Changes in Emission Properties Associated with the Emission Shifts in PSR J0344−0901

Abstract

We investigate changes in the emission properties in association with the emission shifts observed in PSR J0344−0901 and their implications for the underlying emission mechanism. By decomposing the averaged pulse profile into multiple Gaussian components, the observed emission shift can be modeled through the variation in the peak phase of each component in relation to the plasma flow in a pulsar magnetosphere of multiple emission states based on the model by Melrose and Yuen. From the arrangements of the Gaussian components to fit the two averaged profiles, we show that the emission shift is due to (i) shifting of the Gaussian components toward later longitudinal phases and (ii) an increase in the plasma density. We show that the plasma flow is not uniform, which may be the reason for the irregular drifting subpulses observed. In addition, the change in the plasma density can either positively or negatively affect the pulse amplitude, depending on the amount of change. We demonstrate that an emission shift should be more prominent when it occurs at a lower emission height, where the plasma density is higher. This suggests that this phenomenon should be common, but it is more likely detected in pulsars with small impact parameters.

1. Introduction

An increasing number of occasions have been identified where the supposedly stable emission from radio pulsars is disrupted. Such disruptions manifest in different forms and have been observed across different timescales, ranging from individual pulses to the aggregation of an order of several hundred pulses, resulting in a significant change in the averaged pulse profile. Examples include pulse nulling, which exhibits as short-term cessation of the radio emission [1,2,3]; profile mode-changing, manifesting as sudden changes in the averaged pulse profile between two or more different shapes, each lasting for minutes to hours [3,4]; and intermittency, which is similar to pulse nulling but on a much longer scale, up to years [5,6,7,8]. Significant profile changes have also been observed in millisecond pulsars, such as that in PSR J1713+0747 [9,10]. Another type of change in pulsar emissions manifests as a shift in a single pulse emission towards earlier or later longitudinal phases, with accompanying changes in the profile width (e.g., [11,12,13]). Known as swooshing [14], also referred to here as an emission shift, this event is quasi-periodic or non-periodic and appears to be linked to magnetospheric changes. This type of event has been observed in a few pulsars, including PSRs B0919+06 and B1859+07 captured by the Arecibo telescope [11] and PSR J0344−0901 [13] discovered by the Five-hundred-meter Aperture Spherical radio Telescope (FAST; [15]). It appears that the manifestation of such an emission shift is different in different pulsars, and efforts have been made to understand its cause. At present, the physical mechanism responsible for this phenomenon is still unclear, and questions like “Under what conditions would a pulsar be more likely to exhibit frequent and observable swooshing?” remain unanswered.

This paper focuses on the emission shifts recently observed in PSR J0344−0901 [13]. This pulsar is believed to be isolated and possesses a rotation period of 1.23 s and a period derivative of $3.47\times {10}^{-15}$ s/s, indicating a characteristic age of 5.58 million years. The obliquity angle is estimated to be 75.12° ± 3.80° with an impact parameter of −3.17° ± 5.32°. In addition, pulse nulling was not detected, suggesting a continuous emission. Two more distinct phenomena were identified. First, the pulsar exhibits emission shifts, where its pulses temporarily move to later longitudinal phases before returning to the normal position. The event lasted for approximately 216 pulse periods, with an average shift of about 0.7° measured between the peaks of the non-shifted and shifted averaged profiles. This was accompanied by changes in the polarization position angle and an increase in the profile width. Second, an accompanying movement of the subpulses across the profile, called subpulse drifting, was also observable. The drift pattern is usually defined by the longitudinal spacing between successive subpulses within a pulse, $P_2$, and the rate of the repeating pattern, $P_3$. The drifting subpulses in PSR J0344−0901 exhibit different patterns, and each pulse has been identified as belonging to one of four emission modes. Mode A and Mode B display regular subpulse movement patterns across the profile window but in opposite directions. Mode C features irregular subpulse movements, where the drift tracks are curved and change direction continuously, and Mode D exhibits complex subpulse movement behavior. Although $P_3$ was measured only in some drift modes and results were not given for $P_2$ (although the longitudinal spacing between successive subpulses was still observable in the pulse sequences), the subpulse movements maintained an overarching modulation cycle, supporting their classification as drifting subpulses. The co-existence of drifting subpulses with emission shifts observed at multiple instances suggests that the two phenomena are not exclusive from each other and that the latter may be linked to drifting of the subpulses.

The traditional models describe subpulse drifting as an orderly change in the longitudinal arrival phase of the subpulse emission across the profile window in a sequence of pulses [16,17,18]. Interpretation of this phenomenon involves assuming the subpulse emission is confined to locations of discrete emission areas (subbeams) that are placed evenly on a carousel. The carousel rotates about the magnetic axis under the $\mathit{E}\times \mathit{B}$ drift [16,17,18]. The subpulses will appear to drift when the corotation and rotation of the subbeams through a fixed line of sight occur at a relative speed. For the emission shifts in PSR J0344−0901, the emission from the subpulses appears to move gradually towards later longitudinal phases, reaching a maximum longitude, and then return gradually to its usual phases (see Figure 2 in Tedila et al. [13]). Assuming that the discrete emission areas are fixed to the plasma, such phase shifts in the subpulse emissions will be related to the flow rate of the emitting plasma. The main difference between subpulse drifting and emission shifting is that the subpulse movement in the former is usually consecutive, periodic, and mode-dependent, whereas the subpulse movement in the latter is sporadic and non-periodic, and its occurrence is unpredictable. This implies that the variation in the plasma flow rate is continuous in the former but momentary in the latter.

While there is no widely accepted model for variations in the plasma flow rate, a model has been proposed for multiple quasi-stable emission states, each associated with a particular plasma flow rate, for an obliquely rotating pulsar magnetosphere [19]. The model is based on synthesizing the electric field ($\mathit{E}$) from two limiting conditions. One limit corresponds to a corotating magnetosphere, in which $\mathit{E}$ is the corotation electric field, as given in the Goldreich and Julian [20] model. Another limit refers to the inductive electric field, ${\mathit{E}}_{\mathrm{ind}}$, ascribed to a rotating magnetic dipole in the vacuum-dipole model [21]. A class of synthesized electric field models is established, with each emission state being defined by a particular value of the parameter y between 0 and 1. The resulting electric field then takes the form of a combination of y and $(1-y)$ fractions of the corotation electric field (${\mathit{E}}_{\mathrm{cor}}$) from the Goldreich and Julian model and the inductive electric field (${\mathit{E}}_{\mathrm{ind}}$) from the vacuum-dipole model, respectively. It follows that a change in the plasma flow is associated with a change in the electric field, which corresponds to a change in the emission state. Regular subpulse drifting would correspond to the pulsar being in an emission state other than corotation. On the other hand, an emission shift requires the changes in the y value to deviate from the corotation value until the maximum shift in longitude and then the y value to change again for the emission to return to its original position. In order for the emission shift to be detectable, this switching in the emission state must occur at locations that are visible to the observer. For this, we assume an emission geometry in which the pulsed radio radiation is emitted from field-lines whose foot-points are located within the open-field region. We consider a dipolar field-line structure for emissions originating from near the stellar surface [22]. In this geometry, an emission from a source point is visible only if it is emitted tangentially to the dipolar magnetic field-line [23,24,25] and parallel to the direction of the line of sight (see Appendix B). For a pulsar with a known viewing angle, $\zeta$, measured from the rotation axis to the line of sight, and an obliquity angle, $\alpha$, which is the angle between the rotation and magnetic axes, the location of the visible point can be determined exactly at a given rotational phase, $\psi$. As the pulsar rotates, the location of the visible point changes, tracing a closed trajectory every pulsar rotation. Therefore, changes in the emission state can be identified by an observer as long as the changes occur at the locations specified by the trajectory of the visible point.

The purpose of this paper is to investigate the changes in the emission properties that accompanyinh the emission shifts observed in PSR J0344−0901, as revealed by changes in the emission states, assuming the emission model of Melrose and Yuen [19]. In Section 2, we outline the model for visible emission shifts. The emission states as revealed by the changes in profiles are determined in Section 3. An analysis of the changes in the emission properties and their implications for the characteristics of the underlying mechanism of the emission shift are provided in Section 4. We discuss our results and conclude this paper in Section 5. The electromagnetic fields essential to our modeling and the emission geometry for the simulation are given in Appendix A and Appendix B, respectively.

2. The Model

We assume that the apparent shift in an emission is due to variation in the flow rate of the emitting plasma. In this section, we outline a model of changes in the plasma flow in a pulsar magnetosphere with multiple emission states.

2.1. Detectable Variation in the Plasma Flow

The presence of drifting subpulses in PSR J0344−0901 indicates that the conventional models for pulsar radio emissions are applicable. Various observations of drifting subpulses have demonstrated that pulsar radio emissions likely emanate from favored locations (e.g., [17]). It has been suggested that the emission is associated with certain ‘plasma columns’, and they are distributed periodically around the magnetic pole and drift relative to corotation. Such plasma columns may be interpreted as nodes or antinodes in a wave-like structure generated due to large-scale instability in the magnetosphere [26,27,28]. Assuming that the wave grows preferentially at a specific spherical harmonic [29,30], it will give rise to a periodic pattern of overdense (antinodes) and underdense (nodes) plasma. The pattern is arranged in a way such that it is proportional to $cos\left(m{\varphi }_b\right)$ corresponding to the mth antinode, where ${\varphi }_b$ is the azimuthal angle in the magnetic frame. The emission from the subpulses is assumed to be confined to regions of overdense plasma (antinodes) such that a pattern of $m_{\max }$ discrete emission areas proportional to $cos\left(m{\varphi }_b\right)$ is formed around the magnetic axis, where $m=\{1\dots m_{\max }\}$. This yields a carousel-type model with $m_{\max }$ discrete emission areas. We also assume that each discrete area has the same diameter [31] and that their locations are independent of the polar angle (${\theta }_b$) and locally independent of the height, r, measured from the stellar center. When projecting onto a surface with a constant r, the discrete areas will align in the radial direction, forming a system of radial spokes.

Considering the discrete areas fixed to the magnetospheric plasma implies that the former drift with an angular velocity that is equal to that of the plasma flow, which is given by [19]

$${\mathit{v}}_{\mathrm{dr}}={\displaystyle \frac{\mathit{E}\times {\mathit{B}}_{\mathrm{dip}}}{B_{\mathrm{dip}}^2}}=y{\mathit{v}}_{\mathrm{ind}}+(1-y){\mathit{v}}_{\mathrm{cor}},$$

(1)

or

$${\mathit{\omega }}_{\mathrm{dr}}={\displaystyle \frac{{\mathit{v}}_{\mathrm{dr}}}{r}}=y{\mathit{\omega }}_{\mathrm{ind}}+(1-y){\mathit{\omega }}_{\mathrm{cor}}$$

(2)

when expressed in angular frequency. The subscripts “dr”, “dip”, and “ind” indicate plasma drift, dipolar, and inductive, respectively. Here, $\mathit{E}$ is the electric field defined by Equation (A1), and ${\mathit{B}}_{\mathrm{dip}}$ is the dipolar term in the magnetic field equation specified by Equation (A4). The different emission states are indicated by the parameter y. The case for $y=0$ reduces ${\mathit{v}}_{\mathrm{dr}}\to {\mathit{E}}_{\mathrm{cor}}\times {\mathit{B}}_{\mathrm{dip}}={\mathit{v}}_{\mathrm{cor}}$, where ${\mathit{E}}_{\mathrm{cor}}$ is given by Equation (A3). This means that the plasma flow is in corotation, as described in the Goldreich and Julian [20] model, and it corresponds to no subpulse drifting or emission shifts. The case for $y=1$ corresponds to the ‘minimal’ state in the magnetosphere (see Appendix A), in which ${\mathit{v}}_{\mathrm{dr}}={\mathit{v}}_{\mathrm{ind}}$, and the electric field takes the form given by Equation (A2). For $0<y<1$ of an oblique rotator, ${\mathit{v}}_{\mathrm{dr}}$ is a combination of the two cases given by Equation (1). In addition, a charge density is required to screen the electric field along the magnetic field-lines in each state. Expressed in terms of y, it is given by [19]

$${\rho }_{\mathrm{sn}}=y{\rho }_{\min }+(1-y){\rho }_{\mathrm{GJ}},$$

(3)

with

$${\rho }_{\mathrm{GJ}}=-{\epsilon }_0\mathrm{div}{\mathit{E}}_{\mathrm{cor}}$$

(4)

being the corotation charge density [20], and

$${\rho }_{\min }=-{\epsilon }_0\mathrm{div}\left(\mathit{b}{E}_{\mathrm{ind}\Vert }\right)$$

(5)

being the charge density required to screen the parallel inductive electric field in the minimal state. Here, $\mathit{b}$ represents the unit vector along the magnetic field, and “sn” signifies synthesis of the emission states. The condition for corotation ($y=0$) gives ${\rho }_{\mathrm{sn}}={\rho }_{\mathrm{GJ}}$. As y increases, the charge density reduces until it reaches the minimal value, signified by ${\rho }_{\mathrm{sn}}={\rho }_{\min }$ at $y=1$. It follows that a change in the emission state, corresponding to a change in the value of y, results in a change in both the plasma flow and the charge density.

In order for the changes to be detectable, they are required to take place at emission points that are visible to the observer. We assume a geometry as outlined in Appendix B, in which the radio emission originates from source points on open dipolar field-lines whose foot-points are situated in the open-field region. From this, the location of the visible point can be determined by requiring the emission to come from the last closed-field-lines and emit tangentially to the local magnetic field-line [23,24,25] and parallel to the line of sight [32,33]. Equation (A10) specifies the polar coordinates for the location of the visible point as a function of $\psi$. As $\psi$ varies, the visible point moves, tracing a closed trajectory after one pulsar rotation. The value of ${\mathit{v}}_{\mathrm{dr}}$, or ${\mathit{\omega }}_{\mathrm{dr}}$, is determined at the locations defined by the trajectory of the visible point $({\theta }_{b\mathrm{V}},{\varphi }_{b\mathrm{V}})$, and hence it is dependent on $\zeta$ and $\alpha$. An illustration of the geometry is shown in Figure 1. The apparent component of ${\mathit{\omega }}_{\mathrm{dr}}$ to the observer corresponds to its projection onto the trajectory of the visible point. In subsequent calculations, we refer to this particular version of ${\mathit{\omega }}_{\mathrm{dr}}$ unless stated otherwise. Then, visibility requires the emission to come from discrete areas on the radial spokes that are cut by the trajectory inside the open-field region. We refer to discrete emission areas as emission spots.

The assumption that the emission occurs only within the open-field region requires the trajectory of the visible point to lie partly or entirely inside that region. This also introduces the dependence of the visible emission on height. The emission height is commonly estimated based on (i) a relativistic phase shift, which requires the pulse profile to be asymmetric and the core and cone components to be clearly defined [34], or (ii) geometry, with the locations of the emission source on the last closed-field-lines [25]. We assume point (ii) in this paper, and the visible emission height can be estimated from Equation (A13) in Appendix B.

2.2. Measuring the Changes

Observations of drifting subpulses have suggested that there are 20 emission areas in the emission region [17,18]. Here, we assume an arrangement of 20 radial spokes radiating from the magnetic pole. An emission from a discrete area is observable only if the spoke is cut by the trajectory, and the number of intersected spokes is related to the size of the open-field region. Since the profile center is located near the leading edge, we estimate the maximum height for the open-field region by assuming its boundary to coincide with the trailing edge of the profile. The height can be estimated by assuming that the emission originates from the last closed-field-lines using Equation (A13). For $\zeta =71.95°$ and $\alpha =75.12°$ [13], this gives $0.0145r_{\mathrm{L}}$, where $r_{\mathrm{L}}$ is the light cylinder’s radius. The observable emission then comes from emission spots that are intercepted by the trajectory of the visible point over the range of the rotational phase forming the pulse profile.

For a designated y value, a particular emission spot flows around the magnetic pole according to Equation (2) and is intersected by the trajectory of the visible point at a specific longitudinal phase each pulsar rotation. A change in the y value leads to a change in the flow rate, causing a deviation in the intersection from that emission spot to a different longitudinal phase. It follows that once this deviation is known, it is possible to determine the change in y and the associated change in the charge density.

3. Gaussian Fitting and Changes in the Emission States

In this section, we treat the observed shifts in emissions as the result of variations in the plasma flow and other related emission conditions and measure such variations through the changes, as exhibited by the shifted profiles. The two profiles are generated using the dataset from Observation I, as presented in Tedila et al. [13]. We construct the total average profiles based on all single pulses, which we refer to as the non-shifted profile, and the shifted average profile using only the single pulses in the shifted emission.

3.1. Changes of the Profile

Pulsar visibility is defined in a geometry in which an emission occurs tangentially to the local dipolar magnetic field and parallel to the observer’s line of sight. It determines the visible emission points for given $\zeta$ and $\alpha$ values and the trajectory it traces that intersects the emission spots as the pulsar rotates. The emission from a given emission spot is modeled as a Gaussian function such that the intensity varies with the phase according to [35,36,37,38]:

$$I_i\left(\psi \right)=A_{i}exp\left[-{\displaystyle \frac{{(\psi -{\psi }_{\mathrm{P}i})}^2}{2{\sigma }_i^2}}\right],$$

(6)

where $A_i,{\psi }_{\mathrm{P}i}$, and ${\sigma }_i$ are, respectively, the amplitude and phase at the component peak and the standard deviation for the i-th component. The resulting intensity of the profile is the sum of all of the components [36,39]. An emission component is formed from an emission spot that is visible to the observer. For known values of $\zeta$ and $\alpha$ and a given component peak phase, ${\psi }_{\mathrm{P}i}$, Equation (A10) can be used to determine the polar coordinates of the visible point, $\{{\theta }_{b\mathrm{V}},{\varphi }_{b\mathrm{V}}\}$, in the magnetic frame, or, $\{{\theta }_{\mathrm{V}},{\varphi }_{\mathrm{V}}\}$, in the observer’s frame. From this, the plasma charge density can be evaluated using Equation (3), and the emission height at ${\psi }_{\mathrm{P}i}$ is identified based on Equation (A13). The variation in the plasma flow can also be calculated from Equation (2). If we designate the emission states for the non-shifted (subscript N) and shifted profiles (subscript S) as $y_{\mathrm{N}}$ and $y_{\mathrm{S}}$, the shift in the peak phase of a component can be represented by a deviation in the plasma flow between the two states, as given by

$$\delta {\omega }_{\mathrm{dr}}={\omega }_{\mathrm{dr}}\left(y_{\mathrm{S}}\right)-{\omega }_{\mathrm{dr}}\left(y_{\mathrm{N}}\right).$$

(7)

It follows that the value of $\delta {\omega }_{\mathrm{dr}}$ changes as the value of $y_{\mathrm{N}}$ or $y_{\mathrm{S}}$ changes, leading to a shift in the peak phase of the component. From Equation (7), the direction of the shift is dependent on the sign of $\delta {\omega }_{\mathrm{dr}}$. For $\delta {\omega }_{\mathrm{dr}}>0$, the plasma flow in the emission state designated by $y_{\mathrm{S}}$ is ahead of that in the emission state designated by $y_{\mathrm{N}}$, causing forward movement of the component peak to later phases in successive pulses. This reverses for $\delta {\omega }_{\mathrm{dr}}<0$. The amount of shift can be expressed in degrees with $\delta {\psi }_{\mathrm{dr}}=\delta {\omega }_{\mathrm{dr}}(180°P/\pi )$, where $P=1.23\mathrm{s}$ is the rotation period of PSR J0344−0901.

The number of emission spots (Gaussian components) cut by a trajectory of the visible point is dependent on the viewing geometry presented in Appendix B. According to Equation (A10), ${\varphi }_{b\mathrm{V}}$ varies as a function of $\psi$ for an oblique rotator, and the trajectory cuts the emission spots at irregular phase intervals. Hence, an even distribution of emission spots around the magnetic pole does not necessarily result in the same uniform distribution along the trajectory of the visible point. In general, the number of emission spots within a fixed profile width increases as the impact parameter, $\left|\beta \right|=|\zeta -\alpha |$, decreases. Using the same $\alpha =75.12°$ and $\zeta =71.95°$ [13] and the geometry with $m_{\max }=20$ [17,18] presented in Section 2, we obtain seven spokes (seven emission spots) that are cut by the trajectory of the visible point. This gives $i_{\max }=7$ in Equation (6). Since the profile width at 10% of the peak intensity does not exhibit considerable changes before and during the emission shift [13], we therefore assume that seven Gaussian components remain for both profiles.

Table 1. The parameters of the Gaussian components for fitting the non-shifted and shifted profiles are shown in the upper and lower panels, respectively. The components are numbered from the leading to the trailing edges. The azimuthal angle in the magnetic frame for each ${\psi }_{\mathrm{P}}$ is given under the column signified by ${\varphi }_b$. The emission height estimated at the peak of each component is given in units of the light cylinder’s radius in the sixth column. The emission state and the corresponding plasma charge density (in units of Goldreich and Julian [20]’s value) at each component peak are given in the last two columns, respectively.

Comp.	${\mathit{\psi }}_{\mathbf{P}}^{\circ }$	A	$\mathit{\sigma }$	${\mathit{\varphi }}_{\mathit{b}}^{\circ }$	$\mathit{r}(\times {10}^{-3}{\mathit{r}}_{\mathbf{L}})$	y	${\mathit{\rho }}_{\mathbf{sn}}/{\mathit{\rho }}_{\mathbf{GJ}}$
Non-shifted profile
1	−0.60	0.070	0.80	$-16.7$	2.64	0.502 ± 0.014	0.531 ± 0.013
2	0.80	0.685	0.98	8.5	2.58	0.501 ± 0.018	0.532 ± 0.017
3	2.30	0.667	0.80	33.3	3.00	0.415 ± 0.012	0.612 ± 0.011
4	3.20	0.275	0.52	43.6	3.54	0.675 ± 0.030	0.369 ± 0.028
5	4.10	0.580	0.80	50.6	4.28	0.501 ± 0.018	0.532 ± 0.017
6	5.50	0.240	0.90	58.2	5.89	0.364 ± 0.012	0.660 ± 0.011
7	6.90	0.090	1.50	63.5	8.01	0.501 ± 0.018	0.533 ± 0.017
Shifted profile
1	−0.20	0.030	1.00	−3.4	2.57	0.491 ± 0.014	0.541 ± 0.013
2	1.11	0.600	1.15	18.3	2.66	0.492 ± 0.018	0.540 ± 0.017
3	2.63	0.535	0.72	36.8	3.13	0.405 ± 0.012	0.621 ± 0.011
4	3.35	0.153	0.39	44.5	3.62	0.671 ± 0.030	0.372 ± 0.028
5	4.15	0.595	0.75	50.6	4.28	0.500 ± 0.018	0.533 ± 0.017
6	5.60	0.289	0.85	58.7	6.02	0.361 ± 0.012	0.663 ± 0.011
7	6.95	0.106	1.50	63.5	8.01	0.500 ± 0.018	0.534 ± 0.017

lists the parameters for the seven components used to decompose the two profiles. The resulting fit of the simulated profiles to the observed profiles is shown in Figure 2. The goodness of fit is evaluated using ${\chi }^2$ between the total intensity of the observed (fixed-parameter) and corresponding simulated (free-parameter) profiles. To avoid the noise at the baseline, the tests were performed from above 5% of the total intensity of the profiles. The corresponding ${\chi }^2$ values are 0.019 and 0.024. Note that the fiducial plane for both profiles is assumed to be near the leading edge of the profile, as given by Tedila et al. [13].

A comparison of the peak phases between the non-shifted and shifted profiles reveals that all of the Gaussian components move to later longitudinal phases during the emission shift. This is also shown in Figure 3, which indicates that the apparent shift in the emission is the result of a consistent movement of the components in one direction. Significant movements occur in the first three components, whereas the least movement takes place in components 5 and 7. The average shift in the component peaks is about $+0.20°$ towards a later longitudinal phase.

It is possible to determine the emission states for the emissions that form the non-shifted and shifted profiles. This gives an average change in the emission state, which filters out the fluctuations in the single pulses and reveals essential characteristics of the emission shift by comparing the two averaged profiles. Since the normal emission constitutes the larger proportion of the observation, we assume that it represents the ‘primary’ emission state and that the shifted emission is the result of a change in the emission state. We simulate the change in the emission state that is required to produce the shift in each component through variation in the longitudinal phase at each component peak in the shifted profile. In the model presented in Section 2, a pulse profile is detected when the visible point traverses an emission spot. The flow rate of the emission spots is the flow rate of the plasma given by Equation (2). It follows that a shift in the observed peak phase, designated by $\delta {\psi }_{\mathrm{P}}$ in degrees, corresponds to the emission spot being intersected by the trajectory of the visible point at a different longitudinal phase. We suggest that such a shift arises from a change in the plasma flow, due to a change in the emission state, causing the emission spot to be intersected at a different phase. Hence, the $\delta {\psi }_{\mathrm{P}}$ value of a component between the non-shifted and shifted profiles can be used to evaluate the corresponding change in the emission state. Since the average shift in the emission is less than one degree in longitude, we assume that the same set of emission spots is intersected every pulsar rotation. Our simulation does not presume any fixed value for $y_{\mathrm{N}}$ or $y_{\mathrm{S}}$. Instead, we search for all possible y value pairs that will give the required change by iterating both $y_{\mathrm{N}}$ and $y_{\mathrm{S}}$ from 0 to 1, in steps of 0.001. Consider the first component as an example, where the component peaks are at $-0.6°$ and $-0.2°$, giving a phase shift of $\delta {\psi }_{\mathrm{P}}=0.4°$. First, the value for $y_{\mathrm{N}}$ is iterated from 0 to 1 in steps of 0.001. Then, for every $y_{\mathrm{N}}$ value, a value of $y_{\mathrm{S}}$ is iterated over 0 to 1, also in steps of 0.001, to determine the phase shift. This gives $1001\times 1001=1002001$ pairs of y values, each associated with a particular $\delta {\omega }_{\mathrm{dr}}$, or $\delta {\psi }_{\mathrm{dr}}$, for the component. This process is repeated for all components. Since a simulated $\delta {\psi }_{\mathrm{dr}}$ from the $\{y_{\mathrm{N}},y_{\mathrm{S}}\}$ pair that gives a phase shift closer to $\delta {\psi }_{\mathrm{P}}$ is more likely to have greater significance, we evaluate the weight according to the distance of $\delta {\psi }_{\mathrm{dr}}$ from the putative value using a Gaussian likelihood function. This way, a higher weight is assigned to a result that is closer to the observation, and vice versa. The same weight is then assigned to each simulated value from the same calculation. Finally, a weighted average is calculated for each of the parameters in the simulation.

3.2. The Results

The measured average y values for each component are shown in the seventh column in Table 1. Since $y_{\mathrm{N}}$ and $y_{\mathrm{S}}$ are related to the location of a component and the amount of shift in the longitudinal phase, their values are different for different emission components. In general, the y value measured at the peak of a component exhibits a change between the non-shifted and shifted profiles. This means that the occurrence of the emission shift requires a change in the emission state across the profile. We find that the y values are slightly lower when the pulsar is emitting a shifted emission than when it is emitting a normal emission. The average y values as represented by the non-shifted and shifted profiles are $0.494$ and $0.488$, respectively. However, the emission states are not consistent across the two profiles, as illustrated by the y values in Table 1. Figure 4 shows the weighted average y values at all of the component peaks. We find that the values of $y_{\mathrm{N}}$ are similar for components 1, 2, and 7, but noticeable variation is seen across components 3–6. Similar variation is also seen in the values of $y_{\mathrm{S}}$. This indicates that significant variation in the emission conditions occurs around the middle part of the two profiles, but they become more consistent near the two edges. The changes in the y values at each component peak are also shown in Figure 4. Again, the change is not consistent but displays a decreasing trend towards later longitudinal phases, with the largest changes found around the first three components. This is consistent with observations that show the largest emission shift occurring around the leading edge of the profile. The smallest change occurs at components 5 and 7. The magnetic azimuthal angle at each component peak is given in column 5 in Table 1, and the corresponding height is derived from this using Equation (A13) by assuming that the emission comes from the last closed-field-lines. The emission height estimated at each of the component peaks is given in column 6 in Table 1. In five of the seven components, the emission height changes slightly to either lower (component 1) or higher (components 2–4 and 6) in the shifted profile. The average heights are 251 km and 254 km for the emissions that form the non-shifted and shifted profiles, respectively, which are consistent with the overall heights predicted from the observations [13].

4. Properties of the Emission Shift

Based on the arrangements of the Gaussian components shown in Table 1, we explore the implications of a change in the emission state as an interpretative tool for the emission shift in order to reveal the emission conditions underlying the phenomenon.

4.1. Irregular Subpulse Drifting

An emission state equivalent to a non-zero value of y indicates non-corotation, with a larger y value signifying a greater deviation from corotation. According to Equation (7), this causes a change in the longitudinal phase at which an emission spot is intercepted by the trajectory of the visible point, resulting in the associated Gaussian component appearing to be shifted in phase. A related phenomenon that also relies on a deviation in the magnetospheric plasma flow from corotation manifests as subpulse drifting [18,40,41,42,43]. In a similar manner, a constant y value (not equal to zero) results in a constant shift in the phase at which an emission spot is intersected by the trajectory of the visible point each pulsar rotation. The result is that the Gaussian component (or the subpulse) will appear to systematically march across the profile. For PSR J0344−0901, both $y_{\mathrm{N}}$ and $y_{\mathrm{S}}$ deviate from zero across the profile, meaning that each of the Gaussian components does not arrive at a fixed longitudinal phase in consecutive pulsar rotations. This is consistent with the detection of drifting subpulses throughout the observations, even during the emission shifts. Furthermore, the greatest variation in the y value occurs around the middle part of the profile. This gives the largest variation in the drifting subpulses, which appears to be the case, as revealed by the observations. However, the observed subpulse drifting is irregular, implying that the y values are constantly changing. Here, the values of $y_{\mathrm{N}}$ and $y_{\mathrm{S}}$ are determined based on the averaged profiles, which give an average y value at each component peak in each profile.

4.2. Changes in ${\rho }_{\mathrm{sn}}$ and the Emission Height

A notable feature of the emission shifts in PSR J0344−0901 manifests as the shifting of the profile toward later longitudinal phases, as opposed to a shift towards earlier longitudinal phases, as found in other pulsars (e.g., [11]). As the emission components shift, the corresponding plasma charge density also changes, as shown in Table 1. We find that the value of ${\rho }_{\mathrm{sn}}$ is not consistent across the profile. An examination of the variation in ${\rho }_{\mathrm{sn}}$ in units of ${\rho }_{\mathrm{GJ}}$ for both the non-shifted and shifted profiles indicates that each profile may be divided into two parts, similar to the conditions for the y values described in Section 3.2. We find that the part of the profile signified by the peaks of components 1, 2, and 7 exhibits similar ${\rho }_{\mathrm{sn}}/{\rho }_{\mathrm{GJ}}$ values to those in the non-shifted profile. This is identical for the shifted profile but with somewhat higher values. The average ${\rho }_{\mathrm{sn}}/{\rho }_{\mathrm{GJ}}$ values are $0.532$ and $0.539$, respectively. However, significant variation in ${\rho }_{\mathrm{sn}}/{\rho }_{\mathrm{GJ}}$ is seen across the longitudinal phases around the middle part of the profile, indicated by the peaks of components 3–6. For both the non-shifted and shifted profiles, the smallest and largest ${\rho }_{\mathrm{sn}}/{\rho }_{\mathrm{GJ}}$ values occur at the peaks of components 4 and 6, respectively, with an almost two-fold difference.

The emission shifts are also accompanied by a transition in the emission locations to different altitudes. In Table 1, five components display changes in their emission heights, except for components 5 and 7, which remain unchanged, resulting in an overall increase in the emission height. Furthermore, the width of the shifted profile at 10% of the profile peak intensity ($W_{10}$) is $9.1°$, which is $0.3°$ wider than the non-shifted profile, also indicating a slightly wider beam-opening angle. Similar emission shifts associated with changes in the altitudes have also been proposed [12,44]. A possible model for this phenomenon relates the different emissions (non-shifted and shifted) observed at a given frequency to different heights within the same magnetic flux surface [12,45]. This implies that the relativistic plasma may generate broadband radio emissions while streaming out along the same magnetic flux tube [46]. In addition, the different emission heights correspond to different physical conditions, as shown by the increase in ${\rho }_{\mathrm{sn}}$ in all of the components when the pulsar switches to a shifted emission. Such an increase is opposite to the variation in the plasma charge density along a field-line, which decreases according to $r^{-3}$ in a dipolar field structure [20]. This may be an indication that a different mechanism is in action, which is responsible for the changes in the plasma charge density across the emission region at different heights, resulting in the observed emission shifts.

4.3. Changes in the Plasma Density and the Peak Intensity

The different emission properties in the pulsar can be uncovered by examining how a given quantity varies in response to the change in the shifted emission. The emission shift is accompanied by a change in the emission state and an associated change in the plasma charge density. It is known that the charge density is related to the particle acceleration and the radio emission mechanism [16]. To explore the influence of a change in ${\rho }_{\mathrm{sn}}$ on the emission mechanism during the emission shift, we consider the change in ${\rho }_{\mathrm{sn}}$ relative to its original value. Assuming the emission associated with the non-shifted profile is the primary emission state means that the original values are those represented by the non-shifted profile, or $R_{{\rho }_{\mathrm{sn}}}=({\rho }_{\mathrm{sn},\mathrm{S}}-{\rho }_{\mathrm{sn},\mathrm{N}})/{\rho }_{\mathrm{sn},\mathrm{N}}$. The relative change provides a normalized perspective for comparing the changes in the emission region at locations represented by the peaks of different components within the profile. Hence, $R_{{\rho }_{\mathrm{sn}}}$ quantifies how the plasma changes over time and reveals the properties of the radio mechanism. Figure 5 demonstrates the change in $R_{{\rho }_{\mathrm{sn}}}$ at each component peak plotted as a function of the longitudinal phase at each component peak of the non-shifted profile. It is clear that $R_{{\rho }_{\mathrm{sn}}}$ is not a constant, but it exhibits a general decreasing trend towards the trailing part of the profile. Also shown in the figure is the associated normalized relative change in the amplitude $R_A=(A_{\mathrm{S}}-A_{\mathrm{N}})/A_{\mathrm{N}}$. We find that $R_A$ is negative around components 1–4 and then rises to positive in the last three components. A negative $R_A$ means $A_{\mathrm{S}}<A_{\mathrm{N}}$, and the component amplitude reduces around the leading part of the profile when the pulsar is in the shifted emission. On the contrary, $A_{\mathrm{S}}>A_{\mathrm{N}}$ gives a positive $R_A$, and the pulse amplitude increases around the trailing part of the profile. $R_A$ is related to $R_{{\rho }_{\mathrm{sn}}}$ when considering pulsar radio emissions coming from the inner magnetosphere [22], where the pair production is efficient. We then obtain $\lambda {\rho }_{\mathrm{sn}}/e=n=n_{+}+n_{-}$ in our model [47,48], with $n_{\pm }$ denoting the electrons and positrons, respectively. The pair multiplicity is signified by $\lambda$, and it is required to be $\lambda =e(n_{+}+n_{-})/{\rho }_{\mathrm{GJ}}\gg 1$. Assuming an identical $\lambda$ value for all emission states implies that the plasma density is proportional to ${\rho }_{\mathrm{sn}}$. The radio emission is then assumed to be produced in two-stream instability from ultra-relativistic plasma that streams out along the open-field-lines. This leads to the prediction that the pulse intensity is proportional to the plasma density [49,50,51], and a change in the latter leads to a change in the former. This is consistent with the changes in $R_{{\rho }_{\mathrm{sn}}}$ in the last three components, which show that a change in the pulse intensity is positively correlated with a change in the emitting plasma density. However, for large $R_{{\rho }_{\mathrm{sn}}}$ values, such as that at components 1–4, an increase in the plasma density actually reduces the pulse intensity, giving a negative $R_A$ value.

Consider a scenario in which pulsar radio emissions are due to some kind of coherent mechanism [51,52,53,54,55,56,57,58]. A simple model may involve plasma oscillation in two-stream instability that requires the charged particles to move collectively and coherently in each oscillation and convert into radiation. The characteristic frequency of the radio emission is given by the plasma’s frequency, which has the form in the co-moving frame of

$${\omega }_p\propto \sqrt{{\displaystyle \frac{e^{2}n}{m_e{\epsilon }_0}}},$$

(8)

where e and $m_e$ are the electron’s charge and mass, respectively, and n is the plasma number density. Since the free charged particles, say electrons, in the plasma can oscillate at frequencies at or above the plasma’s frequency, the plasma becomes transparent when the emission frequency matches or exceeds ${\omega }_p$. In simple terms, ${\omega }_p$ represents the ‘cut-off’ frequency for the propagation of radio emissions, and the plasma density affects the behavior of the coherent mechanism. In our model, the dependence of the plasma’s density on the value of y means that a change in the latter would lead to a change in ${\omega }_p$. Table 1 shows ${\rho }_{\mathrm{sn},\mathrm{S}}>{\rho }_{\mathrm{sn},\mathrm{N}}$ at most of the component peaks, suggesting that ${\rho }_{\mathrm{sn}}$ is higher in the shifted emission. This implies an increase in n in Equation (8) and thus an increase in ${\omega }_p$. Then, the different correlations between $R_A$ and $R_{{\rho }_{\mathrm{sn}}}$ over the first four and last three components indicate that the response of the pulse intensity can vary with the amount of change in the plasma’s density in the immediate medium (cf. Section 4.4). From Section 4.2, the association of the emission shifts with changes in the emission heights suggests that the non-shifted and shifted emissions come from different heights along the same magnetic flux surface at a fixed observed frequency. Across a wide frequency range, the emission heights corresponding to each emission do not vary greatly within the uncertainty. As the observed emission frequency is not affected by a change in the plasma density, this also implies that the original plasma bunching remains largely unchanged. Then, a low value of $R_{{\rho }_{\mathrm{sn}}}$ leads to a small change in ${\omega }_p$, and the pulse intensity responds positively to a change in the plasma’s density, as seen around the trailing part of the profile. On the contrary, a large value of $R_{{\rho }_{\mathrm{sn}}}$ causes large variation in ${\omega }_p$ so that a reduced amount of plasma, whose emission frequency is higher than ${\omega }_p$, propagates through the plasma, resulting in a decrease in or the disappearance of some of the emission. This may be the case around the leading part of the profile, where greater changes in $R_{{\rho }_{\mathrm{sn}}}$ occur. This drop in the pulse intensity implies that no mechanism for regulating the plasma’s density in the bunches to accommodate the change in ${\omega }_p$ exists. Therefore, the emission shift in this pulsar is the result of two effects: (i) movements of the emission components toward later longitudinal phases and (ii) accompanying increases in the plasma’s density.

4.4. Conditions for the Emission Shift

Emission shifts, or swooshing, have been observed in a few pulsars (e.g., [11,12]), but how this phenomenon is related to the pulsar parameters and the emission properties required to execute the change is uncertain. Here, we measure the impact of a change in the emission state through the changes in $R_{{\rho }_{\mathrm{sn}}}$ across the profile. Figure 6 shows the change in $R_{{\rho }_{\mathrm{sn}}}$ as a function of the emission height at the peak of each component. Each emission height in the figure is an average of the emission heights at the two corresponding component peaks in the non-shifted and shifted profiles. The value of $R_{{\rho }_{\mathrm{sn}}}$ shows a clear decreasing trend as height increases. This can be fitted by a function that contains a term $\propto 1/r^{2.5}$ indicating an exponential relationship. This means that the amount of change in the plasma’s density (in units of ${\rho }_{\mathrm{GJ}}$) relative to its original value reduces rapidly with height. As height increases, the value of $R_{{\rho }_{\mathrm{sn}}}$ approaches zero but never reaches zero. In addition, comparing the decreasing trend for $\Delta y$ towards the trailing edge of the profile, shown in Figure 4, with the change in $R_{{\rho }_{\mathrm{sn}}}$, shown in Figure 5, indicates that changes in the plasma’s density are related to changes in the emission state. When combined with Figure 6, this demonstrates that a large relative change in ${\rho }_{\mathrm{sn}}$ at lower height corresponds to a large change in y, and the change in the emission state reduces as the height increases, where the relative change in ${\rho }_{\mathrm{sn}}$ is also small. It follows that changes in the emission state may still occur at high heights, but the effects are likely diminished. This implies that the emission shift, as interpreted in our model, favors emissions from lower heights. It also suggests that decreases in $R_{{\rho }_{\mathrm{sn}}}$ and $\Delta y$ may be related to a drop in the plasma charge density with height along the field-lines. As mentioned in Section 4.2, the plasma charge density changes as a function of $r^{-3}$ in a dipolar field structure, while $R_{{\rho }_{\mathrm{sn}}}\propto r^{-2.5}$ from our simulation. Hence, the change in $R_{{\rho }_{\mathrm{sn}}}$ with height is roughly similar to the change in ${\rho }_{\mathrm{sn}}$ along a dipolar field-line. This implies that changes in the emission state are related to the plasma density at the location where the changes take place, and the effects of the change are more prominent where the plasma’s density is higher. This is consistent with the suggestion that some observed emission shifts may be related to the emission height [12]. In general, the height along the trajectory of a visible point is lower for a low value of $\alpha$. However, since many radio pulsars possess a duty cycle of about 0.1 [49] or less [59,60], observable emissions from lower heights then require only the trajectory of the visible point to be close to the magnetic pole when traversing the open-field region. Such a condition is met for a low value of $\left|\beta \right|=|\zeta -\alpha |$. It then follows that an emission shift should occur in many pulsars, but the effects are more pronounced in pulsars with small $\beta$ values. In the case of PSR J0344−0901, the duty cycle is less than 0.1, and the values for $\beta$ and $\alpha$ are about $-3.17°$ and $75.12°$, respectively. PSRs B0919+06 and B1859+07 [11] also have profiles with duty cycles that are less than 0.1, and their values of $\beta$ are $5.1°$ and $4.8°$, but with smaller reported $\alpha$ values at ${53}^{\circ }$ and ${31}^{\circ }$, respectively.

5. Discussion and Conclusions

We have investigated the emission shift in PSR J0344−0901 and the accompanying changes in its emission properties based on published results obtained from observations using the FAST telescope. We first decomposed the non-shifted and shifted profiles using seven Gaussian components from a geometry in which the magnetic field was dipolar and the emission was emitted tangentially to the source point and parallel to the line of sight. We further assumed that the emission would appear shifted when the arrival phases of the Gaussian components were changed. The latter was modeled as a change in the emission state in a pulsar magnetosphere of multiple emission states, which led to a change in the electric drift and the plasma flow rate. Based on the arrangements of the Gaussian components that fitted the non-shifted and shifted profiles, we determined the necessary changes in the emission state for the shifts in the longitudinal phase at the peaks of the seven components. From this, the associated changes in the plasma charge density and the emission height were also evaluated. We found that the emission state was not uniform across the profile, which may have been the reason for the irregular drifting subpulses. We examined the implications of our results for the emission shift and concluded that the phenomenon was more readily detectable for pulsars with small impact parameters.

The similarity between the change in $R_{{\rho }_{\mathrm{sn}}}$ with height (due to a change in the emission state) and the change in ${\rho }_{\mathrm{sn}}$ with height (in a dipolar field structure) implies a causal relationship between the change in the charge density and its amount. Considering Goldreich and Julian [20]’s model, a plasma charge density (${\rho }_{\mathrm{sn}}={\rho }_{\mathrm{GJ}}$) is required to support corotation (${\omega }_{\mathrm{dr}}={\omega }_{\mathrm{GJ}}$) in the magnetosphere. However, corotation is not well justified [57]. For an oblique rotator, the screening of the parallel electric field (${\mathit{E}}_{\Vert }=0$) cannot occur everywhere, leading to the existence of a vacuum gap. Within this gap, ${\mathit{E}}_{\Vert }\ne 0$, and so the field-lines that pass through it are not equipotential. This leads to a potential difference that changes through the gap, and the frozen-in condition does not hold. In Ruderman and Sutherland [16]’s model, this leads to variation in the plasma flow across the gap, and the flow rate is different below and above the gap, with the latter ocurring at a lower angular speed than corotation. This implies that ${\omega }_{\mathrm{dr}}={\omega }_{\mathrm{cor}}$, or $y=0$ in our model, from below the gap to ${\omega }_{\mathrm{dr}}\ne {\omega }_{\mathrm{cor}}$, or $y>0$, above the gap. The changes in the y value, and the deviation from corotation, also imply corresponding changes in ${\rho }_{\mathrm{sn}}$ such that it is different from the Goldreich–Julian value across the gap. From Figure 4, the y values at components 1, 2, and 7 are similar in each of the two profiles, but the relative change in ${\rho }_{\mathrm{sn}}$ as a result of a change in the emission state is greater in the former two components, as shown in Figure 5, where the emission heights are lower. This demonstrates that for similar y values, a greater effect due to a change in the emission state comes from locations closer to corotation. Therefore, the state of corotation may serve as a ‘reference’ point, such that a change in the emission state will have a greater effect at locations closer to corotation. This is consistent with the radius-to-frequency mapping (RFM; [23]), and emissions at different heights will possess unique properties. This also implies that the effects of changes in the emission state that take place at different heights are likely unique. For emission shifts, they suggest a relationship between corotation and the effects of a change in the emission state, such that the latter is more prominent when occurring at locations closer to corotation.

The change in the profile associated with the emission shift exhibits small but noticeable differences in the longitudinal phase detected using the FAST telescope–currently the largest and most powerful radio telescope. The shift is also accompanied by detectable changes in the flux density and the width in the profile, suggesting a temporary change in the emission properties. This is similar to the swooshing event observed in PSRs B0919+06 and B1859+07, which manifests as a displacement of the pulses to an earlier phase before returning to their usual position. In addition to the analysis presented in this paper, there are a number of proposals for the origin of the event. One possible explanation is that the pulsar’s magnetosphere undergoes slight structural changes [61,62], akin to a “breathing” effect. The magnetosphere experiences periodic expansions and contractions [61], briefly shifting the emission region and producing the observed profile displacement. This idea aligns with our model for emission state switching, where variations in the plasma flow and charge density lead to abrupt changes in the emission properties. During a glitch-like event [63], the pulsar rotation undergoes a transient change, which could momentarily alter the emission beam relative to the line of sight. However, rotational irregularities tend to manifest as gradual changes in the pulse arrival times rather than sudden shifts in the profile structure. In addition, glitches typically induce long-term recovery as opposed to short-lived swooshing behavior. It is also possible that the pulsar encounters a strong gravitational interaction with a dense object, causing its emission beam to momentarily deflect, resulting in a shift in the observed profile [64]. Swooshing may also be linked to variations in the emission height and field-line distortions. It is suggested that changes in the observed profile’s width and shape are accompanied by changes in height [65]. When the field-lines are also distorted from being purely dipolar [66], a change in the emission height causes the emission to be directed along the field-line in a slightly different direction. This causes the pulse to arrive at a different longitudinal phase, resulting in the apparent emission shift. Other models suggest that magnetospheric reconfiguration, such as changes in the current flow, could lead to temporary shifts in the emission geometry. While these models provide plausible explanations, a definitive mechanism remains elusive. It is expected that studying this phenomenon in a large population would help refine our understanding.

We discuss a few assumptions made for our investigation of the emission shifts. An obvious limitation relates to the set of Gaussian components used to fit the profiles. Gaussian decomposition is inherently degenerate, causing the fit to not be unique. Here, we employ several constraints on the parameters for the decomposition. First, our analysis uses high-quality observational data from the FAST telescope–currently the largest and most sensitive radio telescope. This ensures that the profiles, and their measured widths and intensities, are precise, with a high signal-to-noise ratio. Second, the use of 20 emission spots in the emission region is based on observations [17,18]. From this, the number of Gaussian components within the profile width is determined through the conventional emission geometry using the obliquity and viewing angles obtained from the polarization details in the observations. Lastly, the goodness of the resulting fits is measured using a chi-square test to minimize the fitting errors. In our simulation, we found that the results do vary with slight changes in the number of components. While individual parameters exhibited some dependence on these choices, our results did not change significantly, and the overarching trends remained robust. Therefore, our approach to Gaussian decomposition is rigorous and well constrained, thus enhancing its reliability. We measure the change in the emission state in relation to the emission shift solely based on changes in the peak phases of the components without considering changes in the emission mechanism that may accompany the changes in the emission state. Pulsar radio emissions are thought to arise from ultra-relativistic plasma located along dipolar magnetic field-lines. This emission is confined within a narrow forward cone and slightly aberrated relative to the field-line [67]. Specifically, the emission from a given location is restricted to a cone aligned with the direction tangent to the field-line, and we disregard the size of this cone. Our simulation for the emission shift is based on $m_{\max }$ discrete areas located around the magnetic axis, and the emission comes from the last closed-field-line of the dipolar structure. The latter is consistent with the observed changes in the polarization position angle, which displays an almost S-shape curve (see Figure 6 in Tedila et al. [13]). However, the placement of the fiducial point near the leading edge, resulting in an asymmetric profile window, suggests that the dipolar structure may also be aberrated. A more accurate model should also include all of the terms in the magnetic field equation. When they are included, the field-lines are distorted, and the emission position is dependent on r and ${\varphi }_b$, which varies along a field-line. The number of pulsars with emission shifts (or swooshing) that have been detected in the past is low, thus limiting systematic and statistical studies of this phenomenon. This may be due to their low flux density, making it a challenge to observe them using small telescopes. For example, PSR J0344−0901 has been reported to have an average flux density of 3.5 mJy [13]. However, this situation has changed since the operation of the FAST telescope, as it allows pulses with a low flux density to be detected with high clarity. With the construction of other large telescopes underway, such as the QiTai radio Telescope (QTT; [68,69]) and the Square Kilometer Array (SKA), it is expected that more pulsars with emission shifts will be discovered, which will undoubtedly provide unparalleled information for resolving the mystery of pulsar radio emissions under changing environments.