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A&A
Volume 687, July 2024
Article Number L15
Number of page(s) 12
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202450688
Published online 15 July 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1. Introduction

Harmonics, associated with independent pulsation modes, are a prevalent phenomenon in the constellation of pulsating stars and are present in a wide array of objects, including Cepheids (Rathour et al. 2021), RR Lyrae stars (Kurtz et al. 2016), δ Scuti stars (Breger & Montgomery 2014), γ Dor stars (Kurtz et al. 2015), pulsating white dwarfs (Wu 2001), β Cep stars (Degroote et al. 2009), and slowly pulsating B stars (Pápics et al. 2017). Traditionally, these harmonics have been ascribed to light curves’ non-sinusoidal patterns, a manifestation of the stellar medium’s non-linear response to the propagation of pulsation waves. Consequently, they are not classified as intrinsic stellar pulsation modes (Brickhill 1992; Wu 2001) and are presumed to faithfully replicate the characteristics of their independent parent modes.

Delta Scuti stars, known for their short-period pulsations, have periods ranging from 15 minutes to 8 hours and are classified within spectral types A through F. They are located at the confluence of the main sequence and the lower boundary of the classical Cepheid instability strip on the Hertzsprung-Russell diagram. The pulsations in these stars are self-excited through the κ mechanism, stemming from the partial ionization of helium in the outer layers (Kallinger et al. 2008; Handler 2009; Guenther et al. 2009; Uytterhoeven et al. 2011; Holdsworth et al. 2014; Steindl et al. 2022). The accumulation of high-precision photometric data from space telescopes (such as Kepler and TESS) provides the opportunity to study the amplitude and frequency (and phase) variations in δ Scuti stars in depth, which gives us additional perspectives on the pulsation properties of such stars besides the amplitudes and frequencies (and phases; Bowman & Kurtz 2014; Bowman et al. 2016; Bowman 2016; Guzik et al. 2016).

The high-amplitude δ Scuti (HADS) stars represent a distinct subclass within the δ Scuti stars, characterized by more pronounced amplitudes ( Δ V ≥ 0 · m 1 $ \Delta V \geq 0_{\cdot}^{m}1 $) and slower rotation speeds (v sin i ≤ 30 km s−1), although the accumulation of more HADS samples has blurred these traditional distinctions (Balona et al. 2012). The majority of HADS stars oscillate with one or two radial pulsation modes (Niu et al. 2013, 2017; Xue et al. 2018, 2022; Bowman et al. 2021; Daszyńska-Daszkiewicz et al. 2022), yet a subset exhibits more complex pulsation patterns, including three radial modes (Wils et al. 2008; Niu & Xue 2022; Xue et al. 2023), four radial modes (Pietrukowicz et al. 2013; Netzel et al. 2022; Netzel & Smolec 2022), and even non-radial modes (Poretti et al. 2011; Xue & Niu 2020).

In the case of the SX Phe star (a Population II subclass of HADS stars) XX Cyg, which features a spectrogram dominated by the fundamental pulsation mode (f0) and its 19 harmonics, Niu et al. (2023) discovered that the harmonics exhibit significant and uncorrelated amplitude and frequency variations over time, with these variations becoming more pronounced with increasing harmonic order. Furthermore, for the radial triple-mode HADS star KIC 6382916, the harmonics of the fundamental and first overtone pulsation modes (f0 and f1) display uncorrelated amplitude and frequency variations relative to the independent primary pulsation modes, even for the first and second harmonics (2f0, 3f0, 2f1, and 3f1; Niu & Xue 2024). We refer to these as ‘disharmonized harmonic pulsation modes’, or simply disharmonized harmonics.

These disharmonized harmonics challenge the conventional wisdom regarding the behaviour of harmonics and expose an unexplored aspect of asteroseismology. The question of whether these disharmonized harmonics are a specific feature of certain pulsating stars or represent a widespread phenomenon across certain types of pulsating stars is a critical inquiry in contemporary research.

In this study we scrutinize the amplitude and frequency variations in the harmonics of five HADS stars: DO CMi, GSC 06047-00749 (ASAS J094303-1707.3), V0803 Aur, V1384 Tau, and V1393 Cen. These stars have been identified as radial triple-mode HADS stars based on continuous time-series photometric data from the Transiting Exoplanet Survey Satellite (TESS) space telescope (Xue et al. 2023), whose amplitudes and frequencies for the fundamental, first overtone, and second overtone pulsation modes (A0, A1, A2 and f0, f1, f2) are presented in Table 1; the spectrograms of the five HADS stars are presented in Fig. A.1.

Table 1.

Amplitudes and frequencies of the fundamental, first overtone, and second overtone pulsation modes of the five triple-mode HADS stars.

2. Results

Ultimately, we extracted the variation data (amplitudes and frequencies) for the fundamental and first overtone pulsation modes and their harmonics (denoted as mf0 and nf1, where m, n ∈ ℕ and m, n > 1) for the five HADS stars (see Appendix B for more details on the data reduction, and Appendix E for a demonstration of the validity of the methods). The precise number of harmonics was contingent upon the signal-to-noise ratio within the moving time windows, with no significant signals detected for the harmonics of the second overtone pulsation mode.

The variations in amplitude and frequency for the fundamental and first overtones and their associated harmonic pulsation modes in DO CMi and V1393 Cen are depicted in Fig. 1 (DO CMi: f0 through 3f0 and f1 through 3f1) and Fig. 2 (V1393 Cen: f0 through 3f0 and f1 through 6f1), respectively. They both show evident correlated amplitude and frequency modulations in some of their harmonics. The corresponding data for the remaining three HADS stars are presented in Figs. C.1 (GSC 06047-00749: f0 through 3f0 and f1 through 4f1), C.2 (V0803 Aur: f0 through 4f0 and f1 through 2f1), and C.3 (V1384 Tau: f0 through 3f0 and f1 through 3f1).

thumbnail Fig. 1.

Amplitude and frequency variations in the fundamental and first overtones and their harmonic pulsation modes (f0 through 3f0 and f1 through 3f1) in DO CMi.

To quantitatively illustrate the variations in amplitude and frequency, as well as their interrelations, we calculated the relative amplitude variations ( Δ A / A ¯ ≡ ( A max − A min ) / A ¯ $ \Delta A/\bar{A} \equiv (A_{\mathrm{max}} - A_{\mathrm{min}})/\bar{A} $)1, the absolute frequency variations (Δf ≡ fmax − fmin)2, and the Pearson correlation coefficient (ρA, f) between amplitude and frequency. These quantitative metrics are displayed in Figs. 1, 2, C.1–C.3, providing a detailed account of the observed phenomena.

thumbnail Fig. 2.

Amplitude and frequency variations in the fundamental and first overtones and their harmonic pulsation modes (f0 through 3f0 and f1 through 6f1) in V1393 Cen.

3. Conclusions and discussions

Typically, the amplitudes and frequencies of the fundamental (f0) and first overtone (f1) pulsation modes exhibit a high degree of stability over a timescale of more than ten days. However, two distinct behaviours appear when the order of the harmonics increases: (a) a fluctuation (which deviates from the stability of f0 and f1) grows progressively with the increasing order; and (b) a marked modulation is abruptly injected into a harmonic of a certain order, and all the higher-order harmonics then present amplitude and frequency variations that are highly correlated to it. The former seems to be a general trend in all five HADS stars and is similar to the case of XX Cyg (Niu et al. 2023). The latter is only evident in some cases (such as 2f1 and 3f1 in DO CMi, 3f1 and 4f1 in GSC 06047-00749, and 2f1 through 6f1 in V1393 Cen), which is similar to the case of KIC 6382916 (Niu & Xue 2024). Both behaviours indicate that the disharmonized harmonics are commonly encountered in triple-mode HADS stars and suggest the presence of as yet unidentified mechanisms

On a shorter timescale of approximately 0.2 to 1.0 days, minor frequency modulations become apparent, affecting certain pulsation modes. This effect is notably evident in DO CMi (f0 and f1), V1393 Cen (f1 and 3f1), GSC 06047-00749 (f1), and V1384 Tau (f0). The observation of a similar 0.6570-day modulation in the time-frequency diagram of KIC 6382916 (Niu & Xue 2024) indicates that such modulations may be a common characteristic of triple-mode HADS stars, although the underlying physical processes remain elusive.

The disharmonized harmonics exhibit distinct behaviours for f0 and f1. For f0, the amplitude and frequency variations in its harmonics do not exhibit a strong correlation; this also what is seen with 2f0 and 3f0 in DO CMi. In contrast, for f1, there is a significant correlation in the amplitude and frequency variations, which is particularly evident in DO CMi (2f1 and 3f1) and V1393 Cen (2f1, 3f1, 4f1, 5f1, and 6f1).

An intriguing observation is the significant (anti-)correlated relationships in the variations in amplitude and frequency for certain pulsation modes, as precisely quantified by the Pearson correlation coefficient (ρA, f) and presented in the figures. This is notably observed in DO CMi (3f0, 2f1, and 3f1), V1393 Cen (2f0, 2f1, 3f1, 4f1, 5f1, and 6f1), and GSC 06047-00749 (3f1 and 4f1). These (anti-)correlations are significant as they suggest that the observed amplitude and frequency variations may be attributed to modulations resulting from interactions between pulsation modes (Niu & Xue 2024).

The harmonics of f1 in V1393 Cen, which display a high degree of correlation, provide a valuable context for exploring the detailed relationships among all f0 and f1 harmonics. We analysed these harmonics by constructing an interaction diagram of the frequencies (Fig. 3)3. The frequency variations in the pulsation modes are grouped into two distinct categories: those related to f0 and those related to f1. As depicted in Fig. 3, all f1-related harmonics, with the exception of f1 itself, exhibit significant correlations. This is analogous to the findings for KIC 6382916 (Niu & Xue 2024), where 2f1 is considered an independent parent mode for 3f1 due to their significant correlation. The correct representation of 3f1 should thus be (2f1)+(f1) rather than 3 ⋅ (f1)4.

thumbnail Fig. 3.

Interaction diagram of the frequencies of f0 and f1 and their harmonic pulsation modes in V1393 Cen.

Applying the principle that a harmonic’s parent mode should exhibit the highest correlation in frequency variations, we can deduce the correct representation of all harmonics as follows:

( 6 f 1 ) = ( 5 f 1 ) + ( f 1 ) , ( 5 f 1 ) = ( 4 f 1 ) + ( f 1 ) , ( 4 f 1 ) = ( 3 f 1 ) + ( f 1 ) , ( 3 f 1 ) = ( 2 f 1 ) + ( f 1 ) . $$ \begin{aligned} (6f_1)&= (5f_1) + (f_1), \\ (5f_1)&= (4f_1) + (f_1), \\ (4f_1)&= (3f_1) + (f_1), \\ (3f_1)&= (2f_1) + (f_1). \end{aligned} $$

Each of these high-order harmonics (nf1, n ≥ 3) have two parent modes: f1 and (n − 1)f1 (the latter passes on its variation patterns to nf1). Even more interesting from a global perspective is that these higher-order harmonics seem to be generated step by step, that is to say, nf1 is directly generated by (n − 1)f1 rather than the lower-order harmonics. This interesting generation mechanism of harmonics explains the ubiquitous phenomenon in the spectrogram of many pulsating stars: harmonics appear one by one in order, and their amplitudes decrease as the order increases.

However, the relationship

( 2 f 1 ) ≠ ( f 1 ) + ( f 1 ) $$ \begin{aligned} (2f_1) \ne (f_1) + (f_1) \end{aligned} $$(1)

holds true due to the negative correlation coefficient (−0.16) between f1 and 2f1. Consequently, considering the frequency variations, 2f1 emerges as an independent pulsation mode, the progenitor of the significant frequency variations observed in higher-order harmonics. This pattern is also evident in KIC 6382916 (Niu & Xue 2024) and DO CMi (as shown in Fig. 1). Moreover, a similar case also occurs in GSC 06047-00749, but with a different relationship: (3f1)≠(f1)+(2f1) and (3f1)≠(f1)+(f1)+(f1).

The origin of the frequency variations in 2f1 needs further investigation and may be linked to the nature of f2 as a resonating integration mode, as well as the resonance relationship between f0, f1, and f2 as expressed by the equation

2 f 1 − f 0 = f 2 − Δ ω , $$ \begin{aligned} 2f_1 - f_0 = f_2 - \Delta \omega , \end{aligned} $$(2)

where Δω represents the star’s rotational frequency. Elucidating the details of this relationship is an area earmarked for future research.

In summary, the present study has provided compelling evidence for the prevalence of disharmonized harmonics in HADS stars, challenging the traditional paradigm of harmonic behaviour in pulsating stars. The detailed analysis of five triple-mode HADS stars has unveiled a complex pattern of amplitude and frequency variations that cannot be solely attributed to the independent primary pulsation modes. The discovery of 2f1 as an independent pulsation mode, with its own intrinsic frequency variations, suggests the existence of a multifaceted mechanism governing the pulsations. An intuitive but non-trivial discovery suggests a generation process of harmonics. These findings call for a reassessment of the current models and invite further investigation into the underlying physics, which would potentially lead to a more comprehensive understanding of the pulsation phenomena in δ Scuti stars and other variable stars.

1

Here, Amax and Amin represent the maximum and minimum amplitude values within the moving windows, while A ¯ $ \bar{A} $ denotes the mean amplitude across these windows.

2

fmax and fmin correspond to the maximum and minimum frequency values observed within the moving windows.

3

For the subsequent quantitative analysis, the correlation coefficients between frequency variations are given in the coloured squares for reference. The interaction diagrams of frequencies for the other four stars are presented in Appendix D.

4

The parentheses here indicate that the enclosed pulsation modes should be regarded as a collective entity rather than mere multiples of f1.

6

The choices of the time window length and dataset do not affect the conclusions of the present work, which are tested by f1 and its harmonics of DO CMi in Figs. E.1, E.2, and E.3.

7

The phases of the harmonics on short timescales show violent and irregular fluctuations, and there are no clear correlations between these fluctuations from different harmonics. See, e.g., Fig. E.4 for f1 and its harmonics of DO CMi.

Acknowledgments

We would like to thank Jue-Ran Niu for providing us with an efficient working environment. H.F.X. acknowledges support from the National Natural Science Foundation of China (NSFC) (No. 12303036) and the Applied Basic Research Programs of Natural Science Foundation of Shanxi Province (No. 202103021223320). J.S.N. acknowledges support from the National Natural Science Foundation of China (NSFC) (No. 12005124). The authors acknowledge the TESS Science team and everyone who has contributed to making the TESS mission possible.

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Appendix A: Spectrograms

thumbnail Fig. A.1.

Spectrograms of the five HADS stars. The three independent frequencies for each of them are marked as coloured dots.

Appendix B: Methods

All five HADS stars under investigation were observed by TESS, with their flux measurements obtained from the MAST Portal5 and processed by the TESS Science Processing Operations Center (SPOC; Jenkins et al. (2016)). We converted the normalized fluxes to magnitudes using the TESS magnitude system and eliminated long-term trends for each sector. Subsequently, we selected the light curves with the most data points in a single sector for each star, as detailed in Table 1. Fourier analysis was applied to these light curves, and the spectrograms of the five HADS stars are presented in Fig. A.1. Moreover, the resulting amplitudes and frequencies for the fundamental, first overtone, and second overtone pulsation modes (A0, A1, A2 and f0, f1, f2) are also presented in Table 1.

To elucidate the temporal variations in the amplitudes and frequencies of the harmonics, we employed the short-time Fourier transform, as described in detail by Niu & Xue (2022, 2024) and Niu et al. (2023). A pre-whitening process was executed within an 8-day time window, which was advanced in increments of 0.2 days across the entire dataset.6 At each step, the pre-whitening process was utilized to obtain the amplitudes and frequencies of the independent pulsation modes and their harmonics, with the phase, not the primary focus of this study, left as a free parameter7

The pre-whitening process was performed through Fourier decomposition, as represented by the following equation:

m = m 0 + ∑ A i sin [ 2 π ( f i t + ϕ i ) ] , $$ \begin{aligned} m = m_{0} + \sum A_{i} \sin \left[ 2 \pi (f_{i} t + \phi _{i}) \right], \end{aligned} $$(B.1)

where m0 is the shifted value, Ai the amplitude, fi the frequency, and ϕi the phase. The amplitude uncertainties (σa) were determined as the median amplitude value within a spectral window of 4 c/d, divided equally by the frequency peak, in line with methods described by Niu et al. (2023) and Niu & Xue (2024). Frequency uncertainties (σf) were estimated based on the relationship between amplitude and frequency as proposed by Montgomery & O’Donoghue (1999) and Aerts (2021), as shown in the equation below:

σ f = σ a · 3 π · A · T , $$ \begin{aligned} \sigma _{f} = \sigma _{a} \cdot \frac{\sqrt{3}}{\pi \cdot A \cdot T}, \end{aligned} $$(B.2)

where A is the amplitude and T represents the total time baseline used in the pre-whitening process.

Appendix C: Amplitude and frequency variations

thumbnail Fig. C.1.

Amplitude and frequency variations in the fundamental and first overtones and their harmonic pulsation modes (f0 through 3f0 and f1 through 4f1) in GSC 06047-00749.

thumbnail Fig. C.2.

Amplitude and frequency variations in the fundamental and first overtones and their harmonic pulsation modes (f0 through 4f0 and f1 through 2f1) in V0803 Aur.

thumbnail Fig. C.3.

Amplitude and frequency variations in the fundamental and first overtones and their harmonic pulsation modes (f0 through 3f0 and f1 through 3f1) in V1384 Tau.

Appendix D: Interaction diagrams

thumbnail Fig. D.1.

Interaction diagram of the frequencies of f0 and f1 and their harmonic pulsation modes in DO CMi.

thumbnail Fig. D.2.

Interaction diagram of the frequencies of f0 and f1 and their harmonic pulsation modes in GSC 06047-00749.

thumbnail Fig. D.3.

Interaction diagram of the frequencies of f0 and f1 and their harmonic pulsation modes in V0803 Aur.

thumbnail Fig. D.4.

Interaction diagram of the frequencies of f0 and f1 and their harmonic pulsation modes in V1384 Tau.

Appendix E: Supplementary materials

thumbnail Fig. E.1.

Amplitude and frequency variations in f1 and its two harmonics for DO CMi, for TESS Sector 07 (left panel) and Sector 33 (right panel). We see that the detailed modulation phases in different sectors are different, but the correlation between the harmonics remains the same.

thumbnail Fig. E.2.

Amplitude and frequency variations in f1 and its two harmonics for DO CMi, using half of the data points (left panel) and all of them (right panel) from Sector 33. We see that the modulation periods are not related to the data length.

thumbnail Fig. E.3.

Amplitude and frequency variations in f1 and its two harmonics for DO CMi, using different time window lengths: 4 days (left panel), 8 days (middle panel), and 12 days (right panel). We see that using different time windows can slightly affect the modulation phases and redistribute the lengths of the three segments of data points. However, the variation features are maintained.

thumbnail Fig. E.4.

Frequency and phase variations in f1 and its two harmonics for DO CMi. We see that the phases of the harmonics show violent and irregular fluctuations on short timescales (like the case in this study), and there are no clear correlations between these fluctuations from different harmonics.

All Tables

Table 1.

Amplitudes and frequencies of the fundamental, first overtone, and second overtone pulsation modes of the five triple-mode HADS stars.

In the text

All Figures

thumbnail Fig. 1.

Amplitude and frequency variations in the fundamental and first overtones and their harmonic pulsation modes (f0 through 3f0 and f1 through 3f1) in DO CMi.
In the text
thumbnail Fig. 2.

Amplitude and frequency variations in the fundamental and first overtones and their harmonic pulsation modes (f0 through 3f0 and f1 through 6f1) in V1393 Cen.
In the text
thumbnail Fig. 3.

Interaction diagram of the frequencies of f0 and f1 and their harmonic pulsation modes in V1393 Cen.
In the text
thumbnail Fig. A.1.

Spectrograms of the five HADS stars. The three independent frequencies for each of them are marked as coloured dots.
In the text
thumbnail Fig. C.1.

Amplitude and frequency variations in the fundamental and first overtones and their harmonic pulsation modes (f0 through 3f0 and f1 through 4f1) in GSC 06047-00749.
In the text
thumbnail Fig. C.2.

Amplitude and frequency variations in the fundamental and first overtones and their harmonic pulsation modes (f0 through 4f0 and f1 through 2f1) in V0803 Aur.
In the text
thumbnail Fig. C.3.

Amplitude and frequency variations in the fundamental and first overtones and their harmonic pulsation modes (f0 through 3f0 and f1 through 3f1) in V1384 Tau.
In the text
thumbnail Fig. D.1.

Interaction diagram of the frequencies of f0 and f1 and their harmonic pulsation modes in DO CMi.
In the text
thumbnail Fig. D.2.

Interaction diagram of the frequencies of f0 and f1 and their harmonic pulsation modes in GSC 06047-00749.
In the text
thumbnail Fig. D.3.

Interaction diagram of the frequencies of f0 and f1 and their harmonic pulsation modes in V0803 Aur.
In the text
thumbnail Fig. D.4.

Interaction diagram of the frequencies of f0 and f1 and their harmonic pulsation modes in V1384 Tau.
In the text
thumbnail Fig. E.1.

Amplitude and frequency variations in f1 and its two harmonics for DO CMi, for TESS Sector 07 (left panel) and Sector 33 (right panel). We see that the detailed modulation phases in different sectors are different, but the correlation between the harmonics remains the same.
In the text
thumbnail Fig. E.2.

Amplitude and frequency variations in f1 and its two harmonics for DO CMi, using half of the data points (left panel) and all of them (right panel) from Sector 33. We see that the modulation periods are not related to the data length.
In the text
thumbnail Fig. E.3.

Amplitude and frequency variations in f1 and its two harmonics for DO CMi, using different time window lengths: 4 days (left panel), 8 days (middle panel), and 12 days (right panel). We see that using different time windows can slightly affect the modulation phases and redistribute the lengths of the three segments of data points. However, the variation features are maintained.
In the text
thumbnail Fig. E.4.

Frequency and phase variations in f1 and its two harmonics for DO CMi. We see that the phases of the harmonics show violent and irregular fluctuations on short timescales (like the case in this study), and there are no clear correlations between these fluctuations from different harmonics.
In the text

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Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.