Asteroseismic Analysis of δ Scuti Components of Binary Systems: The Case of KIC 8504570

Abstract

The present work concerns the Asteroseismology of the $Kepler$-detached eclipsing binary KIC 8504570. Particularly, it focuses on the pulsational behaviour of the oscillating component of this system and the estimation of its physical parameters in order to enrich the so far poor sample of systems of this kind. Using spectroscopic observations, the spectral type of the primary component was determined and used to create accurate light curve models and estimate its absolute parameters. The light curve residuals were subsequently analysed using Fourier transformation techniques to obtain the pulsation models. Theoretical models of $\delta$ Scuti stars were employed to identify the oscillation modes of the six detected independent frequencies of the pulsator. In addition, more than 385 combination frequencies were also detected. The absolute and the pulsational properties of the $\delta$ Scuti star of this system are discussed and compared with all the currently known similar cases. Moreover, using a recent(empirical) luminosity–pulsation period relationship for $\delta$ Scuti stars, the distance of the system was estimated.

1. Introduction

The $\delta$ Scuti stars are short-period and multiperiodic pulsating variables. In general, they oscillate in radial and low-order non-radial pulsations due to $\kappa$-mechanism [1,2]. However, recently, it has been proposed that the turbulent pressure in the hydrogen convective zone may explain the observed high-order non-radial modes [3,4]. Their masses typically range between 1.4 and 2.5 $M_{\odot }$ [1], their spectral types between AIII/V and FIII/V, and they are located inside the classical instability strip. Thanks to the $Kepler$ [5,6], the $K2$ [7], the GAIA [8] and the Transiting Exoplanet Survey Satellite (TESS) [9,10] missions, as well as the All-Sky Automated Survey for Supernovae (ASAS-SN) [11] project, many studies, e.g., [12,13,14,15,16], based on large data sets of $\delta$ Scuti stars, have been published, providing new, tremendous knowledge for pulsators of this kind.

The eclipsing binaries (EBs) can be considered as the utmost tools for the calculation of stellar absolute parameters (e.g., masses, radii, luminosities) and the evolutionary stages of their components, particularly in cases when spectroscopy and photometry are combined. However, it should be noted that for systems with large luminosity differences between their components, the radial velocity measurements of the less luminous component are, in general, very difficult to get, because the light of the more luminous component dominates the spectrum.

In addition, given that the phase parts along the quadratures are the most important for the calculation of the amplitudes of the radial velocity curves, the systems with orbital periods longer than 1–2 days need observations in different and, depending on the orbital period, possibly distant dates. Moreover, particularly for the $Kepler$ systems (with a luminosity range of approximately 10–15 mag) at least 2–4 m size telescopes have to be employed. Hence, according to these limitations, telescope time is not easy to be allocated. Therefore, for the aforementioned reasons, in the absence of radial velocity curves, the least-squares minimisation technique has to be applied to the photometric data in order to estimate the parameters of the components. Moreover, another powerful tool of the EBs is the “eclipse timing variations” (ETV) method, which allows one to detect mechanisms (e.g., mass transfer, tertiary component, etc.; c.f. Budding and Demircan [17]-ch.8 and Borkovits et al. [18] and references therein) that modulate the orbital period.

Specifically, the subject of $\delta$ Scuti stars in EBs is extremely interesting because it combines two totally different topics of astrophysics and provides the means for remarkable results. On one hand, these systems host an oscillating component, whose pulsational properties are directly measurable by analysing photometric/spectroscopic data. On the other hand, the geometric phenomena of eclipses that occur during the orbital cycles, can be used to determine the absolute properties of the components of these systems. Therefore, the study of systems of this kind, especially the detached ones with wide orbits, allows for the direct determination of the physical properties of pulsating stars. The latter can be further used to correlate their pulsation properties with their evolutionary stages, and in the future, to constrain further the current evolutionary models of the pulsating stars. Furthermore, the close detached eclipsing systems (i.e., with orbital periods in the order of a few tens of days) and the semi-detached binaries have opened a new window for examining the influences of the binarity and the mass transfer on the pulsations.

Approximately 20 years ago, Mkrtichian et al. [19] suggested the term “$oEAstars$” (oscillating eclipsing binaries of Algol type) for categorising the EBs with a $\delta$ Scuti mass accretor component of (B)A-F spectral type. A few years later, the first connection between orbital ($P_{\mathrm{orb}}$) and dominant pulsation ($P_{\mathrm{pul}}$) periods for these systems was published by Soydugan et al. [20]. Liakos et al. [21] performed a long-scale observational survey on more than 100 candidate systems, which resulted in the publication of a catalogue with 74 cases and updated correlations between fundamental parameters for these systems. The first try for a theoretical justification for the $P_{\mathrm{pul}}-P_{\mathrm{orb}}$ correlation was made by Zhang et al. [22], who derived a similar empirical relation with the previous studies, based on different assumptions (i.e., the coefficient of $log{P}_{\mathrm{orb}}$ is 1), and found that its slope may be a function of the pulsation constant, the filling factor of the oscillating component and the mass ratio of the binary system. Liakos and Niarchos [23,24] announced the existence of a possible boundary in the $P_{\mathrm{orb}}$ (∼13 d); beyond that $P_{\mathrm{pul}}$ and $P_{\mathrm{orb}}$ can be considered uncorrelated. Kahraman Aliçavuş et al. [25], based only on eclipsing systems, suggested an almost doubled value for this boundary. Liakos and Niarchos [26] published the most coherent catalogue for these systems to date (available online1), providing updated correlations between the fundamental parameters of these systems, distinguished according to geometrical status and the reliability of their absolute parameters. An extended review for binaries with pulsating components was published by Murphy [27]. Murphy et al. [28], based on a sample of 2224 $Kepler$$\delta$ Scuti light curves, employed the pulsations timing technique [29,30,31] and identified 341 new binaries with long $P_{\mathrm{orb}}$ ($>100$ d) that each host a $\delta$ Scuti component. Liakos [32], using recently discovered systems, presented updated correlations between $P_{\mathrm{pul}}-P_{\mathrm{orb}}$ and $P_{\mathrm{pul}}-logg$ for close (i.e., $P_{\mathrm{orb}}<13$ d) detached eclipsing binaries with $\delta$ Scuti components.

The data quality of both $Kepler$ and $K2$ missions provided new insights for asteroseismology. Due to their unprecedented accuracy (i.e., order of a tens of mmag), they allow the detections of low amplitude frequencies in the order of a few $\mathsf{\mu }$mag [33]. Moreover, the continuous data acquisition from these missions for relatively long periods of time practically extinguishes the alias effect [34] in frequency detections. The time resolution of the short-cadence data especially (∼1 min) has been proven as extremely useful for the studies of short-period pulsating stars, such as the $\delta$ Scuti stars. Furthermore, the data of these missions have been widely used for the study of EBs. Specifically for the latter systems, an excellent online catalogue, namely, “$Kepler$ Eclipsing Binary Catalog” ($KEBC$) [35], that is publicly available2, has been created and includes the detrended data and other useful information for a few thousands of EBs.

The present work is a series paper on individual EBs with $\delta$ Scuti components; see also [24,32,36,37,38,39,40]. The system KIC 8504570 was selected for the present work because its pulsational behaviour is currently unknown to the community and its detailed analysis contributes to the sample of $Kepler$-detached binaries with a $\delta$ Scuti member (27 systems in total published to date; see Liakos and Niarchos [26] and updated lists in Liakos [32]).

KIC 8504570 (2MASS J19405685+4430276) was discovered by the $Kepler$ mission [41] and has an orbital period of ∼4 d. The only existing references concern mostly its temperature determination at 6874–7390 K [41,42,43,44,45,46,47,48], but Davenport [49] included it in the list of $Kepler$ systems that present flare activity.

Details about the ground-based spectroscopic observations and the estimation of the spectral type of the primary component of the system are given in Section 2. The $Kepler$ light curve (LC) analyses, the modelling results and the absolute parameter calculations are presented in Section 3. Section 4 includes the frequency search of the LC residuals, the pulsation models and the oscillation modes’ identification. Finally, Section 5 contains the summary of this work, a comparison in terms of evolution and properties of this system with other similar cases, discussion, conclusions and future prospects.

2. Spectroscopy

The purpose of the spectroscopic observations was the estimation of the spectral type of the primary component of the system. The spectra of the target were obtained with the 2.3 m Ritchey-Cretien “Aristarchos” telescope at Helmos Observatory in Greece on 6 October 2016. The Aristarchos Transient Spectrometer3 (ATS) instrument [50] using the low resolution grating (600 lines mm${}^{-1}$) was employed for the observations. This set-up provided a resolution of ∼3.2 Å pixel${}^{-1}$ and a spectral coverage between approximately 4000 and 7260 Å. Three successive spectra with 10 min exposures were acquired for KIC 8504570 during the orbital phase 0.81 and added together in order to achieve a better signal-to-noise ratio (S/N). The mean S/N of the individual spectra was ∼13, while that of the final integrated spectrum was ∼18. For the spectral classification, a spectral line correlation technique for the spectra of the variable and standard stars was applied. The selected standard stars, suggested by the Gemini Observatory4, ranged between A0 and K8 spectral classes (one standard star per subclass) and were observed with the same set-up during August–October 2016. All spectra were calibrated (bias, dark, flat-field corrections) using the MaxIm DL software. The data reduction (wavelength calibration, cosmic rays removal, spectra normalisation, sky background removal) was done with the RaVeRe v.2.2c software [51].

The applied correlation method has been described in detail in Liakos [39], but is briefly presented here too. For the comparison between the spectrum of KIC 8504570 and the standard stars, the Balmer and the strong metallic lines between 4000 and 6800 Å were used. The differences of spectral line depths between each standard star and the target star were compared via sums of squared residuals in each case, with the least squares sums indicating the best fit. This method is quite efficient in cases of EBs with large luminosity differences between their components, because the total spectrum is practically dominated by the light of the primary star. In our study we did not use any synthetic spectrum approximation (c.f. [52]) in order to avoid any instrumental effects (e.g., distortion) that cannot be taken into account in a synthetic model. Therefore, using the direct comparison method, and given that all spectra were acquired with the same set-up, any systematic effects were directly removed.

On the other hand, in cases with small luminosity differences between the components, this method does not provide accurate results, and more specifically, it might lead to an underestimation of the spectral type of the primary. Therefore, in order to avoid this, the following method described in [32] was applied. Using the spectra of the standard stars, all the possible combinations were calculated by simply adding and normalising the spectra. Furthermore, for every spectra combination, the spectrum of each component was given a weight between 0 and 1 denoting its light contribution to the combined spectrum. The starting value for the contribution of the primary component was 0.5 and the step was 0.05. Finally, for each spectral combination, ten sub-combinations with different light contributions of the components were derived. Similarly to the previous method, the same spectral lines were used for the comparison of the combined spectra with that of KIC 8504570, again via deriving sums of squared residuals. Hence, the smallest value of these residuals indicated again the best match.

The spectrum of KIC 8504570 was found to be dominated (at least 95%) by the light of the primary component. Therefore, its spectrum was directly compared with those of the standards. The sum of squared residuals against the spectral type for this system is plotted in Figure 1, which shows that the best fit was found with the spectrum of an A9V standard star. The spectrum of the system along with that of best-match standard star are illustrated in Figure 2. It should be noted that for the spectra continuum normalisation, polynomials of various orders were used according to the spectral types of the stars, since each spectral type has a different peak wavelength. However, for the continuum normalisation of the spectra of the standard stars with spectral types close to those of the targets (e.g., between A5–F5), the same polynomials were used. Therefore, since our method is based on direct comparison (i.e., subtraction of spectra), the non-perfect continuum normalisation does not affect the results. The present spectral classification, with an error assumption of one sub-class, corresponds to a temperature $T_{\mathrm{eff}}=7450\pm 150$ K for the primary, based on the relations between $T_{\mathrm{eff}}$ and spectral types of Cox [53]. The present result comes in relatively good agreement with those given in previous studies (see Section 1).

3. Light Curve Modelling and Absolute Parameter Calculations

The system was observed in long- and short-cadence modes by the $Kepler$ mission during various quarters. However, since the primary goal of this study concerns the asteroseismic analysis of the pulsating star of KIC 8504570 (i.e., pulsation modelling and mode identification), only the short-cadence data downloaded from the $KEBC$ [35] were used for the frequency analysis. However, it should be noted that the data obtained for this system during non successive quarters of the $Kepler$ mission provide significant time gaps, something that is crucial for the frequency analysis alias effect [34]. Furthermore, time gaps exist also within the data of a single quarter. Therefore, the selection of data for this system was made according to their continuity and total amount in time in order to include the most compact data sample possible. More specifically, the data of Q13 and a part of Q14 were selected for analysis. In total, 150,458 available points were used. These data were obtained during 106.9 consecutive days and provide 27 full LCs. The level of light contamination for this system is zero (as listed in the Mikulski Archive for Space Telescopes; MAST). The total covering and continuous time of observations is more than three months (with negligible time gaps), which is sufficient for the study of short-period pulsations and for LC modelling. The short-cadence $Kepler$ LCs of the first 40 days of observations for KIC 8504570 are illustrated in Figure 3. The orbital phases and the flux to magnitude conversions for this system were derived using the ephemeris ($T_0=$ 2,454,955.78(3) BJD, $P_{\mathrm{orb}}=4.007705\left(8\right)$ d) and the $Kepler$ magnitude $K_{\mathrm{p}}=13.25$ mag, respectively, as listed in $KEBC$.

The LC analyses were done with the PHOEBE v.0.29d software [54] that is based on the 2003 version of the Wilson–Devinney code [55,56,57]. The temperature ($T_{\mathrm{eff},1}$) of the primary component was given a value as yielded from the spectral classification (see Section 2), and it was kept fixed during the analysis. On the other hand, the temperature of the secondary component ($T_{\mathrm{eff},2}$) was adjusted. The albedos (A) and the gravity darkening coefficients (g) were assigned values according to the spectral types of the components [58,59,60]. The (linear) limb darkening coefficients (x) were taken from the lists of van Hamme [61]. The synchronicity parameters (F) were initially adjusted, but due to the absence of significant changes during the iterations, the system was assumed to be tidally locked (i.e., $F_1=F_2=1$) following the preliminary findings of Lurie et al. [62]. The dimensionless potentials ($\Omega$), the fractional luminosity of the primary component ($L_1$) and the inclination of the system (i) were set as adjustable parameters. Since there is no supporting evidence for the existence of a tertiary component, and additionally, since the light contamination was zero, the third light parameter ($l_3$) was not taken into account. At this point, it should be noted that the R filter (Bessell photometric system—range between 550 and 870 nm and with a transmittance peak at 597 nm) simulated the best spectral response of the CCD sensors of $Kepler$ (410–910 nm with a peak at ∼588 nm). Therefore, it was used for the calculation of the filter depended parameters (i.e., x and L) in PHOEBE.

In the absence of spectroscopic mass ratio (q) for KIC 8504570, the q-search method (for details, see e.g., [63]) was applied. For this, a mean LC exempted from the presence of pulsations was needed. Moreover, in this system, except the short-period pulsations, brightness variations due to magnetic activity (e.g., spots), occurring mostly in the out-of-eclipse phase parts, were also found. Therefore, the mean LC (folded into the orbital period) was calculated from two to four successive LCs; there were no major brightness changes between them. It should be noted that a complete LC of KIC 8504570 contains approximately 5500 data points. The mean LC, using averaged points per phase, contained approximately 300 normal points, and the variations of both the pulsations and the spots almost vanished. The q-search was applied in modes 2 (detached system), 4 (semi-detached system with the primary component filling its Roche lobe) and 5 (conventional semi-detached binary) to find feasible (“photometric”) estimates of the mass ratio. The step of q change during the search was 0.1 starting from q = 0.1. The sums of the squared residuals were systematically lower for all q values in mode 2; therefore, this system can be plausibly considered as a detached EB.

According to the q-search method, the minimum sum of squared residuals was found for q = 0.5 (Figure 4). This value was initially assigned to q, but later on it was adjusted. This system presents remarkable brightness changes from cycle to cycle after the 10 day of observations. It was found that for 40 continuous days after the 10 day, a hot spot on the surface of the secondary component describes the individual LCs very well. The selection of the hot spot was based on the results of Davenport [49] regarding possible flare activity in the system and fits well to a profile of a star with temperature of 5300 K (secondary component). Between 52 and 75 days of observations, no spots were required for the LC model, in contrast with the time range between 76 and 104 days, for which a cool spot was adopted on the surface of the same component. The spot parameters (colatitude $Colat$, longitude $long$, radius and temperature factor $Tf$) were adjusted in the individual LC models. Finally, for this system, one model per LC was obtained; thus, 27 models were totally derived and combined for the final average model.

The analyses of $Kepler$ LCs for EBs require special handling due to light variations caused by magnetic spots between successive LCs; c.f. [32,39,40]. That justifies our choice not to model all the available points folded into the $P_{\mathrm{orb}}$, but to model each LC separately. This method provides more realistic errors for the final model results, since its single parameter (except from those of the spots) is the average from those of the individual models, while its error is the standard deviation of them. Moreover, using this method, the brightness changes due to the spots and other proximity effects are well modelled; hence, the final LC residuals can be considered as free as possible of the binarity, something that is extremely crucial for the subsequent pulsation analysis (Section 4).

The LCs’ modelling results for KIC 8504570 are listed in

Table 1. LC modelling and absolute parameters for KIC 8504570. The errors are given in parentheses alongside values and correspond to the last digit(s).

	Components Parameters			Absolute Parameters
	Primary	Secondary		Primary	Secondary
$T_{\mathrm{eff}}$ (K)	7450(150) ${}^1$	5300(113)	$M$($M_{\odot }$)	1.67(17) ${}^{\mathrm{a}}$	0.87(9)
$\mathrm{\Omega }$	7.76(4)	9.57(4)	$R$($R_{\odot }$)	1.97(9)	0.89(14)
A	1 ${}^{\mathrm{a}}$	0.5 ${}^{\mathrm{a}}$	$L$($L_{\odot }$)	11(1)	0.6(2)
g	1 ${}^{\mathrm{a}}$	0.32 ${}^{\mathrm{a}}$	$logg$ (cm s${}^{-2}$)	4.07(5)	4.48(15)
x	0.455	0.583	a ($R_{\odot }$)	5.07(8)	9.75(4)
L/$L_{\mathrm{T}}$	0.945(1)	0.055(2)	$M_{\mathrm{bol}}$ (mag)	2.17(6)	5.4(2)
$r_{\mathrm{pole}}$	0.133(1)	0.060(1)		System Parameters
$r_{\mathrm{point}}$	0.134(1)	0.060(1)	q ($m_2$/$m_1$)	0.52(1)
$r_{\mathrm{side}}$	0.133(1)	0.060(1)	i (deg)	84.6(1)
$r_{\mathrm{back}}$	0.133(1)	0.060(1)

${}^1$Section 2, ${}^{\mathrm{a}}$ assumed, $L_T=L_1+L_2$.

. Examples of LC modelling and Roche geometry representation are plotted in Figure 5. The LC residuals after the subtraction of the individual models are illustrated below the observed LCs in Figure 3. Moreover, the parameters of the spots for each LC (cycle) are given in

Table A1. Spot parameters for KIC 8504570.

Time	Colat.	long.	Radius	Tf ($\frac{{\mathit{T}}_{\mathbf{spot}}}{{\mathit{T}}_{\mathbf{eff}}}$)	Time	Colat.	long.	Radius	Tf ($\frac{{\mathit{T}}_{\mathbf{spot}}}{{\mathit{T}}_{\mathbf{eff}}}$)
(BJD 2456016.0+)	(${}^{\circ }$)	(${}^{\circ }$)	(${}^{\circ }$)		(BJD 2456016.0+)	(${}^{\circ }$)	(${}^{\circ }$)	(${}^{\circ }$)
12.04	90(6)	140(6)	10(2)	1.18(5)	52.12	74(6)	78(6)	11(2)	1.09(4)
16.05	90(5)	147(5)	10(1)	1.25(2)	76.16	83(2)	20(4)	14(1)	0.81(6)
20.05	90(6)	135(6)	10(2)	1.25(4)	80.17	83(2)	16(1)	14(1)	0.72(12)
24.06	73(5)	131(5)	12(1)	1.24(2)	84.18	83(1)	13(2)	14(1)	0.69(16)
28.07	73(5)	126(5)	12(1)	1.24(3)	88.19	73(2)	13(2)	14(1)	0.65(11)
32.69	75(3)	122(2)	12(1)	1.27(4)	92.19	73(1)	13(2)	14(1)	0.64(11)
36.08	77(1)	107(3)	12(1)	1.25(4)	96.20	73(4)	2(4)	14(4)	0.57(8)
40.09	75(2)	99(3)	12(1)	1.22(5)	100.21	69(3)	−12(2)	15(3)	0.77(5)
44.10	73(9)	87(5)	12(1)	1.17(4)	104.22	68(5)	−13(5)	14(5)	0.76(11)
48.11	73(8)	75(7)	13(1)	1.17(2)

in Appendix A. Figure A1 includes the immigration plots of the spots and their locations on the surface of the secondary component for two different dates of observations.

Although no RV curves exist for this system, the absolute parameters of its components can be estimated making plausible assumptions. The adopted mass (1.67 $M_{\odot }$) of the primary was based on its spectral type according to the spectral type-mass correlations of Cox [53] for main-sequence stars. A fair mass error value of 10% was also adopted. The mass of the secondary component can be directly derived from the calculated (photometric) mass ratio. The semi-major axes a can then be derived from Kepler’s third law. The luminosities (L), gravity’s acceleration ($logg$) and the bolometric magnitude values ($M_{\mathrm{bol}}$) were calculated using the standard definitions. The calculations of the absolute parameters were done with the software AbsParEB [64], and they are listed in Table 1.

4. Pulsation Modelling

The search for pulsation frequencies was done with the software PERIOD04 v.1.2 [65] that is based on classical Fourier analysis. Although the typical frequency range of $\delta$ Scuti stars is 4–80 d${}^{-1}$ [34], the present analysis included the regime 0–4 d${}^{-1}$ too. This selection was based on the fact that it has been noticed (e.g., [32,39]) that these stars may also exhibit longer-period oscillations due either to tidal effects, which are connected to their $P_{\mathrm{orb}}$, or even to the intrinsic hybrid behaviour of $\gamma$-Doradus–$\delta$ Scuti type. Therefore, the present pulsation analysis was done in the range 0–80 d${}^{-1}$ on the LC residuals of the system (Figure 3). Moreover, since the eclipses affect the amplitudes of the pulsations (i.e., variations of the total light) and in order to keep the data sample homogeneous, only the out-of-eclipse data were used. The ranges of orbital phases (${\Phi }_{\mathrm{orb}}$) of the excluded data were 0.97–0.03 and 0.47–0.53. For the signal-to-noise ratio (S/N) calculation of the frequencies, the method for the background noise estimation, as described in detail in Liakos [39], was applied. Particularly, the background noise of the data set was calculated as 7.51 $\mathsf{\mu }$mag in regimes with absence of frequencies, with a spacing of 2 d${}^{-1}$, and a box size of 2. A 4$\mathsf{\sigma }$ limit (i.e.,S/N$=4$) [65] regarding the reliability of the detected frequencies was adopted (0.03 mmag). Hence, after the first frequency computation the residuals were subsequently pre-whitened for the next one until the detected frequency had S/N∼4. The Nyquist frequency and the frequency resolution according to the Rayleigh criterion (i.e., 1/T, where T is the observation time range in days; c.f. Aerts et al. [1], Schwarzenberg-Czerny [66]) for the present data set were 239.5 d${}^{-1}$ and 0.009 d${}^{-1}$, respectively. According to the present spectroscopic and LC modelling results (Section 2 and Section 3), only the primary component of KIC 8504570 adequately simulates the properties of $\delta$ Scuti-type stars (i.e., mass and temperature); hence, it can be plausibly concluded that this star is the pulsator of this system.

After the frequency search, the pulsation constant for each independent frequency (f) was calculated based on the relation of Breger [34]:

$$logQ=-logf+0.5logg+0.1M_{\mathrm{bol}}+log{T}_{\mathrm{eff}}-6.456.$$

(1)

Moreover, the following pulsation constant-density relation was used for the calculation of the density of the pulsators:

$$Q=f_{\mathrm{dom}}^{-1}\sqrt{{\rho }_{\mathrm{pul}}/{\rho }_{\odot }},$$

(2)

where $f_{\mathrm{dom}}$ is the frequency of the dominant pulsation mode (i.e., that with the largest amplitude). At this point it should be noted that the $f_{\mathrm{dom}}$ of the multiperiodic $\delta$ Scuti stars varies over time. Therefore, for a more realistic estimation of the density of this pulsator, the average value of Q of the independent frequencies was used.

The identification of the oscillating modes (i.e., l-degrees and type) employed the theoretical MAD models for $\delta$ Scuti stars [67] in the FAMIAS software v.1.01 [68]. The l-degrees from the closest MAD models (i.e., f, $logg$, M and $T_{\mathrm{eff}}$) to the detected independent frequencies were adopted as the most possible pulsation modes. Moreover, the ratio $P_{\mathrm{pul}}$/$P_{\mathrm{orb}}$ of all independent frequencies was calculated in order to check whether it is less than 0.07, which is the upper value, according to Zhang et al. [22], for the discrimination of p-type modes.

Table 2. Frequency search results and mode identification for the independent frequencies of the pulsating component of KIC 8504570. The errors are given in parentheses alongside values and correspond to the last digit(s).

i	${\mathit{f}}_{\mathbf{i}}$	A	$\mathbf{\Phi }$	S/N	Q	${\mathit{P}}_{\mathbf{pul}}$/${\mathit{P}}_{\mathbf{orb}}$${}^1$	l-Degree	Pulsation Mode
	(d${}^{-1}$)	(mmag)	(${}^{\circ }$)		(d)
1	14.37417(1)	2.550(5)	1.8(1)	339.3	0.032(2)	0.017	3 or 1	NR p
2	14.46668(1)	2.215(5)	59.2(1)	294.7	0.032(2)	0.017	1 or 3	NR p
3	26.19524(1)	1.779(5)	32.1(1)	236.7	0.018(1)	0.010	2	NR p
4	13.91847(2)	1.560(5)	236.5(2)	207.6	0.033(2)	0.018	2 or 3	NR p
8	11.89034(2)	1.070(5)	57.1(2)	142.4	0.039(2)	0.021	3	NR p
10	17.80685(4)	0.565(5)	264.1(5)	75.2	0.026(2)	0.014	1	NR p

${}^1$ Error values are of 10${}^{-7}$–10${}^{-8}$ order of magnitude. NR p non-radial pressure mode.

includes the pulsation modelling results regarding the independent frequencies for KIC 8504570 as well as their respective mode identification. Particularly, this table lists: The frequency value $f_{\mathrm{i}}$, the amplitude A, the phase $\Phi$, the S/N, the Q, the $P_{\mathrm{pul}}$/$P_{\mathrm{orb}}$, the l-degrees and the mode of each detected independent frequency. The rest of the detected frequencies (i.e., dependent/combination frequencies) are given in Appendix B (

Table A2. Combination frequencies of KIC 8504570.

i	${\mathit{f}}_{\mathbf{i}}$	A	$\mathbf{\Phi }$	S/N	Combination	i	${\mathit{f}}_{\mathbf{i}}$	A	$\mathbf{\Phi }$	S/N	Combination
	(d${}^{-1}$)	(mmag)	(${}^{\circ }$)				(d${}^{-1}$)	(mmag)	(${}^{\circ }$)
5	13.96772(2)	1.462(5)	25.8(2)	194.5	$f_1-2f_{\mathrm{orb}}$	44	15.81073(12)	0.195(5)	172.1(1.3)	26	$f_2$ + $f_{39}-f_4$
6	14.26173(2)	1.197(5)	127.1(2)	159.2	$f_1$ + $2f_4-2f_5$	45	0.46827(12)	0.193(5)	315.2(1.4)	25.7	$2f_{12}$
7	13.87519(2)	1.187(5)	304.5(2)	158	$f_1$ + $f_5-f_2$	46	14.30447(12)	0.191(5)	262.1(1.4)	25.4	$f_3-f_8$
9	14.15746(4)	0.620(5)	84.7(4)	82.6	$f_1$ + $f_6-f_2$	47	18.43246(13)	0.190(5)	331.8(1.4)	25.3	$f_{18}$ + $f_5-f_1$
11	14.52475(4)	0.556(5)	315.2(5)	74	$f_2$ + $f_5-f_4$	48	18.21829(13)	0.188(5)	159.1(1.4)	25	$f_{37}-f_{39}$
12	0.23719(4)	0.539(5)	127.8(5)	71.7	$f_9-f_4$	49	10.27324(13)	0.184(5)	131.9(1.4)	24.5	$f_{21}-f_2$
13	25.59738(5)	0.467(5)	196.9(6)	62.1	$f_3$ + $f_7-f_2$	50	25.41796(13)	0.180(5)	185.4(1.5)	23.9	$f_2$ + $f_{22}-f_6$
14	27.16046(5)	0.469(5)	134.7(6)	62.3	$3f_7-f_2$	51	13.41866(14)	0.176(5)	265.4(1.5)	23.4	$f_4-2f_{\mathrm{orb}}$
15	30.46276(5)	0.464(5)	153.0(6)	61.7	$f_{11}$ + $2f_4-f_8$	52	0.28509(14)	0.175(5)	333.8(1.5)	23.3	$f_6-f_5$
16	27.61182(5)	0.517(5)	4.5(5)	68.9	$f_1$ + $f_{14}-f_4$	53	32.03430(14)	0.173(5)	177.3(1.5)	23	$f_{21}$ + $f_{41}$
17	27.11303(5)	0.462(5)	323.8(6)	61.5	$f_{16}$ + $f_5-f_2$	54	1.67956(14)	0.169(5)	153.5(1.6)	22.5	$f_4$ + $f_5-f_3$
18	18.83497(5)	0.441(5)	315.3(6)	58.7	$f_{15}$ + $f_5-f_{13}$	55	36.13223(14)	0.166(5)	97.7(1.6)	22.1	∼$f_{31}$
19	0.49692(6)	0.388(5)	245.8(7)	51.6	$2f_{\mathrm{orb}}$	56	34.65322(14)	0.166(5)	239.4(1.6)	22.2	$f_{25}$ + $f_{28}$
20	21.93578(6)	0.411(5)	94.0(6)	54.7	$2f_3-f_{15}$	57	32.10851(14)	0.167(5)	152.4(1.6)	22.2	$f_{10}$ + $f_{46}$
21	24.73787(6)	0.409(5)	44.4(6)	54.4	$f_{17}$ + $f_8-f_6$	58	22.50879(14)	0.166(5)	266.8(1.6)	22.1	$f_{34}-f_{52}$
22	25.20896(6)	0.402(5)	29.2(7)	53.5	$f_{13}$ + $f_7-f_6$	59	36.09231(15)	0.163(5)	251.8(1.6)	21.7	$f_{20}$ + $f_9$
23	27.65926(6)	0.386(5)	351.2(7)	51.4	$f_{14}$ + $2f_{\mathrm{orb}}$	60	12.96918(15)	0.162(5)	42.6(1.6)	21.5	$f_{17}-f_9$
24	28.33090(6)	0.370(5)	248.9(7)	49.2	$f_1$ + $f_5$	61	27.46012(15)	0.162(5)	65.2(1.6)	21.5	$f_{40}$ + $f_{60}$
25	13.44778(7)	0.354(5)	305.5(7)	47.1	$f_{16}-f_9$	62	15.58199(15)	0.161(5)	230.0(1.6)	21.4	$f_{61}-f_8$
26	24.70969(7)	0.347(5)	0.6(8)	46.1	$f_{22}-2f_{\mathrm{orb}}$	63	29.89821(15)	0.160(5)	223.4(1.7)	21.2	$f_{46}$ + $f_{62}$
27	13.99402(7)	0.341(5)	100.8(8)	45.3	$f_1$ + $f_7-f_6$	64	13.69577(15)	0.157(5)	334.2(1.7)	20.9	$f_{16}-f_4$
28	21.19886(8)	0.306(5)	75.1(9)	40.7	$f_{18}$ + $f_6-f_8$	65	29.36747(15)	0.155(5)	212.2(1.7)	20.7	$f_{42}$ + $f_8$
29	24.98915(8)	0.284(5)	211.7(9)	37.8	$f_{22}$ + $f_9-f_1$	66	0.26866(16)	0.152(5)	22.9(1.7)	20.3	∼$f_{\mathrm{orb}}$
30	17.78666(9)	0.276(5)	127.0(1.0)	36.7	$f_{10}$ + $f_5-f_{27}$	67	36.62210(16)	0.152(5)	336.5(1.7)	20.3	$f_{21}$ + $f_8$
31	36.12330(10)	0.240(5)	139.7(1.1)	32	$f_3$ + $f_{29}-f_2$	68	25.23620(16)	0.148(5)	271.9(1.8)	19.7	$f_{21}$ + $2f_{\mathrm{orb}}$
32	36.06506(10)	0.232(5)	18.6(1.1)	30.8	$f_1$ + $f_{20}-f_{12}$	69	1.75471(17)	0.143(5)	195.6(1.8)	19	$7f_{\mathrm{orb}}$
33	7.06065(10)	0.231(5)	274.2(1.1)	30.7	$f_1$ + $f_7-f_{28}$	70	23.00711(17)	0.138(5)	131.6(1.9)	18.4	$f_{58}$ + $2f_{\mathrm{orb}}$
34	22.78777(10)	0.229(5)	349.1(1.1)	30.5	$f_1$ + $f_3-f_{30}$	71	14.60554(17)	0.137(5)	235.0(1.9)	18.2	$2f_{41}$
35	36.56292(10)	0.228(5)	75.9(1.2)	30.3	$2f_{\mathrm{orb}}$ + $f_{32}$	72	0.24564(18)	0.135(5)	61.3(1.9)	18	∼$f_{12}$
36	0.48424(11)	0.227(5)	125.2(1.2)	30.2	∼$2f_{\mathrm{orb}}$	73	14.41579(18)	0.135(5)	140.8(1.9)	18	$f_4$ + $2f_{\mathrm{orb}}$
37	33.50439(11)	0.223(5)	66.3(1.2)	29.7	$f_1$ + $f_3-f_{33}$	74	30.07058(18)	0.135(5)	341.8(2.0)	17.9	$f_{44}$ + $f_6$
38	28.75971(11)	0.219(5)	74.6(1.2)	29.1	$2f_1$	75	36.59298(18)	0.132(5)	243.6(2.0)	17.5	$f_{26}$ + $f_8$
39	15.27483(11)	0.212(5)	334.8(1.2)	28.2	$f_{14}-f_8$	76	36.14820(18)	0.133(5)	101.8(2.0)	17.7	$f_{67}-f_{36}$
40	14.49281(12)	0.206(5)	320.2(1.3)	27.4	$f_{12}$ + $f_6$	77	1.61146(19)	0.129(5)	94.4(2.0)	17.1	$f_{62}-f_5$
41	7.30582(12)	0.202(5)	98.9(1.3)	26.9	$f_1-f_{33}$	78	11.09517(18)	0.130(5)	327.0(2.0)	17.3	$f_{13}-f_{40}$
42	17.46822(12)	0.199(5)	106.4(1.3)	26.5	$f_{10}$ + $f_4-f_6$	79	29.86627(19)	0.129(5)	249.8(2.0)	17.1	$f_{65}$ + $2f_{\mathrm{orb}}$
43	0.25832(12)	0.199(5)	80.8(1.3)	26.4	$f_{\mathrm{orb}}$	80	29.39894(19)	0.128(5)	19.2(2.1)	17	$f_{63}-2f_{\mathrm{orb}}$
81	29.65398(19)	0.128(5)	5.7(2.1)	17	$f_1$ + $f_{39}$	118	1.88387(27)	0.088(5)	75.6(3.0)	11.7	$f_{44}-f_4$
82	0.50819(19)	0.127(5)	319.2(2.1)	16.9	∼$2f_{\mathrm{orb}}$	119	23.64493(27)	0.088(5)	357.2(3.0)	11.7	$f_{67}-f_{60}$
83	13.62955(19)	0.125(5)	139.7(2.1)	16.6	$f_7-f_{12}$	120	23.41808(27)	0.087(5)	79.5(3.0)	11.6	$f_{119}-f_{12}$
84	31.77175(20)	0.122(5)	241.8(2.2)	16.2	$f_{10}$ + $f_5$	121	0.59273(27)	0.087(5)	226.5(3.0)	11.5	$f_2-f_7$
85	26.41931(20)	0.121(5)	35.6(2.2)	16.1	$f_{12}$ + $f_3$	122	0.45887(27)	0.088(5)	318.0(3.0)	11.7	∼$f_{45}$
86	16.23344(20)	0.119(5)	170.1(2.2)	15.8	$f_2$ + $f_{69}$	123	0.21840(27)	0.088(5)	216.5(3.0)	11.7	$f_2-f_6$
87	0.82146(20)	0.117(5)	183.8(2.2)	15.6	$f_{39}-f_2$	124	27.10128(28)	0.086(5)	248.2(3.1)	11.5	∼$f_{17}$
88	13.94799(20)	0.117(5)	250.4(2.3)	15.6	$f_{24}-f_1$	125	0.03147(28)	0.086(5)	239.7(3.1)	11.4	$f_4-f_7$
89	27.26003(20)	0.117(5)	220.8(2.3)	15.6	$2f_{83}$	126	0.14043(28)	0.087(5)	63.0(3.0)	11.5	$f_{11}-f_1$
90	16.75243(21)	0.116(5)	61.5(2.3)	15.4	$f_{53}-f_{39}$	127	34.64194(28)	0.086(5)	264.6(3.1)	11.4	∼$f_{56}$
91	15.05783(21)	0.113(5)	108.8(2.3)	15.1	$f_{25}$ + $f_{77}$	128	0.94076(28)	0.084(5)	79.9(3.1)	11.2	$2f_{45}$
92	0.52933(21)	0.113(5)	141.6(2.3)	15.1	$2f_{\mathrm{orb}}$	129	36.05708(28)	0.084(5)	156.5(3.1)	11.2	∼$f_{32}$
93	0.34709(21)	0.112(5)	44.7(2.3)	15	$f_6-f_4$	130	10.15488(29)	0.083(5)	350.0(3.2)	11.1	$f_{16}-f_{42}$
94	18.74761(21)	0.111(5)	155.5(2.4)	14.8	$f_{35}-f_{10}$	131	18.94535(29)	0.083(5)	120.1(3.2)	11.1	$f_{33}$ + $f_8$
95	36.63478(22)	0.108(5)	271.3(2.4)	14.4	∼$f_{67}$	132	31.73511(29)	0.083(5)	308.6(3.2)	11	$f_{10}$ + $f_4$
96	12.27641(22)	0.109(5)	16.7(2.4)	14.4	$f_3-f_4$	133	0.73974(29)	0.082(5)	202.6(3.2)	11	$f_1-f_{83}$
97	22.96202(22)	0.106(5)	296.7(2.5)	14.1	$f_{26}-f_{69}$	134	16.30624(30)	0.080(5)	50.4(3.3)	10.7	$f_{15}-f_9$
98	22.46229(22)	0.109(5)	292.1(2.4)	14.5	$f_{67}-f_9$	135	16.02725(30)	0.080(5)	134.9(3.3)	10.7	$f_{63}-f_7$
99	21.69765(23)	0.105(5)	288.0(2.5)	13.9	$f_{32}-f_1$	136	31.16305(30)	0.080(5)	85.4(3.3)	10.6	$2f_{62}$
100	4.37269(23)	0.103(5)	131.4(2.5)	13.8	$f_{18}-f_2$	137	12.97670(30)	0.080(5)	177.2(3.3)	10.6	∼$f_{60}$
101	21.88975(24)	0.101(5)	143.3(2.6)	13.4	$f_{76}-f_6$	138	0.99431(30)	0.080(5)	127.4(3.3)	10.6	$4f_{\mathrm{orb}}$
102	13.65820(24)	0.100(5)	55.8(2.6)	13.3	$f_4-f_{\mathrm{orb}}$	139	27.62122(30)	0.079(5)	208.7(3.3)	10.6	∼$f_{16}$
103	36.64746(24)	0.099(5)	305.0(2.7)	13.2	∼$f_{95}$	140	10.03276(30)	0.079(5)	321.0(3.3)	10.5	$f_{20}-f_8$
104	27.34082(24)	0.099(5)	147.9(2.7)	13.1	$f_1$ + $f_{60}$	141	38.46793(31)	0.078(5)	45.6(3.4)	10.3	$f_3$ + $f_{96}$
105	26.63865(24)	0.099(5)	96.3(2.6)	13.2	$f_1$ + $f_{96}$	142	36.07352(31)	0.078(5)	157.9(3.4)	10.3	∼$f_{32}$
106	32.23532(25)	0.097(5)	127.1(2.7)	12.9	$f_{10}$ + $f_{73}$	143	21.94752(31)	0.078(5)	0.3(3.4)	10.3	∼$f_{20}$
107	36.11673(25)	0.097(5)	107.4(2.7)	12.9	∼$f_{31}$	144	26.66871(31)	0.077(5)	44.1(3.4)	10.3	$f_3$ + $f_{36}$
108	36.57325(25)	0.096(5)	208.1(2.7)	12.8	∼$f_{35}$	145	18.99466(31)	0.076(5)	169.6(3.5)	10.1	$f_{12}$ + $f_{94}$
109	32.22593(25)	0.095(5)	65.5(2.8)	12.6	∼$f_{106}$	146	13.85546(31)	0.076(5)	39.8(3.5)	10.1	$f_{24}-f_2$
110	12.35531(25)	0.095(5)	281.4(2.8)	12.6	$f_{45}$ + $f_8$	147	14.35473(31)	0.078(5)	76.3(3.4)	10.4	$f_{24}-f_5$
111	0.47672(25)	0.094(5)	33.8(2.8)	12.5	∼$f_{36}$	148	21.47033(32)	0.075(5)	150.6(3.5)	10	$f_{41}$ + $f_9$
112	11.69260(26)	0.092(5)	19.3(2.9)	12.2	$f_{13}-f_4$	149	26.14173(32)	0.076(5)	14.0(3.5)	10.1	$f_6$ + $f_8$
113	21.44966(26)	0.092(5)	127.2(2.9)	12.2	$f_{12}$ + $f_{28}$	150	25.03236(32)	0.075(5)	283.6(3.5)	9.9	$f_{21}$ + $f_{52}$
114	36.55494(26)	0.091(5)	211.4(2.9)	12.1	∼$f_{35}$	151	28.36237(32)	0.074(5)	211.9(3.5)	9.9	$f_1$ + $f_{27}$
115	22.08279(26)	0.090(5)	140.7(2.9)	12	$f_{35}-f_2$	152	21.67229(32)	0.074(5)	20.0(3.6)	9.9	$f_{41}$ + $f_1$
116	29.08191(27)	0.089(5)	203.6(2.9)	11.9	$f_{71}$ + $f_2$	153	10.95004(32)	0.074(5)	127.8(3.6)	9.8	$f_{22}-f_6$
117	30.36789(27)	0.089(5)	120.1(2.9)	11.9	$f_{79}$ + $2f_{\mathrm{orb}}$	154	31.12219(33)	0.072(5)	30.7(3.6)	9.6	$f_1$ + $f_{90}$
155	0.72753(33)	0.072(5)	11.2(3.6)	9.6	∼$f_{133}$	192	40.56879(40)	0.060(5)	122.9(4.4)	7.9	$f_3$ + $f_1$
156	17.96654(33)	0.072(5)	292.0(3.7)	9.6	$f_{79}-f_8$	193	23.34011(40)	0.059(5)	104.6(4.4)	7.9	$f_{163}-f_{11}$
157	31.72666(33)	0.072(5)	225.9(3.7)	9.5	∼$f_{132}$	194	24.53309(40)	0.059(5)	54.8(4.5)	7.8	$f_{49}$ + $f_6$
158	31.36642(33)	0.072(5)	21.3(3.7)	9.6	$f_2$ + $f_{154}$	195	0.74773(41)	0.059(5)	131.1(4.5)	7.8	∼$f_{133}$
159	26.18870(34)	0.071(5)	52.9(3.7)	9.4	∼$f_3$	196	1.98861(41)	0.058(5)	212.3(4.5)	7.7	$f_7-f_8$
160	41.07370(34)	0.070(5)	250.4(3.8)	9.3	$f_{14}$ + $f_4$	197	14.38056(41)	0.058(5)	183.0(4.5)	7.7	∼$f_1$
161	16.77450(34)	0.069(5)	135.3(3.8)	9.2	$f_{15}-f_{64}$	198	41.29397(41)	0.058(5)	78.0(4.6)	7.7	$f_{80}$ + $f_8$
162	33.51284(35)	0.069(5)	163.5(3.8)	9.2	∼$f_{37}$	199	0.06059(42)	0.057(5)	1.5(4.6)	7.6	$f_5-f_4$
163	37.86299(35)	0.069(5)	171.6(3.8)	9.1	$f_{13}$ + $f_{96}$	200	13.73006(42)	0.057(5)	265.5(4.7)	7.5	$f_5-f_{12}$
164	36.10311(35)	0.068(5)	194.3(3.9)	9.1	∼$f_{59}$	201	25.09952(42)	0.056(5)	142.7(4.7)	7.5	$f_{13}-2f_{\mathrm{orb}}$
165	37.85124(35)	0.068(5)	324.1(3.9)	9	∼$f_{163}$	202	27.83304(42)	0.057(5)	192.4(4.7)	7.5	$2f_4$
166	36.03547(36)	0.067(5)	275.4(3.9)	8.9	$f_1$ + $f_{152}$	203	0.21229(43)	0.056(5)	71.7(4.7)	7.4	∼$f_{123}$
167	36.53427(34)	0.070(5)	313.3(3.8)	9.3	$f_{71}$ + $f_{20}$	204	13.37545(44)	0.055(5)	311.6(4.8)	7.3	$f_7-2f_{\mathrm{orb}}$
168	27.15624(36)	0.067(5)	43.3(3.9)	8.9	∼$f_{14}$	205	27.87061(44)	0.055(5)	247.7(4.8)	7.3	$f_7$ + $f_{27}$
169	27.71186(36)	0.066(5)	241.6(4.0)	8.8	$2f_{146}$	206	36.62821(44)	0.055(5)	312.0(4.8)	7.3	∼$f_{67}$
170	27.60008(36)	0.066(5)	276.2(4.0)	8.8	∼$f_{16}$	207	40.66602(44)	0.054(5)	243.5(4.9)	7.2	$f_3$ + $f_2$
171	18.39489(36)	0.066(5)	195.7(4.0)	8.8	$f_{10}$ + $f_{121}$	208	32.08314(44)	0.054(5)	324.2(4.9)	7.2	$f_7$ + $f_{48}$
172	0.12212(36)	0.066(5)	320.6(4.0)	8.8	$f_1-f_6$	209	34.06565(43)	0.055(5)	166.9(4.8)	7.3	$f_{56}$ + $f_{121}$
173	26.16757(37)	0.065(5)	250.0(4.0)	8.7	$f_2$ + $f_{112}$	210	0.53590(44)	0.054(5)	260.0(4.9)	7.1	∼$f_{92}$
174	0.16955(37)	0.065(5)	161.0(4.1)	8.6	$f_2-f_{46}$	211	13.27118(45)	0.054(5)	98.1(4.9)	7.1	$f_{23}-f_1$
175	1.45412(37)	0.064(5)	238.5(4.1)	8.6	$f_3-f_{21}$	212	36.65545(45)	0.053(5)	295.0(4.9)	7.1	∼$f_{103}$
176	34.00412(37)	0.064(5)	68.4(4.1)	8.5	$2f_{\mathrm{orb}}$ + $f_{37}$	213	19.21306(45)	0.053(5)	121.4(4.9)	7.1	$f_{37}-f_{46}$
177	14.17625(38)	0.063(5)	215.8(4.2)	8.4	$f_2-f_{52}$	214	17.64669(45)	0.053(5)	359.8(5.0)	7.0	$f_{53}-f_1$
178	31.61018(38)	0.063(5)	119.0(4.2)	8.4	$2f_{44}$	215	24.26679(45)	0.053(5)	326.2(5.2)	7.0	$f_{76}-f_8$
179	31.57402(37)	0.064(5)	155.1(4.1)	8.6	$f_{53}-f_{45}$	216	0.90554(46)	0.052(5)	216.3(5.1)	6.9	$f_{17}-f_3$
180	0.20196(38)	0.063(5)	242.0(4.2)	8.4	$f_2-f_6$	217	18.97682(46)	0.052(5)	3.4(5.1)	6.9	$f_{37}-f_{11}$
181	0.29824(34)	0.070(5)	324.0(3.8)	9.3	∼$f_{52}$	218	25.70775(46)	0.052(5)	220.8(5.1)	6.9	$f_3-2f_{\mathrm{orb}}$
182	18.49634(38)	0.062(5)	290.9(4.2)	8.3	$f_{18}-f_{93}$	219	41.04270(46)	0.052(5)	92.9(5.1)	6.9	$f_{17}$ + $f_4$
183	25.20520(38)	0.062(5)	94.1(4.2)	8.3	∼$f_{22}$	220	0.08877(46)	0.052(5)	110.0(5.1)	6.9	$f_2-f_1$
184	28.10451(39)	0.062(5)	305.5(4.3)	8.2	$f_2$ + $f_{83}$	221	3.05619(46)	0.052(5)	32.8(5.1)	6.9	$f_{29}-f_{20}$
185	30.26831(39)	0.062(5)	199.9(4.3)	8.2	$f_2$ + $f_{44}$	222	14.02595(46)	0.051(5)	47.5(5.1)	6.8	$f_6-f_{12}$
186	16.08924(39)	0.062(5)	146.7(4.3)	8.2	$f_{15}-f_1$	223	10.47896(47)	0.051(5)	73.4(5.2)	6.8	$f_{21}-f_6$
187	3.65737(39)	0.061(5)	46.8(4.3)	8.2	$f_{10}-f_9$	224	30.62339(47)	0.051(5)	314.4(5.2)	6.8	$f_7$ + $f_{90}$
188	3.40985(38)	0.062(5)	305.7(4.3)	8.2	$f_{30}-f_1$	225	22.68914(47)	0.051(5)	177.7(5.2)	6.8	$f_{35}-f_7$
189	17.81296(39)	0.061(5)	136.4(4.3)	8.2	∼$f_{10}$	226	7.57260(47)	0.051(5)	39.1(5.2)	6.7	$f_{20}-f_1$
190	28.39102(40)	0.060(5)	214.7(4.4)	8	$f_2$ + $f_4$	227	38.64171(47)	0.050(5)	98.2(5.2)	6.7	$f_{26}$ + $f_{41}$
191	34.45924(40)	0.060(5)	2.7(4.4)	7.9	$f_{14}$ + $f_{41}$	228	38.14057(42)	0.057(5)	102.1(4.7)	7.5	$f_{215}$ + $f_7$
229	1.59878(48)	0.050(5)	8.5(5.3)	6.7	∼$f_{77}$	266	23.07991(50)	0.043(4)	337.3(5.5)	5.7
230	27.08297(48)	0.050(5)	107.0(5.3)	6.6	$f_{219}-f_5$	267	32.05825(50)	0.042(4)	235.4(5.6)	5.6
231	0.70827(48)	0.049(5)	220.6(5.3)	6.6	$f_1-f_{102}$	268	38.92399(50)	0.042(4)	157.0(5.6)	5.6
232	34.62597(49)	0.048(5)	111.9(5.4)	6.5	$f_{51}$ + $f_{28}$	269	13.99214(50)	0.042(4)	310.0(5.6)	5.6
233	0.38842(49)	0.048(5)	314.4(5.4)	6.4	$f_6-f_7$	270	31.07569(51)	0.042(4)	107.8(5.6)	5.6
234	15.59984(49)	0.048(5)	73.3(5.5)	6.4	$f_{74}-f_2$	271	31.39131(50)	0.042(4)	171.7(5.6)	5.6
235	40.92481(50)	0.048(5)	116.2(5.5)	6.4	$f_6$ + $f_{144}$	272	0.12822(51)	0.042(4)	86.1(5.6)	5.6
236	24.74257(50)	0.048(5)	157.2(5.5)	6.4	∼$f_{21}$	273	18.60718(51)	0.042(4)	318.8(5.6)	5.6
237	0.19398(50)	0.047(5)	117.9(5.6)	6.3	∼$f_{180}$	274	18.47614(50)	0.042(4)	353.0(5.6)	5.6
238	26.20326(50)	0.047(5)	297.0(5.6)	6.3	∼$f_3$	275	40.52840(51)	0.042(4)	229.9(5.6)	5.6
239	0.83696(51)	0.047(5)	323.3(5.6)	6.3	$f_2-f_{83}$	276	18.33665(51)	0.042(4)	287.7(5.7)	5.5
240	0.86374(49)	0.049(5)	336.6(5.4)	6.5	$f_{13}-f_{21}$	277	11.47702(51)	0.042(4)	147.2(5.7)	5.5
241	28.33935(51)	0.047(5)	186.2(5.6)	6.3	∼$f_{24}$	278	26.46018(51)	0.041(4)	1.4(5.7)	5.5
242	30.09078(51)	0.047(5)	216.2(5.6)	6.2	$f_{12}-f_{79}$	279	21.75167(52)	0.041(4)	3.4(5.7)	5.5
243	36.37364(51)	0.047(5)	196.1(5.6)	6.2	$f_{58}$ + $f_7$	280	14.12834(52)	0.041(4)	1.5(5.7)	5.5
244	27.16892(51)	0.047(5)	218.8(5.6)	6.2	∼$f_{14}$	281	27.27976(52)	0.041(4)	6.7(5.7)	5.5
245	22.59802(51)	0.047(5)	183.6(5.6)	6.2	$f_{35}-f_5$	282	52.44645(52)	0.041(4)	116.8(5.8)	5.4
246	41.05115(51)	0.046(5)	38.3(5.7)	6.2	∼$f_{129}$	283	18.66448(53)	0.040(4)	330.3(5.8)	5.4
247	0.01738(52)	0.046(5)	358.1(5.7)	6.1	$f_{27}-f_5$	284	32.63830(53)	0.040(4)	272.2(5.8)	5.4
248	18.44796(52)	0.046(5)	233.8(5.7)	6.1	$f_{48}$ + $f_{12}$	285	32.26773(53)	0.040(4)	144.4(5.8)	5.4
249	28.88230(53)	0.045(5)	324.7(5.8)	6	$f_2$ + $f_{73}$	286	32.13246(53)	0.040(4)	58.5(5.8)	5.4
250	11.91476(53)	0.045(5)	207.0(5.8)	6	$f_{125}$ + $f_8$	287	27.88330(53)	0.040(4)	75.1(5.9)	5.3
251	1.22304(53)	0.045(5)	218.1(5.9)	5.9	$f_{24}-f_{17}$	288	38.65768(53)	0.040(4)	300.7(5.9)	5.3
252	25.08167(54)	0.044(5)	194.0(5.9)	5.9	$f_{13}-f_{82}$	289	17.62462(53)	0.040(4)	300.3(5.9)	5.3
253	18.41978(54)	0.044(5)	313.2(6.0)	5.8	∼$f_{47}$	290	34.28546(54)	0.040(4)	70.5(5.9)	5.3
254	0.41942(55)	0.044(5)	215.3(6.0)	5.8	$f_1-f_5$	291	0.57676(54)	0.040(4)	267.5(5.9)	5.3
255	38.19834(55)	0.044(5)	200.8(6.1)	5.8	$f_{21}$ + $f_{25}$	292	3.91476(53)	0.040(4)	24.5(5.9)	5.3
256	37.87661(49)	0.044(4)	247.1(5.4)	5.8		293	0.18176(53)	0.040(4)	152.7(5.9)	5.3
257	31.67828(49)	0.043(4)	206.9(5.4)	5.8		294	40.69514(54)	0.040(4)	323.4(5.9)	5.3
258	40.59838(49)	0.043(4)	304.0(5.5)	5.7		295	29.56990(54)	0.039(4)	136.9(6.0)	5.2
259	21.17303(50)	0.043(4)	3.0(5.5)	5.7		296	30.29039(54)	0.039(4)	163.6(6.0)	5.3
260	0.78389(50)	0.043(4)	350.5(5.5)	5.7		297	27.21213(54)	0.039(4)	178.6(6.0)	5.2
261	2.67105(50)	0.043(4)	252.9(5.5)	5.7		298	0.98303(54)	0.039(4)	338.9(6.0)	5.2
262	5.59244(50)	0.043(4)	217.8(5.5)	5.7		299	36.43047(55)	0.039(4)	346.3(6.0)	5.2
263	14.36553(50)	0.043(4)	343.9(5.5)	5.7		300	36.04909(54)	0.039(4)	220.3(6.0)	5.2
264	26.13140(50)	0.042(4)	308.0(5.5)	5.6		301	0.49504(55)	0.039(4)	284.1(6.1)	5.1
265	22.87795(50)	0.042(4)	136.7(5.6)	5.6		302	22.02925(55)	0.039(4)	249.9(6.1)	5.1
303	22.84977(56)	0.038(4)	22.1(6.1)	5.1		340	2.33640(61)	0.035(4)	201.6(6.8)	4.6
304	38.82301(56)	0.038(4)	84.4(6.2)	5.1		341	5.79546(61)	0.035(4)	323.3(6.8)	4.6
305	36.08432(56)	0.038(4)	328.8(6.2)	5		342	8.99316(61)	0.035(4)	94.4(6.8)	4.6
306	17.28880(56)	0.038(4)	138.2(6.2)	5		343	40.57713(62)	0.035(4)	49.3(6.8)	4.6
307	32.83745(56)	0.038(4)	283.2(6.2)	5		344	23.28876(62)	0.034(4)	153.6(6.8)	4.6
308	0.44478(57)	0.038(4)	112.8(6.3)	5		345	32.32647(62)	0.034(4)	311.9(6.8)	4.6
309	0.40909(56)	0.038(4)	131.4(6.2)	5		346	38.62485(62)	0.034(4)	117.6(6.9)	4.5
310	20.29755(57)	0.038(4)	256.7(6.3)	5		347	43.33923(62)	0.034(4)	325.0(6.9)	4.5
311	25.51472(57)	0.037(4)	349.6(6.3)	5		348	26.17857(62)	0.034(4)	65.6(6.9)	4.5
312	34.44515(57)	0.037(4)	134.3(6.3)	5		349	24.85296(63)	0.034(4)	62.6(6.9)	4.5
313	31.94882(55)	0.038(4)	28.5(6.1)	5.1		350	1.30086(63)	0.034(4)	146.9(6.9)	4.5
314	34.08021(57)	0.038(4)	81.9(6.3)	5		351	12.47697(63)	0.034(4)	175.2(7.0)	4.5
315	22.79623(57)	0.037(4)	283.8(6.3)	5		352	22.24233(63)	0.034(4)	97.3(7.0)	4.5
316	13.47549(57)	0.037(4)	269.2(6.3)	4.9		353	13.96989(63)	0.034(4)	350.7(7.0)	4.5
317	15.35138(57)	0.037(4)	198.7(6.3)	4.9		354	0.11682(63)	0.034(4)	172.3(7.0)	4.5
318	26.50339(58)	0.037(4)	159.5(6.4)	4.9		355	38.84400(63)	0.034(4)	299.6(7.0)	4.5
319	14.42471(58)	0.037(4)	356.5(6.4)	4.9		356	36.01940(63)	0.034(4)	304.2(7.0)	4.5
320	18.24741(58)	0.037(4)	328.3(6.4)	4.9		357	0.37396(63)	0.034(4)	180.4(7.0)	4.5
321	17.79464(58)	0.037(4)	332.9(6.4)	4.9		358	41.03200(63)	0.034(4)	135.0(7.0)	4.5
322	40.93702(58)	0.037(4)	93.1(6.4)	4.9		359	26.21200(64)	0.033(4)	43.0(7.1)	4.4
323	33.49546(58)	0.037(4)	10.4(6.4)	4.9		360	28.15600(64)	0.033(4)	352.1(7.1)	4.4
324	13.92591(58)	0.037(4)	272.1(6.4)	4.9		361	32.02800(64)	0.033(4)	214.8(7.1)	4.4
325	13.95691(56)	0.038(4)	4.9(6.2)	5.1		362	23.12800(64)	0.033(4)	277.8(7.1)	4.4
326	12.96200(57)	0.038(4)	205.9(6.2)	5		363	23.19200(63)	0.034(4)	310.8(7.0)	4.5
327	3.99150(58)	0.036(4)	29.1(6.5)	4.8		364	1.20800(64)	0.033(4)	27.0(7.1)	4.4
328	25.99003(59)	0.036(4)	39.2(6.5)	4.8		365	0.80800(64)	0.033(4)	326.9(7.1)	4.4
329	32.17410(58)	0.036(4)	314.5(6.5)	4.8		366	15.02400(64)	0.033(4)	183.1(7.1)	4.4
330	0.65755(59)	0.036(4)	164.6(6.5)	4.8		367	23.75200(65)	0.033(4)	63.5(7.1)	4.4
331	37.96485(59)	0.036(4)	288.9(6.5)	4.8		368	30.63200(65)	0.033(4)	44.7(7.2)	4.4
332	40.99194(60)	0.036(4)	219.8(6.6)	4.8		369	22.22400(66)	0.032(4)	57.2(7.3)	4.3
333	16.60626(60)	0.035(4)	150.4(6.6)	4.7		370	0.16000(66)	0.032(4)	22.5(7.3)	4.3
334	18.21105(61)	0.035(4)	45.2(6.7)	4.7		371	25.36400(67)	0.032(4)	286.6(7.4)	4.2
335	36.66564(61)	0.035(4)	256.7(6.7)	4.6		372	32.10000(67)	0.032(4)	114.0(7.4)	4.2
336	38.35755(61)	0.035(4)	267.6(6.7)	4.6		373	0.07200(67)	0.032(4)	258.4(7.4)	4.2
337	32.37003(60)	0.036(4)	13.3(6.6)	4.7		374	30.66400(67)	0.032(4)	223.7(7.5)	4.2
338	36.54486(61)	0.035(4)	273.7(6.7)	4.6		375	30.15200(67)	0.032(4)	227.3(7.4)	4.2
339	38.19024(61)	0.035(4)	288.6(6.8)	4.6		376	36.58400(68)	0.031(4)	146.9(7.5)	4.2
377	21.98800(68)	0.031(4)	337.6(7.5)	4.2		386	27.12000(69)	0.031(4)	155.5(7.7)	4.1
378	0.60400(68)	0.031(4)	199.3(7.6)	4.1		387	26.86800(68)	0.031(4)	150.9(7.6)	4.1
379	24.72800(68)	0.031(4)	174.4(7.6)	4.1		388	26.76000(68)	0.031(4)	31.3(7.5)	4.2
380	11.54800(69)	0.031(4)	139.7(7.6)	4.1		389	40.55200(70)	0.030(4)	165.7(7.8)	4.0
381	12.04400(68)	0.031(4)	251.3(7.5)	4.2		390	33.04000(70)	0.030(4)	245.9(7.7)	4.0
382	31.75600(68)	0.031(4)	5.6(7.6)	4.1		391	26.03200(70)	0.030(4)	307.6(7.8)	4.0
383	22.86000(69)	0.031(4)	263.6(7.6)	4.1		392	30.45600(71)	0.030(4)	253.3(7.8)	4.0
384	24.38800(69)	0.031(4)	210.8(7.6)	4.1		393	27.76000(71)	0.030(4)	47.3(7.8)	4.0
385	38.91200(69)	0.031(4)	195.7(7.7)	4.1

). Figure 6 shows the periodogram of the pulsating star of KIC 8504570 and the distribution of its oscillation frequencies. Representative Fourier fittings on the LC residuals are plotted in Figure 7.

The pulsator of KIC 8504570 oscillates in a total of 393 frequencies. Six of them are independent and were detected in the regime 11.8–26.2 d${}^{-1}$. Among the other 387 depended frequencies, 309 were spread almost uniformly in the range 10–43.4 d${}^{-1}$; 72 had values less than 4.4 d${}^{-1}$; five were found between 5.5–9 d${}^{-1}$; and only one, namely, $f_{282}$, exceeded 50 d${}^{-1}$. As can be seen in Figure 6, one main concentration of frequencies is between 12 and 17 d${}^{-1}$, while a slightly more spread out one is between 23 and 30 d${}^{-1}$. The results based on MAD models show that all oscillations are probably non-radial pressure modes. Although the ratio $f_4$/$f_{10}$ has value ∼0.78, $f_4$ was not identified as a radial mode by the MAD models. Finally, a value of ${\rho }_{\mathrm{pul}}$ = 0.215(4) ${\rho }_{\odot }$ was derived.

5. Summary, Discussion and Conclusions

In the present work, detailed LC and pulsation modellings for KIC 8504570, a neglected $Kepler$-detached EB with an oscillating component, are presented. The spectral classification of its primary component, based on our spectroscopic observations, provided the means for accurate LC analyses, and for the estimation of the absolute parameters and evolutionary stages of both the components of the EB. The primary component was also identified as a $\delta$ Scuti star and its pulsational characteristics (pulsation frequencies model and mode identification) were accurately determined.

The primary component of KIC 8504570 was classified as an A9-type star and pulsates in six independent frequencies in the regime 11.89–26.2 d${}^{-1}$ with the dominant part at 14.37 d${}^{-1}$. These frequencies were identified as non-radial (pressure) modes according to the MAD models. Moreover, this star oscillates in another 387 combination frequencies. During the LC modelling, initially a hot and subsequently a cool spot on the surface of the secondary component were used to overcome the brightness asymmetries in the quadratures. This selection can be justified from the fact that this EB was listed as a possible flare system [49].

For the estimation of the evolutionary stages of the components of KIC 8504570, the locations of its members on the mass-radius ($M-R$) and Hertzsprung–Russell ($HR$) diagrams are illustrated in Figure 8 and Figure 9, respectively. Both components are located inside the main-sequence and follow the theoretical evolutionary tracks of Girardi et al. [69] (see Figure 9) very well according to their derived masses and the corresponding error ranges (see Table 1). Therefore, it seems that they have been evolving without any significant interactions so far. In terms of evolution, the $\delta$ Scuti component of KIC 8504570 has similar absolute properties to other $\delta$ Scuti stars in detached binary systems. It is among the eight less massive and less luminous stars of this sample and it is located closer to red edge of the classical instability strip.

In order to check the accordance of the pulsational properties of the $\delta$ Scuti star of KIC 8504570 with others that belong in similar systems, it was placed on the $P_{\mathrm{pul}}-P_{\mathrm{orb}}$ and $logg-P_{\mathrm{pul}}$ diagrams (Figure 10 and Figure 11, respectively) along with the well established empirical relations of Liakos [32] for $\delta$ Scuti stars in detached binaries with $P_{\mathrm{orb}}<13$ d. The studied star in these plots follows very well both the distributions of the sample stars and the empirical relations.

Using the current dominant oscillation frequency of the pulsator and the pulsation period-luminosity relation for $\delta$ Scuti stars of Ziaali et al. [14],

$$M_{\mathrm{V}}=-2.94\left(6\right)log{P}_{\mathrm{pul}}-1.34\left(6\right),$$

(3)

it is feasible to calculate its absolute magnitude ($M_{\mathrm{V}}$ = 2.06(13) mag). Hence, using the apparent magnitude ($m_{\mathrm{V}}$) and the distance modulus, its distance can be calculated. The $m_{\mathrm{V}}$ of KIC 8504570 is 13.28 mag according to the NOMAD-1 catalogue [71] and the extinction in V band is $A_{\mathrm{V}}=0.336$ mag [47]; thus, its distance is determined as 1502${}_{-87}^{+93}$ pc. This value is in very good agreement with the value 1488 ± 41 pc. as derived by Berger et al. [47] and Bailer-Jones et al. [72], and in slight disagreement with the value 1305 pc of Queiroz et al. [73]. The latter discrepancy is attributed to the different extinction value ($A_{\mathrm{V}}=0.474$ mag) used by Queiroz et al. [73]. It should be noted that the aforementioned $M_{\mathrm{V}}$ is in very good agreement with the $M_{\mathrm{bol},1}$ = 2.17(6) mag, which was calculated based on the LC model;ing (Table 1).

For the future, radial velocity measurements are welcome to validate the present results for the LC model, although the ∼95% light domination of the primary component makes the acquisition of the radial velocities of the secondary an extremely difficult task. At best, we anticipate that only the radial velocities of the primary can be measured, which will only constrain the mass of the primary component, and hence the mass ratio of the system. However, these potential future measurements cannot significantly change the present pulsations models, especially the results for the dominant and the independent frequencies, which were the main goals of the present study. The asteroseismic modelling of other similar systems, especially of those observed by satellite missions, is highly encouraged and recommended because the sample of $\delta$ Scuti stars in binary systems is still small and we lack of enough information. Moreover, systems with $P_{\mathrm{orb}}$ between 10 and 20 d should be prioritised for detailed analysis in order to check the reasons for the existence of the boundary of $P_{\mathrm{orb}}\sim$ 13 d.