Preamble

Vol. 43, No. 2

June 2025

Progress in Astronomy Vol. 43, No. 2 June, 2025 doi: 10.3969/j.issn.1000-8349.2025.02.06

Pulsation Properties and Frequency Analysis of Three High-amplitude δ Scuti Stars

Ayzada Jumahali¹, Lv Chenglong², Ali Esamdin², Yang Taozhi³, Shen Lixian²,⁴

(1. College of Physical Science and Technology, Xinjiang University, Urumqi 830017, China; 2. Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi 830011, China; 3. School of Physics, Xi'an Jiaotong University, Xi'an 710049, China; 4. School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, Beijing 100049, China)

Abstract: High-amplitude δ Scuti stars (HADS) are a subclass of δ Scuti pulsating variables characterized by short pulsation periods, large amplitudes, and single or multiple radial pulsation modes, making them important targets for asteroseismology. Using time-series photometric data from the Transiting Exoplanet Survey Satellite (TESS), we performed frequency analysis of three HADS for the first time. The results show that TIC 355547586 pulsates primarily in the fundamental and first overtone modes, TIC 358502706 in the fundamental, first, and third overtone modes, with non-radial pulsation frequencies also detected in their frequency spectra. TIC 260654645 pulsates in the fundamental and first overtone modes. The phase-folded curve of this source shows a prominent bump in the descending branch near minimum light, exhibiting characteristics of RR Lyrae ab-type variables. Moreover, the period ratio of its first overtone to fundamental frequency exceeds the theoretical prediction for radial pulsation, and its position in the Hertzsprung-Russell diagram deviates from the HADS instability strip, warranting further investigation.

Keywords: high-amplitude δ Scuti star; frequency analysis; pulsation mode

CLC number: P145.2 Document code: A

1 Introduction

Pulsating variable stars are stars whose luminosity varies due to wave propagation within their interiors. Among the various types of pulsating variables, δ Scuti stars are located at the intersection of the main sequence and the classical instability strip in the Hertzsprung-Russell diagram, typically exhibiting short pulsation periods of 1–6 hours and relatively large photometric amplitudes. In 2000, McNamara proposed defining high-amplitude δ Scuti stars (HADS) as those δ Scuti variables with photometric amplitudes exceeding 0.3 mag. HADS typically show asymmetric light curves and have effective temperatures in the range of 7,000–8,000 K. Observational studies reveal that HADS constitute a very small fraction of δ Scuti variables, with fewer than 200 double-mode HADS currently discovered. These HADS are distributed throughout the main sequence evolutionary stage from zero-age to terminal-age main sequence. Confirmed HADS usually exhibit one or two radial pulsation modes, primarily arising from pulsations in the fundamental or first overtone frequencies. In recent years, a few HADS with radial triple-mode pulsations have also been detected.

HADS typically have slow rotation velocities, with v sin i values generally not exceeding 30 km/s. This relatively low rotation helps maintain stability at the stellar surface and highlights the characteristics of pulsation modes. Current research suggests that slow rotation may be a necessary condition for large-amplitude pulsations in variables within the pulsation instability strip. The exact effect of rotation on stellar pulsation still requires further investigation. Small-amplitude δ Scuti stars generally exhibit numerous pulsation frequencies, and their rapid rotation leads to coupling between intrinsic frequencies and rotational splitting frequencies, making identification of pulsation modes extremely difficult. In contrast, HADS have low rotation velocities and display relatively simple pulsation spectra, facilitating easier mode identification and making them valuable targets for constructing stellar evolution models and asteroseismology.

Asteroseismology studies the internal structure of stars through analysis of their pulsation spectra and represents the only effective method for understanding stellar interiors. In frequency spectra, pulsation frequencies appear as distinct peaks. Identifying intrinsic frequencies and confirming their pulsation modes based on these frequencies forms the foundation for constructing and testing theoretical evolution models of pulsating variables. Over the past decade, the Kepler and TESS space telescopes have provided continuous, long-term, high-precision observational data for pulsating variables, greatly improving the accuracy of pulsation spectra and enhancing the reliability of frequency identification and mode certification.

Due to their rarity, our understanding of HADS remains incomplete, necessitating in-depth research. This study analyzes the pulsation frequencies and properties of three HADS using TESS time-series photometric data. Section 2 introduces the data sources and processing methods, Section 3 presents the analysis results and discussion, and Section 4 summarizes our research.

2 Data Sources and Processing

Since its launch in 2018, the TESS space telescope has observed numerous δ Scuti variables. TESS is a NASA astrophysics mission led by MIT, with the primary scientific objective of discovering exoplanets orbiting nearby bright stars—particularly those of appropriate size and temperature for life—by observing photometric variations of stars across the entire sky. TESS's observing strategy divides the sky into 26 sectors, observed sequentially, with each sector observed for approximately 27.4 days, enabling it to cover nearly the entire sky at least once during its two-year primary mission.

This study primarily uses TESS SPOC (Science Processing Operations Center) photometric data, which can be accessed online through the TESS Asteroseismic Science Operations Center (TASOC). SPOC photometric data are based on two processing methods: simple aperture photometry (SAP) and pre-search data conditioning (PDC) processed SAP flux. The PDC SAP method corrects for TESS's onboard system and accounts for contributions from neighboring stars to the total flux, effectively reducing external interference in stellar photometric measurements. Therefore, this study uses PDC SAP processed data.

The data processing workflow includes the following steps. First, trends are removed from each target's sector data. For data from consecutive sectors, they are concatenated before removing NaN values. Additionally, to improve data quality, a 5σ clipping procedure is applied to exclude obvious outliers. Next, the PDC SAP flux time series is converted to relative magnitude time series, and discrete Fourier transform (DFT) is applied to process these magnitude variations. This step converts time-domain data to frequency-domain data, transforming light variations over time into power variations with frequency, making it easier to identify and analyze periodic pulsation patterns.

For HADS pulsation frequency analysis, time-domain photometric data are converted to the frequency domain via DFT, where pulsation frequencies appear as peaks in the power spectrum, corresponding to different pulsation modes. This study uses the software tool PERIOD04 to perform DFT on discontinuous time-series photometric data and extract significant frequencies. Significant frequencies are identified based on their signal-to-noise ratio (S/N). Following Baran et al., the S/N threshold for space telescope data such as TESS is set at 5.2. The extraction process stops when the S/N falls below 5.2. Frequency uncertainties are determined using the method proposed by Montgomery and O'Donoghue.

Within the frequency range of 0–50 d⁻¹, multiple iterations of pre-whitening are typically required. These steps include: (1) performing DFT to convert time-series photometric data to the frequency domain; (2) identifying the highest peak frequency fₙ in the power spectrum; (3) constructing a sinusoidal curve using the extracted frequency fₙ with its amplitude Aₙ and phase ϕₙ to fit the original time-series photometric data; (4) subtracting the fitted sinusoidal curve from the original time-series light curve to obtain the residual light curve; and (5) repeating the above steps on the residual light curve to identify other significant frequencies.

The formula for fitting the time-series light curve is: ΔL = Σ Aₙ sin(2πfₙt + ϕₙ), where ΔL is the luminosity variation, Aₙ is the amplitude, fₙ is the frequency, and ϕₙ is the phase of fₙ.

In asteroseismology, identifying stellar pulsation modes is crucial. Stellingwerf proposed empirical formulas in 1979 for identifying radial pulsation modes, providing theoretical predictions for period ratios between different radial modes (fundamental and overtones): 0.756 ≤ P₁/P₀ ≤ 0.787, 0.611 ≤ P₂/P₀ ≤ 0.632, and 0.500 ≤ P₃/P₀ ≤ 0.518, where P₀, P₁, P₂, and P₃ correspond to the periods of the fundamental, first overtone, second overtone, and third overtone modes, respectively. This study follows these relationships to identify HADS pulsation modes. In practice, if observed period ratios between different modes fall within these ranges, the frequencies can be considered radial pulsation modes; independent frequencies not following these period ratios may represent non-radial pulsation modes. For simplicity in frequency analysis, the fundamental, first overtone, second overtone, and third overtone are labeled as F0, F1, F2, and F3, respectively.

3 Results and Discussion

This chapter reports detailed analysis results for the three HADS, including frequency analysis tables, light curves, amplitude spectra, and phase-folded diagrams. To better understand these targets' properties, their right ascension, declination, magnitude, effective temperature, and other stellar parameters were obtained from the SIMBAD database and the latest TESS catalog (Exoplanet Follow-up Observing Program, ExoFOP). Their TESS absolute magnitudes were calculated using distances from Gaia DR3, with results listed in Table 1 [TABLE:1].

3.1 Basic Stellar Parameters of the Three HADS

Table 1 Basic stellar parameters of the three HADS

Parameter TIC 355547586 TIC 358502706 TIC 260654645 TESS magnitude (mag) 11.7(1) 12.55(1) 12.6(1) T_eff (K) 7,420(190) 7,690(161) 6,967(120) log g (dex) 4.11(2) 4.02(8) 3.6(1) [Fe/H] 0.3(1) 0.25(6) -0.463(9) Luminosity (L_⊙) 1.7(3) 1.8(3) 1.6(3) Density (g cm⁻³) 2.0(2) 2.16(8) 3.6(3) Mass (M_⊙) 1.339(119) 1.97(1) 1.564(49) Distance (pc) 1,339(119) 926(18) 1,564(49) v_R (km s⁻¹) 1.87(1) 1.58(1) 1.58(1) M_TESS (mag) +21h24min20.3s +04h03min20.1s +06h32min45.2s RA (J2000) -83°41′58.6″ -57°48′19.8″ -57°51′53.9″ Dec (J2000) 11.7(1) 12.55(1) 12.6(1)

3.2 TIC 355547586

TIC 355547586 (UCAC4 161-218640) is classified as a HADS in the catalog published by Antoci et al. TESS photometric data are available for this source in Sectors 1 and 27. This study adopts 2-minute cadence data from Sector 27, spanning from Julian Date 2459036.28297938 to 2459060.64568572, totaling 24.4 days. Following the data processing procedure described in the previous chapter, we obtained a light curve comprising 16,469 data points. Figure 1a [FIGURE:1] shows 3 days of continuous data for this source, with an amplitude of approximately 0.35 mag. The light curve shape is consistent with typical HADS characteristics.

Fourier analysis extracted 23 significant frequencies for TIC 355547586 (see Table 2 [TABLE:2]). The period ratio between the two large-amplitude independent frequencies f₃ and f₁ is 0.7723, which according to theoretical predictions identifies f₁ and f₃ as the fundamental and first overtone frequencies, respectively.

For a time-series dataset with time span ΔT, the frequency resolution in the frequency domain is σ_v = 1/ΔT. By accurately identifying and removing alias frequencies, we can extract significant frequencies and their corresponding amplitudes. Table 2 lists two radial frequencies for TIC 355547586: the fundamental frequency f₁ (labeled "F0") and the first overtone f₃ (labeled "F1"), along with combination frequencies (e.g., f₄, f₆, f₇) and harmonics (f₂, f₅, f₈, f₁₃). Frequency f₁₁ may be a non-radial frequency. The table also shows multiple combination frequencies with the fundamental and first overtone (f₁₆, f₁₇, f₁₉, f₂₃), though their exact pulsation modes cannot be determined. Figure 1b shows the phase-folded curve using the fundamental frequency of 10.51243 d⁻¹, Figure 1c displays the Fourier amplitude spectrum, and Figure 1d shows the residuals after pre-whitening all significant frequencies.

3.3 TIC 358502706

TIC 358502706 (TYC 9492-2623-1) is classified as a HADS(B) variable in the AAVSO VSX database, consistent with the classification by Antoci et al. Time-series photometric data for TIC 358502706 are available in TESS Sectors 1, 12, 13, 27, and 39. Sectors 12 and 13 were observed continuously; after merging these segments, we obtained data spanning from Julian Date 2458624.9567046 to 2458682.35844299, totaling 56.3 days. After processing, we obtained a light curve consisting of 33,905 data points. Figure 2a [FIGURE:2] shows 3 days of photometric data for this source, with an amplitude of approximately 0.30 mag. Figure 2b shows the phase-folded diagram using the fundamental frequency of 11.674079 d⁻¹, Figure 2c displays the Fourier amplitude spectrum, and Figure 2d shows the residuals after pre-whitening.

Fourier analysis of the time-series photometric data for TIC 358502706 extracted 19 significant frequencies (see Table 3 [TABLE:3]). The period ratio between f₃ and the fundamental frequency is 0.7740, consistent with results from Khruslov. The period ratio between f₄ and the fundamental frequency is 0.5382, indicating that f₃ and f₄ correspond to the first and third overtone radial modes, respectively. Although the period ratio of the third overtone to the fundamental is slightly higher than the range provided by Stellingwerf for radial pulsation modes, based on the detection of multiple combination frequencies of the fundamental and third overtone (f₉, f₁₁, f₁₅, f₁₇, f₁₈), this study preliminarily classifies TIC 358502706 as a triple-mode HADS. Table 3 lists three radial mode frequencies (f₁ as fundamental "F0", f₃ as first overtone "F1", and f₄ as third overtone "F3"), their combination frequencies (f₅, f₇, f₈), and harmonics (f₂, f₆, f₁₂), plus one detected non-radial frequency f₁₃.

3.4 TIC 260654645

TIC 260654645 (UCAC4-161-008109) is classified as a δ Scuti variable in the AAVSO International Variable Star Index (VSX), and Antoci et al. classified it as a HADS. Table 1 lists its basic parameters, including a TESS absolute magnitude of M_TESS = 1.58(1) mag calculated from distance and apparent magnitude. TIC 260654645 was observed in multiple TESS sectors; this study uses continuous 2-minute cadence data from Sectors 12 and 13, spanning from Julian Date 2458626.46312538 to 2458682.35572725, yielding 36,668 data points after processing. Figure 3a [FIGURE:3] shows 3 days of the light curve for this source, with an amplitude of approximately 0.34 mag, dominated by two radial pulsation frequencies F0 and F1.

Table 4 [TABLE:4] lists 17 significant frequencies detected in TIC 260654645, including the fundamental frequency f₁ (labeled F0), the first overtone frequency f₃ (labeled F1), harmonics (f₄, f₇, f₁₄), and combination frequencies (f₅, f₆, f₈) of these two radial frequencies. The period ratio of F1 to F0 is 0.8044, slightly higher than the theoretical prediction for radial pulsation modes. Based on multiple combination frequencies of f₁ and f₃ and the significant amplitude of f₃, TIC 260654645 is likely a double-mode HADS pulsating in radial fundamental and first overtone modes. Figure 3b shows the phase-folded curve using the fundamental frequency of 10.98476 d⁻¹, Figure 3c displays the Fourier amplitude spectrum, and Figure 3d shows residuals after pre-whitening all significant frequencies.

Figure 3b reveals a prominent bump near minimum light in the descending branch of the phase-folded curve. Additionally, the phase diagram shows amplitude and phase modulation, typical characteristics of RR Lyrae ab-type variables exhibiting the Blazhko effect. However, RR Lyrae variables typically have stellar masses of about 0.7 M_⊙ and absolute magnitudes of about 0.6 mag, with light variation periods ranging from 0.3 to 1.2 days (average ~0.55 days). Based on the fundamental stellar parameters in Table 1, this source differs significantly from typical RR Lyrae variables. Furthermore, RR Lyrae frequency spectra typically show a 1.5F0 frequency, which is not present in TIC 260654645. These considerations essentially rule out the possibility that this source is an RR Lyrae variable, though its amplitude and phase modulation phenomena warrant further detailed study.

The higher-than-predicted period ratio of F0 and F1 may be caused by stellar rotation. According to data from the Galactic Archaeology with HERMES (GALAH) spectroscopic survey, the spectral line broadening velocity v_b for this source is only 20.5 km/s. Although the rotation velocity is low, studies show that even low rotation velocities can affect HADS period ratios. Additionally, stellar metallicity influences HADS period ratios, with metal-poor stars typically having higher period ratios, such as SX Phoenicis pulsating variables. Therefore, based on metallicity, this source may also be a candidate SX Phoenicis variable.

To investigate the evolutionary status of these three HADS, we plotted their positions on the Hertzsprung-Russell diagram based on their TESS absolute magnitudes and effective temperatures (see Figure 4 [FIGURE:4]). In this figure, colored diamond points represent the three HADS, gray solid points represent HADS from other literature, and blue solid points represent seven HADS published by Lv et al. in 2023. The green solid line indicates the zero-age main sequence (ZAMS), two black dashed lines represent the location of the HADS instability strip, and blue and red solid lines mark the blue and red edges of the δ Scuti instability strip, respectively. TIC 260654645 shows clear deviation from other HADS, and combined with its light curve characteristics, suggests that this source may have different physical properties or be in a different evolutionary stage.

4 Summary and Outlook

Based on time-series photometric data from the TESS space telescope, this paper presents an in-depth frequency analysis of three HADS, obtaining for the first time their pulsation frequencies, amplitudes, phases, and mode identifications.

Frequency analysis preliminary identifies TIC 355547586 as a double-mode radially pulsating HADS, with one non-radial pulsation frequency also detected. Its luminosity variations are dominated by radial pulsations in the fundamental and first overtone modes. TIC 358502706 exhibits triple-mode radial pulsation characteristics, with one non-radial pulsation frequency detected, adding a new sample for subsequent studies of triple-mode HADS. The light curves and phase-folded diagrams of TIC 355547586 and TIC 358502706 are consistent with typical HADS characteristics.

TIC 260654645 is a double-mode radially pulsating HADS. Its phase curve shows a prominent bump in the descending branch near minimum light, displaying light variation characteristics of RR Lyrae ab-type variables. However, based on its stellar parameters, it is unlikely to be an RR Lyrae ab-type variable. With its low rotation velocity and high period ratio of first overtone to fundamental frequency, this source may also be an SX Phoenicis variable. Additionally, it shows amplitude and phase modulation phenomena and deviates from the typical HADS pulsation instability band in the HR diagram. This peculiar HADS warrants further observational and theoretical investigation.

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