A characteristic property of pulsars is pulsed periodic radio emission, which has a high stability of periods. Despite the high stability of the emission periods of pulsars, monitoring the time of arrival of pulses (timing) shows the presence of different types of irregularities: variations of residual deviations, changes in the shape of the pulse, switching on and off of radio emission, and rotation discontinuities. Numerous observations of the radio emission of pulsars indicate that they are caused mainly by processes occurring in the pulsar’s magnetosphere. The special interest causes the observations of a pulsar in the Crab Nebula, performed, in particular, at Jodrell Bank and Pushchino Radio Astronomy Observatory of Lebedev Physical Institute. The connection between the scattering of radio pulses and the measure of the pulsar dispersion, which was established earlier in Pushchino together with Jodrell Bank, has been confirmed. The observed variations in the scattering of radio pulses and their partial correlation with the dispersion measure are explained by the eclipse of the pulsar by plasma clouds with electron density fluctuations significantly exceeding the corresponding fluctuations in the interstellar medium. The question of a possible connection between glitches, dispersion measure variations, radio pulses scattering, and gamma-ray flares is discussed.
The current field of research in astrophysics is the study of the effect of rapid rotation on the properties of various physical systems. Such physical systems can be attributed to pulsars. Pulsars are magnetized neutron stars that are formed as a result of supernova explosions (Shklovskii 1984). Observations of pulsars have shown that they are rapidly rotating objects with a period from milliseconds to several tens of seconds. The rapid rotation of pulsars is a consequence of the law of conservation of angular momentum during the collapse of a star. A magnetized rotating neutron star creates a powerful electric field. Moving along closed magnetic lines of force, charged particles create a pulsar magnetosphere that propagates to the light cylinder, where the speed of rotation is equal to the speed of light. Within the light cylinder, the plasma rotates together with the pulsar. In contrast, charges in open lines of force accelerated by an electric field to relativistic energies leave the magnetosphere and stimulate curvature radiation. The angle between the rotation axis and the magnetic axis turns the pulsar into a cosmic beacon. A characteristic property of pulsars is pulsed periodic radio emission, which has a high period stability (Figure 1).
Pulsars gradually slow down as a result of the transition of rotation energy to the energy of charged particles and electromagnetic waves. The simple model uses the rotation period P, the first derivative of the period Ṗ, and the measurement epoch in Julian days JD. The actual moments of arrival of a certain pulse of the observed pulsar to the telescope in the coordinated universal time (UTC) scale are recalculated to the barycenter of the solar system in dynamic barycentric time (TDB) t s, taking into account a number of corrections (relativistic, dispersion, etc.) (Zharov et al. 2019). The theoretical moments of arrival t d of the same pulse in the barycenter, assuming a quadratic dependence of the time of arrival of the pulse on the pulse number N, are calculated by the formula: t d = t o + P o N + 1/2 P o ṖN 2, where P o is the period and Ṗ is the period derivative at the initial time t o, corresponding to the epoch JDo. The P o and other parameters are taken from the pulsar catalog.
In the same model of pulsar rotation frequency υ, the first derivative of the rotation frequency ύ and the second derivative of the rotation frequency ϋ are used. In this case, power-law slowdown has the form ύ = −kυ n . The value of the exponent of υ is known as braking index n, k is usually taken as a constant. k ∼ M 2 sin2 α/3Ic3, where M is magnetic dipole moment, α is the angle of inclination between magnetic and rotation axes, and I is the moment of inertia.
The value of the braking index is n = υ ϋ/ύ 2.
The braking index n is expected to be 3 in the case of braking by pure dipole radiation and 1 in the case of pulsar wind-dominated torque. The slowdown can be represented by a combination of two power laws ύ = −(Aυ 3 + Bυ).
For example, for the Crab pulsar the observed value n obs ∼ 2.5 and the wind accounts ∼ 1/3 of slowdown torque. For the Vela pulsar n obs ∼ 1.4 and wind is responsible for ∼ 4/5 of the slowdown torque (Lyne et al. 2015).
The difference between the actual time of arrival of the pulse (t s) and the calculated time (t d) is called the residual deviation. The normal pulsars are stable to an accuracy of one part in 1011 or more with the taking into the account ύ. Despite the high stability of the pulsar radiation periods, monitoring the pulse arrival time (timing) shows the presence of different types of irregularities: variations in residual deviations, changes in the pulse shape (mode), nonstationary operation (stopping and resuming radio emission), and period failures (glitches). It is shown by Lyne et al. (2010) that all of them are caused by disturbances in the pulsar magnetosphere. We will discuss these types of irregularities in this study. The special attention will be made to the pulsar in the Crab nebula. The discussion will be devoted to the problem of a possible connection between glitches, dispersion measure variations, radio pulses scattering, and gamma-ray flares.
2 Variations of the residual deviations
An exhaustive analysis of the timing irregularities observed for 366 pulsars in the range from 10 to 36 years was performed on the 76 m radio telescope of the Jodrell Bank Observatory (Hobbs et al. 2010). Observations were made at frequencies of 235, 325, 408, 610, 910, 1,410, and 1,630 MHz. The time of arrival of pulses at each frequency was determined according to the standard procedure using a template. Timing was performed according to the TEMPO2 pulsar timing software (Hobbs et al. 2006). The analysis included both normal and recycled pulsars. Figure 2 shows, as an example, the residual deviations for pulsars B1540-06, B1642-03, B1818-04, B1826-17, B1828-11, and B1848 + 63, for which the quasi-periodic structures are most pronounced.
Power spectra were obtained for each pulsar. The power spectra for the above pulsars are shown in Figure 3.
The analysis of the measurement results shows that variations in the arrival time are quasi-periodic with characteristic periods from 1 to 10 years and are not associated with errors in the monitoring and data processing system. As the observation interval increases, new periodicities appear. Quasi-periodic structures in timing residual are clearly visible for pulsars B1540-06, B1826-17, B1828-11, B2148 + 63, and typical for many other pulsars. Hobbs et al. (2010) consider that the observations do not confirm hypothesis than that quasi-periodic structures caused by low-frequency noise processes, planetary companions or free precession of the neutron star.
3 Mode switching pulsars
A separate class consists of pulsars with mode switching, in which the average profile is shown in two types and sometimes in three types. The problem of tipping points is well known in the theory of nonlinear complex systems, for example, in climate science. This is a situation where small disturbances lead to significant changes in the system and lead to a different quasi – stable state. The situation is reversible. It is obvious that the process of radio emission of many pulsars is subject to such disturbances. What causes these perturbations, why do they have two or more stable states, and what determines the duration of their state? Currently, these are unanswerable questions (Wang et al. 2007).
In the study by Lyne et al. (2010), integral profiles of pulses for six pulsars J2043 + 2740, B2035 + 36, B1828-11, B0740-28, B1540-06, and B1822-09 at 1,400 MHz with shape variations are given (Figure 4).
For these pulsars, variations in the derivative of the rotation frequency |ύ| correlate with the width of the pulse profile. That correlation indicates that the causes of these phenomena are linked and, as later will be shown, are magnetospheric in origin (Figure 5).
The cross-correlation function confirms a high degree of correlation between the width of the pulse profile and parameter ύ. Figure 6 shows the Lomb-Scargle power spectra of ύ.
A comparison of Figures 3 and 6 shows the similarity of the spectra of the parameter ύ and power spectra of the residual deviations. It follows that the quasi-periodic variations of the residual deviations, the derivative of the rotation frequency, and the pulse shape are interdependent. This indicates that they are apparently controlled by a single mechanism, magnetospheric in origin. The pulsar PSR B1931 + 24 can be considered as an example of the interaction between the pulsar and the magnetosphere, in which the flow of charged particles controls the braking of the pulsar (Kramer et al. 2006). This was supported by Kou and Tong (2015). They show for the Crab pulsar, when the outflow of particles increases, the effect of particle wind becomes stronger, and the braking index must be smaller and vice versa.
4 Intermittent radio pulsars
The PSR B1931 + 24(J1933 + 2421) emits for 5–10 days as a normal radio pulsar, then turns off for 25–30 days in less than 10 s, and then turns on again. This process is repeated quasi-periodically. It should be noted that during radio emission, there is a ∼50% increase in the spin-down rate of neutron star (Kramer et al. 2006).
According to the existing concepts, the magnetosphere of a neutron star is filled with electron–positron plasma, and radio emission is generated by a stream of these charged particles. The termination of radio emission may be due to the termination of plasma generation in the magnetosphere, and the beginning of radio emission may be due to the resumption of plasma generation (Istomin and Sob’yanin 2010).
In addition to switching off radio pulsars, a group of the so-called nulling pulsars is known, which also do not have radio emission for a certain period of time, but not as regularly as those that turn off, and for which the difference in rotation deceleration has not yet been measured. Nulling lasts from seconds to hours and even days. Wang et al. (2007) suggest that nulling pulsars are a type of mode-switching pulsars in which, after changing the current distribution in the magnetosphere, the radiation direction, and intensity change, and, as a result, the radiation pattern (mode) changes. Most of the nulling pulsars in the P − Ṗ diagram are located near the “line of death” (cessation of radio emission) (Keane 2010). Therefore, the presence of nulling may also indicate the aging of the pulsar, which leads to a failure of the radio emission mechanism (Lyne and Grahm-Smith 2006). According to the estimates of Wang et al. (2007), the flux density in nulling is less than 1% of the average intensity. For example, according to Esamdin et al. (2005), for the PSR B0826-34, the flux density in the passive phase was 2% of the density in the active phase (Figure 7).
In the theory of pulsar radio emission, the angle α between the direction of the magnetic moment vector and the rotation axis of the neutron star plays an important role. We can expect that pulsars with small α angles will have inter-pulse radiation. An example is the same pulsar PSR B0826-34. Malov (2004) found 24 pulsars with small α values. McLaughlin et al. (2006) reported the discovery of radio transients. They are characterized by one or several millisecond pulses, after which there is a break from several minutes to hours. The analysis of the arrival time showed the presence of periodicity in the radio emission of radio transients, indicating that the radio transients belong to neutron stars. The gamma-ray pulsar Geminga B0633 + 17 can also be attributed to the number of transient radio emitters. Maan (2015) reported detecting bursts of radio emission from the B0633 + 17 pulsar at a frequency of 34 MHz using the Gauribidanur radio telescope (India). Observations were made from June 23, 2012, to April 18, 2013. Radio bursts were detected on July 13 and 15, 2012. The strongest radio burst with a high signal-to-noise ratio of over 200, a flux density of 2,350 Jy, and a dispersion measure of ∼2.1 pc/cm3 was recorded on July 13. The dispersion measure for various bursts is changing from 1.4 to 3.6 pc/cm3. According to the author’s estimates, the probability that the bursts come from the Geminga pulsar is almost 100%. The author believes that radio bursts can be giant pulses, since their energy is comparable to the energy of giant radio bursts from the pulsar in the Crab nebula on decameter waves, and the energy distribution obeys a power law.
5 Rotation discontinuities
Discontinuous irregularities in the pulsar rotation are known as glitches. Espinoza et al. (2011) report that, by the time their article was published in 2011 at the Jodrell Bank Observatory, more than 700 pulsars were observed, using the 76 m radio telescope, and 128 new glitches were registered in 63 pulsars. Based on the previously published data, there were 315 glitches in 102 pulsars. Examples of discrete glitches are shown in Figure 8 (Lyne et al. 2009).
Glitches can be of two types: discrete glitches and slow glitches. Both types of glitches cause accelerated rotation of the pulsar, which occurs against the background of secular deceleration of the rotation of the neutron star. In discrete glitches, the frequency increases suddenly, followed by an exponential decrease in frequency to the previous value. Slow glitches are associated with slow frequency fluctuations. Examples of slow and discrete glitches are shown in Figure 9 (Shabanova et al. 2013).
The results of timing of 27 pulsars at the Pushchino Observatory for 33.5 years in the period from 1978 to 2012 and 10 pulsars from the Jet Propulsion Laboratory archive for 43.5 years are presented in this work. The presence of the aforementioned irregularities is confirmed: discrete glitches, slow glitches, and quasi-periodic oscillations. Zou et al. (2004) confirmed the report of Shabanova and Urama (2000) about slow glitches for the pulsar J1825-0935 (B1822-09) and also for J1835-1106. However, since the observed residual deviations of these pulsars during glitches are similar to the timing variations of other pulsars, Hobbs et al. (2010) suggest that slow-glitches are not a unique phenomenon, but they are caused by the same processes as the timing noise seen in pulsars, caused by magnetospheric processes. Cadez et al. (2016) devoted their work to the study of the pulsar in the Crab nebula using radio data from the Jodrell Bank Observatory from 1988 to 2014 together with optical observations of the pulsar using the ultra-fast photon counter installed on the Copernicus telescope at Asiago Observatory (Italy) in October 2008 and on the European Southern Observatory telescope at La Silla (Chile) in 2009. Data analysis has shown that jumps in the braking index n = υϋ/ύ 2 are mainly associated with large glitches when the relative change in the speed of rotation exceeds 10−8. A similar conclusion about the relationship of the braking index with glitches was made by Lyne et al. (2015).
The delay in the variations of the dispersion measure relative to the variations of the braking index is ∼1,010 days and is explained by the time of ionization of the nebula by the pulsar wind (Figure 10).
Glitches and subsequent variations in the braking index are caused by instability in the magnetosphere, which changes the configuration of the magnetic field and the currents in the plasma through which the pulsar interacts with the nebula. Disturbances in the magnetosphere cause powerful flows of relativistic particles in the form of pulsar wind and polar jets. It should be recalled the hypothesis about the connection of glitches with processes in the magnetosphere which was expressed in the book by Manchester and Taylor, “PULSARS”: “there is a possibility that period irregularities occur due to small changes in the structure of the magnetosphere” (Manchester and Taylor 1980). Active processes in the Crab nebula allow us to study the dynamics of radio emission scattering on inhomogeneities of interstellar plasma by receiving and analyzing giant pulses from the B0531 + 21 pulsar at the 111 MHz at the Large Phased Array of Pushchino Radio Astronomy Observatory (LPA PRAO) of the Lebedev Physical Institute, (Moscow region, Puschino). Radio emission from the pulsar in the crab nebula is regularly monitored at the LPA. Giant pulses (GP) of the pulsar in the Crab nebula are analyzed using a special program that allows us to determine the amount of scattering by simulating the passage of the pulse through the scattering medium (Alurkar et al. 1986). The results of the analysis of the GP scattering time scale are compared with the dispersion measure measurements of the Crab pulsar continuously made by Jodrell Bank Observatory (Lyne et al. 1993). The relationship between the scattering time scale and the dispersion measure is clearly seen (Figure 11). The correlation coefficient is 0.80 ± 0.05. Partial correlation with the dispersion measure is explained by the eclipse of the pulsar by plasma clouds with electron density fluctuations significantly exceeding the corresponding fluctuations in the interstellar medium (Losovsky et al. 2019).
There are some models, describing the origin of glitches. One model treats glitches as starquakes, caused by the rearrangement of the flattened crust, which tends to become spherical, as the star’s rotation slows down. Another model considers a neutron star as a reservoir, filled with a superfluid, the mass of which, when the pulsar’s rotation slows down, transmits the angular momentum to the crust, which leads to a glitch (Espinoza et al. 2011). The model of Cadez et al. (2016) suggests that glitches and subsequent variations in the braking index are caused by instability in the magnetosphere, which changes the configuration of the magnetic field and the currents in the plasma, through which the pulsar interacts with the nebula. Perturbations in the pulsar’s magnetosphere can lead to glitches and jumps in the braking index, the ejection of charged particles into the nebula, the formation of filaments, and an increase in the dispersion measure. The release of energetic particles can increase plasma ionization and lead to an increase in the dispersion measure and scattering time scale. Intermediary models have been discussed by Kou and Tong (2015). They consider that glitches are related to changes in the interior of the neutron stars, but may lead to some effects in the outer magnetosphere. In turn, plasma instabilities in the nebula can disrupt the configuration of magnetic field lines, when fields of opposite polarity are pressed together and cause gamma-ray flare (Huang et al. 2021), similar to flares in the solar corona (Priest and Forbes 2000). The same opinion says Buhler and Blandford (2014): “The gamma-ray flares are therefore likely connected to explosive reconnection events triggered by current instabilities.” Striani et al. 2013 also support this idea, but they note: “However, evidence for magnetic field reconnection events in the Crab nebula is elusive, and no optical or X-ray emission in coincidence with the gamma-ray flaring has been unambiguously detected to date.” According to our data (Figure 11), gamma-flares have a tendency to concentrate during the period of enhanced disturbances in the Crab nebula, which can be considered as support of this model. As for the correlation between pulsar glitches and gamma-ray flares, Buhler and Blandford (2014) write: “The time scale of the recurrence of pulsar glitches is similar to the recurrence of the gamma – ray flares, however, there is no obvious correlation in time between these two events.”
The following types of anomalous phenomena of pulsar radio emission are considered: variations of residual deviations, changes in the pulse shape, switching on and off of radio emission, glitches, and changes in the braking index, dispersion measure, and scattering time scale. The connection established between them indicates a common mechanism for their generation in the pulsar’s magnetosphere and propagation of disturbances in the environment. The release of energetic particles can increase plasma ionization and lead to an increase in the dispersion measure and scattering time scale. In turn, plasma instabilities can disrupt the configuration of magnetic field lines and cause gamma-ray flares. This fact closes the chain of communication between glitches, changes in the braking index, variations in the dispersion measure, and scattering time scale of radio waves, and, finally, gamma-ray flares.
This work was supported by the research program of the Presidium of the Russian Academy of Sciences “Non-stationary phenomena in the objects of the Universe.” Many thanks to I.F. Malov for reading the manuscript and useful remarks and A.S. Losovsky for the editing of the manuscript. The author thanks the staff of the meter wave – length technical service laboratory of the Pushchino Radio Astronomy Observatory for providing reliable observations.
Conflict of interest: The author confirms that this article content has no conflict of interest.
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