Jiamusi pulsar observations: III. Nulling of 20 pulsars

The emission or null state of pulsars are analyzed in a few steps. Although most of the intense RFIs have already been exercised in a previous data processing step, some pulsars are still affected by low levels of RFI. To get a flat baseline, we perform a least square fitting of a fifth order polynomial to the off-pulse regions for each sub-integration, and subtract them from the data. An off-pulse window with the same number of bins ($n_{\rm bins}$) as the on-pulse one is selected. The significance of emission from both the on and off pulse windows are calculated for each sub-integration,

Here, $\sigma$ is the standard derivation of off-pulse emission, $W_{\rm eq}$ is the equivalent width of a top-hat pulse with the same area and peak height as the pulse profile integrated from all the subintegrations. The significance sequences are obtained for both the on and off pulses, which are then binned to form the histogram distributions. The distribution of off-pulse significance can be well modeled by a normal function with a mean of zero, an amplitude of $A_{0}$ and the standard deviation of $\sigma_{\rm off}$,

Any excess out of this distribution for the on-pulse significance indicates that some pulses are in nulling state. We fit the normal function with a mean of zero and the standard deviation $\sigma_{\rm off}$ to the on-pulse distribution with significance less than zero to obtain the amplitude $A_{1}$. The ratio is termed as the nulling fraction $f_{\rm n}=A_{1}/A_{0}$, which has an uncertainty of

Here, $\delta A_{0}$ and $\delta A_{1}$ represent the uncertainties obtained from fitting. The scaled off-pulse distribution with $f_{\rm n}$ is then subtracted from the distribution of on-pulse significance. Residual distribution generally follows a lognormal distribution,

Here, $\mu_{\rm on}$ and $\sigma_{\rm on}$ are the mean and standard deviation of the normally distributed logarithm of the on-pulse significance.

The value $3\sigma_{\rm off}$ represents the upper limit for the distribution of off pulse significance. If $\exp(\mu_{\rm on}-3\sigma_{\rm on})>3\sigma_{\rm off}$, it means that the pulses in the emission and null states are separated. The estimated $f_{\rm n}$ represents a real fraction of pulses in nulling states. If $\exp(\mu_{\rm on}-3\sigma_{\rm on})<3\sigma_{\rm off}$, the distributions for the emission and null pulses can not be separated. The obtained $f_{\rm n}$ is then overestimated, which represents an upper limit. When the possible short nulls are ignored, every $2^{n}$ pulses are integrated for the pulse sequence until $\exp(\mu_{\rm on}-3\sigma_{\rm on})>3\sigma_{\rm off}$. The so-obtained $f_{\rm n}$ represents a lower limit.

An example analysis of the nulling fraction is shown in Figure 2 for PSR J0908-1739. The distribution of on-pulse significance is much broader, whose lower 3-sigma limit $\exp(\mu_{\rm on}-3\sigma_{\rm on})$ is smaller than $3\sigma_{\rm off}$. The nulling pulses around zero are not well separated from the emission pulses, and the so estimated $f_{\rm n}$ of 29.2% represents an upper limit. In the bottom panels, every 32 pulses are integrated to improve the significance of pulses. The histograms for subintegrations with significance around zero are separated from those for the emission ones. The so estimated $f_{\rm n}=18.8$% represents a lower limit, because nulls shorter than 32 pulses can not be resolved.

It should be noted that nulling fractions can be estimated either from the distributions of on and off pulse energy (e.g. Ritchings 1976; Gajjar et al. 2012) or from the distributions of their significance (Lynch et al. 2013). In previous researches, the on and off pulse energies were generally scaled by averages of energies for every 200 pulses to compensate for energy variation caused by interstellar scintillation following Ritchings (1976). For long nulls (e.g. PSR J1709$-$1640), such robust correction bias the off-pulse energy distribution from the Gaussian, as noticed by Vivekanand (1995). Identification of the emission or null state of a pulse or subintegration is often based on an artificially setting threshold significance, e.g. S/N=3. If the energy distribution has a $\sigma_{\rm off}>1$, the threshold S/N=3 is not enough to discriminate the emission and null states. Our identification is based on the distribution of pulse significance with the following criteria,

(1) any pulse with $\rm S_{on}\geq 4\sigma_{\rm off}$ is classified as an emission state.

(2) any pulse with $\rm S_{on}\leq 2\sigma_{\rm off}$ is classified as a null for non-detection.

(3) if a pulse is of 2 $\sigma_{\rm off}<\rm S_{on}\leq 3\sigma_{\rm off}$ and the adjacent pulses ahead and behind have $\rm S_{\rm on}$ ¿ 3 $\sigma_{\rm off}$, it is classified as an emission state, otherwise a null.

(4) if a pulse has 3 $\sigma_{\rm off}$ ¡ $\rm S_{on}$ ¡ 4 $\sigma_{\rm off}$ and the adjacent in the two sides have $\rm S_{\rm on}$ ¡ 3 $\sigma_{\rm off}$, it is termed as a null, otherwise an emission state.

The identified emission or null state for the example pulse sequence of PSR J1709-1640 is also shown in Figure 1 with diverse nulling behaviors. The left and middle panels of the observation represent the normal states of the pulsar, which usually stays in an on state with short nulls of several to several tens of pulses. But occasionally, it shows a long null (as of 6368 pulses for $\sim$1.2h) as shown by the right panels for the entire pulse sequence. Moreover, the sequence shows moderate nulls of 640 and 560 pulses before it goes into the long null, similar to those in the observation by Naidu et al. (2018).

By this approach, we analyzed nulling fractions and states of 20 pulsars, as shown in Figure 46 and listed in column 3 of Table 2, with the number of pulses or subintegrations employed for the analysis listed in column 2. For 5 PSRs J0332+5434, J1509+5531, J1932+1059, J2022+5154, and J2313+4253, the $f_{\rm n}$ are obtained since their distributions of on and off-pulse significance are separated with single pulses. For the other pulsars, only upper and lower limits of $f_{\rm n}$ are estimated. The upper limits are obtained with single pulses whose distributions of on and off-pulse significance are not resolved, and the lower limits are obtained with subintegrations with the separated distributions of on and off-pulse significance. The cumulative probability, $P_{\rm cri}$, for the modeled on-pulse distribution at $3\sigma_{\rm off}$ is calculated, as listed in column 4 in Table 2, which is the probability of misidentification of emission to null for the resolved on and off pulses or subintegrations.

The emission and null lengths represent the durations for a pulsar staying contiguously at a given state. A emission-null (1-0) sequence is composed of inter-changing emission and null states with lengths of ($N_{e_{1}}$, $N_{n_{1}}$, $N_{e_{2}}$, $N_{n_{2}}$, $N_{e_{3}}$, …). The number of state changing cycles, $N_{c}$, is listed in column 5 of Table 2 for each pulsar. Its ratio, $N_{c}/N_{\rm sub}$, represents the nulling cadence during one observation. Distributions of emission and null lengths vary a lot, their means, ¡$N_{e}$¿ and ¡$N_{n}$¿, together with standard derivations given in parentheses are listed in columns 6 and 7 of Table 2. Previously, it was suggested that more emission and null states tend to have small lengths and exponential functions were usually employed to model both distributions (e.g. Gajjar et al. 2012).

An example of distributions of emission and null lengths is shown in Figure 3 for PSR J1709-1640, with length pairs of $EN$ and $NE$ represented by red “o”s and blue “$\times$”s in the 2-D length plan (bottom-left). Histograms for the emission and null lengths are shown in the top and right panels, which do not follow exponential functions. Figure 4 shows the distributions of nulling degrees and scales for both the $EN$ and $NE$ length pairs of PSR J1709$-$1640. It is apparent from the distribution of nulling degrees that nulls can have variable portions within a nulling cycle, but more cycles tend to have small nulling degrees. There are more cycles tending to have small nulling scales, but the distribution of which does not decay exponentially with the scale length.

The duration of an emission state might be related to the length of its prior or post nulls. From one emission-null sequence, the emission and the next null length pairs [($N_{e_{1}}$, $N_{n_{1}}$), ($N_{e_{2}}$, $N_{n_{2}}$), …], named as $EN$, can be formed together with the emission and the pre null length pairs [($N_{e_{2}}$, $N_{n_{1}}$), ($N_{e_{3}}$, $N_{n_{2}}$), …], named as $NE$. As introduced by Yang et al. (2014), interaction between the states of emission and null can be demonstrated in terms of nulling degrees, $\alpha_{1}$ and $\alpha_{2}$, and nulling scales, $s_{1}$ and $s_{2}$, that correspond to emission and null length pairs from $EN$ and $NE$. Their means and the standard derivations given in parentheses are listed in columns 8 to 11 of Table 2.

By this approach, the emission and null lengths are analyzed for 10 of the 20 pulsars that have well distinguished emission and null states, as listed in Table 3 and shown in Figure 47.

1) A few pulsars have dominant emission states and nulls generally last for short durations, for example, PSRs J1932+1059 and J2022+5154.

2) The nulling degrees are in the range from about 1.4 to 26.3 degrees.

3) The nulling scales can have lengths from 31 to 353 pulses with comparable deviations.

4) For most pulsars, the distributions of emission and null lengths cannot be well-modeled by the exponential functions for the stochastic Poisson processes except for the emission lengths of PSRs J1509+5531 and J1932+1059.

For nulling pulsars, it was intuitively thought that the duration of an emission state might be dependent on the duration of the preceding nulling state, or vice versa. Correlations between the emission and null lengths are examined, and correlation coefficients are listed in columns 2 and 3 of Table 3 for the $EN$ and $NE$ length pairs. The correlation coefficients for $EN$ vary from -0.04 to 0.62 with a pair number weighted average of 0.03. While for $NE$ length pairs, the coefficients vary from -0.14 to 0.36 with a weighted average of 0.07. It means that the duration of an emission or null state can be correlated with the duration of its preceding null or emission for some pulsar, but only with a marginal significance.

To further quantify the difference between the $EN$ and $NE$, two sample K-S tests are carried out for both the nulling degrees ($\alpha_{1}$ vs $\alpha_{2}$) and nulling scales ($s_{1}$ vs $s_{2}$). Here, the tests are performed on the unbinned data, which return p-values and the statistic deviatives that represent the maximum differences between two cumulative distribution functions, as listed in columns 4 and 7 of Table 3. If a p-value is larger than the significance level 0.1, the hypothesis that length pairs of $EN$ and $NE$ come from the same distribution cannot be rejected. For 10 of the 20 pulsars with $N_{c}>10$, the afore-mentioned statistics and K-S tests are performed. We found:

1) For $\alpha_{1}$ and $\alpha_{2}$, the K-S tests have maximum statistic deviative of 0.19 and p-values larger than 0.51, much larger than the significance level of 0.1.

2) For $s_{1}$ and $s_{2}$, the K-S tests have maximum statistics deviative of 0.19 and p-values larger than 0.71, also larger than the significance level.

In summary, no significant difference is found between $EN$ and $NE$ length pairs from the distributions of $\alpha$ and $s$. Both the $EN$ and $NE$ may come from the same processes.

To examine if the emission and null interact randomly, the statistical analysis is done as following. A sequence with the same size as observations of a pulsar is first simulated from a uniform distribution $[0,1)$. A threshold of nulling fraction is then set to the sequence, any value below that is denoted as null and vice visa for those above the threshold. The $EN$ and $NE$ length pairs are extracted from the simulated 1-0 sequence. If observations of a pulsar have more than 10000 subintegrations or pulses, 1000 sets of randomly distributed 1-0 sequences are simulated to give average estimates for $\alpha^{s}$ and $s^{s}$. Otherwise, 10000 sets of sequences are simulated for the estimation. From the $EN$ and $NE$ length pairs of these simulated sequences, nulling degrees and scales are calculated, which are termed as $\alpha_{1}^{s}$, $\alpha_{2}^{s}$, $s_{1}^{s}$ and $s_{2}^{s}$.

Two sample K-S tests are taken on the observed and simulated sequences for $\alpha_{1}$ vs $\alpha_{1}^{s}$ and $\alpha_{2}$ vs $\alpha_{2}^{s}$ for the occurrence of nulls. The statistic derivatives and p-values are listed in columns 5 and 6 of Table 3 for the $EN$ and $NE$ length pairs. Same is done for $s_{1}$ vs $s_{1}^{s}$ and $s_{2}$ vs $s_{2}^{s}$ to test the duration of nulling cycles, as listed in columns 8 and 9 of Table 3. The simulated distributions are shown in Figure 4 as an example and in Figure 47 for the 10 pulsars.

1) For the occurrence of nulls, the randomness hypothesis is rejected at high significance levels (p-values smaller than 0.1) for 9 pulsars in Table 3 from both $EN$ and $NE$ except for PSR J0304+1932.

2) For the duration of nulling cycles, the randomness hypothesis is rejected at high significance levels (p-values smaller than 0.1) for all the 10 pulsars from both $EN$ and $NE$.

In summary, emission and null do not interact randomly in the aspects of both the occurrence and duration for 9 pulsars. The occurrence of null can be random but its duration is not random for PSR J0304+1932.

Blocks of emission and null pulses are generally arranged orderly within a sequence, some of which show quasi-periodic variations, as demonstrated in Figure 1. To quantify this quasi-periodic pattern, a Fourier transform is performed on the emission-null (1-0) time sequence instead of the sequence for the total intensity. This is because the method concentrates only on the states of emission and eliminates the periodicities caused by sub-pulse drifting (e.g. Herfindal & Rankin 2007; Basu et al. 2017), mode-changing (e.g. Yan et al. 2020) and amplitude modulations (e.g. Basu et al. 2020). The discrete Fourier transformation (DFT) length depends on the length of an observation, which is chosen to ensure that the sequence has more than or equal to two but less than four DFT lengths. For a given sequence, the DFT process repeats by sliding 8, 16, or 32 points until the end. The power of all the DFTs from all the observations of a pulsar are finally averaged to form the entire spectra.

The DFT of the state sequence of one observation of PSR J1709$-$1640 is shown in Figure 5 as an example. It is apparent that the 1-0 sequence is modulated across a broad range of periods. There might be a significant periodicity for part of the sequence. But it could be averaged out for the whole sequence and lead to the most probable modulations of about 200 periods. The 1-0 sequences of the 10 pulsars show three kinds of periodic modulations, as shown in Figure 48.

1) Four PSRs J0304+1932, J1509+5531, J1709-1640 and J2313+4253 have quasi-periodic nulling, with nulling frequencies listed in column 2 of Table 4. These behaviors might be related to the carousel beam patterns, and the nulls represent empty passes of sight lines through the patterns (e.g. Herfindal & Rankin 2007).

2) Emissions of PSRs J1239+2453, J2022+5154 and J2048$-$1616 are modulated by nulls with long periodicities, and the modulation is very strong for PSR J2321+6024. These pulsars occasionally exhibit frequent nulls, which might result from the change of pulsar emission mechanisms. It can be caused either by the failure of particle production in the polar cap region of a pulsar magnetosphere or by the lose of coherence for the relativistic particles (e.g. Filippenko & Radhakrishnan 1982; Zhang et al. 1997).

3) PSRs J1136+1551 and J1932+1059 tend to have featureless modulations, i.e., with a ‘white spectrum’. These pulsars null at a wide range of pulsar periods with no preferred periodicities.

Pulse intensities vary during an emission state. For emission state lasting for more than or equal to 5 pulses or subintegrations, the pulse intensity sequences are first normalized by the averages and sequence lengths, and then averaged across the normalized length. Systematic variation of pulse intensities, if exists, should be revealed. Integrated pulse profiles are also obtained for the first, the second, the last two subintegrations of emission on states, as labeled by $E_{1}$, $E_{2}$, $E_{-2}$ and $E_{-1}$, respectively. Intensity ratios, $<I_{2}>/<I_{1}>$, for $E_{2}$ to $E_{1}$, and $<I_{-1}>/<I_{-2}>$ for $E_{-1}$ with respect to $E_{-2}$, are listed in columns 2 and 3 of Table 4.

The tendencies of intensity variations for the emission states are shown in Figure 6 for PSRs J1709$-$1640 and J2313+4253 as examples. PSR J2313+4253 is a representative of most pulsars, whose pulse intensity increases abruptly from a null at the beginning and diminish quickly to null at the end, and the intensities of these pulses vary little during emission states. PSR J1709$-$1640 exceptionally sets up to a large intensity than the average at the start of the emission state, and gradually diminish to null at the end.

Variations of pulse shapes and intensities during the transitions from emission to null states and vice versa are shown in Figure 49. For PSRs J0304+1932, J1239+2453, J2048$-$1616 and J2321+6024, the first two subintegrations, $E_{1}$ and $E_{2}$, and the last two subintegrations, $E_{-2}$ and $E_{-1}$, are weaker than the average. For PSRs J1136+1551, J1932+1059 and J2022+5154, these pulses near the transitions are stronger than the average. PSRs J1709$-$1640 and J1509+5531 show exceptional transitions, as demonstrated in Figure 7. The intensities of $E_{1}$ and $E_{2}$ of PSR J1709$-$1640 are stronger than the average, and $E_{2}$ is even much stronger than $E_{1}$, but intensities of $E_{-2}$ and $E_{-1}$ are weaker than the average with $E_{-1}$ weaker than $E_{-2}$. Null-to-emission transition of PSR J1509$+$5531 starts with the leading component of the pulse profile emerging first, and emission-to-null transition also takes place first with the leading component, as shown in Figure 8 for the four transitions. The phase dependency of transitions between emission and null implies that nulling of the pulsar is not caused by global change of pulsar magnetosphere, but just by density patch changes of relativistic particles in parts of a magnetosphere.

So far, 214 pulsars have been reported to null by observations from 303 to 4850 MHz, as collected from literatures in appendix C. 146 of them have nulling fraction measurements as listed in Table LABEL:table:null, and nulling fractions of 79 pulsars were estimated at multi-frequencies with some from simultaneous observations (e.g. Gajjar et al. 2014a; Naidu et al. 2017). These nulling fractions are grouped into 5 frequency bands, 303-333, 408-430, 607-645, 1308-1518 and 2000-4850 MHz and then averaged. For most pulsars, these nulling fractions are roughly consistent among observations (within a factor of 2). But nulling fractions vary significantly for PSRs J0630$-$2834, J0820$-$1350, J0828$-$3417, J1559$-$4438, J1709$-$1640 and J1901$-$0906.

Averages of the available nulling fractions at these frequency bands are calculated for each of the 146 nulling pulsars, as listed in Table LABEL:table:null. Its distribution is shown in Figure 9. It is apparent that more than half the pulsars have nulling fractions less than 10 percent.

For 22 pulsars with nulling fractions available at at least 3 frequency bands, their frequency dependencies are shown in Figure 10. It is apparent that the nulling fractions are generally consistent across a wide rage of frequencies for PSRs J0814+7429, J0908$-$1739, J0953+0755 and J2018+2839. While for PSRs J0630-2834, J0820-1350, J1107-5907, J1709-1640 and J1946+1805, nulling fractions vary significantly across frequencies. For PSRs like J1136+1551 and J2048-1616, the so-estimated nulling fractions at high frequencies are smaller than the low-frequency ones. These diverse properties mean that the broad-band nulls are produced by physical processes with a global change of pulsar magnetosphere. Maybe the distributions of relativistic particles are also changed. A combined effects of the distributions of relativistic particles and pulsar geometry can lead to various nulling features.

These effects lie in the sense that emissions of different frequencies originate from different heights of pulsar magnetosphere (e.g. Wang et al. 2013). Relativistic particles emitting at different heights are generated from different polar cap regions through sparking process. If nulls of a pulsar from simultaneous observations are frequency dependent, it means that parts of the polar cap region are inactive so that nulling occurs at certain frequencies. In other words, frequency dependent nulling can help us to distinguish the sparking polar cap regions. Any global change of pulsar magnetosphere will cause simultaneous nulling across a broad frequency band. PSR J1709$-$1640 has nulls lasting for minutes to hours, and observations in so limited time can cause different estimates of nulling fractions. Moreover, different modes might also cause different nulling fractions. For example, PSRs J0826+2637 and J1107$-$5907 have two modes with quite different nulling fractions (Young et al. 2014; Basu & Mitra 2019).

Our observations of 10 pulsars have multiple nulling cycles ($N_{c}$¿10), as listed in Table 2. Figure 11 shows correlations of nulling fraction, the cadence of occurrence of nulling $N_{c}/N_{\rm sub}$, null to emission length ratios $N_{n}$/$N_{e}$, nulling degrees ($\alpha_{1}$ and $\alpha_{2}$) and nulling scales ($s_{1}$ and $s_{2}$) with respect to pulsar age and Edot for these pulsars. Here, nulling fractions and their uncertainties are taken as the mean and half range of upper and lower limits when both limits are available. Pulsars with large ages tend to have large nulling fractions with correlation coefficient of 0.33, and to have small nulling scales with correlation coefficients of -0.30 and -0.35. Little correlations are found for the other nulling parameters with respect to pulsar age. Pulsars with large energy loss rate tend to have small nulling fractions, small nulling cadence and small null-to-emission length ratios with correlation coefficients of -0.69, -0.65 and -0.53, but tend to have large nulling scales with correlation coefficients of 0.49 and 0.61.

Distribution of the nulling pulsars on the period and period derivative diagram is shown in Figure 12. It is obvious that periods of these nulling pulsars range from 0.1 to 7 seconds and period derivatives from $10^{-18}$ to $10^{-12}$. The line with constant rate of loss of rotational kinetic energy (Edot) of $10^{30}\rm erg\leavevmode\nobreak\ s^{-1}$ roughly represents the lower boundary for these nulling pulsars, which is in parallel with the lines with constant accelerating potential above pulsar polar cap. It is generally believed that pulsars are formed from the upper left part of the diagram and move down to the right by crossing the constant Edot lines when pulsars evolve. Particle acceleration process is reduced in the magnetosphere of aged pulsars, so that pulsars start to null with a nulling fraction increasing and eventually reach the off state. This scenario is supported by the distribution of pulsars with different nulling fractions in Figure 12. It is obvious that pulsars with nulling fractions $\leq 1\%$ are generally located in the upper left part of the diagram, pulsars with nulling fractions ¿$30\%$ are located in the lower right, as divided by the Edot line of $10^{32}\rm erg\leavevmode\nobreak\ s^{-1}$. The intermittent pulsars are exceptions (Lyne et al. 2017).

Figure 13 shows the correlations of nulling fraction with respect to pulsar period, period derivative, age and Edot for this most complete sample of nulling pulsars by now. The nulling fraction is positively correlated with pulsar period with correlation coefficients of r=0.26 and $r_{l}$=0.36 for both parameters in linear and logarithmic scales. No correlation is found for period derivatives. A marginal positive correlation is found between nulling fraction and pulsar ages with r=0.15 and $r_{l}$=0.12, which agrees with the understanding that old pulsars tend to have large nulling fraction. Moreover, the conventional understanding that pulsars with large Edot tend to have small nulling fraction is also confirmed with correlation coefficients of r=-0.08 and $r_{l}$=-0.30. In summary, large nulling fractions are more related to long period than to large age and small Edot, in favor of the conclusion obtained by Biggs (1992).

Data for emission-null sequences of all pulsars presented in this paper are available at http://zmtt.bao.ac.cn/psr-jms/.