Preamble

MNRAS 000, 1–11 (2023)
Preprint 12 December 2023
Compiled using MNRAS LATEX style file v3.0

Search for "Ronin" Pulsars in Globular Clusters Using FAST: Discovery of Two New Slow Pulsars in M15

Dengke Zhou,¹★ Pei Wang,²,3† Di Li,¹,²,4 Jianhua Fang,¹ Chenchen Miao,¹ Paulo C. C. Freire,5 Lei Zhang,² Dandan Zhang,6,7 Huaxi Chen,¹ Yi Feng,¹ Yifan Xiao,8 Jintao Xie,¹ Xu Zhang,¹ Chenwu Jin,¹ Han Wang,¹ Yinan Ke,¹ Xuerong Guo,¹ Rushuang Zhao,9 Chenhui Niu,10 Weiwei Zhu,² Mengyao Xue,² Yabiao Wang,11 Jiafu Wu,11 Zhenye Gan,11 Zhongyi Sun,11 Chengjie Wang,11 Junshuo Zhang,2,4 Jinhuang Cao,2,4 Wanjin Lu²,4

¹Research Center for Intelligent Computing Platforms, Zhejiang Laboratory, Hangzhou 311100, China
²National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
³Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China
⁴University of Chinese Academy of Sciences, Beijing 100101, China
⁵Max Planck Institut für Radioastronomie, Auf dem Hügel 69 D-53121, Bonn, Germany
⁶School of Mathematical Sciences, School of Physics and Electronic Sciences, Guizhou Normal University, Guiyang 550001, China
⁷Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Guiyang 550001, China
⁸School of Physics and Electronic Engineering, Qilu Normal University, Jinan 250200, China
⁹Guizhou Normal University, Guiyang 550001, China
¹⁰Central China Normal University, Wuhan 430079, China
¹¹Tencent Youtu Lab, Shanghai 201103, China

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT

Globular clusters harbor numerous millisecond pulsars, yet the detection of long-period pulsars within these clusters has been notably scarce. Searching for long-period pulsars encounters significant challenges due to pronounced red noise interference, necessitating crucial red noise removal during data preprocessing. In this study, we employ running median filtering to mitigate red noise in multiple globular cluster datasets obtained through observations with the Five-hundred-meter Aperture Spherical radio Telescope (FAST). Additionally, we estimate the minimum detectable flux density of pulsars ($S_{\text{min}}$) accounting for this processing step, resulting in a function depicting how $S_{\text{min}}$ varies with different duty cycles and periods. Subsequently, we conduct a systematic search for long-period pulsars in the globular cluster datasets after red noise elimination. Ultimately, we discover two isolated long-period pulsars in the M15 globular cluster, with periods of approximately 1.928451 seconds and 3.960716 seconds, both exhibiting remarkably low pulse duty cycles of around 1%. Using archived data, we obtain timing solutions for these pulsars. Based on the timing results, their positions are found to be close to the center of the M15 cluster. On the $P-\dot{P}$ diagram, they both lie below the spin-up line, suggesting that their recycling process was likely interrupted, leading them to become isolated pulsars. Moreover, in our current search, these very faint long-period pulsars are exclusively identified in M15, and one possible reason for this could be the relatively close proximity and extremely high stellar encounter rate of M15.

As observational data accumulate and search algorithms undergo iterative enhancements, the prospect of discovering additional long-period pulsars within globular clusters such as M15 becomes increasingly promising.

Key words: globular clusters: general – pulsars: general

1 INTRODUCTION

Globular clusters, characterized by their high stellar density, are environments where stellar collisions and interactions are prevalent throughout their evolutionary history, giving rise to a plethora of pulsar binaries, including millisecond pulsars (MSPs) \citep{Clark1975, Bhattacharya1991}. Consequently, globular clusters have long been recognized as crucial birthplaces for MSPs. To date, hundreds of MSPs have been discovered in globular clusters, providing essential samples for studying the evolutionary mechanisms of these pulsars.

In contrast, long-period pulsars (with periods on the order of seconds) are notably rare within globular clusters. Thus far, only two instances of long-period pulsars with periods on the order of seconds have been detected in the literature during searches of globular cluster data. One pulsar has a period of approximately 1 second and resides in a binary system with a companion mass of around 0.1 solar masses \citep{Lyne1993}. The other pulsar, with a period of about 2.5 seconds, is an isolated pulsar, and its affiliation with a globular cluster is yet to be confirmed \citep{Abbate2022}.

The scarcity of long-period pulsars underscores their significance in unraveling the evolutionary pathways of pulsars within globular clusters. The presence of long-period pulsars may imply the existence of distinct dynamical and stellar evolution processes in globular clusters, divergent from the formation mechanisms of MSPs, indicating an alternative pulsar evolutionary path. The rarity of these long-period pulsars emphasizes the urgent need for in-depth investigations. Through more extensive searches and studies of this class of pulsars, we anticipate uncovering the subtle and complex structures within globular clusters, providing more accurate clues to interpret the past and future of these ancient stellar systems.

The FAST telescope, as the world's largest single-dish telescope, possesses extremely high sensitivity in pulsar searches, making the exploration of new pulsars one of its primary objectives \citep{Li2018, Qian2019, Pan2020, Cruces2021, Zhang2019}. In particular, FAST has conducted frequent and prolonged observations of globular clusters, providing ample data for the search of long-period pulsars in globular clusters \citep{Pan2020, Pan2021, Zhang2023b}.

Challenges in searching for long-period pulsars persist, primarily due to the impact of red noise introduced by prolonged telescope integration times \citep{Lazarus2015, Parent2017, Singh2022}. Additionally, the issue of pulse nulling significantly affects the search for periodic signals based on Fourier transform techniques \citep{Backer1970, Singh2023}. In this regard, the fast folding algorithm (FFA) has demonstrated higher sensitivity in searching for periodic signals compared to Fourier transform-based search algorithms \citep{Morello2020}.

In this paper, we utilize FFA to search for long-period pulsars in FAST observations of globular cluster data, hoping to unveil evolutionary branches distinct from MSPs within these ancient celestial systems. The structure of this paper is as follows: Section 2 details the search techniques employed, including algorithmic descriptions and parameter settings. Section 3 presents our search outcomes. Section 4 discusses the two discovered pulsars. Finally, Section 5 provides a summary of the entire article.

2 OBSERVATION AND DATA REDUCTION

We conducted a search on publicly available partial globular cluster data from FAST, and the information for these globular clusters is listed in Table 1. The observation times for these clusters mostly range from one to a few hours, providing us with sufficient signal-to-noise ratio (SNR) for detecting faint pulsars, especially long-period pulsars.

2.1 FFA Algorithm and Search Process

The fast folding algorithm (FFA) differs from the commonly used fast Fourier transform (FFT) for pulsar periodic signal searches. This algorithm folds the time series containing pulsar period signals at regular intervals. During the folding process, the signal is enhanced more than the noise, resulting in an increased signal-to-noise ratio. Ultimately, the signal's most significant pulse profile emerges when folded at the true period. By testing different periods and obtaining the significance of the pulsar profile as a function of the test period, we can identify the optimal period corresponding to the maximum significance. This method exhibits significant advantages over FFT in searching for weak signals, signals with pulse nulling, and long-period signals \citep{Lazarus2015, Kondratiev2009, Morello2020, Singh2023}. However, due to the relatively high computational cost compared to FFT, this algorithm has been used less frequently in recent years in the field of radio pulsar searches.

Recently, \citet{Morello2020} proposed that the sensitivity of FFA in searching for pulsars at any period surpasses that of FFT, especially when searching for long-period pulsars. This prompted us to reconsider the application of this algorithm in the pulsar search process. Especially for target sources like globular clusters, most of which have known dispersions, this can help reduce the number of dispersion measures to search, making it feasible to apply FFA in pulsar searches. Additionally, FAST has relatively long observation durations for these sources (from several hours to several hours), increasing our confidence in the detectability of these pulsars. However, due to fluctuations in telescope system temperature, gain, and celestial background, strong red noise is introduced into the data. This poses challenges for the search of long-period pulsars, especially those that are faint. We will assess this issue in Section 2.2.

FFA folds the time series according to a presumed period, obtaining an averaged time series within one period. Assuming the whitened and normalized time series after folding is represented by vector $\mathbf{x}$, the folded signal profile is represented by vector $a\mathbf{s}$, and the folded background noise is represented by vector $\mathbf{w}$, the equation is given by:
$$
\mathbf{x} = a\mathbf{s} + \mathbf{w}.
$$
Here, $n$ represents the number of points in the folded profile, $\mathbf{s}$ satisfies $\sum_i s_i = 0$ and $\sum_i s_i^2 = 1$. If $a > 0$, it indicates the presence of a pulsar signal in $\mathbf{x}$. To verify the presence of a pulsar signal in $\mathbf{x}$, a direct method is to use a known pulsar profile template $\mathbf{s}$ and perform a dot product with $\mathbf{x}$. The resulting value is called the Z-statistic, given by:
$$
Z = \mathbf{x} \cdot \mathbf{s} = a + \mathbf{w} \cdot \mathbf{s} = a + \sum_i s_i w_i.
$$
As $\mathbf{x}$ is whitened and normalized, its mean value is 0. Assuming that the contribution of the pulsar signal $a\mathbf{s}$ to determining the overall profile mean is much smaller than the contribution of the noise $\mathbf{w}$, then $w_i \sim N(0, 1)$. Thus, $\sum_i s_i w_i \sim N(0, 1)$, considering $\sum_i s_i^2 = 1$. Consequently, $Z \sim N(a, 1)$. In practical searches, the actual pulsar profile template $\mathbf{s}$ is unknown. However, a series of normalized square waves with varying widths and phases can be used to progressively approximate the pulsar profile template $\mathbf{s}$. \citet{Morello2020} suggests continuing to refer to $Z$ as the detection SNR, and SNR can be used to filter candidate signals.

Our search process is as follows. Initially, \textsc{presto}\footnote{https://github.com/scottransom/presto} is utilized to mitigate radio frequency interference (RFI) in the data, followed by batch de-dispersion processing to obtain a large number of time series corresponding to different dispersion measures. In these globular clusters, most have detected several pulsars, enabling us to concentrate the de-dispersion range around a good value. If no pulsars have been detected in a specific globular cluster, making it impossible to determine the average dispersion, we use the YMW16 model to estimate its dispersion measure \citep{Yao2017}. Additionally, we appropriately downsampled the data in time to reduce the data volume and improve the efficiency of searching for long-period pulsars. After obtaining the time series, it is necessary to remove red noise from the data, as our main focus is on searching for long-period pulsars.

The removal of red noise and FFA search are both performed using \textsc{riptide}\footnote{https://github.com/v-morello/riptide}. As revealed in the simulations in the subsequent section, although the red noise in FAST data does not significantly impact the minimum detectable flux density of pulsars, the process of removing red noise weakens the amplitude of the signal itself, thus affecting the minimum detectable flux density. We will discuss this in detail in Section 2.2. Figure 1 shows the flowchart of our search for long-period pulsars in globular clusters.

2.2 Calculating Sensitivity in the Presence of Red Noise

In pulsar searches, to estimate the telescope's sensitivity to pulsars, the radiometer equation proposed by \citet{Dewey1985} and \citet{Lorimer2012} under the assumption of white noise can be used to estimate the average minimum detectable flux density, given by:
\begin{equation}
S_{\text{min}} = \beta \frac{(T_{\text{sys}} + T_{\text{sky}}) \text{SNR}}{\epsilon G \sqrt{n_p t_{\text{obs}} \Delta f}} \sqrt{\frac{W}{P - W}}
\end{equation}
where $P$ is the period of the pulsar signal, $\beta$ represents data digitization losses, $T_{\text{sys}} + T_{\text{sky}}$ is the sum of the telescope system temperature and the sky background temperature, $G$ is the telescope gain, SNR denotes the signal-to-noise ratio of the pulsar signal, $n_p$ is the number of polarization channels for data merging, $\Delta f$ is the frequency bandwidth for data merging, and $t_{\text{obs}}$ is the observation duration.

The original formula does not include the parameter $\epsilon$, but according to the suggestion by \citet{Morello2020}, the algorithm search efficiency $\epsilon$ needs to be introduced to adjust the original radiometer equation. $W$ refers to the width of the observed pulsar signal, encompassing both the intrinsic width of the pulsar signal and the broadening due to sampling time, scattering, and channel dispersion. Using $W_0$ to represent the intrinsic width of the pulsar signal and $t_{\text{sample}}$, $t_{\text{DM}}$, and $t_{\text{scatt}}$ for the sampling time, interstellar medium dispersion in the frequency channel, and interstellar scattering-induced pulse broadening, respectively:
\begin{equation}
W = \sqrt{W_0^2 + t_{\text{sample}}^2 + t_{\text{DM}}^2 + t_{\text{scatt}}^2}
\end{equation}
where
\begin{equation}
\log_{10}(t_{\text{scatt}}(\text{ms})) = -6.46 + 0.154 \log_{10}(\text{DM}) + 1.07(\log_{10}(\text{DM})^2 - 3.86 \log_{10}(f(\text{GHz})))
\end{equation}
\citep{Bhat2004}. For FAST, the typical minimum detectable flux density of pulsars can be calculated by taking $T_{\text{sys}} + T_{\text{sky}} = 25$ K, $G = 16$ K Jy$^{-1}$, $n_p = 2$, $\Delta f = 300 \times 10^6$ Hz (considering the proportion of remaining frequency channels after excluding RFI, approximately 75% of good data), $P = 0.15$, SNR = 6, $\epsilon = 0.93$, and $t_{\text{obs}} = 3600$ s, resulting in an approximate value of 3 $\mu$Jy.

However, Equation 3 assumes that the data is dominated by white noise. In reality, with an increase in the telescope's integration time, variations in the telescope system temperature, sky background temperature, and gain changes will eventually introduce strong red noise into the data. Panel (a) of Fig. 3 displays the power spectral density (PDS) of the time series of globular cluster data we searched, revealing the presence of strong power-law components in the colored noise, collectively referred to as red noise. This disrupts the statistical assumption of Equation 3, rendering the calculation of the minimum detectable flux density of pulsars unreliable. Quantifying the specific impact of red noise on the minimum detectable flux density is exceptionally challenging. The study conducted by \citet{Lazarus2015} utilized simulated pulse injection to obtain the simulated variation of the minimum detectable flux density with pulse period. They found that the original formula significantly underestimated the minimum detectable flux density as the signal period increased.

In our approach, we take a different route to obtain the actual minimum detectable flux density. We assume that the average temperature of the telescope remains nearly constant during the observation period and only fluctuates around this mean temperature. We then employ a multiple-sampling method to establish the relationship between telescope temperature fluctuations and the spectral characteristics of the measured time series, thus further establishing the connection between the time series and the minimum detectable flux density. The specific methodology is as follows:

• Resample the input time series with an observation time of $T$ using a normal distribution. The center of the normal distribution is the measured value of the time series, and the standard deviation is taken as the measured value when the time sampling is 1 second (for other time samplings, it is converted according to the error synthesis formula). Then, obtain $m$ resampled time series.
• Calculate the mean value for each time series to obtain $m$ mean values.
• Calculate the standard deviation of these $m$ mean values, which serves as the measure of the telescope temperature fluctuation, denoted as $\Delta T$.
• For periodic pulse signals, assuming a duty cycle of $\delta$, the telescope temperature fluctuation still follows the following formula:
  \begin{equation}
  \Delta T(t_{\text{obs}} = \delta T)^2 + \Delta T(t_{\text{obs}} = T - \delta T)^2.
  \end{equation}
• Assuming $T_{\text{peak}}$ is the increase in telescope temperature caused by the pulsar signal, the SNR $= T_{\text{peak}}/\Delta T$ can be used to obtain $T_{\text{peak}}$ by the pulsar signal under a given SNR using an iterative method.

Using the above steps, the relationship between the time series and the minimum detectable flux density can be established without the need to ensure that the time series must be white noise. As a validation, we separately simulated white noise and red noise time series. Subsequently, we used our method to estimate the minimum detectable flux density of pulsars under these two time series and compared it with the radiometer equation. The simulation was performed using the \textsc{simulate} module in \textsc{stingray}\footnote{https://github.com/StingraySoftware/stingray} \citep{Huppenkothen2019}, a module based on the algorithm proposed by \citet{Timmer1995} for reverse simulating time series given a PDS. By setting different rms values, we can adjust the strength of the simulated red noise, where smaller rms values make the simulated time series closer to white noise, and larger rms values result in stronger red noise. Fig. 2 displays our simulation results. The top two panels of Fig. 2 depict curves of the simulated minimum detectable flux density as a function of duty cycle and period, considering both white noise and red noise with an rms of 40%. These results are compared with theoretical values from the radiometer equation. It is observed that the simulated values match well with the theoretical values in the case of white noise, but there is some deviation in the case of red noise. The bottom two panels of Fig. 2 display results obtained from the observational data of the globular cluster M15 in 2019. Overall, the minimum detectable flux density shows minimal deviation from theoretical values, with a maximum difference of approximately 10% in the worst-case scenario. This indicates that as long as the average temperature remains nearly constant, the differences in pulsar minimum detectable flux density caused by the red noise itself are negligible.

Although the minimum detectable flux density is not significantly impacted by red noise, not removing the red noise could lead to signal overlap, making it challenging to distinguish the signal. Although the signal is technically detectable once the red noise is removed, the removal process also eliminates signals on the same time scale, artificially reducing the signal amplitude and consequently increasing the actual minimum detectable flux density. In practical pulsar searches, especially for long-period pulsars (where, under constant duty cycle, the pulse width of the signal is larger and can be confused with red noise), it is necessary to eliminate red noise. This process increases the actual minimum detectable flux density, and the specific increase depends on the algorithm and parameters used for red noise removal.

In our pulsar search, we employed a running median filtering algorithm to remove red noise. The use of running median filtering to eliminate red noise directly affects the signal amplitude reduction, leading to an increase in the average minimum flux density required for the original signal at a fixed SNR. The running median filtering algorithm's impact on pulse reduction is related to the pulse width; the algorithm preserves any pulse narrower than half the window width and weakens broader pulses \citep{Gallagher1981}. Therefore, it is evident that the increase in the minimum detectable flux density due to red noise removal is not directly related to the signal period but is directly related to the signal's time scale or pulse width.

Assuming a nearly constant duty cycle for pulsar signals, longer signals will naturally have larger widths, making them more affected by red noise removal. To quantify this, assuming the duty cycle of pulsar signals is nearly constant and at a typical value of 0.1, and the sliding time window used for red noise removal is 5 seconds, red noise removal will significantly impact signals with periods longer than 25 seconds (5/2/0.1 = 25 seconds). The specific impact value may depend on the real data, so we use a simulation method to investigate the relationship between the actual minimum detectable flux density and the period after red noise removal. The specific approach involves injecting signals into each globular cluster dataset, then using running median filtering to remove red noise. By comparing the signal amplitudes before and after removal, we obtain the signal amplitude reduction ratio due to red noise removal, referred to as the correction factor $CF_{\delta10}$. The correction factors for the globular cluster data we searched as a function of pulse period are shown in panel (b) of Fig. 3. It can be seen that, within a relatively short period range, the correction factor is generally close to 1, indicating that no correction is needed for the original minimum detectable flux density. However, for longer periods, the correction factor is noticeably less than 1, indicating the need for correction to the minimum detectable flux density.

For a duty cycle of 10%, the complete formula for the minimum detectable flux density is:
\begin{equation}
S_{\text{min}} = \frac{\beta (T_{\text{sys}} + T_{\text{sky}}) \text{SNR}}{CF_{\delta10} \epsilon G \sqrt{n_p t_{\text{obs}} \Delta f}} \sqrt{\frac{W_{\delta10}}{P - W_{\delta10}}}
\end{equation}
where the dependence of the minimum detectable flux density on the period $P$ is partly included in $CF_{\delta10}$. Since we assume that red noise has been eliminated, there will be no red noise factor in the expression. Panel (c) of Fig. 3 presents a heatmap of the minimum detectable pulsar flux density of the globular cluster data as a function of duty cycle and period after red noise removal, with an SNR of 6. It can be observed that, at any given duty cycle, the minimum detectable flux density increases with an increase in the period. However, the position where the minimum detectable flux density starts to increase is related to the duty cycle. This is because the running median filter is sensitive only to the pulse width of the pulsar signal, so as soon as the product of the duty cycle and the period reaches a threshold, the signal is rapidly attenuated. Panel (d) of Fig. 3 presents a heatmap of the signal-to-noise ratio of pulsars in the globular cluster data as a function of the average pulsar flux density and period after red noise removal, with a duty cycle of 10%. It can be observed that, for shorter periods, a relatively small average flux density is sufficient to achieve a high SNR. However, as the period increases, a larger average flux density is required to achieve the same SNR. Finally, we list the sensitivity of the pulsar search for each globular cluster data in Table 1. As the minimum detectable flux density now depends on multiple parameters, we calculated values for periods of 1 second and 100 seconds at a duty cycle of 10% and a SNR of 6 to obtain a typical range.

2.3 Configuring Search Parameters

From Fig. 3, it can be seen that when the searched period exceeds 100 seconds, the correction factor is almost close to 0, making it extremely difficult to search for longer-period signals. While increasing the width of the running median filter can improve the correction factor for periods exceeding 100 seconds, it simultaneously makes the detection of signals at the second level difficult. Therefore, this is a balance selection problem. Considering that pulsars in globular clusters with periods at the second level are currently very rare, with only two reported \citep{Lyne1993, Abbate2022}, we focus our search on the range of 100 milliseconds to 100 seconds. The width of the running median filter is set to 5 seconds.

As for the dispersion search, as mentioned in Section 2.1, for globular clusters where pulsars have been discovered, a search is conducted within a range of $\pm 20$ pc cm$^{-3}$ around their known dispersion measures. For globular clusters where pulsars have not been discovered, the YMW16 model is used for dispersion estimation, and a search is conducted within a range of twice the estimated dispersion measure. From panel (c) of Fig. 3, it can be seen that under high duty cycles and short periods, the simulated minimum detectable flux density based on the data is still acceptable. Therefore, to avoid missing signals with high duty cycles, we set the minimum pulse duty cycle for the search to 0.001 and the maximum to 0.5. Additionally, to ensure that long-period signals still have sufficient resolution at small duty cycles, we have appropriately adjusted the time resolution based on the size of the searched period.

3 DISCOVERIES

We systematically conducted a periodic search using FFA on the globular cluster data listed in Table 1. Ultimately, we discovered two long-period pulsars in the M15 globular cluster. The discovery of these two pulsars marks the first time that FAST has detected pulsars with periods on the order of seconds in globular clusters. Additionally, one of these pulsars has the longest period among pulsars discovered in globular clusters to date. We will introduce these two pulsars separately.

3.1 M15K

The pulsar M15K was discovered in the archived data from FAST on November 9, 2019. Its period is about 1.928449 seconds, and the best dispersion measure is 66.5 pc cm$^{-3}$. The frequency-phase plot, the pulse profile, and the time-phase plot are shown in the left panel of Fig. 4. The pulse profile is single-peaked, with a duty cycle of approximately 1.09% and an effective pulse width of 21.01 ms. We further searched the archived FAST data for additional observations of this globular cluster (details listed in Table 2), and the pulse profiles for these observations are displayed in Fig. 5. In most cases, the new search confirmed the existence of this pulsar. The lack of signals in some observations may be attributed to the potential effects of interstellar scintillation \citep{Zhang2023a}.

The severe presence of red noise in the data significantly hindered the acquisition of TOAs. Therefore, we applied a running median filter with a width of 0.3 seconds to mitigate the red noise in the data. Finally, utilizing these archived data, and with the assistance of the code determining the rotation count of pulsars ("DRACULA", \citealt{Freire2018}), we determined the phase-connected timing solution for M15K, spanning from 2019 to 2023. The timing solution is shown in Table 3 and the timing residuals are shown in the upper panel of Fig. 6. According to the timing solution, the position of M15K relative to the center of M15 is shown in Fig. 7. We can see that the position of M15K deviates by 0.063 arcmin from the center of the cluster, which is very close to the central location of the cluster.

3.2 M15L

The pulsar M15L was discovered in the archived data from FAST on November 9, 2019. Its period is about 3.960716 seconds, and the best dispersion measure is 66.1 pc cm$^{-3}$. The frequency-phase plot, the pulse profile, and the time-phase plot are shown in the right panel of Fig. 4. The pulse profile is single-peaked, with a duty cycle of approximately 0.86% and an effective pulse width of 33.93 ms.

Similar to M15K, we further searched the archived FAST data for additional observations of this globular cluster (details listed in Table 2), and the pulse profiles for these observations are displayed in Fig. 5. In most cases, the new search confirmed the existence of this pulsar. Consistent with the data processing approach for M15K, we applied a running median filter with a width of 0.3 seconds to mitigate the red noise in the data. Finally, these archived data enable us to determine the phase-connected timing solution for M15L that spans from 2019 to 2023. The timing solution is shown in Table 3 and the timing residuals are shown in the lower panel of Fig. 6. According to the timing solution, the position of M15L relative to the center of M15 is shown in Fig. 7. We can see that the position of M15L deviates by 0.455 arcmin from the center of the cluster but still falls within the cluster's half-mass radius of 1.06 arcmin. Unlike PSR J1823-3022 in NGC 6624 \citep{Abbate2022}, both the dispersion and position of M15L indicate that this pulsar resides in the globular cluster M15, marking it as the pulsar with the longest period discovered in globular clusters to date.

4 DISCUSSION

The left panel of Fig. 8 shows the period distribution of all pulsars currently discovered in globular clusters. It can be seen that the majority of them are millisecond pulsars. The two pulsars, M15K and M15L, discovered in this study, are located at the outskirts of the distribution with long periods.

The formation of long-period pulsars in globular clusters may be influenced by various factors. In the simplest scenario, isolated pulsars lose energy through radiation, eventually leading them to become long-period pulsars. In addition, dynamic interactions among celestial bodies, including tidal effects, collisions, and gravitational interactions \citep{Fabian1975, Bhattacharya1991, Verbunt2014, Abbate2022}, may also play a role in shaping the characteristics of these pulsars. In previous studies, two pulsars with periods in the order of seconds were detected in globular clusters, both associated with core-collapsed clusters.

The first pulsar, PSR B1718-19, with a period of 1.004 seconds, is linked to the core-collapsed NGC 6342 cluster, exhibiting a substantial offset from the cluster center (2.3′). This pulsar has a low-mass non-degenerate star as a companion, further strengthening its association with the cluster. The second pulsar, PSR J1823-3022, with a period of approximately 2.5 seconds, may be associated with the core-collapsed globular cluster NGC 6624. However, its larger offset from the cluster center (3′) and noticeable deviation in DM from other pulsars in the cluster make its association less certain.

As for M15K and M15L, their association with the core-collapsed globular cluster M15 can be confirmed based on the positions obtained from the timing solutions and the typical dispersion measures corresponding to the globular cluster. According to the timing solutions, we obtained period derivatives for M15K and M15L, which are $1.183 \times 10^{-15}$ s s$^{-1}$ and $8.85 \times 10^{-16}$ s s$^{-1}$, respectively. The corresponding surface magnetic fields are $1.53 \times 10^{12}$ G and $1.89 \times 10^{12}$ G, and the associated characteristic ages are 25.82 Myr and 70.92 Myr.

The positions of M15K and M15L on the $P-\dot{P}$ diagram are shown in the right panel of Fig. 8. We also plotted PSR B1718-19 and PSR J1823-3022 for comparison. The magnetic fields of M15K and M15L update the records of measured magnetic fields for pulsars in globular clusters. The characteristic ages of these pulsars are not particularly old, especially for M15K. According to the assumption that young pulsars are recycled old neutron stars, they should lie below the spin-up line on the $P-\dot{P}$ diagram \citep{Verbunt2014}:
\begin{equation}
P_{\text{su}}(\text{s}) \approx 1.6 (10^{15} \dot{P})^{3/4}.
\end{equation}
This line represents the shortest spin period achievable through Eddington accretion during the recycling process. Following the suggestion of \citet{Verbunt2014}, we have plotted the spin-up line in the diagram, with the corresponding $\dot{P}$ values increased by a factor of 7. It is observed that both M15K and M15L are below the spin-up line, suggesting that these pulsars may have been disrupted by a dynamical encounter during the recycling process, as detailed in \citet{Verbunt2014}.

We have also plotted the death lines of four models \citep{Bhattacharya1992, Chen1993, Zhang2000} in the right panel of Fig. 8 and find that M15L is relatively close to the death line. Previous theoretical models of pulsar radiative mechanisms have shown that cascade production is necessary to sustain the observed radio emission. The pulsars M15K and M15L in the region of the $P-\dot{P}$ diagram below the spin-up line are hard to support efficient pair cascade production above the pulsar polar caps in their inner magnetospheres. This is attributed to the unlikely increase in the thickness of the vacuum gap above the pulsar's polar cap over a long spin period, which is necessary to maintain the potential difference required for magnetic pair production. This leads to the disappearance of the radio emission. On the other hand, both M15K and M15L are above the death line of the space-charge-limited flow model, and if a multipolar magnetic field configuration is present, pair cascades can also be supported by the free flow of non-relativistic charges from the polar caps. The clear signature of a multipolar component near the surface of a neutron star can be seen in the magnetar SGR J0418+5729 \citep{Tiengo2013} and in a recent recycled MSP PSR J0030+0045 \citep{Riley2019, Raaaijmakers2019}, suggesting that multipolar magnetic field configurations may be ubiquitous in neutron stars, including M15K and M15L.

It is noteworthy that, as shown in Table 1, the sensitivity of M15 globular cluster with FAST is not outstanding. However, up to now, we have only discovered long-period pulsars in M15 globular cluster. \citet{Hui2010} and \citet{Turk2013} pointed out that in any given globular cluster, the number of potentially detectable pulsars $N_{\text{psr}}$ mainly depends on the cluster's stellar encounter rate $\Gamma$, with $\ln(N_{\text{psr}}) = -1.1 + 1.5 \log \Gamma$. Since M15 is expected to have the highest number of detectable pulsars among the observed globular clusters within FAST field of view \citep{Zhang2016}, the probability of detecting pulsars in the M15 globular cluster is higher compared to other clusters. To facilitate a more intuitive comparison of the discovery scenarios across various globular clusters, we introduced the long-period pulsar discovery rate $F$. This rate is defined as the stellar encounter rate $\Gamma$ divided by the product of the minimum flux sensitivity $S$ and the square of the distance $D$, i.e. $F = \Gamma / (S D^2)$. In Table 1, the values of $F$ have been normalized by dividing each $F$ value by their maximum. By examining Table 1, it is evident that the discovery rate of long-period pulsars in M15 globular cluster is significantly higher than that in other globular clusters. Therefore, the discovery of two long-period pulsars in M15, with no such discoveries in other globular clusters, can be considered a normal phenomenon. Other globular clusters similar to M15 may also potentially harbor a population of missing long-period pulsars, and observations in X-rays hold the promise of aiding in the understanding of the intricate processes involved in the evolution of LMXB-MSP intermediate states.

5 SUMMARY

We systematically searched for long-period pulsars (100 ms $\sim$ 100 s) in globular clusters observed by FAST using FFA. Due to the presence of strong red noise in the data, it was necessary to perform red noise mitigation before searching for these pulsars. We proposed a simulation-based method to estimate the minimum detectable flux density of pulsars in the presence of red noise and compared the simulation results with the radiometer equation. We found that the red noise in FAST has a minimal impact on the minimum detectable flux density. However, for separating pulsar signals from red noise, the removal of red noise is necessary, and this affects the minimum detectable flux density of pulsars. Red noise removal has a greater impact on signals with larger pulse widths. Therefore, for signals with a constant duty cycle, longer-period pulsars are more affected by red noise removal. We provided correction factors for the minimum detectable flux density due to red noise removal, calculated typical values of the corrected minimum detectable flux density for 1 s and 100 s, and listed them in Table 1.

After the search, we discovered two long-period pulsars in M15 globular cluster with periods of approximately 1.9 seconds and 3.9 seconds, respectively. Notably, the period of pulsar M15L even surpasses the longest record among all known pulsars in globular clusters, which is unusual for globular clusters dominated by millisecond pulsars. To date, only two long-period pulsars with periods on the order of seconds have been detected in globular clusters \citep{Lyne1993, Abbate2022}, and the affiliation of one of these pulsars with the respective globular cluster is yet to be confirmed. In addition to the two long-period pulsars discovered in this study, all these pulsars are located in core-collapsed globular clusters. Both M15K and M15L in the $P-\dot{P}$ diagram lie below the spin-up line, suggesting that these pulsars may have been disrupted during the recycling process, becoming isolated pulsars and leading to a gradual increase in their periods. The discovery of M15K and M15L provides a new and significant sample for understanding this type of pulsar.

The positive results obtained by applying FFA instead of FFT for the search of long-period pulsars in globular clusters suggest that FFA indeed has a significant advantage in long-period pulsar searches. Future observations of globular clusters and iterative upgrades of algorithms may provide more opportunities to discover these faint, long-period pulsars. Once more long-period pulsars are found, it will contribute to a more comprehensive understanding of the evolutionary mechanisms of pulsars in globular clusters.

ACKNOWLEDGEMENTS

This work has used the data from the Five-hundred-meter Aperture Spherical radio Telescope (FAST). FAST is a Chinese national mega-science facility, operated by the National Astronomical Observatories of Chinese Academy of Sciences (NAOC). This work is supported by the National SKA Program of China No. 2020SKA0120200, the Youth Innovation Promotion Association CAS (id. 2021055), NSFC No. U2031117.

DATA AVAILABILITY

The majority of the data is already in the public domain and can be accessed in accordance with FAST data policy. A small portion of undisclosed data can be obtained by contacting the author with a reasonable justification.

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