The Discovery of Transitive Phenomenon in the Radio Emission of the mode-switcher PSR B0943+10

LPA is a transit instrument and the duration of observing session can not exceed 3.5 min/$\cos\delta$, where $\delta$ is the declination of the source. For PSR B0943+10, with its spin period $P_{1}=1.0977$ s, this translates to a maximum of 194 pulses available per session. The pulses were recorded using a digital receiver with a bandwidth of 2.5 MHz divided into 512 4.88-kHz spectral channels by a Fast Fourier Transform (FFT) processor. The actual total bandwidth taken into processing was 2.245 MHz. The signal was dedispersed to the frequency of 111.88 MHz (hereafter 112 MHz).

The narrow bandwidth of the channels reduced the effect of pulse broadening caused by interstellar dispersion delay. At 112 MHz the total pulse delay within a single channel for PSR B0943+10’s DM of 15.4 pc cm^-3 is 0.5 ms, much smaller than the time resolution of 2.8672 ms. More detailed description of LPA’s digital receiver and data pre-processing is given in Suleymanova et al. (2012).

With a rotation measure of $15\pm 1$ rad/m^-2 (Suleymanova et al., 1998), the Faraday modulation period at 112 MHz is about 1.6 MHz, which is comparable to the total bandwidth of the receiver. Thus, despite LPA being a linearly polarized array, the recordings provide a reasonable estimate of the pulsar’s total intensity emission (see also Suleymanova & Rankin, 2009).

PSR B0943+10 was observed with the low-band antennas (LBAs) of LOFAR core stations at a centre frequency of 53.8 MHz, and 25600 channels across a 78.1-MHz bandwidth. The time resolution of the raw data was 0.65 ms. Observations were pre-processed with the standard LOFAR pulsar pipeline (Stappers et al., 2011). The signal was dedispersed and folded synchronously with the topocentric pulsar period obtained with the ephemerides from Shabanova et al. (2013). The dedispersion was subsequently refined using a more precise value of the dispersion measure (DM), obtained by measuring the $\nu^{-2}$ lag of the fiducial point in the B-mode profile of the same observations (Bilous et al., 2014). Each period contained 512 longitude bins, corresponding to time resolution of about 2 ms. For further details of data acquisition and pre-processing we refer the reader to Bilous et al. (2014) and Bilous (2018).

For both LOFAR and PRAO data the subsequent analysis was performed on frequency-integrated pulse stacks, 2D arrays of the uncalibrated total intensity as a function of rotational longitude $\phi$ and time $t$, $s(\phi,t)$. In this notation, time is assumed to be constant within each rotational period of pulse stack: $t=[$pulse number$]\times P_{1}$.

Soon after the discovery of PSR B0943+10 in 1968 at PRAO (Vitkevich et al., 1969), a remarkable organized change of its subpulse positions had been noticed by Taylor & Huguenin (1971). Further investigations at 430 MHz demonstrated that subpulses shift within the onpulse window with the rate of approximately $4\aas@@fstack{\circ}4$ per stellar rotation (Backer et al., 1975).

Following Edwards & Stappers (2003) and Bilous (2018), the subpulse position $\mathrm{pos}_{\mathrm{drift}}$ can be mathematically expressed as follows:

Here $\hat{P}_{3}$ is the observed modulation period along the lines of constant $\phi$ and $P_{2}$ is the longitudinal spacing between subpulses recorded within same pulse period (see Fig. 1).

Both $\hat{P}_{3}$ and $P_{2}$ can be positive or negative, and the apparent direction of drift (i.e. whether pulses seem to march towards the leading or trailing edges of the profile) depends on whether $P_{3}$ and $P_{2}$ have the same sign. Following Edwards & Stappers (2003), we assume that $P_{2}$ is always positive and let the sense of drift be determined by the sign of $\hat{P}_{3}$.

$\hat{P}_{3}$ is usually determined from the longitude-resolved fluctuation spectra (LRFS). LRFS consist of 1D Fourier transforms of the pulse stacks, computed over the lines of constant rotational longitude $\phi$. When drift is present, LRFS have two peaks at Fourier frequencies of $\nu_{t}=\pm 1/\hat{P}_{3}$. In this case $P_{2}$ can be assessed from evolution of the complex phase of the Fourier transform with the longitude. If $S(\phi,\nu_{t})$ is the longitude-resolved Fourier transform of $s(\phi,t)$ computed over the lines of constant $\phi$, then from Eq. 1 it follows that

Selecting the peak with the observed positive phase gradient will fix the sign of $\hat{P}_{3}$.

For PSR B0943+10, $\hat{P}_{3}\approx-2.2P_{1}$ and $P_{2}=11\degree$, meaning that subpulses drift to the trailing edge of the onpulse window (Bilous, 2018). However, the absolute value of $\hat{P}_{3}$ is commonly used in the literature (e.g. Deshpande & Rankin, 2001, and other papers in this series). Since $\hat{P}_{3}$ is close to a small multiple of $P_{1}$, the apparent drift is not obvious: the subpulses exhibit even-odd modulation together with a slow shift towards the leading edge of the average profile (see Fig. 2).

Measuring $\hat{P}_{3}$ with LRFS does not solve the problem of aliasing. This problem is caused by the undersampling of the subpulse motion (subpulses can be observed only for a small fraction of pulse period) and by volatile subpulse shapes, which prevent tracing the path of each specific subpulse throughout a pulse sequence (van Leeuwen et al., 2003). The true drift period $P_{3}$ in presence of aliasing is related to observed $\hat{P}_{3}$ as

with the signed integer $n$ being the degree of aliasing. Note that the true direction of drift rate can be different from the observed one.

In the work of Deshpande & Rankin (2001), $n$ was determined to be 1 based on fleeting $37P_{1}$ modulation of subpulse intensity. This modulation was interpreted within the popular carousel model (Ruderman & Sutherland, 1975). In this model, pulsar emission comes from discrete spots located on a ring centered on magnetic pole. The slow rotation of the carousel causes the observed subpulse drift. Deshpande & Rankin argued that in their observations the individual subpulses retained their relative distribution of amplitudes for approximately $256P_{1}$, which allowed to trace individual subpulses and resolve aliasing. Such amplitude modulation is extremely rare, with only two other instances being found despite extensive searches (Backus et al., 2011; Bilous, 2018).

PSR B0943+10 is unique in its slow consistent evolution of the drift properties. Rankin & Suleymanova (2006) have found that PSR B0943+10ś subpulse drift rate changes exponentially by some 5% during several hours after B-mode onset with the characteristic time of about 73 min. The change is faster at B-mode onset. Later it was found that this behavior is independent of observing frequencies between 40 and 327 MHz and over the timescale of several years (Rankin & Suleymanova, 2006; Backus et al., 2011; Suleymanova & Pugachev, 2017; Bilous, 2018; Suleymanova et al., 2021).

In turn, emission in the Q-mode is stable in the sense that no gradual changes in its properties have been recorded. Despite systematic searches, no drifting subpulses have been detected in the Q-mode (Suleymanova & Izvekova, 1984; Backus et al., 2010), except for an occasional feature at 0.0275 cycles/$P_{1}$ corresponding to $\hat{P}_{3}=36.4P_{1}$ (Rankin & Suleymanova, 2006). The origin of this feature is unclear and no similar features have been found since then (Bilous, 2018).

In both modes the average pulse profile is composed of two components of equal width which are moving away from each other at lower frequencies. The full width at half maximum of the components is about $7\aas@@fstack{\circ}2$ for the B-mode and $11\aas@@fstack{\circ}5$ in Q-mode and does not show a strong dependence on frequency between 40 and 80 MHz (Bilous et al., 2014). Recently Suleymanova et al. (2021) have shown that the width of the average pulse components in the B-mode is constant over the wider frequency range of 62$-$1391 MHz with the separation between components increasing rapidly towards lower frequencies in accordance with a power law:

In B-mode the leading component is generally brighter than the trailing one, however the ratio of components’ peak amplitudes changes in a frequency-dependent manner as the mode evolves. For the first few minutes of the B-mode, the trailing component may be brighter, again in a frequency-dependent manner. The fiducial longitude (midpoint between profile components, corresponding to the moment of time when the line-of-sight (LOS) passes closest to the magnetic axis) is systematically delayed by about $1\aas@@fstack{\circ}4$ per mode, resetting at the next Q-to-B transition. The amplitude of delay does not depend on frequency at least between 40 and 327 MHz (Bilous et al., 2014; Suleymanova et al., 2021). No such effect has been identified for Q-mode.

Figure 2 shows examples of individual pulse sequences for some of our observations. In general, the difference between the B- and Q-modes is evident to the naked eye. In the B-mode, individual pulses form two columns corresponding to components of the average profile. The drift of subpulses near central longitudes is not visible due to a decrease in average pulse intensity in the saddle between the components. In the Q-mode subpulses are distributed randomly within the on-pulse window.

Panels 150518, L99010, L102418 and 260520 show unusual behavior of subpulses around Q-to-B mode transitions. When subpulse drift recommences after the end of Q-mode, for a short period of time subpulses drift through entire on-pulse window and, unlike later on, there is no fading of subpulses at central longitudes. During this period the on-pulse window is somewhat narrower than in well-established B-mode sequences.

These features allowed us to identify these particular sequences of pulses as a new mode, which we designated “Transitive B-like” mode, or Tb-modes. Tb-mode has been recorded in all available observations except for session L169237 in Bilous et al. (2014). Comparing with the sessions presented on Fig. 2, the S/N ratio of the pulsed signal is smaller there, and emission is modulated by scintillation, with one of the intensity troughs coinciding with Q-to-B transition (see Fig. 1 in Bilous et al., 2014).

On panel 260520, the subpulse drift pattern characteristic to Tb-mode is observed in the 30-second interval between B-mode pulses. There are no Q-mode pulses in this record, but the shape of the average B-mode pulse profile, with two equally bright components, indicates that the Q-to-B switch occurred a few minutes before the start of observing session (Rankin & Suleymanova, 2006; Suleymanova & Rankin, 2009). The presence of Tb-mode pulses within the B-mode sequence suggests that B-mode emission in the first minutes after mode onset is unstable.

The Tb-mode is not the only transitive phenomenon around Q-to-B switches. Some time before Q-mode cessation there appear groups of pulses that slowly drift towards the trailing edge of the on-pulse window. This drift is present in all but one Q-to-B transitions observed. We have labeled these slowly drifting pulses as Transitive Q-like modes, or Tq-modes.

Both Tb- and Tq-modes exhibit drifting subpulses and the presence of this drift was used to establish mode boundaries. However, the Tb-mode does not follow Tq-mode immediately. Between them there is a gap of several spin periods with very scarce single pulses resembling those of the Q-mode. Interestingly enough, the only observed B-Tb-B transition also exhibits the absence of emission for a few seconds before the Tb-mode starts (Fig. 2, panel 260520).

Because the transit time of PSR B0943+10 through the BSA beam is much smaller than the typical duration of its main modes, the records of mode switching are rare, with only five sessions with Q-to-B transitions being found in the archives. Subpulse intensities for these five records are shown in Fig. 3, with different modes shown by different colors. There are no Q-mode pulses in the 090112 session. The low values of the subpulse drift rate indicated that the first 42 pulses correspond to the Tb-mode, and the remaining pulses to the B-mode. In addition, the shape of the averaged pulse in B-mode with the ratio of components $R(2/1)=2.1$ clearly pointed out to the Q-to-B switch occurring shortly before the start of the observation session (Rankin & Suleymanova, 2006; Suleymanova & Rankin, 2009).

In five cases on Fig. 3 the Tq-mode is much shorter than the Tb-mode, and in one session (230920) it is absent completely. The duration of the Tb-mode varies between sessions, comprising 17$-$66 spin periods or 19$-$69 s. At lower frequencies, the two transitions recorded exhibit shorter Tb-modes – only 13 s, however this may be just a coincidence. More low-frequency observations are needed to explore the Tb-mode duration below 100 MHz.

To our surprise, the opposite, B-to-Q mode transitions were featureless (e.g. panel 221219 on Fig. 2). No transitive phenomena were found in any of our recordings.

Figure 8 shows the average profiles for all four modes in two LOFAR sessions with Q-to-B transition. The shape of the average profile in transitive modes varies between the sessions and is dominated by the individual strong pulses. Given the small number of modes and their relative shortness, it is hard to compare the average profiles in transitive modes with the regular B- and Q-modes, however it can be stated that neither the Tq- nor Tb-mode exhibit two distinct separate components like the B-mode.

PRAO observations offer larger mode samples and, also, longer instances of the Tb-mode. Figure 9 shows the average profiles integrated within B- and Tb-modes for the three PRAO sessions which exhibited prominent B-mode component peaks to facilitate alignment between sessions. The bottom panel presents integrated profiles for these three sessions, comprising 135 individual pulses for the Tb-mode and 390 pulses for the B-mode. It is evident that at 112 MHz the Tb-mode has no separate components and the profile is asymmetric and skewed to earlier spin longitudes.

Interestingly, the shape of the pulse profile in the averaged Tb-mode closely matches the composite Q+B profile recorded close to the B-to-Q transition (session 221219, Fig. 10a, b). In the session 221219, the B-to-Q switch happened in the middle of the observation. For comparison, the average Tb-mode profile from Fig. 9 was shifted by 5.7 ms toward earlier spin longitudes. This close resemblance of the pulse shapes could indicate that around the Q-to-B switch the regions responsible for the Q- and B-modes emit simultaneously and that their contribution is equal. This is possible only under the condition that the B- and Q-pulses are emitted from two independent regions in the pulsar magnetosphere. In the core/cone model (Rankin, 1983), the Q- and B- modes in PSR B0943+10 are associated with the core and conal beams of emission which emit alternately. The discovery of the transitive modes allow us to suggest that over some short time these two independent regions may emit simultaneously.

If the Tb-mode emission is indeed a superposition of the Q- and B-modes, then the 5.7 ms delay is naturally explained by the evolution of the on-pulse window location throughout the B-mode. It is known that during the lifetime of each B-mode instance, the average pulse shifts towards later longitudes. The total shift varies from 4 to 6 ms per B-mode duration (Bilous et al., 2014; Suleymanova & Rodin, 2014; Suleymanova & Pugachev, 2017). Thus, the Tb-mode emission is a superposition of late B-mode and the Q-mode. During the instantaneous Tb-to-B switch the emission window resets at earlier longitudes and the slow shift to the later longitudes continues during new B-mode instance. Within this interpretation it is hard to explain though the B-Tb-B sequence of the 260520 session, where both B-mode sequences framing the Tb-mode exhibited the properties of the early B-mode.

Observational evidence speaks against the Q-mode emission being produced by a core component in the core/cone model of the pulsar radio beam. First of all, this evidence includes the appearance of organized subpulse drift (Tq-mode) within the sequence of Q- mode pulses right before the Q-to-B switch and the crude similarities between the shapes of the average profile in both modes. Also, at frequencies lower than 100 MHz Q-mode profiles exhibit conal components that separate progressively (Suleymanova et al., 1998; Bilous et al., 2014). It was shown that for PSR B0943+10 the sightline makes only a grazing transverse of the polar cap (Deshpande & Rankin, 2001) and the scenario in which the observer misses most of the core radiation because of this peripheral sightline was previously put forward to explain the relative faintness of PSR B0943+10’s radio emission during its bright X-ray mode (Hermsen et al., 2013; Rankin et al., 2020). Another scenario can be proposed in which core radio beam is missed completely and all modes observed in PSR B0943+10 originate from conal beam shape variations. These variations are correlated with E$\times$B drift behaviour of the localized spark discharges.

Observations at 112 MHz show that the width of the on-pulse window stays similar throughout all four modes (Fig. 10). It is considered to be determined by the size of the polar cap with the radius $R_{\mathrm{out}}$. Within the framework of the traditional core/cone model of pulsar radio emission (e.g. Rankin, 1983, and other papers in this series), the width of the conal components in the average pulse profile is set by the thickness of the pair-producing ring in the polar gap: $W_{\mathrm{ring}}=R_{\mathrm{out}}-R_{\mathrm{inn}}$, where $R_{\mathrm{inn}}$ is the inner radius of the ring. We suggest that the transitions between modes are accompanied by variations in the inner radius. This can explain the gradual increasing of the component’s width over the B-mode lifetime and the appearance of drifting subpulses at the central longitudes in the Tb and Tq-modes. At B-mode onset the two components of the average pulse profile are well separated with intensity in the saddle region close to zero below 100 MHz. The width of the components is at its minimum and grows with B-mode age (Bilous et al., 2014). For the leading component 112 MHz, the width can increase by as much as 40% (Suleymanova & Rodin, 2014). This effect is well demonstrated in Fig. 10c, where the profiles of the average pulse for the onset and the end of the B-mode are shown. The width of the first component at the level of half maximum increases from 15 to 24 ms, i.e. by 60%. The same panel shows the profile of the average Tb-mode with the width of 30 ms. The total change in profile width between the Tb- and early B-mode is 100%. With such a significant increase in the width of $W_{\mathrm{ring}}$, central longitudes become illuminated for our line of sight and we observe drifting subpulses close to fiducial longitude in transitive modes.

Figure 10d shows the Tq-mode profile obtained at 112 MHz after averaging 20 pulses using sessions 150518 and 051112. It is worth noting that because of the short duration of the mode its average profile is distorted by bright pulses. However, comparing the Tq-mode to the Q-mode profile (Fig. 10a) reveals similarity in profile shape as far as they are both skewed to the trailing side (unlike B-mode). Similar behaviour is observed in LOFAR data (Fig. 8).

1. 1.

   This mode precedes the B-mode onset by 40$\pm$25 s. The duration of the Tb-modes exhibit larger variations from one instance to another, ranging between $12P_{1}$ and $66P_{1}$.

2. 2.

   Similarly to B-mode, the individual subpulses drift from the leading to trailing edge of the onpulse window, however the drift extends now through the whole onpulse window, including central spin longitudes.

3. 3.

   The amplitude modulation frequency in the Tb-mode varies in the range of 0.428$-$0.439 cycles/$P_{1}$ with the mean value of 0.434$\pm$0.005 cycles/$P_{1}$ that is significantly lower than $f_{3}$ values measured for the B-mode. Nevertheless, these values are in accordance with the power-law $f_{3}$ evolution throughout B-mode.

4. 4.

   The average pulse profile in the Tb-mode is skewed to the leading edge, similar to the B-mode, however it does not have resolved conal components.

5. 5.

   Sometimes the Tb-mode happens within an early B-mode instance.

1. 1.

   The Tq-mode occurs within the late Q-mode and, unlike the latter, has a specific subpulse drift pattern. This mode precedes the Tb-mode onset by 34$\pm$7 s and lasts for about $10-22P_{1}$. In one session no Tq-mode was recorded, however the S/N of the recording was quite low.

2. 2.

   As in the Tb- and B-modes, subpulses in the Tq-mode drift towards the trailing edge of the onpulse window. The drift frequency $f_{3}$ varies from 0.149 to 0.312 cycles/$P_{1}$ from one observation to another, with sometimes large variation of $f_{3}$ within the mode.

3. 3.

   The average pulse profile in the Tq-mode has a tendency to be skewed to the trailing edge which is characteristic of the Q-mode.