KIC 10975348: A Double-mode or Triple-mode High-amplitude δ Scuti Star?

The main scientific goal of the Kepler mission was to search for Earth-like planets outside the solar system by detecting transits of the host star (Borucki et al. 2010; Koch et al. 2010). The high-precision photometric data delivered from the Kepler telescope also provides an unprecedented opportunity to probe into stellar interiors using their natural oscillation modes, and hence greatly expands the field of asteroseismology (Chaplin et al. 2010). Thanks to the ultra-high-precision photometric observations at the level of μmag, our understanding of many types of pulsating stars has significantly improved. For instance, Bedding et al. (2011) reported that in red giants the hydrogen- and helium-burning stars can be distinguished according to the observed period spacings of gravity modes; Giammichele et al. (2018) found that a pulsating white dwarf has a large oxygen-dominated core, which exceeds the predictions of existing models of stellar evolution. As a common group of variable stars, δ Scuti stars are excellent targets for asteroseismic study. At present, more than 2000 δ Scuti stars have been found in the Kepler field (Balona & Dziembowski 2011; Balona 2014; Bowman et al. 2016), however, only several high-amplitude δ Scuti stars are found and investigated in detail so far (Balona et al. 2012; Yang et al. 2018a; Yang & Esamdin 2019).

High-amplitude δ Sct (HADS) stars, as a subgroup of δ Sct stars, are usually recognized by their relatively simple, nonsinusoidal light variations with peak-to-peak amplitude larger than 0.3 mag (Breger 2000). They are slow rotators with v sin i < 30 km s⁻¹ and the slow rotation might be a precondition for their high amplitudes (Breger 2000). In the H-R diagram, HADS stars seem to concentrate in a narrow strip in the δ Sct instability region with a width of about 300 K in temperature (McNamara 2000), but observations from space photometry revealed that some HADS stars can also be found beyond the narrow region (Balona 2016). The light variations of HADS stars are usually dominated by the fundamental and/or first overtone radial mode(s) (Breger 2000; McNamara 2000). Although the AAVSO International Variable Star Index (VSX; Watson et al. 2015) lists almost 600 HADS stars, radial triple-mode HADS stars are particularly rare. Only five HADS stars with three consecutive radial modes, i.e., AC And (Fitch & Szeidl 1976), V823 Cas (Jurcsik et al. 2006), V829 Aql (Handler et al. 1998), GSC 762-110 (Wils et al. 2008), and GSC 03144-595 (Mow et al. 2016), are found at present.

In recent decades, nonradial modes with low amplitude are also detected in some HADS stars, owing to extensive high-precision photometric observations (Poretti et al. 2011). Moreover, with the advent of space asteroseismology, long-term variations of the principal modes and more low-amplitude frequencies are detected in HADS stars. Balona et al. (2012) reported a slight amplitude variation of the dominant modes in the HADS star V2367 Cyg and also found that this star rotates with twice the projected rotational velocity of any other HADS star. With Kepler observations, amplitude modulation in several HADS stars was investigated. Yang et al. (2018a) found a pair of low-amplitude triplet structures in the frequency spectra of KIC 5950759 and the reason for this triplet structure might be the amplitude modulation of stellar rotation. Another HADS star KIC 10284901 also shows a weak amplitude modulation with two frequencies, and analysis suggests that they might be related to the Blazhko effect (Yang & Esamdin 2019). Hence, the identification of these low-amplitude frequencies in HADS stars possesses great potential to improve our understanding of the stellar interior, and the comparison of single-, double-, and triple-mode HADS stars may illuminate what determines the number of radial modes in which a star pulsates. More HADS stars with high-precision photometric observations are needed.

KIC 10975348 (α₂₀₀₀ = 19^h26^m461, δ₂₀₀₀ = +48°25'308, 2MASS: J1926461+4825303) was classified as a δ Sct star with a pulsation period of 2.35 hrs by Ramsay et al. (2014). In that paper, KIC 10975348 was reported as a mid-A type star based on spectrum from Gran Telescopio Canarias. Some basic parameters of this target are listed in Table 1. According to the amplitude of the light curves, KIC 10975348 appears to be a HADS star, yet the exact type is uncertain. In this work, using the high-precision photometric data from the Kepler mission, we further verify its nature and investigate its long-term periodic variation as well.

KIC 10975348 was observed by the Kepler space telescope from BJD 2456107.139 to 2456424.001, including three quarters (i.e., Q14, Q15, and Q16). There are two observation strategies for this star: long-cadence (LC; 29.5 minutes integration time) mode and short-cadence (SC; 58.5 s integration time) mode. To avoid the Nyquist alias peaks, we only used the SC data, i.e., Q14.3, Q15.3, and Q16.3 in this work. More details on how to identify the Nyquist alias peaks in LC data can be found in Murphy et al. (2013) and Yang et al. (2018a). All the SC data are available in the Kepler Asteroseismic Science Operations Center (KASOC) database³  (Kjeldsen et al. 2010), in which two types of data, raw flux and corrected flux, are provided. The former data has been reduced by the NASA Kepler Science pipeline in fact, and the corrected one can be obtained from KASOC Working Group 4 (WG#4: δ Sct stars). The corrected flux was used in this work and we first performed several corrections to the data including the removal of obvious outliers and detrending the light curve with a low-order polynomial. Then the flux data was converted to the magnitude scale, and each quarter was adjusted to zero by subtracting their mean value. The final rectified light curve was obtained with 147,481 data points in total, spanning over about 220 days. Figure 1 shows a portion of the rectified light curve of KIC 10975348 covering 2 days. From this figure, it is clear that the amplitude of the light curve is about 0.7 mag, which is larger than the typical amplitude (>0.3 mag) of a high-amplitude δ Sct star.

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We used the software PERIOD04 (Lenz & Breger 2005) to analyze the pulsating behavior of KIC 10975348. The rectified light curve was fitted with the formula

$$\begin{eqnarray}&&m={m}_{0}+{\rm{\Sigma }}{A}_{i}\sin (2\pi ({f}_{i}t+{\phi }_{i})),\end{eqnarray} \tag{ 1 }$$

where m₀, A_i, f_i, and ϕ_i are the zero-point, amplitude, frequency, and the corresponding phase, respectively.

In order to detect more significant frequencies, we chose a frequency range of 0 < ν < 80 day⁻¹, which is wider than the typical period range of δ Scuti stars. During the extraction of significant frequency, the highest peak was usually selected as a potential significant frequency. Then a multifrequency least-square fit using formula 1 was applied to the light curve with all significant frequencies detected, and obtained the solutions for all the frequencies. A theoretical light curve constructed by the above solutions was subtracted from the rectified data and the residual was obtained for the next search. The above steps were repeated until there was no significant peak in the frequency spectrum. The criterion of a signal-to-noise ratio (S/N) > 4.0 suggested by Breger et al. (1993) was adopted to judge the significance of the detected peaks. The uncertainties of the frequencies were obtained following the method provided by Montgomery & O'donoghue (1999).

Figure 2 shows the amplitude spectra and the prewhitening procedures of the light curve. The last panel shows amplitude spectra after 11 detected frequencies were prewhitened. Following Breger et al. (1993), we draw the significance curves at an S/N = 4 in the last panel for the judgment of a significant peak. There is no significant peak in the residuals and the overall distribution of the residual is typical of noise.

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A total of 11 significant frequencies were detected in this work and a full list was given in Table 2. Among these frequencies, three stronger frequencies were considered to be independent. It is reasonable that the strongest peak f_S1 was assumed to be the fundamental mode, since the light variations were dominated by this frequency. Therefore, we marked f_S1 with F0 in the last column of Table 2. In addition, six harmonics of F0 were also detected and listed in the table. The other two independent frequencies are f_S8 and f_S10, and we marked them as F1 and F2, respectively. The rest of the frequencies are combinations with F0.

For frequencies F0 and F1, we found that the period ratio of P₁/P₀ = F0/F1 = 0.758 was in the typical range (0.756–0.787) of the period ratio of the first overtone and fundamental mode for δ Scuti stars (Petersen & Christensen-Dalsgaard 1996). It seems that F1 can be identified as the first overtone mode. Considering its peak-to-peak amplitude over 0.7 mag from the light curve, KIC 10975348 seems to be classified as a new double-mode HADS star.

To examine the potential long-term period changes of KIC 10975348, a classical O − C diagram was constructed. O is the observed time of maximum light and C is the theoretical value from ephemeris formula (Sterken 2005). For the calculations of O, we first visually inspected the light curve and determined the preliminary times of maximum, and then made a second-order polynomial to fit the part of the light curve around one-third of the full amplitude. It is reasonable to do that since this part of the light curve is almost symmetric for this star. The times of extrememum of the polynomial fit were considered as the observed times of maximum light. The typical uncertainties (about 0.00008 day⁻¹) were estimated by Monte Carlo simulations. In total, 1018 times of maximum light were obtained and a full list of each quarter is given in Tables 3, 4, and 5, respectively.

To obtain the calculated times of maximum light, we used all the observed times of maximum and derived a new ephemeris formula:

$$\begin{eqnarray}\begin{array}{rcl}{T}_{\max } & = & {T}_{0}+P\times E=2456170.241912(0)\\ & & +\ 0.097734(1)\times E,\end{array}\end{eqnarray} \tag{ 2 }$$

where T₀ is the initial epoch, P is the period, and E is the cycle number.

Then the values of O − C were derived based on this new ephemeris. All the O − C values, as well as the corresponding cycle numbers, are listed in Tables 3–5, respectively.

Figure 3 shows the O − C diagram of the three quarters used in this work. It is noteworthy that the O − C of each quarter is almost flat, which suggests that the star seems to have no obvious period change.

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5.1. Double Mode or Multimode?

5.2. O − C

The period changes due to stellar evolution for stars in and across the lower part of the classical instability strip allow an observational test of stellar evolution theory, assuming that other physical reasons for period changes can be excluded (Breger 2000). From a theoretical point of view, an evolutionary change in T_eff and M_bol leads to a period change of a size of (Equation (9) of Breger 2000)

$$\begin{eqnarray}&&\displaystyle \frac{1}{P}\displaystyle \frac{{dP}}{{dt}}=-0.69\displaystyle \frac{{{dM}}_{\mathrm{bol}}}{{dt}}-\displaystyle \frac{3}{{T}_{\mathrm{eff}}}\displaystyle \frac{{{dT}}_{\mathrm{eff}}}{{dt}}+\displaystyle \frac{1}{Q}\displaystyle \frac{{dQ}}{{dt}},\end{eqnarray} \tag{ 3 }$$

where P is the period of a radial pulsation mode in units of days, and Q is the pulsation constant. For a specific mode, the Q value is an essential constant for all δ Scuti stars, hence the term (1/Q)/(dQ/dt) is negligible as it is a very small quantity (Breger 2000). The above relation is then reduced to

$$\begin{eqnarray}&&\displaystyle \frac{1}{P}\displaystyle \frac{{dP}}{{dt}}=-0.69\displaystyle \frac{{{dM}}_{\mathrm{bol}}}{{dt}}-\displaystyle \frac{3}{{T}_{\mathrm{eff}}}\displaystyle \frac{{{dT}}_{\mathrm{eff}}}{{dt}}.\end{eqnarray} \tag{ 4 }$$

In the lower instability strip where the δ Scuti stars are found, stellar evolution leads to increasing periods in most of stars, with predicted increase period changes from 10⁻¹⁰ yr⁻¹ for the stars on the main sequence to 10⁻⁷ yr⁻¹ for the longer-period evolved variable stars (Breger & Pamyatnykh 1998). Such period changes are observable and also have been observed in some radial δ Scuti pulsators. Breger & Pamyatnykh (1998) calculated the theoretical periods of the radial fundamental modes and their changes during late main-sequence and post-main-sequence evolution of the 1.8 M_⊙ model and found that the observations are consistent with the predicted values. Xue et al. (2018) investigated a HADS star VX Hya with the observed period change from O − C and the stellar evolutionary models, and found that the period change can be successfully interpreted by the evolutionary effect.

In pulsating stars, when a pulsation period changes linearly with time, the O − C diagram will present a parabolic form that rises or falls depending on whether the period is increasing or decreasing (Sterken 2005). For HADS stars, the O − C diagram is a powerful tool to investigate their period changes. According to Breger (2000), HADS stars can be divided into two groups depending on whether the period change is increasing or decreasing. For instance, some HADS stars have an increasing period, i.e., YZ Boo (Yang et al. 2018b), XX Cyg (Yang et al. 2012), GP And (Zhou & Jiang 2011), etc., while others pulsate with a decreasing period, such as BS Aqr (Boonyarak et al. 2011), BE Lyn (Boonyarak et al. 2011), DY Peg (Blake et al. 2003), etc. Different values of period change may suggest that stars are in different stages of evolution.

For KIC 10975348 in this work, it was also expected to exhibit an increasing or decreasing period. However, from the O − C diagram in Figure 3, it seems that the shape of O − C is nearly flat and does not show any curved part, which is in contrast to the predictions by Breger (2000). The possible cause for that might be due to the shorter time span of the observations for this star. To verify its period variations, theoretical seismic modeling and regular observations from space with a longer time span in the future are necessary.

We analyzed the pulsating behavior of KIC 10975348 using high-precision photometric observations from the Kepler mission, and detected 11 significant frequencies from SC data. Among these frequencies, three independent frequencies, i.e., F0 = 10.231899 day⁻¹, F1 = 13.4988 day⁻¹, and F2 = 19.0002 day⁻¹ were found. The lower period ratio (=0.758) of the first two stronger frequencies (i.e., F0 and F1) suggests that KIC 10975348 might be a metal-rich double-mode HADS star. The third independent frequency might be a third overtone mode. If confirmed, KIC 10975348 would be a new radial triple-mode HADS star, and hence enrich the rare sample of pulsating stars with three radial modes. We also compared KIC 10975348 with current monoperiod HADS stars, and highlighted the potential of the TESS mission for the study of HADS stars.

The O − C diagram was constructed with 1018 times of maximum light and yielded a new ephemeris epoch: T_max =T₀ + P × E = 2456170.241912(0)+0.097734(1) × E. The O − C analysis indicated that the period of KIC 10975348 shows no obvious change, which is very unusual. The cause of that might be due to the shorter time span of current observations. To verify the evolutionary state of KIC 10975348, regular observations from space in the future are necessary.

We thank the referee for the comments that helped clarify the paper. This research is supported by the National Natural Science Foundation of China (grant No. 11573021, U1938104, and 12003020) and the Fundamental Research Funds for the Central Universities. We would like to thank the Kepler science team for providing such excellent data.