# Investigation of profile shifting and subpulse movement in PSR J0344$-$0901
with FAST

H. M. Tedila  National Astronomical Observatories, Chinese Academy of
Sciences, A20 Datun Road, Chaoyang District, Beijing 100101, China  Key
Laboratory of Radio Astronomy and Technolgoy, Chinese Academy of Sciences, A20
Datun Road, Chaoyang District, Beijing, 100101, P.R. China  Xinjiang
Astronomical Observatory, Chinese Academy of Sciences, 150 Science 1-Street,
Urumqi, Xinjiang, 830011, People’s Republic of China  University of Chinese
Academy of Sciences, 19A Yuquan Road, 100049 Beijing, People’s Republic of
China  Arba Minch University, Arba Minch 21, Ethiopia  R. Yuen  Xinjiang
Astronomical Observatory, Chinese Academy of Sciences, 150 Science 1-Street,
Urumqi, Xinjiang, 830011, People’s Republic of China  Xinjiang Key Laboratory
of Radio Astrophysics, 150 Science1-Street, Urumqi, Xinjiang, 830011, People’s
Republic of China  N. Wang  Xinjiang Astronomical Observatory, Chinese Academy
of Sciences, 150 Science 1-Street, Urumqi, Xinjiang, 830011, People’s Republic
of China  Xinjiang Key Laboratory of Radio Astrophysics, 150 Science1-Street,
Urumqi, Xinjiang, 830011, People’s Republic of China  D. Li  National
Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Road,
Chaoyang District, Beijing 100101, China  NAOC-UKZN Computational Astrophysics
Centre, University of KwaZulu-Natal, Durban 4000, South Africa  CAS Key
Laboratory of FAST, NAOC, Chinese Academy of Sciences, Beijing 100101,
People’s Republic of China  Key Laboratory of Radio Astronomy and Technolgoy,
Chinese Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing,
100101, P.R. China  University of Chinese Academy of Sciences, 19A Yuquan
Road, 100049 Beijing, People’s Republic of China  Z. G. Wen  Xinjiang
Astronomical Observatory, Chinese Academy of Sciences, 150 Science 1-Street,
Urumqi, Xinjiang, 830011, People’s Republic of China  Xinjiang Key Laboratory
of Radio Astrophysics, 150 Science1-Street, Urumqi, Xinjiang, 830011, People’s
Republic of China  W. M. Yan  Xinjiang Astronomical Observatory, Chinese
Academy of Sciences, 150 Science 1-Street, Urumqi, Xinjiang, 830011, People’s
Republic of China  Xinjiang Key Laboratory of Radio Astrophysics, 150
Science1-Street, Urumqi, Xinjiang, 830011, People’s Republic of China  J. P.
Yuan  Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 150
Science 1-Street, Urumqi, Xinjiang, 830011, People’s Republic of China
Xinjiang Key Laboratory of Radio Astrophysics, 150 Science1-Street, Urumqi,
Xinjiang, 830011, People’s Republic of China  X. H. Han  Xinjiang Astronomical
Observatory, Chinese Academy of Sciences, 150 Science 1-Street, Urumqi,
Xinjiang, 830011, People’s Republic of China  P. Wang  National Astronomical
Observatories, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District,
Beijing 100101, China  Key Laboratory of Radio Astronomy and Technolgoy,
Chinese Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing,
100101, P.R. China  W. W. Zhu  National Astronomical Observatories, Chinese
Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing 100101, China
Key Laboratory of Radio Astronomy and Technolgoy, Chinese Academy of Sciences,
A20 Datun Road, Chaoyang District, Beijing, 100101, P.R. China  S. J. Dang
School of Physics and Electronic Science, Guizhou Normal University, Guiyang,
550001, People’s Republic of China  S. Q. Wang  Xinjiang Astronomical
Observatory, Chinese Academy of Sciences, 150 Science 1-Street, Urumqi,
Xinjiang, 830011, People’s Republic of China  J. T. Xie  Research Center for
Intelligent Computing Platforms, Zhejiang Laboratory, Hangzhou, 311100, China
Q. D. Wu  Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 150
Science 1-Street, Urumqi, Xinjiang, 830011, People’s Republic of China
University of Chinese Academy of Sciences, 19A Yuquan Road, 100049 Beijing,
People’s Republic of China  Sh. Khasanov  Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou, 730000, China  University of Chinese Academy of
Sciences, 19A Yuquan Road, 100049 Beijing, People’s Republic of China
Samarkand State University, Samarkand 140104, Uzbekistan  FAST Collaboration

###### Abstract

We report two phenomena detected in PSR J0344$-$0901 from two observations
conducted at frequency centered at 1.25 GHz using the Five-hundred-meter
Aperture Spherical radio Telescope (FAST). The first phenomenon manifests as
shifting in the pulse emission to later longitudinal phases and then gradually
returns to its original location. The event lasts for about 216 pulse periods,
with an average shift of about $0.7^{\circ}$ measured at the peak of the
integrated profile. Changes in the polarization position angle (PPA) are
detected around the trailing edge of the profile, together with an increase in
the profile width. The second phenomenon is characterized by the apparent
movement of subpulses, which results in different subpulse track patterns
across the profile window. For the first time in this pulsar, we identify four
emission modes, each with unique subpulse movement, and determine the pattern
periods for three of the emission modes. Pulse nulling was not detected.
Modeling of the changes in the PPA using the rotating vector model gives an
inclination angle of $75.12^{\circ}\pm 3.80^{\circ}$ and an impact parameter
of $-3.17^{\circ}\pm 5.32^{\circ}$ for this pulsar. We speculate that the
subpulse movement may be related to the shifting of the pulse emission.

profile shifting — pulsars: general — pulsars: individual (PSR J0344$-$0901)

revtex4-1.clsrevtex4-2.cls

##  1 Introduction

A large diversity of emission variation at different timescales has been
observed in radio pulsars. This includes the familiar examples of subpulse
drifting, pulse nulling, and profile mode-changing. More than half of the
radio pulsars sampled (Weltevrede et al., 2006, 2007) was found with
measurable drift of subpulses from one pulse to the next through the
integrated profile window. There is also evidence from both radio and
$\gamma$-ray observations that a pulsar magnetosphere may exhibit some kind of
changes, which affect the subpulse drift pattern (Stairs et al., 2000; Lyne et
al., 2010; Kramer et al., 2006; Allafort et al., 2013; Keith et al., 2013).
From the observations of pulsars with multiple subpulse drift modes (Smits et
al., 2005), each with particular drift rate and pattern, the subpulse drift
rate can vary abruptly and then returns to its initial value. In addition, the
changes between different subpulse drift modes have been observed to correlate
with changes in the integrated profile (Wright & Fowler, 1981; Redman et al.,
2005; Joshi, 2013; Rankin et al., 2013). However, changes may also be gradual.
The observation of PSR B0809+74 reveals that a change in the subpulse drift
rate following a null exhibits a slow adjustment back to its normal drift rate
over tens of pulse periods.

On longer timescales, many individual pulses from a pulsar are averaged to
form the integrated profile whose shape is stable over a long time (Helfand et
al., 1975; Manchester & Taylor, 1977; Rathnasree & Rankin, 1995). Sudden
changes in the profile shape, known as profile mode-changing, have also been
detected in many pulsars. However, variations that occur on individual pulse
level or over several tens of pulses are hidden by this averaging method. One
such variation is the observed shifting of the radio profile in PSR
J0344$-$0901\. During the event, the peak of the profile, obtained from
averaging the associated pulses, exhibits a shift towards later longitudinal
phases to a new location, where the profile remains for tens of pulses, and
then returns to its original position. In addition, the profile intensity
drops, and the profile shape changes. The pulsar was first reported by Cameron
et al. (2020) for its timing properties, but without giving detailed
description on the shifting phenomenon. Radio profile shifts have been
detected in a few pulsars (Rankin et al., 2006; Perera et al., 2015; Han et
al., 2016; Wahl et al., 2016; Shaifullah et al., 2018), where the observed
shifts are all toward earlier longitudinal phases, as opposed to that in PSR
J0344$-$0901\. Profile shifting was suggested in association with relativistic
aberration and retardation effects in conventional geometric models (Cheng et
al., 2000; Dyks et al., 2004; Gangadhara, 2005). However, the effects are
long-lasting in contrast to the sporadic changes observed in PSR J0344$-$0901.

In this paper, we present the emission variation observed in PSR
J0344$-$0901\. The pulsar was discovered in September 2017 during the pilot
scans of the Commensal Radio Astronomy FAST Survey (CRAFTS; Li et al., 2018)
using the ultra-wide-bandwidth (UWB) receiver with the Parkes Digital Filter
Bank backend system (PDFB4), and confirmed by the Parkes 64-m Radio Telescope
(Cameron et al., 2020). It has a pulse period of $P=1.23\,\text{s}$, and a
period derivative of $\dot{P}=3.4708\times 10^{-15}\,\text{s}\,\text{s}^{-1}$.
In addition, the pulsar possesses a surface dipolar magnetic field of $B_{\rm
d}=2.06\times 10^{12}\,\text{G}$, a characteristic age of 5.58 Myr, and a
spin-down energy loss of $\dot{E}=7.44\times
10^{31}\,\text{erg}\,\text{s}^{-1}$. The dispersion measure (DM) is given by
$31\,\text{cm}^{-3}\,\text{pc}$ (Cameron et al., 2020). In Section 2, we
outline the observations and the procedures for data analysis. The
classification of different emission pulses is defined in Section 3. Analysis
of the pulse sequences in relation to profile shifting is presented in Section
4. The examination of the subpulse movement is given in Section 5. We briefly
discuss the implications of our results and the main emission features of the
pulsar in Section 6, and summarize our results in Section 7.

Figure 1: The sequences of single pulses from Observations I (left) and II
(right) are shown, and the integrated profile for each is presented in the
lower panel.

##  2 The observations and data analysis

The observations of PSR J0344$-$0901 were conducted using the FAST telescope
with the 19-beam receiver and the Reconfigurable Open Architecture Computing
Hardware–version 2 (ROACH2) signal processor (Jiang et al., 2019, 2020). The
observations spanned a frequency range from 1.05 to 1.45 GHz. For Observation
I, the time and frequency resolutions were set at 49.152 µs and 0.122 MHz,
respectively. Observation II was conducted with time and frequency resolutions
of 49.152 µs and 0.488 MHz, respectively. The data was recorded in the search-
mode PSRFITS format (Hotan et al., 2004), incorporating four polarizations.
Observation I employed 4096 frequency channels, while Observation II used 1024
frequency channels each. More information for the observations is provided in
Table 1. In addition, flux calibration for Observation II could not be
performed due to absence of the off-track data.

Table 1: The two single-pulse observations for PSR J0344$-$0901\. The total duration (in seconds), the total number of pulses, the number of channels ($n_{\rm chan}$), and the frequency resolutions (chbw) in MHz used for each observation are shown in columns 3, 4, 5, and 6, respectively. Observation | Date | Duration | Pulses | $n_{\rm chan}$ | chbw
---|---|---|---|---|---
|  | (s) |  |  | (MHz)
I | 2019-04-23 | 3450 | 2815 | 4096 | 0.122
II | 2020-09-19 | 1800 | 1466 | 1024 | 0.488
Figure 2: Plot in the upper middle panel shows the pulse sequence from
Observation I with sub-integrations of every 30 pulses. The pulse energy
variation in the leading part of the profile is demonstrated in the upper left
plot, where the energy above and below the threshold is signified by blue and
red, respectively. The shifting events are indicated by regions in gray, and
an accompanying drop in intensity is also apparent in the leading component of
the profile. In the upper right plot, the energy variation in the trailing
part of the profile is shown. The integrated pulse profile obtained from all
pulses in Observation I is given in the lower plot, and the vertical dashed
line in gray demarcates the range for the leading and trailing parts of the
profile.

The first step in data analysis involved the use of the DSPSR software package
(van Straten & Bailes, 2011) to de-disperse and produce the single-pulse
integrations. Spectral band edges were zapped using paz, and the radio
frequency interference (RFI) was removed from the data using pazi in the
PSRCHIVE software package (Hotan et al., 2004). The data was then calibrated
using the PAC program in the PSRCHIVE software package to flatten the bandpass
and transform the polarization products to Stokes parameters. After that, the
frequency channels and polarization of the data were compressed. The
radiometer equation,

$S_{\rm av}=\frac{S_{\rm sys}}{\sqrt{n_{\rm p}f_{\rm b}t_{\rm s}}},$ (1)

was then used to calibrate the observed pulses based on the off-pulse noise
(Dicke, 1982; Wen et al., 2021). Here, $n_{\rm p}=4$, $f_{\rm b}=400\,{\rm
MHz}$ and $t_{\rm s}=49.152\,\mu{\rm s}$ are the number of polarization,
effective frequency bandwidth and the sampling time, respectively, and $S_{\rm
sys}\approx 1\,{\rm Jy}$ is the system equivalent flux density (SEFD) of FAST
(Yu et al., 2017). Flux calibration was then performed on all pulse profiles
by multiplying with $S_{\rm av}$, and the estimated flux density ($S_{\rm
av}$) for this pulsar is about 3.5 mJy by averaging the fluxes in all pulse
phases and time.

##  3 Different pulse emission

The pulse sequences of our observations demonstrate significant emission
fluctuations as shown in Figure 1. The sequences on the left show occasional
shifting in the pulse profile, corroborating the findings by Cameron et al.
(2020). On the contrary, no such shifting is detected in the pulse sequence on
the right. Careful examination also reveals visible movement of the subpulses,
which results in a combination of regular and irregular subpulse tracks along
the pulse sequences. In the following, we analyze and discern the emission
between shifting and non-shifting.

Emission during the events of shifting shows continuous changing in phase
position to later longitudes, with an accompanying drop in the pulse intensity
in the “first half” of the profile. To identify the events, we use data in
Observation I and divide the integrated profile into two parts from its peak
intensity, as indicated by the vertical dashed line in gray in Figure 2. We
refer to the longitudinal phase ranges in the parts of the profile to the left
and to the right of the vertical line as the ‘leading’ and ‘trailing’ parts,
respectively. The pulse energy is calculated using the following equation

$E_{p}=\sum_{p=1}^{n}\eta[p,b:e].$ (2)

Here, $p$ represents a starting pulse number, and $n$ is the total number of
pulses in the observation. The array $\eta$ contains the energy of a pulse
within the on-pulse window whose boundary, in terms of bin number, begins at
$b$ and ends at $e$. The identification of the on-pulse range is based on
$10\%$ of the maximum intensity of the integrated pulse profile. Then, the
total energy across the on-pulse window for a pulse at pulse number $p$ is
stored in the array $E_{p}$. In addition, the values of $b$ and $e$ will
change depending on the range of the profile under consideration. For example,
when the leading part of the profile is concerned, the value of $b$ will have
the bin number at the left boundary of the profile (at 10% intensity) and $e$
has the bin number equivalent to the pulse phase at the peak of the integrated
profile. We first determine a threshold value which is obtained from averaging
the energy from all pulses in the leading part of the profile. Next, we
execute a loop that ranges from 1 to 2785, with a step size of 30, to
calculate the pulse energy for each successive block of 30 pulses. Figure 2
presents the pulse energy variation in Observation I. Pulses are characterized
by their energy levels. A shifting pulse emission is recognized if the pulse
energy in the leading part of the profile drops below the threshold, whereas a
non-shifting pulse emission is signified by the pulse energy above the
threshold. We have identified five shifting (shaded in gray) and six non-
shifting emission modes in the observation.

The pulse energy distributions for the on-pulse and off-pulse windows, as well
as for the pulses during profile shifting, are shown in Figure 3. The on-pulse
energy is computed for each pulse by adding up the intensities within the on-
pulse region after subtracting the baseline noise. The off-pulse energy is
calculated within the off-pulse region in a similar manner, with the same bin
size as the on-pulse range. The histograms of the energy distributions are
plotted by normalizing the $x$-axis with the mean on-pulse energy of the
observation, and the $y$-axis is pulse number in logarithmic scale. Note that
the on-pulse energy distribution for the pulses during profile shifting events
is indicated in green. A small amount of on-pulse energy near zero is
observed, which may be attributed to weak emission. Consequently, we believe
that pulse nulling is not evident (Ritchings, 1976; Wang et al., 2007) in the
two observations.

Figure 3: The energy distributions for Observations I and II are shown in the
left and right panels, respectively. The different colors represent the energy
distributions for the on-pulse region (black), off-pulse region (red),
shifting events (green), and non-shifting events (blue). Note that energy
distributions for the on-pulse (black) and off-pulse (red) regions are shown
for Observation II as shifting event is not present in the observation.

##  4 Profile shifting

In this section, we investigate the profile shifting as revealed in
Observation I.

###  4.1 The shifting events

An instance of profile shifting is identified as an intensity drop in the
leading part of the profile, and the pulse emission shifts toward later
longitudinal phases, where it remains for tens of pulse periods before
returning to its original position. Our analysis reveals different durations
for different shifting events, with the longest spanning 390 pulse periods
whereas the shortest lasting only 90 pulse periods. In addition, the intervals
between these events vary, with the longest being 330 pulse periods and the
shortest just 150 pulse periods. To avoid misidentification with the intrinsic
variation of the single pulses, only instances with more than two consecutive
sub-integrations are included, which are marked by shaded areas in Figure 2.
On average, the duration for a shifting event is approximately 216 pulse
periods, whereas non-shifting emission lasts for about 284 pulse periods.
Shifting emission accounts for 38.78% of the observation.

Figure 4: The integrated profiles for the shifting events are shown in blue (Event 1), green (Event 2), cyan (Event 3), brown (Event 4), and magenta (Event 5). The integrated profile obtained from Observation I is shown in black solid line, whose peak is indicated by the vertical dotted black lines, and all other profiles are normalized by its peak intensity. Table 2: The details for each of the five shifting events in Observation I. Here, ‘Pulse #’ gives the pulse numbers where an event occurs, and its duration in pulse number is given in the column under ‘Span’. The separation between two consecutive events is indicated in the fourth column in units of pulse number. The values for $W_{10}$ and $W_{50}$ are measured in degrees. Event | Pulse # | Span | Separation | $W_{10}$ | $W_{50}$
---|---|---|---|---|---
1 | 150–420 | 270 | 150 | 8.80 | 5.28
2 | 660–810 | 150 | 240 | 9.15 | 3.87
3 | 960–1050 | 90 | 150 | 7.04 | 4.22
4 | 1380–1560 | 180 | 330 | 8.45 | 4.93
5 | 1830–2220 | 390 | 270 | 8.80 | 4.93
Figure 5: Plot showing the integrated profiles obtained from pulses during
shifting (blue) and non-shifting (red), and the profile obtained from all the
pulses from Observation I is shown in black. Green dashed lines mark peak
intensities for non-shifting (left) and shifting (right), while the dotted
black line represents the peak intensity in the Observation I profile.

The emission characteristics observed across various profile shifting events
exhibit marked differences. In addition to different duration, the integrated
profiles for the five events relative to that from the entire observation are
also different as shown in Figure 4. There is an apparent shift of
approximately $0.7^{\circ}$ measured from the profile in black to the peaks of
other profiles. We found that the profile from Event 3 displays the highest
peak intensity, while the profile from Event 5 possesses a peak intensity
nearly equal to the profile in black. However, the peak intensities for the
profiles in the remaining events are all lower than the peak intensity of the
profile in black. Variation in profile width is also seen among the shifting
events as shown in Table 2. Measured at 10 percent intensity ($W_{10}$), Event
2 has the broadest pulse width, with about 4.14% more than the profile in
black, whereas Event 3 has the narrowest profile width. The profile width from
Event 4 mostly matches that from the entire observation, and profiles from
both Events 4 and 5 align with the entire observed profile. However, larger
changes are seen at 50 percent intensity ($W_{50}$). Profile from Event 1
demonstrates the widest width, having increased by about 7.1% compared to the
black profile. Event 2 is the narrowest, with a decrease of 21.50% compared to
the entire observed profile.

Figure 5 shows the integrated profiles for the non-shifting emission pulses
(red), the shifting events (blue), and for the entire observation (black). The
shift between the peak intensities of the non-shifted (left) and shifted
(right) emission profiles (between the two green dashed lines) is
approximately $1^{\circ}$ in longitude. In addition, the peak of the non-
shifting profile, which is the strongest among the three profiles, is located
$0.3^{\circ}$ earlier in longitudes relative to the peak of the integrated
profile from the entire observation period. However, the peak intensity of the
shifted profile is lower than that of the non-shifted profile by about 10%. In
addition, the shifted profile clearly reveals more emission components than
the non-shifted counterpart. The variations in profile width the solid black,
blue, and red profiles are, respectively, $8.45^{\circ}$, $8.8^{\circ}$, and
$8.09^{\circ}$ at $W_{10}$ and $4.93^{\circ}$, $4.93^{\circ}$, and
$4.57^{\circ}$ at $W_{50}$. While the changes are approximately consistent
across the three profiles at the two intensity levels, the changes in the blue
profile relative to the black profile become more pronounced as the intensity
level increases to $W_{75}$ and higher.

Profile shifting has been detected in a few radio pulsars. Rankin et al.
(2006) discovered the first profile shifting event, later referred to as
‘swooshing’, in PSRs J1901$+$0716 (B1859$+$07) and J0922$+$0638 (B0919$+$06)
using the 305-m Arecibo telescope. The profiles appear to displace about
$5^{\circ}$ from their original locations to earlier longitudinal phases
together with an increase in the profile width. However, the profile shifting
in PSR J0344$-$0901 is different in that the profile tends to shift toward
later longitudinal phases. Cameron et al. (2020) reported that the event was
present in two out of three observing sessions using the FAST telescope. We
detected the shifting events only in one of two observations, which made up
about two thirds of the total observing time. In addition, the fact that the
shifting event did not occur at all in Observation II, whose duration is
longer than the average separation between two consecutive shifting events in
Observation I, would imply that the event likely takes place in groups.

###  4.2 Changes in the polarization and the position angle

Variations in the polarization and the position angle across the shifted
profile as compared with that from the whole observation are shown in Figure
6. In the lower panel of the figure, the Stokes parameters $I$ (black), the
linear polarization $L=\sqrt{Q^{2}+U^{2}}$ (red), and the circular
polarization $V$ (blue) are shown for the profiles obtained from all the
pulses and from the shifting events. The polarization position angle, $\psi$,
is calculated using $\psi=\tan^{-1}(U/Q)/2$, where $U$ and $Q$ are Stokes
parameters for the two directions of linear polarization. The pulsar possesses
only modest linear polarization. During profile shifting, the circular
polarization remains mostly unchanged whereas the peak of the linear
polarization shifts slightly towards later longitudinal phases. In addition,
the position angle exhibits a noticeable change around the trailing half of
the shifted profile as compared with that across the same section of the
profile from all pulses. The overall range of pulse phase for changes in the
PA is also shortened from the leading edge of the shifted profile resulting in
a lesser symmetric curve.

Figure 6: Changes in the polarization and the position angle from Observation
I. The upper panel shows the changes in the polarization position angle (PPA)
in red and blue across, respectively, profiles from all the pulses (black
solid in the lower panel) and from the shifting events (black dashed in the
lower panel). The curve in green represents the best fit to the PA swing, and
the crossing between the vertical and horizontal gray dashed lines illustrates
the point of the steepest gradient (SG) of the RVM curve. In the lower panel,
the linear (L) and circular (V) polarizations across the two profiles are
shown. Figure 7: Samples of pulse sequence containing the different emission
features observed in Observation I. The modes A, B and D are shown,
respectively (see main text for description of the different emission modes).
The gray lines illustrate the apparent movement of the subpulses. Figure 8:
The different emission modes in Observation II. From left to right are the
modes A, B, C and D, respectively. The gray lines illustrate the apparent
movement of the subpulses.

The swing of the position angle across the integrated profile of the entire
observation is close to an $S$-shape, as shown by the red dots in the upper
panel of Figure 6. Therefore, we assume the rotating vector model (RVM;
Radhakrishnan & Cooke, 1969) for fitting the PA swing. The polarization
position angle ($\psi$) of the linearly polarized emission can be expressed as
a function of pulse phase $\phi$ (Lyne & Manchester, 1988),

$\tan(\psi-\psi_{0})=\frac{\sin\alpha\sin(\phi-\phi_{0})}{\sin\zeta\cos\alpha-\cos\zeta\sin\alpha\cos(\phi-\phi_{0})}.$
(3)

Here, $\alpha$ represents the angle between the magnetic and the rotation
axes, and $\zeta$ signifies the angle between the line of sight and the
rotation axis. We obtain the best-fit parameter of $75.12^{\circ}\pm
3.80^{\circ}$ for $\alpha$ and $71.95^{\circ}\pm 3.73^{\circ}$ for $\zeta$,
with the reduced $\chi^{2}$ being 4.02. The vertical and the horizontal gray
dotted lines at $\phi_{0}$ = $246.50^{\circ}\pm 0.14^{\circ}$ and $\psi_{0}$ =
$10.70^{\circ}\pm 2.54^{\circ}$ in the upper panel represent the maximum
steepest gradient (SG) of the RVM curve given by

$\Big{(}\frac{d\psi}{d\phi}\Big{)}_{\rm max}=\frac{\sin\alpha}{\sin\beta},$
(4)

where $\beta$ = $\zeta-\alpha=-3.17^{\circ}\pm 5.32^{\circ}$ is the impact
parameter of the pulsar. From the fitting using the RVM model, the negative
$\beta$ corresponds to the positive gradient in the PA swing. The SG point is
estimated at approximately $-17.48^{\circ}\pm 29.34^{\circ}$, which is located
near the leading edge of the profile. The location of the SG point at one edge
of a profile is the characteristic of ‘partial cone’ pulsars as identified by
Lyne & Manchester (1988). These pulsars possess profiles that show one steeply
rising edge and another slowly falling edge, and also feature a PA traverse
with SG point positioning toward one edge of the profile. Typically, the SG
point on the ‘S-shaped’ curve, which indicates the emission geometry of a
pulsar, coincides with the profile center in long-period pulsars (Lyne &
Manchester, 1988). However, in PSR J0344$-$0901, the non-central SG point
suggests that it may be a ‘partial cone’ pulsar. The influence of aberration
and retardation effects on the emission geometry have been identified in
slower rotating pulsars (Blaskiewicz et al., 1991; Mitra & Rankin, 2011).
Given the large inclination angle of our pulsar, the magnetic field in the
emission region is likely distorted from a purely dipolar structure and the
shape of the polar cap becomes asymmetric (Cheng et al., 2000). Assuming
emission originating from the last open dipolar field-lines, the estimated
emission height is 288.12 kilometers measured from the center of the star, or
about 0.5% of the radius of the light cylinder. This indicates that for long-
period pulsars with large inclination angle, such as PSR J0344$-$0901, the
impact of aberration and retardation effects can still be significant.

##  5 The emission variation

In this section, we explore the emission variation detected in Observations I
and II.

###  5.1 The different emission features

An inspection of the pulse sequence in Figures 7 and 8 reveals significant
changes in the longitudinal phase where the subpulse emission is detected.
This gives rise to variation in the tracks traced by the subpulses in the
pulse sequence. We categorize the emission features into four distinct modes,
each is characterized by unique pattern of the subpulse tracks.

Figure 9: Example pulse sequences showing the fluctuation analysis for Modes A
(left column) and B (right column) from Observation I. Top: Pulse profiles
with modulation index. Middle: LRFS with side panel showing horizontally
integrated power, highlighting differences between Modes A and B. Bottom: 2DFS
for each mode, with side panels indicating horizontal and vertical integrated
power. The pulse sequence for Mode A is taken from pulse numbers between 20
and 130 in total of 110 pulses. A 64-point Fourier transform is employed, and
then averaged over the blocks of the entire pulse sequence. Similar for Mode
B, which encompasses 40 pulses from pulse numbers 840 to 880, a 32-point
Fourier transform is employed, and also averaged over the blocks of the whole
pulse sequence.

Figure 10: Similar to Figure 9, the plots present the LRFS and 2DFS from
Observation II. Distinct periodic modulations in the pulsar’s emission are
evident in both Modes A (left column) and B (right column). An example of Mode
A spanning 110 pulses is taken from pulse number 790 to 900, while Mode B
captures 59 pulses from 1381 to 1440. In the top panel in each plot, the
integrated power over frequency is depicted. A 64-point and 32-point Fourier
transform were employed and averaged over the blocks of the entire pulse
sequence for Modes A and B, respectively.

Mode A: The tracks appear tilted consistently towards later longitudinal
phase. An example is shown in the leftmost pulse sequence in Figures 7 and 8,
in which the subpulse tracks are indicated by the gray lines showing the
direction of the subpulse movement. In addition, the subpulse movement can be
seen across almost the whole profile, and the slope of the tracks may or may
not be constant. Furthermore, variation in the track separation is also
observed, as shown between pulse numbers 20–130 in Observation I and 760–920
in Observation II, with the tracks in the former being closer to each other.
This demonstrates that changes in subpulse movement occur even within the same
mode. This mode is the second most abundant in our observation, which covers
around 42% and 28% of the single pulses in Observations I and II,
respectively.

Mode B: Similar to Mode A, the subpulse tracks in this mode are also seen
across the whole profile except that they appear tilted toward earlier
longitudinal phases. This indicates that the subpulse movement is also toward
the same direction. Examples are illustrated in the middle plot in Figure 7
and the second plot (from the left) in Figure 8. This mode covers nearly 14%
and 11% of the single pulses in Observations I and II, respectively, making it
the third common emission mode in the observations.

Mode C: The subpulse pattern in Mode C differs distinctly from that in both
Mode A and Mode B. The subpulse tracks in this mode exhibit a continuous
change in direction, resulting in a curved pattern. This is apparent in the
third pulse sequence (from the left) in Figure 8. The tracks can be short, as
shown between pulse numbers 1030 and 1060, or long, as observed between pulse
numbers 1090 and 1150. This mode occurs less frequently, representing about 6%
of the single pulses. Subpulse movement in this mode occurs across the entire
profile, similar to Modes A and B, and changes in the direction generally take
place concurrently across adjacent tracks. Mode C was detected only in
Observation II, indicating that it may not be a prevalent mode in this pulsar.

Mode D: This mode is marked by irregular track patterns that do not correspond
to any of the three categories above. The subpulses appear to move randomly in
both directions or follow a curved path without recurring patterns. Examples
are provided in the rightmost pulse sequence in Figures 7 and 8. Despite these
irregularities, Mode D represents the most common subpulse movement in the two
observations covering 44% and 55% in Observations I and II, respectively.

###  5.2 Periodicity of the subpulse track patterns

Since the subpulse tracks are repeating in modes A and B, it is possible to
use the phase-averaged power spectrum (PAPS) method (Smits et al., 2005; Wen
et al., 2016; Basu & Mitra, 2018) to examine the subpulse movement. The method
measures the vertical separation between consecutive tracks, which is also
commonly designated as $P_{3}$ in subpulse drifting. Subpulse drifting
demonstrates as systematic and (usually) long-lasting marching of subpulses
across the pulse window (Manchester & Taylor, 1977; Weltevrede et al., 2006).
In the following, we also refer to the separation between subpulse tracks as
$P_{3}$. Note that the continuously changing subpulse tracks in Mode C pose a
challenge for identifying the associated $P_{3}$ values, and hence we exclude
them from the following calculation. For a selected pulse sequence, Fourier
transform is performed to determine the absolute values of the pulse flux
density at each longitudinal phase across the pulse window. We then compute
the PAPS value by averaging the resulting transforms over the pulse phase,
which yields a frequency resolution that ranges from 0 to 0.5 cycles per pulse
period, $c/P$, which is equivalent to $P/P_{3}$. Our analysis of fluctuation
spectra indicates that any low-frequency structure observed in the PAPS, with
a value less than 0.05, can be considered negligible. We then obtain the
$P_{3}$ value by taking the reciprocal of the significant peak value from the
frequency resolution. Finally, we utilized the Longitude Resolved Fluctuation
Spectrum (LRFS; Backer, 1973), akin to PAPS, to characterize and estimate
$P_{3}$.

Our observations clearly depict periodic modulations in both Mode A and Mode
B, as showcased in the two example pulse sequences in Figures 9 (Observation
I) and 10 (Observation II). These modes are prominently displayed in the
middle panel of each plot, highlighting periodic characteristics with a
distinct surplus of power within the respective pulse phase range. The
vertical axis on the left panel of each plot indicates $P/P_{3}$. For both
modes, we observed that $P_{3}$ is not consistent and exhibits varying values
across different pulse sequences. The upper panel of each plot presents the
pulse profile, accompanied by a comprehensive depiction of the modulation
index for Modes A and B in both Observations I and II. This modulation index
measures the pulse intensity fluctuation and can be mathematically expressed
as $m_{i}$ = $\sigma_{i}/\mu_{i}$ (Weltevrede et al., 2006). Here,
$\sigma_{i}$ represents the longitude-resolved standard deviation, while
$\mu_{i}$ is the mean intensity for the designated longitude bin $i$. This
metric provides insights into pulse intensity variability at particular pulse
longitudes based on the LRFS methodology (Edwards & Stappers, 2003; Weltevrede
et al., 2012). Error bars for the modulation index are determined through
bootstrapping the data. Each bootstrap iteration introduces random noise
corresponding to the rms of the off-pulse region.

For Observation I, as shown in Figure 9, the modulation displays a peak
frequency at roughly 0.14 $P/P_{3}$ for Mode A, while it is approximately 0.16
$P/P_{3}$ for Mode B. The pattern periodicity for the two modes corresponds to
$P_{3}$ $\sim$ 7.14 $P$ for Mode A, and $P_{3}$ $\sim$ 6.25 $P$ for Mode B.
After conducting a comparative analysis, as shown in Figure 11, the mean
$P_{3}$ values for Modes A and B in Observation I are approximately 8.73 $\pm$
0.5$P$ and 7.88 $\pm$ 0.46$P$, respectively. As for Observation II, as
depicted in Figures 10, the modulation for Mode A indicates a peak frequency
of approximately 0.1 $P/P_{3}$ or $P_{3}$ $\sim$ 10 $P$. For Mode B, the
frequency is around 0.14 $P/P_{3}$ or $P_{3}$ $\sim$ 7.14 $P$. Likewise, in
Observation II, the average $P_{3}$ values for Modes A and B are around 11.76
$\pm$ 0.11$P$ and 7.88 $\pm$ 0.3$P$.

Figure 11: Distribution of $P_{3}$ values for Observation I (upper and middle
plots) and Observation II (lower plot). Modes A and B are shown by red and
blue rectangles, respectively. Mode C is shown in green using a guessed
$P_{3}$ value. Empty areas indicate Mode D, without a well-defined $P_{3}$
value. Figure 12: The normalized integrated profiles from Observation II
(black) and Observation I (red).

Figure 13: The integrated profiles for the different modes obtained from
Observation I (upper panel) and Observation II (lower panel).

###  5.3 Emission mode transition

The different emission modes each manifested with particular $P_{3}$ value.
The distribution of $P_{3}$ values for the two observations are shown in
Figure 11. It demonstrates that changes from one mode to another are common
throughout the two observations. However, such transitions appear random
without showing discernible orders. Furthermore, the transitions do not
accompany with any nulls. We found that Mode A possesses the longest duration
in both observations. The longest duration reaches 121 pulse periods, and the
shortest duration is 18 pulse periods. Nevertheless, the duration for this
mode fluctuates as indicated by $P_{3}$. Mode B occurs less frequently, with
the duration ranging from 23 to 90 pulse periods giving an average duration of
42 pulse periods. As for Mode C, its track pattern is less periodic than Mode
B.

###  5.4 Profile variation

The integrated profiles obtained from all pulses in Observations I and II is
shown in Figure 12 for comparison. The numbers of pulses that formed the two
profiles are 2785 (Observation I) and 1466 (Observation II). It is generally
suggested that an integrated profile with a stable shape could be obtained
with about 500 pulses (Manchester & Taylor, 1977). However, the number of
pulses required for such stabilization has also been shown to relate to the
observed emission features (Helfand et al., 1975). In the case of PSR
J0344$-$0901, the emission variation is significant, so that a longer
stabilization time is required. This is evident from the different profile
shapes shown in Figure 12, even though the number of pulses used to form each
profile is a few times more than that suggested.

It is expected that the different modes of subpulse movement will have impacts
on the profile morphology. When delving into the profiles at $W_{10}$, a
uniformity in the profile width at $W_{10}$ is evident across all modes, as
shown in Figure 13. This is consistent with the argument that certain width
parameters may remain relatively consistent across different modes due to the
overarching pulsar emission geometry (Basu & Mitra, 2018). However, this
consistency disrupts at $W_{50}$, where the profile of Mode B is noticeably
narrower. In addition, Mode A consistently displays the highest peak intensity
in both observations. This agrees with previous studies that highlights the
prominence of certain modes over others due to intrinsic pulsar properties
(Smits et al., 2005). In contrast, Modes B and D show similarities in their
peak intensities. However, Mode C possesses an intensity that is markedly
lower than the other modes.

##  6 Discussion

In this section, we discuss similar events detected in other pulsars, and
comment on several properties of the two phenomena in relation to the
traditional models for drifting subpulses.

###  6.1 Emission variations and the subpulse movement

In the traditional models, subpulse drifting is described as originating from
subbeams located on a carousel that is rotating at a rate determined by the
$\vec{E}\times\vec{B}$ drift (Ruderman & Sutherland, 1975). The subbeams
rotate through the fixed line of sight giving rise to a systematic pattern of
subpulse movement across the profile window with the drift characteristics
being different for different pulsars. The model is effective for interpreting
the emission geometry and subpulse drift behavior in many pulsars
(Bhattacharyya et al., 2007; Rankin, 2017; Basu & Mitra, 2018). In this model,
the subpulse drift-rate and drift direction of a pulsar do not change meaning
that the carousel rotates consistently. However, the model is under challenge
as increasing number of pulsars have been found to exhibit unusual drifting
subpulses demonstrating as changes between different drift-rates with such
changes being pulsar-specific. It shows that the subpulse drift-rate of these
pulsars is time dependent. To address these issues, alternative models have
been proposed. This includes the suggestion of variation in the number of
subbeams on the rotating carousel in different drift regions leading to
different emission modes (Gupta et al., 2004; Bhattacharyya et al., 2009;
McSweeney et al., 2019, 2023). Other proposals involve the partially screened
gap model (Gil et al., 2003; Szary et al., 2015) and the ion-proton radio
pulsar polar cap model (Jones, 2020). There are also proposals for changes in
the magnetospheric geometry as the cause for the observed changes in the
emission properties (Timokhin, 2010), and the introduction of multiple
emission states in the pulsar magnetosphere and switching between different
emission states corresponds to changes in the electric drift resulting in the
different subpulse drift patterns (Yuen, 2019). Nevertheless, the rotating
carousel is assumed in most of these models. Since drifting subpulses are
closely related to the emission properties in the emission region, the
different subpulse movements in PSR J0344$-$0901, and its variations, as
revealed by the different $P_{3}$ values, indicate that the rotation of the
carousel is not fixed but varies with time. It may be that the magnetosphere
of PSR J0344$-$0901 also contains different emission states each with a unique
rotation rate for the carousel. Then, a switch in the rotation rate will
result in a change in the subpulse drift pattern leading to the observed
variation in the subpulse movement and the profile shifting.

Subpulse movement is visible throughout our observations, even before and
after a profile shifting, as shown in the left panel of Figure 1. In the
traditional models, subpulses drift when a relative motion exits between the
plasma flow and the corotation in the magnetosphere. The variation of the
subpulse movement in PSR J0344$-$0901 suggests that the plasma flow, denoted
by $r_{\rm P}$, can either be greater or smaller than the corotation rate,
signified by $r_{\rm C}$, of the star. For plasma flow greater than the
corotation rate, $\delta r=r_{\rm P}/r_{\rm C}>1$ and the plasma flow is ahead
of corotation and the subpulses appear to arrive at earlier longitudinal
phases. For $\delta r<1$, the plasma flow lags behind corotation resulting in
the movement of the subpulses toward later longitudinal phases. It is
difficult to tell whether the changes in the electric drift can be associated
with changes in the electric field or in the magnetic field, or both. Electric
field in pulsar magnetospheres broadly originates from two models of vacuum
and corotation. Furthermore, observations show that the magnetosphere of a
pulsar can abruptly change between the two models (Kramer et al., 2006),
implying that the electric field can also change. In addition, there are
proposals that the magnetic field near the stellar surface is complex and
deviates from dipolar structure (Asseo & Khechinashvili, 2002; Pétri, 2015).
This may account for the changing direction of the subpulse movement. However,
changes in the magnetic field may affect the current flow on a larger scale in
the global magnetosphere. Another feature as revealed by the subpulse movement
is that the sign change of $\delta r$ appears random. However, during profile
shifting, the apparent movement of subpulses is in-phase with the direction of
the profile shifting when the emission returning to its original longitudinal
phases. The value of $\delta r$ is less than one shortly after a shifting
event resulting in the observed subpulses to move toward the leading edge of
the profile. This suggests that the profile shifting may be the result of a
sudden increase in the flow rate of the subpulses relative to the rotation of
the star causing the subpulses to move in and out of the profile window.
Identification of the profile shifting in the paper by Cameron et al. (2020)
was based on deviation in the timing residuals above 2.5 times the weighted
RMS. This also implies that the pulse arrival times (ToAs) during profile
shifting deviate from average. Zero ToAs would mean that the pulses arrive at
the time as predicted, and the presence of timing residuals indicates the
difference between predicted and measured pulse arrival times. From Figure 5
in Cameron et al. (2020), the deviation is about 3 milliseconds above the
weighted RMS. Interpreting the difference as due to delay (or advance) in the
pulse arrival, then this translates to about $0.9^{\circ}$ for a rotation
period of 1.23 s. This agrees with the average amount of shifting in the
profile peak detected in our observations.

###  6.2 Implications for the emission geometry

It is suggested that the underlying mechanism for drifting subpulses is
related to the obliquity angle (Weltevrede et al., 2008). The variation in the
sign of $\delta r$ in PSR J0344$-$0901 implies that the direction of the
subpulse movement is not likely dependent only on the obliquity angle of a
pulsar. The characteristics of pulsar emission are closely related to the
physical properties and geometric aspects of the emission beam. The two-lobed
appearance in the fluctuation spectra, as opposed to other seemingly complex
single profiles, may be an indication of specific physical characteristics or
the geometric configuration of the pulsar’s emission beam (Rankin et al.,
2006, 2013). Notably, pulsars are known to have intricate magnetic field
structures that channel their emissions into narrow beams (Asseo &
Khechinashvili, 2002; Gupta & Gangadhara, 2003). If our line of sight
intersects two distinct regions of the beam, such as an inner core and an
outer cone, this could manifest as a two-lobed profile (e.g., B1944+17;
Kloumann & Rankin, 2010). The situation may be further compounded by numerous
factors, including relativistic beaming effects (Dyks & Rudak, 2003), plasma
instabilities in the emission region (Roy & Gangadhara, 2019; Benáček et al.,
2023), or propagation effects within the pulsar’s magnetosphere (Petrova &
Lyubarskii, 2000). Consequently, our pulsar provides a unique exemplar of how
analyzing emission features can contribute to estimating the radiation
structure and its changes within the emission region. Understanding these
changes can offer explanations for unconventional emission variations, such as
profile shifting and the different subpulse movements observed in PSR
J0344$-$0901, thereby providing new information for the interaction between
pulsar emission properties and the geometric configurations in the emission
beams.

###  6.3 Comparison with others radio pulsars

Recent observations using the FAST telescope have significantly expanded our
knowledge of pulsar emission dynamics through discoveries of variations in
drifting subpulses and the associated changes in the subpulse drift patterns.
For example, PSR J1926$-$0652 (Zhang et al., 2019) displays complex subpulse
drifting across multiple profile components, with irregular $P_{3}$ and
different drift bands, indicating intricate magnetospheric activities. PSR
J1631+1252 (Wen et al., 2022) and the nulling pulsar PSR B2111+46 (Zhi et al.,
2023) both demonstrate modulated drifting subpulses in their leading
components, revealing the local emission properties. In addition, with the
switching in the subpulse drift-rate and curved drift-bands, PSR J1857+0057
(Yan et al., 2023) highlights the temporal variation in pulsar emission.

In comparison, PSR J2007+0910 (Xu et al., 2024) presents an interesting case
with its varying drift-bands, suggesting that diverse emission modes an occur
in the same pulsar. This contrasts with the more systematic and predictable
curved drift-bands observed in PSR B0809+74 (Hassall et al., 2013) and PSR
B0826$-$34 (Esamdin et al., 2005). Furthermore, discoveries by the Arecibo
telescope, such as that in PSR B0919+06 and PSR B1859+07, reveal profile
shifting towards earlier longitudinal phases (Rankin et al., 2006; Perera et
al., 2015; Han et al., 2016; Wahl et al., 2016; Shaifullah et al., 2018). The
phenomenon was suggested in connection with the relativistic aberration and
retardation effects in the conventional geometric models (Cheng et al., 2000;
Dyks et al., 2004; Gangadhara, 2005).

In contrast, the unique observations of the sporadic and irregular subpulse
movements in PSR J0344$-$0901, which are accompanied with profile shifting,
challenge these conventional models. The observation of PSR J0344$-$0901, with
emission potentially extending beyond the profile boundary (Rankin et al.,
2006), shows an exceptional aspect of pulsar behavior and a significant
difference from previous findings.

Collectively, observations from the FAST telescope, together with the Arecibo
findings, reveal a broader range of pulsar behaviors that challenge existing
pulsar emission models. Distinct characteristics from each pulsar, especially
from the unprecedented observations of PSR J0344$-$0901, contribute to a
deeper and more nuanced understanding of pulsar emission processes and the
complex dynamics of pulsar magnetospheres.

##  7 Summary

We have reported on the emission properties in PSR J0344$-$0901 based on
single-pulse observations performed at a frequency range centered at 1.25 GHz
on two different epochs using FAST. The pulsar is classified as a normal
pulsar, and possesses a surface dipole magnetic field strength of
approximately 2.06 $\times$ 1012 G. Applying the rotating vector model to the
changes in the polarization position angle gave an estimation for the
inclination angle of $75.12^{\circ}\pm 3.80^{\circ}$ with an impact parameter
of $-3.17^{\circ}\pm 5.32^{\circ}$ between the line-of-sight and the magnetic
axis. This agrees with the suggestion that the inclination angle of a pulsar
is correlated with the strength of its magnetic field (Zhang et al., 1998). In
addition, the pulsar exhibits significant variation in single-pulse emission,
characterized by profile shifting and different subpulse movement patterns.
Key emission characteristics of the pulsar include:

  * •

Rotation period of 1.23 s.

  * •

Pulse nulling was not detected in the pulsar.

  * •

Shifting of profile to later longitudinal phases was detected, with an average
shift of about 0.7∘ measured at the profile peak from the original position.

  * •

On average, the profile shifting lasts for about 216 pulse periods or
approximately 265 seconds.

  * •

The occurrence frequency and the duration for different events of profile
shifting vary significantly, with shifting emission accounting for about
38.78% of observation time.

  * •

Noticeable changes in the polarization position angle were detected during
profile shifting.

  * •

We identify four distinct emission modes, referred to as Modes A, B, C, and D,
each with unique subpulse movement feature and different subpulse track
pattern. Mode D is the most abundant mode in our observations.

  * •

Subpulses in Modes A and B exhibit movement toward later and earlier
longitudinal phases, respectively. Mode C features consistently curved
subpulses tracks, while Mode D displays irregular subpulse tracks.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 11988101, U2031117, U1838109, 11873080, 12041301) and by the
Alliance of International Science Organizations, Grant No. ANSO-VF-2024-01. We
are grateful to the anonymous referee for valuable comments that have improved
the presentation of this paper. H.M.T. acknowledges Arba Minch University, the
University of Chinese Academy of Sciences, and the CAS-TWAS President’s
Fellowship Programme for providing funding and a PhD Scholarship (No.
2019A8016609001). R.Y. is supported by the National SKA Program of China (No.
2020SKA0120200), the National Key Program for Science and Technology Research
and Development (No. 2022YFC2205201), the National Natural Science Foundation
of China (NSFC) project (Nos. 12041303, 12288102), the Major Science and
Technology Program of Xinjiang Uygur Autonomous Region (No. 2022A03013-2), the
Open Program of the Key Laboratory of Xinjiang Uygur Autonomous Region (No.
2020D04049), the Major Science and Technology Program of Xinjiang Uygur
Autonomous Region No. 2022A03013-2, and is partly supported by the Operation,
Maintenance, and Upgrading Fund for Astronomical Telescopes and Facility
Instruments, which is budgeted from the Ministry of Finance of China (MOF) and
administered by the CAS. N.W. is supported by the National Natural Science
Foundation of China (NSFC Grant No. 12041304). D.L. is supported by the
National Natural Science Foundation of China (NSFC Grant No. 11988101) and the
2020 project of Xinjiang Uygur Autonomous Region of China for flexibly
fetching in upscale talents. P. Wang is supported by the National SKA Program
of China (Grant No. 2020SKA0120200), the Youth Innovation Promotion
Association CAS (id. 2021055), and the CAS Project for Young Scientists in
Basic Research (grant YSBR-006). We thank the XAO pulsar group for valuable
discussions. We also express our gratitude to all members of the FAST
telescope collaboration for establishing the projects (project numbers:
ZD2020-06 and PT2020-0045), which made the observations possible.

This work made use of 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, Chinese Academy of
Sciences.

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