Investigation of Periodic Modulation Behaviors from Pulsar J2022+5154

We have carried out a detailed study of individual pulse emission from the pulsar J2022+5154 (B2021+51), observed at 2250 MHz using the Jiamusi 66 m radio telescope. We have investigated the modulations in single-pulse behavior using fluctuation spectral analysis, which shows the presence of two prominent periodicities, around 5 and 40 rotation periods, respectively. The shorter periodicity is associated with the phenomenon of subpulse drifting. In the absence of aliasing, the emission pattern is demonstrated to consist of eight subbeams, which rotate around the magnetic axis in about 45 periods. In addition to subpulse drifting, the pulsar also shows the presence of periodic amplitude modulation with a longer periodicity in the single-pulse sequence. The pulsar joins a select group that shows the presence of periodic phase-modulated drifting as well as amplitude-modulated drifting. This provides further evidence for the two phenomena being distinct from each other with different physical origins.


Introduction
Radio emission from pulsars with spin periods on the order of a second presents periodic modulations in their single pulses over timescales ranging from a few seconds to several minutes.Such periodic behaviors have usually been associated with the phenomena of subpulse drifting, periodic nulling, and periodic amplitude modulation.Drifting subpulses is a common behavior in pulsars, which is a progressive shift of a subpulse phase in successive pulses.At least one-third of pulsars have been reported to possess drifting subpulses (Weltevrede et al. 2006) since the discovery in 1968 (Drake & Craft 1968).The subpulse drifting is characterized by two periodicities: P 2 (the longitudinal separation between two adjacent drift bands) and P 3 (the interval over which the subpulses repeat at a specific location within the pulse window).
The subpulse structures provide valuable insights into the emission regions in the pulsar magnetosphere, which are associated with columns of localized bundles of relativistic charges in the magnetosphere.The mechanism of subpulse drifting seems to be associated with the inclination angle α between the magnetic axis and the rotation axis, and the impact angle β between the line of sight and the magnetic axis (Weltevrede et al. 2008).More specific physical characteristics related to subpulse drifting have been revealed in recent studies (Basu et al. 2016).Subpulse drifting is dependent on profile type and spindown luminosity.Highly organized subpulse modulation is exclusively observed in conal profiles, with no evidence of drifting detected in the central core component (Rankin 1986;Basu et al. 2020).Tu et al. (2022) employed a model that described the magnetospheres of obliquely rotating pulsars with multiple emission states and accounted for the observed diversity in subpulse drifting behavior.A clear association was indicated between the subpulse drifting behavior observed in different pulse profile components and the specific emission states.Furthermore, the physics of the emission mechanism between periodic amplitude modulation and phase drifting is distinct (Basu et al. 2020).
Pulsar J2022+5154 (B2021+51) is an isolated normal pulsar discovered with a single-pulse search technique using the Mark I radio telescope at Jodrell Bank and a center frequency of 408 MHz (Davies & Large 1970).The spin properties of the pulsar are not in any way extraordinary, as it has a rotational period of P 1 = 0.53 s and a first-period derivative of  P 3.06 10 1 15

=
´s s −1 , giving the characteristic age of τ c = 2.74 Myr and the surface magnetic field of B s = 1.29 × 10 12 G.Table 1 presents the basic parameters and derived quantities of the pulsar.PSR J2022+5154 has dominant emission states and nulls generally last for short durations.Ritchings (1976) reported that the nulls lasted for one or two periods with a nulling fraction below 5% at 408 MHz.At 610 MHz, the nulling fraction was estimated to be 1.4% (Gajjar et al. 2012).The extremely low nulling fraction was estimated to be 0.12% at 2250 MHz (Wang et al. 2020).The quasiperiodic nulling was detected to modulate the emission across the entire profile.The pulses during the transitions were stronger than the average pulse profile.Additionally, the first glitch of PSR J2022+5154 was detected with the Shanghai Tianma 65 m radio telescope at the S/X bands simultaneously around MJD 58289.1 (Liu et al. 2021).The pulse profile variations were reported to be associated with the glitch, which suggested that the movements of flux tubes in the emission zone might be caused by the glitch activity.
In this paper, we present the results of the periodic modulation properties of PSR J2022+5154 at 2250 MHz observed with the Jiamusi 66 m radio telescope.The details of Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
the observations and data processing are presented in Section 2. The results and discussion are presented in Sections 3 and 4, respectively.The summary and conclusions of the study are presented in Section 5.

Observations
The observations were conducted using the Jiamusi 66 m radio telescope at the Jiamusi Deep Space Station, China Xi'an Satellite Control Center.A full description of the observational setup was presented by Han et al. (2016).The cryogenic receiver system consisted of twin-channel preamplifiers, which were used to receive the left-and right-hand circular polarizations at a center frequency of 2250 MHz with a bandwidth of 148.48 MHz.The digital backend subdivided the total bandwidth into 256 spectral channels with a frequency resolution of 0.58 MHz and a time resolution of 0.2 ms.PSR J2022+5154 was observed in three sessions in the searching mode.The detailed information of the radio observations is listed in Table 2.And a detailed explanation of the process of recording and forming the single-pulse data can be found in Wang et al. (2020).Each single-pulse integration was folded into 2048 longitude bins and recorded in the PSRFITS (Hotan et al. 2004) format with 8 bit quantization for the following analysis.

Single-pulse Sequence
Figure 1 shows a section of the color-coded single-pulse sequence from PSR J2022+5154 in the main panel for visual examination of the drifting behavior.The pulse-to-pulse variations in both intensity and longitude are significant.It is apparent that the emission exhibits a regular modulation pattern in both amplitude and longitude between pulse numbers 60 and 100, which visually resembles a set of discrete diagonally oriented burst regions called drift bands.There is no conventional nulling detected in the single-pulse sequences.The average pulse profile integrated across all of the observations is illustrated in the lower panel, where the abscissa is the pulse longitude in degrees and the ordinate is the pulse intensity normalized to the peak intensity.PSR J2022 +5154 shows two prominent peaks in the average pulse profile.The pulse widths at 10% and 50% intensity of the observed pulse profile are measured to be 17°.11 and 9°.32, respectively.Liu et al. (2021) decomposed the mean pulse profile into three Gaussian components at the S, C, and X bands.The third component resembles the core emission, which has a steeper spectrum than the conal components (Rankin 1983).

Pulse Energy Distribution
The investigation of the pulse energy distribution provides insights into the radio pulsar emission mechanism and the physical states of pulsar magnetospheres.Burke-Spolaor et al. (2012) reported on the pulse-to-pulse energy distributions for 315 pulsars in the southern High Time Resolution Universe intermediate-latitude survey.They showed that the energy distribution for the majority of the pulsars could be well described by a lognormal distribution.In addition, the study of the energy distribution from the mode-changing and nulling pulsars revealed that they exhibit multiple energy states (Wen et al. 2016a(Wen et al. , 2020)).The energy distributions of giant pulses, which follow a power-law distribution, suggest that the giant pulse phenomenon is distinct from the regular pulsar radio emission (Chen et al. 2020;Wen et al. 2021).
Figure 2 presents the energy distribution for the on-pulse window, which is calculated using the standard method described by Wen et al. (2016b).The on-pulse window, ranging from 158°to 202°, is determined by an inspection of the integrated pulse profile.The off-pulse window is

Notes.
a The numbers in the parentheses indicate the uncertainty in the last decimal of the measured value.b These quantities are derived from the basic observables of P 1 and  P 1 by assuming that the pulsar spindown is dominated by dipole braking (Lorimer & Kramer 2012).c The pulsar distance is derived using the NE2001 model of the Galactic free electron density of Cordes & Lazio (2002).determined in the off-pulse region with the size equal to the onpulse window.The normalized on-pulse and off-pulse energies are then calculated for each pulsar rotation by integrating the energy in the on-pulse and off-pulse windows, respectively, and then dividing by the mean on-pulse energy obtained from all integrations.The off-pulse energy can be well modeled with a Gaussian distribution centered around zero, representing the Gaussian random noise contributed by the system temperature.It appears that the observed on-pulse energy distribution follows a lognormal distribution with a probability density function defined as 8 where σ is the shape parameter (and represents the standard deviation of the logarithm of the distribution), μ is the location parameter, and m is the scale parameter (which also serves as the median of the distribution; Sangal & Biswas 1970).It is noted that this equation deviates slightly from the standard lognormal distribution, which is the case where μ = 0 and m = 1.A Kolmogorov-Smirnov hypothesis test is used to check whether or not the observed distribution can be described as lognormal.The p-value of the lognormality hypothesis test is about 0.96, which is much greater than the threshold value 0.05, indicating that the observed distribution is consistent with a lognormal distribution.The observation data are fitted to the lognormal and normal models using a least-squares fitting method.Among these two functional forms, the lognormal distribution yields the optimal performance.The red dashed curve in Figure 2 is the best-fitting lognormal distribution, with the best-fitting parameters shown in the legend.

Fluctuation Spectra
In order to characterize whether the subpulse modulation occurs in a systematic or a disordered fashion, the most widely used technique is fluctuation spectral analysis using the Fourier transform method, where the frequencies of drifting are characterized by the peaks in the spectrum.There are three principal analysis schemes: the longitude-resolved fluctuation spectra (LRFS; Backer 1970), the harmonic-resolved fluctuation spectra (HRFS; Deshpande & Rankin 2001), and the two-dimensional fluctuation spectra (2DFS; Edwards & Stappers 2002).

LRFS
Examining the single pulses and their longitude-resolved modulation can provide insights into the state of the plasma emission.The longitude-resolved modulation index is calculated as m I , 2 where σ I (f) and I(f) are the rms and the mean intensity computed at longitude f and σ off is the rms intensity computed from an offpulse window.The modulation index measures the degree to which the intensity varies from pulse to pulse and can be an indicator of the presence of subpulses (Weltevrede et al. 2006).The energies are normalized by the mean on-pulse energy.The red dashed line indicates the lognormal energy model fit for the on-pulse window with the bestfitting parameters shown in the legend.The best-fit normal distribution is presented with the green dashed line as well.Among these two functional forms, the lognormal distribution yields the optimal performance. 8https://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm In Figure 3, the black dotted curve in the left panel is the longitude-resolved standard deviation σ I (f) and the red dashed line corresponds to the longitude-resolved modulation index m I (f) for all observations combined.It is noted that the modulation index flares up at the trailing edge of the profile, implying that the most variable pulse intensities are in the corresponding emission region.
The LRFS technique involves selecting an appropriate number of consecutive single pulses and performing a Discrete Fourier Transform (DFT) across each longitude bin in the pulse stack.It allows one to explore how the drifting feature varies across the pulse window.The pulse sequence is separated into adjacent blocks with a fixed length of 256 successive pulses, which is high enough to obtain sufficient resolution in the frequency domain.The starting point is then shifted by 10 periods and the process is repeated until the end of the observations (Basu & Mitra 2018).To detect and analyze the subpulse modulation, integrating multiple fluctuation spectra obtained from consecutive blocks of pulses increases the signal-to-noise ratio of the resulting spectrum.The timeaveraged LRFS is produced to determine the frequency peaks of the fluctuations, which are presented in Figure 3. Pulsars exhibiting periodically modulated subpulses have a region, the so-called feature, of enhanced spectral power that is visible as a bright region in the LRFS.The fluctuation frequency is given in cycles per period (cpp) and its inverse corresponds to the pattern periodicity, P 3 , expressed in pulsar periods P 1 .The position of the feature along the ordinate denotes the pulse longitude at which the modulation occurs.The left panel shows the integrated pulse profile of PSR J2022+5154, normalized to the peak intensity.The lower panels show the averaged power spectra over the on-pulse window for three observations.In the entire data set, the LRFS is dominated by two frequency peaks of fluctuation, which most fully modulate the power across the whole profile.Following the method described by Basu et al. (2016), the peak frequency ( f p ), the strength of the drifting feature (S) and the corresponding drift periodicity (P 3 = 1/f p ) are determined and listed in Table 3.
The peaks in the LRFS caused by the two modulation features are broad.Hence, the periodicities do not appear to be particularly stable.To explore the stability and temporal evolution of the modulation features on shorter timescales, the sliding fluctuation spectrum (SFS) is used (Maan & Deshpande 2014).The SFS is a stack of fluctuation spectra, each of which is collapsed over all the longitudes.Figure 4 shows the temporal changes in drifting, where one can easily see the drift tracks (regions of enhanced power visible as a bright region).The fluctuation spectra are dominated by two features.The left panel shows the time variation of the modulation power by integrating the two features.The SFS shows the constantly changing frequency of the modulation pattern and no fixed periodicity is present.And the time variability of the modulation pattern is short, with a timescale of a few hundred pulses or less.
In order to investigate the potential impact of different Fourier transform sizes on the measurements of the modulation frequency, we conducted the LRFS analysis by using 512, 1024, and 2048 points as the size of the Fourier transform, as shown in Figure 5.The fluctuation power spectra integrated from the whole emission window at three epochs appear very similar, which implies that the modulation features cannot be resolved with longer fast Fourier transforms.This further confirms the measured fluctuation periodicities.

HRFS
Although the LRFS yields a wealth of information regarding the fluctuations present in the pulse sequences, it suffers from a serious aliasing drawback, due to poor sampling of the fluctuations at a given longitude only once every pulse period.In order to reveal whether the observed fluctuation feature is caused by an alias of fluctuations at frequencies larger than 0.5 cycles per P 1 , the fluctuations are required to be sampled faster than once a period.This can be achieved by exploiting the fact that the fluctuations are sampled only within the bounds of the pulse's finite width.For this purpose, the time sequence is reconstructed from the pulse stack by retaining the values for the on-pulse samples while filling the rest of the longitude regions with zeros, before unfolding the single-pulse stack.By unfolding, a continuous time sequence is obtained by just successively placing periods one after another.An unfolded fluctuation spectrum is obtained by performing a DFT on the the entire continuous sequence.The HRFS is achieved by simply stacking successive slices of 1 cycle per P 1 from the unfolded fluctuation spectrum, forming a series of overlapping bands.Therefore, the HRFS represents the fluctuation spectra integrated over different longitude bins with appropriate phases corresponding to the longitude separation.The fluctuations are sampled more than once in the finite duration of the pulse, which overcomes the basic limitation of sampling and provides an alternative way of The abscissa is the frequency of the drifting periodicity and the ordinate is the pulse longitude in degrees.A 256-point Fourier transform is used and averaged over the blocks of the whole pulse sequence.The power of the drifting periodicities as a function of pulse longitude is plotted in the color contours.The left panel shows the integrated pulse profile (blue solid line), the longituderesolved modulation index (red dashed line), and the longitude-resolved standard deviation (black dotted line) for all observations combined.The lower panels display the overall fluctuation spectra from the whole emission window at three epochs.
determining the phase behavior of the fluctuation feature.The frequency range of f p mapped in the LRFS is within 0−0.5 cycles per P 1 , while the modulation peaks in the HRFS are mapped between 0 and 1.0 cycles per P 1 .The HRFS's extra frequency space (0.5−1.0 cycles per P 1 ) enables direct realization of the phase behavior seen in the LRFS.Therefore, the HRFS can distinguish between amplitude and phase modulations.Amplitude modulations appear as symmetric feature pairs in the integral spectrum, whereas phase modulations are asymmetric.The feature in the HRFS lies symmetrically about the center point of 0.5 cycles per P 1 , which indicates that the responses largely represent amplitude modulation, as expected.
Figure 6 presents the HRFS for PSR J2022+5154 computed for the 2250 MHz observations to decipher the phase nature of the subpulse drifting.The left panel shows the spectral power at a frequency of 1 cycle per P 1 and its harmonics.The harmonic number represents the successive sections of 1 cycle per P 1 .The color scale plot in the central panel shows the full fluctuation power spectrum, and the integrated power spectrum is shown in the bottom panel.The low-frequency fluctuation is predominantly amplitude-modulated drifting, as shown by its symmetrical HRFS, indicating that the subpulses do not move across the pulse window, but exhibit periodic changes in intensity.The high-frequency fluctuation in the LRFS appears in the HRFS with the corresponding peak shifted to f 0.82 p ¯= cpp, (1 − f p ), which indicates that the subpulses show a gradual shift toward the trailing edge.And the primary feature from the LRFS analysis does seem to disappear entirely in the HRFS analysis.The asymmetry around 0.5 cpp is so pronounced that the primary feature can be identified as the hallmark of pure phase modulation.Additionally, it is noted that the amplitude of the high-frequency feature reaches a maximum for about a harmonic number of 45, which means the P 2 is derived as 8°a pproximately.According to the phase behavior, the highfrequency modulation in PSR J2022+5154 is classified as a   positive subpulse drifting population (Basu et al. 2016).The values of peak frequency, the strength of the drifting feature, and the corresponding drift periodicity are listed in Table 3.

2DFS
While the LRFS is a sensitive method for determining the time-averaged properties of periodic subpulse modulation, it is still uncertain whether the above-determined modulation originated from amplitude or longitude variation.In order to turn uncertainty into reality, the 2DFS is calculated to obtain P 2 , P 3 and hence the drift rate.This technique is similar to the calculation of the LRFS, but the DFT is applied twice.First, the DFT of the pulse sequence along the constant pulse longitude is recorded.Then the DFT across each row of the complex LRFS is obtained.Figure 7 shows the 2DFS for observations of PSR J2022+5154 at 2250 MHz.The vertical axis of the resulting spectra is the same as the horizontal axis in the LRFS.The pattern repetition frequency along the pulse longitude is denoted in the horizontal axis of the 2DFS, which is expressed as P 1 /P 2 .If the drifting subpulses have a preferred drift direction, then a feature is seen offset from the vertical axis.The 2DFS is vertically (between dashed lines) and horizontally integrated, resulting in the side and bottom panels.Estimates of P 2 , P 3 , and the drift rate (Δf = P 2 /P 3 ), quoted in Table 4, are calculated using the centroid of a rectangular region in the 2DFS containing the feature.
In order to rule out the possibility that occasional occurrences of strong subpulses are dominating the spectra and therefore leading to misleading conclusions, a comparable technique is employed to further demonstrate the authenticity of the periodic fluctuations (Weltevrede et al. 2006).The order of the pulse sequence is randomized and then the LRFS, HRFS, and 2DFS are calculated from the newly formed pulse stack.No well-defined P 3 in this process is detected in the emission window and noise window, which proves the significance of the drift features.Furthermore, it should be noted that the baselines of the fluctuation power spectra are not flat, which could be attributed to low-lying radio frequency interference, lacking flux calibration and insufficient bandpass calibration.

Discussion
The phenomenon of subpulse drifting can be explained in the seminal polar gap theory originally proposed by Ruderman & Sutherland (1975).The pattern of radio waves is characterized by a set of discrete pockets of quasi-stable electrical activity Figure 6.The HRFS for PSR J2022+5154 computed for the 2250 MHz observations.The color-coded plot in the main panel shows the full spectrum, and the lower panels show the harmonic integrated spectra at three epochs.Three modulation features clearly stand out.The low-frequency feature is close to 0.03 cpp and its symmetric counterpart is close to 0.97 cpp.The highfrequency feature appears at 0.82 cpp.The left panel shows the amplitude of the harmonics at the integral multiple of the pulsar rotation frequency.
Figure 7.The 2DFS for the pulsar J2022+5154 computed for the 2250 MHz observations.The vertical axis has the same units as the LRFS, but now the units of the horizontal axis are cpp, which corresponds to P 1 /P 2 (where P 2 is the horizontal drift band separation in time units).The fluctuation power in the 2DFS is horizontally integrated, producing the left power spectrum.The power in the 2DFS is vertically integrated between the red and blue dashed lines, producing the bottom panel of the spectra with the corresponding colors.(sparks) localized very near the pulsar surface.The particles are accelerated through the polar gap region at the sites of sparks, generating the secondary pair plasma.The curvature radiation is produced by an avalanche of secondary particles that stream along the magnetic field lines at relativistic speeds.The electric potential along the magnetic field at the stellar surface is partially screened by the charged particles, which leads to the relative motion of the sparks to the polar gap surface around the magnetic axis.The magnetic azimuthal arrangement of sparks on the polar gap reflects the spatial structure of the emission beam, which is comprised of discrete beamlets.With each rotation, an Earth-bound observer perceives a different intensity pattern as the line of sight cuts through a slightly rotated emission cone.This is known as the rotating carousel model of pulsar emission, and it is often used for interpreting the drifting of subpulses.Thus, the drift properties are expected to be linked directly to the physical conditions in the polar gap.
The carousel geometry can be constrained by assuming that the surface sparks are equally spaced in the magnetic azimuth.The behavior of drift bands is determined by the number of sparks (n) and the emission-pattern circulation period (P 4 ).The theoretical value of P 4 is predicted to be 48.97P 1 by using the equation where α and β are the inclination angle and impact angle, respectively (Deshpande & Rankin 2001).For PSR J2022 +5154, the magnetic azimuth angle is calculated to be η = 43°.68,with α = 23°and β = 5°.6 (quoted from Rankin 1993).In the absence of aliasing, the number of subbeams is derived to be 360°/η ∼ 8.The emission-pattern circulation period P 4 is estimated to be n × P 3 = 44.50P 1 , which corresponds closely to the theoretical value.In order to visualize the repeating pattern in the pulse stack, each of the single pulses is assigned a phase within the modulation cycle.Subsequently, pulses at the same phase are summed and averaged.Figure 8 shows the folded pulse profiles at the identified period of P 4 for observations on 2015 June 15 (MJD 57188.9), in which each subpulse is associated with a subbeam rotating around the magnetic axis.Similar to the pulse stack illustrated in Figure 1, the horizontal axis of the average drift band represents the pulse longitude, while the vertical axis displays the phase of the modulation period P 4 , expressed as a percentage of P 4 .It is noted that there are eight subpulses that correspond to eight subbeams in the magnetosphere.PSR J2022+5154 shows the presence of two prominent modulation features.The longer-modulation periodicity is demonstrated to be associated with periodic amplitude modulation, and the shorter periodicity is associated with subpulse drifting.Basu et al. (2016) found clear differences between the physical parameters of phase-modulated drifting and amplitude-modulated drifting.Subpulse drifting is profiledependent, phase modulation is only shown in the conal components and absent in the central core emission, and amplitude modulation is seen across all components.Pulsars with lower spindown luminosity (  E 2 10 32 < ´erg s −1 ) possess a higher possibility of showing phase-modulated drifting.Furthermore, the phase-modulated drifting periodicity is anticorrelated with spindown luminosity.No clear dependence between amplitude-modulated drifting and spindown luminosity was shown.The periodic modulations observed in the single-pulse sequences of the pulsar J2022+5154 provide further evidence that the periodic amplitude modulation and subpulse drifting originate from different physical mechanisms.

Conclusions
We have carried out a detailed analysis of the single-pulse emission phenomenology from pulsar J2022+5154, observed at 2250 MHz using the Jiamusi 66 m radio telescope.The single-pulse sequence shows the presence of periodic modulations at two different timescales, 5P 1 corresponding to the phenomenon of subpulse drifting and 40P 1 arising due to periodic amplitude modulation.The possible physical mechanisms are discussed in the context of the rotating carousel model.Further simultaneous observations in multiple frequency bands are required to carry out an in-depth analysis of the polarization properties, which are vital in understanding the physical emission mechanisms.Studying subpulses can be used to construct a polar emission map at the frequency of observation.Moreover, it is observed that the frequency of the radio emission varies with the height in pulsar magnetosphere.Higher frequencies are emitted closer to the surface of the neutron star.This relation is the well-accepted radius-tofrequency mapping relation and it can be used to determine the height of the emission map.The temporal evolution of these polar emission maps and correlation between them can help us get a better understanding of the changes in the magnetospheric current distribution.

Figure 1 .
Figure 1.Top panel: a section of a single-pulse sequence comprising 100 consecutive periods for PSR J2022+5154 observed using the Jiamusi 66 m radio telescope at 2250 MHz.Time runs from left to right across the pulse window (pulse longitude) and from bottom to top with pulse number.The subpulses appear earlier with decreasing pulse number and are arranged into so-called drift bands, which are significant between pulse numbers 60 and 100.Left panel: the energy sequence averaged by all the longitude bins inside the on-pulse window.Bottom panel: the integrated pulse profile from the entire observation.

Figure 2 .
Figure2.Relative pulse energy distribution for the on-pulse (blue) window.The energies are normalized by the mean on-pulse energy.The red dashed line indicates the lognormal energy model fit for the on-pulse window with the bestfitting parameters shown in the legend.The best-fit normal distribution is presented with the green dashed line as well.Among these two functional forms, the lognormal distribution yields the optimal performance.8https://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm

Figure 3 .
Figure 3.The LRFS for the pulsar J2022+5154.The longitude-resolved fluctuation power spectrum for the entire set of observations (top right panel) shows evidence for periodic modulations of PSR J2022+5154 at 2250 MHz.The abscissa is the frequency of the drifting periodicity and the ordinate is the pulse longitude in degrees.A 256-point Fourier transform is used and averaged over the blocks of the whole pulse sequence.The power of the drifting periodicities as a function of pulse longitude is plotted in the color contours.The left panel shows the integrated pulse profile (blue solid line), the longituderesolved modulation index (red dashed line), and the longitude-resolved standard deviation (black dotted line) for all observations combined.The lower panels display the overall fluctuation spectra from the whole emission window at three epochs.

Figure 4 .
Figure 4. Display showing the time evolution of the LRFS in the pulsar J2022 +5154.The main panel is a color-coded representation with the longitudeaveraged LRFS along the x-axis and increasing starting period number along the y-axis.The two modulation features are distinguished with the dashed lines.The left panel shows the time variation of the modulation power by integrating low-frequency (red) and high-frequency (blue) features.The bottom panel shows the aggregate LRFSs along the y-axis.

Figure 5 .
Figure 5.The fluctuation power spectra integrated from the whole emission window at three epochs.For clarity, the sizes of the Fourier transform are set to be 512, 1024, and 2048, respectively.
-Sutherland model.For a very simple spherical geometry, the angular separation between two adjacent subbeams along the magnetic axis (η) can be derived from

Figure 8 .
Figure8.Main panel: the averaged drift band produced by P 4 folding for the pulsar J2022+5154 at 2250 MHz.The horizontal axis is the pulse longitude, while the vertical axis shows the phase of the modulation period P 4 , expressed as a percentage of P 4 .Note that there are eight subpulses that can be mapped to eight subbeams in the pulsar magnetosphere.The average pulse profile is shown in the lower panel.

Table 3
Details of the Measurement of Drifting Features in PSR J2022+5154