K2 Photometry on Oscillation Mode Variability: The New Pulsating Hot B Subdwarf Star EPIC 220422705

EPIC 220422705 had been observed by K2 in short-cadence (SC) mode over a period of 78.72 days during Campaign 8. Its assembled light curves were downloaded from Mikulski Archive for Space Telescopes ⁷ (MAST). These archived data were processed through the EPIC Variability Extraction and Removal for Exoplanet Science Targets (EVEREST) pipeline. ⁸ 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 first shifted to the relative fraction of their mean value. We then used a sixth-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 piecewise where such gaps occurred for our fitting. This detrending method will flatten the light curves and dismiss signals with a period (≳2 days) in Fourier transform, which will not have an impact on the modulating patterns for our prime aim. We note that those signals are not concerned 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 days 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.

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We used the specialized software FELIX ⁹ 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 days of 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 × Δf (Δf = 1/T, and T ∼76.43 days). We have detected 66 independent frequencies and 13 linear combination frequencies, with the highest (1172 ppm) frequency at 279.767 μHz, which are listed in Table 1.

Pulsating stars with both acoustic p-mode oscillations 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 modes is by their pulsation period. In general, theoretical sdB star calculations suggest that dipole p-modes typically have frequency >2500 μHz (P < 400 s), whereas g-modes <1000 μHz (P > 1000 s) (Fontaine et al. 2003; Charpinet et al. 2005, 2011). But p-mode frequencies can decrease below 1700 μHz (periods can increase beyond 600 s) as T_eff and decreases (Charpinet 1999; Charpinet et al. 2001, 2002).

Figure 2 shows preliminary classification for the p- and g-mode regions based only on the period. We detect 53 independent frequencies in the range [∼80–1000] μHz, which are clearly g-mode (P > 1000 s) pulsations (top two panels). Another seven independent frequencies are found in the high-frequency p-mode (P < 400 s) region, [∼2600–3800] μHz (bottom panel). There are six independent frequencies in the region of [∼1100–1400] μHz or [∼715–910] s, which might be low-order high-degree (ℓ > 3) g-modes or mixed modes that need further classification (see, e.g., Charpinet et al. 2011, 2019). Those frequencies, hardly directly classified to be p or g mode by merely 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 nonlinear effects from the linear eigenfrequencies (Brassard et al. 1995).

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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 ℓ, and the azimuthal order m. When a star rotates, the degenerated m components will split into 2ℓ + 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,ℓ is the Ledoux constant. For acoustic p mode, C_n,ℓ is very near to zero and can be ignored, whereas it is estimated as C_n,ℓ ∼ 1/ℓ(ℓ + 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 1 yr, with most having periods a bit longer than 1 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 μHz, which we consider to be dipole modes, i.e., f₅ ∼ 123.09 μHz, f₁₄ ∼ 180.39 μHz, f₄₁ ∼ 198.89 μHz, f₂₃ ∼ 288.56 μHz, f₁₉ ∼ 324.16 μHz, and f₃₀ ∼ 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, f₁₈ ∼ 89.48 μHz, f₅₄ ∼ 622.79 μHz, and f₂₄ ∼ 697.63 μHz, have close frequency spacings of ∼0.23 μHz, which could be rotational quintuplets, considering frequency uncertainties. We also resolve three ℓ = 1 p-mode multiplets, f₅₂ ∼ 3706.12 μHz, f₃₉ ∼ 3720.77 μHz, and f₆₂ ∼ 3741.96 μHz, with low-amplitude peaks at a frequency distance of ∼0.4 μHz.

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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 distribution, , 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 ℓ = 1, and 2 g modes and p modes, respectively. The corresponding rotation periods are , , and days, respectively.

Our result suggests that a slightly differential rotation occurs in EPIC 220422705 as g and p modes probe the stellar interior under different depths (see, e.g., Kurtz 2022). We note that our results are completely based on only marginally resolved frequencies with low amplitudes near the detection limit. We do not detect multiplets in the four highest-amplitude 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) claim 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 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 a radiative envelope first synchronized and then gradually proceeds to the inner part (Goldreich & Nicholson 1989). In combination with the poorly determined orbital and rotation period (because of low-amplitude multiplets), it would be unwise to speculate too much on this star. There are other sdBV stars that 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.

For g-mode pulsations in the asymptotic regime, consecutive high-radial orders (n ≫ ℓ) 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 Π₀ defined as,

where N is the Brunt–Väisälä frequency and r is the radial coordinate. For the period spacing of ℓ = 2 sequence, it is related to the ℓ = 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 quadrupole 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 data set (Kawaler 1988). We first apply the KS test to the six rotational (incomplete) triplets, which have the 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).

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Based on the preliminary period spacings, we have identified nine modes as dipole, 13 as quadrupole, and additional eight frequencies which fit both period sequences. We note that the four identified modes include one of the above six dipole modes, f₅ ∼ 123.1 μHz. We obtained the average period spacing of 268.5 ± 2.8 s and 159.4 ± 0.6 s for ℓ = 1 and ℓ = 2 modes, respectively, via linear fitting to 17 (4 + 5 + 8) dipole modes and 21 (13 + 8) quadrupole modes (Figure 4(b)). We list ℓ 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 . This agrees well with a low derived for EPIC 220422705. However, atmospheric parameters, with much higher precision, are encouraged for EPIC 220422705 to test the results of Uzundag et al. (2021) in the future.

The échelle diagrams for two sequences are presented in Figure 4(d), where the ℓ = 2 sequence is more consistent than the ℓ = 1 sequence. There are two ℓ = 1 frequencies with larger period deviations, f₁₉ ∼ 324.16 μHz, f₃₀ ∼ 501.25 μHz, whereas only one occurs in the ℓ = 2 sequence, f₈ ∼ 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 the échelle diagram is a vertical ridge for central components of a star with an internally homogeneous 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 with a trapped mode at the fifth order. Indeed, seismic models suggest that strong trapping is more likely found for the lower-order (higher-frequency or shorter-period) modes than the higher-order g modes (Charpinet et al. 2014).

A quick look at all AM/FM patterns occurring in these 22 frequencies suggests 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 subset data, σ₀. 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,

Here k = 1, 2, 3, 4, 5. N denotes the number of data points, and m_i is the measured value of amplitude or frequency.

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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%(D₂ − D₁, N − D₂). 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 are listed in Table 2, where most frequencies have a sinusoidal component, G₃(t), or with an additional linear fitting, G₄(t).

Table 2. AM/FM detected in EPIC 220422705, Sorted by Order of Increasing Frequency

ID	Freq.	Corr.	AM/FM	A	T=2 π/ω	ϕ	a	b	c	Fitting	Comment
	(μHz)			(ppm/nHz)	(d)	[0, 2π)	(10−3)
f7	80.5936	−0.35	AM				⋯			G4	I
			FM				⋯	⋯	⋯	G3
* f5	123.0877	0.86	AM				⋯			G4	B
			FM				⋯			G4
f16	139.9180	0.27	AM				⋯			G4	I
			FM							G5
f9	143.4188	−0.37	AM	⋯	⋯	⋯	⋯			G1	I
			FM				⋯	⋯		G3
f12	166.8356	0.21	AM	⋯	⋯	⋯				G2	B
			FM				⋯	⋯	⋯	G3
f15	173.4053	0.68	AM				⋯			G4	I
			FM				⋯	⋯	⋯	G3
* f14	180.3917	0.73	AM				⋯			G4	I
			FM				⋯	⋯		G3
f10	187.4747	0.14	AM							G5	I
			FM				⋯	⋯		G3
f2	207.2400	−0.15	AM				⋯			G4	B
			FM				⋯	⋯	⋯	G3
f8	218.3630	0.6	AM				⋯	⋯		G3	I
			FM				⋯	⋯	⋯	G3
f4	261.2680	−0.45	AM	⋯	⋯	⋯	⋯			G1	I
			FM				⋯			G4
f1	279.7670	−0.35	AM				⋯			G4	I
			FM				⋯			G4
* f24	288.5611	−0.88	AM				⋯			G4	I
			FM				⋯			G4
f13	299.0409	0.67	AM	⋯	⋯	⋯				G2	B
			FM				⋯			G4
f6	306.2220	−0.82	AM	⋯	⋯	⋯				G2	I
			FM	⋯	⋯	⋯				G2
f11	311.9932	0.12	AM	⋯	⋯	⋯				G2	I
			FM				⋯	⋯	⋯	G3
* f18	324.1648	0.74	AM				⋯	⋯		G3	I
			FM				⋯	⋯		G3
f3	328.7461	−0.45	AM				⋯			G4	I
			FM				⋯			G4
* f24	697.6268	0.07	AM	⋯	⋯	⋯	⋯	⋯	⋯	⋯	I
			FM	⋯	⋯	⋯	⋯	⋯	⋯	⋯
f17	2768.5293	0.53	AM	⋯	⋯	⋯	⋯			G1	I
			FM				⋯			G4
f20	2781.6577	0.19	AM				⋯			G4	B
			FM				⋯			G4
* f44	3720.7735	0.52	AM				⋯			G4	I
			FM				⋯			G4

Note. Columns (1) and (2) The ID and frequency are taken from Table 1. (3) The value of correlation between the amplitude and frequency modulations. (4) The indication of amplitude modulation (AM) or frequency modulation (FM). (5), (6), and (7) The fitting coefficients amplitude (A), period (T = 2π/ω), and phase (ϕ) of sinusoidal function if periodic patterns are found. (8), (9), and (10) The coefficients of polynomial fitting up to second order. (11) The index of fitting function to the modulation pattern. (12) The comments are discussed for the modulation patterns in Section 4.1. B and I denote the beating effect and intrinsic modulations of amplitude and frequency, respectively.

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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 autocorrelation 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 16th and 84th percentiles of the distributions.

Figure 5 is an example of the MCMC application for AM and FM occurring in the frequency f₃ ∼ 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 and 20.4 ± 0.2 days, 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.

In this section, we describe the modulation patterns in AM/FM of all 22 significant frequencies except the frequency of f₃, 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 a timescale of months.

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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 f₉ ∼ 143.4 μHz decreases from 200 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 f₆ ∼ 306.2 μHz, where both the amplitude and frequency reach their vertex values near the time BJD = 2457420 days. 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 f₈ ∼ 218.4 μHz exhibit a completely sinusoidal pattern and evolve in phase. Whereas the frequency of f₁₃ ∼ 299.0 μHz demonstrates a sinusoidal pattern with an additional linear fitting. We only find the frequency of f₁₆ ∼ 139.9 μHz and amplitude of f₁₀ ∼ 187.5 μHz show a sinusoidal plus a parabolic pattern. We note that the amplitude of f₁₄ ∼ 180.434 μHz decreases below the detection threshold from BJD − 24554000 = 200 days to 225 days, 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, f₂₀ ∼ 2781.672 μHz, f₁₉ ∼ 324.164 μHz and f₂₃ ∼ 288.547 μHz. The variations of amplitude can span up to a few hundred ppm (f₁ ∼ 279.767 μHz) or down to a few ppm (f₄ ∼ 261.269 μHz). We note that the amplitudes of AM and FM are not strictly proportional to each other. For instance, the amplitude of f₁ increases from about 50 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 f₂₃ ∼ 288.561 μHz and f₆ ∼ 306.223 μHz all exhibit a strong correlation with a coefficient ∣ρ_a,f∣ > 0.8 between their amplitude and frequency variation. For instance, f₂₃ and f₆ show clear anticorrelation between AM and FM, with derived coefficients of −0.867 and −0.824, respectively. For f₂₃, the amplitude exhibits a slightly decreasing trend, whereas the frequency is the opposite. For f₆, its AM and FM are both represented by parabolic fits but with antiphase evolutions. We also note several frequencies with correlated AM and FM patterns showing sinusoidal variations, such as f₁₅ ∼ 173.4 μHz and f₈ ∼ 218.3 μHz.

Figure 7 shows another 4 frequencies, f₅ ∼ 123.087 μHz, f₇ ∼ 80.595 μHz, f₁₁ ∼ 312.0 μHz, and f₃₉ ∼ 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 functions, i.e., three fitting functions to present FM patterns. For a strong correlation frequency, f₅, with ρ_a,f = 0.862, exhibits regular AM and FM with periods of and days, respectively, derived from EMCEE. The other three frequencies, f₇, f₁₁, and f₃₉ 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 f₇ are calculated as and 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 f₇, f₅, f₁₁, and f₃₉ after extracting G_k (t), respectively. We note that the FM periods of these residuals are shorter than the periods fitted by G_k (t).

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Figure 8 presents the AM and FM of the frequency f₂₄ ∼ 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 experienced an increasing trend. The amplitude reaches its maximum with a time interval of about 30 days but passes across one stationary point, whereas the frequency generally goes up to around the average value with a back and forth trend.

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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, Δf ≈ 1/T. This finding suggests that close frequencies, not only the multiplets we found, may be present in EPIC 220422705 with a 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,

If we consider two comparable amplitudes, A₁ ∼ A₂ = A, and a very small frequency separation, Δω ≪ ω₀, the Equation (8) will be reduced to

The above equation indicates that two close frequencies will generate amplitude modulations if they are not resolved. We thus have to evaluate this effect on the observed AMs in EPIC 220422705.

A series of simulations was performed to quantitatively compare the observed and calculated AMs. The simulation process is 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 A₁, A₂, and Δω to be constant in the 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 A₁, A₂ ∈ [0.5, 0.6, ..., 1.5] × A₀ and Δω ∈ [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, f₉ ∼ 143.4 μHz, f₁₃ ∼ 299.0 μHz, f₁₁ ∼ 312.0 μHz, f₂₄ ∼ 697.6 μHz, and f₂₀ ∼ 2781.6 μHz. We can conclude that the modulations of f₁₃ and f₂₀ are potentially induced by two close frequencies as revealed by no significant residuals. The modulation of f₁₁ and f₂₄ 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 f₉ 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 effects such as the close multiplets discovered, but their AMs cannot be completely represented by those beating effects. In summary, we found that five frequency modulations can be attributed to beating, f₅ ∼ 123.1 μHz, f₁₂ ∼ 166.8 μHz, f₂ ∼ 207.2 μHz, f₁₃ ∼ 299.0 μHz, and f₂₀ ∼ 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.

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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 duration. These kinds of variations have previously been investigated for several sdBV stars (Zong et al. 2016a, 2018, 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., f_i + f_j ∼ f_k ), the frequency mismatch, δ f = f_i + f_j − f_k , 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 these 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). The magnetic field is very rare or completely absent in sdB stars (Landstreet et al. 2012), except for 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 traveling time (Silvotti et al. 2007, 2018; Murphy & Shibahashi 2015). This kind of FM for all frequencies has to be found with an 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. Montgomery et al. (2020) recently proposed that temporal changes in 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.