Investigation of the properties of four rotating radio transients at 111 MHz

ABSTRACT

We present an analysis of the individual pulses of four rotating radio transients (RRATs), previously discovered in a monitoring survey running for 5.5 yr at the frequency of 111 MHz. At a time interval equivalent to 5 d of continuous observations for each RRAT, 90, 389, 206 and 157 pulses were detected in J0640+07, J1005+30, J1132+25 and J1336+33, respectively. The investigated RRATs have different distributions of their pulse amplitudes. For J0640+07 and J1132+25, the distribution is described by a single exponent over the entire range of flux densities. For J1005+30 and J1336+33, it is a lognormal function with a power-law tail. For J0640+07 and J1005+30, we have detected pulses with a signal-to-noise ratio (S/N) of a few hundred. For J1132+25 and J1336+33, the S/N of the strongest pulses reaches several tens. These RRATs show a strong change in their emission. When the strengths of their pulse amplitudes are significantly changed, we see long intervals of absence of emission or its strong attenuation. The analysis carried out in this work shows that it is possible that all the studied RRATs are, apparently, pulsars with giant pulses.

pulsars: general, pulsars: individual: J0640+07; J1005+30; J1132+25; J1336+33

1 INTRODUCTION

Rotating radio transients (RRATs) were discovered in 2006 (McLaughlin et al. 2006) as sources of sporadic bursts of dispersed pulses followed by no detectable emission for many rotations – sometimes minutes to hours (Keane 2016; Bhattacharyya et al. 2018). They are Galactic neutron stars with extreme emission variability. Single pulse rates are in the range of a few pulses to a few hundred pulses per hour. The nulling fraction of RRATs can exceed 99 per cent. The average magnetic fields of RRATs and the average periods are higher than those of ordinary second pulsars (McLaughlin et al. 2009; Cui et al. 2017). Other qualities of RRATs are similar to those of pulsars with similar periods. According to Burke-Spolaor & Bailes (2010), the Galactic z-distribution and pulse width distributions are the same as for ordinary pulsars.

Long-term studies of RRATs are still limited (Palliyaguru et al. 2011; Cui et al. 2017; Mickaliger et al. 2018; Shapiro-Albert, McLaughlin & Keane 2018; Bhattacharyya et al. 2018; Brylyakova & Tyul’bashev 2020). Palliyaguru et al. (2011) analysed the short- and long-term variability of eight RRATs studied over a 5.5-yr interval at a frequency of 1400 MHz. They show that there are periodicities from 30 min to years in pulse arrival time for six investigated RRATs and that RRAT pulses appear randomly at small time intervals.

Cui et al. (2017) used a series of observations from 1.5 to 15 yr for eight RRATs to obtain accurate estimates of the rotational parameters, as well as to obtain histograms of the pulse amplitude distributions. It turned out that the typical pulse distribution is lognormal, as for ordinary pulsars. A power-law tail in the pulse energy distributions, typical for pulsars with giant pulses, has been observed for several RRATs. Results of research by Mickaliger et al. (2018) into the pulse distribution by energy are the same as those of Cui et al. (2017), but for 14 RRATs. A study of three RRATs over an 11-yr series of observations was presented by Shapiro-Albert et al. (2018). They show that the energy distribution for two RRATs is lognormal, and for another, it is lognormal and an additional power component is observed.

Using simultaneous observations of single pulses of RRAT J2325−0530 at the Murchison Widefield Array (MWA; 154 MHz) and Parkes radio telescopes (1.4 GHz), Meyers et al. (2019) measure the spectral index α = 2.2 ± 0.1. Shapiro-Albert et al. (2018) provided the single-pulse-based spectral index (α) for three studied RRATs, which are between 0.6 and 1.2. Taylor et al. (2016) measure α at about 0.7 across the band of LWA1 (35–80 MHz) for RRAT J2325−0530, which is substantially shallow than in Meyers et al. (2019). It is possible that a flattening may occur at f< 150 MHz.

The 8-yr observations for three RRATs are considered in Bhattacharyya et al. (2018), who show that there may be long-term trends of changes in the observed arrival rate of pulses. The average number of observed pulses varies by 1.5–2 times, in the direction of both decreasing and increasing the number of observed pulses over the full observation interval. It is noted in this paper that in the absence of strong pulses, one from the investigated RRATs shows weak periodic emission.

The 5-yr observation interval at the frequency of 111 MHz has been used to study the RRAT J0139+33 (Brylyakova & Tyul’bashev 2020). We have shown that the energy distribution of pulses in the RRATs is described by a bimodal (broken) power law, which is typical for some pulsars with giant pulses (Smirnova 2012; Kazantsev & Potapov 2017).

Summarizing the above-mentioned observations, we can conclude that the number of observed pulses can significantly change for RRATs at certain time intervals; trends can be observed that show an increase or decrease in the number of observed pulses over the entire observation interval. The pulse energy distribution function is usually lognormal, although a power-law tail is sometimes observed. There is also one case of a broken power-law distribution.

In 2018, when processing semi-annual daily observations obtained at the Large Phased Array (LPA) radio telescope of the Lebedev Physical Institute (LPI), 33 RRATs were detected by their individual dispersed pulses (Tyul’bashev et al. 2018a; Tyul’bashev, Tyul’bashev & Malofeev2018b). Out of this sample, we have selected for further studies four RRATs that have long nullings or strong individual pulses. These RRATs were discovered in the monitoring survey carried out at the LPA LPI at a wavelength of 2.7 m. They have not yet been confirmed by observations on other instruments. In particular, they were not detected in the search for pulsars using the Low Frequency Array (LOFAR; Sanidas et al. 2019) at the wavelength of 2.2 m.

2 OBSERVATIONS

Round-the-clock daily monitoring observations have been carried out on the upgraded antenna of the LPA LPI since 2014 August. The LPA LPI consists of 16 384 wave dipoles. There are 256 lines, each of which has 64 dipoles. We have used Butler matrices to form a beam in the meridian plane. Details about the modernization of the LPA LPI are given by Shishov et al. (2016) and Tyul’bashev et al. (2016).

The main purpose of the monitoring programme is daily observations under the Space Weather programme, in which daily observations of about 5000 compact (scintillating) radio sources are carried out (Shishov et al. 2016). Observations made for the Space Weather programme can also be used to study pulse emission (Brylyakova & Tyul’bashev 2020).

The antenna operates at a central frequency of 110.3 MHz. The full band 2.5 MHz is divided into 32 frequency channels with a channel width of 78 kHz. The sampling interval is equal to 12.5 ms. Observations are done simultaneously in 96 antenna beams covering declinations from −9° to +42°. The data obtained are used to search for new RRATs and pulsars.

The digital recorders created for these observations do not allow us to obtain more frequency channels and readout time less than 12.5 ms. The accuracy of the time stamps are determined by the quartz oscillator. Therefore, raw data are not optimal for pulsar and RRAT investigations.

Six times a day, a signal of known temperature (calibration signal) is applied to the antenna input in the form of ‘OFF–ON–OFF’, which allows us to equalize the gain between frequency channels (Tyul’bashev et al. 2020). As the data analysis showed, during a 2-h observation, the change in the calibration signal amplitude does not exceed |$5{{\ \rm per\ cent}}$|⁠. Because the closest calibration signal to the source is used for session calibration, the detected pulse amplitudes have errors of less than |$5{{\ \rm per\ cent}}$|⁠. The width of the beam directivity pattern of the LPA LPI at half power is equal to 3.5 min/cos (δ), where δ is a source declination.

The typical sensitivity of the LPA LPI telescope when observing ordinary pulsars in a single observation session is 6–8 mJy, if the pulsar is outside the plane of the Galaxy, and 15–20 mJy, if the pulsar is in the plane of the Galaxy (Tyul’bashev et al. 2016). The detection limit for single pulses with the selected sampling interval 12.5 ms is 2.1 Jy at the signal to noise ratio (S/N) = 7 (Tyul’bashev et al. 2018a). For 5.5 yr of continuous monitoring, the equivalent continuous observation time is approximately 5 d for each point in the sky that falls within the observation area.

Additional information about observation modes with the LPA LPI, the implementation of independent radio telescopes based on a single antenna field and other technical information about the capabilities of the LPA LPI after its modernization can be found in Tyul’bashev et al. (2016) and Shishov et al. (2016).

3 DATA ANALYSIS

The LPA LPI has many specifics as an antenna array. Therefore, special software was created to process the observations, taking into account these features. We have hosted this software on Github.¹

We followed the standard way of searching for RRAT pulses in the obtained data. After equalizing the amplification between the frequency channels, we incoherently dedispersed the data to the known dispersion measure (DM) of the studied RRATs, subtracted the baseline and removed radio frequency interference (RFI), and detected a single pulse from the frequency averaged time series. Details of RFI removal are described in Tyul’bashev et al. (2020).

To get the baseline, we took a recording section containing a pulse and having a duration of 4 s. This time interval is much longer than the duration of the RRAT pulse, which makes it possible to adequately determine the baseline (Brylyakova & Tyul’bashev 2020). First, the frequency channels were summed up without shifts, that is, assuming a DM is equal zero. The resulting points were fit by a polynomial. Then the frequency channels were summed up with shifts corresponding to the DM of the RRAT. The resulting polynomial was subtracted from the resulting series of points. The root-mean-square (rms) deviation (σ_n) of the noises was determined by the area outside the pulse and the S/N was taken as the value of the peak pulse amplitude divided by the estimate of σ_n (see details in Brylyakova & Tyul’bashev 2020). For J1005+30, J1132+25 and J1336+33, the pulse detection threshold was set at S/N = 6. For J0640+07, the threshold was increased to S/N ≥ 7 because a lower threshold resulted in an overwhelming number of false positive detections.

For further work, pulses were selected that were located at an interval of ±1.5 min from the pulsar passing through the centre of the beam. For each pulse, corrections were made to the detection, related to the features of LPA LPI as an antenna array (Shishov et al. 2016; Tyul’bashev et al. 2016). These corrections attempt to mitigate the fact that the direction of the formed radiation pattern does not coincide with the direction to the source (for details, see Brylyakova & Tyul’bashev 2020). For all detected signals, the average profile and dynamic spectrum files were saved for each observation session. The final selection of pulses was done manually. The average profile was determined by summing up a signal with a given period, if known.

As a result of processing, data series were obtained in which the time of arrival of the RRAT pulse and its amplitude (I_k), normalized to σ_n, were recorded for each date over the entire observation interval. These series were analysed to study time variations, the number of pulses and their amplitudes. As the data are calibrated based on the calibration signal, it reflects the correct distribution of amplitudes over a long time interval.

4 RESULTS

Figs 1 and 2 show the values of the amplitudes in units of S/N, and the number of recorded pulses per session for the selected objects, depending on the sequence of days starting from the first day indicated in Table 1.

Figure 1.

The vertical axis shows the pulse amplitude in S/N units (top) and the number of registered pulses (bottom). The horizontal axis is defined as the number of the corresponding days. The first day with number 0 corresponds to JD = 245 6920 for J0640+07 and JD = 245 6898 for J1005+30.

Figure 2.

The horizontal and vertical axes are similar to the axes in Fig. 1. The first day with number 0 corresponds to JD = 245 6896 for J1132+25 and for J1336+33.

Table 1.

Summary of recorded pulse data.

PSR	P1 (s)	DM (pc cm−3)	First date	Nall (d)	Np	fint	Nmax (d)	〈I〉1	〈I〉2	Imax	m
J0640+07	–	52	2014/09/19	1927	90	0.044	107	14	0.62	289	2.2
J1005+30	–	17.5	2014/09/24	1942	389	0.188	31	38.3	7.2	448	1.25
J1132+25	1.002	23	2014/08/26	1937	206	0.075	265	7.13	0.53	16.2	0.2
J1336+33	3.013	8.5	2014/08/26	1896	157	0.069	256	10.4	0.71	41.3	0.5

Table 1 shows information about the observed pulses. The first column gives the name of the RRAT. Columns 2 and 3 show the period (P₁) and the dispersion measure of the RRAT. Columns 4–8 show the date when the first pulse was detected, the total number of observation days between the first and last recorded pulse (N_all), the number of detected pulses (N_p), the ratio of the number of days in which the pulses were detected to the number of all days of observations (i.e. the intermittency factor, f_int), and the maximum continuous duration absence of pulses in days (N_max). Columns 9–12 show the average values of amplitudes (in S/N units) for all detected pulses exceeding the specified threshold, 〈I〉₁, the average values taking into account intermittency factor, 〈I〉₂ = 〈I〉₁ × f_int, the maximum value of S/N for the entire observation period, I_max, and the modulation index |$m = \sum_{i=1}^N [(I_i - \langle I \rangle _1)^2 /(N - 1)]^{1/2}/ \langle I \rangle _1$|⁠, where N is the number of pulses and I_i is the amplitude of pulse i.

The sources in Table 1 can be divided into two samples. In the first sample, there are sources with high modulation and relatively short intervals when no pulses were observed; in the second sample, there are sources with low modulation and intervals when pulses are not detected for many months.

4.1 J0640+07 and J1005+30: RRATs with giant pulses?

The number of known pulsars with giant pulses (GPs) is very small compared to the nearly 3000 pulsars published in the Australia Telescope National Facility (ATNF)² catalogue. According to Kazantsev & Potapov (2018, see their table 1), who give citations of original discoveries of GPs, there are 16 such pulsars. The original papers consider a number of features that distinguish pulsars with GPs from ordinary pulsars (Sutton, Staelin & Price 1971; Kinkhabwala & Thorsett 2000; Soglasnov et al. 2004; Hankins & Eilek 2007; Kazantsev & Potapov 2018). If we consider the radio range, such features may include the following (Kazantsev & Potapov 2018): high peak flux densities in the pulse compared with the peak flux density accumulated in the average profile (a ratio more than 30); the power-law distribution of pulse energy; the narrowness of the GP compared with the average profile, sometimes, an extreme narrowness of the pulse; a high degree of polarization; extremely high brightness temperatures for the narrowest (nanosecond) pulses. However, not all of these features are simultaneously observed in every pulsar with GPs. The issue of the ratio of the amplitudes (or energies) of pulses and the average profile for determining the ‘gigantism’ of pulses is also not strictly defined. So, Karuppusamy, Stappers & Lee (2012) considered that for GPs, their energy must exceed the energy of the average pulse by more than 10 times.

From the set of listed ‘gigantism’ features in our monitoring observations, we can check the distribution of pulses by amplitudes or by energies, and determine the peak flux densities. It is also necessary to obtain the average profile of the pulsar and compare its amplitude with the amplitude of the strongest observed pulses. If the ratio of amplitudes is at least 30 and the ratio of energies is more than 10, then the studied pulsar is a candidate for pulsars with GPs. For RRATs J0640+07 and J1005+30, the period has not yet been determined. The average profile with the lack of timing and a small number of observed pulses per session (one or two) could not be obtained for either J0640+07 or J1005+30 and this is the reason why the ratio of amplitudes was determined from the detected pulses as I_max/〈I〉₂. As shown below, for J1132+25 and J1336+33, the value 〈I〉₂ corresponds well enough to the amplitude of the average profile. This gives us reason to believe that, for the RRATs considered here, the obtained value of 〈I〉₂ also determines the amplitude of the average profile. As follows from Table 1, the ratio I_max/〈I〉₂ is equal to 466 (J0640+07) and 62 (J1005+30) for the strongest pulses. Fig. 1 shows the dependence of the signal amplitude in S/N units (top) and the number of registered pulses (bottom) versus time in days. These amplitude ratios and their distribution functions (analysed in Section 4.3) are GP-like features.

The RRATs J0640+07 and J1005+30 are characterized by a high modulation index associated with strong deviations of the amplitude from the mean value. The emission of J0640+07 and J1005+30 is sporadic. For J0640+07, before the pulse with the maximum amplitude S/N = 289 (Fig. 1, left panels), 8 d before it and another 9 d after it, not a single pulse was registered. In this session, it was also the only one. For J1005+30, on the days of the largest amplitudes (Fig. 1, right panels), no pulses above the threshold were registered for several days before it and after it. Fig. 3 shows the strongest pulses for J0640+07 and J1005+30. The average rate of the occurrence of pulses per observational session is 0.05 (J0640+07) and 0.20 (J1005+30). The average number of pulses during a 3-min session of observation N_p/(f_int × N_all) is equal to 1.05 and 1.06 if we registered pulses during sessions (Fig. 1, bottom panels).

Figure 3.

The vertical axis (amplitude) is shown in arbitrary units. The horizontal axis is a time in points (δt = 12.5 ms). The panels show the strongest pulses for J1132+25 (top right) and J1336+33 (top left) and the average profiles of these RRATs (bottom panels).

4.2 J1132+25 and J1336+33: highly intermittent RRATs

RRATs J1132+25 and J1336+33 differ from RRATs J0640+07 and J1005+30. They have a sharp change in amplitude, accompanied by the absence of emission or its sharp weakening for a long time, like intermittent pulsars. For RRAT J1132+25, there is a long continuous time interval (of 265 sessions, marked by arrows in Fig. 2) when there are no pulses exceeding the specified threshold. The same figure also shows the highly sporadic nature of the emission of these RRATs, when long periods of quiescence are replaced by a significant increase in both the amplitude and the number of observed pulses. It turned out that, as for J0640+07 and J1005+30, they have features of giant pulses: I_max/〈I〉₂ = 30.4 for J1132+25 and I_max/〈I〉₂ = 57.9 for J1336+33.

In contrast to RRATs J0640+07 and J1005+30, for which we were not able to obtain average profiles, for J1132+25 and J1336+33, the usual pulsar-type periodic emission was detected previously (the periods shown in Table 1 obtained by Tyul’bashev et al.2018b). Their average profiles and the strongest pulses are shown in Fig. 4. When obtaining average profiles, at least 20 sessions were randomly selected for each source, during which no strong pulses were observed to exclude their influence on the profile. For these days, average profiles were obtained after averaging the signal with a known period. It is indicated that weak emission exists even though we do not see strong pulses. Because the individual strong pulses and the resulting average profiles were normalized by the calibration signal, it is possible to determine the ratio of the amplitude in the pulse to the amplitude of the average profile.

Figure 4.

The vertical axis (amplitude) is shown in arbitrary units. The horizontal axis is a time in points (δt = 12.5 ms). The panels show the strongest pulses for J0640+07 (left) and J1005+30 (right).

As can be seen from Fig. 4, the width of the individual pulse and of the average profile at the FWHM level for J1132+25 and J1336+33 are comparable, and the pulse shapes are similar, so we can assume that the ratio of their amplitudes corresponds to the ratio of their energies. We calculated energies as the integral over the pulse or average profile up to the 6σ noise level. Using the obtained shapes of the profile and of the individual pulse, we obtained the ratio of the energy in the pulse to the energy of the average profile: E = E_pulse/E_profile. These values for the strongest pulses are equal to E = 25.3 (J1132+25) and E = 50.1 (J1336+33). We note that these energies are close (the difference is of the order of |$10{{\ \rm per\ cent}}$|⁠) to the quantity I_max/〈I〉₂, and so we can assume that 〈I〉₂ approximates the amplitude of the average profile. These ratios of the energy in the pulse to the energy of the average profile and also the amplitude distribution analysed in Section 4.3 tell us that RRATs could be GP emitters.

The average profiles of pulsars J1132+25 and J1336+33, obtained at a 3-min interval and then averaged over several sessions, perhaps do not reflect the real average profile, which should be determined on an interval of at least half an hour. In addition, the average profile we use is affected by polarization, because the antenna receives linearly polarized emission. If the period of the Faraday rotation of the polarization plane is comparable to or longer than the receiver band, then the amplitude of such a profile can vary by several times from session to session. Such behaviour shows PSR B0950+08 (Smirnova 2012) observed at 112 MHz. This pulsar has about 70 per cent linear polarization of emission and a 15-MHz period of the Faraday rotation, which considerably exceeds the receiver bandwidth. This pulsar has one of the smallest rotation measures (RM = 1.35 rad m⁻²), and so the largest profile amplitude instability from session to session compared with other pulsars. The polarization study of 22 known RRATs by Caleb et al. (2019) with the Parkes telescope has shown that there is an average linear polarization fraction of 40 per cent. Individual single pulses were observed to be up to 100 per cent linearly polarized. At our low frequency, the level of polarization can be about this value. However, the value 〈I〉₂ obtained from many pulses will correctly reflect the amplitude of the average profile.

An influence of diffraction scintillations on the profile is very small in our case, since the de-correlation bandwidth for the selected RRATs, according to our estimates (less than a few kHz), is significantly narrower than the receiver band (2.5 MHz). Refractive scintillations for selected objects at our frequency have a scale of the order of a year or longer, so they also do not affect the value of the average profile amplitude. Our estimation of the diffractive scintillation bandwidths f_dif and the time-scale of refractive scintillation is based on the YMW16 Galactic electron density distribution model (Yao, Manchester & Wang 2017). Our measurements are in agreement with previous low-frequency estimates from low-DM pulsars (Malofeev et al. 1995).

For these two sources, as well as for J0640+07 and J1005+30, we have a small value of the ratio of the number of days in which pulses were detected to the total number of all observation days: f_int  = 0.075 and 0.069, respectively. We also have a small number of pulses exceeding the threshold in the observation session. The average rate of occurrence of pulses per minute is 0.037 (J1132+25) and 0.027 (J1336+33). The average number of pulses per minute is equal to 0.47 (J1132+25) and 0.4 (J1336+33) if we registered pulses during sessions (Fig. 2, bottom panels).

The modulation index for J1132+25 and J1336+33 is smaller than 1. As can be seen from Fig. 2, the emission has a flashy characteristic, when the signal amplitude changes dramatically. Especially strong is the change in amplitude for J1336+33, when, after 524 d of high activity, only weak and single pulses are recorded with large intervals of their complete absence (265 and 256 d) up to the last day of observations.

4.3 Distribution function of pulses

The number of detected pulses in the studied RRATs is low. Early observations of four RRATs (McLaughlin et al. 2006) with a small number of detected pulses (from 11 to 229) showed a power-law distribution of the pulses over the amplitudes. Mickaliger et al. (2018) have shown the dependences of the same RRATs by the sum of two lognormal distributions.

We checked different fits (i.e. power law, broken power law, lognormal distribution, lognormal distribution with a power tail) in the resulting histograms,

$$\begin{eqnarray*} f(x)= q/x \, {\rm e}^{-[\ln(x)-w]^2/2c^2} \end{eqnarray*}$$

$$\begin{eqnarray*} f(x)=a\, x^b, \end{eqnarray*}$$

where q, w, c, a andb are fitting parameters of lognormal and power-law distributions.

For the fitting, we used the Levenberg–Marquardt non-linear least-squares method (see the Python library, lmfit). Table 2 contains results of model testing using the Akaike information criterion (AIC³) corrected for small sample size (AICc; see details in Burnham & Anderson 2002). Bold text in the table indicates the best fit, which corresponds to the lowest number in Table 2 for different models.

Table 2.

Tests of different laws of pulse amplitude distribution. The values in the table correspond to the AICc score for each model per pulsar.

RRAT	Power law	Broken	Lognormal +	Lognormal
name		power law	power tail
J0640+07	9	36	39	53
J1005+30	85	62	16	49
J1132+25	5	14	30	21
J1336+33	65	51	31	37

For RRAT J0640+07, the number of detected pulses is less than 100. Therefore, there are no pulses in the majority of the defined S/N bins. We have increased the size of the bin by nine times, to build a smoothed distribution.

Fig. 5 gives integral distribution functions for all four sources in a logarithmic scale. It is clear that J0640+07 and J1132+25 have a power-law spectrum. J1005+30 (up to S/N = 72) and J1336+33 (up to S/N = 12) have a lognormal distribution and a power-law distribution above these S/N values.

Figure 5.

Integral distribution functions of the number of pulses (vertical axis) versus the pulse amplitude in S/N units (horizontal axis) in a logarithmic scale. The data points are approximated by different laws (see Table 2) using the least-squares method. Obtained power coefficients n with their errors are indicated in the figure.

Power or power tail dependences for distribution functions are additional indirect signs of pulsars with GPs. However, because of the small number of detected pulses, we do not have complete confidence to say that the investigated RRATs are pulsars with GPs.

5 DISCUSSION

All four RRATs have a number of common features: (i) a flashy characteristic when the amplitude of the signal increases by several times compared with adjacent sessions (for J1132+25) and even one to two orders of magnitude (for J0640+07, J1005+30 and J1336+33); (ii) a large number of days when there are no pulses above the specified detection threshold (long quiescence periods), f_int < 0.2; (iii) a significant predominance of sessions with only one pulse for 3 min of the source passing through the beam of the LPA LPI at half power; (iv) the maximum number of pulses registered for each 3 min does not exceed four (J1132+25, J1336+33), and even two (J0640+07, J1005+30); (v) power law (J0640+07, J1132+25) and lognormal with a power-law tail (J1005+30, J1336+33) distribution of pulse amplitudes; (vi) an excess of peak amplitudes and energies by tens and hundreds of times relative to the corresponding values for the average profiles, which is typical for pulsars with GPs.

The strongest change in activity was observed for J1336+33. For 524 d, it showed high activity when S/N was up to 41, and then for 1372 d, only solitary weak (at the detection level) pulses were observed with long time intervals between them. It is also possible that nulling is characterized by very weak emission and the sensitivity of the LPA LPI is not enough to detect it. Anyway, J1336+33 is significantly distinguished from all known RRATs by rapid changing of activity to a long interval of (around 3 yr) absence of emission or its strong attenuation. It would be interesting to study this object using timing to see if this rapid change is caused by changing of the period and period derivative. Unfortunately we were not able to do this. Although the observation session lasts only 3 min, the appearance of single pulses is a random process and the absence of pulses over a long time interval during daily observations of sources could reflect a systematic and persistent change in the emission process.

The modulation index reflects amplitude variability in time and it is significantly different for J0640+07 and J1005+30 (m = 2.2 and 1.25) and for J1132+25 and J1336+33 (m = 0.2 and 0.5). In this case, a small value of m is caused mainly by long intervals of nulling or strong decrease of emission. For J0640+07 and J1005+30, we have rare pulses but they are more smoothly distributed in time (see Fig. 1) and the ratio of the amplitude of the strongest pulse to their mean amplitude I_max/〈I₂〉 is much larger than for the other two pulsars. Both of these qualities collectively lead to m > 1.

A strong change in the character of emission can be associated with both external and internal causes. RRATs can be extinct pulsars that are re-activated due to the interaction of the pulsar’s magnetosphere with the matter falling on to it, or the interaction of the magnetosphere with the surrounding matter (Li 2006; Luo & Melrose 2007). Internal causes can be determined by a sharp change in the primary plasma density or the conditions of emission propagation in the magnetosphere (Timokhin 2010).

The broken power-law character of the distribution was previously obtained for RRAT J0139+33 (Brylyakova & Tyul’bashev 2020), which was also observed at the frequency of 111 MHz. This distribution pattern is one of the main features of GPs. Karuppusamy et al. (2012) defined GPs as pulses with an energy exceeding the energy of the average profile by more than 10 times. This ratio is arbitrary, and the RRATs for which the average profile was obtained (J1132+25, J1336+33) satisfy this condition. Studies at higher frequencies, such as 1400 and 820 MHz (Cui et al. 2017; Mickaliger et al. 2018), have shown that for most RRATs the distribution of amplitudes and energies is mostly lognormal, as for ordinary pulsars, but a few RRATs had a lognormal distribution with a power-law tail.

There is only one other investigation with simultaneous observations for RRATs at metre and decimetre wavelengths (Meyers et al. 2019), which finds that the single-pulse amplitude distribution is a truncated exponential at 150 MHz and a power law at 1400 MHz.

Other studies of the pulse amplitude distribution in more than 24 RRATs discovered in the decimetre wavelength range (Cui et al. 2017; Mickaliger et al. 2018; Shapiro-Albert et al. 2018) showed a lognormal or lognormal with power-law tail distribution. Meyers et al. (2019) estimated the pulse energy distribution for J2325−0530 at 150 MHz and 1400 MHz, but because of the small sample of single pulses they were unable to draw concrete conclusions. Because our sample is small and does not have observations at higher frequencies, it is impossible to conclude that the distribution of the pulse amplitude may depend on the frequency of observations for all RRATs. The analysis shows that RRATs are not a homogeneous sample, but a mixture of different kinds of pulsars. Long-term studies of large samples of RRATs are needed, which will allow us to judge more accurately the validity of different hypotheses.

ACKNOWLEDGEMENTS

The PYTHON lmfit package⁴ was used for data analysis. The authors express their appreciation to V. M. Malofeev, V. A. Potapov and A. N. Kazantzev for useful discussions in the course of the work. Special thanks to Yu Yu Kovalev for a donated server that was used to process the observation data.

DATA AVAILABILITY

The raw data underlying this article will be shared on reasonable request to the corresponding author. The tables with the signal-to-noise ratios of pulses can be found at http://prao.ru/online per cent20data/onlinedata.html.

Footnotes

1

https://github.com/Elinxt/rrat_pulsars

2

https://www.atnf.csiro.au/people/pulsar/psrcat/

3

https://github.com/lmfit/lmfit-py/

4

https://zenodo.org/record/4516651