1. Introduction
The
Scuti stars are short-period and multiperiodic pulsating variables. In general, they oscillate in radial and low-order non-radial pulsations due to
-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
[
1], their spectral types between AIII/V and FIII/V, and they are located inside the classical instability strip. Thanks to the
[
5,
6], the
[
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
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
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 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 “
” (oscillating eclipsing binaries of Algol type) for categorising the EBs with a
Scuti mass accretor component of (B)A-F spectral type. A few years later, the first connection between orbital (
) and dominant pulsation (
) 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
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
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
(∼13 d); beyond that
and
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 online
1), 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
Scuti light curves, employed the pulsations timing technique [
29,
30,
31] and identified 341 new binaries with long
(
d) that each host a
Scuti component. Liakos [
32], using recently discovered systems, presented updated correlations between
and
for close (i.e.,
d) detached eclipsing binaries with
Scuti components.
The data quality of both
and
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
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
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, “
Eclipsing Binary Catalog” (
) [
35], that is publicly available
2, 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
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
-detached binaries with a
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
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
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
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
) was employed for the observations. This set-up provided a resolution of ∼3.2 Å pixel
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 Observatory
4, 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
K for the primary, based on the relations between
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
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
[
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
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
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 (
2,454,955.78(3) BJD,
d) and the
magnitude
mag, respectively, as listed in
.
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 (
) 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 (
) 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.,
) following the preliminary findings of Lurie et al. [
62]. The dimensionless potentials (
), the fractional luminosity of the primary component (
) 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 (
) 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
(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
, longitude
, radius and temperature factor
) 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
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
, 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. 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 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
) 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 (
) and the bolometric magnitude values (
) 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
Scuti stars is 4–80 d
[
34], the present analysis included the regime 0–4 d
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
, or even to the intrinsic hybrid behaviour of
-Doradus–
Scuti type. Therefore, the present pulsation analysis was done in the range 0–80 d
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 (
) 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
mag in regimes with absence of frequencies, with a spacing of 2 d
, and a box size of 2. A 4
limit (i.e.,S/N
) [
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
and 0.009 d
, 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
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]:
Moreover, the following pulsation constant-density relation was used for the calculation of the density of the pulsators:
where
is the frequency of the dominant pulsation mode (i.e., that with the largest amplitude). At this point it should be noted that the
of the multiperiodic
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
Scuti stars [
67] in the
FAMIAS software v.1.01 [
68]. The
l-degrees from the closest MAD models (i.e.,
f,
,
M and
) to the detected independent frequencies were adopted as the most possible pulsation modes. Moreover, the ratio
/
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 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
, the amplitude
A, the phase
, the S/N, the
Q, the
/
, 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).
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
. Among the other 387 depended frequencies, 309 were spread almost uniformly in the range 10–43.4 d
; 72 had values less than 4.4 d
; five were found between 5.5–9 d
; and only one, namely,
, exceeded 50 d
. As can be seen in
Figure 6, one main concentration of frequencies is between 12 and 17 d
, while a slightly more spread out one is between 23 and 30 d
. The results based on MAD models show that all oscillations are probably non-radial pressure modes. Although the ratio
/
has value ∼0.78,
was not identified as a radial mode by the MAD models. Finally, a value of
= 0.215(4)
was derived.
5. Summary, Discussion and Conclusions
In the present work, detailed LC and pulsation modellings for KIC 8504570, a neglected -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 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
with the dominant part at 14.37 d
. 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 (
) and Hertzsprung–Russell (
) 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
Scuti component of KIC 8504570 has similar absolute properties to other
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
Scuti star of KIC 8504570 with others that belong in similar systems, it was placed on the
and
diagrams (
Figure 10 and
Figure 11, respectively) along with the well established empirical relations of Liakos [
32] for
Scuti stars in detached binaries with
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
Scuti stars of Ziaali et al. [
14],
it is feasible to calculate its absolute magnitude (
= 2.06(13) mag). Hence, using the apparent magnitude (
) and the distance modulus, its distance can be calculated. The
of KIC 8504570 is 13.28 mag according to the NOMAD-1 catalogue [
71] and the extinction in
V band is
mag [
47]; thus, its distance is determined as 1502
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 (
mag) used by Queiroz et al. [
73]. It should be noted that the aforementioned
is in very good agreement with the
= 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 Scuti stars in binary systems is still small and we lack of enough information. Moreover, systems with between 10 and 20 d should be prioritised for detailed analysis in order to check the reasons for the existence of the boundary of 13 d.