K2 photometry on oscillation mode variability: the new pulsating hot B subdwarf star EPIC 220422705

We present an analysis of oscillation mode variability in the hot B subdwarf star EPIC~220422705, a new pulsator discovered from $\sim78$~days of {\em K}2 photometry. The high-quality light curves provide a detection of 66 significant independent frequencies, from which we identified 9 incomplete potential triplets and 3 quintuplets. Those {\sl g-} and {\sl p-}multiplets give rotation periods of $\sim$ 36 and 29 days in the core and at the surface, respectively, potentially suggesting a slightly differential rotation. We derived a period spacing of 268.5\,s and 159.4\,s for the sequence of dipole and quadruple modes, respectively. We characterized the precise patterns of amplitude and frequency modulations (AM and FM) of 22 frequencies with high enough amplitude for our science. Many of them exhibit intrinsic and periodic patterns of AM and FM, with periods on a timescale of months as derived by the best fitting and \texttt{MCMC} test. The nonlinear resonant mode interactions could be a natural interpretation for such AMs and FMs after other mechanisms are ruled out. Our results are the first step to build a bridge between mode variability from {\em K}2 photometry and nonlinear perturbation theory of stellar oscillation.


INTRODUCTION
Hot B subdwarf (sdB) stars are burning helium in the core and typically wrapped in a thin hydrogen envelop at the surface. Their compact (log g = 5.2 ∼ 6.2 dex) and hot (T eff = 20000 ∼ 40000 K) properties place them to the extreme horizontal branch (EHB) in the Hertzsprung-Russell diagram (see Heber 2009Heber , 2016, for a review). A fraction of those blue faint objects have luminosity variations which can be attributed to oscillations of gravity (g-) or pressure (p-) modes or both (Green et al. 2003;Kilkenny et al. 1997;Schuh et al. 2006). Those modes are driven by the classical κ-mechanism due to an opacity bump produced by ionization of iron group elements (Charpinet et al. 1996(Charpinet et al. , 1997Fontaine et al. 2003). Due to their rich oscillations, sdB variables (sdBV) are good candidates to probe their interior via the tool of asteroseismology (Charpinet et al. 2005). weikai.zong@bnu.edu.cn; jnfu@bnu.edu.cn As advanced by observations from space, for instance, Kepler/K2 and TESS (Borucki et al. 2010;Howell et al. 2014;Ricker et al. 2015), oscillation frequencies in sdBV stars can be sharply resolved to unprecedented high precision, which leads to fruitful achievements for probing the interior of sdB stars (see, e.g., Charpinet et al. 2019;Reed et al. 2014;Van Grootel et al. 2010). There are 18 sdBV stars discovered in the original Kepler field (Østensen et al. 2010Pablo et al. 2011;Reed et al. 2012), among which most stars had been continuously observed after they were discovered to pulsate. In contrast to ground-based photometry, several sdBV stars are found with more than 100 frequencies such as KIC 03527751 (Foster et al. 2015;Zong et al. 2018). A preliminary mass survey on sdBV stars established that they are distributed around the canonical value ∼ 0.47M in a narrow region (Fontaine et al. 2012) with rotational periods distributed from a few days up to even hundreds of days Silvotti et al. 2021;Reed et al. 2021). In individual analyses, many sdBVs show clear variations in amplitude with a timescale much longer than their os-cillation periods Zong et al. 2016a;Kern et al. 2017). Focusing on amplitude modulations, Zong et al. (2016a) found that frequencies are not stable for many rotational components in KIC 10139564. They concluded that the amplitude and frequency modulations (AM/FM) can be attributed to nonlinear interactions of resonant mode coupling (Goupil & Buchler 1994;Buchler et al. 1995Buchler et al. , 1997, a mechanism of intense focus in other pulsators, for instance pulsating white dwarfs (Zong et al. 2016b) and slowly pulsating B stars (Van Beeck et al. 2021). Observational AM/FM variations provide strong constraints for the development of nonlinear stellar oscillation theory.
However, the Kepler space telescope had to begin the reborn mission with pointing using only two reaction wheels. This so-called K2 phase provided nearly-uninterrupted photometry for almost three months but could observe a larger spatial coverage than the original Kepler mission. Therefore, K2 offers a higher chance to finding more sdBV stars. In the 20 campaigns of K2, nearly 200 sdBV candidates were observed to search for pulsations or transits, leading to 10 sdBV stars already published (see, e.g., Reed et al. 2019; Baran et al. 2019;Silvotti et al. 2019). These ∼ 80 d K2 observations could also be helpful to characterize the amplitude modulations of pulsation modes in sdBVs (see, e.g., Silvotti et al. 2019). Similar to Kepler results, K2 photometry will shed new light on AM/FM oscillations in sdBV stars on shorter-term timescales.
As demonstrated by a series of works from Zong et al. (2016aZong et al. ( ,b, 2018, evolved compact pulsators, including pul-sating white dwarfs and sdBVs, could be excellent candidates to provide observational constraints to develop the nonlinear amplitude equations which describe how amplitudes and frequencies modulate. Gained from those experiences, we initiated a new survey of AM/FM in sdBV stars from K2, on relatively shorter modulation timescales appropriate for K2. In this paper, we concentrate on the bright sdB star, EPIC 220422705, or PG 0039+049, which has K p = 12.875 and is located at α = 00 h 42 m 06 s .124 and δ = +05 d 09 m 23 s .376. This star was originally identified as a faint blue star by Berger & Fringant (1980) and then was classified as an sdB star with spectra (Kilkenny et al. 1988). Moehler et al. (1990) derived atmospheric parameters of T eff = 26700 K and log g = 4.7 dex for EPIC 220422705, with a distance of d = 1050 ± 400 pc and refined by GAIA EDR3 to d = 916.6 +69.2 −81.5 pc (Gaia Collaboration et al. 2021). It is a binary system containing a cool companion as disclosed by Copperwheat et al. (2011) and further confirmed as a G2V dwarf star with a preliminary period of 150-300 (±220) days (Barlow et al. 2012). The structure of the paper is organised as follows: we analyze the photometric data from K2 and analyze the asteroseismic properties in Sect. 2. We then characterized the amplitude and frequency modulations of 22 frequencies in Sect. 3, followed by a discussion of those modulation details in Sect. 4. Finally, we summarize our findings in Sect. 5.
2. FREQUENCY CONTENT 2.1. Photometry and frequency extraction Table 1. All frequencies above the 5.2σ threshold detected in EPIC 220422705, by order of increasing frequency. Column (1) the identification (ID in order of decreasing amplitude), (2) and (3) lists frequencies in µHz and errors, (4) and (5) periods in seconds and errors, (6) and (7) amplitudes in ppm (parts per million) and errors, (8) signal-to-noise ratio (SNR) above the local noise level, (9) -(11) the quantum number identified by the asymptotic regime (see Sec. 2.4), and (12) comments on whether amplitude or frequency modulations were explored or not.   Note-AM/FM/AFM indicates that the frequency has modulation of amplitude (AM), frequency (FM) or both (AFM). '×' means the mode identified with period spacing but close to the value identified with potential splitting frequencies.

ID
EPIC 220422705 had been observed by K2 in shortcadence (SC) mode over a period of 78.72 days during Campaign 8. Its assembled light curves were downloaded from Mikulski Archive for Space Telescopes 1 (MAST). These archived data were processed through the EVEREST 1 https://archive.stsci.edu/k2 pipeline 2 . The photometry corrected by EVEREST has comparable precision to the original Kepler mission for targets brighter than K p ≈ 13 (Luger et al. 2018). EPIC 220422705 is within this brightness range.
The EVEREST flux was firstly shifted to the relative fraction to their mean value. We then used a six-order polynomial fitting to detrend the whole light curve due to residual instrumental drifts. To avoid discontinuities in the light curve across gaps longer than 0.02 days, we separated the light curve piece-wise where such gaps occurred for our fitting. This detrending method will flatten the light curves and dismiss signals with period ( 2 d) in Fourier transform, which will not have impact on the modulating patterns for our prime aim. We note that those signals are not con-cerned here due to the fact that the EVEREST pipeline may not recover those signals correctly. Then the light curves were iteratively clipped of a few outliers three times by filtering at 4.5σ around the light curve before we produced a Fourier transformation. Figure 1 (top panel) shows the final light curve of EPIC 220422705 which contains 106,444 data points over a duration of 76.43 days with a 1.2 d gap in the middle. The amplitude scatter clearly reveals multiperiodic signals of hours in a close-up view (middle panel). The corresponding Lomb-Scargle periodogram (LSP;Lomb 1976;Scargle 1982) up to the Nyquist frequency is shown in the bottom panel where the g-mode frequencies are clearly dominant in a region of [∼ 100 − 1000] µHz.
We used the specialized software FELIX 3 to perform frequency extraction from the light curves. The frequencies were prewhitened in order of decreasing amplitude until the value of 5.2 times the local noise level, a value that is the median amplitude in the LSP (Zong et al. 2021). This detection threshold is adopted as a compromise between 2-yr Kepler and 27-d TESS photometry (Zong et al. 2016b;Charpinet et al. 2019). The highest peak will be extracted in the case where there are several close frequencies of < 0.4 µHz, i.e., about 3 × ∆ f (∆ f = 1/T , and T ∼ 76.43 days). We have detected 66 independent frequencies and 13 linear combination frequenciess, with the highest (1172 ppm) frequency at 279.767 µHz, which are listed in Table 1.

p-and g-modes
Pulsating stars with both acoustic p-and gravity g-mode oscillations are excellent candidates to probe their internal profiles since acoustic and gravity waves propagate in different regions of the stellar structure (Aerts et al. 2010;Kurtz 2022). From spaceborne photometry, some g-mode dominated sdBVs are found with low-amplitude p-mode pulsations (see, e.g., Baran et al. 2017;Zong et al. 2018;Sahoo et al. 2020). A direct and easy way to distinguish the two different types of mode is by their pulsation period. In general, theoretical sdB star calculations suggest that dipole pmodes typically have frequency > 2500 µHz ( P < 400 s), whereas g-modes < 1000 µHz (P > 1000 s) Charpinet et al. 2005Charpinet et al. , 2011. But p-mode frequencies can decrease below 1700 µHz (periods can increase beyond 600 s) as T eff and log g decreases (Charpinet 1999;Charpinet et al. 2001Charpinet et al. , 2002. Figure 2 shows preliminary classification for the p-and gmode regions based only on the period. We detect 53 inde- Figure 2. Close-up views of the LSP of EPIC 220422705. The entire periodogram is divided into two different ranges: the low frequency g-mode region (two top panels) and the high frequency pmode region (two bottom panels). The horizontal dashed line denotes the 5.2σ detection threshold and the (blue) vertical segments at the top of each panel indicates locations of extracted frequencies.
pendent frequencies in the range [∼ 80 − 1000] µHz which are clearly g-mode (P > 1000 s) pulsations (two top panels). Another 7 independent frequencies are found in the high frequency p-mode (P < 400 s) region, [∼ 2600 − 3800] µHz (bottom panel). There are six independent frequenciesin the region of [∼ 1100 − 1400] µHz or [∼ 715 − 910] s, which might be low-order high-degree ( > 3) g-modes or mixed modes that need further classification (see, e.g., Charpinet et al. 2011Charpinet et al. , 2019. Those frequencies, hardly directly classified to be p-or g-mode by merely of their frequency value, can be used to penetrate a much larger portion of stellar interior or to detect the differential rotation in radial or longitude. However, determining the exact modes requires an exploration of seismic models. We note that a few frequencies were detected in this intermediate region in sdB stars observed with Kepler photometry, for instance, KIC 3527751 and KIC 10001893 (Foster et al. 2015;Uzundag et al. 2017).
In addition, we have resolved 13 linear combinations with frequencies < 1400 µHz which could be intrinsic resonant modes (Zong et al. 2016a) or non-linear effects from the linear eigenfrequencies (Brassard et al. 1995).

Rotational multiplets
From linear perturbation theory, an eigenmode of oscillation can be characterized by spherical harmonics that are described by three quantum numbers: the radial order n, the degree , and the azimuthal order m. When a star rotates, the degenerated m components will split into 2 + 1 multiplets. Referring to Ledoux (1951), their frequencies are related by, where ν n,l,0 is the frequency of the central m = 0 component, Ω is the solid rotational frequency, and C n, is the Ledoux constant. For acoustic p-mode, C n, is very near to zero and can be ignored, whereas it is estimated as C n, ∼ 1/ ( + 1) for high-radial order gravity g-modes.
To resolve any rotational split multiplet from spaceborne photometry, a minimum criterion is that the observations should cover at least twice the rotation periods. Charpinet et al. (2018) and Silvotti et al. (2021) present the distribution of rotation periods for sdB stars determined from Kepler photometry. Frequency multiplets found rotation periods from a few days to near one year with most having periods a bit longer than one month. This indicates that K2 photometry can likely resolve frequency multiplets in sdB stars. Figure 3 shows the frequency spacings of 12 groups of frequencies. We first consider six g-mode frequency groups that are detected with close frequency spacings around 0.17 µ Hz, which we consider to be dipole modes, i.e., f 5 ∼ 123.09 µHz, f 14 ∼ 180.39 µHz, f 41 ∼ 198.89 µHz, f 23 ∼ 288.56 µHz, f 19 ∼ 324.16 µHz and f 30 ∼ 501.25 µHz. The weighted (by 1/σ f i ) average value is 0.168 ± 0.016 µHz. Those dipole modes give a rotational frequency of 0.33 µHz which would mean quintuplet splitting of 0.28 µHz using C n,1 = 1/2 and C n,2 = 1/6. Three g-modes, f 18 ∼ 89.48 µHz, f 54 ∼ 622.79 µHz and f 24 ∼ 697.63 µHz, have close frequency spacings of ∼ 0.23 µHz, which could be rotational quintuplets, considering frequency uncertainties. We also resolve three = 1 p-mode multiplets, f 52 ∼ 3706.12 µHz, f 39 ∼3720.77 µHz and f 62 ∼3741.96 µHz, with low-amplitude peaks at a frequency distance of ∼ 0.4 µHz.
In order to determine the rotational period in a quantitative way, we propose a new approach that defines the rotation period associated with errors by their probability of occurrence. As shown in Figure 3, we adopt a Gaussian distributions, N ∼ (δ f i , σ f 2 i ), to represent the probability of a group of resolved frequencies at their shifted values. Here δ f i is the relative frequency to the central component. The probability of 68.27% (i.e., 1σ in N ∼ (0, 1)) was calculated to define the values and the uncertainties of frequency spacings. We obtained the values of 0.170 ± 0.049 µHz, 0.234 ± 0.035 µHz and 0.399 ± 0.132 µHz for = 1, and 2 g-modes and pmodes, respectively. The corresponding rotation periods are 34.04 +13.78 −7.12 d, 41.21 +7.22 −5.35 d and 28.86 +14.34 −7.21 d, respectively. Our result suggests that a slightly differential rotation occurs in EPIC 220422705 as g-and p-modes probe stellar interior under different depth (see, e.g., Kurtz 2022). We note that our results are completely based on only marginallyresolved frequencies with low-amplitudes near the detection limit. We do not detect multiplets in the four highestamplitude frequencies or 12 of the 13 highest-amplitude frequencies. Nevertheless, the rotation period we determine is consistent with that of a typical sdB star (see details in Charpinet et al. 2018;Silvotti et al. 2021). Foster et al. (2015) claims to detect a differential rotation in KIC 3527751 whose core rotates slower than the envelope, which, however, was challenged by an independent analysis of the same photometry by Zong et al. (2018), citing that the claimed rotational p-mode multiplets had missing components under significant confidence. In reverse, Kawaler & Hostler (2005) suggest that the core might rotate faster than the envelope of sdB stars from evolutionary models. In the case of EPIC 220422705, a binary system but with a poorly measured orbital period (Barlow et al. 2012), a slightly faster rotating envelope could be interpreted that the orbital companion has accelerated it via the tidal force. Theory predicts that angular momentum transportation leads to radiative envelope first synchronized then gradually proceeds to the inner part (Goldreich & Nicholson 1989). In combination with the poorly-determined orbital and rotation period (because of low-amplitude multi-plets), it would be unwise to speculate too much on this star. There are other sdBV stars which would be better for such work. To be cautious, EPIC 220422705 can still be a rigid object if the uncertainties of rotational periods are fully considered.

Period spacing
For g-mode pulsations in the asymptotic regime, consecutive high-radial orders (n ) follow a pattern of equal period spacing (see, e.g., Aerts et al. 2010), which depends on the structure. Seismology theory provides the following relationship, with Π 0 defined as, where N is the Brunt-Väisälä frequency and r is the radial coordinate. For the period spacing of = 2 sequence, it is related to the = 1 sequence as, Previous analysis of sdBV stars from Kepler and TESS reveals that the period spacing is about 250 s and 150 s for dipole and quadruple modes, respectively (see, e.g., Reed et al. 2011;Sahoo et al. 2020). To find the spacing periods in EPIC 220422705, we performed the popular Kolmogorov-Smirnov (KS) test on the independent g-mode frequencies. The KS test returns spacing correlations as highly-negative values for the most common spacings in a dataset (Kawaler 1988). We first apply the KS test to the six rotational (incomplete) triplets, which has a deepest trough at 63 s = 252/4 (Figure 4 a). In addition, we perform a linear fitting to those 6 periods with a result of ∼265.5 s. Then we applied another KS test for the 53 independent frequencies lower than 1000 µHz as probable g-modes, which gives a value around 276.8 s for dipole modes (Figure 4 a). All values are consistent with that of dipole modes in sdB stars (Reed et al. 2011;Sahoo et al. 2020).
Based on the preliminary period spacings, we have identified nine modes as dipole, 13 as quadruple, and additional 8 frequencies which fitted both period sequences. We note that the four identified modes include one of the above 6 dipole modes, f 5 ∼ 123.1 µHz. We obtained the average period spacing of 268.5 ± 2.8 s and 159.4 ± 0.6 s for = 1 and = 2 modes, respectively, via linear fitting to 17 (4 + 5 + 8) dipole modes and 21 (13 + 8) quadruple modes (Figure 4 b). We list and relative n values in Table 1. We note that the real radial order can only be obtained through seismic modeling. Our results suggest that EPIC 220422705 has a somewhat large period spacing among the known sdB variables, in a range of [220,270] s for the dipole mode (see, e.g., Reed et al. 2011;Sahoo et al. 2020;Uzundag et al. 2021). As stellar models presented in Uzundag et al. (2021), the evolution tracks suggest that the lower value of period spacing, the lower value of log g. This agrees well to a low log g derived for EPIC 220422705. However, atmospheric parameters, with a much higher precision, are encouraged for EPIC 220422705 to test the results of Uzundag et al. (2021) in future.
Theéchelle diagrams for two sequences are presented in Figure 4 (d) where the = 2 sequence is more consistent than the = 1 sequence. There are two = 1 frequencies with larger period deviations, f 19 ∼ 324.16 µHz, f 30 ∼ 501.25 µHz whereas only one occurs in the = 2 sequence, f 8 ∼ 218.36 µHz. Asymptotic theory indicates that period spacings are determined by the size of the pulsation resonant cavity (see e.g. Tassoul 1980) and could be affected by the extent of the convective core (Smeyers & Moya 2007). The ideal pattern for a period sequence in theéchelle diagram is a vertical ridge for central components of a star with internallyhomogeneous composition. Some deviations from the mean period spacing are to be expected in g-mode pulsating sdB stars -and indeed have already been unambiguously detected in some cases (Østensen et al. 2014) -due to the phenomenon of mode trapping by steep composition gradients. Such deviations reflect properties of the stellar interior, and in particular could reveal the character of the mixing processes at work near the core boundary (see, e.g., Charpinet et al. 2011;Ghasemi et al. 2017). Figure 4 (c) shows the deviation of period spacing as a function of the reduced pulsation period. We only observe a large deviation that might be associated to a trapped mode at the fifth order. Indeed, seismic models suggest that strong trapping are more likely found for the lower order (higher frequency or shorter period) modes than the higher order g-modes (Charpinet et al. 2014).

AMPLITUDE AND FREQUENCY MODULATIONS
This section provides our methodology and characterization of amplitude and frequency modulations (AM/FM) for the most significant frequencies. In practice, we follow the processes as described in Zong et al. (2018) to extract frequency information of subsets of the entire light curve. Here the time interval and window width are 1 d and 30 d, respectively. If close peaks within the frequency resolution ( ∼ 0.4 µHz) are detected, we keep the highest peak as the measured value for that frequency. In order to measure AF/FM significantly, frequencies should have amplitudes above 8.8σ of the local noise in each piece of the light curve. This ultimately leads to 22 frequencies that could be analyzed for AMs/FMs, which are marked in the last column of Table 1.

The fitting method
A quick look of all AM/FM patterns occurring in these 22 frequencies suggest that most of them exhibit simple or quasi-regular variations (Figures 5, 6, 7 and 8). In order to quantitatively characterize the modulation patterns, we apply three simple types of fittings: linear, parabolic, and sinusoidal waves. We first calculate standard deviations of our sub-set data, σ 0 . Then we adopt a simple AM/RM fitting, typically linear first and then a second type on the residuals. This fitting method may be iterated several times until the residuals present no clear structure and look like random noise. For each fitting, a standard deviation of residuals will be calculated as: where G k (t) defines as, G k (t) = g 1 (t), g 2 (t), g 3 (t), g 1 (t) + g 3 (t), g 2 (t) + g 3 (t) , Here k = 1, 2, 3, 4, 5 and N denotes number of data points, m i are the measured values of amplitude or frequency. In principle, the freedom degree of G k (t) increases as k increases, which in turn results in a lower σ k . However, to avoid overfitting of the modulation patterns, for instance, by including a linear or parabolic fitting, we follow the statistical test by Pringle (1975) who defines λ as, Here D i is the freedom degree of the fitting function, e.g., 2 and 3 for linear and parabola fitting, respectively. A significant improvement for the higher freedom degree fitting should have the parameter that meets the F−distribution, λ > F P=99.75% (D 2 − D 1 , N − D 2 ). For each AM/FM, we prefer to keep the fitting function with a lower freedom degree as indicated by the parameter λ. The results of our fitting for each AM/FM is listed in Table 2 where most frequencies have a sinusoidal component, G 3 (t), or with an additional linear fitting, G 4 (t). After the fitting function was set, we adopted the posterior distributions based on the Bayesian frame to estimate the best fitting parameters and their uncertainties. The posterior distributions of parameters are sampled by the Markov Chain Monte Carlo (MCMC), which is implemented by the EMCEE code (Foreman-Mackey et al. 2013). The MCMC sampling is performed with more than 22 chains, according to the number of free parameters. It proceeds until the chains converge to values that are inferred by the auto-correlation time module of the EMCEE. Parameters of the best fittings are given by the medians of marginalized posterior distributions, associated with the corresponding errors that are calculated by the half-widths between the 16 th and 84 th percentiles of the distributions. Figure 5 is an example of the MCMC application for AM and FM occurring in the frequency f 3 ∼ 328.75 µHz. We clearly see that both AM and FM exhibit periodic variations but with different periods and phases. In general, we find that almost all measurements are consistent with the fitting curves, accounting for uncertainties. The best fittings return A = 8.61 ± 0.54 ppm and 6.1 ± 0.3 nHz, T = 14.4 ± 0.1 d and 20.4 ± 0.2 d, φ = 5.2 ± 0.1 and 1.4 ± 0.1, b = −0.91 ± 0.05 and −0.08 ± 0.02, and c = 514 ± 1 and 2.4 ± 0.4 for AM and FM, respectively.   (1) and (2) the ID and frequency taken from Table 1, Column (3) the value of correlation between the amplitude and frequency modulations, Column (4) the indication of amplitude modulation (AM) or frequency modulation (FM), Column (5), (6) and (7) the fitting coefficients amplitude (A), period (T = 2π/ω) and phase (φ) of sinusoidal function if periodic patterns found, (8), (9) and (10) the coefficients of polynomial fitting up to second order, Column (11) the index of fitting function to the modulation pattern and (12)      Note-B and I denote the beating effect and intrinsic modulations of amplitude and frequency, respectively.

Characterization of modulation patterns
In this section, we describe the modulation patterns in AM/FM of all 22 significant frequencies except the frequency of f 3 which has already been described. Figure 6 is a gallery of 16 frequencies with various modulation patterns, which are identified by different fitting function G k (t) and summarized in Table 2. In general, the fitting periods of AMs and FMs are on timescale of months.
We now specifically describe these AMs and FMs. A few modes have been observed with simple modulation patterns of only linearly decreasing or increasing in amplitude or frequency. For instance, the amplitude of f 9 ∼ 143.4 µHz decreases from 200 ppm to 100 ppm over the time interval of about 50 days. In other observed minor cases, a few modes can be characterized by a simple parabolic fitting, such as the AM/FM of f 6 ∼ 306.2 µHz where both the amplitude and frequency reach to their vertex values near the time BJD = 2457420 d. The majority of modulations we observed have quasi-sinusoidal patterns, most with extra linear and a few with parabolic fittings. For example, the amplitude and frequency of f 8 ∼ 218.4 µHz exhibit completely sinusoidal pattern and evolve in phase. Whereas the frequency of f 13 ∼ 299.0 µHz demonstrates a sinusoidal pattern with an additional linear fitting. We only find f 16 ∼ 139.9 µHz whose frequency and f 10 ∼ 187.5 µHz whose amplitude show a sinusoidal plus a parabolic pattern. We note that the amplitude of f 14 ∼ 180.434 µHz decreases below the detection threshold from BJD − 24554000 = 200 d to 225 d, which holds even if our subsets span 70 d. For most frequencies, a variation of frequency scale is around 20 nHz but there are a few exceptions with large values around 100 nHz.
For instance, f 20 ∼ 2781.672 µHz, f 19 ∼ 324.164 µHz and f 23 ∼ 288.547 µHz. The variations of amplitude can span up to a few hundred ppm ( f 1 ∼ 279.767 µHz) or down to a few ppm ( f 4 ∼ 261.269 µHz). We note that the amplitudes of AM and FM are not strictly proportional to each other. For instance, the amplitude of f 1 increases from about 50 ppm to 250 ppm, but the frequency only varies around 1 nHz.
A very interesting feature of the observed AM/FM is that several frequencies are found with (anti-) correlations between AM and FM. We thus calculate the values of correlation for all frequencies as listed in Table 2. The frequencies f 23 ∼ 288.561µHz and f 6 ∼ 306.223 µHz all exhibit strong correlation with a coefficient |ρ a,f | > 0.8 be-tween their amplitude and frequency variation. For instance, f 23 and f 6 show clear anti-correlation between AM and FM, with derived coefficients of -0.867 and -0.824, respectively. For f 23 , the amplitude exhibits a slightly decreasing trend whereas the frequency is opposite. For f 6 , its AM and FM are both represented by parabolic fits but with anti-phase evolutions. We also note several frequencies with correlated AM and FM patterns showing sinusoidal variations, such as f 15 ∼ 173.4 µHz and f 8 ∼ 218.3 µHz. Figure 7 shows another 4 frequencies, f 5 ∼ 123.087 µHz, f 7 ∼ 80.595 µHz, f 11 ∼ 312.0 µHz and f 39 ∼ 3720.7 µHz, whose FM patterns still present modulation structures after fitting by a G k (t) function with k = 4, 5. To remove those FM residuals, they needed additional sinusoidal function, i.e., three fitting functions to present FM patterns. For a strong correlation frequency, f 5 , with ρ a,f = 0.862, exhibits regular AM and FM with period of 101 +14 −14 and 40 +1 −1 days, respectively, derived from EMCEE. The other three frequencies, f 7 , f 11 and f 39 are determined by EMCEE with either different fitting types of functions or different periods of sinusoidal patterns between AM and FM. For instance, the periods of AM and FM in f 7 are calculated as 23 +1 −1 and 49 +1 −1 days, respectively, almost with a ratio of 1 : 2. The EMCEE returns periods with values of ∼ 10.3, 19.0, 17.4 and 11.3 days for the additional sinusoidal fittings of the FM residuals of f 7 , f 5 , f 11 and f 39 after extracting G k (t), respectively. We note that FM periods of these residuals are shorter than the periods fitted by G k (t). Figure 8 presents the AM and FM of the frequency f 24 ∼ 697.62 µHz that shows a quasi-regular behavior but cannot be described by simple fittings of G k functions, which is also suggested by the fine profile of the LSP and the sliding LSP (sLSP). Thus we do not perform fitting and EMCEE on its AM and FM. We observe that both the AM and FM began with a decreasing trend: the amplitude went down from ∼ 80 ppm to a local minimum value of ∼ 60 ppm and the frequency varied from +120 nHz down to -100 nHz relative to its averaged frequency. Then the frequency and amplitude had experienced an increasing trend. The amplitude reaches to its maximum with a time interval of about 30 days but passing across one stationary point, whereas the frequency generally went up to around the average value with a back and forth trend. In this section, we will discuss the potential interpretation of the observed AM and FM for all 22 frequencies of EPIC 220422705. Most of those modulations can be fitted with simple functions, G k (t), with most having sinusoidal fittings. They could be induced by the resonant coupling mechanism that predicts periodic amplitude and frequency modulation as a consequence of nonlinear weak interaction between different coupling modes (see, e.g., Dziembowski 1982;Buchler et al. 1995). However, the relative short dura-tion of K2 photometry may suffer from other effects on the observed AMs and FMs, such as beating between unresolved frequencies. We therefore generate a series of quantitative simulations of close signals and compare the modulation patterns of those unresolved frequencies with the observed patterns of AM/FM.  As presented in Section 2.3, we detected several potential rotation multiplets with frequency spacings of about 0.2 to 0.4 µHz, which is comparable to the frequency resolution determined by the observation duration, ∆ f ≈ 1/T . This finding suggests that close frequencies, not only the multiplets we found, may be present in EPIC 220422705 with very high probability. In literature, many sdBV stars are resolved with such nearby frequencies from the original 4-yr Kepler observations (see .e.g, Zong et al. 2018;Foster et al. 2015;Baran et al. 2012). Those close frequencies will induce amplitude variations not frequency variations, called beating, if they are unresolved, as demonstrated in Zong et al. (2018). Here we only take two close frequencies as an example, z = A 1 sin[2π(ω 0 t + φ 1 )] + A 2 sin[2π(ω 0 t + ∆ωt + φ 2 )]. (8) If we consider two comparable amplitudes, A 1 ∼ A 2 = A, and a very small frequency separation, ∆ω ω 0 , the equa- tion (8) will be reduced to,

Presence of closely-spaced frequencies
The above equation indicates that two close frequencies will generate amplitude modulations if they were not resolved. We thus have to evaluate this effect on the observed AMs in EPIC 220422705. A series of simulations had been performed to quantitatively compare the observed and calculated AMs. The simulation process are similar to that of Zong et al. (2018), including light curve construction and frequency injection, but adopted to K2 observations. In practice, we set three parameters A 1 , A 2 and ∆ω to be constant in construction of each light curve based on equation (8) and phase to be zero for simplicity. We then make all parameters as variants in a series of 1331 light curves, with A 1 , A 2 ∈ [0.5, 0.6, ..., 1.5] × A 0 and ∆ω ∈ [0.1, 0.12, 0.14, ...0.3] µHz. Then we transform the simulated light curves into sLSP and directly compare them with the observed sLSP via direct subtraction. Both types of sLSPs have to be transformed into the same time step and window length and scaled to the same maximum amplitude. The flatness of the residual sLSP defines the goodness of similarity between the two sLSPs. We finally select an optimal sLSP for the simulation and return the parameters. Figure 9 shows the comparison results for five representative frequencies, f 9 ∼ 143.4 µHz, f 13 ∼ 299.0 µHz, f 11 ∼ 312.0 µHz, f 24 ∼ 697.6 µHz and f 20 ∼ 2781.6 µHz. We can conclude that the modulations of f 13 and f 20 are potentially induced by two close frequencies as revealed by no significant residuals. The modulation of f 11 and f 24 can hardly be simulated by two close frequencies, which indicates the AM and FM to be intrinsic, as revealed by the complex structure of the residual sLSP. The sLSPs of f 9 suggest that it is dominated by the beating effect but also experienced intrinsic AM and FM. We list the results of simulations in the last column in Table 2 as: 'B' for the completely beating effect and 'I' for intrinsic modulation. Here we note that the index 'I' may also contain beating effect such as the close multiplets discovered but their AMs cannot completely represented by those beating effect. In summary, we found that five frequency modulations can be attributed to beating, f 5 ∼ 123.1 µHz, f 12 ∼ 166.8 µHz, f 2 ∼ 207.2 µHz, f 13 ∼ 299.0 µHz and f 20 ∼ 2781.1 µHz. Their modulations will not be discussed later. However, the other 17 frequencies are not well-represented by beating and must suffer from intrinsic modulations in amplitude and frequency.

Potential interpretation of intrinsic modulation
As stated above, in EPIC 220422705 many frequencies exhibit intrinsic AM and FM even though they may be contaminated by unresolved frequencies limited by K2 observing du-ration. These kinds of variations have previously been investigated for several sdBV stars (Zong et al. 2016a(Zong et al. , 2021, showing several characteristics of their modulation patterns: stable, regular, irregular or complex features. In contrast to Kepler sdBV stars, the observed AMs and FMs here can only be determined for shorter temporal modulations or suffering from the beating of close frequencies. For instance, Zong et al. (2016a) found that in KIC 10139564 the modulating period is about 600 days for the dominant triplet which is much longer than the derived periods of months in EPIC 220422705.
From a theoretical perspective, we expect various types of AMs and FMs when perturbation theory is extended to nonlinear orders where different resonant modes can have weak interactions governed by amplitude equations (see, e.g., Dziembowski 1982;Buchler & Goupil 1984;Buchler et al. 1995;Moskalik 1985;Buchler et al. 1997). Both in multiplet resonance and direct parent-daughter resonance (e.g., is a key parameter to determine the modulating timescale together with the coupling coefficients and linear damping and growth rates. The latter ones cannot be directly obtained from observation while δ f can be measured if the consecutive subset light curves are long enough. The values of those quantities determine the exact modulation patterns of AMs and FMs.
We do not provide direct calculations of nonlinear amplitude equations constrained by the observed results since many of the physical quantities are not currently available. The linear growth/damping rates and coupling coefficients need sophisticated seismic models before they can be determined. Once those physical quantities are available, we could constrain the growth/damping rates using the observed modulation periods provided in this analysis. At least we can conclude that oscillation modes are unstable and this characteristic is a ubiquitous phenomenon in oscillation modes of this pulsating sdB stars.
Other mechanisms can also produce frequency modulations but in a systematic trend. For instance, magnetic cycles generally lead all frequencies to shift with a similar pattern (Salabert et al. 2015). Magnetic field is very rare or completely absent in sdB stars (Landstreet et al. 2012), except one particular object claimed to be produced through the merger channel (Vos et al. 2021). In addition, sdB stars have very stable radiative envelopes and are not known to show magnetic cycles. Frequency or phase modulations can be induced by orbital companions through periodic variations of light travelling time (Silvotti et al. 2007(Silvotti et al. , 2018Murphy & Shibahashi 2015). This kind of FM for all frequencies has to be found with identical orbital period and phase. EPIC 220422705's FMs and AMs cannot be well explained by the above two mechanisms in terms of their modulation patterns. Mont-gomery et al. (2020) recently proposed that temporal changes of depth of a surface convective zone can distort the coherent pulsations in hydrogen-atmosphere white dwarfs. That could produce AMs and FMs in sdB stars, but there is no significant convective zone near the surfaces of sdB stars. They also claim a much wider frequency width than what we observed in EPIC 220422705.

CONCLUSION
We have analyzed the nearly consecutive K2 photometry spanning ∼76 d on EPIC 220422705, a g−mode dominated hybrid pulsating sdB star. A rich frequency spectrum with 66 independent frequencies are detected above the 5.2 σ threshold. We attribute 12 frequencies to unresolved rotational multiplets. Rotational periods of 34.04 +13.78 −7.12 d, 41.21 +7.22 −5.35 d and 28.86 +14.34 −7.21 d were derived based on six dipole g-modes, three quadruple g-modes, and three dipole p-modes, respectively. This suggests that EPIC 220422705 has a differential rotation with a slightly slower core than the envelope. The period spacings within the asymptotic regime are derived with ∼ 268.5 s and 159.4 s for dipole and quadruple modes on average, respectively. We thus identified 9 dipole modes and 13 quadruple modes with eight additional periods that could fit both sequences.
We characterize 22 significant frequencies with amplitude and frequency modulations. All those frequencies show clear modulation patterns which are then fitted with simple functions and their uncertainties were tested by MCMC simulations. Most of those AMs and FMs can be fitted with periodic patterns with periods on a timescale of months which is relatively shorter than that found from Kepler sdBV stars . A notable feature of the modulations we detect is that they exhibit (anti-) correlations between their amplitude and frequency, a similar result to that in Kepler sdBV stars. Limited by the duration of K2 photometry, we have not performed any detailed characterization of the relationship between resonant modes of those modulations since the frequency resolution is not precise enough.
To quantitatively determine whether the discovered modulations are intrinsic or result from two close frequencies, a series of close-frequency simulations were produced and sliding LSPs were compared for each of the 22 frequencies.
Only five frequencies are well-represented by two close frequencies. Thus we conclude that 17 frequencies have AMs and FMs which must be intrinsic modulations.
A natural interpretation for such mode variability is the nonlinear mode interactions through resonance (see, e.g., Buchler et al. 1995). Depending on the physical quantities in the amplitude equations, resonant modes can have various types of modulation patterns both in amplitude and frequency. We have expelled other mechanisms account for our findings although they can also generate AM and FM, for instance, phase variations as depth-of-convective-zone changes (Montgomery et al. 2020). Finally, our results are the first step to precisely characterize the patterns of mode modulations in sdB stars from K2 photometry. Similar to recent results from Kepler (e.g., Zong et al. 2021), as well as several compact pulsators in the continuous view zones of TESS to be analyzed, these AMs and FMs will open a new avenue to develop nonlinear stellar oscillation theory in the near future.