Investigation of Nulling and Subpulse-drifting Properties of PSR J1649+2533 with FAST

We report the nulling and subpulse-drifting properties of PSR J1649+2533 with observations of the Five-hundred-meter Aperture Spherical Radio Telescope (FAST). The FAST observations reveal that the nulling fraction of this pulsar is about 20.9% ± 0.8% at 1250 MHz. The statistical study of the durations of the nulling and burst states shows that the burst states can persist for extended pulse periods, exceeding 100 periods in some cases, while the shortest lasts only a few pulse periods. The null states have a much shorter duration, with a maximum duration of less than 30 pulse periods. A comparative study between the pulse profiles of the first active pulse (FAP) and the last active pulse (LAP) shows that the pulse profiles of LAPs are stronger and wider than those of FAPs. An analysis of the two-dimensional fluctuation spectrum for the single-pulse stack indicates that the pulsar exhibits clear amplitude modulation and subpulse drifting. The periods are a P 3 = 2.5 ± 0.1 spin period and P 2 = 17.°0 ± 0.°5 at 1250 MHz, respectively. In addition, the multiband investigation shows that P 2 seems to increase with the increase of the observing frequency, i.e., P 2 ∝ ν 0.8±0.1. This seems to be caused by the increasing pulse-profile width with the frequency. The high-sensitivity FAST observations will enable a detailed understanding of the emission of this pulsar and provide important clues for theoretical studies of the radiation mechanism.


Introduction
Pulsars are known for their highly periodic pulse emission.The shape of a pulse profile averaged over hundreds or thousands of single pulses is apparently very stable for most pulsars (Helfand et al. 1975;Kaspi & Stinebring 1992;Kramer et al. 1994;Kramer 1994; Qiao et al. 2001;Shang et al. 2017Shang et al. , 2020Shang et al. , 2021)).However, many observations show that the shape and phase of single pulses vary randomly with time.Since the single pulses represent the fundamental emission process, their observation and study are very important and hot in current pulsar astronomy.The radiation mechanism of pulsars has remained a mystery since its discovery over five decades ago.Pulsar emission is intriguing due to its prominent modulation phenomena, especially the "big three" effects of subpulse drifting, mode changing, and pulse nulling.This paper focuses on the subpulse-drifting and pulse-nulling phenomena.
Pulse nulling is a phenomenon where the pulse energy drops suddenly to zero or near zero, followed by an abrupt return to its normal state.This is also an extreme mode change in pulsar emission (Backer 1970;Basu et al. 2016).The phenomenon was initially detected in four pulsars (Backer 1970).Subsequent studies have revealed that pulse nulling is relatively common in pulsars.More than 100 pulsars have been observed to exhibit this phenomenon (Hesse & Wielebinski 1974;Ritchings 1976;Biggs 1992;Vivekanand 1995;Wang et al. 2007).In pulse-nulling studies, the nulling fraction (NF) measures the degree of nulling in a pulsar emission, representing the fraction of pulses with no detectable emission.However, the NF does not specify the durations of individual nulling states, nor does it specify how they are spaced in time.
Although some previous studies have attempted to characterize patterns in pulse nulling (Backer 1970;Ritchings 1976;Janssen & van Leeuwen 2004;Kloumann & Rankin 2010), few pulsars have been systematically studied for their pulse-nulling emission patterns, due to the need for high-sensitivity and long-term observations.
Pulse nulling is often considered a stochastic phenomenon, where pulsars randomly exhibit nulling behavior (Ritchings 1976;Biggs 1992).For instance, Redman & Rankin (2009) observed random nulling in at least four of the 18 pulsars in their sample.Recent studies suggest that some classical nulling pulsars exhibit periodic nulling behavior, as opposed to nonrandom nulling behavior (Kloumann & Rankin 2010).In the radiation of some pulsars, the switch between the null and burst states is periodic (Herfindal & Rankin 2007;Rankin & Wright 2007;Basu & Mitra 2018;Basu et al. 2019Basu et al. , 2020a)).Despite the periodic nulling phenomena of some pulsars that have been observed and studied, the mechanism behind the periodic pulse-nulling behavior in the magnetosphere of pulsars remains unclear.Further observational and theoretical studies with high-sensitivity telescopes are necessary.
The phenomenon of subpulse drifting in pulsars was discovered by Drake & Craft (1968).Traditionally, this phenomenon can be explained by the carrousel model, which suggests that the rotating carrousel of sub-beams is due to the B × E drift in the inner acceleration region.In general, subpulse drifting can be represented by three drift parameters: the vertical band spacing at the phase of the same pulse (P 3 ), the horizontal time interval between successive drift bands (P 2 ), and the drift rate (Δf = P 2 /P 3 ).Since its discovery, various unusual behaviors of subpulse drifting have been reported, which have challenged the traditional carrousel circulation models (e.g., Wen et al. 2016;Dang et al. 2022;Xu et al. 2024;Zhi et al. 2023).Several studies have found some evidence of a link between nulling and subpulse drifting-for example, the drift rate changes after the null state (van Leeuwen et al. 2003;Janssen & van Leeuwen 2004).These require deep research with high-sensitivity observations.PSR J1649+2533 was discovered in a high-galactic-latitude pulsar survey of the arecibo sky by Foster et al. (1995).It is a pulsar that exhibits the behaviors of subpulse drifting and pulse nulling.Its spin period is 1.015 s and its period derivative is 5.594 × 10 −15 s s −1 .The characteristic surface magnetic field is ∼7.63 × 10 11 G.The derived distance from dispersion measures with the electron density YMW16 mode (Yao et al. 2017) is 25.0 kpc.The periodic pulse-nulling behavior of PSR J1649 +2533 was first reported in Herfindal & Rankin (2009).On 2019 June 29, we carried out a high-sensitivity observation of this pulsar with the 19-beam receiver of the Five-hundredmeter Aperture Spherical Radio Telescope (FAST) and obtained many data sets at the center frequency 1250 MHz with a 400 MHz bandwidth, bringing opportunities to study the single pulses as well as the radiation physics of this pulsar.In this paper, we aim to investigate the pulse-nulling phenomenon of PSR J1649+253, utilizing the FAST observations.The paper will be structured as follows.In Section 2, we will present the observations and data processing techniques.In Section 3, we will provide an analysis and the results for single pulses.Finally, in Section 4, we will conclude with relevant discussions and conclusions.

Observations
FAST is a Chinese megascience project located in the karst depression in Guizhou, China.It is the world's largest singledish radio telescope, with a total aperture of 500 m and a 300 m effective aperture.5Some of the scientific goals of the telescope are to discover and observe pulsars, establish a pulsar timing array, and participate in pulsar navigation and gravitationalwave detection in the future.Its longitude and latitude are 106.9°Eand 25.7°N, respectively.The main structure of FAST was completed on 2016 September 25, before it entered the commissioning phase (Jiang et al. 2019).During the initial commissioning phase, from 2016 September to 2018 May, FAST used an ultrawideband receiver with a bandpass of 270-1620 MHz (Lu et al. 2019).However, after 2018 May, FAST switched to a 19-beam receiver with a bandpass of 1.0-1.5 GHz for its observations (Jiang et al. 2020).
In this paper, the archived observational data of pulsars from FAST will be used for the investigation of the nulling and subpulse-drifting properties of PSR J1649+2533.The observation of this pulsar was carried out with the central beam of the FAST 19-beam receiver.The duration of the 19-beam observation is 3600 s, including 10 minutes of extra slew time of the telescope, with 3000 single pulses being recorded.The observing bandpass is from 1000 to 1500 MHz, with 4096 channels, and the time resolution is 49.152 μs.In the actual data analysis, frequencies ranging from 1050 to 1450 MHz are used.All observational data are recorded in the search mode and in PSRFITS format (Hotan et al. 2004).The observation information for the pulsar is shown in Table 1.With the timing ephemeris provided by the ANTF pulsar catalog V1.706 (Manchester et al. 2005; PSRCAT), we used the DSPSR package (Hotan et al. 2004;van Straten & Bailes 2011) to fold the search-mode data and obtain the single-pulse sequence of PSR J1649+2533.In observations, radio signals from pulsars are usually more or less contaminated by narrowband nonpulsar radio radiation at a certain frequency.Such instances of narrowband nonpulsar radio frequency interference were automatically and manually flagged and removed by using the PAZI and PAZ commands provided by the software package PSRCHIVE (Hotan et al. 2004;van Straten et al. 2012).Finally, the single pulses were analyzed by using the PSRSALSA package7 (Weltevrede 2016).

Null and Burst States Extraction
PSR J1649+2533 has been observed for the existence of "null" pulse sequences, and it spent approximately 30% of its time in a null state during observations with the Arecibo telescope at 430 MHz (Lewandowski et al. 2004).This paper aims to investigate the null phenomenon of the radio pulse emission from PSR J1649+2533.To achieve this, we make use of FAST observations conducted at a central frequency of 1250 MHz. Figure 1 displays a partially intercepted singlepulse stack of PSR J1649+2533.The main panel of Figure 1 clearly illustrates that there are several null states interspersed within the intercepted single-pulse stack.For the single-pulse stack, we calculate the energy of each single pulse, as shown in the right panel of Figure 1.It is plainly seen in the right panel of Figure 1 that the total energy of each single pulse is variable.To effectively separate the single pulses of the burst and null states, we calculated the pulse energy of the on-pulse and offpulse windows, respectively, and the calculated pulse energy in the on-pulse and off-pulse regions is studied statistically.In Figure 2, we present the distributions of pulse energy for the on-pulse and off-pulse windows.The pulse energy for the onpulse and off-pulse windows is defined as the cumulative intensity of all phase bins for each window.The on-pulse and off-pulse windows are defined as the longitude ranges 308°-330°and 128°-150°, respectively.It is important to note that the number of phase bins used to calculate the pulse energy is the same for both the off-pulse and on-pulse windows.As shown in Figure 2, the energy distribution of the on-pulse window exhibits a bimodal distribution that peaks at zero energy and approximately 1.2 times the mean pulse energy〈E〉.Based on the bimodal distribution, the first peak at zero times the mean pulse energy suggests that the single-pulse stack in the on-pulse window may contain several nulling pulses, making the energy distribution of the first peak similar to that of the off-pulse window, indicating the existence of the nulling phenomenon in PSR J1649+2533.
Previous studies have shown that for a pulsar with the nulling phenomenon, the energy distribution of the on-pulse window follows a bimodal distribution and can be fitted with a Gaussian function plus a lognormal function.Among them, the energy distribution of single pulses in the null states follows a Gaussian distribution, with the mean value at zero, while the energy distribution of single pulses in the burst states follows a lognormal distribution, with a mean value greater than zero (Rejep et al. 2022).In Figure 2, we fit the pulse energy distribution of the on-pulse window with a Gaussian function plus a lognormal function.The fitting curve is represented as a black line.It is clear that the pulse energy distribution of the null states can be well fitted by a Gaussian function centering around zero, while the pulse energy distribution of the burst states can be well fitted with a lognormal function peaking around 1.2E/〈E〉.To distinguish the null pulses and the burst pulse, we calculated the uncertainty of the energy of each single pulse of the on-pulse window with the method proposed by Bhattacharyya et al. (2010).The uncertainty is defined as , where N on is the number of phase bins of the on-pulse window and σ off is the rms of the pulse energy in the off-pulse region.Following the method of Bhattacharyya et al. (2010), the single pulses with energies lower than a threshold of 5σ ep (the threshold is plotted as a green line in the right panel of Figure 2) are classified as the nulling pulses, and the others are classified as the burst pulses.In order to verify whether the null state that has been extracted contains any burst pulses, as shown in Figure 3, we plotted the integrated pulse profiles of the extracted burst and null states, respectively.It can be seen from the upper panel of Figure 3 that an integrated pulse profile with a high signal-to-noise ratio was obtained by integrating all the single pulses in the extracted burst state, while the integrated pulse profile shows obvious white noise for the extracted null state, which means that the extracted burst states do not contain any identifiable emission.

The NF and Length
PSR J1649+2533 was found to have an NF of 30% by Lewandowski et al. (2004), 25% by Herfindal & Rankin (2009), and 20% by Wright et al. (2012).Here, we will estimate the NF of this pulsar with the FAST observation.We first distinguish the burst and null states by fitting the energy distributions of single pulses of the on-pulse window with a Gaussian function plus a lognormal function, then create a pulse intensity (or energy) sequence plot, as shown in the right panel of Figure 2. The pulses with intensity below the threshold value σ ep in the pulse intensity sequence are designated as nulls.To determine the NF and length, we set the intensities of the pulses in the burst and null states to 1 and zero, respectively.This is shown in the upper panel of Figure 4. Here, we detected a total of 628 null pulses.Then we count the number of pulses with intensity equal to zero and divide this number by the total number of pulses to obtain the NF, i.e., the NF is simply given by n p /N, where n p is the number of null pulses and N is the total number of null pulses plus the burst pulses (Wang et al. 2007).Finally, based on the current FAST data, the estimated NF is 20.9% ± 0.8% at frequency 1.25 GHz, where the uncertainty of the NF is simply given by n N p .Our estimated overall average NF is basically the same as the 20% given by Wright et al. (2012) and is slightly lower than the NFs given by Lewandowski et al. (2004) and Herfindal & Rankin (2009).
The power spectra analysis by Herfindal & Rankin (2009) has shown that the pulse nulling of PSR J1649+2533 exhibits quasiperiodic features.In order to investigate whether the switch between null and burst states is periodic, we use the method given by Basu et al. (2017) to carry out a onedimensional discrete Fourier transform (DFT) on the timeseries data.Similarly, we used 256 consecutive points for carrying out the DFT.If the peak frequency was too close to the edge, the number of points used for the DFT was increased accordingly, to resolve the periodicity.The starting position was shifted by 10 pulses and the process was repeated until the end.Finally, all the individual DFTs were averaged.The results are shown in Figure 4, where the bottom panel is the averaged DFTs, where less than 0.02c/P are set to zero.We estimate the null periodicity by identifying the frequency of maximum amplitude (as the red point shown in the bottom panel of Figure 4) in the power spectrum, giving a null period of approximately 63.8 P, where P is the spin period of PSR J1649+2533.
The time-varying Fourier transform for this pulsar is shown in Figure 4, which reveals that the null phenomenon of PSR J1649+2533 is complex.A possible factor is that the pulse emission may contain various lengths of nulls.In Figure 5, the lengths of each null (top panel) and burst (bottom panel) state are statistically studied.The length distribution of this pulsar reveals that the burst states can persist for extended periods of time, exceeding a hundred pulse periods in some cases.On the other hand, the shortest burst states are quite brief, lasting only for a only a few pulse periods.In comparison, the null states have a much shorter duration, with the maximum duration being limited to just 30 pulse periods.

The Pulse Profile
Several studies have noticed a significant difference in the mean pulse distribution between the first active pulse (FAP)  and the last active pulse (LAP; Vivekanand 1995; Wen et al. 2016), where the is immediately after a null and the LAP is just before a null.Wen et al. (2016) studied the shape of the integrated pulse profiles of the FAPs and LAPs of PSR J1727-2739.They found that the integrated pulse profile of the LAPs has a stronger trailing component, while the the integrated pulse profile of the FAPs has the opposite.The observed differences in the LAPs and FAPs were attributed to different emission conditions at the beginnings and ends of bursts.We conducted a comparative study between the integrated pulse profiles of FAPs and LAPs extracted from the pulse sequence of PSR J1649+2533.The pulse profiles of FAPs, LAPs, and the averaged profile for the entire pulse sequence are plotted in Figure 6, represented by the red, blue, and black lines, respectively.To intuitively show the difference between the integrated pulse profiles of the FAPs, LAPs, and the total integrated pulse profile, we show the integrated pulse profile normalized by the maximum value (left panel) and the integrated pulse profile without normalization (right panel) separately, in Figure 6, where the pulse number of the integrated pulse profiles of both the FAPs and LAPs is 80.The analysis of the integrated pulse-profile shapes of the LAPs and FAPs reveals that both pulse profiles are single components with different overall widths and show a slight kink or bend near the trailing side of the profile along the longitude range 325º.The integrated pulse profile of the FAPs has a narrower width overall and a weaker trailing side, while the pulse profile of the LAPs is wider and stronger in the same aspects, which may be due to the significant difference in energy and shapes between the FAPs and LAPs (Wen et al. 2016).Here, the leading side is roughly considered the one corresponding to the lower value of the longitude, which is between 310º and 315º in Figure 3, while the trailing side is roughly identified with the later longitude range, which is between 320º and 330º.Additionally, as shown in the left panel of Figure 6, comparing the shapes of the integrated pulse profiles of LAPs and FAPs with that of the total integrated pulse profile, the profile shapes of the integrated profile of LAPs and the total integrated profile are more similar.

Subpulse Drifting
Drifting subpulses show an organized emission behavior of a single pulse within the pulsar magnetosphere.The parameters used to describe subpulse-drifting patterns are typically the separation between subpulses (P 2 ) and the spacing between adjacent subpulse bands (P 3 ; Backer 1973).Lewandowski et al. (2004) have analyzed the pulse intensity distribution in the pulse number/pulse-phase plane of PSR J1649+2533 using two-dimensional autocorrelation functions and identified the presence of drifting subpulses with a high drift rate, which is difficult to detect in the single-pulse data.The measured values of the parameters P 2 and P 3 of PSR J1649+2533 by Lewandowski et al. (2004) are 7°.0 and 2.2P at the central observation frequency 430 MHz.Likewise, we also study the subpulse-drifting behavior of this pulsar using the FAST observation at 1.25 GHz.As shown in Figure 7, we try our best to intercept a portion of the single-pulse sequence to clearly show the subpulse-drifting behavior of this pulsar.However, the subpulse-drifting pattern of this pulsar is visually blurred, so needs to be analyzed using mathematical methods, such as the Fast Fourier Transform technique.
In this paper, we perform an analysis of the longituderesolved fluctuation spectrum (LRFS) for the single-pulse plane to measure the radiation modulation of PSR J1649+2533.As shown in Figure 8, the plot of the LFRS clearly shows the presence of amplitude fluctuation (see the red shaded area in the bottom panel of Figure 8), which reveals the presence of periodic radiation modulation.To measure the modulation period, we fit the amplitude fluctuation of the LFRS with a Gaussian function and take the frequency corresponding to the maximum peak of the Gaussian function as the modulation period P 3 .The measured P 3 of PSR J1649+2533 at 1250 MHz is 2.5 ± 0.1P, where P is the spin period of the pulsar and 0.1 is the uncertainty of one σ.It should be noted that periodic modulation is only observed in the trailing sides, while no modulation is seen in the leading sides.To determine the P 2 , we carried out two-dimensional fluctuation spectrum (2DFS; Edwards & Stappers 2002) analysis for the single-pulse sequence of J1649+2533, and the results are shown in Figure 9, where the horizontal integrated power and vertical  integrated power in the LRFS and 2DFS are in the left and bottom panels, respectively.To show more information about the subpulse behavior across the emission window, the phase variations of the subpulse drifting corresponding to the peak frequency in the LRFS are also shown in the top panels of Figures 9 and 10.From the bottom panel showing the 2DFS of Figure 9, it can be seen that the vertical integrated power has a significant peak.This means that the subpulse of PSR J1649 +2533 has a drifting phenomenon, and the drifting effect becomes less prominent after the longitude range 325º on the trailing side of the pulse profile.To obtain the P 2 value, we fitted a single Gaussian function to the interval where the vertical integrated power is higher than half the peak value, and present the period P 2 as 17°.0 ± 0°. 5.The value for P 2 in this paper is almost 2 times that of Lewandowski et al. (2004).

The Frequency Dependence of Subpulse Drifting
The analysis of the 2DFS for the 400 MHz bandwidth FAST observation shows that the modulation period P 3 and subpulsedrifting period P 2 are 2.5 ± 0.1P and 17°.0 ± 0°. 5, respectively.Both the values of P 3 and P 2 are larger than those given by Lewandowski et al. (2004).To check if the amplitude modulation and subpulse drifting are frequency-dependent, the variations of P 3 and P 2 with the observing frequency are investigated by splitting the 400 MHz bandwidth FAST observation into four subbands, with a bandwidth of 100 MHz and central frequencies at 1100, 1200, 1300, and 1400 MHz.Then we carried out an analysis of the 2DFS results for the four subband single-pulse sequences, as shown in Figure 10, and obtained the corresponding P 3 and P 2 values.The variations of P 3 and P 2 with the frequency are plotted in Figure 11.It is obvious that P 2 increases with the increase of the observing frequency.Combining our measured P 2 values with the low-frequency measurement at 430 MHz presented by Lewandowski et al. (2004), the variation of P 2 with the observing frequency is fitted using a power-law function, to obtain P 2 ∝ ν 0.8±0.1 , where ν is the observing frequency.For P 3 , the analysis of the 2DFS results at the four subbands shows that no significant changes were observed in the P 3 of PSR J1649+2533 between the frequencies 1100 MHz to 1400 MHz, as shown in the second column of Table 2.
The variation of the subpulse-drifting period P 2 with frequency may be related to the frequency dependence of the pulse profile.Based on the split four subbands of the FAST observations, we presented the pulse profiles at the four subbands.Then the frequency dependencies of the pulse profiles from frequencies of 430 to 1400 MHz are presented in Figure 12, and the values of W 10 and W 50 of PSR J1649+2533 are presented in the fourth and fifth columns of Table 2, respectively.Here, the error in pulse width is employed as proposed by Zhao et al. (2019).It is found that the pulse width seems to increase with increasing frequency.The variation of P 2 with the observing frequency seems to be related to the change of pulse width.However, we should point out that the frequency change either in W 50 or in W 10 is not significant, once the measurement errors are taken into account.Therefore, it is necessary to observe and study more pulsars to verify whether the variation of P 2 with frequency really exists and There is a distinct peak at 0.4P/P 3 , corresponding to a quasiperiodic modulation of 2.5 ± 0.1 pulse periods.The transverse high-resolution spectrum shows that the periodic modulation only occurs in the trailing side.The leading side does not show such periodic modulation.The frequency spectrum less than 0.1P/P 3 is the period switching between null and burst states.whether the variation of P 2 is caused by changes in pulse profiles.

and Discussions
In this paper, we have reported the pulse-nulling and subpulse-drifting properties of PSR J1649+2533 with FAST observations at a central frequency of 1250 MHz.Based on the FAST observations, we give the measured average NF of this pulsar at 1250 MHz as about 20.9% ± 0.8%.We calculated statistics on the durations of the null and burst states, respectively.It is found that the duration of the burst state is longer than that of the null state.The longest duration of the burst state exceeds 100 pulsar periods.We carried out a comparative study between the integrated pulse profiles of the FAP and the LAP.It is found that the integrated pulse profile of the FAPs has a narrower width overall and weaker trailing side, while the integrated pulse profile of the LAPs is wider and stronger in the same aspects.Additionally, the shape of the integrated pulse profile of the LAPs is more similar to that of the averaged pulse profile.We also carried out a 2DFS analysis Figure 10.The four subband LRFS and 2DFS results for PSR J1649+2533.From left to right are the LRFS and 2DFS results for the subbands 1100 MHz, 1200 MHz, 1300 MHz, and 1400 MHz, respectively.P 2 as well as its error are determined by using a Gaussian function (the blue dashed lines) to fit the points around the peak of the 2DFS.For the four subbands, the positions of P 2 are represented by the red vertical lines.The results show that P 3 at the four subbands is ∼2.5 ± 0.1P and the corresponding P 2 are 16.35 ± 0.54,17.96 ± 0.92,18.11 ± 0.92,and 18.78 ± 0.81,respectively. Figure 11.The variation of P 2 with the observing frequency for PSR J1649 +2533.Here the black line is a power-law fitting with an index of 0.8.The red circles represent the P 2 values derived from the FAST observations.The blue pentagram is cited from the published result at 430 MHz provided by Lewandowski et al. (2004).The result shows that the index of P 2 is 0.8 ± 0.1. .The variations of pulse widths at 10% (W10) and 50% (W50) of the peak with the observing frequency for PSR J1649+2533.Here the black and red lines are the power-law fitting curves, while α 1 and α 2 are the corresponding power-law indexes of W 10 and W 50 , respectively.The red and blue circles represent the W 50 and W 10 of the pulse profiles derived from the FAST observations.The blue pentagram is cited from the published result at 430 MHz provided by Lewandowski et al. (2004).
on the single-pulse sequence of J1649+2533 to investigate subpulse-drifting behavior.The modulation period P 3 and subpulse-drifting period P 2 are 2.5 ± 0.1P and P 2 = 17°.0± 0°. 5, respectively, for the FAST observation of 400 MHz at full bandwidth.The value of P 3 given by this paper is larger than that given by the previous study (Lewandowski et al. 2004), while the value of P 2 is almost two times the value given by Lewandowski et al. (2004), which means that the drift rate of this pulsar is faster at 1250 MHz than at 430 MHz.The frequency dependence of the period P 2 is investigated by splitting the FAST observation of 400 MHz bandwidth into four subbands.Combining the measured P 2 at the four subbands with the P 2 measured at 430 MHz, we found that P 2 increases with the increase of the observing frequency, obtaining P 2 ∝ ν 0.81±0.08 .The investigation of the frequency dependence of the pulse profile for this pulsar suggests that the variation of P 2 with frequency may be caused by the increasing width of the pulse profile with frequency.It should be noted that the determination of P 2 and its error is greatly impacted by the resolution of the 2DFS.The error value of P 2 measured by using a Gaussian function to fit the points around the peak of the 2DFS may be underestimated.Therefore, further observations and research are required to determine if there is indeed a significant frequency evolution behavior of P 2 in more pulsars.At present, numerous studies have suggested that the emission beams of a radio pulsar can be divided into two (core and inner conal) or three (plus an outer conal) emission components (e.g., Rankin 1983;Lyne & Manchester 1988;Qiao & Lin 1998).And the frequency dependence of the inner and outer cone components is opposite (Xu et al. 2021;Zhi et al. 2022).Mitra & Rankin (2002) suggest that the pulse width exhibits constant behavior, with frequency appearing to reflect the inner cone emission.This means that the spectral index of the pulse width of the pulse profile dominated by the inner cone is almost zero.Combined with the Lewandowski et al. (2004) observations at 430 MHz, it is found that the pulse width of this pulsar seems to be wider at high frequencies than at low frequencies, and both the W 10 and W 50 seem to show a flat-frequency evolution behavior, which implies that its radio emission is provided by the inner cone component.This makes it understandable why our calculated P 2 values are larger than those of Lewandowski et al. (2004).
Because the number of pulse-nulling pulsars that have been observed and studied is relatively small compared to all known pulsars, the physical origin of pulse nulling in pulsars is not yet fully understood.Several previous studies have suggested that certain factors may lead to the occurrence of pulse nulling (Jones 1981;Filippenko & Radhakrishnan 1982;Zhang et al. 1997;Geppert et al. 2003Geppert et al. , 2021;;Timokhin 2010;Jones 2020), including the failure of particle production in the polar cap region, the loss of coherence for relativistic particles, the instability of magnetic fields, and changes in the magnetosphere configurations and radiation mechanisms of the pulsar.It should be pointed out that the periodic pulse nulling may be a common occurrence in pulse-nulling pulsars.The pulse-nulling behavior of many known pulse-nulling pulsars may also follow a periodic or quasiperiodic pattern.However, in the past, due to the telescopes used for observations not having enough sensitivity to detect the single pulse necessary to find more periodic pulse-nulling pulsars, there have been relatively few pulsars with periodic pulse nulling observed to date.Presently, the largest sample of periodic pulse-nulling pulsars has been given by Basu et al. (2020b).However, the sample includes only 29 cases.The number of pulsars in this sample is far less than 1% of all known pulsars (to date, more than 4000 pulsars have been discovered).Such a small number of periodicnulling pulsars makes it difficult to conduct a statistical study of periodic pulse nulling to limit radiation models of pulsars.The construction of highly sensitive radio telescopes, such as FAST, is expected to open up new opportunities for observing and studying the single pulses of pulsars.By observing the single pulses of pulsars with FAST, the number of periodicnulling pulsars that can be detected will significantly increase.This will enable further statistical studies of pulse nulling as well as the testing and constraining of radiation theoretical models of pulsars.

Figure 1 .
Figure1.A single-pulse stack of PSR J1649+2533, which contains a singlepulse sequence of 500 periods, shows an example of the obvious nulling phenomenon.The bottom panel shows the average profile over the entire observation, with the intensity normalized using the peak's value.The total energy of each single pulse is plotted in the right-hand panel, where the energy has been normalized by the average energy.The constant switching between the obvious null and burst states can be seen.

Figure 2 .
Figure 2. Pulse energy histograms for the on-pulse (red) and off-pulse (blue) windows.The energy is normalized by the mean on-pulse energy.The black curve represents the result of fitting the on-pulse energy distribution, based on a combination of a Gaussian component and a lognormal component.The vertical green line is the threshold of 5σ ep .The different components are indicated by the black dashed lines.The clearly double-peaked structure is due to the null and burst states.

Figure 3 .
Figure 3.The upper panel and bottom panel show the integrated pulse profiles of the burst and nulling states, respectively, for PSR J1649+2533.

Figure 4 .
Figure 4.The time-varying Fourier transform of the null/burst (0/1) timeseries data for PSR J1649+2533.The left panel shows the identified emission states corresponding to each period of this observation.The main panel shows the time evolution of the DFT corresponding to the null/burst time series.The bottom panel shows the average DFT over the entire sequence.The red dots indicate the most prominent modulation frequencies.Those points with spectrum amplitude less than 0.02 c/P are set to zero.

Figure 5 .
Figure 5.The distribution of the durations of the null and burst states for PSR J1649+2533.

Figure 6 .
Figure 6.Integrated pulse profiles of the FAP (red) and LAP (blue) of the burst state.The black solid line represents the total integrated pulse profile of all bursts.The left panel shows all integrated pulse profiles normalized by their respective peak intensities, while the right panel shows the integrated pulse profiles that were not normalized.

Figure 7 .
Figure 7. Example of a single-pulse stack with 35 consecutive pulses.Here, the x-axis and the y-axis show the pulse phase and number, respectively.Obvious subpulse-drifting behavior with very high drift rates can be seen.

Figure 8 .
Figure 8. Frequency spectrum of PSR J1649+2533 at 1250 MHz.The lighter colors indicate stronger modulation.There is a distinct peak at 0.4P/P 3 , corresponding to a quasiperiodic modulation of 2.5 ± 0.1 pulse periods.The transverse high-resolution spectrum shows that the periodic modulation only occurs in the trailing side.The leading side does not show such periodic modulation.The frequency spectrum less than 0.1P/P 3 is the period switching between null and burst states.

Figure 9 .
Figure 9.The LRFS and 2DFS of PSR J1649+2533.The two panels show the LRFS and 2DFS.The side panels show the horizontal (left) and vertical (top or bottom) integrated power.P 2 as well as its error are determined by using a Gaussian function (the blue dashed line) to fit the points around the peak of the 2DFS.The position of P 2 is represented by the red vertical line.The results show that P 3 ∼ 2.5 ± 0.1P and P 2 ∼ 17°.0 ± 0°. 5.

Table 2
The Derived Parameters of the Pulse Profiles and Subpulse Drifting of PSR J1649+2533