Polarimetric Observations of PSR J0614+2229 and PSR J1938+2213 Using FAST

The single pulse stack of PSR J0614+2229 is shown in the top left panel of Figure 1, in which the pulses show significant pulse-to-pulse variations in both intensity and phase. We folded the data with the subintegration of 30 single pulses and the subintegration sequence and the energy of the on-pulse window are shown in the left and right panels of Figure 2, respectively. The pulse energy was calculated by summing the intensities within the on-pulse region after subtracting the baseline noise. We can clearly see that this pulsar exhibits two states with different emission phases. Following Mahajan et al. (2018), we take a quantitative metric Δχ_i to classify the two states:

$$\begin{eqnarray}&&{\rm{\Delta }}{\chi }_{i}^{2}=\sum _{\phi }\displaystyle \frac{{\left({p}_{i}(\phi )-{p}_{a}(\phi )\right)}^{2}-{\left({p}_{i}(\phi )-{p}_{b}(\phi )\right)}^{2}}{\sigma {\left(\phi \right)}^{2}},\end{eqnarray} \tag{ 1 }$$

where ϕ is phase bin, _pi (ϕ) is the intensity of the ith subintegrated pulse profile at phase ϕ, p_a (ϕ) and p_b (ϕ) are the intensities of the average pulse profiles at phase ϕ for two states and σ(ϕ) is the standard deviation from the mean of the entire data set. The mode metric Δχ² corresponding to subintegrated profiles are shown in Figure 3, which follows a bimodal distribution and the two Gaussian-like components are corresponding to state A and state B, respectively. We used Gaussian functions to fit these two components. The intersection of the two fitted Gaussian functions is about 47.6 (the vertical dashed line in Figure 3). If the Δχ² of a subintegration is smaller than 47.6, we classify that subintegration as being in state A; otherwise, it belongs to state B. We found this pulsar spends 62% of the time in state A and 38% of the time in state B. The duration of state A ranges from 5 to 54 subintegrations (50.2 to 542.7 s) and that of state B ranges from 14 to 30 subintegrations (140.7 to 301.5 s).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We measured an RM of 66.84 ± 0.02 rad m⁻² for PSR J0614+2229, which is more accurate than the psrcat value of 66.0 ± 0.3 rad m⁻² (Johnston et al. 2007). The Faraday rotation effect in our data has been corrected by using 66.84 rad m⁻². The polarization profiles of the two states are shown as the red (state A) and blue (state B) lines in subplot (a) of Figure 4. The peak flux density of state A is 204.945 mJy, which is slightly smaller than that of state B of 207.527 mJy. This is consistent with the results at 1400 MHz of Seymour et al. (2014). We estimated the fractional linear polarization by calculating the ratio between the mean linear polarization intensity 〈L〉 and the mean flux density S. The fractional linear polarization (〈L〉/S) for state A is 69.60%, which is smaller than that for state B of 85.32%. The fractional circular polarization (〈V〉/S) for state A is 19.28%, which is larger than that of state B of 13.49%. The absolute circular polarization fraction (〈∣V∣〉/S) for states A and B are 19.28% and 13.51%, respectively. The 〈L〉/S, 〈V〉/S, and 〈∣V∣〉/S for the average profile of PSR J0614+2229 are 75.17%, 17.16%, and 17.16% at 1250 MHz, respectively. By comparing the polarization results that obtained at about 325 and 610 MHz (Olszanski et al. 2019; Mitra et al. 2016), we find that PSR J0614+2229 shows decreasing fractional linear polarization and increasing fractional circular polarization with increasing frequency, like most of pulsars (Sobey et al. 2021). The PA swings of the average profile of the two states are also different. The profile of state A is slightly earlier than that of state B in longitude. The offset of the profile peaks between the two emission states is about 105 in our data. The profile of state A is slightly wider than that of state B. The profile widths of states A and B at the peak intensity levels of 10% are 1477 ± 035 and 1336 ± 035, while at the peak intensity level of 50%, the profile widths of states A and B are 703 ± 035 and 668 ± 035, respectively.

Zoom In Zoom Out Reset image size

As shown in Figure 2, the emissions of PSR J0614+2229 become much brighter during the transition between the two states. The red and blue lines in the right panel of Figure 2 indicate state A and state B, respectively. These brighter subintegrations are detected in both states. To further study these brighter subintegrations, we selected subintegration sequences belonging to each state whose durations are longer than four subintegrations. The first two and last two pulses of each sequence were folded to construct an average pulse profile for the mode transition phases, namely the AB-bright phase. The average polarization profile for the AB-bright phase is shown in the subplot (b) of Figure 4. Just as expected, the profile of the AB-bright phase is brighter than that of both state A and state B. The peak flux density of the AB-bright phase is 222.08 mJy, which is 17.13 mJy larger than that of state A and is 14.55 mJy larger than that of state B. The W10 of the AB-bright phase is 1406 ± 035, which falls in between that of state A and state B. The W50 of the AB-bright phase is 703 ± 035, which is equal to that of state A, but larger than that of state B. The 〈L〉/S, 〈V〉/S, and 〈∣V∣〉/S of the AB-bright phase are 78.60%, 15.47%, and 15.49%, respectively, which are all fall in between that of state A and state B. The PA swings of the state A, state B, and AB-bright phase are shown in Figure 5. The PA swings of state A (red line) and state B (blue line) exhibit different variations. The PA curve of the AB-bright phase (black line) lies between that of state A and state B, but closer to state B. Our results suggest that the emissions of the two states of PSR J0614+2229 may be generated in different regions in the pulsar magnetosphere. The AB-bright emission may be generated in the region that lies between state A and state B.

Zoom In Zoom Out Reset image size

The PA variation of a pulsar usually exhibits an S-shaped sweep through the profile, which has been reported in numerous earlier studies (e.g., von Hoensbroech & Xilouris 1997). In our observations, high signal-to-noise ratio profiles are obtained for states A and B. Using the PSRMODEL plug-in in the PSRCHIVE package, we fitted the PA swings for both states and found that the χ² value of the fitting of state B is smaller than that of state A. Therefore, we refitted the PA swing of state A with the same α and β values as state B, where α is the angle between the rotation axis and the magnetic axis and β is the angle between the rotation axis and the observers line of sight. We perform Markov Chain Monte Carlo fitting (Johnston & Kramer 2019) using the python package EMCEE (Foreman-Mackey et al. 2013) to determine the values for the steepest point phases (ϕ₀) and PA offset (ψ₀) of different states. The ϕ₀ of states A and B are ${287.31}_{-0.02}^{+0.02}$° and ${288.72}_{-0.01}^{+0.01}$° at the 16 and 84 percentiles of the distribution, respectively, which are significantly different. The fitting results of the two states are shown in subplot (a) of Figure 4, in which the red and blue curves represent state A and state B, respectively, and the fitted parameters are shown in Table 1.

Table 1. List of the PA-fitting Parameters of PSR J0614+2229

		α	β	ϕ0	ψ0
		(°)	(°)	(°)	(°)
J0614+2229	State A	44.84	−3.78	${287.31}_{-0.02}^{+0.02}$	${24.98}_{+0.09}^{-0.09}$
	State B	44.84	−3.78	${288.72}_{-0.01}^{+0.01}$	${28.00}_{+0.03}^{-0.04}$

Note. α is the angle between the rotation axis and the magnetic axis, β is the angle between the rotation axis and the observers line of sight, and ψ₀ and ϕ₀ are the PA and pulse phase of the steepest gradient point of S-shape curve, respectively. δ ϕ₀ is the phase difference between the steepest gradient points traversed by PA in different states.

Download table as:  ASCIITypeset image

The steepest gradient (SG) point of the PA traverse lies toward on the trailing edges of the profiles due to the aberration retardation effects (Blaskiewicz et al. 1991; Johnston & Weisberg 2006; Mitra & Rankin 2011). By considering the aberration retardation effects, Blaskiewicz et al. (1991) estimated the emission height of PSR J0614+2229 to be 350 ± 100 km at 1.4 GHz. Note that the model presented by Blaskiewicz et al. (1991) works well if there is a full S-shape swing of the PA and the SG is somewhere in the middle of a pulsar window (see, e.g., Rahaman et al. 2021). For PSR J0614+2229, the SG point is supposed to be at the trailing edge of the pulse window and the S-shape swing is not full. Therefore, the emission heights may be not constrained well. However, considering the signal-to-noise ratio of PSR J0614+2229 in our observation is much larger than that of Blaskiewicz et al. (1991), we try to reestimate the emission heights of different states of PSR J0614+2229. The emission height difference between the two states of the pulsar can be estimated from the phase difference (δ ϕ₀) between the SG point of their PA traverses (Blaskiewicz et al. 1991): Δh_em = Pc δ ϕ₀/8π, where c is the speed of the light and P is the rotation period of a pulsar. Using the parameters in Table 1, we obtained that the derived altitude difference between states A and B for PSR J0614+2229 is ∼90 km. Note that there may be a large uncertainty in the value.

Pulsar radiation is characterized by short timescales of about several microseconds to a few hundred microseconds (Craft et al. 1968; Hankins 1972), called microstructures. The microstructure is an important characteristic of pulsar radiation, which is closely related to the emission process (Buschauer & Benford 1980; Lange et al. 1998). Therefore, it can promote or restrict the emission model. Microstructures have not been reported in PSR J0614+2229 up to now. In our observations, the microstructures of single pulses for this pulsar are detected. According to Johnston et al. (2001), the R parameter can be used for the identification of giant micropulse, in which ${R}_{i}=({\mathrm{MAX}}_{i}-{m}_{i})/{\sigma }_{i}$, where MAX_i , σ_i , and m_i are the maximum intensity, the rms, and the mean intensity in the ith bin, respectively. We identified three pulses with R > 20, and the maximum value of R is 21.77. These three single pulses are shown in the upper panel of Figure 6. The brightest of these pulses (the red line) has a peak flux density of 1659.85 mJy, which is 8.32 times stronger than that of the mean pulse profile, and 19.73 times stronger than the mean flux density at that specific pulse phase. We performed an analysis of the microstructure emission timescales (P_μ ) using the method of Mitra et al. (2015). The estimated P_μ of the three brightest giant micropulses of PSR J0614+2229 are ∼2.0, 2.9, and 2.9 ms using a smoothing bandwidth 0.075.

Zoom In Zoom Out Reset image size

In the right subplot of Figure 1, we showed the single pulse stack and average pulse profile of PSR J1938+2213. It is noticeable that the emission of the pulsar consists of a weak state superposed by brighter bursts, and the weak emission is always present, which is consistent with the results of Lorimer et al. (2013). This suggests that the emissions for the two states may be generated in different regions in the pulsar magnetosphere. These bright pulses show dramatic variations, where the pulse energy can vary up to 57 times larger than the average pulse energy, and the peak flux density of the burst state is 857.8 mJy, which is about 41 times larger than that of the mean pulse profile. The pulse energy is calculated by summing the intensities within the on-pulse window after subtracting the baseline noise.

We named these bright single pulses the burst state. To identify whether the burst state exists in any particular pulse, we used the same method as in Section 3.1.1. If the Δχ² of an individual pulse is smaller than 410 (the vertical black dashed line in Figure 7), we classify it as the burst state. We found that 271 pulses in PSR J1938+2213 were in burst state, with the duration ranging from 1 to 63 spin periods (0.166 to 379.5 s).

Zoom In Zoom Out Reset image size

Using the RMFIT program of PSRCHIVE, the RM we measured for this pulsar is 142.67 ± 0.07 rad m⁻² and the Faraday rotation effect has been corrected in our data. The polarization profiles of the two states are shown in Figure 8. The average profile of the weak state (blue line) has two main separated components in which the second component shows two peaks and is much stronger than the first component. The average profile of the burst state (red line) has three main components (separated by the vertical dashed lines in Figure 8) with a stronger central component. The peak flux density of the burst state is about 56 times that of the weak state. The polarization properties of the two states are also different. The 〈L〉/S of the weak state is about 91.33%, which is much larger than that of the burst state of 46.80%. The 〈V〉/S of the weak state is 11.19%, which is also larger than that of the burst state of 1.33%.

Zoom In Zoom Out Reset image size

The PA swings of the two states of PSR J1938+2213 show remarkable differences (the red and blue dots with error bars in the upper panel of Figure 8). The PA swing of the weak state profile shows a smooth variation. The PA swings of the first and third components of the burst state profile basically overlap with the PA curve of the weak state, while PA of the second component shows a sudden jump of about 90^◦ (labeled by the vertical dashed lines in Figure 8), known as the orthogonal polarization mode (OPM) phenomenon. Surprisingly, the OPM is not present in the weak state.

We then separated the OPMs for the average profile of the burst state. We assumed that the observed Stokes I, Q, U, and V are the incoherent sum of the 100% polarized OPMs. The two orthogonal modes should have opposite senses of circular polarization. Using the method presented by Karastergiou et al. (2003), we decomposed the average profile of burst state and the mode-separated pulses profiles are shown in Figure 9, in which the positive values of V are associated with Mode 1, and negative V with Mode 2. These two orthogonal polarization modes have PAs separated by 90^◦ and each mode is almost completely linearly polarized.

Zoom In Zoom Out Reset image size

The microstructures of single pulses were also detected in PSR J1938+2213. As presented in Section 3.1.2, R is an important parameter for the identification of giant micropulse. We estimated the value of R for PSR J1938+2213, and the measured maximum R is 39.45. There are 41 pulses with R > 25. Three individual pulses with R > 37 are shown in the lower panel of Figure 6. The peak flux density of the brightest pulses (the red line) is 2372.54 mJy, whose peak flux density is about 115 times larger than that of the average pulse at the specific pulse phase. The P_μ of the three brightest giant micropulses of PSR J1938+2213 are ∼1.6, 2.6, and 2.9 ms. As suggested by Mitra et al. (2015), P_μ increases with increasing pulsar spin period, although a large spread of P_μ values exists for all pulsars. Our results basically agree with the conclusion of Mitra et al. (2015).