Absolute Properties of the Eclipsing γ Dor Star V404 Lyrae

Double-lined eclipsing binary (EB) stars allow accurate and direct determination of their fundamental parameters, such as the mass and radius of each component at the percent level. Further, the binary components can be assumed to be coeval and to share the same evolutionary history. These constraints are very useful for asteroseismical studies of pulsating stars in binaries. Various types of pulsating stars are known as members of binary systems (see the review by Murphy 2018). We placed V404 Lyr (KIC 3228863, Gaia DR2 2051369705923204224) on our spectroscopic observation program for EBs in 2016. This paper is the sixth contribution in determining the physical properties of pulsating EBs by time-series spectroscopy and photometry (Hong et al. 2015, 2017, 2019; Koo et al. 2016; Lee et al. 2018).

V404 Lyr is one of many A–F spectral type stars observed by the Kepler satellite. Lee et al. (2014) reviewed the history of the binary star observations and presented the most recent comprehensive study from detailed analyses of the Kepler time-series data and all available eclipse timings. They reported that the EB system is a quadruple star exhibiting γ Dor pulsations and that the primary component is responsible for the high-order low-degree gravity (g) modes. However, their binary parameters were preliminary because no spectroscopic observations were available at the time. Here, we present the results from analyzing our high-resolution spectra and the Kepler light curve supporting our binary star model and pulsational characteristics for V404 Lyr, consisting of a γ Dor-type pulsator and its lobe-filling companion. The rest of the paper is organized as follows. Sections 2 and 3 describe the observations and spectral analysis, respectively. In Section 4, we present the binary modeling and pulsation frequencies. Finally, Section 5 summarizes and discusses our conclusions.

2.1. Kepler Photometry

2.2. Ground-based Spectroscopy

As in our previous papers (Hong et al. 2015; Lee et al. 2018), we chose isolated absorption lines of Fe i λ 4957.61 for the RV measurements of both components. The Fe i lines were fitted about 10 times using two Gaussian functions with the splot routine and deblending option in IRAF. The upper panel of Figure 1 shows the trailed spectra of V404 Lyr in the spectral region, and the lower panel presents the observed spectrum and its Gaussian fitting at an orbital phase of 0.74. The average RVs and the standard deviations (σ) of the primary and secondary stars are listed in Table 1 and displayed in Figure 2.

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The measured RVs of each star were applied to a sine curve of V(t) = γ + K sin[2π(t − T₀)/P_orb] to obtain the orbital parameters of V404 Lyr. Here, γ is the systemic velocity, K is the velocity semi-amplitude, t is the observation time of each RV, and T₀ is the reference epoch of the orbital ephemeris. The orbital period was held fixed to be P_orb = 0.73094326 days obtained with the quadratic plus two light-travel-time (LTT) ephemeris in the paper of Lee et al. (2014). We used the Levenberg–Marquart technique (Press et al. 1992) to evaluate the unknown parameters of γ, K, and T₀. The circular orbital solutions (e = 0) are given in Table 2, together with related quantities, and are plotted in Figures 1 and 2. The rms residuals from each fit were calculated to be 11.7 km s⁻¹ and 18.1 km s⁻¹, respectively, for the primary and secondary components.

To obtain the atmospheric parameters of our target star, we extracted the separate spectra for each component using the spectral disentangling code FDBinary (Ilijić et al. 2004).⁶ Unfortunately, the disentangling spectrum of the faint secondary has a low S/N, so it is difficult to measure its atmospheric parameters reliably. Generally, γ Dor pulsators are main-sequence or subgiant stars with spectral types of A7−F5 (Kaye et al. 1999), which correspond to temperatures ranging from 7800 to 6500 K (Pecaut & Mamajek 2013). For the effective temperature (T_eff,1) and projected rotational velocity (v₁ sin i) of the primary component, we selected four absorption lines (Ca i λ4226, Fe i λ4271, Fe i λ4383, and Mg ii λ4481) that are useful in the temperature classification of dwarf A−F stars according to the Digital Spectral Classification Atlas given by R. O. Gray. We followed a procedure almost identical to that applied by Hong et al. (2017) and Lee et al. (2018).

First of all, the surface gravity was set to be log g₁ = 4.2 based on the binary star modeling discussed in the next section. A microturbulent velocity of 2.0 km s⁻¹ and a solar metallicity of [Fe/H] = 0 were assumed. Second, we constructed about 38,000 synthetic spectra with ranges of 6000 ≤ T_eff,1 ≤ 8500 K (in steps of 10 K) and 50 ≤ v₁ sin i ≤ 200 km s⁻¹ (in steps of 1 km s⁻¹) from the ATLAS9 atmosphere models (Kurucz 1993; Castelli & Kurucz 2003). Third, the atmospheric parameters of the pulsating primary were found from a grid search minimizing the χ² values between the synthetic models and the disentangling spectrum. Figure 3 shows the resulting χ² diagrams for T_eff,1 and v₁ sin i. We obtained the best-fitting parameters of T_eff,1 = 7330 ± 150 K and v₁ sin i = 148 ± 18 km s⁻¹. The T_eff,1 value is about 770 K hotter than the effective temperature (6561 K) in the Kepler Input Catalogue (KIC; Kepler Mission Team 2009), which is a much larger difference than the average difference between the KIC and spectroscopic temperatures for dwarfs (Pinsonneault et al. 2012). We examined how changes in metallicity (±0.1 in [Fe/H]) can affect our derived quantities (Katz et al. 1998; Torres et al. 2017). The result indicates that they have a minimal effect on T_eff,1 and v₁ sin i. Figure 4 presents four selected regions from the disentangling spectrum of the primary star, together with the synthetic spectra of 7180, 7330, and 7480 K.

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The mass ratio (q) is one of the most important parameters in binary modeling, because many quantities are sensitive to the q value. Lee et al. (2014) analyzed the Kepler light curve of V404 Lyr with the photometric q-search procedure. Their result indicated that the eclipsing pair is in a semi-detached configuration with a mass ratio of q_ph = 0.382. However, the photometric q_ph is very far from the spectroscopic mass ratio of q_sp = 0.660 ± 0.015 presented in the previous section. This discrepancy may originate from partial eclipses and the possible existence of a third light source (Pribulla et al. 2003; Terrell & Wilson 2005). To derive a unique set of binary parameters, we simultaneously modeled our RVs and the Kepler light curve of V404 Lyr by considering third-body and starspot effects. The 2007 version of the Wilson–Devinney synthesis code (Wilson & Devinney 1971; Van Hamme & Wilson 2007; hereafter W-D) was used.

The binary modeling of V404 Lyr was conducted in a method analogous to that for the double-lined EBs OO Dra (Lee et al. 2018) and KIC 6206751 (Lee & Park 2018). The effective temperature of the pulsating primary star was fixed at 7330 ± 150 K as measured by our spectral analysis in Section 3. The logarithmic limb-darkening coefficients were calculated from the tables of Van Hamme (1993). The bolometric albedos (A) and the gravity-darkening exponents (g) were assumed to be A₁ = 1.0 and g₁ = 1.0 for the primary component, and A₂ = 0.5 and g₂ = 0.32 for its companion. Also, synchronous rotations for both components (F₁ = F₂ = 1.0) were considered, and a circular orbit was adopted after several trials. The differential correction program of the W-D code was run until the correction of each parameter became smaller than its standard deviation. The result from this synthesis is given as Model 1 in the second to third columns of Table 3. The synthetic RV curves are displayed as red solid curves in Figure 2, and the synthetic light curves are presented as a blue curve in the top panel of Figure 5, where the corresponding residuals are plotted in the middle panel. The parenthesized errors in Table 3 are the 1σ values of the parameters obtained from the Kepler data in each quarter.

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Although some previous studies (e.g., Lee et al. 2017; Lee & Park 2018) have shown that the light-curve parameters are immune to the light variations due to the pulsations, we cannot exclude this possibility for V404 Lyr. On the contrary, the pulsation frequencies could be affected by changes in the derived binary parameters. As the first step for frequency analyses, we split the Kepler data into intervals of 10 orbital periods. A total of 195 light curves were individually analyzed by simply adjusting both the ephemeris epoch (T₀) and the spot parameters except the colatitude in Model 1 of Table 3. The results are presented in Table 4, and the corresponding residuals are displayed as magnitude versus BJD in Figure 6. Then, the PERIOD04 program (Lenz & Breger 2005) was applied to the outside-eclipse light residuals from the whole data set. According to the successive prewhitening procedure for each frequency peak (Lee et al. 2014), we detected at least 45 frequencies with S/N amplitude ratios larger than 4.0 (Breger et al. 1993), which are listed in Table 5 and shown in Figure 7. The synthetic curve computed from the 45 frequency fit is displayed in the lower panel of Figure 6.

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We removed the pulsation signatures from the observed Kepler data. The pulsation-subtracted light curve was solved using the Model 1 parameters as the initial values. The final results are listed as Model 2 in the fourth to fifth columns of Table 3. The binary parameters from both data sets (Model 1 and Model 2) are a good match to each other within their errors. This indicates that the light-curve parameters of V404 Lyr are almost unaffected by multiperiodic oscillations. The pulsation-subtracted data and synthetic light curve are plotted as black circles and a red line, respectively, in the top panels of Figure 5. The light-curve residuals from Model 2 are displayed in the bottom panel of this figure.

Our binary modeling indicates that the eclipsing pair of V404 Lyr is a semi-detached binary with a Roche-lobe-filling secondary component, and it allowed us to compute the absolute dimensions for each star. The results are listed in Table 6, together with those of Lee et al. (2014) for comparison. We determined the masses and radii of the EB system with an accuracy of about 2.6% and 1.2%, respectively. The distance to the system was calculated to be 671 ± 41 pc, which is in satisfactory accord with the Gaia distance of 634 ± 14 pc taken from Gaia DR2⁷ (1.578 ± 0.033 mas; Gaia Collaboration et al. 2018). The fundamental parameters of V404 Lyr presented in this paper are quite different from those of Lee et al. (2014), which were obtained from photometric data alone. We prefer our present results, which include the double-lined RVs and atmospheric parameters from our high-resolution spectra.

In this article, we report the first spectroscopic observations of the eclipsing γ Dor star V404 Lyr. From these spectra, double-lined RVs were measured. The effective temperature and projected rotational velocity of the pulsating primary were determined to be T_eff,1 = 7330 ± 150 K and v₁ sin i = 148 ± 18 km s⁻¹, respectively. We analyzed the observed and pulsation-subtracted Kepler data, respectively, with our RV measurements. As seen in Table 3, the pulsation frequencies do not affect the binary parameters of the eclipsing system. The light and RV solutions indicate that V404 Lyr is a classical Algol system, with a mass ratio of q = 0.653, a semimajor axis of a = 5.23 R_⊙, an inclination angle of i = 785, a temperature difference of Δ(T₁–T₂) = 1712 K, and a third light of l₃ = 4.1%. The detached primary star fills its inner Roche lobe by about 91%. The masses, radii, and luminosities of each component are M₁ = 2.17 M_⊙, M₂ = 1.42 M_⊙, R₁ = 1.91 R_⊙, R₂ = 1.79 R_⊙, L₁ = 9.4 L_⊙, and L₂ = 2.9 L_⊙. In the Hertzsprung–Russell diagram (see Lee 2016), the primary component is located close to the blue edge of the γ Dor instability strip inside the main-sequence band, while the lobe-filling secondary lies above this band.

The synchronized rotations for the primary and secondary components were computed to be v_1,sync = 132.2 ± 1.6 km s⁻¹ and v_2,sync = 124.0 ± 1.5 km s⁻¹, respectively. The v_1,sync value suggests that the primary star with the rotational velocity of v₁ sin i = 148 ± 18 km s⁻¹ is super-synchronous, but its does not provide sufficient evidence because the difference between them is within margins of errors. We examined the sensitivity of the physical parameters of the binary system to the axial-to-mean orbital rotation ratio F₁. We find no notable differences between synchronous rotation F₁ = 1.0 or asynchronous F₁ = 1.1. However, the close separation of the components of V404 Lyr lead us to believe that the components are in synchronous rotation or nearly so.

The eclipse timing analyses of Lee et al. (2014) indicated that V404 Lyr is probably a quadruple star system, with LTT periods of P₃ = 649 days and P₄ = 2154 days and minimum masses of M₃ = 0.47 M_⊙ and M₄ = 0.047 M_⊙, where M₁ + M₂ = 1.87 M_⊙ was assumed. If we use the new value of M₁ + M₂ = 3.59 M_⊙, these masses become M₃ = 0.71 M_⊙ and M₄ = 0.073 M_⊙, respectively. In this paper, the improved physical parameters of the third body were obtained by assuming its orbit to be coplanar with that of the eclipsing pair. As a result, the tertiary component has a period of P_3b = 642 days, an eccentricity of e_3b = 0.21, and a semimajor axis of a_3b = 509 R_⊙, and its mass becomes M₃ = 0.71 M_⊙. The third-body mass corresponds to a spectral type of about K4.5V, and the bolometric luminosity is calculated to be L₃ = 0.21 L_☉ (Pecaut & Mamajek 2013), which contributes about 2% to the total light of this system. The third light of l₃ ≈ 4% detected in this work may be attributed to the K-type circumbinary object gravitationally bound to V404 Lyr and/or the contamination due to the nearby star. It is nearly impossible to find such a companion from our spectroscopic observations due to its faintness and low S/N. Instead, the predicted semi-amplitude of the systemic RV change for the eclipsing pair by the circumbinary object is about 6 km s⁻¹, which may be detected with high-resolution spectroscopy extending over the course of a few years. On the other hand, the (V − K_s) color index for such K-type stars is about +2.8 mag (Pecaut & Mamajek 2013), so the third body can be as bright as K_s ∼ 12 mag. However, because the maximum angular separation between the eclipsing pair and its companion is less than 4 mas, the circumbinary object cannot be revealed easily using interferometry.

The PERIOD04 periodogram for the eclipse-subtracted light residuals revealed 45 oscillation signals with S/N > 4.0, which is five more than those detected by Lee et al. (2014). The main frequencies of f₁ − f₆ and f₉ found in this paper are exactly the same as the γ Dor pulsations identified in the previous paper. In contrast, most of the other frequencies may be orbital harmonics or combination frequencies. γ Dor stars pulsate in high-order g modes with multiple periods between 0.4 and 3 days, and they have the pulsation constants of Q > 0.23 days (Henry et al. 2005). To obtain the Q values for f₁ − f₆ and f₉, we used the relation between the pulsation period (P_pul) and the mean density (ρ) defined as Q = ${P}_{\mathrm{pul}}\sqrt{\rho /{\rho }_{\odot }}$. From this equation and the ρ₁ value given in Table 6, we obtained 0.268 < Q < 0.307 for the period range of P_pul = 0.473–0.539 day. The results confirm that the primary component is a γ Dor-type pulsating star. The physical properties of V404 Lyr match well the possible correlations (P_pul − P_orb, P_pul − Q, and P_pul − R/a) for eclipsing γ Dor stars presented by Ibanoglu et al. (2018). Because the primary components in semi-detached EBs have evolved with the tidal force and mass accretion from their companions, the pulsating characteristics of our target star may be affected by the binary effects, unlike those of single pulsators. As in case of KIC 620751 (Lee & Park 2018), the ratios of the orbital-to-pulsational frequencies for V404 Lyr are f_orb:f_1–6,9 ≃ 2:3. These indicate that the γ Dor pulsations may be excited by the tidal interaction between the component stars. The pulsating EBs with reliable absolute properties can be applied to study the influence of tides on stellar structure.

The authors wish to thank the staff of BOAO for assistance during our observations. This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). We appreciate the careful reading and valuable comments of the anonymous referee. This research has made use of the Simbad database maintained at CDS, Strasbourg, France, and was supported by the KASI grant 2019-1-830-03. K.H. was supported by the grant nos. 2017R1A4A1015178 and 2019R1I1A1A01056776 of the National Research Foundation (NRF) of Korea, respectively.