KIC 12602250: A Low-amplitude Double-mode Delta Scuti Star with Amplitude Modulation

The high-precision photometric data provided by Kepler provides an unprecedented opportunity to explore stellar interiors by using the natural oscillation mode of stars, thus greatly expanding the research field of asteroseismology (Chaplin et al. 2010). The ultrahigh precision photometric observations at μ mag level have greatly improved our understanding of many types of pulsating variables (Balona et al. 2012). Bedding et al. (2011) proposed that the observed period spacings of gravity modes could be applied to distinguish the hydrogen and helium burning stars in red giants, while Giammichele et al. (2018) suggested that there may be a suppressed oxygen-dominated core in pulsating white dwarfs. As a group of conventional variable stars, the intrinsic homogenous pulsating modes of δ-Scuti stars make them excellent targets in the study of asteroseismology (Balona & Dziembowski 2011; Breger et al. 2011; Yang et al. 2018). The fundamental mode, the first, second, even the third and fourth radial pulsation mode could be indicators to the interiors burning mechanisms of δ-Scuti stars (Breger 2000b).

The δ-Scuti-type pulsating stars lie inside the classical Cepheid instability strip main sequence on the Hertzsprung Russell diagram. δ-Scuti stars typically range from A2–F2 in the spectral type with luminosity classes from iii–v (e.g., Lopez de Coca et al. 1990; Breger 2000b; Rodríguez & Breger 2001), and within the effective temperature range of 6300 K ≤ T_eff ≤ 8600 K (Uytterhoeven et al. 2011). The pulsating amplitudes are in the range of 0.003–0.9 mag in the V band, with periods usually between 0.02 and 0.3 days (Breger 1979). Some stars show amplitude modulation of pulsation modes caused by different reasons, e.g., beating, mode coupling, rotation, and the Blazhko effect (Bowman & Kurtz 2014; Bowman et al. 2016; Yang et al. 2018; Yang & Esamdin 2019). These targets are excellent samples for asteroseismic study, as they could improve our knowledge of the stellar structure and evolution of stars.

As a subclass of δ-Scuti stars, the HADS stars usually pulsate with a light amplitude larger than 0.3 mag and generally rotate slowly with v sin i ≤ 30 km s⁻¹ (Breger 2000b). Compared with the low-amplitude δ-Scuti stars, the HADS stars possess a more restrictive instability strip with a width in temperature of about 300 K and tend to shift to a lower temperature with evolution (McNamara 2000). Lee et al. (2008) reveal that only ∼0.24 percent of the stars suited in the δ-Scuti region belong to HADS stars. The majority of HADS stars are typically young and metal-rich Population i stars. Some have been confirmed to be SX Phe variables and are Population ii metal-deficient stars (Breger 2000b; Balona & Nemec 2012). In general, the HADS stars pulsate with only one or two modes (e.g., AE UMa, Niu et al. 2017; YZ Boo, Yang et al. 2018; etc.) and most of their pulsations belong to radial modes.

Kepler Input Catalog (KIC) 12602250 is classified as a δ-Scuti star with a pulsation period of 2.07 hr by Debosscher et al. (2011). According to the frequency analysis, we propose for the first time that KIC 12602250 is a low-amplitude radial double-mode δ-Scuti star with no nonradial modes. The frequency spectra of KIC 12602250 is shown in Figure 2 while the basic parameters are tabulated in Table 1.

Table 1. KIC 12602250 Observational (Photometry) Data Characteristics

Parameters	Value in Catalog
Kepler ID	12602250
2MASS ID	J19213193+5140320
Gaia ID	2139231542654343552
R.A., Decl.	+19h:21m:31.9s, +51°:40':321
BJD0	2454953.5388
Rayleigh$\,{f}_{\mathrm{res}}$	0.000679	d−1
Period	2.07	hr
Kmag	13.277
Contamination	0.024
Teff	6879	K
$\mathrm{log}g$	4.131	cgs
$\tfrac{R}{{R}_{\odot }}$	1.597
$\tfrac{{Fe}}{H}$	−0.138	cgs

Note. These parameters are available in the KASOC: https://kasoc.phys.au.dk/.

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In this paper, the observations of KIC 12602250 are introduced in Section 2. Section 3 presents the frequency analysis while Section 4 discusses the radial triple modes and amplitude modulation of this star. We summarize in Section 5.

KIC 12602250 was observed from BJD 2454953.538 to 2456424.001 by a Kepler space telescope, including eighteen quarters (i.e., Q0–Q17). There are only long cadence (LC) photometric observations of KIC 12602250 available through the Kepler Asteroseismic Science Operations Center (KASOC) database⁴ (Kjeldsen et al. 2010) with two types: one is the raw flux, which is reduced by the NASA Kepler Science pipeline, and the other is corrected flux, which is provided by KASOC Working Group 4 (WG # 4: δ-Scuti targets). The second type has been corrected for systematic errors such as the cooling down, cooling up, outliers, and jumps. We use the corrected flux and convert it to magnitudes. Then the mean value of each quarter is subtracted to obtain the rectified time series. The final rectified light curve was obtained with 65,264 data points, spanning over about 1471 days.

The light curves of variable stars from the Kepler data (Gilliland et al. 2010) could contain gaps due to the operating procedure or the influence of the environment (Pascual-Granado et al. 2018). The discrete Fourier transform of a gapped time series is the convolution of the Fourier transform of the original time series of observations with a spectral window function determined by the observation times (Deeming 1975). The amplitudes of the peaks might be reduced due to the modulation caused by periodic gaps (Pascual-Granado et al. 2015) that result in spurious combination frequencies in the power spectrum. While this is not always true when the time series is unevenly sampled, it is so when the sampling rate changes simply due to data loss in evenly sampled light curves (Appourchaux et al. 2008).

Kepler data for this star shows a duty cycle of 90.69% with 791 gaps. The largest gap has 773 loss data points, which corresponds to 16 days approximately, but most of the gaps ( 98%) have a size of 39 data points, which is 0.81 days approx. These gaps have larger size than the typical pulsation period for a star of this class. On the other side, the gaps are not periodically distributed so they do not cause interference issues with the signal. However, the amplitudes might be affected by the loss of cycles. Hence, the signal-to-noise ratio (S/N) might be reduced, leading to an unreliable frequency detection and parameter estimation and this could affect the detection limit. Therefore, a proper gap treatment is necessary in order to perform a reliable frequency analysis of this star.

Gaps in the time series cause spurious frequencies in the power spectra and in light curves of pulsating stars. This hampers the identifying of theoretical oscillation modes. We used a forward-backward predictor based on autoregressive moving-average modeling (ARMA) in the time domain. The algorithm MIARMA is particularly suitable for replacing invalid data (Pascual-Granado et al. 2015, 2018). We have used this algorithm to fill the gaps in the light curve of KIC 12602250. Figure 1 shows a portion of the rectified light curve of KIC 12602250 covering seven days. From this figure, the peak-to-peak amplitude of KIC 12602250 obtained from the rectified light curve is ∼0.06 mag, suggesting that this star belongs to the low-amplitude class of δ-Scuti stars.

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The software PERIOD04 (Lenz & Breger 2005) is employed to analyze the pulsating behavior of KIC 12602250. The rectified light curve was fitted with the following formula:

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

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

Nyquist frequency sets the minimum sampling frequency that needs to be defined to prevent alias frequencies, while some potential significant frequencies could beyond Nyquist frequency (Southworth et al. 2011; Bowman et al. 2016; Yang et al. 2018). The Nyquist frequency of LC observations is f_N = 24.469 d⁻¹ (Murphy et al. 2013; Holdsworth et al. 2014). In order to detect all potential significant frequencies, we choose the frequency range 0 < ν < 50 d⁻¹, which is somewhat wider than the typical pulsation frequency of δ-Scuti stars (Breger 1979, 2000b; Balona & Dziembowski 2011; Uytterhoeven et al. 2011). As LC observations apply a longer integration time, the signal-to-noise data of LC observations are much higher than that of SC observations, so the noise in the LC spectrum is lower than that in the SC spectrum (Gilliland et al. 2010; Yang et al. 2018). Owing to the combined effects of a high signal to noise and a nonregular sampling in LC light curves, we might detect signal beyond the Nyquist limit and discriminate it from aliases.

We consider two frequencies to be resolved if the difference between them is larger than the resolution frequency, f_res = 1/T, which is 0.000679 d⁻¹ for the LC light curve. The highest peak is usually identified as a potential significant frequency while extracting significant frequencies. Then a multifrequency least-square fit using Equation (1) applies to the light curve with all significant frequencies detected and obtained the solutions for all the frequencies. A theoretical light curve constructed using the above solutions is subtracted from the rectified data while the residual is obtained for the next search. The above steps are repeated until no significant peak is detected in the frequency spectrum. The criterion of S/N > 5.0 suggested by Baran et al. (2015) is adopted to determine the significance of the detected peaks. The uncertainties of frequencies were obtained following the method proposed by Montgomery & O'donoghue (1999). Figure 2 shows the amplitude spectra and the prewhitening procedures of the light curve. The last panel shows the residual amplitude spectra after prewhiten eleven detected frequencies. No significant peak could be detected in the residual spectrum, which displays an overall distribution of typical noise.

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The gaps in the time series could cause a reduced amplitude of the peaks when noninterpolated data are analyzed. The amplitudes of the spurious frequencies will be significantly prevented or removed from the gap-filled power spectrum. The spurious frequency 18.99 d⁻¹ can be detected from the gapped data while it is removed from the gap-filled data (Pascual-Granado et al. 2015, 2018). Moreover, the amplitudes of the extracted frequencies are increased significantly. A total of eight significant frequencies are detected in the spectra of KIC 12602250 and a full list is given in Table 2. Among these frequencies, two of them are considered to be independent. It is reasonable that the strongest peak f₁ was assumed to be the fundamental mode, since the light variations were dominated by this frequency. Therefore, we marked f₁ with "F0" in the last column of Table 2 and the other independent frequency f₂ is labeled as "F1." In addition, harmonics (i.e., f₄, f₈) of 'F0','F1' and three combination frequencies (i.e., f₃, f₅, f₆) of 'F0', 'F1' are also detected. The frequency f₇ is considered as an alias frequency.

4.1. The Double-mode in KIC 12602250

4.2. Amplitude Growth in KIC 12602250

Bowman et al. (2016) propose that the amplitude of KIC 12602250 might increase somewhat with time. We then investigate the amplitude variation of the three frequencies with the highest amplitude. Following the method described by Murphy et al. (2012), we employ a nonlinear least-squares fit of the frequencies to the data of each quarter using the software PERIOD04 (Lenz & Breger 2005). The frequency, amplitude, and phase of the peak from the high-resolution amplitude spectrum for all data are utilized as input parameters, while the fixed frequency of each mode used for tracking amplitude and phase of the whole data set can be obtained from the least-square fitting. The fixed frequency is then used to optimize the amplitude and phase of each time bin, and the same method has been applied to track the amplitude and phase for the other frequency.

Figure 3 shows slow growth in the amplitude of F1 and f₃ (F1–F0), while a relatively flat evolution in the amplitude of F0. According to Murphy et al. (2012), two potential scenarios have been proposed for the apparent amplitude growth: (1) the amplitude growth is in some way instrumental and (2) the amplitude growth is an intrinsic or extrinsic effect to the star.

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The slight variation of F1 and f₃ might somewhat be a result of a systematic error from the instrument. So we check the contamination of KIC 12602250 from the KIC (Brown et al. 2011). The contamination reported for KIC 12602250 is 0.024, significantly smaller than KIC 3429637 (Murphy et al. 2012), suggesting that the amplitude increasing of F1 is not an instrumental signal. Besides, Murphy et al. (2012) demonstrate that it could not be an instrumental effect, as all modes would decrease or increase with the same functional form if the modulation is instrumental, and instrument trends generally affect low frequencies only.

The 4 CVn is the only one δ-Scuti star with detailed studies of amplitude modulation based on ground-based telescopes (Breger 2000a, 2009, 2016). Bowman et al. (2016) utilize observations of ∼1000 δ-Scuti stars from the Kepler Space Telescope (Borucki et al. 2010) to investigate the amplitude modulation as well as its possible mechanism. The reason for δ-Scuti stars with variable pulsation amplitudes (and/or frequencies) could roughly be classified as the intrinsic and extrinsic, i.e., physical interior variations to the star and external effects (Bowman et al. 2016). The beating from pairs (or groups) of close frequency (Breger & Bischof 2002; Breger & Pamyatnykh 2006) and nonlinearity or mode coupling (Bowman et al. 2016) could be considered intrinsic effects.

The amplitude variation of KIC 12602250 may be inconsistent with beating effects since no pairs of close frequency have been detected. Breger et al. (2012) propose that three frequencies within a family follow the frequency, amplitude, and phase relations described in equations ν₁ ≃ ν₂ ± ν₃, A₁ = μ_c (A₂ A₃), ϕ₁ = ϕ₂ ± ϕ₃, where A_i and ϕ_i represent the amplitude and phase of the child and parent modes, respectively, and μ_c is the coupling factor. The μ_c could be small values for combination frequencies from the nonlinear distortion model. For KIC 12602250, the f₃ is a combination frequency of F0 and F1 with a μ_mc estimated as 0.0025, which might be consistent with the nonlinear distortion model. However, Dziembowski (1982) suggest that two linearly unstable low-frequency parent modes can damp a high-frequency unstable child mode once they reach the critical amplitude. We propose that the amplitude modulation of the f₃ detected in KIC 12602250 could not originate from the nonlinearity of either F0 or F1, which are low-frequency modes. Lares-Martiz et al. (2020) utilize a different approach that is based on a Volterra expansion to describe the nonlinearities. Murphy et al. (2012) propose that stellar evolution is the primary reason for the amplitude variation of modes in KIC 3429637. The amplitude variation of KIC 12602250 could be a result of stellar evolution.

We have analyzed the pulsating behavior of KIC 12602250 using high-precision photometric observations from the Kepler mission and eight significant frequencies are detected. While two of them are independent frequencies, i.e., F0 = 11.6141 d⁻¹ and F1 = 14.9741 d⁻¹. The ratio of the P1/P0 of KIC 12602250 is measured to be 0.776, suggesting that this target could be a new low-amplitude radial double-mode δ-Scuti star. This may indicate the modification of the definition of HADS is necessary. The low-amplitude, double-mode, and slow amplitude growth of F1 and f₃ might suggest that KIC 12602250 could be a post-main-sequence δ Scuti crossing the instability strip for the first time (Poleski et al. 2010). The slow amplitude growth detected in F1 might be an indicator of stellar evolution. Spectroscopic observations provide us with opportunities to probe the elemental abundance of KIC 12602250, which in turn provide additional constraints on the determination of its pulsation modes and the origin of the amplitude growth. In order to confirm that the pulsation modes detected using Kepler photometry are indeed overtones and to reveal the nature of the amplitude variation found in KIC 12602250, follow-up spectroscopic observations and full seismic modeling are necessary.

We thank the anonymous referee for the suggestive comments, which improved the manuscript. This research is supported by the National Natural Science Foundation of China (grant No. U2031209 and 12003020). J.P.G. acknowledge funding support from Spanish public funds for research from project PID2019-107061GB-C63 from the "Programas Estatales de Generación de Conocimiento y Fortalecimiento Científico y Tecnológico del Sistema de I+D+i y de I+D+i Orientada a los Retos de la Sociedad," and from the State Agency for Research through the "Center of Excellence Severo Ochoa" award to the Instituto de Astrofísica de Andalucía (SEV-2017-0709), all from the Spanish Ministry of Science, Innovation and Universities (MCIU). We would like to thank the Kepler science team for providing such excellent data.