The Mode Switching in Pulsar J1326–6700

With the benefit of Parkes high sensitivity, PSR J1326−6700 is bright enough to study its single pulse sequences, and a color-coded pulse sequence displays several of the pulsar's emission behaviors in the top-left panel of Figure 1. An intriguing combination of nulling and mode changing is clearly shown. The pulses with no detectable emission from central and trailing components are clearly visible. During the partial nulls, an obvious abnormal mode emission appears to flicker on at the leading edge the profile for typically less than a minute. The sporadic emission in the abnormal mode is frequently separated by short nulls. On several occasions, the two modes are separated by a complete null interval where the coherent radio emission in the whole pulse period ceases.

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In order to characterize the pulse nulling properties statistically, the pulse energy distributions for the on-pulse region and off-pulse region with similar lengths, after a normalization by the mean pulse energy, are presented in Figure 2. The off-pulse energy histogram, centering around zeros, represents a Gaussian random noise contributed by the telescope noise. While the on-pulse energy distribution shows the presence of two distinct regions, which corresponds to bistable emission modes. The two peaks as determined from a fit with the sum of two normal distributions are obviously larger than zero, which implies that the apparent nulls are in an abnormal emission mode rather than real null pulses.

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The single pulse energy distribution indicates the existence of mode switching in PSR J1326−6700, with the pulse profile switching on timescales of seconds between two quasistable modes, which differ in intensity. In order to determine mode switching properties such as the timescale and polarization, a quantitative metric is defined to distinguish whether an individual pulse belongs to the normal or the abnormal emission mode, which is given by Mahajan et al. (2018)

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

where _pi(ϕ) is an individual pulse profile, p_n(ϕ) and p_a(ϕ) are the average pulse profiles for normal and abnormal modes, respectively, and σ(ϕ) is the standard deviation from the mean of the entire data set. The mode metric Δχ² corresponding to individual pulses are indicated in the top right panel of Figure 1.

A histogram of the mode metric for our whole data set of ∼44,000 pulses is shown in the bottom right panel of Figure 1, which clearly shows a bi-modal distribution. We fit the histogram with a sum of two normal distributions, finding centers for the normal and abnormal mode of Δχ² = 0.442 ± 0.003 and −0.238 ± 0.009, respectively, and associated widths of ${\sigma }_{{\rm{\Delta }}{\chi }^{2}}=0.315\pm 0.004$ and 0.138 ± 0.009. The Δχ² distribution of the normal mode is considerably wider than that of the abnormal mode, which may indicate that the abnormal mode is more stable than the normal mode, and the timescale for the normal mode to become stable is longer than that for the abnormal mode.

It is noted that some contamination by mode transitions is clearly presented, and it is not always obvious to which mode an individual pulse belongs due to low signal-to-noise ratio (S/N). As suggested, the durations of transitions do not necessarily represent the true time spans for which mode changes occur, a Δχ² threshold is determined as the center of the inner 1σ boundaries of normal and abnormal modes. Thus, all of the individual pulses are associated with either the normal or the abnormal mode.

The average pulse profiles for the normal and abnormal modes are shown in Figure 3, where the pulse peaks are normalized to unity. A combination of three Gaussian components is adopted to fit the observed profiles. The best-fitted parameters determined using the Levenberg-Marquardt algorithm (Press et al. 1992) are presented in Table 2 with R-square of 0.996 and 0.998 for the normal and abnormal modes, respectively. This demonstrates the existence of three emission components in the abnormal mode. The relative amplitudes of the central and trailing components decrease with respect to the leading component, and the three components shift earlier in pulse phase.

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In our observations, PSR J1326−6700 spends  85% of the time in the normal mode, and  15% in the abnormal mode. The probability density functions (PDFs) for the lengths of the normal and abnormal modes taken from all observations together are presented in the upper and lower panels of Figure 4, respectively. They show that the occurrence of both the modes decreases at longer timescales. The PDFs of the two modes are well fitted by a Weibull distribution:

$$\begin{eqnarray}&&P({\rm{\Delta }}t)=\displaystyle \frac{k}{\lambda }{\left(\displaystyle \frac{{\rm{\Delta }}t-\theta }{\lambda }\right)}^{k-1}{e}^{-{\left(\tfrac{{\rm{\Delta }}t-\theta }{\lambda }\right)}^{k}},\end{eqnarray} \tag{ 2 }$$

where k is the shape parameter, λ is the scale parameter, and θ is the location parameter. Considering the asymmetric distributions, the fittings are performed on unbinned data by maximum likelihood estimation. The best-fitting coefficients are estimated to be k = 0.79 ± 0.03, λ = 67 ± 4, θ = 6.6 ± 0.5 and R-square = 0.991 for the normal mode, and k = 0.81 ± 0.03, λ = 11.8 ± 0.7, θ = 2.81 ± 0.04 and R-square = 0.998 for the abnormal mode. To validate the goodness-of-fit of the Weibull distribution fits, a one-sample nonparametric Anderson–Darling test is carried out. The durations of the normal and abnormal emission modes are both drawn from Weibull distributions at the 75% confidence level. For both normal and abnormal modes, the Weibull distribution is applied to all observations. The same parameters are obtained at the lower significance level, implying that the underlying physical processes that produce the moding are fixed and unchanging.

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The Weibull distribution is commonly used to analyze life data. A k < 1 implies that the probability of a mode change occurring decreases with time, the shorter the pulsar is in a mode. If k = 1, an exponential distribution is produced, which indicates that the probability of a mode change occurring is time-invariant. If k > 1, the occurrance of mode changing increases with time. The best-fitted power-law distributions for both modes are shown in Figure 4 as well, which do not fit the observed duration distributions well. For PSR B0919+06, the abnormal emission events occur randomly about every 1000–3000 periods (Han et al. 2016), which is much longer than PSR J1326−6700.

In order to understand the emission properties further, sets of normal and abnormal integrated polarimetric profiles are constructed from 2012 January 15, 2014 May 25, and 2014 May 30 observations, and shown in Figure 5. The bottom panels present the total intensity (black solid curves), total linear polarization (red dashed lines) and circular polarization (blue dotted lines). The upper panels give the linear polarized position angle (hereafter PA) histograms for all single pulses, as well as the average PA traverses for normal and abnormal modes. To limit the effects of measurement uncertainties, the PAs for each single pulse are estimated at longitudes where the linear polarization power is more than 6 times the noise rms in the off pulse region.

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The asymmetric total power profile of the normal mode has three main emission components, with the trailing one being the strongest. During the abnormal emission state, however, the mean pulse profile is very different in amplitude and pulse phase. The emission from the normal profile ceases and the abnormal emission emerges at a new leading component. The abnormal profile possesses a single emission component with sharp leading and gradual trailing edges, and the profile position shifts markedly earlier. The mean flux density of the normal mode is around 4 times brighter than that of the abnormal mode. Table 3 lists the mean flux density, W₅₀ (width of pulse at 50% of peak) and W₁₀ (width of pulse at 10% of peak) of both normal and abnormal modes. Note that the strong linear polarization under the trailing feature simply disappears during the abnormal mode. Significant right-circular polarization is observed in the center of the normal profile. The abnormal mode shows a marginal circular polarization over the whole profile.

The PA exhibits a fast swing across the profile, and an "S"-shaped sweep is presented, which may indicate that we are seeing the whole conal beam. Then the line-of-sight geometry can be assessed by fitting the rotating vector model (RVM; Radhakrishnan & Cooke 1969) to the PA traverse. It is noted that for both emission modes a jump of around 60° is shown in the PA under the leading component, which is also associated with a substantial dip in the linear polarization. For the normal mode, a bimodal distribution of PAs separated by around 80° predominates in the trailing emission region as well. The two preferred PAs at the leading and trailing components manifest the presence of two orthogonal polarization states. The classical RVM curve could be severely corrupted by the variations in strength between the two modes. In order to refrain the modal effects, only the pulse longitudes where the average PA traverse is in good agreement with the single pulse PA in the central component are considered. The best fits to the RVM are shown as red curves in the upper panels of Figure 5. The viewing geometries for normal and abnormal modes are specified with the values of magnetic inclination angle (α), impact angle (β), position angle offset (ψ₀), and fiducial plane angle (ϕ₀), which are listed in Table 3. As is shown, the quantities of ϕ₀ and ψ₀ are significantly constrained. However, the α and β values obtained from these fits are extremely covariant and unreliable due to the limited duty cycle of the profile (Mitra & Li 2004). For instance, Figure 6 shows the reduced χ² values of the fit as a function of α and β for the normal mode. Here, the best fit occurs where χ² reaches a minimum. Evidently, there are a number of combinations of α and β that provide equally acceptable fits. Therefore, the actual geometry of the system cannot be necessarily represented by the derived angles. From the RVM fit alone, we can conclude that 0° < β < 10°, corresponding to a positive gradient of the PA swing. Nevertheless, the fact that α is practically unconstrained. More constraints on these parameters are described in Section 4.

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As shown in Figure 1, the mode switching events seem to appear frequently with a rough regularity of several hundred pulses in PSR J1326−6700. The longitude-resolved fluctuation spectra (Backer 1973) are calculated to investigate whether the mode switching occurs with regularity. This involves performing discrete fast Fourier transforms along each longitude bin within the pulse window. Any periodicity will be indicated as a peak in the Fourier spectrum. The spectrum of the 2014 October 15 observation of PSR J1326−6700, shown in the top panel of Figure 7, remarkably displays a well-defined periodic feature with a peak frequency at 0.004 cycles period⁻¹ associated most strongly with the central and trailing components of the profile. The latter property ties this principal fluctuation to the mode switching events, since a longitudinal shift of emission would be most evident at these longitudes. A more accurate value for the total fluctuation frequency can be determined with the peak value with an uncertainty estimated as FWHM/$2\sqrt{2\mathrm{ln}(2)}$, where FWHM is the full width at half maximum of the peak feature and $2\sqrt{2\mathrm{ln}(2)}$ is the scaling for Gaussian approximation (Basu et al. 2019), giving P_f = 256 ± 30 rotation periods. The total fluctuation spectra of all observations are shown in the following lower panels of Figure 7. The somewhat long 2014 May 25 and 30 spectra present periods of 205 ± 18 and 341 ± 48, respectively. The short 2014 December 2 observation shows a strong feature at 0.0059 cycles period⁻¹ with a period of 171 ± 20 stellar rotations. Three strong peaks at 113, 147, and 333 periods are shown in the 2012 January 15 spectrum. The somewhat different spectra presented may be caused by the wide spreading of fluctuation power in these observations. Similar periodicities have also been identified in PSR B0919+06 with an approximate period of 150 rotation periods and PSR B1859+07 with a rough period of 700 rotations (Wahl et al. 2016). The quasiperiodicity found in the five separate fluctuation spectra is time variant, it is a consequence of taking five separate observational samples of unchanging Weibull distributions for the normal and abnormal modes.

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In order to determine whether the periodic modulations originated from occasional phase drifting as suggested by Wang et al. (2007), the two-dimensional fluctuation spectra (Edwards & Stappers 2002) are calculated. No evidence of drifting subpulses is shown in our observations.

The location of the radio emission region in the pulsar magnetosphere remains a major uncertainty in our understanding of pulsar emission physics. The emission altitude calculated using a geometrical approach involves a dipole file configuration where the viewing geometry has to be specified (Phillips 1992). However, α and β values cannot be unambiguously determined from the RVM fitting as shown in Figure 6. An independent procedure with significant advantages for estimating emission heights has been developed by Blaskiewicz et al. (1991). The steepest gradient point of the PA traverse is expected to have a time lag with respect to the centroid of the total intensity profile due to relativistic effects such as aberration and retardation. Then the absolute emission height at the frequency of observation can be converted from this time delay (Δϕ), and is given by

$$\begin{eqnarray}&&{h}_{\mathrm{em}}=\displaystyle \frac{{Pc}{\rm{\Delta }}\phi }{8\pi },\end{eqnarray} \tag{ 3 }$$

where P is the rotation period of the star and c is the speed of light. The profile centroid ϕ₁ is identified with the pulse longitude midway between the outermost edges of the pulse intensity profile. The uncertainty in ϕ₁ is estimated by $\sigma ({\phi }_{1})=2\sigma (I)/| {dI}/d\phi |$, where σ(I) represents the noise level, and dI/dϕ is the gradient of the profile in the vicinity of the edge. The pulse longitude at which the steepest slope of the PA curve occurs can be measured at the fiducial plane. The derived emission heights for normal and abnormal modes are given in Table 4.

Subsequently, the half opening angle of the radio emission beam can be determined from the emission height under the assumption that the beam is bounded by tangents to the last open field lines of a dipolar magnetic field. The half opening angle is obtained using the formula

$$\begin{eqnarray}&&\rho ={\theta }_{{PC}}+\arctan \left(\displaystyle \frac{1}{2}\tan {\theta }_{{PC}}\right),\end{eqnarray} \tag{ 4 }$$

where θ_PC denotes the angular radius of the open field line region, and is given by

$$\begin{eqnarray}&&{\theta }_{{PC}}=\arcsin \left(\sqrt{\displaystyle \frac{2\pi {h}_{\mathrm{em}}}{{Pc}}}\right)\end{eqnarray} \tag{ 5 }$$

(Lyne & Graham-Smith 2012).

As a result, the values of α and β can be constrained from

$$\begin{eqnarray}&&\cos \rho =\cos \alpha \cos (\alpha +\beta )+\sin \alpha \sin (\alpha +\beta )\cos \left(\displaystyle \frac{{W}_{\mathrm{open}}}{2}\right)\end{eqnarray} \tag{ 6 }$$

(Gil et al. 1984), where W_open is the range of rotational longitude for which the line-of-sight samples the open field line region. ${W}_{\mathrm{open}}$ is taken to be twice the difference in phase between the fiducial plane position and the pulse edge furthest from it (Rookyard et al. 2015). The green areas shown in Figure 6 show the favored viewing geometry that can produce a pulse of the measured width. The magnetic axis is relatively aligned with the pulsar's rotation axis with α < 45°.

As shown in Figure 1, the time-dependent mode changing presents a gradual earlier shift pattern of the emission longitude, and occasional nulls interrupt the abnormal emission state. The W₁₀ remains constant for both emission modes. However, the W₅₀ during the abnormal mode narrows (see Table 3). Furthermore, in PSR J1326−6700 the profile changes occur quasiperiodically, implying the existence of a different rotation frequency from that of the star. It is suggested that the magnetosphere does not corotate with the star, and the structure of the magnetosphere changes in a quasi-periodic pattern. For instance, recently, the swooshes were identified with a quasi-periodicity in pulsar B1859+07 and possibly B0919+06 (Wahl et al. 2016).

The origin of mode changing has remaining a mystery since the first discovery of mode switching in PSR B1237+25 (Backer 1970). There is increasing evidence that the mode changing is possibly caused by changes of magnetospheric particle current flow (Lyne et al. 1971). Nevertheless, the underlying physical mechanism for the change of magnetosphere state is not clear yet. Lyne et al. (2010) presented the correlated changes in the pulse profile and in the timing noise for six pulsars. The switching between the radio-quiet "off" and radio-loud "on" states in the intermittent pulsar B1931+24 was found to be correlated with the slowing down rate of the pulsar Kramer et al. (2006). The abrupt changes of the emission mode suggest that the pulsar jumps between two different magnetospheric states, and a causal connection between the radio emission and the torque on the star is physically established. A purely magnetospheric model for observed abrupt changes in the pulsar radio profile is applied to the swooshing in PSR B0919+06 (Yuen & Melrose 2017). Therefore, the phase shift during the abnormal emission mode in PSR J1326−6700 can be described by the shifts in the intersection points of the trajectory with the emission spot as well. A long term monitoring is suggested to investigate whether the spin-down rate correlates with the shifting between the two profile states.

The delay emission heights for both modes derived using the relativistic beaming model based on effects of aberration and retardation are given in Table 4. It is found that the abnormal emission arises at higher altitude from the surface of the neutron star while the normal mode arises closer to the stellar surface, corresponding to a jump from 0.015 to 0.04 light cylinder radius. Furthermore, the beam opening angle of the abnormal mode is wider than that of the normal mode. A possible model is constructed, wherein the emission of both modes comes from the same magnetic flux surface, but from different heights at a fixed frequency (van Leeuwen et al. 2003). As shown in Figure 8, the altitude at a fixed frequency increases when the emission changes from normal mode to abnormal mode. According to the traditional vacuum gap model (Ruderman & Sutherland 1975), the transition from normal mode to abnormal mode occurs by an increase in height of the voltage gap. As a result the altitude of emission increases.

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Alternative possible origins of the anomalous variations in the on-pulse phase have been proposed. Rankin et al. (2006) suggested that the swooshes could be caused by partial conal emission, where the emission region was partly obscured. The binary companions in the light cylinder orbits was adopted to interpret the inter-swoosh quasi-period (Wahl et al. 2016).