Profile Variation in PSR B0355+54 over a Narrow Frequency Range

Abstract

We investigate changes in the shape of the averaged pulse profile in PSR B0355+54 (PSR J0358+5413) based on data obtained at the center frequency of 1250 MHz using the Five-hundred-meter Aperture Spherical radio Telescope (FAST). Our dataset consists of 12 non-consecutive observations, each lasting between 1 and 2 h. Considerable variation is observed in the averaged profiles across the observations even though each is folded from thousands of single pulses. Changes in the profile are measured through the ratio (R) between the peak intensities of the leading and trailing components. We find that the averaged pulse profile exhibits significant variation across observations, but distinctive from typical profile mode-changing. By dividing the frequency bandwidth into eight sub-bands, we demonstrate that the shape of the averaged profile undergoes significant evolution with frequency. In general, the changes in R across the sub-bands are different in different observations, but its value is uniform at low frequencies implying a more consistent emission. We demonstrate that the profile stabilization timescale for this pulsar is much longer than commonly suggested for ordinary pulsars, which is likely due to non-uniform and varying arrangement of the emission sources in the emission region.

1. Introduction

A well-known approach for the investigation of pulsar emissions involves folding a large amount of single pulses, which exhibit varying intensity, shape, and the arrival phase [1]. The result is an averaged pulse profile, which serves an important purpose of revealing the distinctive radio emission properties and polarization details specific to the pulsar. Averaged profiles are also key to understanding the radiation processes and the associated emission geometry in pulsar magnetospheres [2,3,4,5]. Owing to the stable shape and specific features, the averaged profiles are sometimes referred to as the fingerprint of pulsars. In general, such stability in shape can be achieved with an accumulation of around 500 single pulses for an ordinary pulsar [6]. However, Helfand et al. [2] show that the timescale for obtaining a stable averaged profile may vary, and it depends on the emission features of the pulsar. While some pulsars show rapid convergence to a stable profile, such as PSR B1919+21, which needs only about 150 single pulses, others may require an accumulation of up to (and more than) 5000 single pulses, as in the case of PSR B1237+25 [2]. One such emission feature that could affect the stabilization timescale is known as the profile mode-changing, which is observed in some pulsars. During the event, the averaged profile exhibits sudden changes between two (or more) quasi-stable shapes [7,8]. A typical example is PSR B0329+54, whose profile shape exhibits changes between two modes of normal and abnormal emissions [9,10,11]. The changes in the profile can be measured through changes in the relative peak intensity between different components. Another example relates to PSR B1322−66, whose averaged profile undergoes changes at 1369 MHz, with the emission from the center and the tailing components being extremely weak in the abnormal mode [12]. It is apparent that the required information for determining whether the structure of an averaged pulse profile is stable or not is already present in the magnetosphere of the pulsar. However, such information is unknown, and a widely accepted model for changes in the averaged profiles is still lacking.

Previous multi-frequency observations revealed that the shape of some averaged profiles also exhibits evolution with the observing frequencies [13,14,15,16,17,18,19]. For example, the profile width of most pulsars becomes narrower as the observing frequency increases [20,21,22]. The phenomenon can be explained based on the broadening of the field-line separation as the height increases in a dipolar field structure. Thus, the profile width increases as the height increases, corresponding to a decrease in the frequency, leading to the suggestion of radius-to-frequency mapping (RFM; Manchester & Taylor [6], Cordes [23], Gil & Kijak [24], Phillips [25]). The findings indicate that W10 (pulse width at 10% of peak energy intensity) is significantly influenced by pulse shape evolution. In addition, the intensity of certain profile components may diminish or disappear at high frequencies [26]. Usually, these phenomena are more prominent as the separation between the observing frequencies is larger. The study of the change in the profile components with frequency, such as variations in the width of each profile component and the spacing between components, revealed previously unrecognized emission features.

This paper investigates the changes in the averaged profile of PSR B0355+54 (PSR J0358+5413) across the frequency range of 1.05–1.45 GHz using the latest data from the Five-hundred-meter Aperture Spherical radio Telescope (FAST). The pulsar1 possesses a rotation period of 156.4 milliseconds, a magnetic field strength of $8.39\times {10}^{11}$ G, and a characteristic age of 546 kyr [27]. Its emission is detectable across different frequency bands, encompassing radio and X-ray domains. The pulse profile of PSR B0355+54 manifests as a single-peak structure at frequencies below about 1.0 GHz, and evolves to a double-peak configuration between 1.0 and 4.8 GHz frequency range, and reverts to a single-peak structure for frequencies above 4.8 GHz. The paper is organized as follows. Section 2 introduces the data source and the data processing. In Section 3, we present our results, and our discussion and conclusions are provided in Section 4.

2. The Observations and Data Processing

The FAST project is a significant scientific endeavor in China, with the aim of constructing the world’s largest single-dish radio telescope. It utilizes the pits in the Karst Region of Guizhou as the telescope site, and is equipped with highly sensitive radio instruments. The telescope has an effective aperture of 300 m2. The observations were conducted using the center beam of the 19-beam FAST receiver at frequency between 1.0 and 1.5 GHz. The data were recorded using the pulsar backend, which has bilinear polarization (XX and YY) and relatively high temporal resolution of ∼49.152 µs. The FAST data used in this study consist of 12 observation sessions, each lasting between 1 and 2 h, giving a total observing time of approximately 19 h. Details of the observations are given in

Table 1. Details of the 12 observations. The second column represents the date of observation in Modified Julian Date (MJD), the third column gives the observation duration, the fourth column contains the number of single pulses (NSP), and the fifth and sixth columns outline the frequency bandwidth of the observation and the telescope used, respectively.

No.	MJD	Duration (s)	NSP	Bandwidth (GHz)	Telescope
1	58650	5374.184	34412	1.05–1.45	FAST
2	59363	3536.422	22608
3	59371	3499.226	22347
4	59373	3471.105	22194
5	59414	7071.104	45310
6	59415	7071.104	45310
7	59436	7071.104	45310
8	59508	7054.352	45192
9	59870	3471.105	22194
10	59881	7071.104	45310
11	59883	7071.104	45310
12	59981	7071.104	45310

.

The data were recorded in the PSRFITS format [28]. We used DSPSR [29] to fold the pulses to generate 1024 phase bins for each pulse profile. The frequency ranges from 1000 to 1500 MHz, with 500 MHz bandwidth and divided into 8192 channels. Due to the inability to remove the gain of radio interference at both ends of the bandwidth, it was necessary to assign a weight of zero to the 10% frequency channels at each end of the frequency bandwidth. This gives a practical frequency range from 1050 to 1450 MHz [30,31]. After that, we used the pazi and paz commands provided by the PSRCHIVE [28] software ([jiang@mu01] psrchive—version PSRCHIVE 2022-01-14(2022-04-04 005306c)) package to remove the radio interference in the time and the frequency domains. With FAST’s large aperture, we have obtained observations of good quality. Subsequent data processing and analysis was performed using the Python and Shell scripting languages.

3. Results

3.1. Changes in the Averaged Pulse Profile in Time

Since all of the profiles possess two apparent peaks, we divide each profile into two parts from the point of minimum intensity of the bridge emission. We refer to the left part located at earlier longitudinal phases as the leading component, and the right part located at later longitudinal phases as the trailing component. We normalize each of 12 averaged pulse profiles using the peak of the trailing component, and they are shown with different colors in the bottom row in Figure 1, Figure 2 and Figure 3. Examination of the averaged profiles from the four observations in Figure 1 reveals that they vary across the observations. On MJD 59371, the leading component of the profile exhibits the greatest relative intensity. In contrast, the leading components of the profiles on MJDs 59373 and 58650 have nearly similar intensity, while the relative intensity of the leading component of the profile on MJD 59363 is the weakest. In Figure 2, the relative intensity of the profile components varies across the observations. On MJD 59508, the leading component of the profile has the highest relative intensity, approaching that of the trailing component. Additionally, an extra structure is visible between the two profile components on MJD 59508, which is not present in the other profiles. In Figure 3, although variation across the averaged profiles is not as significant, differences in the relative intensity of the profile components are still observable. Specifically, the leading component of the profile on MJD 59981 has a weaker relative intensity than the averaged profile from the other observations in the figure.

For a better illustration of the differences in the averaged pulse profiles from the different observations, the normalized profiles from the 12 observations are plotted together in Figure 4. As demonstrated in the right subplot of the figure, the relative intensity of the leading component fluctuates significantly across different profiles, with the lowest and highest values at approximately 0.75 and >0.93, respectively, giving a change of about 24%. In addition, the minimum intensity between the two components also varies across the profiles, ranging from 0.49 to 0.60. Furthermore, the profile obtained on MJD 59508 has widths measured at 50% and 25% of the peak intensity that are significantly wider than that of the other profiles. It is evident from Figure 4 that the shape of the averaged profiles varies in time.

We compare changes in the averaged profiles based on differences in the relative intensity between the leading and trailing components. A ratio, R, is calculated by dividing the peak intensity of the leading component by that of the trailing component for an observation. The variation in R is shown in Figure 5. It demonstrates that changes in the value of R range from approximately 0.745 (on MJD 59363) to 0.925 (on MJD 59508). This result further confirms the instability of the profile shape across different observations.

3.2. Changes in the Averaged Pulse Profile in Frequency

In this section, we examine the intensity variation in the two components of the normalized averaged profile obtained from different frequency sub-bands.

We divided the frequency band from each observation into eight sub-bands, as shown in

Table 2. Details of the frequency sub-bands. Each sub-band has a bandwidth of 50 MHz, and the center frequency of each is given in column 4.

Number	Bands (MHz)	Bandwidth (MHz)	Center Frequency
1	1050–1100	50	1075
2	1100–1150	50	1125
3	1150–1200	50	1175
4	1200–1250	50	1225
5	1250–1300	50	1275
6	1300–1350	50	1325
7	1350–1400	50	1375
8	1400–1450	50	1425

. We then extracted the data from the frequency sub-bands and generated a normalized averaged pulse profile for each. This gives a total of eight profiles for each observation, and they are shown in the upper eight rows in Figure 1, Figure 2 and Figure 3. In general, the relative peak intensity of the leading component is lower than that of the trailing component at low frequencies for all observations. However, it changes as frequency increases, and the pattern of change can be different for different observations. For the profiles on MJDs 59363, 59371, and 59373 in Figure 1, the relative peak intensity of the leading component increases as the central frequency of the sub-bands increases, and eventually exceeds that of the trailing component at higher frequencies. Similar pattern is also seen in majority of the observations, such as those on MJDs 59414, 59415, and 59436 in Figure 2, and for the four observations shown in Figure 3. This is different for observation on MJD 58650. Here, the relative peak intensity of the leading component also increases as the frequency increases, reaching that of the trailing component, but then gradually decreases as frequency increases. Another different pattern is seen from observation on MJD 59508, where the peak intensity of the leading component exhibits a rapid increase, reaching that of the trailing component at around 1275 MHz, and remains nearly unchanged with increasing frequency. Our results show that, in general, a small change in frequency could lead to significant variation in the relative intensity between the leading and trailing components, and hence a change in the profile shape.

The changes in intensity between the two component peaks across different frequency sub-bands can also be measured using the R ratio described in Section 3.1. Figure 6 presents the variation in R calculated at the center frequency of each sub-band for the 12 observations. As previously elucidated, except for observations on MJDs 58650 and 59508, the profiles across the sub-bands exhibit tendency of similar variations manifesting in the R value as a gradual increase from about 0.6 to 1 or higher. Among them, the maximum value of R reaches about 1.36 at 1425 MHz, which occurs on MJD 59414, and the minimum is about 0.42 on MJD 58650. It also means that the same R value will appear at different sub-bands in different observations. This is illustrated with the two horizontal lines, at $R=0.65$ and $R=1.12$, which cut the curves at different frequencies. In addition, Figure 6 clearly emphasizes the distinct variations in R on MJDs 58650 (blue curve) and 59508 (gray curve). Observation on MJD 58650 shows that the R value increases rapidly from about 0.43 to about 1.02, and then decreases to about 0.62, with the maximum change of 137%. On the other hand, the change in the R value as revealed by the observation on MJD 59508 exhibits a rapid increase from about 0.6 to about 1.03 and shows only small fluctuations thereafter, with a maximum change of 72%.

Figure 6 shows the value of the intensity ratio R at the same frequency also varies across different observations, but they fall within specific ranges. This is confirmed with Figure 7, which demonstrates the variation in the R value for three different sub-bands. The variance calculated for the three sets of data indicate that Data 1 (1075 MHz, green curve) and Data 2 (1275 MHz, orange curve) exhibit relatively low variance, as shown in

Table 3. Different values of the variance and standard deviation for data from sub-bands at 1075 MHz, 1275 MHz and 1425 MHz, signified by Data 1, Data 2, and Data 3, respectively.

Date Number	Freq	Variance	Std Deviation
Data 1	1075 MHz	0.00162	0.04025
Data 2	1275 MHz	0.00178	0.04222
Data 3	1425 MHz	0.02414	0.15538

. This means that their variations are close to the average R value of the respective data, and there is little spread in the data. However, Data 3 (1425 MHz, blue curve) demonstrates greater dispersion, with the maximum and minimum variance differing by more than ten times. This indicates that the R values in the dataset tend to spread out over a wider range and they are relatively far from the average value.

4. Discussion

It is well known that pulsars’ single pulses are highly variable in shape and intensity. Therefore, the averaged pulse profile can generally be used as a significant and unique tool that allows us to distinguish one pulsar from another. For most pulsars, there exists a specific timescale over which averaging the single pulses results in a stable profile shape. Typically, this requires an accumulation of several hundred single pulses [6]. From Table 1, the number of single pulses from any one of the observations is more than twenty thousand. Even with this amount of single pulses, the resulting averaged profiles from the 12 observations remain unstable. For example, the observations indicated by no. 5–7 and 10–12 contain the greatest number of single pulses, yet the shape of the resulting averaged profiles are still different from each other, as shown in Figure 4 and Figure 5. It follows that the profile stability timescale for this pulsar is likely longer than the longest duration of the 12 observations. We also showed that the shape of the averaged profile changes with time, which is reminiscent of another common phenomenon known as the profile mode-changing. The phenomenon refers to the discontinuous, but reversible, switching between two or more quasi-steady states in the profile shape [8]. The first confirmation of this phenomenon came from the observations of the pulsar PSR B1237+25 [7]. Typically, these changes occur within one rotation period, and the new profile persists for hundreds of periods before returning to the original shape or switching to another shape. Based on our analysis in Section 3, there are significant differences in the profile component intensity across the sub-bands for 10 of the 12 observations, even though such differences tend to show a similar trend. The remaining two observations (MJDs 58650 and 59508) give profiles that exhibit changes in completely different properties. This implies that, if interpreting the profile variation of this pulsar in terms of profile mode-changing, we are unable to identify a specific profile mode or determine the timescale of the mode change. This is consistent with Morris et al. [21], who reported that the changes in the average pulse shape at 2650 MHz were different from conventional profile mode-changing. Therefore, it is reasonable to assume that more single pulses are needed to obtain a stable averaged pulse profile for PSR B0355+54.

PSR B0355+54 resembles the millisecond pulsar PSR J1022+1001 [32], and several other millisecond pulsars [33,34], in the way that they all show similar unusual profile changes. Usually, significant changes in the averaged profiles occur over a wider range of frequency bands, from a few hundred to a thousand megahertz [35]. Variation in the profile across a narrow frequency band suggests a rather active emission region. However, millisecond pulsars are older pulsars and tend to show high level of stability in their profiles because of the fast rotation rate gained from the recycling process [36,37,38]. On the contrary, the magnetospheres in young ordinary pulsars are believed to be relatively more active and, hence, these pulsars should display greater diversity in the changes in their emission features. However, the changes in profiles observed in PSR B0355+54 are not reported in other ordinary pulsars, but more in millisecond pulsars. This may indicate that such profile change is not related to the pulsar age or the evolutionary path, but it is intrinsic to the pulsar.

Previous extensive multi-frequency observations have revealed the evolution of averaged profiles with frequency in many pulsars [13,16,17,18,39,40]. Chen & Wang [13] studied the profile width of 150 pulsars, and found that most of them vary in accordance with the RFM. Among them, an investigation of the change in W10 has been conducted for B0355+54 within the 0.92 to 10.7 GHz frequency range. Here, we identify the profile-frequency evolution in PSR B0355+54 over a narrow frequency band. For most pulsars, the profile evolution with frequency is coherent regardless of the observations. For our pulsar, the profile shows a significant evolution in the relative intensity between the two profile components across different frequencies and at different times. A commonly used model for pulsar emission, derived from drifting subpulses, which exhibit as a systematic movement of subpulses across the profile window [41,42,43,44,45,46], involves discrete emission sub-beams placed evenly on a carousel that rotates about the magnetic axis. The motion of the carousel, under the $E\times B$ drift [42,47,48,49], relative to co-rotation causes the sub-beams to rotate through the fixed line of sight. Given that the observed emission features are closely related to the emission properties in the emission region [50], the different shapes of an averaged profile may be considered as an indication for changes in the underlying arrangement of the emission sub-beams. In our analysis, a profile shape possesses a particular R value, and hence the changes in the R value across the frequency sub-bands in one observation signify changes in the corresponding subbeam arrangement. From Figure 6, the R value is relatively stable at 1075 MHz, but its variation increases as frequency increases, suggesting that the changes in the R value is dependent on frequency. For higher frequencies, the shape of the pulse profile in the same frequency sub-band is unstable over time even though most of the observed profile changes seem to follow a similar pattern, as discussed in Section 3.2, except for the observations on MJDs 58650 and 59508. According to the RFM, pulses radiating in the same frequency band originate from similar height, with similar emission characteristics. Hence, the detection of a similar profile shape (similar R) in different frequency sub-bands at different MJDs implies that the particular emission arrangement, which gives rise to the profile shape, ‘shifts’ across frequency over time. This also indicates that the arrangement of the emission sub-beams on a carousel in the emission region is not fixed, but dependent on time. However, the profile variation in PSR B0355+54 is different from drifting subpulses in that the locations of the emission sub-beams on a carousel are fixed in the latter but they change in the former.

The degree of variation in the averaged profiles can be determined based on the variance in the intensity. For a longitudinal phase, it is evaluated from the spread of the intensity over the eight frequency sub-bands, and the calculation is repeated for each phase across the profile. The result is shown in Figure 8. In each of the subplots, the blue curve represents the variation in the normalized variance across the averaged pulse profile (yellow). The variance represents the degree of dispersion in the data, with a larger variance indicating more pronounced variation in the intensity over the frequency sub-bands for a given phase. All 12 observations in Figure 8 demonstrate variation in the intensity at different phases. Furthermore, none of the variance curves are the same, which implies that the variation pattern with frequency is not consistent across the observations. Additionally, the variance curves in Figure 8 appear to have two main components, indicating that the emission at different phases varies inconsistently with frequency and changes over time.

5. Conclusions

We have demonstrated the instability in the averaged pulse profile of PSR B0355+54 using observational data collected by the FAST telescope. The number of single pulses for each of the 12 observations ranges from 22,194 to 45,310. We found that the shape of the averaged pulse profiles for the 12 observations are not consistent and they vary across different observations performed on different MJDs. We measured the variation in the relative intensity between the peaks of the leading and trailing components, denoted by the R value, across the 12 observations and show that they exhibit instability. Furthermore, the changes in R appear random, which suggest that there is no quasi-stable state for the profile. In addition, we divided the frequency bandwidth (1.05–1.45 GHz) into eight equal sub-bands and demonstrated that the relative intensity of the two components exhibited significant changes with frequency. We found that the profile is more stable in the lower frequency sub-bands compared to that in the higher frequency sub-bands. Furthermore, we showed that the R value changes significantly as frequency increases, and the pattern of this change varies across different observations. We have shown that the profile change is similar to, but different from, the characteristics of a typical profile mode-changing. This indicates that a stable profile does not exist, and the stability timescale is not apparent for this pulsar.

A related question concerns how common it is for pulsars without a stable averaged profile. With only a handful of such candidates identified among more than 3000 known pulsars, it is likely that these pulsars do not exist in large numbers. However, their existence challenges the conventional approach to investigate pulsar emission and highlights the deficiency in our current understanding of radio pulsars. An important initial tactic to decode the properties of pulsar emission involves constructing an averaged profile, from which subsequent analyses can be performed. An example regards the conduction of pulsar timing and its related disciplines for determining the pulsar properties and the surrounding medium. The approach is fruitful, which has led to many discoveries. It is based on the assumption that a stable profile can be obtained for each pulsar within an ‘adequate’ timescale. Pulsars with unstable profiles make it difficult to achieve precise pulsar timing and many of its applications. Conversely, the phenomenon offers a unique opportunity to study the complex and variable emission conditions in the magnetosphere. Quantifying the changes in the profile across the frequency bandwidth can reveal the emission arrangement and the variation of it across the emission region in three-dimensional representation. To do that would require more observations at regular time interval for establishing a picture of a continuously changing distribution of the sub-beams. The results will provide additional information to modify the traditional models, which currently do not distinguish pulsars with or without a stable profile. It is clear that, while unstable profiles pose challenges, they also open up new avenues for radio pulsar research which will lead to a better understanding of the radio emission mechanism.