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

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.

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.

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.

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),

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

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.

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.

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$P$.

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.

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.

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.

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.

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.