# Oscillation Mode Variability in Evolved Compact Pulsators from *Kepler* Photometry. I. The Hot B Subdwarf Star KIC 3527751

Weikai Zong<sup>1,2</sup>①, Stéphane Charpinet<sup>2</sup>①, Jian-Ning Fu<sup>1,7</sup>①, Gérard Vauclair<sup>2</sup>, Jia-Shu Niu<sup>3,4</sup>①, and Jie Su<sup>1,5,6</sup>①

## Abstract

We present the first results of an ensemble and systematic survey of oscillation mode variability in pulsating hot B subdwarf (sdB) and white dwarf stars observed with the original *Kepler* mission. The satellite provides uninterrupted high-quality photometric data with a time baseline that can reach up to 4 yr collected on pulsating stars. This is a unique opportunity to characterize long-term behaviors of oscillation modes. A mode modulation in amplitude and frequency can be independently inferred by its fine structure in the Fourier spectrum, from the sLSP, or with prewhitening methods applied to various parts of the light curve. We apply all these techniques to the sdB star KIC 3527751, a long-period-dominated hybrid pulsator. We find that all the detected modes with sufficiently large amplitudes to be thoroughly studied show amplitude and/or frequency variations. Components of three identified quintuplets around 92, 114, and 253 µHz show signatures that can be linked to nonlinear interactions according to the resonant mode coupling theory. This interpretation is further supported by the fact that many oscillation modes are found to have amplitudes and frequencies showing correlated or anticorrelated variations, a behavior that can be linked to the amplitude equation formalism, where nonlinear frequency corrections are determined by their amplitude variations. Our results suggest that oscillation modes varying with diverse patterns are a very common phenomenon in pulsating sdB stars. Close structures around main frequencies therefore need to be carefully interpreted in light of this finding to secure a robust identification of real eigenfrequencies, which is crucial for seismic modeling. The various modulation patterns uncovered should encourage further developments in the field of nonlinear stellar oscillation theory. It also raises a warning to any long-term project aiming at measuring the rate of period change of pulsations caused by stellar evolution, or at discovering stellar (planetary) companions around pulsating stars using timing methods, as both require very stable pulsation modes.

**Key words:** stars: individual (KIC 3527751) – stars: oscillations (including pulsations) – techniques: photometric

## 1. Introduction

Hot subdwarf and white dwarf stars, hereafter referred to as evolved compact stars, are typically faint blue objects with high surface gravity due to their small radii. White dwarfs correspond to the final stage of stellar evolution for most (~95%) stars in the universe. They cover a large region of the H-R diagram with effective temperatures ranging from ~3000 to 250,000 K and surface gravity from ~5.5 to 9.0 dex. When cooling down, they cross several instability strips, where they develop observable nonradial g-mode pulsations (Fontaine & Brassard 2008; Winget & Kepler 2008). Hot B subdwarfs (sdB) belong to the extreme horizontal branch, burning helium in the core. They have typical masses of ~0.47 M<sub>☉</sub>, effective temperatures ranging from ~20,000 to 40,000 K, and surface gravities log g ~ 5.0–6.2 dex (Heber 2009, 2016; Fontaine et al. 2012). A fraction of the sdB stars also develop nonradial oscillations in either acoustic (p-) or gravity (g-) modes, or both (Kilkenny et al. 1997; Green et al. 2003; Schuh et al. 2006), which are driven by the κ-mechanism induced by iron-group elements accumulating in the Z-bump region owing to radiative levitation (Charpinet et al. 1996, 1997; Fontaine et al. 2003). The oscillations observed in these

compact stars make them excellent candidates for probing their interior with asteroseismology. Most recent results on this front include the derivation of the core chemical stratification inside white dwarf stars and hot subdwarfs, as well as the determination of their fundamental parameters (see, e.g., Van Grootel et al. 2010, 2013; Giammichele et al. 2017).

Current seismic models rely on the linear (most often adiabatic) stellar oscillation theory, which proves very accurate in connecting the stellar structure with mode measured eigenfrequencies. In this framework, however, amplitudes and frequencies are predicted to be stable (e.g., Unno et al. 1989; Aerts et al. 2010), while significant variations of these quantities have long been suspected, e.g., in several white dwarfs and hot subdwarfs from ground-based photometric campaigns spanning different seasons (see, e.g., Kilkenny 2010; Vauclair et al. 2011; Barlow et al. 2017; Hutchens et al. 2017; Silvotti et al. 2017). However, measuring precisely these variations has remained out of reach until recently with high-precision, long-time-baseline observations carried out from space. The NASA spacecraft *Kepler* monitored more than 200,000 stars within a 105 deg<fnref n="2" /> field in the Cygnus–Lyrae region, some of them for up to ~4 yr continuously, providing unprecedented high-quality photometric data for asteroseismology (Gilliland et al. 2010) and exoplanets (Borucki et al. 2010).

<footnote n="1">¹Department of Astronomy, Beijing Normal University, Beijing 100875, People's Republic of China; weikai.zong@bnu.edu.cn, stephane.charpinet@irap.omp.eu, jnfu@bnu.edu.cn</footnote>

<footnote n="2">²IRAP, Université de Toulouse, CNRS, UPS, CNES, 14 avenue Edouard Belin, F-31400, Toulouse, France</footnote>

<footnote n="3">³Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China</footnote>

<footnote n="4">⁴School of Physical Sciences, University of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049, People's Republic of China</footnote>

<footnote n="5">⁵Yunnan Observatories, Chinese Academy of Sciences, P.O. Box 110, Kunming 650216, People's Republic of China</footnote>

<footnote n="6">⁶Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, P.O. Box 110, Kunming 650216, People's Republic of China
Received 2017 November 18; revised 2018 January 1; accepted 2018 January 2; published 2018 January 26</footnote>

<footnote n="7">⁷ Corresponding author.</footnote>
It has also offered new opportunities to identify and characterize mode amplitude and frequency modulations (FMs) in pulsating stars (see, e.g., Zong et al. 2016a, 2016b).

During *Kepler*'s survey phase, 113 evolved compact stars were monitored, and low-amplitude oscillations were originally detected in 15 sdB stars: one short-period, V 361 Hya type pulsator, and 14 long-period, V 1093 Her type pulsators (Østensen et al. 2010, 2011b). Three additional long-period (g-mode) pulsating sdB stars were later identified in the open cluster NGC 6791 (Pablo et al. 2011; Reed et al. 2012), and short-period (p-mode) oscillations were confirmed in another sdB star, which was formerly identified as a short-period transient object (Østensen et al. 2010). Therefore, the number of sdB pulsators monitored by *Kepler* is 18. Quite notably, there was no pulsating white dwarfs found among the 113 evolved compact objects proposed for the survey phase. The first pulsating white dwarf star, a hot hydrogen-deficient one (a DB-type white dwarf), was discovered about 2 yr after *Kepler*'s launch (Østensen et al. 2011a). The second one is a hydrogen-atmosphere (DA) pulsating white dwarf, first identified from ground-based observations by Hermes et al. (2011). Four additional pulsating DA white dwarfs were discovered afterward by *Kepler* (see Greiss et al. 2016, and references therein), raising the count to only six objects of this type.

Most of the compact pulsators have been continuously monitored by *Kepler* for longer than 2 yr, leading to a series of interesting breakthroughs, such as the use of three independent methods to determine the orbital parameters in the sdB star KIC 11558725 (Telting et al. 2012), the discovery of small Earth-size planet candidates in tight orbits around the sdB stars KIC 05807616 (Charpinet et al. 2011) and KIC 10001893 (Silvotti et al. 2014), the longest light curve obtained so far for a pulsating white dwarf (Østensen et al. 2011a), or the detection of the subsynchronous sdB+dM close binary system KIC 7664467 (Baran et al. 2016). Moreover, the long and consecutive photometric data now allow us to study precisely amplitude and frequency variations, using sliding Fourier transform (sFT) or sliding Lomb–Scargle periodogram (sLSP) techniques (see, e.g., Reed et al. 2014; Zong et al. 2016a, 2016b). Among the various mode behaviors reported, Kern et al. (2017) found, for instance, that frequency convergence occurs in all rotationally split $l \ge 2$ multiplets of the sdB pulsator KIC 2697388, i.e., a long-term trend of unknown origin decreases the frequency spacing in those multiplets.

Baran et al. (2015) attempted to measure the rate of period change of an apparently stable mode in the sdB binary KIC 8302197, but they could not find any correlations with the presence of the companion or with expected secular evolution from the O–C technique. Many frequencies have also been thoroughly analyzed in the sdB star KIC 10670103 by Reed et al. (2014), who found that many rotational multiplets show clear amplitude variations. Deeper analyses of amplitude and frequency variations occurring in rotation triplets observed in the pulsation spectra of the white dwarf star KIC 08626021 and the sdB star KIC 10139564 suggest that these variations are linked to nonlinear resonant mode coupling mechanisms (Zong et al. 2016a, 2016b), which in turn could be a natural explanation for various types of AMs and FMs (see, e.g., Goupil et al. 1998). In this framework, the modulations should exhibit diverse patterns due to the complexity of nonlinear couplings between the interacting modes. Hence, at this stage, we feel that a detailed investigation aiming at measuring

precisely the variability of all pulsation modes seen in evolved compact pulsators is a necessary step to clarify the mechanisms triggering these variations and to eventually provide further constraints on nonlinear stellar pulsation theories.

In this paper, we present the first results of our ongoing investigation of mode variability in all pulsating white dwarf and sdB stars observed with *Kepler*. We focus on a particular sdB star, KIC 3527751 (or 2MASS J19033701+3836126), which has the advantage of being free of light contamination from nearby stars. This star has a mean brightness of $Kp = 14.859$ (*Kepler* magnitude), and data were taken from the Mikulski Archive for Space Telescopes (MAST).<fnref n="8" /> This star was discovered to be a hybrid pulsator with 38 g- and 3 p-mode oscillations detected in the initial, 1-month long, *Kepler* survey data (Q2.3; Reed et al. 2010). Atmospheric parameters given by Foster et al. (2015) indicate a star with effective temperature $T_{eff} = 27, 818 \pm 163$ K and surface gravity $\log g = 5.35 \pm 0.03$ dex. The full data set of ~1148 days was initially studied by Foster et al. (2015), who found a very rich pulsation spectrum with 251 periodicities. Furthermore, identified rotationally split multiplets could indicate a differential rotation between the envelope of the star (~15 days) and its core (~43 days). They also emphasize that many frequencies, in particular for some suspected high-$l$ modes, show amplitude and frequency variations as revealed by sFT diagrams. KIC 3527751 is therefore an excellent target to start describing more quantitatively the pulsation variability in evolved compact pulsators.

The paper is organized as follows: We present in Section 2 a recap of the mathematical framework allowing us to understand the effect of amplitude, frequency, and phase modulations (PMs) on periodic signals analyzed in Fourier space. We provide in Section 3 various simulations along these lines that readily connect with real observations with *Kepler*. In Section 4, we provide our detailed analysis of the modulations detected in the dominant pulsation modes of KIC 3527751, which we discuss in Section 5. A summary and conclusions are provided in Section 6.

## 2. Amplitude, Frequency, and Phase Modulations

The Fourier analysis is the prevalent and standard method to analyze periodic stellar pulsation signals. However, techniques based on this method, such as the widely used prewithening approach, are usually based on the assumption that this signal is strictly periodic and nonvarying in phase and amplitude. Since current evidence shows that this is very often not the case, we investigate briefly the signature of departures from strict periodicity. In this section, we recall some mathematical properties of sinusoidal signals (e.g., pulsation modes) with amplitude modulation (AM), FM, and PM when analyzed in Fourier space.

Oscillation modes, within the framework of linear stellar oscillation theory, have stable frequencies and amplitudes (e.g., Cox 1980; Unno et al. 1989). The observed flux variations can be represented by a sum of sinusoidal waves,

$$z = \sum_{j} A_{j} \sin[2\pi(\omega_{j}t + \phi_{j})], \quad (1)$$

where $A_j$, $\omega_j$, and $\phi_j$ are the (constant) amplitudes, frequencies, and phases of the oscillation modes, respectively.

<footnote n="8">⁸ http://archive.stsci.edu/kepler</footnote>
However, if one considers second- and higher-order terms of the perturbation treatment to derive the equations of stellar oscillation, thus going beyond the linear theory, (nonlinear) oscillations may be described as

$$z = \sum_{j} A_{j}(t)\sin[2\pi(\omega_{j}t + \phi_{j})], \qquad (2)$$

where the amplitude $A(t)$ is no longer constant (e.g., Goupil & Buchler 1994). In nonlinear oscillation theories involving resonant modes, the amplitude equations suggest that stable amplitudes, or periodic, irregular, or even chaotic modulations, can be produced (Buchler et al. 1995, 1997).

Concretely, we first consider the simpler case where AMs are regular mathematically, meaning that the AM can be represented as an infinite sum of sinusoidal waves:

$$A_{j}(t) = \sum_{i=1}^{\infty} A_{ji} \sin[2\pi(\omega_{i}t + \phi_{i})] + A_{j}. \qquad (3)$$

Equation (3) is the Fourier series expansion of the AM itself. For simple AMs, the infinite sum of the Fourier series can be truncated, for instance, in the case of a single-tone sinusoidal AM (see, e.g., AMs in a DB white dwarf star; Zong et al. 2016b). Furthermore, AMs are inevitably observed with a finite precision related to observational constraints. Therefore, in practice, we can always use a series of finite terms to represent stellar oscillations with regular AMs. These can then be rewritten, using sum and difference rules of trigonometry, as

$$z = \sum_{j} \sum_{k} A_{jk} \sin[2\pi(\omega_{jk}t + \phi_{jk})]. \qquad (4)$$

Here, $\omega_{jk} = \omega_j + \omega_k$ and $\omega_k$ is very small compared to $\omega_j$ (the oscillation frequency) because the timescale of AMs is usually much longer than that of the oscillation itself. Equation (4) indicates that oscillation modes with regular AMs should produce structures in Fourier space where main peaks are accompanied by close sidelobes. This means that the AMs themselves can create significant signal. In the case of complex AMs, such as irregular or chaotic patterns, Equation (4) is still applicable, but it will require a much larger number of terms in the Fourier expansion to be reproduced down to the observation noise level, hence increasing the number of sidelobes.

In the context of stellar oscillations, modes can also exhibit FMs (or “chirps”), which may also be induced by, e.g., nonlinear resonant mode interactions (see, e.g., Moskalik 1985; Buchler et al. 1997; Zong et al. 2016a) or stellar activity on the surface (see, e.g., Salabert et al. 2015). Oscillations with FMs can be described as a temporal function,

$$\begin{aligned} z &= \sum_{j} A_{j} \sin \left\{ 2\pi \left[ \int_{0}^{t} \omega_{j}(\tau) d\tau + \phi_{j} \right] \right\} \\ &= \sum_{j} A_{j} \sin \left\{ 2\pi \left[ \int_{0}^{t} (\omega_{j} + \omega_{j,\Delta} x_{j}(\tau)) d\tau + \phi_{j} \right] \right\} \\ &= \sum_{j} A_{j} \sin \left[ 2\pi\omega_{j}t + \phi_{j} + 2\pi\omega_{j,\Delta} \int_{0}^{t} x_{j}(\tau) d\tau \right], \end{aligned} \qquad (5)$$

where the frequency $\omega_j(t)$ is the instantaneous frequency and $\omega_j$ is the linear stable (carrier) frequency, $\omega_{j,\Delta}$ defines the scale of frequency variations, and $x_j(t)$ is the pattern of frequency

variation for mode j. Usually, $\omega_j \gg \omega_{j,\Delta}$ and all the energy of mode j is contained within $\omega_j \pm \omega_{j,\Delta}$. In Fourier space, a continuous range of frequencies is required to accurately represent the frequency-modulated modes. Strictly speaking, the spectrum of an FM oscillation has components expanding infinitely, although their amplitudes decrease and higher-order components are often neglected (see, e.g., Thomas & Sekhar 2005).

In order to discuss specific examples of interest, we first introduce oscillations with a single-tone FM. In this case, the carrier frequency follows a sinusoidal variation over time. The integral can then be expressed as

$$\int_{0}^{t} x_{j}(\tau) d\tau = \frac{\cos(2\pi\omega_{j,m}t)}{2\pi\omega_{j,m}}, \qquad (6)$$

where $\omega_{j,m}$ is the modulation frequency. A set of stellar oscillation modes can now be written as

$$z = \sum_{j} A_{j} \sin \left[ 2\pi\omega_{j}t + \phi_{j} + \frac{\omega_{j,\Delta}}{\omega_{j,m}} \cos(2\pi\omega_{j,m}t) \right]. \qquad (7)$$

The above equation can be further developed into Bessel functions of the sidelobe number and modulation index $h = \omega_{j,\Delta}/\omega_{j,m}$, which allow us to mathematically describe this type of FM in the Fourier domain. If the modulation index $h \ll 1$, the problem corresponds to narrowband FMs; otherwise, it is wideband FMs.

Another simple type of FM is the linear changing frequency, also called quadratic sine waves,

$$\int_{0}^{t} x_{j}(\tau) d\tau = \frac{1}{2T} t^{2}, \qquad (8)$$

where $T$ is the entire observation duration. The spectrum of this linear FM (chirp) signal can be calculated with the help of Fresnel integrals. The linear chirp spectrum shows a profile of Fresnel ripples when the time-bandwidth product, $T \cdot \omega_{j,\Delta}$, is small, while it tends to a more rectangular profile for larger values. The Fourier signature of the two simple FMs discussed above can be seen in Figures 2 and 3, showing simulations discussed in the next section.

An intrinsic property of FMs in stellar oscillations is that the modulation timescale is typically much longer than the pulsation period, i.e., $\omega_{j,m} \ll \omega_j$. FMs can therefore be expanded around the carrier frequencies, $\omega_j$, into a sum of infinite terms. Using sum and difference rules of trigonometry, and considering noise limits inherent to observations, many kinds of FMs can ultimately be truncated into finite terms and rearranged into forms very similar to Equation (4). This suggests that, in Fourier space, FMs can also induce many significant signals clustered in a narrow frequency range.

PMs, the last type of modulation pattern for oscillation modes, can be induced by external causes, such as the presence of companion objects around pulsating stars (e.g., Silvotti et al. 2007). When adding a time-dependent phase term, $\phi_{j,m}(t)$, into Equation (1), one has

$$z = \sum_{j} A_{j} \sin\{2\pi[\omega_{j}t + \phi_{j} + \phi_{j,m}(t)]\}. \qquad (9)$$

In the above equation, the PMs can be treated as special kinds of FMs, since an FM is given by the time derivative
<figure><img src="image_1.png" /><figcaption>**Figure 1.** Simulations of eight frequencies with AMs in Fourier space (see text for details). Top panels: sLSP. Bottom panels: LSP of amplitude modulated signals, where the horizontal dashed lines denote the 5.6σ detection threshold and the dashed (blue) curve is the residual after prewhitening the most significant signal. Note that the frequency $F_4$ is stable (no amplitude variation) and is used as a reference.</figcaption></figure>

of the phase, $\omega_m = d\phi_m/dt$. Comparing Equation (5) with Equation (9), it is obvious that frequency and phase variations cannot be disentangled, if we have no a priori knowledge on any of those modulations.

The above equations describe simple AMs, FMs, and PMs of oscillation modes taken separately. In real situations, the modulations can result from a combination of those and can be expressed by the general formula

$$z = \sum_j A_j(t) \sin \left\{ 2\pi \left[ \int_0^t \omega_j(\tau) d\tau + \phi_j(t) \right] \right\}, \quad (10)$$

where all the frequencies, amplitudes, and phases can be time-dependent or constant physical quantities. In the Fourier domain, modulated oscillations are thus represented by many sidelobes around the carrier frequency, i.e., the linear stable frequency. Sidelobes may be very close to each other and overlap in frequency range if they cannot be well resolved considering the usually finite duration of observations, generating a broadened frequency structure.

### 3. Simulations

In this section, we illustrate through a series of simulations the various imprints of amplitude-, frequency-, and phase-modulated signals in Fourier space. By setting these simulations to mimic real data from *Kepler*, the main idea is to provide a set of examples that can be directly comparable, qualitatively, with the behavior of real oscillations modes. All artificial light curves were constructed with added white Gaussian noise with $\sigma = 4$ ppt (parts per thousand). This corresponds to the typical photometric precision for a faint sdB star. The sampling time was also set to 58.85 s, identical to *Kepler*'s short-cadence mode. The generated light curves were set to be about 1000 days long.

We, again, begin with the simplest case where only AMs are being considered. We injected seven signals with amplitude

variations and one stable frequency (as a reference) into the first generated light curve. The software FELIX<fnref n="9" /> was then used to perform the Fourier analysis and extraction of the frequencies, as one would normally do with real observations.

Figure 1 shows the Lomb–Scargle periodograms, both in time-frequency analysis mode (sLSP) and in full light-curve analysis mode (LSP), for the eight injected frequencies. Frequencies $F_1, F_2$, and $F_3$ are pure sinusoidal with different varying amplitude patterns, as shown from the sLSP. The sLSP is constructed with time steps of 20 days and a window width of 200 days (see details for setting these parameters in Zong et al. 2016a, 2016b). For periodic AMs, all the AM periods are set to 502 days, about half of the simulated observations. These are therefore resolved in these artificial data and produce the small frequency spacing seen in the LSP. Amplitudes of sidebands reflect the magnitude of the AM itself. The frequency $F_4$, for its part, is set to be stable with a constant amplitude of 1 ppt and provides a reference. The other four frequencies $F_5, F_6, F_7$, and $F_8$ are injected with various AM patterns: $F_5$ has a linearly increasing amplitude, $F_6$ amplitude first linearly decreases and then increases, and $F_7$ and $F_8$ have parabolic amplitude variations, with their amplitude decreasing under the detection threshold in the last half and first half of the light curve, respectively. These variation patterns are all illustrated in the sLSP. We find that the residuals (except for $F_3$ and $F_4$), after prewhitening the main peak, show symmetric structures near the extracted frequency. Hence, close frequencies often detected from Fourier analyses may be an indicator of AMs of one real carrier frequency instead of real signal. The injected frequencies represent rather simple AMs that can be described by a few sinusoidal waves. The modulations may be more complex in real pulsating stars and could possibly require many sinusoidal harmonics (see, e.g., Zong et al. 2016a).

<footnote n="9">⁹ Frequency Extraction for LIghtcurve eXploitation is developed by one of us, S.C., and is optimized to analyze high-quality data from space missions (see, e.g., Charpinet et al. 2010, 2011; Zong et al. 2016a, 2016b).</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 2.** Similar to Figure 1, but for simulations of five signals with FMs (see the text for details).</figcaption></figure>

The signature of FMs in five injected oscillations is shown in Figure 2. Two “artificial” modes, $F_9$ and $F_{10}$, are set with linearly increasing frequencies at different scales, 0.1 and 0.4 µHz, respectively. These clearly appear in the sLSP and influence the width of the hyperfine frequency structure in the LSP. We note that the amplitudes of injected “modes” as measured in the LSP are significantly reduced by the broadening due to the FMs. Moreover, the signal has a structure similar to Fresnel ripples, as expected. Different FM patterns are injected in another two “modes,” $F_{11}$ and $F_{12}$, which incorporate a resolved sinusoidal modulation of the frequency of period 502 days with different scales, 0.1 and 0.3 µHz, respectively. The FMs are clearly resolved from the sLSP. However, their corresponding LSPs present more complicated structures reminiscent of the spectrum profile of Bessels’s functions, and the amplitudes, again, are reduced by the broadening of FMs. The frequency $F_{13}$ has an FM pattern of polynomial form that can be well fitted by a series of sinusoidal waves. All these simple FMs can generate many close sidebands covering very narrow regions whose width indicates the amplitude scale of the FMs. Real FMs detected in pulsating stars may be the combination of several simple patterns or may be complex. Therefore, the resolved structure in LSP may present very complicated configurations.

We do not show simulations of PMs here because integrals of FMs can be treated as PMs. We instead finish this section with some examples having both AMs and FMs. Figure 3 shows four “artificial” modes with simple patterns of AMs and FMs. The $F_{14}$ frequency is a signal with a linearly increasing amplitude and a sinusoidal FM. The corresponding structure in the LSP contains many close sidelobes, among which some are well resolved. The $F_{15}$ frequency has a parabolic AM pattern at first and vanishes in the late portion of the light curve. The $F_{16}$ frequency is set with a linearly increasing frequency and a sinusoidal AM. The $F_{17}$ frequency has a polynomial FM and a sinusoidal AM. Both of the corresponding structures in the LSP are well resolved as the composition of two close modulated frequencies. We find that modes with combined AM and FM show complex configurations where the width of the frequency bands possibly indicates the scale of the FM.

To complete these simulations, we emphasize that the injected artificial signals can also be recovered using the usual prewhitening process based on the technique of Fourier analysis, as

<figure><img src="image_3.png" /><figcaption>**Figure 3.** Similar to Figure 1, but for simulations of four signals with AMs and FMs (see the text for details).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 4.** Example of the prewhitening process applied to an injected frequency with AM and FM patterns similar to $F_{14}$. Top panel: sLSP for that artificial signal; lower panels (from top to bottom) present residuals after prewhitening of significant frequencies, whose positions are indicated by filled circles. The red solid (blue dashed) line refers to the detection threshold of 5.6 × (5.0×) the local median noise.</figcaption></figure>

implemented in the software FELIX. To illustrate this point, we again take an injected frequency with AM and FM patterns similar to that of $F_{14}$, but with a somewhat longer FM timescale of $P_m \sim 950$ days, as shown in the top panel of Figure 4. Lower panels of this figure show the prewhitening sequence of 12 close
<figure><img src="image_5.png" /><figcaption>**Figure 5.** Comparison between the measured amplitude and frequency as a function of time (black circles; see the text for details) and the injected signal (see Figure 4). Measurement errors are comparable to the size of the symbols themselves.</figcaption></figure>

peaks in a frequency range of ±0.04 µHz generated by this mode and its AM/FM modulations, all being above the 5.6σ threshold. All these peaks must indeed be removed in order to leave no significant residual in this frequency range. Figure 5 reveals that the measured values of both amplitude and frequency as a function of time match very well the injected modulation pattern. These measured values are obtained, following Zong et al. (2016a, 2016b; see also Section 4), by applying a running filter window of 120-day width translated by time steps of 30 days throughout the entire light curve. Each window is then analyzed as an independent light-curve subset in which signal is prewhitened and its characteristics (frequency, amplitude, and phase) measured. As a final check, we have also reconstructed different synthetic light curves composed with the first 1, 4, 8, and 12 prewhitened frequencies (by order of their amplitude; see lower panels of Figure 4) and compared them with the injected one. Figure 6 illustrates that the reconstructed light curves are gradually approaching the injected signal as the number of frequencies considered increases. In this figure, the comparison is done at three different epochs, namely, at the beginning, middle, and end of the mimicked observations. These tests demonstrate that (1) the AMs and FMs can indeed be precisely measured by prewhitening various parts of the entire light curve and (2) a sum of close frequencies can accurately represent the light curve of a modulated signal.

The simulations presented in this section involve simple modulation patterns of the amplitude and/or frequency (phase). We could easily create further simulations with more complicated variations. Our main point, however, was to illustrate that close sidebands or side peaks surrounding frequencies in real data can be induced by such amplitude/FMs and may not correspond to additional independent oscillation modes. As will be documented in the following section and in subsequent papers of the series, such patterns are found to be very common and diverse in compact pulsators. In this context, the sLSP provides very important clues for identifying real frequencies in the LSP, in particular when dealing with the long duration of *Kepler* photometry. Extracting securely the real frequencies associated with eigenmodes is of course essential, as these are key input data

<figure><img src="image_6.png" /><figcaption>**Figure 6.** Comparison of the light curve with the injected signal (see Figure 4), with the light curves reconstructed by using 1, 4, 8, and 12 frequencies extracted with FELIX. The comparison is done at different epochs corresponding to the beginning (top panel), middle (middle panel), and end (bottom panel) of the mimicked observations.</figcaption></figure>

for seismic modeling. Furthermore, the identification and characterization of the modulations themselves are important for the study of nonlinear interactions between modes, such as those triggered, for instance, by resonant couplings. With the output of these simulations in mind, we can now examine the mode behaviors observed in real pulsators.

# 4. The Pulsating Hot B Subdwarf Star KIC 3527751

## 4.1. *The Frequencies Revisited*

The pulsating sdB star KIC 3527751 was observed by *Kepler* in short-cadence (SC) exposure mode during quarter Q2.3 and from Q5.1 to Q17.2. The assembled light curves were obtained from MAST, from which raw data, corrected data, and target pixel data can be downloaded. These archived data were processed through the standard *Kepler* Science Processing Pipeline provided by Jenkins et al. (2010). In the following, we consider the main campaign of continuous observations from Q5.1 to Q17.2 and neglect the (1-month) Q2.3 light curve, which is well disconnected in time and would introduce a large and detrimental gap for our analysis.

The full time series was constructed from the “raw” light curve of each quarter instead of the “corrected” data used previously in Zong et al. (2016a, 2016b). Corrected data include a notable correction of the flux from the estimated contamination of nearby stars (a factor that varies from one star to another; this factor is actually zero for KIC 3527751). As it turns out from some tests that we have carried out, this correction can slightly distort the data distribution for some targets and can have undesirable effects for our specific purposes. Our tests show that amplitude differences between raw and corrected data can reach up a few percent in some extreme cases (W. Zong et al. 2018, in preparation). Although this does not affect the general form of AM patterns, precise measurements are necessary to feed future
<figure><img src="image_7.png" /><figcaption>**Figure 7.** Top panel: condensed representation of the full *Kepler* light curve of KIC 3527751 (amplitude as the residual relative to the mean brightness intensity of the star vs. time in Barycentric Julian Date) covering from Q5.1 to Q17.2. Bottom panel: close-up view showing 10 days of the *Kepler* light curve by 1-day slices. At this scale the oscillations are clearly seen.</figcaption></figure>

works on nonlinear modeling. We thus recommend using raw data as the best starting point for extracting the modulation patterns from the data. Each 1-month light curve was individually detrended to correct for residual instrumental drifts by performing a sixth-order polynomial fit. Then, data points differing significantly from the local standard deviation of the light curve were removed by applying a running 3.5σ clipping filter three times. We note that the latter operation decreases very slightly the overall noise level in the Fourier spectrum but has no incidence on the measured frequencies.

The top panel of Figure 7 shows the fully assembled light curve of KIC 3527751, starting from BJD 2,455,276.5 and ending at BJD 2,456,424, hence covering ~1147.5 days with a duty cycle of ~91%. The bottom panel of that figure expands a 10-day portion of the data, where the multiperiodic low-amplitude variations dominated by periodicities of a few hours are clearly visible. The corresponding LSP (Scargle 1982) confirms the presence of these modulations. The LSP shows that significant signal is found (Figure 8) from the *g*-mode region at low frequency (from ~80 µHz) to the *p*-mode domain at high frequency (up to ~3700 µHz). This identifies KIC 3527751 as a hybrid sdB pulsator whose oscillations are dominated by *g*-modes. The formal frequency resolution achieved with the light curve is about 0.010 µHz (1/T, T ~ 1147.5 days).

The frequencies were extracted using a standard prewhitening and nonlinear least-squares fitting technique (Deeming 1975) that is implemented in FELIX. We extracted signal until amplitudes decrease below a given threshold, which is set to 5.6σ above the local average noise level (see details in Zong et al. 2016a, 2016b). In practice, suspected frequencies were also prewhitened down to 5σ, to allow for a full comparison

with the frequencies provided by Foster et al. (2015) for that star. Our list of extracted signals is provided in Table 2, along with their fitted attributes, which include frequency in µHz, period in seconds, amplitude in ppm (parts per million) of the mean brightness, and signal-to-noise ratio (S/N) of the detection, as well as their associated errors σ<sub>f</sub>, σ<sub>p</sub>, and σ<sub>A</sub>. Some of this information is repeated in Table 1 for the relevant modes, since we hereafter focus on a particular subset of the observed frequencies whose amplitudes are sufficiently high for our purpose. The “ID” column (Tables 1 and 2) uniquely identifies a detected frequency with the number that indicates its rank by order of decreasing amplitude.

We have detected 204 clear frequencies that come out well above the 5.6σ threshold, including many rotational multiplets (the exact number is not given here, due to the difficulty in identifying the degree number *ℓ* for all frequencies). For the purpose of comparing with Foster et al. (2015), we also extracted peaks whose amplitude is above 5.0σ, leading to an additional set of 43 frequencies, but which we consider much less secure. Among these 247 frequencies, we found 22 periodicities that could be linked to other independent frequencies through linear combinations. In summary, we have securely identified 211 frequencies that are located in the low-frequency *g*-mode region, six are in the high-frequency *p*-mode domain, and eight are found in between.

## 4.2. Amplitude and Frequency Modulations

The detected modes with relatively large amplitudes are excellent candidates to characterize AMs and FMs and their correlations, as well as the physical mechanisms behind them. We, in practice, focus on modes with average amplitudes down to ~20 ppm, corresponding to ~8σ detection (see the full list provided in Table 1). This concerns about 60% of all frequencies. Most of these frequencies are components belonging to multiplets caused by rotation, with an associated frequency spacing of ~0.12 µHz (for *ℓ* = 1 modes). Therefore, in order to have the resolution to well separate components of these multiplets, the sliding and filtering windows were chosen to be 200 days in width, and time steps for building the sLSP were set to 20 days. The sLSP provides an overall view of any modulation in amplitude and/or frequency occurring for the interesting modes. The precise pattern of the AM and FM can then be obtained by prewhitening various parts of the light curves, as illustrated in the previous section. For that purpose, we divide the 38-month data of KIC 3527751 into 101 segments, with beginning time changing by steps of 10 days, each containing 200 days of data, except for the last six segments at the end of the observation. This setup is the optimal compromise to obtain relative smooth and precise measurements of the variation patterns, with their associated errors. These parameters (window size and time steps) may be set differently for the various stars that we will study in this series of papers.

Each light-curve segment is then analyzed with FELIX and frequencies prewhitened down to a threshold of 6σ. If close peaks within the corresponding frequency resolution (~0.05 µHz) are detected, we keep the highest peak as the measured value for this mode. The extracted AM and FM are presented as follows: three quintuplets are found to show signatures representative of nonlinear mode interactions, six independent modes are used for an attempt to measure rates of period changes, and 128 modes are just presented with their precise AM and FM patterns, but without clear association with a particular mechanism. Before proceeding to the detailed
<figure><img src="image_8.png" /><figcaption>**Figure 8.** LSP of the *Kepler* light curve of KIC 3527751. The presented range covers both the high-frequency *p*-mode and low-frequency *g*-mode domains where the frequencies are found. The two dashed vertical lines between 1200 and 2400 µHz indicate a region where low-amplitude signal is detected. In that region, amplitudes are multiplied by 10 to improve visibility. In the region between 2400 and 3780 µHz, amplitudes are increased a factor of 2 for similar reasons. The dashed (blue) horizontal lines denotes the 5.6σ detection threshold. Note that a few well-known *Kepler* instrumental artifacts can easily be recognized and are of course identified as such in our analysis.</figcaption></figure>

description of our results, we acknowledge the efforts from Foster et al. (2015), who showed detailed AMs and FMs by providing sliding Fourier spectra around their detected frequencies. Our present work is pushing further this effort of characterization of the modes in KIC 3527751.

### 4.2.1. The Rotation-induced Quintuplets

We here focus on the three quintuplets $Q_1$, $Q_2$, and $Q_3$ at 92, 253, and 114 µHz, respectively, which all show clear AM and FM occurring in their components (see Figures 9–11). We find many additional modes forming multiplets associated with the rotation of the star, but these will be presented in another section because of their very complex modulation patterns (see Figure 14).

Figure 9 shows the AMs and FMs occurring in components of the $Q_1$ incomplete quintuplet, near 92 µHz, with the $m = \pm 1$ modes missing. The clearly broadened structure of $Q_1$ indicates that the components are modulated, which is further illustrated from the sLSP. We find that amplitudes of all components show quasi-periodic modulations. Moreover, the amplitude of the summed energy of these modes $\sqrt{\sum A_m^2}$ is also varying. The timescale involved is of the order of a few years. The quantity $\sqrt{\sum A_m^2}$ is mainly determined by the largest-amplitude mode among the involved components of the multiplet. Correlated variations possibly happen between these three components, in particular between the $m = -2$ and $m = 0$ modes, whose amplitudes evolve in phase during the first half of the observation run and in anti-phase during the second half. The frequencies also show (quasi-)periodic variations of the order of a few percent of a µHz. We also note that correlated behaviors could somewhat occur between these FMs. For instance, both the $m = -2$ and $m = 0$ components seem to evolve in anti-phase with the $m = +2$ component, although the variations occur on different scales.

Figure 10 shows the AMs and FMs occurring in three components of the $Q_2$ quintuplet, near 253 µHz. In $Q_2$, the two prograde modes ($m = +1$ and $m = +2$) have amplitudes too low to be consistently extracted from sectioning the light curve (i.e., too few measured points), but can be resolved with the entire data set (see Table 2). The broadened frequency structures again suggest modulations in all shown components.

We find that the $m = -2$ mode suggests a periodic shape for the AM, with a possible modulation period slightly longer than 3 yr. The lower amplitude $m = 0$ mode shows an AM pattern similar to the $m = -2$ component, but with larger errors and lagging behind in phase. In contrast, the suggested quasi-periodic AM for the $m = -1$ component experiences an abrupt amplitude decrease for a relatively short duration of ~100 days during the first-half observations. The AM of the quantity $\sqrt{\sum A_m^2}$ is very similar to that of the highest-amplitude $m = -2$ component. Frequencies, for their part, all show simple patterns. The $m = -2$ component exhibits a seemingly periodic variation of ~0.04 µHz. Interestingly, the $m = -1$ mode experiences a short pulse shape FM that occurs at the same time as the rapid amplitude decrease mentioned above. The $m = 0$ component also varies in frequency, but at a smaller scale of ~0.01 µHz.

Figure 11 shows the AMs and FMs occurring in the detected components forming the incomplete quintuplet $Q_3$ at 114 µHz. The broadened peaks, as with the above two multiplets, again indicate modulations. Amplitudes of the central ($m = 0$) and prograde ($m = +2$) components clearly show a periodic modulation with a timescale of ~500 days that occur in phase. The retrograde ($m = -2$) mode shows a possibly periodic amplitude variation, but the modulation period in this case would be somewhat longer than the ~1000 days of observation. Interestingly, we clearly find that this component had its frequency reaching its maximum when the amplitude decreased down to its minimum. The anticorrelation between AM and FM for this mode is further illustrated in Figure 12, where we also find that there is a slight phase lag between the two modulations (this phase difference corresponds to about 40 days). We quantitatively check this fact by calculating the correlation coefficient $\rho(a, f) = -0.857$ (i.e., strong anticorrelation) and $\rho(a, f)_{\text{lag}} = -0.904$ (when we shift ahead the FM in time by 40 days). The precise AM pattern of the quantity $\sqrt{\sum A_m^2}$, again, is dominated by the $m = +2$ mode that has the largest amplitude. The frequency of the $m = +2$ component exhibits quasi-periodic modulations on the same timescale as its AM. A periodic FM is also found in the central component, with a timescale of ~500 days and a frequency variation scale of 0.03 µHz. The $m = +2$ component shows a
<table><caption>Table 1
List of 143 Frequencies with AM and FM Securely Detected in KIC 3527751, by Order of Increasing Frequency</caption><thead><tr><th>ID</th><th>Frequency (μHz)</th><th>σ<sub>f</sub> (μHz)</th><th>Period (s)</th><th>σ<sub>p</sub> (s)</th><th>Amplitude (ppm)</th><th>σ<sub>A</sub> (ppm)</th><th>S/N</th><th>Comment<sup>a</sup></th></tr></thead><tbody><tr><td>f<sub>078</sub></td><td>77.636099</td><td>0.000569</td><td>12880.605947</td><td>0.094480</td><td>32.04</td><td>3.28</td><td>9.8</td><td>C</td></tr><tr><td>f<sub>127</sub></td><td>85.430839</td><td>0.000870</td><td>11705.374864</td><td>0.119209</td><td>20.82</td><td>3.26</td><td>6.4</td><td>...</td></tr><tr><td>f<sub>099</sub></td><td>88.029992</td><td>0.000707</td><td>11359.764774</td><td>0.091251</td><td>25.61</td><td>3.26</td><td>7.9</td><td>...</td></tr><tr><td>f<sub>007</sub></td><td>91.703981</td><td>0.000134</td><td>10904.651949</td><td>0.015922</td><td>134.91</td><td>3.25</td><td>41.5</td><td>Q1</td></tr><tr><td>f<sub>014</sub></td><td>91.933601</td><td>0.000177</td><td>10877.415733</td><td>0.020970</td><td>101.90</td><td>3.25</td><td>31.4</td><td>Q1</td></tr><tr><td>f<sub>005</sub></td><td>92.160502</td><td>0.000120</td><td>10850.635391</td><td>0.014131</td><td>150.45</td><td>3.25</td><td>46.3</td><td>Q1</td></tr><tr><td>f<sub>084</sub></td><td>94.570334</td><td>0.000583</td><td>10574.140491</td><td>0.065179</td><td>30.92</td><td>3.24</td><td>9.5</td><td>...</td></tr><tr><td>f<sub>131</sub></td><td>101.580288</td><td>0.000877</td><td>9844.429615</td><td>0.085004</td><td>20.46</td><td>3.23</td><td>6.3</td><td>...</td></tr><tr><td>f<sub>044</sub></td><td>105.258447</td><td>0.000395</td><td>9500.425219</td><td>0.035636</td><td>45.30</td><td>3.22</td><td>14.1</td><td>...</td></tr><tr><td>f<sub>058</sub></td><td>105.439576</td><td>0.000489</td><td>9484.104882</td><td>0.043943</td><td>36.60</td><td>3.22</td><td>11.4</td><td>...</td></tr><tr><td>f<sub>010</sub></td><td>105.605349</td><td>0.000144</td><td>9469.217329</td><td>0.012896</td><td>124.32</td><td>3.22</td><td>38.7</td><td>...</td></tr><tr><td>f<sub>076</sub></td><td>107.413554</td><td>0.000541</td><td>9309.812062</td><td>0.046853</td><td>33.03</td><td>3.21</td><td>10.3</td><td>...</td></tr><tr><td>f<sub>117</sub></td><td>108.083570</td><td>0.000816</td><td>9252.100042</td><td>0.069820</td><td>21.88</td><td>3.21</td><td>6.8</td><td>C</td></tr><tr><td>f<sub>093</sub></td><td>111.169058</td><td>0.000648</td><td>8995.308718</td><td>0.052407</td><td>27.47</td><td>3.20</td><td>8.6</td><td>...</td></tr><tr><td>f<sub>086</sub></td><td>111.336596</td><td>0.000604</td><td>8981.772689</td><td>0.048728</td><td>29.45</td><td>3.20</td><td>9.2</td><td>...</td></tr><tr><td>f<sub>054</sub></td><td>111.900323</td><td>0.000468</td><td>8936.524718</td><td>0.037402</td><td>37.96</td><td>3.20</td><td>11.9</td><td>...</td></tr><tr><td>f<sub>020</sub></td><td>114.061601</td><td>0.000215</td><td>8767.192397</td><td>0.016506</td><td>82.74</td><td>3.20</td><td>25.9</td><td>Q3</td></tr><tr><td>f<sub>023</sub></td><td>114.368056</td><td>0.000237</td><td>8743.700214</td><td>0.018146</td><td>74.84</td><td>3.19</td><td>23.4</td><td>Q3</td></tr><tr><td>f<sub>011</sub></td><td>114.666811</td><td>0.000153</td><td>8720.919253</td><td>0.011655</td><td>115.86</td><td>3.19</td><td>36.3</td><td>Q3</td></tr><tr><td>f<sub>015</sub></td><td>118.864601</td><td>0.000176</td><td>8412.933643</td><td>0.012437</td><td>100.78</td><td>3.18</td><td>31.6</td><td>...</td></tr><tr><td>f<sub>075</sub></td><td>119.083162</td><td>0.000534</td><td>8397.492855</td><td>0.037666</td><td>33.15</td><td>3.18</td><td>10.4</td><td>...</td></tr><tr><td>f<sub>119</sub></td><td>119.297491</td><td>0.000820</td><td>8382.405959</td><td>0.057593</td><td>21.60</td><td>3.18</td><td>6.8</td><td>...</td></tr><tr><td>f<sub>135</sub></td><td>122.161303</td><td>0.000871</td><td>8185.898269</td><td>0.058374</td><td>20.28</td><td>3.18</td><td>6.4</td><td>...</td></tr><tr><td>f<sub>138</sub></td><td>122.423063</td><td>0.000881</td><td>8168.395516</td><td>0.058813</td><td>20.04</td><td>3.18</td><td>6.3</td><td>...</td></tr><tr><td>f<sub>144</sub></td><td>125.615714</td><td>0.000895</td><td>7960.787445</td><td>0.056716</td><td>19.68</td><td>3.17</td><td>6.2</td><td>...</td></tr><tr><td>f<sub>016</sub></td><td>126.097688</td><td>0.000180</td><td>7930.359508</td><td>0.011296</td><td>98.01</td><td>3.17</td><td>31.0</td><td>...</td></tr><tr><td>f<sub>031</sub></td><td>126.337263</td><td>0.000305</td><td>7915.321099</td><td>0.019101</td><td>57.72</td><td>3.16</td><td>18.2</td><td>...</td></tr><tr><td>f<sub>088</sub></td><td>135.606439</td><td>0.000605</td><td>7374.281116</td><td>0.032878</td><td>28.90</td><td>3.14</td><td>9.2</td><td>...</td></tr><tr><td>f<sub>090</sub></td><td>135.721394</td><td>0.000611</td><td>7368.035141</td><td>0.033183</td><td>28.58</td><td>3.14</td><td>9.1</td><td>...</td></tr><tr><td>f<sub>081</sub></td><td>135.840474</td><td>0.000556</td><td>7361.576194</td><td>0.030125</td><td>31.42</td><td>3.14</td><td>10.0</td><td>...</td></tr><tr><td>f<sub>046</sub></td><td>141.098734</td><td>0.000412</td><td>7087.235807</td><td>0.020716</td><td>42.14</td><td>3.13</td><td>13.5</td><td>...</td></tr><tr><td>f<sub>056</sub></td><td>140.547433</td><td>0.000465</td><td>7115.035664</td><td>0.023534</td><td>37.40</td><td>3.13</td><td>12.0</td><td>...</td></tr><tr><td>f<sub>032</sub></td><td>140.712570</td><td>0.000307</td><td>7106.685620</td><td>0.015486</td><td>56.70</td><td>3.13</td><td>18.1</td><td>...</td></tr><tr><td>f<sub>025</sub></td><td>140.857081</td><td>0.000266</td><td>7099.394612</td><td>0.013424</td><td>65.27</td><td>3.13</td><td>20.9</td><td>...</td></tr><tr><td>f<sub>004</sub></td><td>140.994047</td><td>0.000098</td><td>7092.498042</td><td>0.004945</td><td>176.83</td><td>3.13</td><td>56.6</td><td>...</td></tr><tr><td>f<sub>110</sub></td><td>142.520537</td><td>0.000753</td><td>7016.532647</td><td>0.037059</td><td>23.06</td><td>3.12</td><td>7.4</td><td>...</td></tr><tr><td>f<sub>051</sub></td><td>145.682883</td><td>0.000435</td><td>6864.224402</td><td>0.020480</td><td>39.82</td><td>3.11</td><td>12.8</td><td>...</td></tr><tr><td>f<sub>115</sub></td><td>146.341833</td><td>0.000784</td><td>6833.316087</td><td>0.036621</td><td>22.06</td><td>3.11</td><td>7.1</td><td>...</td></tr><tr><td>f<sub>068</sub></td><td>146.522480</td><td>0.000504</td><td>6824.891285</td><td>0.023460</td><td>34.35</td><td>3.11</td><td>11.0</td><td>...</td></tr><tr><td>f<sub>057</sub></td><td>146.635780</td><td>0.000464</td><td>6819.617954</td><td>0.021600</td><td>37.25</td><td>3.11</td><td>12.0</td><td>...</td></tr><tr><td>f<sub>082</sub></td><td>151.937843</td><td>0.000550</td><td>6581.638785</td><td>0.023814</td><td>31.39</td><td>3.10</td><td>10.1</td><td>...</td></tr><tr><td>f<sub>162</sub></td><td>153.688622</td><td>0.000963</td><td>6506.662559</td><td>0.040767</td><td>17.90</td><td>3.10</td><td>5.8</td><td>...</td></tr><tr><td>f<sub>128</sub></td><td>156.244480</td><td>0.000832</td><td>6400.226115</td><td>0.034098</td><td>20.68</td><td>3.10</td><td>6.7</td><td>...</td></tr><tr><td>f<sub>008</sub></td><td>158.839630</td><td>0.000137</td><td>6295.658085</td><td>0.005420</td><td>125.75</td><td>3.09</td><td>40.7</td><td>...</td></tr><tr><td>f<sub>033</sub></td><td>158.951690</td><td>0.000304</td><td>6291.219686</td><td>0.012013</td><td>56.65</td><td>3.09</td><td>18.3</td><td>...</td></tr><tr><td>f<sub>027</sub></td><td>159.084558</td><td>0.000272</td><td>6285.965232</td><td>0.010735</td><td>63.29</td><td>3.09</td><td>20.5</td><td>...</td></tr><tr><td>f<sub>063</sub></td><td>165.633485</td><td>0.000488</td><td>6037.426554</td><td>0.017773</td><td>35.21</td><td>3.09</td><td>11.4</td><td>...</td></tr><tr><td>f<sub>134</sub></td><td>177.856270</td><td>0.000837</td><td>5622.517558</td><td>0.026472</td><td>20.36</td><td>3.07</td><td>6.6</td><td>...</td></tr><tr><td>f<sub>067</sub></td><td>180.755584</td><td>0.000488</td><td>5532.332554</td><td>0.014928</td><td>34.88</td><td>3.06</td><td>11.4</td><td>...</td></tr><tr><td>f<sub>021</sub></td><td>181.015942</td><td>0.000224</td><td>5524.375305</td><td>0.006831</td><td>76.00</td><td>3.06</td><td>24.8</td><td>...</td></tr><tr><td>f<sub>153</sub></td><td>182.651384</td><td>0.000897</td><td>5474.910599</td><td>0.026884</td><td>18.95</td><td>3.06</td><td>6.2</td><td>...</td></tr><tr><td>f<sub>069</sub></td><td>183.240569</td><td>0.000506</td><td>5457.306785</td><td>0.015059</td><td>33.60</td><td>3.06</td><td>11.0</td><td>...</td></tr><tr><td>f<sub>125</sub></td><td>167.784735</td><td>0.000817</td><td>5960.017748</td><td>0.029023</td><td>20.99</td><td>3.08</td><td>6.8</td><td>...</td></tr><tr><td>f<sub>149</sub></td><td>168.011829</td><td>0.000887</td><td>5951.961876</td><td>0.031431</td><td>19.33</td><td>3.08</td><td>6.3</td><td>...</td></tr><tr><td>f<sub>024</sub></td><td>171.907064</td><td>0.000248</td><td>5817.096598</td><td>0.008405</td><td>68.91</td><td>3.08</td><td>22.4</td><td>...</td></tr><tr><td>f<sub>030</sub></td><td>172.129870</td><td>0.000280</td><td>5809.566943</td><td>0.009441</td><td>61.18</td><td>3.08</td><td>19.9</td><td>...</td></tr><tr><td>f<sub>045</sub></td><td>172.321559</td><td>0.000400</td><td>5803.104427</td><td>0.013468</td><td>42.79</td><td>3.08</td><td>13.9</td><td>...</td></tr><tr><td>f<sub>035</sub></td><td>172.529402</td><td>0.000310</td><td>5796.113520</td><td>0.010409</td><td>55.23</td><td>3.08</td><td>17.9</td><td>...</td></tr><tr><td>f<sub>098</sub></td><td>172.740448</td><td>0.000667</td><td>5789.032093</td><td>0.022353</td><td>25.65</td><td>3.08</td><td>8.3</td><td>...</td></tr><tr><td>f<sub>085</sub></td><td>173.003923</td><td>0.000566</td><td>5780.215732</td><td>0.018905</td><td>30.23</td><td>3.08</td><td>9.8</td><td>...</td></tr><tr><td>f<sub>064</sub></td><td>173.180128</td><td>0.000488</td><td>5774.334569</td><td>0.016273</td><td>35.05</td><td>3.08</td><td>11.4</td><td>...</td></tr><tr><td>f<sub>012</sub></td><td>173.319754</td><td>0.000153</td><td>5769.682766</td><td>0.005085</td><td>111.97</td><td>3.08</td><td>36.4</td><td>...</td></tr><tr><td>f<sub>009</sub></td><td>189.569349</td><td>0.000136</td><td>5275.114379</td><td>0.003787</td><td>124.43</td><td>3.05</td><td>40.9</td><td>...</td></tr></tbody></table>
<table><caption>**Table 1**
(Continued)</caption><thead><tr><th>ID</th><th>Frequency (μHz)</th><th>σ<sub>f</sub> (μHz)</th><th>Period (s)</th><th>σ<sub>P</sub> (s)</th><th>Amplitude (ppm)</th><th>σ<sub>A</sub> (ppm)</th><th>S/N</th><th>Comment<sup>a</sup></th></tr></thead><tbody><tr><td>f<sub>038</sub></td><td>225.200200</td><td>0.000342</td><td>4440.493402</td><td>0.006752</td><td>48.39</td><td>2.98</td><td>16.2</td><td>...</td></tr><tr><td>f<sub>006</sub></td><td>252.976914</td><td>0.000112</td><td>3952.929871</td><td>0.001755</td><td>146.39</td><td>2.96</td><td>49.5</td><td>Q2</td></tr><tr><td>f<sub>017</sub></td><td>253.184267</td><td>0.000173</td><td>3949.692500</td><td>0.002699</td><td>95.02</td><td>2.96</td><td>32.1</td><td>Q2</td></tr><tr><td>f<sub>049</sub></td><td>253.384608</td><td>0.000406</td><td>3946.569633</td><td>0.006319</td><td>40.51</td><td>2.96</td><td>13.7</td><td>Q2</td></tr><tr><td>f<sub>066</sub></td><td>187.732597</td><td>0.000486</td><td>5326.725446</td><td>0.013782</td><td>34.90</td><td>3.05</td><td>11.4</td><td>...</td></tr><tr><td>f<sub>114</sub></td><td>187.992303</td><td>0.000761</td><td>5319.366729</td><td>0.021535</td><td>22.27</td><td>3.05</td><td>7.3</td><td>...</td></tr><tr><td>f<sub>091</sub></td><td>211.618337</td><td>0.000586</td><td>4725.488409</td><td>0.013088</td><td>28.39</td><td>2.99</td><td>9.5</td><td>...</td></tr><tr><td>f<sub>028</sub></td><td>211.800334</td><td>0.000267</td><td>4721.427876</td><td>0.005947</td><td>62.36</td><td>2.99</td><td>20.8</td><td>...</td></tr><tr><td>f<sub>002</sub></td><td>211.883063</td><td>0.000060</td><td>4719.584401</td><td>0.001339</td><td>276.68</td><td>2.99</td><td>92.5</td><td>...</td></tr><tr><td>f<sub>040</sub></td><td>211.956796</td><td>0.000350</td><td>4717.942609</td><td>0.007794</td><td>47.50</td><td>2.99</td><td>15.9</td><td>...</td></tr><tr><td>f<sub>080</sub></td><td>212.203991</td><td>0.000528</td><td>4712.446708</td><td>0.011716</td><td>31.53</td><td>2.99</td><td>10.5</td><td>...</td></tr><tr><td>f<sub>052</sub></td><td>212.485237</td><td>0.000431</td><td>4706.209312</td><td>0.009536</td><td>38.63</td><td>2.99</td><td>12.9</td><td>...</td></tr><tr><td>f<sub>105</sub></td><td>212.790917</td><td>0.000670</td><td>4699.448698</td><td>0.014805</td><td>24.81</td><td>2.99</td><td>8.3</td><td>...</td></tr><tr><td>f<sub>036</sub></td><td>213.048095</td><td>0.000311</td><td>4693.775838</td><td>0.006859</td><td>53.42</td><td>2.99</td><td>17.9</td><td>...</td></tr><tr><td>f<sub>126</sub></td><td>205.357948</td><td>0.000802</td><td>4869.546117</td><td>0.019009</td><td>20.90</td><td>3.01</td><td>6.9</td><td>...</td></tr><tr><td>f<sub>055</sub></td><td>217.833157</td><td>0.000443</td><td>4590.669354</td><td>0.009333</td><td>37.46</td><td>2.98</td><td>12.6</td><td>...</td></tr><tr><td>f<sub>022</sub></td><td>218.093223</td><td>0.000221</td><td>4585.195195</td><td>0.004648</td><td>75.04</td><td>2.98</td><td>25.2</td><td>...</td></tr><tr><td>f<sub>018</sub></td><td>218.280206</td><td>0.000176</td><td>4581.267436</td><td>0.003689</td><td>94.39</td><td>2.98</td><td>31.6</td><td>...</td></tr><tr><td>f<sub>029</sub></td><td>218.505811</td><td>0.000268</td><td>4576.537329</td><td>0.005605</td><td>62.00</td><td>2.98</td><td>20.8</td><td>...</td></tr><tr><td>f<sub>074</sub></td><td>218.754070</td><td>0.000497</td><td>4571.343511</td><td>0.010388</td><td>33.37</td><td>2.98</td><td>11.2</td><td>...</td></tr><tr><td>f<sub>065</sub></td><td>234.421228</td><td>0.000472</td><td>4265.825284</td><td>0.008588</td><td>35.02</td><td>2.97</td><td>11.8</td><td>...</td></tr><tr><td>f<sub>059</sub></td><td>234.599763</td><td>0.000452</td><td>4262.578894</td><td>0.008207</td><td>36.59</td><td>2.97</td><td>12.3</td><td>...</td></tr><tr><td>f<sub>113</sub></td><td>234.811136</td><td>0.000740</td><td>4258.741806</td><td>0.013418</td><td>22.34</td><td>2.97</td><td>7.5</td><td>...</td></tr><tr><td>f<sub>111</sub></td><td>235.004101</td><td>0.000718</td><td>4255.244885</td><td>0.013004</td><td>23.01</td><td>2.97</td><td>7.7</td><td>...</td></tr><tr><td>f<sub>092</sub></td><td>227.149113</td><td>0.000596</td><td>4402.394471</td><td>0.011547</td><td>27.79</td><td>2.98</td><td>9.3</td><td>...</td></tr><tr><td>f<sub>001</sub></td><td>255.686462</td><td>0.000029</td><td>3911.040081</td><td>0.000440</td><td>571.74</td><td>2.96</td><td>193.4</td><td>...</td></tr><tr><td>f<sub>048</sub></td><td>262.149998</td><td>0.000402</td><td>3814.609983</td><td>0.005845</td><td>40.88</td><td>2.95</td><td>13.8</td><td>...</td></tr><tr><td>f<sub>077</sub></td><td>262.462766</td><td>0.000498</td><td>3810.064249</td><td>0.007225</td><td>32.99</td><td>2.95</td><td>11.2</td><td>...</td></tr><tr><td>f<sub>083</sub></td><td>263.025718</td><td>0.000523</td><td>3801.909594</td><td>0.007564</td><td>31.37</td><td>2.95</td><td>10.6</td><td>...</td></tr><tr><td>f<sub>041</sub></td><td>263.241401</td><td>0.000349</td><td>3798.794555</td><td>0.005036</td><td>47.03</td><td>2.95</td><td>15.9</td><td>...</td></tr><tr><td>f<sub>139</sub></td><td>263.521574</td><td>0.000824</td><td>3794.755716</td><td>0.011863</td><td>19.92</td><td>2.95</td><td>6.7</td><td>C</td></tr><tr><td>f<sub>095</sub></td><td>274.698355</td><td>0.000615</td><td>3640.356710</td><td>0.008146</td><td>26.57</td><td>2.94</td><td>9.0</td><td>...</td></tr><tr><td>f<sub>037</sub></td><td>275.139416</td><td>0.000329</td><td>3634.521056</td><td>0.004349</td><td>49.61</td><td>2.94</td><td>16.9</td><td>...</td></tr><tr><td>f<sub>060</sub></td><td>275.355450</td><td>0.000453</td><td>3631.669534</td><td>0.005972</td><td>36.06</td><td>2.94</td><td>12.3</td><td>...</td></tr><tr><td>f<sub>097</sub></td><td>286.905571</td><td>0.000628</td><td>3485.467345</td><td>0.007624</td><td>25.80</td><td>2.91</td><td>8.9</td><td>...</td></tr><tr><td>f<sub>070</sub></td><td>299.299917</td><td>0.000478</td><td>3341.130226</td><td>0.005335</td><td>33.47</td><td>2.88</td><td>11.6</td><td>...</td></tr><tr><td>f<sub>019</sub></td><td>366.416573</td><td>0.000174</td><td>2729.134196</td><td>0.001297</td><td>85.58</td><td>2.68</td><td>31.9</td><td>...</td></tr><tr><td>f<sub>034</sub></td><td>389.015084</td><td>0.000263</td><td>2570.594409</td><td>0.001738</td><td>55.53</td><td>2.63</td><td>21.1</td><td>...</td></tr><tr><td>f<sub>026</sub></td><td>441.504922</td><td>0.000220</td><td>2264.980413</td><td>0.001131</td><td>64.53</td><td>2.56</td><td>25.2</td><td>...</td></tr><tr><td>f<sub>071</sub></td><td>459.993053</td><td>0.000421</td><td>2173.945876</td><td>0.001991</td><td>33.43</td><td>2.53</td><td>13.2</td><td>...</td></tr><tr><td>f<sub>148</sub></td><td>565.904241</td><td>0.000718</td><td>1767.083418</td><td>0.002241</td><td>19.45</td><td>2.51</td><td>7.7</td><td>...</td></tr><tr><td>f<sub>013</sub></td><td>566.395082</td><td>0.000137</td><td>1765.552053</td><td>0.000426</td><td>102.14</td><td>2.51</td><td>40.7</td><td>...</td></tr><tr><td>f<sub>062</sub></td><td>566.917800</td><td>0.000395</td><td>1763.924152</td><td>0.001228</td><td>35.36</td><td>2.51</td><td>14.1</td><td>...</td></tr><tr><td>f<sub>147</sub></td><td>567.450732</td><td>0.000714</td><td>1762.267530</td><td>0.002218</td><td>19.54</td><td>2.51</td><td>7.8</td><td>...</td></tr><tr><td>f<sub>141</sub></td><td>567.776009</td><td>0.000706</td><td>1761.257934</td><td>0.002190</td><td>19.76</td><td>2.51</td><td>7.9</td><td>...</td></tr><tr><td>f<sub>121</sub></td><td>743.340177</td><td>0.000661</td><td>1345.279093</td><td>0.001197</td><td>21.38</td><td>2.54</td><td>8.4</td><td>...</td></tr><tr><td>f<sub>181</sub></td><td>743.476145</td><td>0.000836</td><td>1345.033068</td><td>0.001513</td><td>16.91</td><td>2.54</td><td>6.7</td><td>...</td></tr><tr><td>f<sub>103</sub></td><td>748.390627</td><td>0.000569</td><td>1336.200593</td><td>0.001016</td><td>24.83</td><td>2.54</td><td>9.8</td><td>...</td></tr><tr><td>f<sub>039</sub></td><td>748.699303</td><td>0.000293</td><td>1335.649701</td><td>0.000523</td><td>48.19</td><td>2.54</td><td>19.0</td><td>...</td></tr><tr><td>f<sub>176</sub></td><td>749.018036</td><td>0.000826</td><td>1335.081336</td><td>0.001473</td><td>17.11</td><td>2.54</td><td>6.7</td><td>...</td></tr><tr><td>f<sub>047</sub></td><td>749.331023</td><td>0.000340</td><td>1334.523688</td><td>0.000606</td><td>41.54</td><td>2.54</td><td>16.3</td><td>...</td></tr><tr><td>f<sub>166</sub></td><td>749.651593</td><td>0.000797</td><td>1333.953012</td><td>0.001418</td><td>17.74</td><td>2.54</td><td>7.0</td><td>...</td></tr><tr><td>f<sub>100</sub></td><td>773.066022</td><td>0.000556</td><td>1293.550579</td><td>0.000930</td><td>25.43</td><td>2.54</td><td>10.0</td><td>...</td></tr><tr><td>f<sub>073</sub></td><td>773.434573</td><td>0.000423</td><td>1292.934187</td><td>0.000707</td><td>33.40</td><td>2.54</td><td>13.1</td><td>...</td></tr><tr><td>f<sub>050</sub></td><td>773.817252</td><td>0.000350</td><td>1292.294786</td><td>0.000584</td><td>40.38</td><td>2.54</td><td>15.9</td><td>...</td></tr><tr><td>f<sub>167</sub></td><td>774.151054</td><td>0.000799</td><td>1291.737569</td><td>0.001333</td><td>17.69</td><td>2.54</td><td>7.0</td><td>...</td></tr><tr><td>f<sub>108</sub></td><td>774.475248</td><td>0.000597</td><td>1291.196849</td><td>0.000995</td><td>23.68</td><td>2.54</td><td>9.3</td><td>...</td></tr><tr><td>f<sub>123</sub></td><td>779.419432</td><td>0.000672</td><td>1283.006246</td><td>0.001106</td><td>21.04</td><td>2.54</td><td>8.3</td><td>...</td></tr><tr><td>f<sub>142</sub></td><td>779.698913</td><td>0.000716</td><td>1282.546357</td><td>0.001177</td><td>19.75</td><td>2.54</td><td>7.8</td><td>...</td></tr><tr><td>f<sub>079</sub></td><td>774.669373</td><td>0.000441</td><td>1290.873287</td><td>0.000736</td><td>32.01</td><td>2.54</td><td>12.6</td><td>...</td></tr><tr><td>f<sub>087</sub></td><td>774.859849</td><td>0.000483</td><td>1290.555965</td><td>0.000804</td><td>29.27</td><td>2.54</td><td>11.5</td><td>...</td></tr><tr><td>f<sub>102</sub></td><td>775.247158</td><td>0.000568</td><td>1289.911211</td><td>0.000945</td><td>24.87</td><td>2.54</td><td>9.8</td><td>...</td></tr><tr><td>f<sub>061</sub></td><td>932.333607</td><td>0.000395</td><td>1072.577448</td><td>0.000454</td><td>35.92</td><td>2.55</td><td>14.1</td><td>...</td></tr></tbody></table>
<table><caption>**Table 1**
(Continued)</caption><thead><tr><th>ID</th><th>Frequency (μHz)</th><th>σ<sub>f</sub> (μHz)</th><th>Period (s)</th><th>σ<sub>P</sub> (s)</th><th>Amplitude (ppm)</th><th>σ<sub>A</sub> (ppm)</th><th>S/N</th><th>Comment<sup>a</sup></th></tr></thead><tbody><tr><td>f<sub>043</sub></td><td>932.995201</td><td>0.000310</td><td>1071.816875</td><td>0.000356</td><td>45.76</td><td>2.55</td><td>17.9</td><td>...</td></tr><tr><td>f<sub>129</sub></td><td>969.424430</td><td>0.000683</td><td>1031.539921</td><td>0.000727</td><td>20.68</td><td>2.54</td><td>8.1</td><td>...</td></tr><tr><td>f<sub>140</sub></td><td>969.887148</td><td>0.000714</td><td>1031.047790</td><td>0.000759</td><td>19.79</td><td>2.54</td><td>7.8</td><td>...</td></tr><tr><td>f<sub>133</sub></td><td>1002.646638</td><td>0.000690</td><td>997.360348</td><td>0.000687</td><td>20.40</td><td>2.53</td><td>8.1</td><td>...</td></tr><tr><td>f<sub>122</sub></td><td>1002.932567</td><td>0.000663</td><td>997.076008</td><td>0.000659</td><td>21.25</td><td>2.53</td><td>8.4</td><td>...</td></tr><tr><td>f<sub>146</sub></td><td>1012.924293</td><td>0.000717</td><td>987.240613</td><td>0.000699</td><td>19.64</td><td>2.53</td><td>7.8</td><td>...</td></tr><tr><td>f<sub>137</sub></td><td>1014.114739</td><td>0.000702</td><td>986.081713</td><td>0.000682</td><td>20.06</td><td>2.53</td><td>7.9</td><td>...</td></tr><tr><td>f<sub>106</sub></td><td>1014.590863</td><td>0.000578</td><td>985.618969</td><td>0.000561</td><td>24.35</td><td>2.53</td><td>9.6</td><td>...</td></tr><tr><td>f<sub>094</sub></td><td>1015.000200</td><td>0.000516</td><td>985.221481</td><td>0.000501</td><td>27.29</td><td>2.53</td><td>10.8</td><td>...</td></tr><tr><td>f<sub>112</sub></td><td>1015.466213</td><td>0.000621</td><td>984.769347</td><td>0.000602</td><td>22.65</td><td>2.53</td><td>9.0</td><td>...</td></tr><tr><td>f<sub>104</sub></td><td>1015.875675</td><td>0.000567</td><td>984.372424</td><td>0.000549</td><td>24.82</td><td>2.53</td><td>9.8</td><td>...</td></tr><tr><td>f<sub>130</sub></td><td>1016.894687</td><td>0.000687</td><td>983.386002</td><td>0.000665</td><td>20.47</td><td>2.53</td><td>8.1</td><td>...</td></tr><tr><td>f<sub>161</sub></td><td>1508.399851</td><td>0.000766</td><td>662.954189</td><td>0.000337</td><td>18.01</td><td>2.48</td><td>7.3</td><td>C</td></tr><tr><td>f<sub>155</sub></td><td>1696.132130</td><td>0.000734</td><td>589.576710</td><td>0.000255</td><td>18.65</td><td>2.46</td><td>7.6</td><td>...</td></tr><tr><td>f<sub>003</sub></td><td>2767.518348</td><td>0.000066</td><td>361.334551</td><td>0.000009</td><td>204.32</td><td>2.41</td><td>84.8</td><td>...</td></tr><tr><td>f<sub>101</sub></td><td>2782.428575</td><td>0.000537</td><td>359.398264</td><td>0.000069</td><td>24.95</td><td>2.41</td><td>10.4</td><td>...</td></tr><tr><td>f<sub>042</sub></td><td>2816.458893</td><td>0.000290</td><td>355.055777</td><td>0.000037</td><td>46.22</td><td>2.41</td><td>19.2</td><td>...</td></tr></tbody></table>

**Note.**

a C represents linear combination frequency, and Q indicates rotational quintuplet.

more complicated pattern, which seems to be composed of two linear discontinuous segments. A leap in frequency, of about 0.1 μHz, happens around the mid-observation run, when the amplitude of the mode decreased down to its minimum.

### 4.2.2. The Most Stable Frequencies

Among all extracted modes, we identified six of them having relatively stable amplitudes and frequencies. These are usually sought in various contexts, typically when one wants to use stellar pulsations to detect the presence of a companion (stellar or substellar) or to measure rates of period change triggered by stellar evolution. The residual, after fitting a single sinusoidal wave to these modes, is far below the main signal. However, a close look suggests that these six frequencies may still exhibit shallow AMs and narrowband FMs, as illustrated in Figure 13.

The 189.5689 μHz frequency ($f_0$) possibly has a periodic amplitude variation of ~±10 ppm and completes about one cycle during the observation run. Its FM is possibly composed of a long-term variation with several shorter ones superimposed, with a timescale of a few months. This mode varies around its averaged frequency by ~±0.01 μHz.

The 225.2002 and 389.015 μHz modes ($f_{38}$ and $f_{34}$) both exhibit small, possibly periodic, AMs with a variation amplitude of ~20 ppm. The FMs show slight wiggles around their averaged values, but overall they seem quite stable within the estimated errors.

The 366.4166 μHz mode ($f_{19}$) shows marginal quasi-periodic behaviors for both AM and FM. During the 1150-day observation, the amplitude almost completes three cycles, with a scale of variation of about 20 ppm. This mode may also vary around its average frequency by ~±0.005 μHz, but this detection is just marginally above the 1σ limit.

The 441.5051 μHz mode ($f_{26}$) has an amplitude that first decreases slightly and then increases during the second half of observation. The frequency shows marginal modulations.

The 255.6864 μHz mode ($f_1$) has the largest amplitude among all the detected frequencies. The amplitude of $f_1$ first

increases from 560 to ~620 ppm in about 300 days, exhibits a marginal quasi-periodic variation in the next 400 days, then abruptly decreases down to ~400 ppm and increases rapidly to 600 ppm (the former value), and finally shows marginal quasi-periodic variation again. In contrast, the frequency exhibits periodic modulations on a timescale of about 4 months over the entire observation but within a very small scale of ±2 nHz, which was measurable thanks to the high precision obtained with the *Kepler* data combined with the relatively large amplitude of this mode.

### 4.2.3. The Other Frequencies with Relative Large Amplitude

In this section, we provide, for completeness, the measured AMs and FMs for all other modes with large enough amplitudes, typically above 20 ppm (see Figure 14). Most of these 128 frequencies are components that belong to rotational multiplets, such as the frequencies $f_2$, $f_{28}$, and $f_{40}$ forming an $l = 1$ triplet at 212 μHz. The software FELIX has automatically measured amplitudes and frequencies from various parts of the light curve for all these modes if they are above a 6σ threshold (a limit that ensures that each measurement has acceptable errors). We describe these patterns in more detail, but in terms of groups, which means that we classify similar patterns as one kind of modulation.

We find several modes with amplitudes very low at the beginning of *Kepler* observations, even with very few measured points, while their amplitudes appear to be significant in the last half of the observation run. The pattern of AMs can be as simple as linearly increasing with time, such as for the modes $f_{58} \sim 105.4$ μHz, $f_{71} \sim 460$ μHz, $f_{76} \sim 107.4$ μHz, $f_{78} \sim 77.6$ μHz, and $f_{110} \sim 142.5$ μHz, or it may present seemingly periodic variations, such as the modes $f_{42} \sim 2816.5$ μHz, $f_{54} \sim 111.9$ μHz, $f_{93} \sim 111.2$ μHz, $f_{113} \sim 234.8$ μHz, and $f_{146} \sim 1012.9$ μHz. Their corresponding frequencies exhibit simple linear increases or decreases ($f_{76}$ and $f_{113}$), or feature periodic modulations (e.g., $f_{71}$). In this group, we note that frequencies and amplitudes may have clear correlations or anticorrelations, such as for the mode $f_{113}$, which shows a
<table><caption>**Table 2**
Full List of Frequencies Detected in KIC 3527751 by Order of Increasing Frequency</caption><thead><tr><th>ID</th><th>Frequency (μHz)</th><th>σ<sub>f</sub> (μHz)</th><th>Period (s)</th><th>σ<sub>p</sub> (s)</th><th>Amplitude (ppm)</th><th>σ<sub>A</sub> (ppm)</th><th>S/N</th><th>Comment<sup>a</sup></th></tr></thead><tbody><tr><td>f<sub>078</sub></td><td>77.636099</td><td>0.000569</td><td>12880.605947</td><td>0.094480</td><td>32.04</td><td>3.28</td><td>9.8</td><td>C</td></tr><tr><td>f<sub>127</sub></td><td>85.430839</td><td>0.000870</td><td>11705.374864</td><td>0.119209</td><td>20.82</td><td>3.26</td><td>6.4</td><td>...</td></tr><tr><td>f<sub>150</sub></td><td>87.846652</td><td>0.000944</td><td>11383.473083</td><td>0.122361</td><td>19.18</td><td>3.26</td><td>5.9</td><td>...</td></tr><tr><td>f<sub>099</sub></td><td>88.029992</td><td>0.000707</td><td>11359.764774</td><td>0.091251</td><td>25.61</td><td>3.26</td><td>7.9</td><td>...</td></tr><tr><td>f<sub>169</sub></td><td>90.134639</td><td>0.001029</td><td>11094.513900</td><td>0.126624</td><td>17.58</td><td>3.25</td><td>5.4</td><td>...</td></tr><tr><td>f<sub>007</sub></td><td>91.703981</td><td>0.000134</td><td>10904.651949</td><td>0.015922</td><td>134.91</td><td>3.25</td><td>41.5</td><td>Q1</td></tr><tr><td>f<sub>014</sub></td><td>91.933601</td><td>0.000177</td><td>10877.415733</td><td>0.020970</td><td>101.90</td><td>3.25</td><td>31.4</td><td>Q1</td></tr><tr><td>f<sub>005</sub></td><td>92.160502</td><td>0.000120</td><td>10850.635391</td><td>0.014131</td><td>150.45</td><td>3.25</td><td>46.3</td><td>Q1</td></tr><tr><td>f<sub>145</sub></td><td>92.336426</td><td>0.000918</td><td>10829.962110</td><td>0.107713</td><td>19.66</td><td>3.25</td><td>6.1</td><td>...</td></tr><tr><td>f<sub>195</sub></td><td>93.664144</td><td>0.001107</td><td>10676.444101</td><td>0.126158</td><td>16.30</td><td>3.24</td><td>5.0</td><td>...</td></tr><tr><td>f<sub>084</sub></td><td>94.570334</td><td>0.000583</td><td>10574.140491</td><td>0.065179</td><td>30.92</td><td>3.24</td><td>9.5</td><td>...</td></tr><tr><td>f<sub>191</sub></td><td>96.815945</td><td>0.001097</td><td>10328.877088</td><td>0.117018</td><td>16.39</td><td>3.23</td><td>5.1</td><td>...</td></tr><tr><td>f<sub>168</sub></td><td>99.288715</td><td>0.001020</td><td>10071.638061</td><td>0.103452</td><td>17.62</td><td>3.23</td><td>5.5</td><td>C</td></tr><tr><td>f<sub>183</sub></td><td>99.410551</td><td>0.001075</td><td>10059.294392</td><td>0.108741</td><td>16.72</td><td>3.23</td><td>5.2</td><td>...</td></tr><tr><td>f<sub>182</sub></td><td>99.694700</td><td>0.001071</td><td>10030.623492</td><td>0.107708</td><td>16.79</td><td>3.23</td><td>5.2</td><td>C</td></tr><tr><td>f<sub>131</sub></td><td>101.580288</td><td>0.000877</td><td>9844.429615</td><td>0.085004</td><td>20.46</td><td>3.23</td><td>6.3</td><td>...</td></tr><tr><td>f<sub>203</sub></td><td>101.838704</td><td>0.001115</td><td>9819.449363</td><td>0.107529</td><td>16.09</td><td>3.23</td><td>5.0</td><td>C</td></tr><tr><td>f<sub>179</sub></td><td>102.082066</td><td>0.001058</td><td>9796.039988</td><td>0.101497</td><td>16.96</td><td>3.23</td><td>5.3</td><td>...</td></tr><tr><td>f<sub>044</sub></td><td>105.258447</td><td>0.000395</td><td>9500.425219</td><td>0.035636</td><td>45.30</td><td>3.22</td><td>14.1</td><td>...</td></tr><tr><td>f<sub>058</sub></td><td>105.439576</td><td>0.000489</td><td>9484.104882</td><td>0.043943</td><td>36.60</td><td>3.22</td><td>11.4</td><td>...</td></tr><tr><td>f<sub>010</sub></td><td>105.605349</td><td>0.000144</td><td>9469.217329</td><td>0.012896</td><td>124.32</td><td>3.22</td><td>38.7</td><td>...</td></tr><tr><td>f<sub>143</sub></td><td>106.798441</td><td>0.000907</td><td>9363.432599</td><td>0.079541</td><td>19.69</td><td>3.21</td><td>6.1</td><td>...</td></tr><tr><td>f<sub>076</sub></td><td>107.413554</td><td>0.000541</td><td>9309.812062</td><td>0.046853</td><td>33.03</td><td>3.21</td><td>10.3</td><td>...</td></tr><tr><td>f<sub>117</sub></td><td>108.083570</td><td>0.000816</td><td>9252.100042</td><td>0.069820</td><td>21.88</td><td>3.21</td><td>6.8</td><td>C</td></tr><tr><td>f<sub>093</sub></td><td>111.169058</td><td>0.000648</td><td>8995.308718</td><td>0.052407</td><td>27.47</td><td>3.20</td><td>8.6</td><td>...</td></tr><tr><td>f<sub>086</sub></td><td>111.336596</td><td>0.000604</td><td>8981.772689</td><td>0.048728</td><td>29.45</td><td>3.20</td><td>9.2</td><td>...</td></tr><tr><td>f<sub>054</sub></td><td>111.900323</td><td>0.000468</td><td>8936.524718</td><td>0.037402</td><td>37.96</td><td>3.20</td><td>11.9</td><td>...</td></tr><tr><td>f<sub>020</sub></td><td>114.061601</td><td>0.000215</td><td>8767.192397</td><td>0.016506</td><td>82.74</td><td>3.20</td><td>25.9</td><td>Q3</td></tr><tr><td>f<sub>023</sub></td><td>114.368056</td><td>0.000237</td><td>8743.700214</td><td>0.018146</td><td>74.84</td><td>3.19</td><td>23.4</td><td>Q3</td></tr><tr><td>f<sub>011</sub></td><td>114.666811</td><td>0.000153</td><td>8720.919253</td><td>0.011655</td><td>115.86</td><td>3.19</td><td>36.3</td><td>Q3</td></tr><tr><td>f<sub>116</sub></td><td>118.624734</td><td>0.000806</td><td>8429.945161</td><td>0.057311</td><td>21.96</td><td>3.18</td><td>6.9</td><td>...</td></tr><tr><td>f<sub>015</sub></td><td>118.864601</td><td>0.000176</td><td>8412.933643</td><td>0.012437</td><td>100.78</td><td>3.18</td><td>31.6</td><td>...</td></tr><tr><td>f<sub>075</sub></td><td>119.083162</td><td>0.000534</td><td>8397.492855</td><td>0.037666</td><td>33.15</td><td>3.18</td><td>10.4</td><td>...</td></tr><tr><td>f<sub>119</sub></td><td>119.297491</td><td>0.000820</td><td>8382.405959</td><td>0.057593</td><td>21.60</td><td>3.18</td><td>6.8</td><td>...</td></tr><tr><td>f<sub>193</sub></td><td>119.591096</td><td>0.001081</td><td>8361.826554</td><td>0.075602</td><td>16.37</td><td>3.18</td><td>5.1</td><td>...</td></tr><tr><td>f<sub>135</sub></td><td>122.161303</td><td>0.000871</td><td>8185.898269</td><td>0.058374</td><td>20.28</td><td>3.18</td><td>6.4</td><td>...</td></tr><tr><td>f<sub>157</sub></td><td>122.273806</td><td>0.000960</td><td>8178.366540</td><td>0.064219</td><td>18.40</td><td>3.18</td><td>5.8</td><td>...</td></tr><tr><td>f<sub>138</sub></td><td>122.423063</td><td>0.000881</td><td>8168.395516</td><td>0.058813</td><td>20.04</td><td>3.18</td><td>6.3</td><td>...</td></tr><tr><td>f<sub>144</sub></td><td>125.615714</td><td>0.000895</td><td>7960.787445</td><td>0.056716</td><td>19.68</td><td>3.17</td><td>6.2</td><td>...</td></tr><tr><td>f<sub>016</sub></td><td>126.097688</td><td>0.000180</td><td>7930.359508</td><td>0.011296</td><td>98.01</td><td>3.17</td><td>31.0</td><td>...</td></tr><tr><td>f<sub>031</sub></td><td>126.337263</td><td>0.000305</td><td>7915.321099</td><td>0.019101</td><td>57.72</td><td>3.16</td><td>18.2</td><td>...</td></tr><tr><td>f<sub>109</sub></td><td>126.483299</td><td>0.000755</td><td>7906.182132</td><td>0.047196</td><td>23.30</td><td>3.16</td><td>7.4</td><td>...</td></tr><tr><td>f<sub>201</sub></td><td>128.297855</td><td>0.001087</td><td>7794.362588</td><td>0.066052</td><td>16.16</td><td>3.16</td><td>5.1</td><td>...</td></tr><tr><td>f<sub>088</sub></td><td>135.606439</td><td>0.000605</td><td>7374.281116</td><td>0.032878</td><td>28.90</td><td>3.14</td><td>9.2</td><td>...</td></tr><tr><td>f<sub>090</sub></td><td>135.721394</td><td>0.000611</td><td>7368.035141</td><td>0.033183</td><td>28.58</td><td>3.14</td><td>9.1</td><td>...</td></tr><tr><td>f<sub>081</sub></td><td>135.840474</td><td>0.000556</td><td>7361.576194</td><td>0.030125</td><td>31.42</td><td>3.14</td><td>10.0</td><td>...</td></tr><tr><td>f<sub>056</sub></td><td>140.547433</td><td>0.000465</td><td>7115.035664</td><td>0.023534</td><td>37.40</td><td>3.13</td><td>12.0</td><td>...</td></tr><tr><td>f<sub>032</sub></td><td>140.712570</td><td>0.000307</td><td>7106.685620</td><td>0.015486</td><td>56.70</td><td>3.13</td><td>18.1</td><td>...</td></tr><tr><td>f<sub>025</sub></td><td>140.857081</td><td>0.000266</td><td>7099.394612</td><td>0.013424</td><td>65.27</td><td>3.13</td><td>20.9</td><td>...</td></tr><tr><td>f<sub>004</sub></td><td>140.994047</td><td>0.000098</td><td>7092.498042</td><td>0.004945</td><td>176.83</td><td>3.13</td><td>56.6</td><td>...</td></tr><tr><td>f<sub>046</sub></td><td>141.098734</td><td>0.000412</td><td>7087.235807</td><td>0.020716</td><td>42.14</td><td>3.13</td><td>13.5</td><td>...</td></tr><tr><td>f<sub>151</sub></td><td>141.232443</td><td>0.000909</td><td>7080.526118</td><td>0.045567</td><td>19.12</td><td>3.13</td><td>6.1</td><td>C</td></tr><tr><td>f<sub>107</sub></td><td>141.367520</td><td>0.000730</td><td>7073.760673</td><td>0.036512</td><td>23.81</td><td>3.12</td><td>7.6</td><td>...</td></tr><tr><td>f<sub>110</sub></td><td>142.520537</td><td>0.000753</td><td>7016.532647</td><td>0.037059</td><td>23.06</td><td>3.12</td><td>7.4</td><td>...</td></tr><tr><td>f<sub>174</sub></td><td>144.359410</td><td>0.001006</td><td>6927.154921</td><td>0.048268</td><td>17.23</td><td>3.12</td><td>5.5</td><td>C</td></tr><tr><td>f<sub>152</sub></td><td>145.413463</td><td>0.000908</td><td>6876.942329</td><td>0.042921</td><td>19.08</td><td>3.11</td><td>6.1</td><td>...</td></tr><tr><td>f<sub>051</sub></td><td>145.682883</td><td>0.000435</td><td>6864.224402</td><td>0.020480</td><td>39.82</td><td>3.11</td><td>12.8</td><td>...</td></tr><tr><td>f<sub>115</sub></td><td>146.341833</td><td>0.000784</td><td>6833.316087</td><td>0.036621</td><td>22.06</td><td>3.11</td><td>7.1</td><td>...</td></tr><tr><td>f<sub>068</sub></td><td>146.522480</td><td>0.000504</td><td>6824.891285</td><td>0.023460</td><td>34.35</td><td>3.11</td><td>11.0</td><td>...</td></tr><tr><td>f<sub>057</sub></td><td>146.635780</td><td>0.000464</td><td>6819.617954</td><td>0.021600</td><td>37.25</td><td>3.11</td><td>12.0</td><td>...</td></tr><tr><td>f<sub>082</sub></td><td>151.937843</td><td>0.000550</td><td>6581.638785</td><td>0.023814</td><td>31.39</td><td>3.10</td><td>10.1</td><td>...</td></tr><tr><td>f<sub>162</sub></td><td>153.688622</td><td>0.000963</td><td>6506.662559</td><td>0.040767</td><td>17.90</td><td>3.10</td><td>5.8</td><td>...</td></tr><tr><td>f<sub>128</sub></td><td>156.244480</td><td>0.000832</td><td>6400.226115</td><td>0.034098</td><td>20.68</td><td>3.10</td><td>6.7</td><td>...</td></tr></tbody></table>
<table><caption>**Table 2**
(Continued)</caption><thead><tr><th>ID</th><th>Frequency (μHz)</th><th>σ<sub>f</sub> (μHz)</th><th>Period (s)</th><th>σ<sub>P</sub> (s)</th><th>Amplitude (ppm)</th><th>σ<sub>A</sub> (ppm)</th><th>S/N</th><th>Comment<sup>a</sup></th></tr></thead><tbody><tr><td>f<sub>008</sub></td><td>158.839630</td><td>0.000137</td><td>6295.658085</td><td>0.005420</td><td>125.75</td><td>3.09</td><td>40.7</td><td>...</td></tr><tr><td>f<sub>033</sub></td><td>158.951690</td><td>0.000304</td><td>6291.219686</td><td>0.012013</td><td>56.65</td><td>3.09</td><td>18.3</td><td>...</td></tr><tr><td>f<sub>027</sub></td><td>159.084558</td><td>0.000272</td><td>6285.965232</td><td>0.010735</td><td>63.29</td><td>3.09</td><td>20.5</td><td>...</td></tr><tr><td>f<sub>186</sub></td><td>159.729759</td><td>0.001032</td><td>6260.574159</td><td>0.040460</td><td>16.65</td><td>3.09</td><td>5.4</td><td>...</td></tr><tr><td>f<sub>202</sub></td><td>165.441422</td><td>0.001067</td><td>6044.435458</td><td>0.038985</td><td>16.09</td><td>3.09</td><td>5.2</td><td>...</td></tr><tr><td>f<sub>063</sub></td><td>165.633485</td><td>0.000488</td><td>6037.426554</td><td>0.017773</td><td>35.21</td><td>3.09</td><td>11.4</td><td>...</td></tr><tr><td>f<sub>154</sub></td><td>165.776512</td><td>0.000914</td><td>6032.217649</td><td>0.033261</td><td>18.78</td><td>3.09</td><td>6.1</td><td>...</td></tr><tr><td>f<sub>089</sub></td><td>165.913722</td><td>0.000595</td><td>6027.229016</td><td>0.021612</td><td>28.85</td><td>3.09</td><td>9.3</td><td>...</td></tr><tr><td>f<sub>125</sub></td><td>167.784735</td><td>0.000817</td><td>5960.017748</td><td>0.029023</td><td>20.99</td><td>3.08</td><td>6.8</td><td>...</td></tr><tr><td>f<sub>149</sub></td><td>168.011829</td><td>0.000887</td><td>5951.961876</td><td>0.031431</td><td>19.33</td><td>3.08</td><td>6.3</td><td>...</td></tr><tr><td>f<sub>160</sub></td><td>168.214390</td><td>0.000950</td><td>5944.794599</td><td>0.033560</td><td>18.06</td><td>3.08</td><td>5.9</td><td>...</td></tr><tr><td>f<sub>024</sub></td><td>171.907064</td><td>0.000248</td><td>5817.096598</td><td>0.008405</td><td>68.91</td><td>3.08</td><td>22.4</td><td>...</td></tr><tr><td>f<sub>030</sub></td><td>172.129870</td><td>0.000280</td><td>5809.566943</td><td>0.009441</td><td>61.18</td><td>3.08</td><td>19.9</td><td>...</td></tr><tr><td>f<sub>045</sub></td><td>172.321559</td><td>0.000400</td><td>5803.104427</td><td>0.013468</td><td>42.79</td><td>3.08</td><td>13.9</td><td>...</td></tr><tr><td>f<sub>035</sub></td><td>172.529402</td><td>0.000310</td><td>5796.113520</td><td>0.010409</td><td>55.23</td><td>3.08</td><td>17.9</td><td>...</td></tr><tr><td>f<sub>098</sub></td><td>172.740448</td><td>0.000667</td><td>5789.032093</td><td>0.022353</td><td>25.65</td><td>3.08</td><td>8.3</td><td>...</td></tr><tr><td>f<sub>085</sub></td><td>173.003923</td><td>0.000566</td><td>5780.215732</td><td>0.018905</td><td>30.23</td><td>3.08</td><td>9.8</td><td>...</td></tr><tr><td>f<sub>064</sub></td><td>173.180128</td><td>0.000488</td><td>5774.334569</td><td>0.016273</td><td>35.05</td><td>3.08</td><td>11.4</td><td>...</td></tr><tr><td>f<sub>012</sub></td><td>173.319754</td><td>0.000153</td><td>5769.682766</td><td>0.005085</td><td>111.97</td><td>3.08</td><td>36.4</td><td>...</td></tr><tr><td>f<sub>134</sub></td><td>177.856270</td><td>0.000837</td><td>5622.517558</td><td>0.026472</td><td>20.36</td><td>3.07</td><td>6.6</td><td>...</td></tr><tr><td>f<sub>196</sub></td><td>178.930570</td><td>0.001046</td><td>5588.759925</td><td>0.032661</td><td>16.29</td><td>3.06</td><td>5.3</td><td>...</td></tr><tr><td>f<sub>190</sub></td><td>179.017457</td><td>0.001030</td><td>5586.047399</td><td>0.032132</td><td>16.54</td><td>3.06</td><td>5.4</td><td>...</td></tr><tr><td>f<sub>067</sub></td><td>180.755584</td><td>0.000488</td><td>5532.332554</td><td>0.014928</td><td>34.88</td><td>3.06</td><td>11.4</td><td>...</td></tr><tr><td>f<sub>021</sub></td><td>181.015942</td><td>0.000224</td><td>5524.375305</td><td>0.006831</td><td>76.00</td><td>3.06</td><td>24.8</td><td>...</td></tr><tr><td>f<sub>170</sub></td><td>181.551077</td><td>0.000968</td><td>5508.091801</td><td>0.029382</td><td>17.56</td><td>3.06</td><td>5.7</td><td>...</td></tr><tr><td>f<sub>164</sub></td><td>182.059708</td><td>0.000955</td><td>5492.703521</td><td>0.028798</td><td>17.81</td><td>3.06</td><td>5.8</td><td>...</td></tr><tr><td>f<sub>163</sub></td><td>182.143352</td><td>0.000953</td><td>5490.181160</td><td>0.028723</td><td>17.84</td><td>3.06</td><td>5.8</td><td>...</td></tr><tr><td>f<sub>153</sub></td><td>182.651384</td><td>0.000897</td><td>5474.910599</td><td>0.026884</td><td>18.95</td><td>3.06</td><td>6.2</td><td>...</td></tr><tr><td>f<sub>120</sub></td><td>182.956751</td><td>0.000793</td><td>5465.772608</td><td>0.023690</td><td>21.43</td><td>3.06</td><td>7.0</td><td>...</td></tr><tr><td>f<sub>069</sub></td><td>183.240569</td><td>0.000506</td><td>5457.306785</td><td>0.015059</td><td>33.60</td><td>3.06</td><td>11.0</td><td>...</td></tr><tr><td>f<sub>066</sub></td><td>187.732597</td><td>0.000486</td><td>5326.725446</td><td>0.013782</td><td>34.90</td><td>3.05</td><td>11.4</td><td>...</td></tr><tr><td>f<sub>114</sub></td><td>187.992303</td><td>0.000761</td><td>5319.366729</td><td>0.021535</td><td>22.27</td><td>3.05</td><td>7.3</td><td>...</td></tr><tr><td>f<sub>197</sub></td><td>188.249471</td><td>0.001043</td><td>5312.099926</td><td>0.029445</td><td>16.24</td><td>3.05</td><td>5.3</td><td>...</td></tr><tr><td>f<sub>009</sub></td><td>189.569349</td><td>0.000136</td><td>5275.114379</td><td>0.003787</td><td>124.43</td><td>3.05</td><td>40.9</td><td>...</td></tr><tr><td>f<sub>184</sub></td><td>193.031220</td><td>0.001012</td><td>5180.509154</td><td>0.027157</td><td>16.70</td><td>3.04</td><td>5.5</td><td>...</td></tr><tr><td>f<sub>200</sub></td><td>193.235012</td><td>0.001045</td><td>5175.045616</td><td>0.027988</td><td>16.17</td><td>3.04</td><td>5.3</td><td>C</td></tr><tr><td>f<sub>187</sub></td><td>193.436993</td><td>0.001018</td><td>5169.641985</td><td>0.027217</td><td>16.59</td><td>3.04</td><td>5.5</td><td>...</td></tr><tr><td>f<sub>173</sub></td><td>193.665072</td><td>0.000974</td><td>5163.553710</td><td>0.025974</td><td>17.34</td><td>3.04</td><td>5.7</td><td>...</td></tr><tr><td>f<sub>180</sub></td><td>205.037751</td><td>0.000989</td><td>4877.150651</td><td>0.023534</td><td>16.94</td><td>3.01</td><td>5.6</td><td>...</td></tr><tr><td>f<sub>126</sub></td><td>205.357948</td><td>0.000802</td><td>4869.546117</td><td>0.019009</td><td>20.90</td><td>3.01</td><td>6.9</td><td>...</td></tr><tr><td>f<sub>205</sub></td><td>205.807784</td><td>0.001064</td><td>4858.902719</td><td>0.025117</td><td>15.74</td><td>3.01</td><td>5.2</td><td>...</td></tr><tr><td>f<sub>185</sub></td><td>206.548337</td><td>0.001002</td><td>4841.481722</td><td>0.023490</td><td>16.69</td><td>3.01</td><td>5.5</td><td>...</td></tr><tr><td>f<sub>091</sub></td><td>211.618337</td><td>0.000586</td><td>4725.488409</td><td>0.013088</td><td>28.39</td><td>2.99</td><td>9.5</td><td>...</td></tr><tr><td>f<sub>028</sub></td><td>211.800334</td><td>0.000267</td><td>4721.427876</td><td>0.005947</td><td>62.36</td><td>2.99</td><td>20.8</td><td>...</td></tr><tr><td>f<sub>002</sub></td><td>211.883063</td><td>0.000060</td><td>4719.584401</td><td>0.001339</td><td>276.68</td><td>2.99</td><td>92.5</td><td>...</td></tr><tr><td>f<sub>040</sub></td><td>211.956796</td><td>0.000350</td><td>4717.942609</td><td>0.007794</td><td>47.50</td><td>2.99</td><td>15.9</td><td>...</td></tr><tr><td>f<sub>080</sub></td><td>212.203991</td><td>0.000528</td><td>4712.446708</td><td>0.011716</td><td>31.53</td><td>2.99</td><td>10.5</td><td>...</td></tr><tr><td>f<sub>052</sub></td><td>212.485237</td><td>0.000431</td><td>4706.209312</td><td>0.009536</td><td>38.63</td><td>2.99</td><td>12.9</td><td>...</td></tr><tr><td>f<sub>105</sub></td><td>212.790917</td><td>0.000670</td><td>4699.448698</td><td>0.014805</td><td>24.81</td><td>2.99</td><td>8.3</td><td>...</td></tr><tr><td>f<sub>036</sub></td><td>213.048095</td><td>0.000311</td><td>4693.775838</td><td>0.006859</td><td>53.42</td><td>2.99</td><td>17.9</td><td>...</td></tr><tr><td>f<sub>055</sub></td><td>217.833157</td><td>0.000443</td><td>4590.669354</td><td>0.009333</td><td>37.46</td><td>2.98</td><td>12.6</td><td>...</td></tr><tr><td>f<sub>022</sub></td><td>218.093223</td><td>0.000221</td><td>4585.195195</td><td>0.004648</td><td>75.04</td><td>2.98</td><td>25.2</td><td>...</td></tr><tr><td>f<sub>018</sub></td><td>218.280206</td><td>0.000176</td><td>4581.267436</td><td>0.003689</td><td>94.39</td><td>2.98</td><td>31.6</td><td>...</td></tr><tr><td>f<sub>029</sub></td><td>218.505811</td><td>0.000268</td><td>4576.537329</td><td>0.005605</td><td>62.00</td><td>2.98</td><td>20.8</td><td>...</td></tr><tr><td>f<sub>074</sub></td><td>218.754070</td><td>0.000497</td><td>4571.343511</td><td>0.010388</td><td>33.37</td><td>2.98</td><td>11.2</td><td>...</td></tr><tr><td>f<sub>172</sub></td><td>221.956482</td><td>0.000955</td><td>4505.387678</td><td>0.019394</td><td>17.35</td><td>2.98</td><td>5.8</td><td>...</td></tr><tr><td>f<sub>038</sub></td><td>225.200200</td><td>0.000342</td><td>4440.493402</td><td>0.006752</td><td>48.39</td><td>2.98</td><td>16.2</td><td>...</td></tr><tr><td>f<sub>207</sub></td><td>226.978042</td><td>0.001060</td><td>4405.712509</td><td>0.020575</td><td>15.62</td><td>2.98</td><td>5.2</td><td>...</td></tr><tr><td>f<sub>092</sub></td><td>227.149113</td><td>0.000596</td><td>4402.394471</td><td>0.011547</td><td>27.79</td><td>2.98</td><td>9.3</td><td>...</td></tr><tr><td>f<sub>065</sub></td><td>234.421228</td><td>0.000472</td><td>4265.825284</td><td>0.008588</td><td>35.02</td><td>2.97</td><td>11.8</td><td>...</td></tr><tr><td>f<sub>059</sub></td><td>234.599763</td><td>0.000452</td><td>4262.578894</td><td>0.008207</td><td>36.59</td><td>2.97</td><td>12.3</td><td>...</td></tr><tr><td>f<sub>113</sub></td><td>234.811136</td><td>0.000740</td><td>4258.741806</td><td>0.013418</td><td>22.34</td><td>2.97</td><td>7.5</td><td>...</td></tr><tr><td>f<sub>111</sub></td><td>235.004101</td><td>0.000718</td><td>4255.244885</td><td>0.013004</td><td>23.01</td><td>2.97</td><td>7.7</td><td>...</td></tr></tbody></table>
<table><caption>**Table 2**
(Continued)</caption><thead><tr><th>ID</th><th>Frequency (μHz)</th><th>σ<sub>f</sub> (μHz)</th><th>Period (s)</th><th>σ<sub>P</sub> (s)</th><th>Amplitude (ppm)</th><th>σ<sub>A</sub> (ppm)</th><th>S/N</th><th>Comment<sup>a</sup></th></tr></thead><tbody><tr><td>f<sub>118</sub></td><td>239.421458</td><td>0.000761</td><td>4176.735075</td><td>0.013280</td><td>21.65</td><td>2.96</td><td>7.3</td><td>...</td></tr><tr><td>f<sub>006</sub></td><td>252.976914</td><td>0.000112</td><td>3952.929871</td><td>0.001755</td><td>146.39</td><td>2.96</td><td>49.5</td><td>Q2</td></tr><tr><td>f<sub>017</sub></td><td>253.184267</td><td>0.000173</td><td>3949.692500</td><td>0.002699</td><td>95.02</td><td>2.96</td><td>32.1</td><td>Q2</td></tr><tr><td>f<sub>049</sub></td><td>253.384608</td><td>0.000406</td><td>3946.569633</td><td>0.006319</td><td>40.51</td><td>2.96</td><td>13.7</td><td>Q2</td></tr><tr><td>f<sub>096</sub></td><td>253.571567</td><td>0.000632</td><td>3943.659819</td><td>0.009829</td><td>26.01</td><td>2.96</td><td>8.8</td><td>Q2</td></tr><tr><td>f<sub>219</sub></td><td>253.797749</td><td>0.001111</td><td>3940.145275</td><td>0.017246</td><td>14.80</td><td>2.96</td><td>5.0</td><td>Q2</td></tr><tr><td>f<sub>072</sub></td><td>255.524679</td><td>0.000492</td><td>3913.516315</td><td>0.007532</td><td>33.42</td><td>2.96</td><td>11.3</td><td>C</td></tr><tr><td>f<sub>053</sub></td><td>255.603453</td><td>0.000427</td><td>3912.310213</td><td>0.006539</td><td>38.47</td><td>2.96</td><td>13.0</td><td>...</td></tr><tr><td>f<sub>001</sub></td><td>255.686462</td><td>0.000029</td><td>3911.040081</td><td>0.000440</td><td>571.74</td><td>2.96</td><td>193.4</td><td>...</td></tr><tr><td>f<sub>048</sub></td><td>262.149998</td><td>0.000402</td><td>3814.609983</td><td>0.005845</td><td>40.88</td><td>2.95</td><td>13.8</td><td>...</td></tr><tr><td>f<sub>077</sub></td><td>262.462766</td><td>0.000498</td><td>3810.064249</td><td>0.007225</td><td>32.99</td><td>2.95</td><td>11.2</td><td>...</td></tr><tr><td>f<sub>178</sub></td><td>262.707089</td><td>0.000961</td><td>3806.520804</td><td>0.013919</td><td>17.09</td><td>2.95</td><td>5.8</td><td>C</td></tr><tr><td>f<sub>083</sub></td><td>263.025718</td><td>0.000523</td><td>3801.909594</td><td>0.007564</td><td>31.37</td><td>2.95</td><td>10.6</td><td>...</td></tr><tr><td>f<sub>041</sub></td><td>263.241401</td><td>0.000349</td><td>3798.794555</td><td>0.005036</td><td>47.03</td><td>2.95</td><td>15.9</td><td>...</td></tr><tr><td>f<sub>139</sub></td><td>263.521574</td><td>0.000824</td><td>3794.755716</td><td>0.011863</td><td>19.92</td><td>2.95</td><td>6.7</td><td>C</td></tr><tr><td>f<sub>095</sub></td><td>274.698355</td><td>0.000615</td><td>3640.356710</td><td>0.008146</td><td>26.57</td><td>2.94</td><td>9.0</td><td>...</td></tr><tr><td>f<sub>037</sub></td><td>275.139416</td><td>0.000329</td><td>3634.521056</td><td>0.004349</td><td>49.61</td><td>2.94</td><td>16.9</td><td>...</td></tr><tr><td>f<sub>060</sub></td><td>275.355450</td><td>0.000453</td><td>3631.669534</td><td>0.005972</td><td>36.06</td><td>2.94</td><td>12.3</td><td>...</td></tr><tr><td>f<sub>171</sub></td><td>286.168421</td><td>0.000925</td><td>3494.445670</td><td>0.011294</td><td>17.51</td><td>2.91</td><td>6.0</td><td>...</td></tr><tr><td>f<sub>097</sub></td><td>286.905571</td><td>0.000628</td><td>3485.467345</td><td>0.007624</td><td>25.80</td><td>2.91</td><td>8.9</td><td>...</td></tr><tr><td>f<sub>165</sub></td><td>298.960621</td><td>0.000901</td><td>3344.922143</td><td>0.010083</td><td>17.75</td><td>2.88</td><td>6.2</td><td>...</td></tr><tr><td>f<sub>070</sub></td><td>299.299917</td><td>0.000478</td><td>3341.130226</td><td>0.005335</td><td>33.47</td><td>2.88</td><td>11.6</td><td>...</td></tr><tr><td>f<sub>225</sub></td><td>312.216182</td><td>0.001103</td><td>3202.908943</td><td>0.011317</td><td>14.25</td><td>2.83</td><td>5.0</td><td>...</td></tr><tr><td>f<sub>124</sub></td><td>358.699130</td><td>0.000714</td><td>2787.851759</td><td>0.005546</td><td>21.02</td><td>2.70</td><td>7.8</td><td>...</td></tr><tr><td>f<sub>019</sub></td><td>366.416573</td><td>0.000174</td><td>2729.134196</td><td>0.001297</td><td>85.58</td><td>2.68</td><td>31.9</td><td>...</td></tr><tr><td>f<sub>034</sub></td><td>389.015084</td><td>0.000263</td><td>2570.594409</td><td>0.001738</td><td>55.53</td><td>2.63</td><td>21.1</td><td>...</td></tr><tr><td>f<sub>189</sub></td><td>428.094211</td><td>0.000862</td><td>2335.934412</td><td>0.004703</td><td>16.55</td><td>2.57</td><td>6.5</td><td>C</td></tr><tr><td>f<sub>026</sub></td><td>441.504922</td><td>0.000220</td><td>2264.980413</td><td>0.001131</td><td>64.53</td><td>2.56</td><td>25.2</td><td>...</td></tr><tr><td>f<sub>071</sub></td><td>459.993053</td><td>0.000421</td><td>2173.945876</td><td>0.001991</td><td>33.43</td><td>2.53</td><td>13.2</td><td>...</td></tr><tr><td>f<sub>175</sub></td><td>460.457147</td><td>0.000820</td><td>2171.754757</td><td>0.003866</td><td>17.18</td><td>2.53</td><td>6.8</td><td>...</td></tr><tr><td>f<sub>148</sub></td><td>565.904241</td><td>0.000718</td><td>1767.083418</td><td>0.002241</td><td>19.45</td><td>2.51</td><td>7.7</td><td>...</td></tr><tr><td>f<sub>013</sub></td><td>566.395082</td><td>0.000137</td><td>1765.552053</td><td>0.000426</td><td>102.14</td><td>2.51</td><td>40.7</td><td>...</td></tr><tr><td>f<sub>062</sub></td><td>566.917800</td><td>0.000395</td><td>1763.924152</td><td>0.001228</td><td>35.36</td><td>2.51</td><td>14.1</td><td>...</td></tr><tr><td>f<sub>147</sub></td><td>567.450732</td><td>0.000714</td><td>1762.267530</td><td>0.002218</td><td>19.54</td><td>2.51</td><td>7.8</td><td>...</td></tr><tr><td>f<sub>141</sub></td><td>567.776009</td><td>0.000706</td><td>1761.257934</td><td>0.002190</td><td>19.76</td><td>2.51</td><td>7.9</td><td>...</td></tr><tr><td>f<sub>188</sub></td><td>614.603086</td><td>0.000845</td><td>1627.066349</td><td>0.002237</td><td>16.55</td><td>2.52</td><td>6.6</td><td>...</td></tr><tr><td>f<sub>194</sub></td><td>663.235867</td><td>0.000859</td><td>1507.759229</td><td>0.001952</td><td>16.36</td><td>2.53</td><td>6.5</td><td>...</td></tr><tr><td>f<sub>220</sub></td><td>663.652297</td><td>0.000950</td><td>1506.813138</td><td>0.002156</td><td>14.79</td><td>2.53</td><td>5.9</td><td>C</td></tr><tr><td>f<sub>221</sub></td><td>664.425967</td><td>0.000955</td><td>1505.058577</td><td>0.002162</td><td>14.71</td><td>2.53</td><td>5.8</td><td>...</td></tr><tr><td>f<sub>239</sub></td><td>716.967387</td><td>0.001061</td><td>1394.763580</td><td>0.002064</td><td>13.29</td><td>2.54</td><td>5.2</td><td>...</td></tr><tr><td>f<sub>192</sub></td><td>719.537727</td><td>0.000862</td><td>1389.781192</td><td>0.001665</td><td>16.37</td><td>2.54</td><td>6.5</td><td>...</td></tr><tr><td>f<sub>227</sub></td><td>719.896837</td><td>0.001002</td><td>1389.087921</td><td>0.001933</td><td>14.08</td><td>2.54</td><td>5.6</td><td>...</td></tr><tr><td>f<sub>222</sub></td><td>743.008917</td><td>0.000972</td><td>1345.878869</td><td>0.001761</td><td>14.55</td><td>2.54</td><td>5.7</td><td>...</td></tr><tr><td>f<sub>217</sub></td><td>743.222238</td><td>0.000954</td><td>1345.492572</td><td>0.001726</td><td>14.83</td><td>2.54</td><td>5.8</td><td>...</td></tr><tr><td>f<sub>121</sub></td><td>743.340177</td><td>0.000661</td><td>1345.279093</td><td>0.001197</td><td>21.38</td><td>2.54</td><td>8.4</td><td>...</td></tr><tr><td>f<sub>181</sub></td><td>743.476145</td><td>0.000836</td><td>1345.033068</td><td>0.001513</td><td>16.91</td><td>2.54</td><td>6.7</td><td>...</td></tr><tr><td>f<sub>156</sub></td><td>743.570018</td><td>0.000768</td><td>1344.863262</td><td>0.001389</td><td>18.41</td><td>2.54</td><td>7.2</td><td>...</td></tr><tr><td>f<sub>229</sub></td><td>743.790449</td><td>0.001007</td><td>1344.464697</td><td>0.001820</td><td>14.04</td><td>2.54</td><td>5.5</td><td>...</td></tr><tr><td>f<sub>215</sub></td><td>743.999350</td><td>0.000939</td><td>1344.087196</td><td>0.001697</td><td>15.05</td><td>2.54</td><td>5.9</td><td>...</td></tr><tr><td>f<sub>198</sub></td><td>744.155577</td><td>0.000871</td><td>1343.805020</td><td>0.001573</td><td>16.23</td><td>2.54</td><td>6.4</td><td>...</td></tr><tr><td>f<sub>177</sub></td><td>744.540215</td><td>0.000827</td><td>1343.110794</td><td>0.001491</td><td>17.10</td><td>2.54</td><td>6.7</td><td>...</td></tr><tr><td>f<sub>247</sub></td><td>747.071566</td><td>0.001119</td><td>1338.559846</td><td>0.002005</td><td>12.63</td><td>2.54</td><td>5.0</td><td>...</td></tr><tr><td>f<sub>103</sub></td><td>748.390627</td><td>0.000569</td><td>1336.200593</td><td>0.001016</td><td>24.83</td><td>2.54</td><td>9.8</td><td>...</td></tr><tr><td>f<sub>039</sub></td><td>748.699303</td><td>0.000293</td><td>1335.649701</td><td>0.000523</td><td>48.19</td><td>2.54</td><td>19.0</td><td>...</td></tr><tr><td>f<sub>176</sub></td><td>749.018036</td><td>0.000826</td><td>1335.081336</td><td>0.001473</td><td>17.11</td><td>2.54</td><td>6.7</td><td>...</td></tr><tr><td>f<sub>047</sub></td><td>749.331023</td><td>0.000340</td><td>1334.523688</td><td>0.000606</td><td>41.54</td><td>2.54</td><td>16.3</td><td>...</td></tr><tr><td>f<sub>166</sub></td><td>749.651593</td><td>0.000797</td><td>1333.953012</td><td>0.001418</td><td>17.74</td><td>2.54</td><td>7.0</td><td>...</td></tr><tr><td>f<sub>100</sub></td><td>773.066022</td><td>0.000556</td><td>1293.550579</td><td>0.000930</td><td>25.43</td><td>2.54</td><td>10.0</td><td>...</td></tr><tr><td>f<sub>073</sub></td><td>773.434573</td><td>0.000423</td><td>1292.934187</td><td>0.000707</td><td>33.40</td><td>2.54</td><td>13.1</td><td>...</td></tr><tr><td>f<sub>050</sub></td><td>773.817252</td><td>0.000350</td><td>1292.294786</td><td>0.000584</td><td>40.38</td><td>2.54</td><td>15.9</td><td>...</td></tr><tr><td>f<sub>167</sub></td><td>774.151054</td><td>0.000799</td><td>1291.737569</td><td>0.001333</td><td>17.69</td><td>2.54</td><td>7.0</td><td>...</td></tr><tr><td>f<sub>199</sub></td><td>774.320672</td><td>0.000872</td><td>1291.454608</td><td>0.001455</td><td>16.20</td><td>2.54</td><td>6.4</td><td>C</td></tr><tr><td>f<sub>108</sub></td><td>774.475248</td><td>0.000597</td><td>1291.196849</td><td>0.000995</td><td>23.68</td><td>2.54</td><td>9.3</td><td>...</td></tr></tbody></table>
<table><caption>**Table 2**
(Continued)</caption><thead><tr><th>ID</th><th>Frequency (μHz)</th><th>σ<sub>f</sub> (μHz)</th><th>Period (s)</th><th>σ<sub>P</sub> (s)</th><th>Amplitude (ppm)</th><th>σ<sub>A</sub> (ppm)</th><th>S/N</th><th>Comment<sup>a</sup></th></tr></thead><tbody><tr><td>f<sub>079</sub></td><td>774.669373</td><td>0.000441</td><td>1290.873287</td><td>0.000736</td><td>32.01</td><td>2.54</td><td>12.6</td><td>...</td></tr><tr><td>f<sub>087</sub></td><td>774.859849</td><td>0.000483</td><td>1290.555965</td><td>0.000804</td><td>29.27</td><td>2.54</td><td>11.5</td><td>...</td></tr><tr><td>f<sub>102</sub></td><td>775.247158</td><td>0.000568</td><td>1289.911211</td><td>0.000945</td><td>24.87</td><td>2.54</td><td>9.8</td><td>...</td></tr><tr><td>f<sub>232</sub></td><td>778.012553</td><td>0.001015</td><td>1285.326305</td><td>0.001676</td><td>13.93</td><td>2.54</td><td>5.5</td><td>C</td></tr><tr><td>f<sub>230</sub></td><td>778.293486</td><td>0.001008</td><td>1284.862353</td><td>0.001664</td><td>14.02</td><td>2.54</td><td>5.5</td><td>...</td></tr><tr><td>f<sub>132</sub></td><td>778.855337</td><td>0.000692</td><td>1283.935478</td><td>0.001140</td><td>20.44</td><td>2.54</td><td>8.0</td><td>...</td></tr><tr><td>f<sub>136</sub></td><td>779.136677</td><td>0.000699</td><td>1283.471861</td><td>0.001152</td><td>20.22</td><td>2.54</td><td>8.0</td><td>...</td></tr><tr><td>f<sub>123</sub></td><td>779.419432</td><td>0.000672</td><td>1283.006246</td><td>0.001106</td><td>21.04</td><td>2.54</td><td>8.3</td><td>...</td></tr><tr><td>f<sub>142</sub></td><td>779.698913</td><td>0.000716</td><td>1282.546357</td><td>0.001177</td><td>19.75</td><td>2.54</td><td>7.8</td><td>...</td></tr><tr><td>f<sub>234</sub></td><td>813.158821</td><td>0.001032</td><td>1229.772062</td><td>0.001561</td><td>13.74</td><td>2.55</td><td>5.4</td><td>...</td></tr><tr><td>f<sub>213</sub></td><td>823.704592</td><td>0.000940</td><td>1214.027467</td><td>0.001385</td><td>15.15</td><td>2.56</td><td>5.9</td><td>...</td></tr><tr><td>f<sub>228</sub></td><td>829.647777</td><td>0.001013</td><td>1205.330777</td><td>0.001472</td><td>14.05</td><td>2.56</td><td>5.5</td><td>...</td></tr><tr><td>f<sub>233</sub></td><td>887.126992</td><td>0.001021</td><td>1127.234330</td><td>0.001297</td><td>13.92</td><td>2.56</td><td>5.4</td><td>C</td></tr><tr><td>f<sub>158</sub></td><td>892.889763</td><td>0.000775</td><td>1119.959082</td><td>0.000972</td><td>18.35</td><td>2.56</td><td>7.2</td><td>...</td></tr><tr><td>f<sub>236</sub></td><td>893.502989</td><td>0.001041</td><td>1119.190436</td><td>0.001304</td><td>13.66</td><td>2.56</td><td>5.3</td><td>C</td></tr><tr><td>f<sub>218</sub></td><td>893.800898</td><td>0.000958</td><td>1118.817404</td><td>0.001200</td><td>14.83</td><td>2.56</td><td>5.8</td><td>C</td></tr><tr><td>f<sub>208</sub></td><td>931.646137</td><td>0.000925</td><td>1073.368912</td><td>0.001065</td><td>15.33</td><td>2.55</td><td>6.0</td><td>...</td></tr><tr><td>f<sub>245</sub></td><td>931.977262</td><td>0.001113</td><td>1072.987551</td><td>0.001282</td><td>12.73</td><td>2.55</td><td>5.0</td><td>C</td></tr><tr><td>f<sub>224</sub></td><td>932.121381</td><td>0.000981</td><td>1072.821653</td><td>0.001129</td><td>14.45</td><td>2.55</td><td>5.7</td><td>...</td></tr><tr><td>f<sub>061</sub></td><td>932.333607</td><td>0.000395</td><td>1072.577448</td><td>0.000454</td><td>35.92</td><td>2.55</td><td>14.1</td><td>...</td></tr><tr><td>f<sub>043</sub></td><td>932.995201</td><td>0.000310</td><td>1071.816875</td><td>0.000356</td><td>45.76</td><td>2.55</td><td>17.9</td><td>...</td></tr><tr><td>f<sub>129</sub></td><td>969.424430</td><td>0.000683</td><td>1031.539921</td><td>0.000727</td><td>20.68</td><td>2.54</td><td>8.1</td><td>...</td></tr><tr><td>f<sub>140</sub></td><td>969.887148</td><td>0.000714</td><td>1031.047790</td><td>0.000759</td><td>19.79</td><td>2.54</td><td>7.8</td><td>...</td></tr><tr><td>f<sub>133</sub></td><td>1002.646638</td><td>0.000690</td><td>997.360348</td><td>0.000687</td><td>20.40</td><td>2.53</td><td>8.1</td><td>...</td></tr><tr><td>f<sub>122</sub></td><td>1002.932567</td><td>0.000663</td><td>997.076008</td><td>0.000659</td><td>21.25</td><td>2.53</td><td>8.4</td><td>...</td></tr><tr><td>f<sub>146</sub></td><td>1012.924293</td><td>0.000717</td><td>987.240613</td><td>0.000699</td><td>19.64</td><td>2.53</td><td>7.8</td><td>...</td></tr><tr><td>f<sub>209</sub></td><td>1013.139431</td><td>0.000923</td><td>987.030975</td><td>0.000899</td><td>15.25</td><td>2.53</td><td>6.0</td><td>...</td></tr><tr><td>f<sub>231</sub></td><td>1013.376176</td><td>0.001006</td><td>986.800385</td><td>0.000980</td><td>13.99</td><td>2.53</td><td>5.5</td><td>...</td></tr><tr><td>f<sub>214</sub></td><td>1013.626213</td><td>0.000929</td><td>986.556964</td><td>0.000904</td><td>15.15</td><td>2.53</td><td>6.0</td><td>...</td></tr><tr><td>f<sub>137</sub></td><td>1014.114739</td><td>0.000702</td><td>986.081713</td><td>0.000682</td><td>20.06</td><td>2.53</td><td>7.9</td><td>...</td></tr><tr><td>f<sub>106</sub></td><td>1014.590863</td><td>0.000578</td><td>985.618969</td><td>0.000561</td><td>24.35</td><td>2.53</td><td>9.6</td><td>...</td></tr><tr><td>f<sub>244</sub></td><td>1014.818751</td><td>0.001104</td><td>985.397638</td><td>0.001072</td><td>12.74</td><td>2.53</td><td>5.0</td><td>...</td></tr><tr><td>f<sub>094</sub></td><td>1015.000200</td><td>0.000516</td><td>985.221481</td><td>0.000501</td><td>27.29</td><td>2.53</td><td>10.8</td><td>...</td></tr><tr><td>f<sub>237</sub></td><td>1015.216276</td><td>0.001033</td><td>985.011789</td><td>0.001002</td><td>13.62</td><td>2.53</td><td>5.4</td><td>...</td></tr><tr><td>f<sub>112</sub></td><td>1015.466213</td><td>0.000621</td><td>984.769347</td><td>0.000602</td><td>22.65</td><td>2.53</td><td>9.0</td><td>...</td></tr><tr><td>f<sub>104</sub></td><td>1015.875675</td><td>0.000567</td><td>984.372424</td><td>0.000549</td><td>24.82</td><td>2.53</td><td>9.8</td><td>...</td></tr><tr><td>f<sub>212</sub></td><td>1016.255729</td><td>0.000924</td><td>984.004293</td><td>0.000894</td><td>15.23</td><td>2.53</td><td>6.0</td><td>...</td></tr><tr><td>f<sub>206</sub></td><td>1016.716183</td><td>0.000899</td><td>983.558653</td><td>0.000870</td><td>15.65</td><td>2.53</td><td>6.2</td><td>...</td></tr><tr><td>f<sub>130</sub></td><td>1016.894687</td><td>0.000687</td><td>983.386002</td><td>0.000665</td><td>20.47</td><td>2.53</td><td>8.1</td><td>...</td></tr><tr><td>f<sub>240</sub></td><td>1133.915058</td><td>0.001050</td><td>881.900274</td><td>0.000817</td><td>13.21</td><td>2.50</td><td>5.3</td><td>...</td></tr><tr><td>f<sub>241</sub></td><td>1136.241019</td><td>0.001066</td><td>880.094965</td><td>0.000825</td><td>13.02</td><td>2.49</td><td>5.2</td><td>...</td></tr><tr><td>f<sub>161</sub></td><td>1508.399851</td><td>0.000766</td><td>662.954189</td><td>0.000337</td><td>18.01</td><td>2.48</td><td>7.3</td><td>C</td></tr><tr><td>f<sub>242</sub></td><td>1518.289633</td><td>0.001079</td><td>658.635861</td><td>0.000468</td><td>12.77</td><td>2.48</td><td>5.2</td><td>...</td></tr><tr><td>f<sub>235</sub></td><td>1573.083735</td><td>0.001005</td><td>635.694069</td><td>0.000406</td><td>13.70</td><td>2.48</td><td>5.5</td><td>...</td></tr><tr><td>f<sub>204</sub></td><td>1598.435967</td><td>0.000862</td><td>625.611548</td><td>0.000337</td><td>15.98</td><td>2.48</td><td>6.5</td><td>...</td></tr><tr><td>f<sub>159</sub></td><td>1650.709874</td><td>0.000752</td><td>605.799975</td><td>0.000276</td><td>18.32</td><td>2.48</td><td>7.4</td><td>...</td></tr><tr><td>f<sub>210</sub></td><td>1666.172440</td><td>0.000903</td><td>600.177974</td><td>0.000325</td><td>15.24</td><td>2.48</td><td>6.2</td><td>C</td></tr><tr><td>f<sub>155</sub></td><td>1696.132130</td><td>0.000734</td><td>589.576710</td><td>0.000255</td><td>18.65</td><td>2.46</td><td>7.6</td><td>...</td></tr><tr><td>f<sub>211</sub></td><td>1756.314565</td><td>0.000898</td><td>569.374086</td><td>0.000291</td><td>15.23</td><td>2.46</td><td>6.2</td><td>C</td></tr><tr><td>f<sub>226</sub></td><td>1786.454770</td><td>0.000956</td><td>559.767881</td><td>0.000299</td><td>14.25</td><td>2.45</td><td>5.8</td><td>...</td></tr><tr><td>f<sub>216</sub></td><td>1846.180859</td><td>0.000913</td><td>541.658741</td><td>0.000268</td><td>14.83</td><td>2.44</td><td>6.1</td><td>...</td></tr><tr><td>f<sub>246</sub></td><td>1853.772567</td><td>0.001071</td><td>539.440500</td><td>0.000312</td><td>12.65</td><td>2.44</td><td>5.2</td><td>...</td></tr><tr><td>f<sub>003</sub></td><td>2767.518348</td><td>0.000066</td><td>361.334551</td><td>0.000009</td><td>204.32</td><td>2.41</td><td>84.8</td><td>...</td></tr><tr><td>f<sub>101</sub></td><td>2782.428575</td><td>0.000537</td><td>359.398264</td><td>0.000069</td><td>24.95</td><td>2.41</td><td>10.4</td><td>...</td></tr><tr><td>f<sub>223</sub></td><td>2783.927382</td><td>0.000925</td><td>359.204772</td><td>0.000119</td><td>14.49</td><td>2.41</td><td>6.0</td><td>...</td></tr><tr><td>f<sub>042</sub></td><td>2816.458893</td><td>0.000290</td><td>355.055777</td><td>0.000037</td><td>46.22</td><td>2.41</td><td>19.2</td><td>...</td></tr><tr><td>f<sub>238</sub></td><td>3702.743363</td><td>0.000998</td><td>270.070027</td><td>0.000073</td><td>13.59</td><td>2.44</td><td>5.6</td><td>...</td></tr><tr><td>f<sub>243</sub></td><td>3702.864590</td><td>0.001062</td><td>270.061185</td><td>0.000077</td><td>12.77</td><td>2.44</td><td>5.2</td><td>...</td></tr></tbody></table>

**Note.**

<footnote>a C represents linear combination frequency, and Q indicates rotational quintuplet.</footnote>
<figure><img src="image_9.png" /></figure>

**Figure 9.** Frequency and AMs in the $Q_1$ quintuplet at 92 µHz. The top left panel presents the fine structure of the quintuplet with near-symmetric frequency spacing and two components missing. The long-dashed (blue) horizontal segments measure the exact value (the associated number) of frequency spacings between different components whose frequencies are marked by the short vertical segments. The dashed horizontal line in red represents the 5.6σ detection threshold. The middle left panel shows the sLSP (giving the amplitude in units of ppm as a function of frequency in µHz and time in days) of $Q_1$ as a whole, where three components clearly vary around their averaged frequencies (the vertical solid lines). The corresponding color bar is shifted to the top right. The bottom left panel shows expanded views around the average frequency (the solid vertical lines) of each component, with a precise measuring of the frequencies as functions of time. The lower right panels (from top to bottom) provide the measured amplitudes as functions of time for the sum and three components with determination of *m* number. More details are given in the text.

linearly increasing amplitude and frequency, or for the mode $f_{76}$, which has a linearly increasing amplitude but a decreasing frequency.

We identified another group of modes in which the amplitudes are relative high at the beginning but disappear afterward (or become lower than the 6σ threshold that we use for selecting modes). In this group, several AM and FM patterns are clearly periodic, such as for $f_{35} \sim 172.5$ µHz, which shows two cycles of about 700 days, or $f_{67} \sim 180.7$ µHz and $f_{75} \sim 119.1$ µHz, which experience one maximum. However, several modes also show periodic AMs but with linearly changing frequencies, such as $f_{104} \sim 1015.9$ µHz. As for the previous group, we, again, see correlations or anticorrelations happening between the AMs and their corresponding FMs.

A third group of apparently transient modes, with “lifetimes” of a few hundred days, are visible only in the middle of the *Kepler* observation run. Here the amplitude can vary periodically, while the frequency is either stable or variable, such as for the modes $f_{12} \sim 173.3$ µHz, $f_{85} \sim 173$ µHz, and $f_{117} \sim 108.1$ µHz. We note that $f_{12}$ and $f_{85}$ are components of the same multiplet and show similar AM and FM patterns. An anticorrelation is found between the amplitude and frequency variations of the mode $f_{121} \sim 743.3$ µHz.

Finally, we find that most of the modes are clearly visible over the entire observation (e.g., $f_{37} \sim 275.1$ µHz), while some modes only disappear below detection for short periods of time (e.g., $f_{66} \sim 187.8$ µHz). The modulations, for this group, present various patterns and may be similar or different. The AM and FM can be periodic (e.g., $f_{101} \sim 2782.4$ µHz) or quasi-periodic (e.g., $f_3 \sim 2767.5$ µHz). We note that periodic AM and FM with similar patterns are generally found in components forming a multiplet (e.g., $f_{50} \sim 773.8$ µHz and $f_{73} \sim 773.4$ µHz). Several modes show amplitudes and frequencies evolving in phase or anti-phase (correlation and anticorrelation; e.g., $f_{16} \sim 126.1$ µHz, $f_{68} \sim 183.2$ µHz, and $f_{103} \sim 748.4$ µHz). Several modes experience AM and FM on a timescale of a few months, such as the two components $f_2$ and $f_{28}$ at 212 µHz. For components of the same multiplet, whose amplitudes show correlated behaviors, their frequencies do as well (see, e.g., the components $f_{13} \sim 566.4$ µHz, $f_{62} \sim 566.9$ µHz, and $f_{123} \sim 565.9$ µHz and the components $f_{50} \sim 773.8$ µHz, $f_{73} \sim 773.4$ µHz, and $f_{100} \sim 773.1$ µHz). We point out that a few modes have frequencies showing very complex variations on the scale of a few hundredths (or tenths) of a µHz. These large FMs are observed, for example, in $f_{87} \sim 774.8$ µHz, or in $f_{10} \sim 105.6$ µHz, which has anti-phased AM and FM during the first 500 days of observation and then evolves in phase during the next 500 days.
<figure><img src="image_10.png" /><figcaption>**Figure 10.** Same as Figure 9, but for the Q₂ quintuplet at 253 µHz.</figcaption></figure>

**5. Discussion**

Most pulsation modes in KIC 3527751 are found to show various modulation patterns in both amplitude and frequency. These can be periodic or quasi-periodic changes, quasi-linear drifts, or in some cases more complicated patterns. We have not detected modes that could be considered as strictly stable, since even the relatively stable ones still exhibit some frequency and amplitude variations, even though these occur on a very small scale.

A natural interpretation for some of these modulations was originally proposed by Zong et al. (2016b), who argued that weak nonlinear interactions between resonant modes can induce the AMs and FMs. It was shown that such interactions indeed likely occur in the rotation-induced triplets observed in the DB white dwarf star KIC 08626021, based on continuous observations spanning over 2 yr. This was further confirmed in another star, the short-period-dominated hybrid sdB pulsator KIC 10139564 (Zong et al. 2016a), which shows similar AM and FM patterns occurring in several multiplets, based on 38 months of *Kepler* photometry. However, modulations observed in single modes, without obvious resonance conditions found with other identified modes, is still an open question. We indeed observe modulations in several independent apparently single modes in KIC 3527751. The various modulation patterns encountered, in particular when they are very complex, may induce pseudo-frequencies in the Fourier spectrum. Therefore, one should be extremely careful when identifying and interpreting frequencies extracted from the *Kepler* photometry. In particular, a robust identification of the

true eigenfrequencies that are really associated with the oscillation modes of the star is essential for seismic modeling (see, e.g., Charpinet et al. 2005a, 2005b).

**5.1. Comparison of Frequency Content with Previous Results**

The pulsations in KIC 3527751 were first discovered from an initial 1-month *Kepler* run, which led to the identification of 41 low-amplitude signals (Reed et al. 2010). Compared to this first analysis, our frequency extraction agrees on most of these 41 periodicities, except that one frequency detected at that time, at 894.394 µHz, is not above our secure 5.6σ detection threshold. This weak signal (97 ppm, 5.1σ in the Q2.3 data) has also been missed in Foster et al. (2015, hereafter F15), who also analyzed the complete 38-month light curve of KIC 3527751 and found 251 periodicities. The amplitude of this mode possibly decreased from Q2.3 afterward, since it can still be spotted at around 4σ in Q5, while completely disappearing later.

We made a meticulous comparison between the 251 periodicities found by F15 and ours, since we have analyzed completely independently the same data set. Because many peak structures are very complicated owing to the modulations described previously, neither F15 nor us can provide a frequency list that is completely unambiguous. A third-part analysis or forthcoming seismic modeling may provide an independent check. We first compare the 143 periodicities with measured AMs and FMs (Table 2) and find that they are in very good agreement with those from F15, except for eight frequencies that are discrepant by ~5.5%. The frequency
<figure><img src="image_11.png" /><figcaption>Figure 11. Same as Figure 9, but for the Q<sub>3</sub> quintuplet at 114 µHz.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 12.** Evidence of a correlation happening between AMs and FMs in the *m* = −2 component of *Q*<sub>3</sub>. The frequency has been shifted around the averaged value indicated by the horizontal line (in green) in the bottom panel. The two vertical arrows indicate the epoch when the amplitude reaches its minimum and the frequency its maximum.</figcaption></figure>

$f_{51}$ = 145.683 µHz from F15 (adopting for convenience our nomenclature) is resolved as two close frequencies at 145.696 and 145.761 µHz, which is likely induced by the observed modulations for that mode (Figure 14). The frequency $f_{58}$ = 105.439 µHz from F15 is extracted at 105.552 µHz, while the latter one should be the modulation-induced

frequency of $f_{10}$ = 105.605 µHz (Figure 14). Three frequencies $f_{28}$ = 211.8004 µHz, $f_2$ = 211.8831 µHz, and $f_{41}$ = 211.9568 µHz are well prewhitened, but F15 have not detected the two side frequencies (Figure 14). Similar differences are also found for the frequencies $f_1$ = 255.6864 µHz, $f_{56}$ = 255.6035 µHz, and $f_{72}$ = 255.5247 µHz because the highest peaks here are too significant, in contrast to their close frequencies. Due to very complex modulations in the frequencies $f_{41}$ = 263.241 µHz and $f_{83}$ = 263.0257 µHz (Figure 14), the latter one was not prewhitened in F15. The complicated pattern of AMs and FMs is the main factor that triggers some differences in frequencies from those of F15, in particular when several close frequencies are involved. We also find two frequencies $f_{31}$ = 126.3372 µHz and $f_{32}$ = 140.7126 µHz that are not listed in the F15 frequency list, probably due to the same reasons mentioned above.

Comparison with other frequencies, whose amplitudes are from ~8σ down to 5σ, shows some differences, in particular for the lowest-amplitude ones. We first mention that the four low-amplitude components of the *p*-mode multiplets near 2815 µHz identified as such by F15 are all missed (actually the S/N of their amplitudes, if they exist, would range from about 3σ to 4.5σ; see Figure 15). For the other *p*-mode multiplet near 2783 µHz, we only detect the two *l* = ±1 (if they are) side components without the central one mentioned in F15 (Figure 15). We miss another 30 frequencies whose amplitudes are typically about 13–17 ppm in F15. In fact, several of those low-amplitude peaks can be recognized in our analysis if we reduce our detection threshold down to 3σ–4σ. However, as discussed at length in Zong et al. (2016b), the false-alarm
<figure><img src="image_13.png" /><figcaption>**Figure 13.** FMs and AMs in the six relatively (at first sight) stable frequencies. Left panels: LSP profile centered on the main frequency peak. The (red) dashed lines indicate the $5.6\sigma$ detection threshold. The middle and right panels provide a precise measuring of the amplitudes and frequencies as functions of time. The (green) solid horizontal line represents the averaged frequency. Note that the top five frequency panels use the same scale, but not the bottom one.</figcaption></figure>

probability of choosing a peak that is not true signal is very high at such S/N thresholds with the very long time series dealt with. In terms of the total frequency number, comparing their 251 frequencies with our 247, this means that around 30 low-amplitude frequencies in our list are also missed by F15, which inversely suggests that discrepancies from our two lists are probably the result of noise fluctuations, since those frequencies have amplitudes just above ~$5\sigma$ (still below our secure threshold set to $5.6\sigma$ according to Zong et al. [2016b] analysis).

## 5.2. Is KIC 3527751 Really Rotating Differentially?

Foster et al. (2015) suggested that KIC 3527751 is rotating differentially based on the difference in frequency spacing between the low-frequency *g*-mode multiplets and the high-frequency *p*-modes that they identified as multiplets (see Figure 15). These authors claim that KIC 3527751 has an envelope rotating faster, with a period of ~15 days, than the

core whose rotation period is estimated to be ~43 days. This unusual configuration is at odds with classical expectations that would favor a core rotating similarly to, or more rapidly (but not slower) than, the envelope if angular momentum is conserved during the previous stage of stellar evolution (Kawaler & Hostler 2005). While an external mechanism, such as tidal forces to accelerate the rotation speed of the envelope, may be considered to explain the situation, the presence of a close companion has not been detected in KIC 3527751 thus far.

We argue here that the claim of differential rotation is in fact questionable, because we do not recover in our analysis the signature of rotation seen by F15 in the *p*-mode multiplets. We note that all peaks associated by F15 with *p*-modes forming multiplet components, if we specifically search for them in our data, would have very low amplitudes, at best with an S/N of $3\sigma-4\sigma$ (see Figure 15). This suggests that the low-amplitude
<figure><img src="image_14.png" /><figcaption>**Figure 14.** AMs and FMs of 128 modes presented individually in 128 groups of panels. Each group consists of four panels: the top left panel shows the LSP around that mode, where the dashed horizontal line (in red) denotes the 5.6σ threshold; the bottom left panel provides the sLSP around that frequency, and the color bar (indicating amplitude) is shifted to the leftmost side, where the vertical line indicates the averaged frequency; the precise measurements of the amplitude and frequency variations are given in the top right and bottom right panels, respectively. The frequencies are shifted to their average value as indicated by the horizontal line. We note that all amplitudes, frequencies, and times are in units of ppm, µHz, and days (from the date BJD 2,455,000), respectively.</figcaption></figure>

components of the claimed rotation multiplets at 2783 and 2815 µHz are at a very high risk of just being noise fluctuations (Zong et al. 2016b). This makes the actual detection of rotational splittings among the *p*-modes doubtful, in which case the claimed measurement of the rotation period of the stellar envelope cannot be trusted. In contrast, the identification of rotational splittings among the *g*-modes is well established, and the estimated rotation timescale for the core is secure. Hence, overall the conclusion from F15 that the envelope of KIC 3527751 is rotating more rapidly than the core should be taken with care.

### 5.3. Mode Modulations Possibly Linked to Nonlinear Interactions

In light of previous works by Zong et al. (2016a, 2016b), we tentatively interpret some of the described mode behaviors, in particular for the rotational multiplets, as the signature of nonlinear weak interactions when the modes involved satisfy the condition of being resonant. Several other mechanisms, such as magnetic activity, can also be responsible for mode modulations, but we will not describe them here (see details in Zong et al. 2016a, 2016b), in particular since the large diversity of observed AMs and FMs does not favor such interpretations that should affect modes similarly.
<figure><img src="image_15.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

The theoretical background that describes nonlinear interactions through the amplitude equation formalism can be found in a series of papers (e.g., Dziembowski 1982; Goupil & Buchler 1994;

Van Hoolst 1994; Buchler et al. 1995, 1997; Wu & Goldreich 2001), while observational signatures of such effects are described in Goupil et al. (1998) and Zong et al. (2016a, 2016b).
<figure><img src="image_16.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

### 5.3.1. *The Quintuplets in the Intermediate Regime*

We observed very suggestive periodic amplitude variations occurring in the three $ℓ = 2$ incomplete quintuplets $Q_1, Q_2$, and

$Q_3$. These, similarly to $ℓ = 1$ triplets, can be associated with the so-called intermediate regime of a nonlinear resonant coupling mechanism (e.g., Zong et al. 2016a). All the AMs, within these
<figure><img src="image_17.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

three-mode systems, vary on timescales of a few years, as illustrated in Figures 9–11, which is similar to that observed in the sdB star KIC 10139564 and the DB white dwarf

KIC 08626021, both observed with ~600-day modulation periods in several triplets (Zong et al. 2016a, 2016b). However, in Zong et al. (2016a, 2016b), triplets in different resonant
<figure><img src="image_18.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

regimes (namely, the frequency-locked regime, the narrow transitory regime, and the intermediate regime) have been found.

The modulation period of the intermediate regime is determined somewhat by the frequency mismatch between the three resonant modes (assuming here that an incomplete
<figure><img src="image_19.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

quintuplet is indeed reduced to a three-component resonance). However, the measured mismatch is at least one order of magnitude smaller than that estimated from the inverse

of the modulation period, e.g., comparing $\delta\nu \sim 0.006$ µHz in $Q_3$ (the maximum among the three quintuplets, where $\delta\nu = \nu_1 + \nu_2 - 2\nu_3$) with $1/P_{\text{obs}} \sim 0.02$ µHz. We note that
<figure><img src="image_20.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

frequency shifts caused by nonlinear corrections have values similar to the inverse of the modulation period. All of them are found varying on a frequency scale of a few hundredths of a

μHz, such as $Q_1 \sim 0.04, 0.06$, and $0.08$ μHz for the $m = -2, 0$, and +2 components of this multiplet, respectively. These variations in frequency generate uncertainties for
<figure><img src="image_21.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

measuring the true average frequency of the modes. Therefore, to some extent, the frequency mismatch may still be consistent with the modulation period observed in all the quintuplets, as is

expected from nonlinear stellar oscillation theory. This small frequency asymmetry, possibly forced by the nonlinear interactions, was also observed in the two stars
<figure><img src="image_22.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

KIC 10139564 and KIC 08626021. We note in particular that a very precise seismic solution for the latter star was recently obtained by Giammichele et al. (2018), who recovered the

eight observed independent *m* = 0 frequencies with an accuracy reaching the precision of the *Kepler* photometry. It is therefore probable that the observed frequencies are very
<figure><img src="image_23.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

close to the eigenfrequencies expected from linear pulsation theory, meaning that the nonlinear corrections to the frequencies are very small.

Another hint pointing toward nonlinear couplings is that we indeed observe correlations or anticorrelations between the involved modes, a signature of energy transfers occurring from
<figure><img src="image_24.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

complex nonlinear interacting processes between modes. The total “energy” $E = \sum |A_m|^2$, mainly reflecting the largest-amplitude component, has been observed below a certain

(maximum) amplitude during the observation, which may give further constraints on nonlinear coupling coefficients, if the total energy is indeed bounded. This amplitude saturation arises
<figure><img src="image_25.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

under certain conditions when the real parts of a few nonlinear coefficients are larger than zero (Buchler et al. 1995). However, calculating nonlinear coefficients is currently not within our

reach and would require a specific seismic model to be obtained for KIC 3527751 first. This would allow more quantitative comparisons of the observed modulation patterns
<figure><img src="image_26.png" /><figcaption>Figure 14. (Continued.)</figcaption></figure>

with theory. We point out that solving this problem could lead, for the first time, to relatively direct measurements of mode linear growth rates. Such measurements would constitute

interesting constraints on the driving mechanism itself. Finally, we note that it is not unexpected to observe that AM patterns are somewhat more regular than FM patterns in $Q_1$, $Q_2$, and $Q_3$.
<figure><img src="image_27.png" /><figcaption>**Figure 15.** Frequency spacing in the four multiplets used by Foster et al. (2015) to suggest radial differential rotation. The two top and bottom panels show multiplets in two low-frequency g-modes and two high-frequency p-modes, respectively. The (red) solid and (blue) dashed lines represent the 5.6σ and 5.0σ detection threshold, respectively. Filled and open circles indicate the position where frequencies forming the multiplets are claimed to be detected in Foster et al. (2015). In particular, open circles show insecure frequencies in our analysis that are below the detection threshold.</figcaption></figure>

This is because the frequency variations are affected by the AMs, while the AMs depend on the amplitudes themselves (see, e.g., Equation (28) and (32) in Buchler et al. 1995).

### 5.3.2. Interpretation for the Other Modes

Zong et al. (2016a, 2016b) mainly focused on modes under clear resonant conditions, such as three-mode resonance of the type $2\nu_0 \sim \nu_1 + \nu_2$ and $\nu_0 \sim \nu_1 + \nu_2$ that naturally occurs between components of rotational multiplets when slow rotation is involved. In KIC 3527751, we also observe many other modes whose amplitude and frequency show modulations despite the fact that they are not obviously paired with other modes. F15 identified most of the detected frequencies to be components of rotation multiplets. Since our frequency list is mainly consistent with the results of F15, we keep their identification for most modes, except eight frequencies, for simplicity. The observed AMs and FMs can exhibit very complicated patterns if they are components forming multiplets with degree $\ell \ge 2$, since more interacting terms should be considered in such multiresonant nonlinear systems. Eight relatively large amplitude components forming a (possibly high degree, $\ell > 3$) multiplet near 1015 µHz show different patterns of variations in frequency and amplitude. However, this kind of multiplet can also show somewhat regular variation patterns, such as in the presumably incomplete $\ell = 4$ nonuplet near 566 µHz. At present, we do not link these FMs and AMs occurring in such multiplets directly to nonlinear coupling effects, since the theory for such complex potentially resonant

systems has yet to be developed. Many-mode ($\ge 5$ modes) interactions are also suspected in KIC 10139564, where a complete quintuplet shows a diversity of mode behaviors from one component to the other (Zong et al. 2016a).

Among the relatively high amplitude modes, many are identified as components of $\ell = 2$ quintuplets (F15). Even if nonlinear resonant mode coupling theory does not currently cover the case of $\ell = 2$ multiplets, it is interesting to note that many modes show clear correlations or anticorrelations between their amplitude variation and their corresponding FMs (Figure 14). This is the first time that such an intriguing phenomenon is identified. This would indicate that frequencies are connected, through nonlinear effects, to the AMs, as is in fact predicted by nonlinear amplitude equations. Based on existing theoretical explorations, one can associate the observed modulations with three main regimes depending on the involved modes seen as a “triplet” component, or a “doublet” component, or even a “singlet.” The AMs and FMs that we have identified will provide constraints for future theoretical developments. Another important nonlinear resonance, the parametric instability, can also induce AMs and FMs and occurs between linearly growing (driven) modes and stable (damped) modes (see, e.g., Wu & Goldreich 2001). This mechanism may provide an alternative explanation for some of the single modes observed in KIC 3527751, such as the six relatively stable modes that still show small modulations (Figure 13). However, the exact interpretation for all the modulations seen in this star remains very complex and will require future dedicated work on the subject.

### 5.3.3. On Measuring Rates of Period Change

We briefly discuss here the pulsation timing technique that offers the opportunity to measure rates of period change induced by stellar evolution (see, e.g., Kepler et al. 2005; Hermes et al. 2013) or by the presence of stellar or planetary companions (see, e.g., Silvotti et al. 2007). In practice, this method requires that the measured frequencies are intrinsically stable, since variations could bring contamination on the phase measurement (a mode varying in frequency can induce shifts in phase; see, e.g., Equation (5)). For KIC 3527751 we find that the only six relatively stable modes still show slight AMs and FMs. Therefore, we cannot provide an unambiguous estimate of the secular rate of period change induced by stellar evolution effects. Radial velocity measurements from spectroscopy suggest that KIC 3527751 may have companion objects, but the orbital parameters cannot be constrained owing to the large uncertainties (Foster et al. 2015). Unfortunately, the small frequency variations in these six modes also prevent us from providing constraints on the determination of orbital parameters from the timing method. For this to become feasible, the FM effects should be modeled and removed beforehand. These observed FMs may also connect with their AMs as an intrinsic effect induced by the oscillation itself (e.g., Wu & Goldreich 2001). We stress that the observed AMs cannot be the consequence of instrumental effects from the *Kepler* satellite, which would cause a similar AM pattern for all modes (see, e.g., Bányai et al. 2013).

## 6. Conclusion

In this paper, the first of a series dedicated to a systematic survey of pulsations in evolved compact stars (hot B subdwarf
and white dwarf stars) as seen from *Kepler* photometry, we focus on characterizing the AMs and FMs of the oscillation modes detected in KIC 3527751, a long-period g-mode sdB pulsator that has been monitored continuously for more than 3 yr. Some pulsation modes in this star show clear AMs and FMs of periodic nature, a signature that can be associated with nonlinear mode interactions described by the amplitude equation formalism. Following the results of Zong et al. (2016a, 2016b), along with the present work, it becomes clear that oscillation mode varying in amplitude and frequency is a very common phenomenon in pulsating sdB and white dwarf stars. These can usually only be revealed by long and continuous high-quality photometry. In the forthcoming papers of the series, we will explore the other compact stars listed in Table 3.4 of Zong (2017) to unravel their mode properties and AMs/FMs.

A mode modulated in frequency and amplitude can widely broaden the imprint it has in the Fourier domain. We recalled that a varying mode can be represented mathematically by a (possibly infinite, but in practice often finite or truncated) series of sinusoidal waves that can be close together in frequency, thus appearing as one wide-structure signal. The complexity of a varying signal depends on how complicated the modulation pattern is, in particular for FMs. A series of simulations using artificial light curves including white noise and injected signals with various AM and FM patterns (periodic, linear, or parabolic) have been conducted. These simulations confirm that a modulated signal cannot be described by a well-defined peak in Fourier space (see Figures 1–3). In some cases when relatively large scale FMs induce many well-resolved peaks in Fourier domain, the interpretation of the extracted frequencies (usually assumed to be real eigenmodes) can become fuzzy and should be done with extra care. It is clearly necessary to identify close peaks by combining sLSPs (or any time-frequency analysis method), if modes are found with high enough amplitude to apply such techniques. These considerations are particularly important to extract frequencies with wide imprint from long-term high-quality spaceborne time-series photometry.

To obtain an accurate set of eigenfrequencies, we first reanalyzed the 38-month *Kepler* photometry collected on the sdB star KIC 3527751. We detected 204 frequencies above a conservative false-alarm 5.6σ threshold and an additional 43 suspected frequencies if we reduce the criterion down to 5σ. Our identified frequencies are in perfect agreement with those of F15 for most large-amplitude signals, except for eight frequencies that have very complex modulation patterns. In contrast, we find significant difference for more than 30 low-amplitude frequencies, particularly for six *p*-modes being components of two multiplets. As a consequence, we estimate that the differential rotation of KIC 3527751 claimed by F15 may not be reliable and, if present, needs to be confirmed by further observations.

We measured precise AMs and FMs for 102 pulsation modes with amplitudes ≳20 ppm (see Figures 9–11, 13, and 14). Among these modes, three identified incomplete quintuplets at 92, 114, and 253 µHz all exhibit seemingly (quasi-)periodic AMs and FMs that would be on a timescale of a few years. It is interesting to note that all components within the quintuplets clearly evolve in phase or anti-phase in terms of their amplitudes and corresponding frequencies. The total energy, $\sum A_m^2$, of $Q_1$, $Q_2$, and $Q_3$ has been observed to be bounded

during the 3 yr of observations. Various types of modulation patterns have been observed in the other 128 frequencies, including in six relatively stable ones. Some modes can appear or disappear in any period during the *Kepler* observations. However, most of the highest-amplitude frequencies are always found at any time over the entire observation run. The modulation patterns can be (quasi-)periodic as simple sinusoidal functions, quasi-linear (increasing or decreasing), or very complex. A striking character of those modulations is that many of them show (anti)correlations in the way their amplitude and frequency vary. We find that all the analyzed oscillations (143 frequencies) experience clear amplitude and frequency variations, and even the six relatively stable modes undergo small-scale amplitude and frequency variations.

The periodic modulations in amplitude and frequency occurring in the quintuplets $Q_1$, $Q_2$, and $Q_3$ can be explained as modes being in the intermediate regime of a nonlinear resonant mode coupling mechanism. We recall that these nonlinear mode interactions can induce various patterns in the amplitude variations and can weakly affect the linear eigenfrequencies. A similar phenomenon was previously found in observed triplets of two compact pulsators: the sdB star KIC 10139564 and the helium-atmosphere DB white dwarf KIC 08626021 (Zong et al. 2016a, 2016b). The modulation periods indicate that the frequency mismatch of $Q_1$, $Q_2$, and $Q_3$ should be of the order of a few hundredths of a µHz, while the observed one is at least one order of magnitude lower (typically a few n/µHz). The modulations in frequency themselves are of the order of a few hundredths to a tenth of a µHz. We have extracted six relatively stable modes that could normally be candidates for determining rates of period change and constrain an eventually orbital modulation in KIC 3527751. However, due to the remaining low AMs and FMs for these modes, unambiguous constraints on such variations may only be possible if the phase contamination is removed beforehand. The other 128 frequencies with AMs and FMs may have also been induced by nonlinear resonant couplings, since most of them have been identified as components of rotation multiplets. Resonances involving many modes can lead the modes to develop different types of AMs and FMs (possibly with much more complicated patterns). However, theoretical developments have yet to be done for such cases, which would involve much more nonlinear terms and coupling coefficients in the nonlinear amplitude equations. The most stringent evidence suggesting nonlinear weak interactions is that (anti)correlations have been found between the amplitude variations and the corresponding FMs in many of these modes, a signature that nonlinear frequency corrections steadily occur while the amplitude varies.

We emphasize that our investigation will help clarify identification of real eigenfrequencies in compact pulsators observed with *Kepler*. The ultrahigh precision and long time baseline indeed provide an unprecedented resolution and accuracy when measuring frequencies, but they also complexify mode extraction owing to the many close peaks or broadened structures created by long-term modulations. To avoid misidentifying frequencies, we find it important to combine sLSPs (sFT) (or equivalent time-frequency analysis techniques) with standard prewhitening approaches. It is remarkable that in KIC 3527751 no mode is found stable in amplitude over the entire observation. This is in contrast with KIC 10139564 and KIC 08626021, which showed very stable
components in several triplets. In the following papers of this
series, we will explore further such differences by analyzing the
other pulsating sdB and white dwarf stars from the *Kepler*
sample. This may allow us to estimate the fraction of varying
versus stable modes in compact pulsators.

We end this paper by emphasizing the prospect that other
instruments may further encourage research in this field, such
as the upcoming *TESS* satellite that will monitor brighter stars,
including compact pulsators, distributed over the whole sky
(Ricker et al. 2014). Such observations may lead to the
discovery of shorter variation timescales in bright pulsating
stars, if present. On the ground, opportunities for such projects
aiming at characterizing the precise modulation patterns also
exist. In particular, telescopes installed on the Antarctic plateau
can continuously make observations for a period of more than 3
months during polar nights (see, e.g., Zong et al. 2015).

W.Z., S.C., and G.V. acknowledge financial support from
Programme National de Physique Stellaire (PNPS) of CNRS/
INSU, France, and from the Centre National d'Etudes Spatiales
(CNES, France). W.Z. acknowledges the LAMOST fellowship
as a Youth Researcher and J.S. the LAMOST postdoctoral
fellowship, supported by the Special Funding for Advanced
Users, budgeted and administrated by the Center for Astro-
nomical Mega-Science, Chinese Academy of Sciences
(CAMS). J.-N.F. acknowledges the support from the National
Natural Science Foundation of China (NSFC) through the grant
11673003 and the National Basic Research Program of China
(973 Program 2014CB845700). The authors gratefully
acknowledge the *Kepler* team and all who have contributed
to making this mission possible. Funding for the *Kepler*
mission is provided by NASA's Science Mission Directorate.

Facility: Kepler.

Software: FELIX (Charpinet et al. 2010).

ORCID iDs

Weikai Zong https://orcid.org/0000-0002-7660-9803

Stéphane Charpinet https://orcid.org/0000-0002-6018-6180

Jian-Ning Fu https://orcid.org/0000-0001-8241-1740

Jia-Shu Niu https://orcid.org/0000-0001-5232-9500

Jie Su https://orcid.org/0000-0001-7566-9436

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