frac{m}{{P}_{\mathrm{rot}}}\end{eqnarray} \tag{ 1 }$$

(Aerts et al. 2010). In the equation, ℓ, n, and m characterize the oscillation modes, and δ ν_ℓ,n is the splitting frequency. According to Equation (1), the effect of the rotation on pulsation can be completely dictated by the constant β_ℓ,n . Aerts et al. (2010) derived the general expression of β_ℓ,n to be

$$\begin{eqnarray}&&{\beta }_{{\ell },n}=\displaystyle \frac{{\int }_{0}^{R}({\xi }_{r}^{2}+{L}^{2}{\xi }_{h}^{2}-2{\xi }_{r}{\xi }_{h}-{\xi }_{h}^{2}){r}^{2}\rho {dr}}{{\int }_{0}^{R}({\xi }_{r}^{2}+{L}^{2}{\xi }_{h}^{2}){r}^{2}\rho {dr}},\end{eqnarray} \tag{ 2 }$$

where ξ_r is the radial displacement, ξ_h is the horizontal displacement, ρ is the local density of the star, and L² = ℓ(ℓ + 1). According to the above analyses, each dipole mode splits into three different frequencies with m = −1, 0, and +1, respectively, and each quadrupole mode splits into five different frequencies with m = −2, −1, 0, +1, and +2, respectively.

Then, we compare frequencies between models and observations according to

$$\begin{eqnarray}&&{S}^{2}=\displaystyle \frac{1}{k}\sum (| {\nu }_{\mathrm{mod},{\rm{i}}}-{\nu }_{\mathrm{obs},\ {\rm{i}}}{| }^{2}),\end{eqnarray} \tag{ 3 }$$

in which ${\nu }_{\mathrm{mod},{\rm{i}}}$ and ν_obs,i represent a pair of matched model-observed frequencies, and k is the number of the observed frequencies. Because we have no preconceived idea of the identification of the δ Scuti frequencies, we adopt the random fitting algorithm, and consider the model frequency nearest to observations as the most probably matched model counterpart.

Figures 3 and 4 show plots of the fitting results ${S}_{{\rm{m}}}^{2}$ versus various physical parameters. In the figures, circles in red, blue, and black correspond to the theoretical models with Y = 0.220, 0.249+1.33Z, and 0.300, respectively. Similar to Chen et al. (2016), we find that the solution concentrates on a small parameter space for a given evolutionary track. Thus, we choose the one model with the minimum value of S² along the evolutionary track, and denote the minimum value with the symbol ${S}_{{\rm{m}}}^{2}$. We use horizontal lines to mark the position of ${S}_{{\rm{m}}}^{2}$ = 0.10, which corresponds to the square of the frequency resolution 1.5/ΔT.

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Figures 3(a) depicts the plot of ${S}_{{\rm{m}}}^{2}$ versus the stellar mass M. It can be clearly seen in the figure that theoretical models with poorer helium abundances have higher mass, i.e., 1.80−1.86 M_⊙ for Y = 0.300, 1.80−1.92 M_⊙ for 0.249 + 1.33Z, and 1.90−2.04 M_⊙ for Y = 0.220, respectively. Lee et al. (2018) determined the mass of the primary star to be 2.03 ± 0.06 M_⊙. We therefore consider the 61 models with Y = 0.220 shown above the horizontal line as our preferred models, and list them in Table 2. Among these models, Model A44 has the minimum value of ${S}_{{\rm{m}}}^{2}$, we then choose it as the optimal single-star evolutionary model in this work.

Table 2. Preferred Models with ${S}_{{\rm{m}}}^{2}$ ≤ 0.10

Model	Prot	Z	M	Y	Teff	logg	R	L	τ0	Xc	Age	${S}_{{\rm{m}}}^{2}$
	(day)		(M⊙)		(K)	(cgs)	(R⊙)	(L⊙)	(hr)		(Myr)
A01	1.15	0.013	1.94	0.220	8172	4.091	2.076	17.27	2.308	0.767	7.93	0.098
A02	1.15	0.013	1.96	0.220	8241	4.093	2.083	17.98	2.311	0.767	7.75	0.094
A03	1.15	0.013	2.00	0.220	8376	4.096	2.097	19.44	2.317	0.767	7.41	0.099
A04	1.15	0.014	1.94	0.220	8065	4.091	2.076	16.38	2.305	0.766	8.22	0.090
A05	1.15	0.014	1.96	0.220	8131	4.093	2.083	17.04	2.307	0.766	8.04	0.088
A06	1.15	0.015	1.96	0.220	8020	4.093	2.082	16.12	2.302	0.765	8.34	0.097
A07	1.16	0.011	2.00	0.220	8593	4.096	2.097	21.54	2.324	0.769	6.88	0.098
A08	1.16	0.012	1.94	0.220	8277	4.091	2.076	18.18	2.313	0.768	7.64	0.094
A09	1.16	0.012	1.96	0.220	8347	4.093	2.083	18.93	2.316	0.768	7.47	0.090
A10	1.16	0.012	1.98	0.220	8417	4.094	2.090	19.69	2.317	0.768	7.30	0.091
A11	1.16	0.012	2.00	0.220	8486	4.096	2.097	20.48	2.320	0.768	7.14	0.082
A12	1.16	0.012	2.02	0.220	8554	4.097	2.104	21.30	2.322	0.768	6.99	0.090
A13	1.16	0.013	1.90	0.220	8034	4.088	2.062	15.91	2.303	0.767	8.30	0.095
A14	1.16	0.013	1.92	0.220	8103	4.090	2.069	16.58	2.305	0.767	8.11	0.081
A15	1.16	0.013	1.94	0.220	8172	4.091	2.076	17.27	2.308	0.767	7.93	0.077
A16	1.16	0.013	1.96	0.220	8241	4.093	2.083	17.98	2.311	0.767	7.75	0.076
A17	1.16	0.013	1.98	0.220	8309	4.094	2.090	18.70	2.313	0.767	7.58	0.079
A18	1.16	0.013	2.00	0.220	8376	4.096	2.097	19.44	2.317	0.767	7.41	0.084
A19	1.16	0.013	2.02	0.220	8443	4.097	2.104	20.20	2.318	0.767	7.25	0.090
A20	1.16	0.013	2.04	0.220	8509	4.099	2.110	20.98	2.321	0.767	7.09	0.095
A21	1.16	0.014	1.94	0.220	8065	4.092	2.075	16.37	2.304	0.766	8.22	0.074
A22	1.16	0.014	1.96	0.220	8131	4.093	2.083	17.04	2.307	0.766	8.04	0.076
A23	1.16	0.014	1.98	0.220	8198	4.095	2.089	17.71	2.309	0.766	7.86	0.083
A24	1.16	0.014	2.00	0.220	8263	4.096	2.097	18.41	2.313	0.766	7.69	0.096
A25	1.16	0.015	1.96	0.220	8020	4.093	2.082	16.11	2.301	0.765	8.34	0.088
A26	1.17	0.011	1.94	0.220	8379	4.091	2.076	19.08	2.317	0.769	7.36	0.096
A27	1.17	0.011	1.96	0.220	8451	4.093	2.083	19.88	2.318	0.769	7.20	0.088
A28	1.17	0.011	1.98	0.220	8522	4.094	2.090	20.70	2.321	0.769	7.04	0.086
A29	1.17	0.011	2.00	0.220	8593	4.096	2.097	21.54	2.324	0.769	6.88	0.083
A30	1.17	0.012	1.90	0.220	8135	4.088	2.062	16.73	2.306	0.768	8.00	0.099
A31	1.17	0.012	1.92	0.220	8206	4.090	2.069	17.44	2.309	0.768	7.82	0.092
A32	1.17	0.012	1.94	0.220	8278	4.091	2.075	18.17	2.312	0.768	7.64	0.089
A33	1.17	0.012	1.96	0.220	8348	4.093	2.083	18.92	2.315	0.768	7.47	0.085
A34	1.17	0.012	1.98	0.220	8417	4.094	2.090	19.69	2.317	0.768	7.30	0.079
A35	1.17	0.012	2.00	0.220	8486	4.096	2.097	20.48	2.320	0.768	7.14	0.076
A36	1.17	0.012	2.02	0.220	8554	4.097	2.104	21.30	2.322	0.768	6.99	0.092
A37	1.17	0.013	1.90	0.220	8034	4.089	2.061	15.90	2.302	0.767	8.30	0.083
A38	1.17	0.013	1.92	0.220	8103	4.090	2.069	16.58	2.305	0.767	8.11	0.078
A39	1.17	0.013	1.94	0.220	8172	4.091	2.076	17.27	2.308	0.767	7.93	0.078
A40	1.17	0.013	1.96	0.220	8241	4.093	2.083	17.98	2.311	0.767	7.75	0.079
A41	1.17	0.013	1.98	0.220	8309	4.094	2.090	18.70	2.313	0.767	7.58	0.075
A42	1.17	0.013	2.00	0.220	8376	4.096	2.097	19.44	2.317	0.767	7.41	0.091
A43	1.17	0.013	2.02	0.220	8443	4.097	2.104	20.20	2.318	0.767	7.25	0.098
A44	1.17	0.014	1.94	0.220	8065	4.092	2.075	16.37	2.304	0.766	8.22	0.072
A45	1.17	0.014	1.96	0.220	8132	4.093	2.082	17.03	2.306	0.766	8.04	0.085
A46	1.17	0.014	1.98	0.220	8198	4.095	2.089	17.71	2.309	0.766	7.86	0.086
A47	1.17	0.015	1.96	0.220	8020	4.093	2.082	16.11	2.301	0.765	8.34	0.095
A48	1.18	0.011	1.96	0.220	8451	4.093	2.083	19.88	2.318	0.769	7.20	0.094
A49	1.18	0.011	1.98	0.220	8522	4.094	2.090	20.70	2.321	0.769	7.04	0.088
A50	1.18	0.011	2.00	0.220	8593	4.096	2.097	21.54	2.324	0.769	6.88	0.088
A51	1.18	0.012	1.92	0.220	8207	4.090	2.068	17.44	2.308	0.768	7.82	0.098
A52	1.18	0.012	1.94	0.220	8278	4.091	2.075	18.17	2.312	0.768	7.64	0.091
A53	1.18	0.012	1.96	0.220	8348	4.093	2.083	18.92	2.315	0.768	7.47	0.088
A54	1.18	0.012	1.98	0.220	8417	4.094	2.090	19.69	2.317	0.768	7.30	0.087
A55	1.18	0.012	2.00	0.220	8486	4.096	2.097	20.48	2.320	0.768	7.14	0.089
A56	1.18	0.013	1.90	0.220	8034	4.089	2.061	15.90	2.302	0.767	8.30	0.089
A57	1.18	0.013	1.92	0.220	8104	4.090	2.068	16.57	2.304	0.767	8.11	0.090
A58	1.18	0.013	1.94	0.220	8173	4.092	2.075	17.26	2.307	0.767	7.93	0.090
A59	1.18	0.013	1.96	0.220	8241	4.093	2.082	17.97	2.310	0.767	7.75	0.092
A60	1.18	0.013	1.98	0.220	8309	4.094	2.090	18.70	2.313	0.767	7.58	0.091
A61	1.18	0.014	1.94	0.220	8065	4.092	2.075	16.37	2.304	0.766	8.22	0.090

Note. P_rot represents the rotation period, τ₀ represents the acoustic radius, and X_c represents the mass fraction of hydrogen in the center of the star.

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Figures 3(b) and (c) depict plots of ${S}_{{\rm{m}}}^{2}$ versus the metallicity Z and the rotational period P_rot, respectively. In the figures, it can be seen that values of Z and P_rot show excellent convergence. Values of Z converge well to ${0.014}_{-0.003}^{+0.001}$, and those of P_rot converge well to ${1.17}_{-0.02}^{+0.01}$ days.

Figure 3(d) depicts the plot of ${S}_{{\rm{m}}}^{2}$ as a function of the age of the star. As illustrated in the figure, the ages of all preferred models converge well to ${8.22}_{-1.33}^{+0.12}$ Myr. Similar to KIC 10736223 (Chen et al. 2020), the primary component of OO Dra also looks like an almost unevolved star near the zero-age main sequence. We therefore identify OO Dra to be another binary system that has just undergone the rapid mass-transfer stage.

Figures 4(a)–(d) present plots of ${S}_{{\rm{m}}}^{2}$ as a function of various global stellar parameters, $\mathrm{log}g$, R, T_eff, and L, respectively. Values of $\mathrm{log}g$ and R converge to ${4.092}_{-0.004}^{+0.007}$ and ${2.075}_{-0.014}^{+0.035}$ R_⊙, respectively. The convergence of T_eff and L is relatively worse, i.e., T_eff = ${8065}_{-45}^{+528}$ K and L = ${16.37}_{-0.47}^{+5.17}$ L_⊙.

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Finally, the fundamental physical parameters of the primary star yielded by asteroseismic fittings of single stars are listed in the second column of Table 3. Table 4 lists the model frequencies of the best-fitting single-star evolutionary model, and Table 5 shows comparisons between the observed frequencies and their corresponding model counterparts. Based on the comparisons, f₂, f₇, and f₁₁ are identified as three dipole modes, and f₁, f₃, f₄, and f₁₂ as four quadrupole modes. Furthermore, we find that f₇ and f₂ are m = −1 and 0 components of one triplet, and f₁₂ and f₁ are m = +1 and +2 components of one quintuplet.

Table 3. Fundamental Parameters of the Primary Component of OO Dra Yielded by Asteroseismology

Parameters	Single-star Models	Mass-accreting Models	Lee et al. (2018)
Z	0.011−0.015 (${0.014}_{-0.003}^{+0.001}$)	0.010−0.017 (${0.011}_{-0.001}^{+0.006}$)	⋯
M (M⊙)	1.90−2.04 (${1.94}_{-0.04}^{+0.10}$)	⋯	2.031±0.058
Y	0.220	0.249+1.33Z	⋯
M1 (M⊙)	⋯	0.60-0.80 (0.70 ± 0.10)	⋯
M2 (M⊙)	⋯	1.90−2.02 (${1.92}_{-0.02}^{+0.10}$)	⋯
Prot(day)	1.15−1.18 (${1.17}_{-0.02}^{+0.01}$)	1.15−1.18 (${1.16}_{-0.01}^{+0.02}$)	⋯
Teff (K)	8020−8593 (${8065}_{-45}^{+528}$)	8026−8590 (${8346}_{-320}^{+244}$)	8260±210
$\mathrm{log}g$ (cgs)	4.088−4.099 (${4.092}_{-0.004}^{+0.007}$)	4.088−4.100 (${4.090}_{-0.002}^{+0.010}$)	4.110±0.017
R (R⊙)	2.061−2.110 (${2.075}_{-0.014}^{+0.035}$)	2.061−2.118 (${2.068}_{-0.007}^{+0.050}$)	2.078±0.026
L (L⊙)	15.90−21.54 (${16.37}_{-0.47}^{+5.17}$)	15.83−21.96(${18.65}_{-2.82}^{+3.31}$)	18.0±1.9
τ0 (h)	2.301−2.324 (${2.304}_{-0.003}^{+0.020}$)	2.302−2.324 (${2.315}_{-0.013}^{+0.009}$)	⋯
Xc	0.765−0.769 (${0.766}_{-0.001}^{+0.003}$)	0.696−0.726 (${0.723}_{-0.027}^{+0.003}$)	⋯
Age (Myr)	6.89−8.34 (${8.22}_{-1.33}^{+0.12}$)	⋯	⋯
Tge (Myr)	⋯	0.14−0.67 (0.49${}_{-0.35}^{+0.18}$)	⋯

Note. P_rot is the rotation period. τ₀ represents the acoustic radius. X_c denotes the mass fraction of hydrogen in the center of the star. Tge denotes the evolutionary time of mass-accreting models since the rapid mass accretion ended. The stellar parameters of Lee et al. (2018) using the binary model are listed in the last column.

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Table 4. Theoretical Frequencies of the Best-fitting Single-star Evolutionary Model (Model A44)

${\nu }_{\mathrm{mod}}({\ell },n)$	βℓ,n	${\nu }_{\mathrm{mod}}({\ell },n)$	βℓ,n	${\nu }_{\mathrm{mod}}({\ell },n)$	βℓ,n
(μHz)		(μHz)		(μHz)
166.800(0,0)		124.265(1,0)	0.487	164.889(2,0)	0.918
215.551(0,1)		171.134(1,1)	0.985	191.521(2,0)	0.861
265.700(0,2)		222.486(1,2)	0.997	226.527(2,1)	0.996
315.399(0,3)		276.840(1,3)	0.998	261.773(2,2)	0.878
365.904(0,4)		332.122(1,4)	0.996	305.360(2,3)	0.879
417.999(0,5)		387.732(1,5)	0.993	357.115(2,4)	0.925
470.551(0,6)		442.375(1,6)	0.990	411.472(2,5)	0.951
523.603(0,7)		496.263(1,7)	0.989	465.420(2,6)	0.964
577.884(0,8)		550.456(1,8)	0.988	519.169(2,7)	0.972
633.391(0,9)		605.631(1,9)	0.987	573.737(2,8)	0.977
689.930(0,10)		661.736(1,10)	0.986	629.321(2,9)	0.980

Note. ${\nu }_{\mathrm{mod}}$ is the model frequency. ℓ and n are its spherical harmonic degree and radial order, respectively. β_ℓ,n is the rotational parameters defined as Equation (2).

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Table 5. Comparisons between Theoretical Frequencies of the Best-fitting Single-star Evolutionary Model (Model A44) and Observations

ID	νobs	${\nu }_{\mathrm{mod}}$	(ℓ, n, m)	$| {\nu }_{\mathrm{obs}}-{\nu }_{\mathrm{mod}}|$
	(μHz)	(μHz)		(μHz)
f1	484.571	484.493	(2, 6, +2)	0.078
f2	387.417	387.732	(1, 5, 0)	0.315
f3	402.092	402.064	(2, 5, −1)	0.028
f4	375.622	375.416	(2, 4, +2)	0.206
f7	377.678	377.909	(1, 5, −1)	0.231
f11	442.740	442.375	(1, 6, 0)	0.365
f12	474.546	474.957	(2, 6, +1)	0.411

Note. ν_obs denotes the observed frequency. ${\nu }_{\mathrm{mod}}$ represents its model counterpart. $| {\nu }_{\mathrm{obs}}-{\nu }_{\mathrm{mod}}|$ shows the difference between the observed frequency and the model counterpart.

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4.2. Mass-accreting Models

In Section 4.1, we analyzed the fitting results of single-star evolutionary models and found that OO Dra might be another binary system that has just undergone the rapid mass-transfer stage. Given that there are many uncertainties in binary evolution models, especially for the mass-transfer process, we model the mass-transfer process using a similar approach to that used by Chen et al. (2020). We evolve a single star to the position where the rapid mass-transfer process may occur (Case A or Case B binary evolution, Han et al. 2000; Chen et al. 2017), and then artificially accrete mass at a rate of 10⁻⁶ M_⊙ yr⁻¹ until masses of the accretor up to the given values. We considered the initial mass M₁ of the accretor between 0.40 and 1.00 M_⊙ in intervals of 0.1 M_⊙, and the final mass M₂ of the accretor between 1.90 and 2.20 M_⊙ in intervals of 0.02 M_⊙. For each pair, we adopt two schemes of the mass accretion, i.e., starting at the age of 0.5 Gyr (the donor on the main-sequence stage, Case A binary evolution) and 1.0 Gyr (the central hydrogen of the donor exhaustion, Case B binary evolution). Moreover, the fitting results of the single-star evolutionary models indicate that the primary component of OO Dra is a helium-poor star. The primordial cosmological helium abundance is determined by Planck Collaboration et al. (2016) to be 0.249. Therefore, we choose the moderate helium abundance 0.249+1.33Z as the base helium abundance of the accretor, and adopt a helium-poor mass accretion, i.e., accreting 0.85 times the base helium abundance to the surface of the accretor. In addition, we consider Z between 0.005 and 0.030 in intervals of 0.001, and P_rot between 1.00 and 1.50 days in intervals of 0.01 day.

We compare frequencies between model and observations according to Equation (3), and show changes in ${S}_{{\rm{m}}}^{2}$ as a function of various physical parameters of the mass-accreting models in Figures 5 and 6. In the figures, circles in red and blue correspond to the accreting models of Case A and Case B binary evolutionary scenario, respectively. It can be seen in the figures that the Case A evolutionary scenario reproduces the δ Scuti frequencies better than the Case B evolutionary scenario. The circles above the horizontal lines correspond to the 85 preferred models of the Case A evolutionary scenario in Table 6.

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Table 6. Preferred Mass-accreting Models with ${S}_{{\rm{m}}}^{2}$ ≤ 0.10

Model	Prot	Z	M1	M2	Teff	logg	R	L	τ0	Xc	Tge	${S}_{{\rm{m}}}^{2}$
	(day)		(M⊙)	(M⊙)	(K)		(R⊙)	(L⊙)	(hr)		(Myr)
B01	1.15	0.010	0.70	1.94	8558	4.091	2.076	20.78	2.322	0.726	0.43	0.096
B02	1.15	0.011	0.70	1.90	8257	4.089	2.061	17.74	2.311	0.723	0.52	0.098
B03	1.15	0.011	0.70	1.92	8346	4.090	2.068	18.65	2.315	0.723	0.49	0.084
B04	1.15	0.011	0.70	1.94	8433	4.092	2.075	19.57	2.318	0.723	0.46	0.095
B05	1.15	0.011	0.70	1.96	8522	4.093	2.083	20.57	2.322	0.723	0.43	0.091
B06	1.15	0.012	0.60	1.90	8416	4.088	2.062	19.17	2.316	0.722	0.18	0.094
B07	1.15	0.012	0.60	1.92	8501	4.090	2.070	20.09	2.319	0.722	0.17	0.082
B08	1.15	0.012	0.60	1.94	8590	4.091	2.077	21.10	2.322	0.722	0.16	0.079
B09	1.15	0.012	0.70	1.90	8142	4.089	2.061	16.77	2.306	0.721	0.56	0.081
B10	1.15	0.012	0.70	1.92	8227	4.090	2.068	17.60	2.309	0.721	0.53	0.085
B11	1.15	0.012	0.70	1.94	8312	4.091	2.076	18.47	2.313	0.721	0.50	0.092
B12	1.15	0.012	0.70	2.00	8563	4.096	2.097	21.24	2.322	0.721	0.41	0.094
B13	1.15	0.013	0.60	1.94	8476	4.091	2.076	19.99	2.318	0.720	0.14	0.099
B14	1.15	0.013	0.70	1.90	8026	4.089	2.061	15.83	2.302	0.719	0.59	0.087
B15	1.15	0.013	0.70	1.92	8108	4.090	2.067	16.60	2.305	0.719	0.56	0.100
B16	1.15	0.013	0.70	1.94	8191	4.092	2.075	17.41	2.309	0.719	0.53	0.091
B17	1.16	0.010	0.70	1.90	8375	4.089	2.061	18.78	2.314	0.726	0.49	0.091
B18	1.16	0.010	0.70	1.92	8467	4.090	2.068	19.75	2.318	0.726	0.46	0.084
B19	1.16	0.010	0.70	1.94	8558	4.091	2.076	20.78	2.322	0.726	0.43	0.085
B20	1.16	0.011	0.70	1.90	8257	4.089	2.061	17.74	2.311	0.723	0.52	0.078
B21	1.16	0.011	0.70	1.92	8346	4.090	2.068	18.65	2.315	0.723	0.49	0.069
B22	1.16	0.011	0.70	1.94	8433	4.092	2.075	19.57	2.318	0.723	0.46	0.075
B23	1.16	0.011	0.70	1.96	8522	4.093	2.083	20.57	2.322	0.723	0.43	0.089
B24	1.16	0.012	0.60	1.90	8416	4.088	2.062	19.17	2.316	0.722	0.18	0.083
B25	1.16	0.012	0.60	1.92	8501	4.090	2.070	20.09	2.319	0.722	0.17	0.075
B26	1.16	0.012	0.60	1.94	8590	4.091	2.077	21.10	2.322	0.722	0.16	0.073
B27	1.16	0.012	0.70	1.90	8142	4.089	2.061	16.77	2.306	0.721	0.56	0.072
B28	1.16	0.012	0.70	1.92	8227	4.090	2.068	17.60	2.309	0.721	0.53	0.073
B29	1.16	0.012	0.70	1.94	8312	4.091	2.076	18.47	2.313	0.721	0.50	0.094
B30	1.16	0.012	0.70	1.96	8396	4.093	2.082	19.35	2.315	0.721	0.47	0.090
B31	1.16	0.012	0.70	2.00	8563	4.096	2.097	21.24	2.322	0.721	0.41	0.089
B32	1.16	0.013	0.60	1.94	8476	4.091	2.076	19.99	2.318	0.720	0.14	0.100
B33	1.16	0.013	0.70	1.90	8026	4.089	2.061	15.83	2.302	0.719	0.59	0.088
B34	1.16	0.013	0.70	1.92	8108	4.090	2.067	16.60	2.305	0.719	0.56	0.090
B35	1.16	0.013	0.70	1.94	8191	4.092	2.075	17.41	2.309	0.719	0.53	0.092
B36	1.16	0.016	0.80	1.96	8179	4.093	2.084	17.45	2.309	0.698	0.60	0.096
B37	1.16	0.016	0.80	1.98	8262	4.094	2.091	18.30	2.312	0.698	0.57	0.094
B38	1.16	0.016	0.80	2.00	8344	4.095	2.098	19.16	2.315	0.698	0.53	0.090
B39	1.16	0.016	0.80	2.02	8427	4.097	2.105	20.07	2.318	0.698	0.50	0.089
B40	1.16	0.017	0.80	1.96	8049	4.093	2.083	16.37	2.303	0.696	0.65	0.080
B41	1.16	0.017	0.80	1.98	8128	4.094	2.090	17.13	2.306	0.696	0.62	0.085
B42	1.16	0.017	0.80	2.00	8207	4.096	2.097	17.93	2.309	0.696	0.58	0.095
B43	1.16	0.017	0.80	2.02	8285	4.097	2.104	18.74	2.312	0.696	0.55	0.097
B44	1.17	0.010	0.70	1.90	8375	4.089	2.061	18.78	2.314	0.726	0.49	0.085
B45	1.17	0.010	0.70	1.92	8467	4.090	2.068	19.75	2.318	0.726	0.46	0.079
B46	1.17	0.010	0.70	1.94	8558	4.091	2.076	20.78	2.322	0.726	0.43	0.096
B47	1.17	0.011	0.70	1.90	8257	4.089	2.061	17.74	2.311	0.723	0.52	0.079
B48	1.17	0.011	0.70	1.92	8346	4.090	2.068	18.65	2.315	0.723	0.49	0.075
B49	1.17	0.011	0.70	1.94	8433	4.092	2.075	19.57	2.318	0.723	0.46	0.076
B50	1.17	0.011	0.70	1.96	8522	4.093	2.082	20.55	2.320	0.723	0.44	0.093
B51	1.17	0.012	0.60	1.90	8416	4.088	2.062	19.17	2.316	0.722	0.18	0.093
B52	1.17	0.012	0.60	1.92	8501	4.090	2.070	20.09	2.319	0.722	0.17	0.089
B53	1.17	0.012	0.60	1.94	8590	4.091	2.077	21.10	2.322	0.722	0.16	0.088
B54	1.17	0.012	0.70	1.90	8142	4.089	2.061	16.77	2.306	0.721	0.56	0.085
B55	1.17	0.012	0.70	1.92	8227	4.090	2.068	17.60	2.309	0.721	0.53	0.083
B56	1.17	0.012	0.70	1.94	8311	4.092	2.075	18.45	2.312	0.721	0.50	0.098
B57	1.17	0.012	0.70	1.96	8396	4.093	2.082	19.35	2.315	0.721	0.47	0.097
B58	1.17	0.015	0.80	1.98	8389	4.094	2.091	19.46	2.316	0.700	0.52	0.097
B59	1.17	0.015	0.80	2.02	8561	4.097	2.106	21.39	2.323	0.700	0.46	0.091
B60	1.17	0.016	0.80	1.92	8013	4.090	2.069	15.86	2.302	0.698	0.67	0.094
B61	1.17	0.016	0.80	1.94	8097	4.091	2.076	16.65	2.305	0.698	0.63	0.085
B62	1.17	0.016	0.80	1.96	8179	4.093	2.084	17.45	2.309	0.698	0.60	0.081
B63	1.17	0.016	0.80	1.98	8262	4.094	2.091	18.30	2.312	0.698	0.57	0.080
B64	1.17	0.016	0.80	2.00	8344	4.095	2.098	19.16	2.315	0.698	0.53	0.079
B65	1.17	0.016	0.80	2.02	8427	4.097	2.105	20.07	2.318	0.698	0.50	0.084
B66	1.17	0.016	0.80	2.04	8509	4.098	2.111	20.99	2.320	0.698	0.47	0.097
B67	1.17	0.016	0.80	2.06	8590	4.100	2.118	21.96	2.324	0.698	0.45	0.095
B68	1.17	0.017	0.80	1.96	8049	4.093	2.083	16.37	2.303	0.696	0.65	0.081
B69	1.17	0.017	0.80	1.98	8128	4.094	2.090	17.13	2.306	0.696	0.62	0.081
B70	1.17	0.017	0.80	2.00	8206	4.096	2.097	17.91	2.308	0.696	0.58	0.096
B71	1.17	0.017	0.80	2.02	8285	4.097	2.104	18.74	2.312	0.696	0.55	0.099
B72	1.18	0.010	0.70	1.90	8375	4.089	2.061	18.78	2.314	0.726	0.49	0.099
B73	1.18	0.010	0.70	1.92	8467	4.090	2.068	19.75	2.318	0.726	0.46	0.094
B74	1.18	0.011	0.70	1.90	8257	4.089	2.061	17.74	2.311	0.723	0.52	0.100
B75	1.18	0.011	0.70	1.94	8433	4.092	2.075	19.57	2.318	0.723	0.46	0.097
B76	1.18	0.015	0.80	1.98	8389	4.094	2.091	19.46	2.316	0.700	0.52	0.098
B77	1.18	0.015	0.80	2.02	8561	4.097	2.106	21.39	2.323	0.700	0.46	0.098
B78	1.18	0.016	0.80	1.92	8013	4.090	2.069	15.86	2.302	0.698	0.67	0.094
B79	1.18	0.016	0.80	1.94	8097	4.091	2.076	16.65	2.305	0.698	0.63	0.090
B80	1.18	0.016	0.80	1.96	8179	4.093	2.084	17.45	2.309	0.698	0.60	0.086
B81	1.18	0.016	0.80	1.98	8262	4.094	2.091	18.30	2.312	0.698	0.57	0.087
B82	1.18	0.016	0.80	2.00	8344	4.095	2.098	19.16	2.315	0.698	0.53	0.087
B83	1.18	0.016	0.80	2.02	8427	4.097	2.105	20.07	2.318	0.698	0.50	0.099
B84	1.18	0.017	0.80	1.96	8049	4.093	2.083	16.35	2.302	0.696	0.65	0.094
B85	1.18	0.017	0.80	1.98	8128	4.094	2.090	17.13	2.306	0.696	0.62	0.097

Note. P_rot represents the rotation period. τ₀ represents the acoustic radius. X_c represents the mass fraction of hydrogen in the center of the star. Tge denotes the evolutionary time of mass-accreting models since the rapid mass accretion ended.

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Figures 5(a)–(d) present changes in the fitting results ${S}_{{\rm{m}}}^{2}$ as a function of Z, M₁, M₂, and P_rot, respectively. It can clearly be seen in the figures their values converge well to ${0.011}_{-0.001}^{+0.006}$, 0.70 ± 0.1 M_⊙, ${1.92}_{-0.02}^{+0.10}$ M_⊙, and ${1.16}_{-0.01}^{+0.02}$ days, respectively.

Figures 6(a)–(d) depict changes in ${S}_{{\rm{m}}}^{2}$ as a function of the global physical parameters $\mathrm{log}g$, R, T_eff, and L, respectively. Therein, $\mathrm{log}g$ and R exhibit good convergence to ${4.090}_{-0.002}^{+0.010}$ and ${2.068}_{-0.007}^{+0.050}$ M_⊙, respectively. While T_eff and L cover a wide range. Values of T_eff vary from 8026–8590 K and those of L vary from 15.83–21.96 L_⊙. We list the parameters in the third column of Table 3, and find them corresponding well with those of the single-star evolutionary models.

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Figure 7 presents changes in ${S}_{{\rm{m}}}^{2}$ as a function of the Tge of the mass-accreting models, where Tge is the evolutionary time since the rapid mass accretion ended. In the figure, the Tge of the preferred models converge well to ${0.49}_{-0.35}^{+0.18}$ Myr, further confirming that OO Dra is an Algol system that has just undergone the rapid mass-transfer stage.

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Finally, we list comparisons between the model frequencies of the optimal mass-accreting model and observations in Table 7. In the table, we note that the identifications of the δ Scuti frequencies are the same as those of the single-star evolutionary scenario.

Table 7. Comparisons between Model Frequencies and Observations for the Best-fitting Model (Model B21) in the Mass-accreting Evolutionary Grid

ID	νobs	${\nu }_{\mathrm{mod}}$	(ℓ, n, m)	$| {\nu }_{\mathrm{obs}}-{\nu }_{\mathrm{mod}}|$
	(μHz)	(μHz)		(μHz)
f1	484.571	484.464	(2, 6, +2)	0.107
f2	387.417	387.756	(1, 5, 0)	0.339
f3	402.092	401.898	(2, 5, −1)	0.194
f4	375.622	375.656	(2, 4, +2)	0.034
f7	377.678	377.848	(1, 5, −1)	0.170
f11	442.740	442.294	(1, 6, 0)	0.446
f12	474.546	474.846	(2, 6, +1)	0.300

Note. ν_obs denotes the observed frequency. ${\nu }_{\mathrm{mod}}$ denotes the model frequency. $| {\nu }_{\mathrm{obs}}-{\nu }_{\mathrm{mod}}|$ is the difference between the observed frequency and its model counterpart.

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In this work, we presented simultaneous frequency analyses and asteroseismic modeling for the Algol-type eclipsing binary OO Dra. We performed a multi-frequency analysis of the eclipse-subtracted TESS light residuals, and obtained seven confident independent frequencies (f₁, f₂, f₃, f₄, f₇, f₁₁, and f₁₂). Therein, f₁ and f₃ have been detected by Zhang et al. (2014) from B- and V-band pulsational light curves, while f₂, f₄, f₇, f₁₁, and f₁₂ are five new frequencies not detected before. These frequencies range from 32.4538–41.8669 cycle days⁻¹, the δ Sct-type pulsational nature of OO Dra is further confirmed.

Two grids of theoretical models, the single-star evolutionary model grid and the mass-accreting model grid, are computed to reproduce the seven δ Scuti frequencies. Due to no preconceived idea of mode identifications of the δ Scuti frequencies, a random fitting algorithm is adopted in this work. The fitting results of the mass-accreting models are found to agree well with those of the single-star evolutionary models. The fundamental physical parameters of the primary star yielded by asteroseismology are M = ${1.92}_{-0.02}^{+0.10}$ M_⊙, Z = ${0.011}_{-0.001}^{+0.006}$, R = ${2.068}_{-0.007}^{+0.050}$ R_⊙, $\mathrm{log}g$ = ${4.090}_{-0.002}^{+0.010}$, T_eff = ${8346}_{-320}^{+244}$ K, and L = ${18.65}_{-2.82}^{+3.31}$ L_⊙, which match well the results of the binary model given by Lee et al. (2018).

The primary component of OO Dra is found to be an almost unevolved star near the zero-age main sequence. The ages of the primary star are determined to be ${8.22}_{-1.33}^{+0.12}$ Myr for the single-star evolutionary models. The values of Tge are determined to be ${0.49}_{-0.35}^{+0.18}$ for the mass-accreting models. Considering that OO Dra is a detached system with the cool secondary almost filling its Roche lobe (Zhang et al. 2014; Lee et al. 2018), OO Dra may be another Algol system that has just undergone the rapid mass-transfer stage.

Besides, our asteroseismic results show that OO Dra is an Algol-type eclipsing binary formed through a helium-poor mass accretion in the Case A binary evolutionary scenario. Figures 8(a) and (b) show profiles of Brunt–Väisäl frequency N, characteristic acoustic frequencies S_ℓ (ℓ = 1 and 2), and hydrogen abundance X_H in the best-fitting single-star model (Model A44) and mass-accreting model (Model B21), respectively. As shown in the figures, the main difference between the two models is the three bumps in N inside the mass-accreting model. In particular, we find that bumps 1 and 2 result from the evolution of the star, while the highest bump, bump 3, is caused by the elemental abundance gradient left behind during the process of helium-poor mass accretion. The seven δ Scuti frequencies of OO Dra are found to be p-modes, which mainly propagate in the envelope of the star. However, for the three bumps, they are not located in the p-mode propagation zones where ω² > N² and ${\omega }^{2}\,\gt \,{S}_{{\ell }}^{2}$. No frequency can reach the bumps, that is to say, the δ Scuti frequencies are not qualified to distinguish those bumps. Furthermore, we introduce an important asteroseismic parameter, the acoustic radius τ₀, which is the sound travel time between the surface and the center of the star. The acoustic radius τ₀ is usually used to characterize the properties of the stellar envelope (Ballot et al. 2004; Miglio et al. 2010; Chen et al. 2016), and defined by Aerts et al. (2010) as

$$\begin{eqnarray}&&{\tau }_{0}={\int }_{0}^{R}\displaystyle \frac{{dr}}{{c}_{s}},\end{eqnarray} \tag{ 4 }$$

where c_s is the adiabatic sound speed and R is the stellar radius. The acoustic radius τ₀ yielded by asteroseismic fittings of the mass-accreting models is ${2.315}_{-0.013}^{+0.009}$ hr, which agree well with the value ${2.304}_{-0.003}^{+0.020}$ hr of the single-star evolutionary models. In the Hertzsprung–Russell diagram, the instability strip of δ Scuti stars largely overlaps with that of γ Dor stars (Balona 2011; Henry et al. 2011; Uytterhoeven et al. 2011; Xiong et al. 2016). The γ Dor-type pulsations allow us to probe the interiors of the star, such as chemical mixing (Miglio et al. 2008) and rotation (Van Reeth et al. 2015, 2016, 2018). The δ Sct-γ Dor hybrid behavior is found to be common for A- and F-type stars (Grigahcène et al. 2010; Balona et al. 2015). A number of eclipsing binaries with hybrid pulsators have been identified and analyzed, such as KIC 9592855 (Guo et al. 2017), KIC 7385478 (Guo & Li 2019), KIC 8113154 (Zhang et al. 2019), and KIC 9850387 (Zhang et al. 2020). Due to the presence of both p and g modes, eclipsing binaries with hybrid δ Sct-γ Dor pulsators will be very promising objects to understand the influences of mass-transfer processes.

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Finally, the rotation period P_rot of the primary star is determined to be ${1.17}_{-0.02}^{+0.01}$ days for the single-star evolutionary models and ${1.16}_{-0.01}^{+0.02}$ days for the mass-accreting models. The spin of the primary star is slightly faster than the synchronous value. Saio (1981) and Dziembowski & Goode (1992) derived that the first-order effect of rotation on pulsation is proportional to 1/P_rot, and the second-order effect on pulsation is proportional to $1/({P}_{\mathrm{rot}}^{2}{\nu }_{{\ell },n})$. The ratio is deduced to be on the order of 1/(P_rot ν_ℓ,n ). For OO Dra, ν_ℓ,n varies from 32.4538–41.8669 day⁻¹. The second-order effect of rotation on pulsation is much less than that of the first-order one. Therefore, the second-order effect of rotation on pulsation is not included in this work.

We are sincerely grateful to the anonymous referee for instructive advice and productive suggestions. This paper includes data collected with the TESS mission, obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the TESS mission is provided by the NASA Explorer Program. The authors sincerely acknowledge them for providing such excellent data. This work is supported by the B-type Strategic Priority Program No. XDB41000000 funded by the Chinese Academy of Sciences. The authors acknowledge the support of the National Natural Science Foundation of China (grant No. 11803082 to X-H.C., grant Nos. 11833002 and 11973053 to X-B.Z., grant Nos. 11333006 and 11521303 to Y.L, grant No. 11803050 to C-Q.L., and grant No. 11833006 to J.S.). We acknowledge the science research grants from the China Manned Space Project with grant Nos. CMS-CSST-2021-B06 and CMS-CSST-2021-B07. X-H.C. also gratefully acknowledges the support of the Yunnan Fundamental Research Projects and the West Light Foundation of The Chinese Academy of Sciences. X-B.Z. acknowledges the support of the Sichuan Science and Technology Program (2020YFSY034). The authors appreciate the computing time granted by the Yunnan Observatories, and provided by the facilities at the Yunnan Observatories Supercomputing Platform. They also gratefully acknowledge the "PHOENIX Supercomputing Platform" jointly operated by the Binary Population Synthesis Group and The Stellar Astrophysics Group at Yunnan Observatories, Chinese Academy of Sciences.

A.1. The Inlsit File for Single-star Evolution

A.2. The Inlsit File for Mass Accretion from M₁–M₂

A.3. The Inlsit File for the Evolution of the Accreted Models