arXiv subjects

Qi-Bin Sun

Publications and source records attributed to Qi-Bin Sun.

At least 19 recordsLinked to original sources

arXiv2026

Eclipse Properties and Superhump Evolution in the SU UMa-Type Dwarf Nova Z Cha

The advent of large-scale time-domain surveys provides both opportunities and challenges for understanding accretion disk evolution in cataclysmic variables (CVs). Using high-cadence photometry from the Transiting Exoplanet Survey Satellite (TESS), we investigate the eclipsing SU UMa-type dwarf nova Z Cha. Leveraging eclipses as a natural probe, we examine the evolution of the accretion disk through variations in eclipse depth, O--C of eclipse minima, and positive superhump (PSH) amplitude. During superoutbursts, all three quantities exhibit quasi-periodic modulations with a common period of $\sim$2 days, consistent with the precession period of an eccentric disk. We interpret these correlated variations as evidence of an eccentric, precessing disk: O--C traces the periodic shift of the system's brightness center, while eclipse depth and PSH amplitude vary with the orientation of the disk bulge relative to the line of sight. In quiescence (Sectors 13 and 93), PSHs with periods of $\sim$0.0762 days show linearly decreasing amplitudes and periods, indicating gradual shrinkage of the eccentric disk and a slowing precession. Remarkably, a coherent signal with a period of $\sim$0.0729~days ($\epsilon^{-}\approx-0.02$) appears in the same quiescent intervals. This signal may represent negative superhumps (NSHs) coexisting with PSHs, although an orbital sideband of the PSH cannot presently be excluded with the available data. If confirmed as NSHs, their coexistence with PSHs would challenge the classical tilted-disk model, and could be explained by retrograde apsidal precession of an eccentric disk, where the inner disk precesses retrogradely (NSHs) and the outer disk progradely (PSHs); this interpretation remains to be tested by further observations.

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arXiv2026

Universal scaling between magnetar field and initial spin period for short gamma ray bursts

The $B_p$--$P_0$ correlation serves as a critical probe of magnetar engine physics. Although this scaling relation has been firmly established for long gamma-ray bursts (lGRBs), systematic investigations for short GRBs (sGRBs) remain absent, leaving the physical differences between the two populations poorly constrained. Here we analyze 33 Swift sGRBs exhibiting prominent X-ray plateaus from newborn millisecond magnetar spin-down, and derive their initial spin period $P_0$ and polar magnetic field $B_p$. sGRB magnetars span $P_0 \in [1.73,\,18.28]\ \mathrm{ms}$ and $B_p \in [0.06,\,2.82] \times 10^{17}\ \mathrm{G}$ ($\langle B_p \rangle = 7.05 \times 10^{16}\ \mathrm{G}$), significantly more magnetized than lGRB magnetars ($B_p \in [0.39,\,23.08] \times 10^{15}\ \mathrm{G}$; $\langle B_p \rangle = 3.69 \times 10^{15}\ \mathrm{G}$). For the first time, we derive consistent power-law $B_p$--$P_0$ correlations for GRBs : the scaling for sGRBs is $\log B_p = (0.84\pm0.07)\log P_0 + (15.79\pm0.07)$, whose slope is highly consistent with that of lGRBs, $\log B_p = (0.83\pm0.09)\log P_0 + (14.92\pm0.06)$. The near-identical slopes imply a universal magnetar spin-down mechanism, while the vertical offset between intercepts traces divergent progenitor channels. This scaling relation thus offers a new diagnostic to disentangle the formation pathways of GRB. Within the framework of the standard spin-up model, the mass accretion rates of sGRBs ($\dot{M} \sim 1 \times 10^{-1}$ to $3 \times 10^{-1}\,M_\odot\,\mathrm{s}^{-1}$) are substantially higher than those of lGRBs ($\dot{M} \sim 10^{-4}$ to $1 \times 10^{-1}\,M_\odot\,\mathrm{s}^{-1}$). Our work completes the missing $B_p$--$P_0$ statistics for sGRBs, quantitatively unifies their magnetar physics with lGRBs, and provides new observational constraints on the origin diversity of relativistic transients.

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arXiv2025

Tidally Trapped Two-pole Pulsations Discovered in a Close Binary with a Massive $β$ Cephei Star

Tidally tilted pulsators (TTPs), whose pulsation axis aligns with the binary's semi-major axis, represent a newly established class of oscillators in binary systems. While all previously known TTPs are either $δ$ Scuti or subdwarf B-type stars, their existence has remained unidentified in more massive $β$ Cephei variables. Here, we report the discovery of tidally trapped pulsations in the massive ellipsoidal variable HD~329379, based on photometry from the Transiting Exoplanet Survey Satellite (TESS). Our analysis reveals a $β$ Cephei pulsator in a 2.25-day orbit whose pulsation mode amplitude is strongly modulated with the orbital frequency. Based on our analysis, we concluded that this modulation can be explained by pulsations with significantly larger amplitude near the star's two tidal poles (apsides). We interpret this as a tidally distorted quadrupole pulsation chariacteristiced by trapped two-pole pulsations, with a pulsation axis aligned with the tidal axis. This represents the first identification of such a pulsation mode in a $β$ Cephei star, which differs from single-sided pulsations observed in previous works, marking a rare and important discovery. Our work extends the family of TTPs beyond $δ$ Scuti and subdwarf B-type stars to include more massive $β$ Cephei variables. In particular, the two-pole pulsator HD~329379 stands out as the prototype of a new class of TTPs in massive stars. These results not only provide a new insight to probe the interior structure and evolutionary state for massive stars but also offer a unique opportunity to study the interaction between pulsations and strong tidal distortions.

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arXiv2025

Asteroseismology and Dynamics Reveal Interior Structure and Coeval Evolution in the Triply Post-Main-Sequence system DG Leo

$δ$ Scuti stars in binary or multiple systems serve as crucial probes for studying stellar pulsation and evolution. However, many such systems are not ideal for asteroseismology due to uncertainties in mass transfer with close companions and the challenges of dynamically measuring all components' physical properties. The triple system DG~Leo, comprising an inner binary and a distant $δ$ Scuti star, is an ideal target due to its well-separated pulsator. By combining new \textit{TESS} photometry with archival spectroscopy, our dynamical analysis shows that the system's three components share similar masses, radii, and luminosities within errors, occupying coincident Hertzsprung--Russell diagram positions, indicative of coeval evolution. By fitting seven observed $δ$ Scuti frequencies through asteroseismic modeling with dynamically constrained theoretical grids, we simultaneously trace the pulsating star's evolution and constrain the triple system's evolutionary stage, with the derived fundamental parameters showing consistency with the dynamical solutions. Our analysis reveals that all three components of DG~Leo are in the post-main-sequence phase, with a system age of $0.7664^{+0.1402}_{-0.1258}$~Gyr. Additionally, the $δ$ Scuti component shows multiple non-radial modes with significant mixed-character frequencies, providing precise constraints on its convective core extent ($R_{\mathrm{cz}}/R = 0.0562^{+0.0137}_{-0.0021}$).

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arXiv2025

New Evidence for Extragalactic Einstein Probe Transients associated with Long Gamma-ray Bursts

The origin of extragalactic fast X-ray transients (EFXTs) remains a fundamental open question in high-energy astrophysics. The Einstein Probe (EP) mission provides a transformative opportunity to investigate their nature. While mounting observations of EP-discovered EFXTs (EP-EFXTs) suggest a possible connection to long gamma-ray bursts (lGRBs), an in-depth comparative analysis between them remains lacking. Here, we present a comparative analysis of their cosmic formation histories, revealing that EP-EFXTs and lGRBs share a similar evolutionary trend-showing a marked decline at $z<1.0$ and a plateau beyond $1.0<z<5$-which clearly distinguishes them from short GRBs. This result is derived from a rigorously selected sample of EP-EFXTs, using Lynden-Bell's $c^{-}$ method to reconstruct, for the first time, the luminosity function and formation rate of EP-EFXTs without any assumptions. Our findings provide independent evidence that EP-EFXTs and lGRBs may originate from a common progenitor channel.

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arXiv2025

The Cosmic Star Formation History: Insights from Kilonova-Associated Gamma-Ray Bursts

The origin of the Universe and its material content remains one of the most fundamental questions in science. Gamma-ray bursts (GRBs), with their extreme luminosities and high-redshift detectability, provide a unique window into the history of cosmic formation and chemical evolution. Consequently, the GRB formation rate (FR) has been employed to trace the star formation rate (SFR) across cosmic time. GRBs are conventionally classified into long and short categories (lGRBs and sGRBs) based on their $ T_{90} $ duration. sGRBs are widely employed as tracers of the delayed SFR, owing to their origin linked to the inspiral timescales of compact binary systems. However, some studies suggest that the detection of supernova-associated sGRBs may indicate potential contamination by core-collapse events. In this work, we move beyond the $ T_{90} $ classification and focus exclusively on GRBs with confirmed kilonova signatures, which provide unambiguous evidence of binary compact star mergers, to reassess their connection with the delayed SFR. Through analysis of a kilonova-associated GRB (KN/GRBs) sample, we find that even within this robust subset, the KN/GRB FR displays a trend contrary to that of the delayed SFR at low redshifts ($ z < 1 $). This result challenges the conventional theory by indicating that low-redshift KN/GRBs may not accurately trace the delayed SFR, independent of core-collapse contamination, while further validation with larger KN/GRB samples is essential to determine the reliability of compact binary mergers as probes of delayed SFR.

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arXiv2025

Orbital and Pulsation Analysis of 42 Heartbeat Stars Discovered in TESS Data

Heartbeat stars (HBSs) are ideal laboratories for studying the formation and evolution of binary stars in eccentric orbits and their mutual tidal interactions. We present 42 new HBSs discovered based on TESS-SPOC and QLP data. Their physical parameters have been obtained through modeling with appropriate models. Subsequently, Tidally excited oscillations (TEOs) are detected in ten systems, and their pulsation phases and modes are identified. Most pulsation phases can be explained by the dominant being spherical harmonic degree $l=2$ and azimuthal order $m=0$ or $\pm2$. For TIC 156846634, the harmonic with large deviation ($>3σ$) from the expected adiabatic phase can be expected to be a traveling wave or significantly nonadiabatic. The harmonic numbers $n$ = 16 in TIC 184413651 may not be considered as a TEO candidate due to its large deviation ($>2σ$) from the adiabatic expectation. Moreover, TIC 92828790 shows no TEOs but exhibits a significant $γ$\,Dor-type pulsation. The eccentricity-period ($e-P$) relation also shows a positive correlation between eccentricity and period, as well as the existence of orbital circularization. The Hertzsprung-Russell diagram shows that TESS HBSs have higher temperatures and greater luminosities than Kepler HBSs, possibly due to selection effects. This significantly enhances the detectability of massive HBSs and those containing TEOs.

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arXiv2025

Heartbeat Stars Recognition Based on Recurrent Neural Networks: Method and Validation

Since the variety of their light curve morphologies, the vast majority of the known heartbeat stars (HBSs) have been discovered by manual inspection. Machine learning, which has already been successfully applied to the classification of variable stars based on light curves, offers another possibility for the automatic detection of HBSs. We propose a novel feature extraction approach for HBSs. First, the orbital frequencies are calculated automatically according to the Fourier spectra of the light curves. Then, the amplitudes of the first 100 harmonics are extracted. Finally, these harmonics are normalized as feature vectors of the light curve. A training data set of synthetic light curves is constructed using ELLC, and their features are fed into recurrent neural networks (RNNs) for supervised learning, with the expected output being the eccentricity of these light curves. The performance of the RNNs is evaluated using a test data set of synthetic light curves, achieving 95$\%$ accuracy. When applied to known HBSs from the OGLE, Kepler, and TESS surveys, the networks achieve an average accuracy of 86$\%$. This method successfully identifies four new HBSs within the eclipsing binary catalog of Kirk et al. The use of orbital harmonics as features for HBSs proves to be a practical approach that significantly reduces the computational cost of neural networks. RNNs show excellent performance in recognizing this type of time series data. This method not only allows efficient identification of HBSs but can also be extended to recognize other types of periodic variable stars.

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arXiv2025

Detection of Low-Redshift Excess in Supernova-Linked Gamma-Ray Bursts

Gamma-ray bursts (GRBs) are traditionally classified into long (lGRBs) and short (sGRBs) durations based on their $T_{90}$, with lGRBs widely used as tracers of the cosmic star formation rate (SFR) due to their observed association with core-collapse supernovae. However, recent detections of kilonovae accompanying some lGRBs challenge this assumption, suggesting potential contamination from compact binary mergers. Here, we move beyond the conventional $T_{90}$-based classification and focus exclusively on GRBs directly associated with supernovae - the most direct signatures of massive stellar collapse - to reassess their connection to the SFR. Using a sample of SN/GRBs, we construct the luminosity - redshift ($L$-$z$) plane and uncover a significant correlation between these variables. To account for observational biases, we apply the $τ$ statistic and Lynden-Bell's $C^{-}$ method to derive the intrinsic luminosity function and formation rate. Our analysis reveals that even among this well-defined subsample, the SN/GRB formation rate still exceeds the SFR at low redshifts ($z < 1$). These findings suggest that GRBs at low redshift may not serve as reliable tracers of the SFR, and that larger samples are required to further investigate this discrepancy.

astro-ph.HE↗

arXiv2025

Superoutbursts and Positive Superhumps Occurred During the Standstill of a Z Cam-type Dwarf Nova

Dwarf novae are semi-detached binaries, where a white dwarf accretes material from a cool main-sequence companion via an accretion disk, and are known for their intermittent outbursts, making them key systems for studying accretion physics. The accumulation of large survey datasets has challenged traditional models, which assumed that the disk remains hot and cannot produce superoutbursts during the standstill of Z Cam-type dwarf nova and that superoutbursts require a mass ratio of $ q = M_2/M_1 \leq 0.25 - 0.33 $. Here we report the detection of superoutbursts and positive superhumps (PSHs) during a standstill in the Z Cam-type star AT Cnc with a mass ratio larger than 0.33. Notably, the PSHs evolve gradually before the superoutburst begins, suggesting that an eccentric, precessing disk forms first, with the superoutburst occurring as the disk radius continues to expand. These findings provide the first detailed observational evidence of superoutbursts and PSHs occurring during standstill, offering important new insights into the classification of dwarf novae and the underlying mechanisms of outbursts

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arXiv2025

Comparison of the Origin of Short Gamma-Ray Bursts with or without Extended Emission

The merger of compact binary stars produces short gamma-ray bursts (sGRBs), involving channels such as neutron star - neutron star (BNS) and neutron star - black hole (NS-BH). The association between sGRB 170817A and gravitational wave GW 170817 provides reliable evidence for the BNS channel. Some speculations suggest that sGRBs with extended emission (EE) may represent another distinct population. The offset is the distance between the GRB sky localization and the host galaxy center. We compared the offset distributions of these two types of samples (46 sGRBs with EE and 9 without EE samples) and found that they follow the same distribution. Utilizing non-parametric methods, we examined the luminosity function and formation rate of sGRBs without any assuming. The luminosity function can be described as $ψ(L_{0}) \propto L_{0}^{-0.12 \pm 0.01}$ for $L_{0} < L_0^b$ ($ψ(L_{0}) \propto L_{0}^{-0.73 \pm 0.02}$ for $L_{0} > L_0^b$) for sGRB without EE and $ψ(L_{0}) \propto L_{0}^{-0.13 \pm 0.003}$ for $L_{0} < L_0^b$ ($ψ(L_{0}) \propto L_{0}^{-0.61 \pm 0.01}$ for $L_{0} > L_0^b$) for sGRBs with EE. The formation rate is characterized as $ρ(z) \propto (1 + z)^{-3.04 \pm 0.10}$ for $z < 1$ and $ρ(z) \propto (1 + z)^{-0.29 \pm 0.38}$ for $1 < z < 3$ for sGRB without EE, while for sGRBs with EE, it is $ρ(z) \propto (1 + z)^{-3.85 \pm 0.15}$ for $z < 1$ and $ρ(z) \propto (1 + z)^{-0.40 \pm 1.11}$ for $1 < z < 3$. Our findings suggest no significant difference in the progenitors of sGRBs with and without EE when considered in terms of spatial offsets, formation rates, and luminosity function.

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arXiv2024

KIC 7914906: An Eclipsing Heartbeat Star with Tidally Excited Oscillations and Gamma Doradus/Delta Scuti Hybrid Pulsations

We present the eclipsing Heartbeat Star KIC 9704906 with tidally excited oscillations (TEOs) and gamma Doradus/delta Scuti hybrid pulsations. The derived parameters show that it has an orbital period of $P$=8.7529108(1) days, a high eccentricity of $e$=0.467(3), and a high inclination of $i$=78$^{\circ}$.81(6). The mass ratio $q$=0.981(5), the relative radii (radius divided by semi-major axis) $r_1$=0.0639(2), and $r_2$=0.0715(4) indicate that the secondary component has a less mass and a larger radius, and may have evolved off the main sequence. The eight derived TEO candidates, $n$ = 3, 4, 5, 6, 7, 12, 40, and 44 harmonics, are consistent with or close to the dominant spherical harmonic $l=2$, $m=0$, or $\pm2$, assuming that the spin and orbital axes are aligned, and the pulsations are adiabatic and standing waves. We also identify ten independent frequency candidates, but one of them, $\mathit{f_{7}}$, is more like a modulation of a quasi-periodic signal and the orbits. According to the g-mode frequencies, we find that the rotation period of one component is 11.52(29) days. Although the masses and radii cannot be further constrained due to the lack of sufficient high-precision spectra, the fascinating phenomena in the Fourier spectra are evident and valuable in this system.

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arXiv2024

A New IW And-Type Star: Karachurin 12 with Tilted Disks and Diverse cycles

The IW And-type phenomenon in cataclysmic variables presents a significant challenge to the accretion disk instability model. Using photometric data from the All-Sky Automated Survey for Supernovae, the Zwicky Transient Facility, and the Transiting Exoplanet Survey Satellite, we identify Karachurin 12 as a new non-eclipsing IW And-type object with a cycle period of 35.69(3) days. We also report for the first time that Karachurin 12 is a negative superhump (NSH) system featuring a precessing tilted disks, with precession, orbital, and NSH periods of 4.9588(2) days, 0.3168895(13) days, and 0.2979861(8) days, respectively. Our analysis, using dips as index and NSHs as probe, reveals diverse cycle patterns in Karachurin 12, with NSH amplitude varying throughout the cycle. These findings offer new insights for studying tilted disks and the IW And-type phenomenon. The mass-transfer burst model has difficulty explaining the observed variations in NSH amplitude, especially given the uncertainty surrounding the origin of the mass transfer burst. Meanwhile, the tilted thermally unstable disk model indicates a possible connection to the IW And-type phenomenon, but it also struggles to account for the detailed variations in Karachurin 12. Therefore, a wider range of factors must be considered to fully understand the complex changes in Karachurin 12.

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arXiv2024

Evolutionary states and triplicity of four massive semi-detached binaries with long-term decreasing orbital periods in the LMC

The massive semi-detached binary with a long-term decreasing orbital period may involve a rapid mass-transfer phase in Case A, and thus they are good astrophysical laboratories for investigating the evolution of massive binary stars. In this work, by using the long-term observational light curves from the OGLE project and other data in the low-metallicity LMC, four semi-detached massive binaries with long-term decreases in the orbital periods are detected from 165 EB-type close binaries. It is found that the more massive component in S07798 is filling its Roche lobe where the period decrease is caused by mass transfer from the primary to the secondary. However, the other three (S03065, S12631, S16873) are semi-detached binaries with a lobe-filling secondary where the mass transfer between the components should cause the period to increase if the angular momentum is conservative. The long-term period decreases in these three systems may be caused by the angular momentum loss. Additionally, the orbital periods of three systems (S03065, S07798, S16873) are detected to show cyclic variation with periods shorter than 11 years, which can be plausibly explained by the presence of close-in third bodies in these massive binaries. Based on all of these results, it is suggested that the detected four semi-detached binaries almost have multiplicity. The companion stars are crucial for the origin and evolution of these massive close binaries.

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arXiv2024

Twenty-three New Heartbeat Star Systems Discovered Based on TESS Data

Heartbeat stars (HBSs) are ideal astrophysical laboratories to study the formation and evolution of binary stars in eccentric orbits and the internal structural changes of their components under strong tidal action. We discover 23 new HBSs based on TESS photometric data. The orbital parameters, including orbital period, eccentricity, orbital inclination, argument of periastron, and epoch of periastron passage of these HBSs are derived by using a corrected version of Kumar et al.'s model based on the Markov Chain Monte Carlo (MCMC) method. The preliminary results show that these HBSs have orbital periods in the range from 2.7 to 20 days and eccentricities in the range from 0.08 to 0.70. The eccentricity-period relation of these objects shows a positive correlation between eccentricity and period, and also shows the existence of orbital circularization. The Hertzsprung-Russell diagram shows that the HBSs are not all located in a particular area. The distribution of the derived parameters suggests a selection bias within the TESS survey towards massive HBSs with shorter orbital periods, higher temperatures and luminosities. These objects are a very useful source to study the structure and evolution of eccentricity orbit binaries and to extend the TESS HBS catalog.

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arXiv2024

Tilted Disk Precession and Negative Superhumps in HS 2325+8205: A Multi-Window Analysis

Tilted disk precession exists in different objects. Negative superhumps (NSHs) in cataclysmic variable stars (CVs) are believed to arise from the interaction between the reverse precession of a tilted disk and the streams from the secondary star.Utilizing TESS photometry, we present a comprehensive investigation into the tilted disk precession and NSHs in the dwarf nova (DN) HS 2325+8205, employing eclipse minima, eclipse depths, NSH frequencies, and NSH amplitudes and the correlation between them as the windows. We identified NSHs with a period of 0.185671(17) days in HS 2325+8205. The NSH frequency exhibits variability with a period of 3.943(9) days, akin to the tilted disk precession period validated in novae-like stars (NLs, SDSS J0812) and intermediate polars (IPs, TV Col).The O-C of eclipse minima were similarly found to vary cyclically in period 4.135(5) days, characterized by a faster rise than fall. Furthermore, the NSH amplitude exhibits complex and diverse variations, which may be linked to changes in the disk radius, mass transfer rate, and the apparent area of the hot spot. For the first time in DNe, we observe bi-periodic variations in eclipse depth (P1= 4.131(4) d and P2= 2.065(2) d ~ Pprec/2), resembling those seen in IPs, suggesting that variations with P2 are not attributable to an accretion curtain, as previously suspected. Moreover, NSH amplitude and eclipse depth decrease with increasing NSH frequency, while NSH amplitude correlates positively with eclipse depth.These complex variations observed across multiple observational windows provide substantial evidence for understanding of tilted disk precession and NSHs.

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arXiv2024

Five New Heartbeat Star Systems with Tidally Excited Oscillations Discovered Based on TESS Data

Heartbeat stars (HBSs) with tidally excited oscillations (TEOs) are ideal astrophysical laboratories for studying the internal properties of the systems. In this paper, five new HBSs exhibiting TEOs are discovered using TESS photometric data. The orbital parameters are derived using a corrected version of Kumar et al.'s model based on the Markov Chain Monte Carlo (MCMC) method. The TEOs in these objects are examined, and their pulsation phases and modes are identified. The pulsation phases of the TEOs in TIC 266809405, TIC 266894805, and TIC 412881444 are consistent with the dominant $l=2$, $m=0$, or $\pm2$ spherical harmonic. For TIC 11619404, although the TEO phase is close to the $m=+2$ mode, the $m = 0$ mode cannot be excluded because of the low inclination in this system. The TEO phase in TIC 447927324 shows a large deviation ($>2σ$) from the adiabatic expectations, suggesting that it is expected to be a traveling wave rather than a standing wave. In addition, these TEOs occur at relatively low orbital harmonics, and we cautiously suggest that this may be an observational bias. These objects are valuable sources for studying the structure and evolution of eccentricity orbit binaries and extending the TESS HBS catalog with TEOs.

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arXiv2024

Evidence for a close-in tertiary orbiting around the Algol-type system HZ Dra with tidal splitting and spots activities

We reported a cyclic variation of $O-C$ diagram with a semi-amplitude of 0.0033 days and a period of 1.05 years for the pulsating eclipsing binary HZ Dra. The cyclic variation can be explained by the light travel-time effect via the presence of a close-in third body orbiting around HZ Dra in an elliptical orbit with a maximum semi-major axis of 0.92 au. Based on the W-D code, the contribution of the third light to the total system is determined to be 29 $\%$, which is in agreement with the estimated value. Our light curve modelling indicates an evolving hot and cool spot on the surface of the primary and secondary components, respectively. Their positions are roughly symmetrical to the inner Lagrangian point L1, which could be used to explain the variation in the O$^{'}$Connell effect. Our frequency analysis detects 1 radial p-mode, 7 non-radial p-modes and 1 non-radial g-mode. In addition, a total of 6 multiplets are identified, spaced by the orbital frequency, which can be explained as a tidally split mode caused by the equilibrium tides of the close binary system with a circular orbit. These pulsating features suggest that the primary of HZ Dra is a $δ$ Scuti star, pulsating in both p- and g-mode and influenced by tidal forces.

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