The X-ray emissions from low-mass X-ray binaries (LMXBs) arise from the accretion processes surrounding compact objects such as neutron stars (NS) or black holes. In the case of NS, the accreted hydrogen, helium, or mixture of them can be consumed via unstable nuclear burning on the stellar surface, leading to the observed type I X-ray bursts (Galloway et al., 2008).

X-ray burst oscillations (XBOs), identified through timing analysis, are periodic signals observed during bursts originating from the NS rotation (Strohmayer et al., 1996; Galloway and Keek, 2021). A type I X-ray burst can form a hot spot on the NS surface, leading to an uneven temperature distribution. The star’s rotation modulates this uneven distribution, producing periodic signals in the soft X-ray band (Strohmayer et al., 1997b; Goodwin et al., 2021). The standard methods for XBO detection include a fast Fourier transform (FFT) or Z n 2 statistics (see Watts, 2012, and references therein). During a burst, the oscillation frequency of XBOs evolves (Muno et al., 2002). To effectively search for and track XBO signals, the dynamic power density spectrum method is commonly used to analyze type I X-ray bursts (e.g., Strohmayer et al., 2008).

However, the mechanisms behind the formation and evolution of these hot spots remain incompletely understood. Proposed surface modes, such as the Rossby model, suggest that type I X-ray bursts can excite oscillations in the NS ocean (Chambers and Watts, 2020). A temperature gradient forms at different heights on the star’s surface, and the star’s rotation modulates this gradient to produce XBOs (Cumming and Bildsten, 2000; Watts, 2012; Mahmoodifar and Strohmayer, 2016). Nonetheless, these models only partially explain the observed XBO phenomena, necessitating further study into their physical processes and model interpretations.

Detecting XBO signals requires high-energy X-ray telescopes with high timing resolution, large effective areas for substantial photon accumulation, and minimal dead-time effects. Previous searches for XBO signals have utilized data from RXTE (Strohmayer, 1999; 2001; Bilous and Watts, 2019) and NICER (Mahmoodifar et al., 2018; Li et al., 2022). Observations have shown that oscillation signals in most bursts exhibit an upward frequency drift and generally occur during the burst’s tail. In some burst samples, oscillation signals have such high amplitudes that the accretion pulsar’s oscillating frequency diverges from the pulsar’s rotation frequency by a few hertz (e.g., Chakrabarty et al., 2003).

Up until now, 349 galactic NS LMXBs have been found and the number is increasing (Avakyan et al., 2023). However, less than 10% of these sources have had their spin frequency and burst oscillation measured. Detecting burst oscillations from sources with unknown spin frequencies, or from newly discovered sources, will expand the sample of accreting pulsars (Patruno et al., 2017). This will allow for a more credible study of the spin frequency distribution, and for searching coherent X-ray pulsation during outbursts. Additionally, among these unknown sources, there may be neutron stars spinning faster than 716 Hz, which would impose stronger constraints on the equation of state of compact stellar objects.

In’t Zand et al. (2019) carried out a comprehensive searching and spectral analysis of type I X-ray bursts from Galactic NSs observed by Swift BAT/XRT. They identified 28 X-ray bursts for which BAT event data were available. In this paper, we search the Swift/BAT archives for triggered events of LMXBs from 2005 to April 2024 and perform timing analysis to search burst oscillation. We introduce the observations and data analysis methods in Sect. 2. We provide the results of the type I X-ray burst oscillation in Sect. 3. In Sect. 4, we discuss and summarize the results.

The Swift/BAT triggering system is designed to identify gamma-ray bursts (GRBs) and other fast transients in high energy from various sources, including black holes, NSs, and magnetars. In this study, we found 50 non-GRB events from the GCN Notice Archive. These events are from 37 distinct sources, with 22 triggers related to NS LMXBs exhibiting X-ray burst oscillations or coherent pulsations, and the remaining samples involving NS LMXBs without known spin frequency. The absolute timing accuracy, 0.1 m of the recorded event files allows us to perform timing analysys. The Swift/BAT data were processed using the batgrbproduct command, and 1-s binned light curves were extracted in the 13–20 keV energy band. Subsequently, the batbinevt command was employed to transform the event data for each X-ray burst into a mask-weighted (background-subtracted) light curve in the 13–20 keV band. The burst peak flux and duration were determined by analyzing these light curves, with the duration, t 90 , defined as the time interval during which the cumulative photon count increases from 5% to 95% of the total count (Kouveliotou et al., 1993). The light curves of all 50 bursts are displayed in Figure1.

Type I X-ray burst sources were categorized based on the presence or absence of previously detected XBOs or coherent pulsations. For the sources without known spin frequency, our analysis of bursts is explained as follows.

Initially, we applied FFT to the event files from each burst, segmenting the analysis into 4-s intervals with a step size of 0.5 s. For each window, statistically independent FFT was recorded between 10 and 2000 Hz with a step of 0.25 Hz. The light curve of duration T is evenly divided into N channels, where N = 2 m , and m is an integer. The time series, x k , represents the number of photons in the k th channel, where k ranges from 0 to N − 1 . Following Leahy et al. (1983), we computed the Leahy-normalized power spectrum P j for each segment, P j = 2 N tot ∑ k = 1 N x k ⁡ cos ⁡ 2 ⁡ π ν j t k 2 + ∑ k = 1 N x k ⁡ sin ⁡ 2 ⁡ π ν j t k 2 , (1) where N tot is the total number of photons. We then obtain the frequency of the oscillation signal according to the power spectrum.

Oscillation signal confirmation is applied using Z n 2 statistic. We employ the Z n 2 statistic on the 13–20 keV event data to confirm the presence of oscillation signals. This method is more computationally intensive than FFT but offers higher frequency precision. The Z n 2 statistic is defined as follows (Buccheri et al., 1983): Z n 2 = 2 N tot ∑ k = 1 n ∑ j = 1 N tot cos ⁡ k ϕ j 2 + ∑ j = 1 N tot sin ⁡ k ϕ j 2 , (2) where n is the harmonic number. Since the first harmonic provides the strongest signal strength and contains most of the dynamic information, we adopt n = 1 . And ϕ j is the photon phase defined as ϕ j = 2 π ∫ t 0 t j ν t d t , (3) where t 0 is the reference time, t j is the arrival time of the photon relative to t 0 , and ν ( t ) represents a constant frequency model. When ν ( t ) does not change with time, ϕ j = 2 π ν t j . The maximum value of Z 1 2 on the dynamic power spectrum and its corresponding frequency were identified.

For sources with previously identified XBOs or coherent pulsations, a dynamic power spectrum was generated by applying the Z 1 2 statistic around the known oscillation frequency ( ± 2 Hz), with analysis intervals set to 4 s and a 0.125-s step size.

We evaluated the confidence level for all burst oscillation signals. The total number of trials is N = N t × N ν , with N t representing the number of time bins and N ν representing the number of frequency bins. According to Roy et al. (2021), the single-trial chance probability is given by the survival function e − Z n 2 / 2 , where Z n 2 is the maximized Z 1 2 or P m . The confidence level is determined as X σ ( X = 2 e r f − 1 ( 1 − x ) ). For the Z 1 2 statistics, we assessed the probability P r o b that the signal measured in N trials was solely produced by noise (Ootes et al., 2017).

In addition to the oscillation frequency, the power spectrum contains information about the pulse amplitude. For the Leahy-normalized power spectrum, the root mean square (RMS) amplitude is defined as follows, A r m s = Z n 2 N γ N γ N γ − B , (4) where N γ and B represent the total and the background photon counts, respectively.

Based on Equations 1–4, we carried out the XBO searching. The results of the two categories of burst sources are summarized in Tables 1, 2. Definitions of burst parameters follow those provided by in’t Zand et al. (2019).

This study provides an extensive analysis of Swift/BAT observations of NS LMXBs over nearly two decades, from 2005 to April 2024. Our dataset comprised 50 type I X-ray bursts, with 21 previously analyzed for burst oscillation signals and 29 newly found. The analysis focused on both sources with previously detected XBOs or coherent pulsations and sources without prior detections, applying FFT and Z 1 2 statistical techniques to identify potential oscillation signals.

For the bursts with known oscillation signals, our analysis confirmed previously reported results, such as the case of 4U 0614 + 09 in 2007 exhibited a significant oscillation signal (Strohmayer et al., 2008). However, most of the bursts showed Z 1 2 values between 10 and 20, indicating low confidence levels for the detected oscillation signals.

For sources without known spin frequencies, we employed FFT analysis to search for burst oscillation signals. The detected oscillation candidates spanned a wide frequency range from 40 to 950 Hz, with maximum P m values varying between 28 and 34. Most of these signals had significance levels below 3 σ . Exceptions included MAXI J1421–613, XTE J1701–407, XMM J174457–2850.3, Swift J1734.5–3027, IGR J17473–2721, Swift J174805.3–244637, Swift J181723.1–164300, and X 1832–330, all of which showed significance levels greater than 3 σ . Of particular interest is the burst from Swift J181723.1–164300 observed in 2007, which exhibited a significance level exceeding 4 σ at a frequency of 946 Hz. The oscillation signal in this burst was prominently observed during its initial rise and subsequent decay phases. If the burst oscillation frequency is confirmed in the future, this source hosts the highest spinning NS.

While theoretical models exist to explain X-ray burst oscillations (XBOs), they do not fully account for all observed phenomena. This highlights the necessity for continued observational efforts, enhanced theoretical models, and comprehensive analyses. Recent advancements in observational technology and analytical methodologies have considerably improved our capability to detect and analyze XBOs.

Future research will benefit from combining data from multiple observatories and utilizing advanced statistical methods to enhance the sensitivity and reliability of burst oscillation detection. Considering the low occurrence rate of burst oscillations, observing more bursts will help increase the number of burst oscillation detection. The anticipated contributions from next-generation telescopes, such as eXTP, THESEUS, ATHENA (Zhang et al., 2019; Amati et al., 2018; Barret et al., 2018), are expected to significantly enhance our observational capabilities. These telescopes, equipped with instruments featuring large effective collecting areas, wide fields of view, and high timing accuracy in the soft X-ray band, will allow for more precise measurements of neutron star properties. These advantages will enable more precise measurements of NS properties, thereby providing tighter constraints on the equation of state of dense matter. Such efforts are providing unprecedented opportunities to study XBOs with greater precision and are essential for refining theoretical models of XBOs and advancing our understanding of these complex phenomena.