KIC 9845907: A δ Scuti Star with the First Overtone as the Dominant Frequency and with Many Equidistant Structures in Its Spectrum

Xiao-Ya Sun; Zhao-Yu Zuo; Tao-Zhi Yang; Antonio García Hernández

doi:10.3847/1538-4357/ace905

1. Introduction

One of the long-standing goals in astronomy is to improve our understanding of stellar internal structure and evolution. Asteroseismology, by comparing observations with theoretical models, constrains the interior physics of stars, such as convection, rotation, etc. (Aerts 2021). In recent years, asteroseismology has entered its golden age thanks to the continuous, long-term and high-resolution data from space telescopes such as Kepler (Koch et al. 2010) and Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015). Significant progress has been made in the study of various types of pulsators, such as red giant stars (Yu et al. 2020; Li et al. 2022), solar-type stars (Chaplin & Miglio 2013), pulsating white dwarfs (Zong et al. 2016), and δ Scuti stars (Chen & Li 2019; Yang et al. 2022) as well. Among the more than 150,000 observed targets in the Kepler field, over 2000 δ Scuti stars have been discovered (Balona 2014; Bowman et al. 2016), which provides a large number of samples for asteroseismic studies.

δ Scuti stars are a class of short-period variables with periods in range of 0.02–0.25 days. They are intermediate-mass stars and have spectral types from A to F (Breger 2000). Their masses are generally between 1.5 M_⊙ and 2.5 M_⊙, which place them in the transition region between the lower-mass stars with thick convective envelope and radiative core and the massive ones with thin or none convective shell but a convective core (Aerts et al. 2010). They are located in the classical Cepheid instability strip where it crosses the main sequence (MS) in the Hertzsprung–Russell (H-R) diagram. Most δ Scuti stars are on the MS or post-MS stages although there are also some on the pre-MS stage (Breger 2000; Zwintz 2008). These variables exhibit both radial and non-radial modes (Qian et al. 2018; Sun et al. 2021), excited mainly by the κ mechanism (Aerts et al. 2010), usually identified as low-radial-order (n) low-angular-degree (l) pressure (p) modes (Uytterhoeven et al. 2011; Holdsworth et al. 2014). Moreover, Uytterhoeven et al. (2011) discovered that many of the δ Scuti stars show also gravity(g) modes. These characteristics make them excellent targets for asteroseismology study.

The pulsation spectra of δ Scuti stars are generally very rich and messy, which challenges the mode identification (Goupil et al. 2005; Bedding et al. 2023). Recently, there have been many studies on complex spectra from different perspectives, such as the low-order large separation (Rodríguez-Martín et al. 2020), rotational splitting (Ramón-Ballesta et al. 2021), (near-)equidistant frequency spacing structures (such as triplets and/or quintuplets; Kolenberg et al. 2011), and the relation between the low-order large separation and the stellar mean density (Suárez et al. 2014; García Hernández et al. 2015). For instance, in KIC 5950759, two pairs of triplet structures centered on the dominant frequency were detected in its frequency spectra, and the cause of that is inferred to be the amplitude modulation of stellar rotation (Yang et al. 2018). Another δ Scuti star KIC 10284901 shows two pairs of quintuplet structures, which might be related to the Blazhko effect (Yang & Esamdin 2019). Moreover, Chen et al. (2017) analyzed the frequency spectrum of CoRoT 102749568 based on the rotational splitting of the oscillation mode and determined the stellar parameters and helium core size using asteroseismology. These multiplet structures and the modes identification might improve our knowledge of the δ Scuti stars and offer new clues for probing the stellar interior and physical processes.

KIC 9845907 was classified as a δ Scuti star by Uytterhoeven et al. (2011). The Kepler magnitude of this star is K_P = 11.64 mag, and its effective temperature and radius are T_eff = 7936 ± 200 K and R = 1.954 R_⊙, respectively (Brown et al. 2011). Based on the data of short-cadence (SC) observations in the Kepler field, Balona (2016) found a large number of combination frequencies in δ Scuti stars, with the number in KIC 9845907 being 10. Some basic parameters of KIC 9845907 are listed in Table 1. In this work, we used the high-precision photometric data (including SC and Long Cadence (LC) data) provided by Kepler to further study the pulsating behavior of KIC 9845907.

Table 1. Basic Properties of KIC 9845907

Parameters	KIC 9845907	References
K_P	11.64 mag	a
TESS magnitude	11.41 mag	c
B	11.81 mag	b
V	11.12 mag	b
J	11.03 mag	b
I	11.23 mag	b
K	10.94 mag	b
H	10.98 mag	b
g	11.67 mag	b
i	11.72 mag	b
z	11.79 mag	b
Gaia	11.57 mag	c
M/M_⊙	1.87 ± 0.29	c
R/R_⊙	1.954	b
	1.928 ± 0.067	c
T_eff	7936 ± 200 K	a
	7861 ± 147 K	c
log g	4.029 ± 0.5 dex	b
	4.140 ± 0.077 dex	c
[Fe/H]	−0.039 ± 0.5 dex	b
Parallax (mas)	1.25 ± 0.01	(Gaia),d

Note. (a) Balona (2016); (b) KASOC: https://kasoc.phys.au.dk/; (c) TASOC: https://tasoc.dk; (d) Gaia (McDonald et al. 2017).

Download table as: ASCII Typeset image

2. Observations and Data Reduction

KIC 9845907 was observed by Kepler from BJD 2,454,953.539 to 2,456,424.001 (Q0–Q6, Q8–Q10, Q12–Q14, and Q16–Q17, total of 15 quarters) in 29.5 minute cadence (i.e., LC mode) and from BJD 2,454,953.529 to 2,455,833.279 (Q0, Q5.1–Q5.3, Q8.1–Q8.3, Q9.1–Q9.3 and Q10.1–Q10.3, total of five quarters) in 58.5 s cadence (i.e., SC mode). All the data were downloaded from Kepler Asteroseismic Science Operations Center (KASOC) database.³ KASOC provides asteroseismological data from the NASA Kepler and K2 missions to astronomers. The KASOC archive classifies the pulsation stars into subcategories and δ Scuti stars corrected by Working Group 4. And KASOC provides two types data: the "raw" and corrected flux data. In this work, we converted the corrected flux to stellar magnitude and performed corrections, including eliminating the outliers and detrending the light curves. Then the mean value of each quarter was subtracted, and the rectified time series were obtained with 49551 data points in LC and 176899 in SC, spanning over about 1471 and 880 days, respectively. Figure 1 shows a section of the SC light curves of KIC 9845907, where the amplitude is about 0.1 mag.

**Figure 1.** A 2 day section of the Kepler SC light curve of KIC 9845907. The amplitude of the light curve is about 0.1 mag. BJD is barycentric Julian date.
Download figure:
Standard image High-resolution image

3. Frequency Analysis

In this work, the Kepler SC data of KIC 9845907 were analyzed to search for significant frequencies using the software PERIOD04 (Lenz & Breger 2005), which allows a pre-whitening procedure spotting the frequencies through a Fourier transform (FT) of the data. The rectified light curve was fitted with the following formula:

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

where m₀ is the zero point and A_i, f_i, ϕ_i are the amplitude, frequency, and phase of each mode, respectively. We chose a range of 0 < ν < 100 day⁻¹ to search for frequencies since it covers the pulsation regime of δ Scuti stars. The upper limit of this range is well below the Nyquist frequency (f_Ny = 1/(2Δt) = 734 day⁻¹, where Δt is the sampling interval between consecutive points). The criterion of a signal-to-noise ratio (S/N) > 4.0 was adopted as stopping criterion for the pre-whitening process (Breger et al. 1993). The noise for each significant frequency was calculated in a box size of 2 day⁻¹ centered in the extracted peak. We selected a step rate "high," corresponding to an oversampling of 20 and a frequency spacing of 0.00005683 day⁻¹. The uncertainties of all frequencies were calculated following the method provided by Montgomery & Odonoghue (1999).

A total of 85 significant frequencies with S/N > 4.0 were extracted in this work. A full list of the detected frequencies (i.e., f₁ to f₈₅), with their corresponding amplitudes and S/N, is given in Table A1 in the Appendix. Figure 2 shows the Fourier spectra of 85 significant frequencies in two different ways: one is the amplitude spectrum of KIC 9845907 in the 0–100 day⁻¹ range (upper panel), and the other is the distribution of detected 85 frequencies (lower panel). As can be seen clearly in Figure 2, the frequencies with higher amplitudes were found in the range of 15–35 day⁻¹, and the other smaller peaks (i.e., f₈ and f₉) were in the range of 0–2 day⁻¹.

**Figure 2.** Amplitude spectra of KIC 9845907 using the SC data. The upper panel shows the amplitude spectrum in the 0–100 c/day range. The lower panel represents the distribution of detected 85 frequencies.
Download figure:
Standard image High-resolution image

The δ Scuti stars usually pulsate in radial mode, and theoretically the radial modes have higher visibility than non-radial (Dziembowski 1977). Therefore, by visually inspecting the highest amplitude of the frequency spectra, we considered f₁ as a radial mode. Further evidence comes from the investigation of the pulsation constant Q (Breger & Bregman1975), as described in Equation (2).

Radial oscillations can be estimated by the pulsation constant, Q, as given in following,

$\begin{eqnarray}&&\mathrm{log}Q=\mathrm{log}P+\displaystyle \frac{1}{2}\mathrm{log}g+\displaystyle \frac{1}{10}{M}_{\mathrm{bol}}+\mathrm{log}{T}_{\mathrm{eff}}-6.454,\end{eqnarray} \tag{ 2 }$

where P is the pulsation period, g is the surface gravity, M_bol is the absolute bolometric magnitude, and T_eff is the effective temperature. And the value of M_bol can be determined below (Driling & Landolt 2000),

$\begin{eqnarray}&&{M}_{\mathrm{bol}}=42.36-5\mathrm{log}R/{R}_{\odot }-10\mathrm{log}{T}_{\mathrm{eff}},\end{eqnarray} \tag{ 3 }$

where R and R_⊙ are the stellar radius and radius of the Sun, respectively. Therefore, the determination of the Q value depends on the accuracy of stellar parameters such as T_eff, log g, and M_bol. In this work, for KIC 9845907, TESS mission provides the parameters as T_eff = 7861.0 ± 147 K, R = 1.928 ± 0.067 R_⊙, and log g = 4.140 ± 0.077 dex (Stassun et al. 2019). Using these physical parameters, we found that the pulsation constant (Q) for f₁ was determined to be 0.029 ± 0.002, which is in the range of values for the fundamental and first overtone modes obtained by Lovekin & Guzik (2017) in a grid of rotating models representative of δ Scuti stars. Hence, we suggest f₁ is a radial mode (and further identified to be the first overtone in Section 4.6), as marked in Table A1 in the Appendix.

The frequencies f₂ ∼ f₇ are within the frequency range of typical δ Scuti stars. They are neither combination nor harmonic frequencies, so we marked them as independent frequencies in Table A1 in the Appendix. Figure 3 shows four frequencies (f₁, f₃, f₈, and f₉), where f₁ and f₃ are two real frequencies. We found that the shape of f₈ and f₉ are similar to that of f₁ and f₃, with no multiplet structure around the peaks. This suggests that f₈ and f₉ are not instrumental artifacts. For more detailed description about the real and alias frequencies of the Kepler data, we refer to Murphy et al. (2013). Moreover, the S/N of f₈ and f₉ in SC spectrum are 28.7 and 20.0 respectively, which is far greater than the standard of S/N = 4 given by Breger et al. (1993). Hence, f₈ and f₉ are two real frequencies.

**Figure 3.** Typical spectra of the real frequencies. Panels (a) and (b) show two real frequencies f₁ = 17.597 day⁻¹ and f₃ = 31.428 day⁻¹; panels (c) and (d) show two modulation terms f₈ = 0.065 day⁻¹ and f₉ = 1.693 day⁻¹, respectively.
Download figure:
Standard image High-resolution image

It is intriguing that 42 of the 85 frequencies form fourteen pairs of equidistant triplet structures. They are labeled as "T" in Figures 4 and 5, where the blue vertical dotted lines indicate the locations of the extracted peaks. It is clear that these equidistant frequency triplets consist of the equal spacing frequency f₈ (=0.065 day⁻¹) in Figure 4 and f₉ (=1.693 day⁻¹) in Figure 5, respectively. Moreover, f₈ and f₉ are not in the typical p-mode range of δ Scuti stars, and they are not combinations of other independent frequencies, either. Hence, we marked the frequencies f₈ and f₉ as f_m1 and f_m2 in Table A1 in the Appendix, respectively. In Table 2, we marked the triplets with the equidistant frequency f_m1 as T1 to T12 and the group with f_m2 as T13 and T14, sorted according to the increasing value of the central frequency, respectively.

**Figure 4.** Twelve pairs of equidistant frequency-triplet structures in the SC spectrum of KIC 9845907. The vertical blue dotted lines indicate the locations of the detected frequencies, where f_m1 = 0.065 day⁻¹ represents the spacing. We marked the triplets with f_m1 as T1 to T12 sorted according to the increasing value of the central frequency.
Download figure:
Standard image High-resolution image

**Figure 5.** Two pairs of equidistant frequency-triplet structures in SC spectrum of KIC 9845907. The vertical blue dotted lines indicate the locations of frequencies, where f_m2 = 1.693 day⁻¹ represents the equally spaced frequency. We marked the triplets with f_m2 as T13 and T14 according to Table 2.
Download figure:
Standard image High-resolution image

Table 2. Dominant Frequencies of the "T" Family in the SC Data (Denoted by f_i)

f_i	Frequency	Identification	Comment	f_i	Frequency	Identification	Comment
	(day⁻¹)				(day⁻¹)
8	0.0649(1)	f₈	f_m1	9	1.6927(5)	f₉	f_m2
34	5.3091(2)	f₃₄	T1	17	62.8562(6)	f₁₇	T8
33	5.3741(4)	f₃₄ + f_m1	T1	48	62.9211(7)	f₁₇ + f_m1	T8
58	5.2443(3)	f₃₄ − f_m1	T1	64	62.7913(6)	f₁₇ − f_m1	T8
21	13.8308(6)	f₂₁	T2	76	75.1444(1)	f₇₆	T9
22	13.8957(6)	f₂₁ + f_m1	T2	79	75.2092(7)	f₇₆ + f_m1	T9
28	13.7659(5)	f₂₁ − f_m1	T2	81	75.0793(9)	f₇₆ − f_m1	T9
15	29.7353(7)	f₁₅	T3	40	78.7607(8)	f₄₀	T10
36	29.8002(8)	f₁₅ + f_m1	T3	77	78.8256(8)	f₄₀ + f_m1	T10
32	29.6703(6)	f₁₅ − f_m1	T3	78	78.6958(8)	f₄₀ − f_m1	T10
3	31.42813(3)	f₃	T4	52	90.8987(7)	f₅₂	T11
10	31.4930(3)	f₃ + f_m1	T4	80	90.9639(1)	f₅₂ + f_m1	T11
11	31.3631(1)	f₃ − f_m1	T4	84	90.8339(8)	f₅₂ − f_m1	T11
23	49.0254(5)	f₂₃	T5	82	94.2843(9)	f₈₂	T12
29	49.0903(1)	f₂₃ + f_m1	T5	83	94.3493(4)	f₈₂ + f_m1	T12
38	48.9605(1)	f₂₃ − f_m1	T5	85	94.2194(9)	f₈₂ − f_m1	T12
46	57.5470(3)	f₄₆	T6	19	61.1635(1)	f₁₉	T13
53	57.6120(4)	f₄₆ + f_m1	T6	17	62.8562(6)	f₁₉ + f_m2	T13
72	57.4821(2)	f₄₆ − f_m1	T6	18	59.4707(5)	f₁₉ − f_m2	T13
19	61.1635(1)	f₁₉	T7	40	78.7607(8)	f₄₀	T14
35	61.2284(1)	f₁₉ + f_m1	T7	47	80.4535(8)	f₄₀ + f_m2	T14
54	61.0984(9)	f₁₉ − f_m1	T7	25	77.0680(3)	f₄₀ − f_m2	T14

Note. "T", f_m1, and f_m2 represent equidistant frequency-triplet structures and two equally spaced frequencies, respectively. We marked the equidistant frequency triplets of the group with f_m1 as T1 to T12 and the group with f_m2 as T13 and T14, sorted according to the increasing value of the central frequency, respectively.

Download table as: ASCII Typeset image

Moreover, we also found two pairs of quintuplet structures ("Q") in the SC frequency spectra. In Table 3, we marked the two pairs as Q1 and Q2, sorted according to the increasing value of the central frequency. Figure 6 shows these two pairs of quintuplet structure with the blue dotted lines representing the location of each extracted frequency. It is clear that the side peaks around f₁₉ (and f₄₀) in the frequency spectra of KIC 9845907 have two pairs of uniformly spaced triplets with intervals of f_m1 = 0.065 day⁻¹ and f_m2 = 1.693 day⁻¹, i.e., Q1 includes T7 and T13, while Q2 includes T10 and T14, respectively. To see clearly the central triplets with f_m1 in Figure 6, we refer to T7 and T10 in Figure 4. Note that four out of fourteen pairs of triplets can form quintuplet structures, which is very interesting and recognized for the first time in δ Scuti stars. We suggest these phenomena deserve further study, and we give a discussion on the possible explanations in Section 4.

**Figure 6.** Two pairs of quintuplet structures in SC spectrum of KIC 9845907. The vertical blue dotted lines indicate the locations of frequencies. We marked the two pairs as Q1 (upper panel) and Q2 (lower panel), sorted according to the increasing value of the central frequency, respectively. The intervals from the side peaks to the center are marked with f_m1 = 0.065 day⁻¹ and f_m2 = 1.693 day⁻¹, respectively. In the upper and lower panels, the zoomed-in view of the central triplets can be seen in Figure 4 (i.e., T7 and T10) for clarity.
Download figure:
Standard image High-resolution image

4. Discussion

In recent years, more and more regular frequency spacings have been found in several δ Scuti stars, thanks to the high-precision continuous photometric data provided by space telescopes. The identification of these pulsation frequency modes provides us with new clues to study these variable stars. For example, Bedding et al. (2020) studied 60 δ Scuti stars with regular frequency spacings and enabled mode identification by comparing with theoretical models.

The most interesting features for KIC 9845907 in this study are the regular frequency spacings, including equidistant frequency-triplet and quintuplet structures, with frequency intervals of f_m1 = 0.065 day⁻¹ and f_m2 = 1.693 day⁻¹. Two pairs of triplet structure overlap to form a quintuplet structure, hence, with the same frequency spacings (of f_m1 and f_m2). To explore the nature of these frequency structures, we first need to consider whether they arise from the known instrumental effects of Kepler, including: (1) the frequency of Kepler orbital period f_orb = 0.00268 day⁻¹, (2) the momentum desaturation of the reaction wheel f_reaction = 0.336 day⁻¹ (Van Cleve & Caldwell 2016), (3) the frequency induced by the data downlink f_downlink = 0.031 day⁻¹, and (4) that of Kepler rotation f_rot = 0.011 day⁻¹. The Rayleigh frequency resolution of the SC data is 0.001136 day⁻¹ for KIC 9845907. We found that the value of f_m1 was not equal to (1) and (2) within the Rayleigh frequency resolution; therefore, these two instrumental effects are excluded first. For the frequency f_downlink (=0.031 day⁻¹), f_m1 is about twice of it. To verify if f_m1 is caused by the data downlink, we only used a part of 32 day (=1/f_downlink) data to extract frequencies and still detected the f_m1, so this instrumental effect is excluded. For the frequency f_rot, f_m1 seems to be 6 times of it. If the equidistant structure in the spectra is caused by the instrument effect, the height of the side lobe frequencies on both sides of the peak should be the same. But the triplets with f_m1 (T1–T12) of KIC 9845907 are not like this. In addition, instrumental effect should be applied at all significant independent frequencies. For KIC 9845907, we only detected the equidistant frequency-triplet structure at one independent mode (i.e., f₃) and none at the others. And when using the software PERIOD04 for frequency extraction, we did not detect 1–5 times of the Kepler rotation frequency; these signals should be stronger than 6f_rot. Hence, f_m1 ≈ 6f_rot is probably a coincidence. For f_m2, we found that none of the known frequencies of instrumental effects from Kepler were equal to it within the Rayleigh frequency resolution.

4.1. Beating?

Breger & Bischof (2002) found that pairs of close-frequency pulsation modes with spacings less than 0.01 day⁻¹ were common by studying seven well-known δ Scuti stars. Breger et al. (2009) found that pulsation frequencies in δ Scuti stars are not distributed randomly and that many non-radial modes had frequencies near radial mode frequencies. These regularities were explained by mode trapping in the stellar envelope (Dziembowski & Krolikowska 1990), which explained the regularities in the amplitude spectrum in the δ Scuti star FG Vir (Breger et al. 2009).

However for KIC 9845907, the spacings of the close frequencies are f_m1 = 0.065 day⁻¹ and f_m2 = 1.693 day⁻¹, both of which are much larger than the typical value (i.e., 0.01 day⁻¹) in Breger & Bischof (2002). Moreover, there is no unresolved peak around the dominant frequencies in the residuals, and the overall distribution of the residuals is typical of noise. Hence, the equidistant triplet structures of KIC 9845907 are unlikely caused by beating.

4.2. Blazhko Effect?

Equidistant structures are often shown in the Fourier spectra of the Blazhko RR Lyrae stars, and the interval of the triplets is the same as the modulation frequency, which can be directly detected (Jurcsik et al. 2005; Kolenberg et al. 2006). For KIC 9845907 in our case the equidistant quintuplet structures in the SC spectrum are similar to that in Blazhko RR Lyr stars, and two modulation frequencies are detected clearly. These features imply that the quintuplet structures in KIC 9845907 may be related to the Blazhko effect.

Using the high-precision photometric data provided by Kepler, Blazhko-like effect has been found in some δ Scuti stars. For example, the double-mode HADS star KIC 10284901 shows two pairs of quintuplet structure, and its analysis suggests that the modulation term of the quintuplet structures might be related to the Blazhko effect (Yang & Esamdin 2019). The main characteristic of this effect is that the quintuplet structures appear around the fundamental and first overtone pulsation modes (i.e., F0 and F1). However, for KIC 9845907, the two pairs of quintuplet structures are not the case, that is to say, they appear around other modes, which are not similar to that in KIC 10284901. Moreover, in KIC 10284901 the ratio of the two modulation frequencies (f_m1 and f_m2) is nearly 1:2. However, in the case of KIC 9845907, the ratio of the two modulation frequencies (f_m1 = 0.065 day⁻¹ and f_m2 =1.693 day⁻¹) of the quintuplet structures is seriously deviated from 1:2. From these aspects, we rule out the possibility that the quintuplet structures are caused by the Blazhko effect, as observed in RR Lyrae stars.

4.3. Combination Mode Hypothesis?

For the equidistant frequency-triplet structures in δ Scuti stars, Breger & Kolenberg (2006) provided an explanation named the "Combination Mode Hypothesis." In this hypothesis, the highest amplitude mode v₁ and a real second mode v₂ are excited; then, the harmonics as well as the combinations of v₁ and v₂ may also occur. The simplest combination (e.g., v₃ = v₂ − v₁) is usually observed, and then a pair of frequency-triplet (i.e., v₃, v₁, and v₂) is formed around v₁ in the frequency spectra.

To test this possibility, we chose the triplet structure T4 (i.e., f₃, f₁₀ and f₁₁) in KIC 9845907 for further analysis since the amplitude of the central component (i.e., f₃) of T4 is the strongest. Under this hypothesis, the right component f₁₀ = 31.493 day⁻¹ in the equidistant frequency-triplet structure in the SC spectrum of KIC 9845907 is considered as a new independent mode. However, the ratio of f₁/f₁₀ = 0.559 is not within the typical range of the continuous radial period ratios (Stellingwerf 1979). This seems to rule out the possibility that f₁₀ belongs to a radial mode. If f₁₀ is assumed to be a non-radial mode, it would split into 2ℓ + 1 frequencies when the star rotates. However, it is not like this case, either, as f₁₀ does not exhibit any splitting structure in the frequency spectra. Consequently, f₁₀ in KIC 9845907 is not an independent mode.

4.4. Nonlinear Mode Coupling?

Different nonlinear effects can generate combination frequencies in the spectrum of pulsating stars (Bowman et al. 2016). Combination frequencies are common in δ Scuti stars, and it is of great importance to identify which peak is combination frequency and which is a real frequency as it can greatly simplify the frequency spectra (Kurtz et al. 2015). However, combination frequencies differ from the mode coupling, which are excited by the resonant interaction of pulsation modes in stars. Coupled frequencies are grouped into two families: child and parent modes, and this coupling of modes promotes energy exchange between different modes of the family (Nowakowski 2005). The frequencies and amplitudes of pulsating modes change with time due to nonlinear mode coupling. Hence, when we distinguish coupled modes from combination frequencies, we need to study the changes of frequency, amplitude, and phase for each family.

In the theory of nonlinear mode coupling, the child and parent modes are directly related in frequency and phase. The way to distinguish the parent and coupled modes is to examine the correlations in amplitude variability. Breger & Montgomery (2014) investigated the dominant modes excited in KIC 8054146, which they called the T family. They computed changes of amplitudes and phases and correlations between different frequencies that show qualitative similarities between Triplets 1 and 2 (see the Figures 2 and 3 in Breger & Montgomery 2014). Finally, they identified the coupled modes of the T family in KIC 8054146. In this work, following the method by Breger & Montgomery (2014), we investigated the correlations in amplitude and phase variability between different dominant pairs of equidistant triplet structures.

The 4 yr of available LC data of KIC 9845907 allow us to examine the correlations of the amplitude and phase changes of the dominant frequencies and help to detect their physical relationship and origin. This near-continuous LC data from Q0 to Q17 span 1471 days in total. To make a compromise between studying short-term amplitude and phase changes and obtaining excellent frequencies from larger time intervals, a 75 day interval was chosen. In the next step, average frequencies covering the entire Q0–Q17 quarters were determined. Each value of the frequencies is labeled in the upper left corner of each panel in Figure 7. Using these average frequencies, the amplitudes and phase were calculated for different modes in each 75 day interval. This is shown in Figures 7 and 8, respectively. In the last row we also show the results for modes at f_m1 = 0.065 day⁻¹ and f₁ = 17.597 day⁻¹ for comparison. The error bars are also plotted, obtained by Monte Carlo simulation in PERIOD04.

Figure 7 shows the amplitude variations of KIC 9845907. The panels in the top three rows represent the low, central, and high components (from top to bottom) of T2 (left) and T4 (right), respectively. The bottom panels are the amplitude variations of f_m1 and f₁, respectively. For the three components of T2, before 1000 days, the amplitude variations were relatively stable but then changed significantly (i.e., increasing or decreasing). However, for T4, the changes occur much earlier, at the time of 700 days. For f_m1, the amplitude change is relatively stable as a whole without obvious changes, and the frequency f₁ breaks the stability after 700 days. Then it tends to be stable after a significant decline. We note there is no similarity and synchronization of amplitude variations between T2 and T4, which is not similar to the case of KIC 8054146.

Figure 8 shows the phase variations of KIC 9845907 with the panels labeled the same as in Figure 7. For the low and high components in T2, there is no obvious change in the overall phase, but the central component has a significant decline after 600 days. For the phase variations of T4, the three components have different changes: the low component can increase and/or decrease, while the central is gradually increasing, and the high is relatively stable. f_m1 and f₁ are in a slowly increasing and stable state, respectively. Therefore, there is no similarity and synchronization between T2 and T4, either.

We also made a further quantitative calculation following the method provided by Breger & Montgomery (2014; see their Equation (5)) and confirmed that T4 is not the result of mode coupling between T2 and the radial mode f₁. We conclude that there is no sufficient evidence so far to support the possibility that the triplets are caused by nonlinear mode coupling.

4.5. Large Separation?

Observations have shown that many δ Scuti stars have regular frequency spacings in their rich pulsation spectra. There have been several studies searching for large separations (e.g., García Hernández et al. 2013; Bedding et al. 2020). This development follows the establishment of the relationship between the large separation in the low-order regions and the average density of the star from the modelings (e.g., Suárez et al. 2014) and the observations (García Hernández et al. 2015, 2017).

We used the methodology provided by García Hernández et al. (2013) and Ramón-Ballesta et al. (2021) to determine the large separation, Δν, of KIC 9845907. Three techniques were applied to the frequencies: (1) the FT, (2) the autocorrelation function (AC), and (3) the histogram of frequency differences (HFD). The result of KIC 9845907 is shown in the left panel of Figure 9, where the black, gray, and blue lines represent FT, HFD, and AC, respectively. All these transformations have been made using just the 30 highest amplitude modes and discarding those frequencies below 5 day⁻¹, and the amplitudes of the frequencies have been normalized to unity when doing all these transformations. If there exists a periodicity in the form of Dirac comb, then we will see a peak corresponding to Δν and others at the multiples in the AC (2 Δν, 3Δν, etc). On the contrary, in the FT we expect to find Δν and the submultiples (Δν/2, Δν/3, etc), just because of how the transform works. In conclusion, in the FT, we could not find Δν but just the submultiples (Δν/2 and/or Δν/3) while Δν appears in the AC and HFD.

For KIC 9845907, the left panel of Figure 9 shows a peak at around 5.25 day⁻¹ (=60.8 μHz, the red dotted–dashed line) in the AC and HFD. However, in the FT, it does not show a peak around Δν, but we can see Δν/2 (the high peak around 30 μHz) and Δν/3 (the high peak around 20 μHz). The absence of the peak of Δν in the FT is expected, although to identify it at least one submultiple is mandatory (e.g., Garcia Hernandez et al. 2009), and the presence of more submultiples indicates that the possible value of large separation (Δν) is around 60.8 μHz and not 30 μHz. Moreover, the large separation (∼5.1 day⁻¹) we obtained from the models (see Section 4.6 for more detail) is close to the observed one.

Using this value (60.8 μHz), the echelle diagram is shown in the right panel of Figure 9. Some frequencies appear darker as they are composed of two very close frequencies. It can be seen that the distribution of these modes is irregular, with additional ridges at various angles, indicating that the analysis method of using simple pulsation spectra (with ℓ = 0 and 1) is insufficient to interpret this echelle diagram. In fact, this phenomenon also appears in other δ Scuti stars. For example, in four δ Scuti stars (i.e., HD 37286, SAO 150524, TYC 8533-329-1, and β Pic), complex patterns were also detected in the echelle diagrams. Bedding et al. (2020) concluded that the complex echelle diagram might be related to the azimuthal order m. Therefore, only using the large separation 60.8 μHz cannot provide a reasonable explanation for the triple structures and the rich spectrum for KIC 9845907.

4.6. Stellar Rotational Splitting?

The theory of stellar oscillations in the spherical approximation states that each oscillation mode can be characterized by three spherical harmonics: number of nodes along the radius direction n, the spherical harmonic degree ℓ, and the azimuthal order m. If a star is rotating, the rotation can cause the non-radial oscillation mode to split into 2ℓ + 1 components in the inertial frame. And at first order in the perturbation theory, in a slowly rotating star, the 2ℓ + 1 frequencies of the non-radial mode are separated by almost the same spacing, corresponding to an average rotation frequency. In addition, if a star is in a binary system, the binarity would split all the modes, even the radial, because of the motion around the center of masses (Shibahashi & Kurtz 2012). This is not the case for KIC 9845907, as the frequency f₁, a radial order and the highest amplitude peak, did not split, implying that this star should be a single star. Hence, the rotational splitting of a single star was considered to account for the triplet structures in KIC 9845907. In T1–T14, the highest amplitude of the middle peak is f₃ (in T4), which is an independent frequency, while all other middle peaks are combination frequencies, so only T4 will be discussed in detail in this section. To verify whether T4 is caused by the rotational splitting of f₃, we used asteroseismic models to estimate the value of l for f₃.

We constructed a grid of evolutionary models of KIC 9845907 and calculated their corresponding frequencies using the submodule "pulse_adipls" of the Modules for Experiments in Stellar Astrophysics (MESA v10398; Paxton et al. 2011, 2013, 2015, 2018, 2019). In the grid, the stellar mass ranges from 1.5 M_⊙ to 2.0 M_⊙ with a step of 0.01 M_⊙, and metallicities range from 0.006 to 0.015 with a step of 0.001. For the helium abundance Y, we adopted Y = 0.249 + 1.33 Z as a function of Z. In addition, the classical mixing length theory of Böhm-Vitense (1958) with α = 1.9 (Paxton et al. 2013) was adopted. Each model in the above grid was evolved from the zero-age MS to the post-MS stage to calculate the pulsation frequencies of ℓ = 0, 1, and 2. Then, by using the method from Chen & Li (2019; i.e., Equation (5), χ² method), the goodness of fit can be obtained by comparing model frequencies with the observed frequencies f₁ and f₃. To select the best-fitting models, we chose a threshold of χ² = 0.0056 (corresponding to about 5 times the frequency resolution 1/ΔT) and limited the model results by combining the observed effective temperature (7700 K < T_eff < 8100 K, covering the values given by Kepler and TESS) and luminosity (14.45 < L/L_⊙ < 15.85, estimated from Gaia parallax). Finally, four candidate models are obtained, as described in the following.

Figure 10 shows the evolutionary tracks of the four candidate models on the H-R diagram, where different colored lines represent different models. The black rectangle marks the 1σ error box of the effective temperature T_eff and the luminosity L, while the diamonds denote the seismic models that can best reproduce the frequencies of the two modes f₁ and f₃. Table 4 lists the parameters of the four candidate models. The (ℓ, n) of f₁ of all candidate models is (0, 2), indicating that f₁ is the first overtone, which makes KIC 9845907 a δ Scuti star with the first overtone as the dominant frequency. For f₃, the (ℓ, n) of all candidate models is (1, 6), indicating that f₃ is a non-radial oscillation mode with ℓ = 1. Then, it may result into 2ℓ + 1 (i.e., 3) frequencies due to the rotational splitting, as shown in T4. Thus, we suggest equidistant triplet structures with f_m1 = 0.065 day⁻¹ (i.e., T1–T12) might be related to rotational splitting. For the T13 and T14 (as well as Q1 and Q2), we suggest that they are derived from combination of independent frequencies (see Table A1 in the Appendix). The mode identification for other independent frequencies deserves a thorough investigation, which is, however, beyond the scope of this paper.

With the value of the rotational splitting (i.e., f_m1 =0.065 day⁻¹), the rotational period of KIC 9845907 can be calculated, i.e., P_rot = 7.69 day. Using the formula ${v}_{\mathrm{rot}}=\tfrac{2\pi R}{{P}_{\mathrm{rot}}}$ and the radius of KIC 9845907 provided by TESS, R = 1.928 ± 0.067 R_⊙ (Stassun et al. 2019), the rotation rate of KIC 9845907 at the equator can be obtained, i.e., v_rot (v sin i) = 12.70 ${}_{-0.46}^{+0.42}$ km s⁻¹. Note that our result is similar to the rotational velocity (10.859 km s⁻¹) measured by Xiang et al. (2022) using low-resolution spectra and the analysis by Niemczura et al. (2017) using high-resolution spectroscopy. When compared with the average rotation (∼150 km s⁻¹) of the typical δ Scuti stars (Breger 2000), the rotation of KIC 9845907 suggests that it is a slow rotator. This is also in agreement with a symmetric structure of the split frequencies.

5. Summary

Based on the high-precision time-series photometric SC data from the Kepler, we analyzed the pulsations of KIC 9845907 and detected 85 significant frequencies (see Figure 2 and Table A1 in the Appendix), including the radial frequency f₁ = 17.597 day⁻¹, the non-radial frequency f₃ = 31.428 day⁻¹ (ℓ = 1), and two modulation terms (f_m1 = 0.065 day⁻¹ and f_m2 = 1.693 day⁻¹). In addition, f₁ is the first overtone mode, which makes KIC 9845907 a δ Scuti star with the first overtone as the dominant frequency.

Although equidistant frequency-triplet structure has been seen in other δ Scuti stars, the number of such occurrences in KIC 9845907 is unusual (fourteen pairs; see Figures 4 and 5 and Table 2). The modulation frequencies do not arise from known instrumental effects of Kepler. We also found quintuplet structures in the frequency spectra in KIC 9845907 (see Figure 6 and Table 3). We discussed several potential explanations, i.e., beating, the Blazhko effect, combination mode hypothesis, nonlinear mode coupling, large separation, and stellar rotational splitting for the equidistant structures in frequency spectra. Numerical results of asteroseismic models indicate this modulation with f_m1 might be related to the rotational splitting. We suggest searching for more δ Scuti stars with triplet and/or quintuplet structures; using high-precision space photometry would be helpful to explore its origin.

Table 3. Dominant Frequencies of the "Q" Family in the SC Data (Denoted by f_i)

f_i	Frequency	Identification	Comment	f_i	Frequency	Identification	Comment
	(day⁻¹)				(day⁻¹)
19	61.1635(1)	f₁₉	Q1	40	78.7607(8)	f₄₀	Q2
17	62.8562(6)	f₁₉ + f_m2	Q1	47	80.4535(8)	f₄₀ + f_m2	Q2
35	61.2284(1)	f₁₉ + f_m1	Q1	77	78.8256(8)	f₄₀ + f_m1	Q2
54	61.0984(9)	f₁₉ − f_m1	Q1	78	78.6958(8)	f₄₀ − f_m1	Q2
18	59.4707(5)	f₁₉ − f_m2	Q1	25	77.0680(3)	f₄₀ − f_m2	Q2

Note. "Q" represents quintuplet structures. We marked the two pairs as Q1 and Q2, sorted according to the increasing value of the central frequency, respectively.

Download table as: ASCII Typeset image

Table 4. Candidate Models with χ² ≤ 0.0056

Mass	Z	T_eff	log(L/L_⊙)	f₁ (ℓ, n)	f₃ (ℓ, n)	χ²
(M_⊙)	(dex)	(K)		(c days⁻¹)	(c days⁻¹)
1.65	0.007	7885	1.164	17.5405 (0,2)	31.4926 (1,6)	0.0037
1.68	0.008	7881	1.168	17.5386 (0,2)	31.4876 (1,6)	0.0035
1.70	0.008	7958	1.188	17.5364 (0,2)	31.4752 (1,6)	0.0031
1.72	0.009	7916	1.182	17.5170 (0,2)	31.4582 (1,6)	0.0037

Note. (ℓ, n) are the spherical harmonic degree and the radial order of the model frequency, respectively.

Download table as: ASCII Typeset image

Acknowledgments

We would like to thank the Kepler science team for providing such excellent data. This research is supported by the program of the National Natural Science Foundation of China (grant Nos. U1938104 and 12003020). A.G.H. acknowledges support from "FEDER/Junta de Andalucía-Consejería de Economía y Conocimiento" under project E-FQM-041-UGR18 by Universidad de Granada and from the Spanish State Research Agency (AEI) project PID2019-107061GB-064.

Appendix

In Table A1, we listed the 85 extracted frequencies (i.e., f₁ to f₈₅), their corresponding amplitudes, and S/N, as well.

Table A1. Multi-frequency Solution of the SC Light Curves of KIC 9845907 (Denoted by f_i)

f_i	Frequency	Amplitude	S/N	Comment
	(day⁻¹)	(mmag)
	Independent frequencies
1	17.597272(1)	39.049	99.4	radial
2	26.11889(7)	9.780	82.3	Independent
3	31.42813(3)	8.127	56.1	non-radial
4	21.2524(1)	5.606	67.8	Independent
5	29.0141(5)	3.219	51.2	Independent
6	27.0100(6)	2.435	22.8	Independent
7	24.4713(8)	2.119	39.7	Independent
8	0.0649(1)	1.321	28.7	f_m1
9	1.6927(5)	0.551	20.0	f_m2
	Combination frequencies
10	31.4930(3)	7.418	40.9	f₃ + f₈
11	31.3631(1)	4.787	26.1	f₃ − f₈
12	21.1456(6)	2.182	23.1	f₂ + 2f₅ − 2f₃ − 2f₈
13	35.1945(4)	1.855	54.7	2f₁
14	23.7585(6)	1.733	31.5	f₂ + f₅ − f₃ − f₈
15	29.7353(7)	1.623	25.1	f₃ − f₉
16	8.5216(2)	1.280	39.6	f₂ − f₁
17	62.8562(6)	1.248	58.3	2f₃
18	59.4707(5)	1.241	54.5	2f₃ − 2f₉
19	61.1635(1)	1.166	53.2	2f₃ − f₉
20	28.9573(1)	0.845	13.6	f₅ − f₈
21	13.8308(6)	0.799	18.9	f₃ − f₁
22	13.8957(6)	0.785	18.2	f₃ − f₁ + f₈
23	49.0254(5)	0.582	27.9	f₁ + f₃
24	26.8801(3)	0.581	23.8	f₆ − f₈
25	77.0680(3)	0.571	45.8	f₁ + 2f₃ − 2f₉
26	47.3326(5)	0.553	29.0	f₁ + f₃ − f₉
27	43.7161(6)	0.473	24.6	f₁ + f₂
28	13.7659(5)	0.472	12.1	f₃ − f₁ − f₈
29	49.0903(1)	0.459	21.5	f₁ + f₃ + f₈
30	24.3602(1)	0.439	8.3	f₇ − 2f₈
31	52.2379(9)	0.416	21.6	2f₂
32	29.6703(6)	0.403	6.3	f₃ + f₇ − f₂ − 2f₈
33	5.3741(4)	0.392	12.6	f₃ + f₈ − f₂
34	5.3091(2)	0.391	12.7	f₃ − f₂
35	61.2284(1)	0.389	18.5	2f₃ + f₈ − f₉
36	29.8002(8)	0.379	8.2	f₃ + f₇ − f₂
37	10.1756(1)	0.376	13.6	f₃ − f₄
38	48.9605(1)	0.374	17.5	f₁ + f₃ − f₈
39	24.4160(2)	0.364	6.8	f₂ − f₉
40	78.7607(8)	0.359	31.7	f₁ + 2f₃ − f₉
41	38.8497(8)	0.348	17.6	f₁ + f₄
42	11.4168(7)	0.335	12.3	f₅ − f₁
43	54.1515(2)	0.308	20.4	f₃ + 2f₇ − f₂ − 2f₈
44	27.8969(3)	0.299	9.6	f₁ + 3f₃ + 2f₈ − f₂ − 2f₅
45	44.6073(3)	0.297	17.7	f₁ + f₆
46	57.5470(3)	0.296	16.8	f₂ + f₃
47	80.4535(8)	0.293	28.7	f₁ + 2f₃
48	62.9211(7)	0.288	13.0	2f₃ + f₈
49	13.9535(1)	0.287	7.5	2f₁ − f₄
50	9.41278(8)	0.276	10.1	f₆ − f₁
51	4.4830(9)	0.275	9.9	f₃ + f₈ − f₆
52	90.8987(7)	0.266	32.0	3f₃ − 2f₉
53	57.6120(4)	0.262	15.2	f₂ + f₃ + f₈
54	61.0984(9)	0.261	12.2	2f₃ − f₈ − f₉
55	54.8187(5)	0.258	15.9	f₂ + f₆ + f₉
56	58.3731(7)	0.244	16.5	f₃ + f₆ − f₈
57	3.6551(2)	0.241	9.9	f₄ − f₁
58	5.2443(3)	0.238	7.5	f₃ − f₂ − f₈
59	10.2405(2)	0.236	7.8	f₃ + f₈ − f₄
60	55.1330(4)	0.228	13.7	f₂ + f₅
61	41.8733(7)	0.219	11.5	2f₃− f₁ − 2f₉
62	52.6805(3)	0.213	10.8	f₃ + f₄
63	3.2188(6)	0.212	7.5	f₇ − f₄
64	62.7913(6)	0.220	10.0	2f₃ − f₈
65	1.7576(5)	0.209	7.4	f₂ + 2f₈ − f₇
66	55.6641(9)	0.198	12.5	2f₁ + 4f₃ + 2f₈ − f₂ − f₄ − 2f₅
67	42.3981(8)	0.189	9.5	f₂ + f₄ + 2f₅ − 2f₃ − 2f₈
68	2.4788(8)	0.183	6.5	f₃ + f₈ − f₅
69	7.6695(6)	0.182	7.3	2f₃ + f₈ − f₂ − f₅
70	55.8442(7)	0.177	9.9	f₃ + f₇ − f₈
71	42.5049(1)	0.176	10.1	2f₄
72	57.4821(2)	0.176	8.2	f₂ + f₃ − f₈
73	55.8995(1)	0.175	9.8	f₃ + f₇
74	6.9567(4)	0.169	6.8	f₃ − f₇
75	45.2589(9)	0.167	9.4	2f₃ − f₁
76	75.1444(1)	0.128	14.2	f₁ + f₂ + f₃
77	78.8256(8)	0.126	11.3	f₁ + 2f₃ + f₈ − f₉
78	78.6958(8)	0.123	10.6	f₁ + 2f₃ − f₈ − f₉
79	75.2092(7)	0.096	10.8	f₁ + f₂ + f₃ + f₈
80	90.9639(1)	0.085	9.8	3f₃ + f₈ − 2f₉
81	75.0793(9)	0.075	8.0	f₁ + f₂ + f₃ − f₈
82	94.2843(9)	0.069	9.8	3f₃
83	94.3493(4)	0.069	9.7	3f₃ + f₈
84	90.8339(8)	0.062	7.3	3f₃ − f₈ − 2f₉
85	94.2194(9)	0.048	6.9	3f₃ − f₈

Download table as: ASCIITypeset images: Typeset image 1

KIC 9845907: A δ Scuti Star with the First Overtone as the Dominant Frequency and with Many Equidistant Structures in Its Spectrum

Article metrics

Author e-mails

Author affiliations

ORCID iDs

Dates

Abstract

1. Introduction

2. Observations and Data Reduction

3. Frequency Analysis