Statistical study of neutron star glitches

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(1)STATISTICAL STUDY OF NEUTRON STAR GLITCHES. Thesis by. José Rafael Fuentes Baeza Thesis Advisors: Professor Andreas Reisenegger Doctor Cristóbal Espinoza. In Partial Fulfillment of the Requirements for the Degree of Master of Astrophysics. INSTITUTE OF ASTROPHYSICS PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE Santiago, Chile. 2018 Defended July 12, 2018.

(2) ii. © 2018 All rights reserved.

(3) iii. ACKNOWLEDGEMENTS Agradezco a mis profesores Cristóbal Espinoza y Andreas Reisenegger, los cuales desde hace más de dos años han contribuido de forma invaluable a mi formación como científico. Sin duda, sin sus exigencias, críticas constructivas, disposición, y sobre todo paciencia, este trabajo jamás habría sido posible. Al Concurso de Investigación para Pregrado, que en el año 2015 que me permitió comenzar este trabajo. Finalmente, agradezco a mi pareja, Constanza, por acompañarme y soportarme a lo largo de toda esta etapa. This work was funded by ALMA-CONICYT Astronomy/PCI Project 31140029, FONDECYT Regular Projects 1150411 and 1171421, CONICYT Basal Funding Grant PFB-06, CONICYT grant PIA ACT1405, and a SOCHIAS travel grant through ALMA/Conicyt Project #31150039. We thank the observers at Jodrell Bank Observatory. Pulsar research at Jodrell Bank Centre for Astrophysics (JBCA) is supported by a consolidated grant from the UK Science and Technology Facilities Council (STFC)..

(4) iv. TABLE OF CONTENTS. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter II: The glitch activity of neutron stars . . . . . . . . . . . . . . . . . . . . 2.1 THE NEW DATABASE . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 THE GLITCH SIZE DISTRIBUTION . . . . . . . . . . . . . . . . . . . 2.3 GLITCH ACTIVITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter III: Time series of neutron star glitches . . . . . . . . . . . . . . . . . . . 3.1 THE PULSARS WITH MORE GLITCHES DETECTED . . . . . . . . . 3.2 TIME SERIES CORRELATIONS: GLITCH SIZE AND TIME TO THE FOLLOWING GLITCH . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 OTHER CORRELATIONS . . . . . . . . . . . . . . . . . . . . . . . . . Chapter IV: SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Model selection: Akaike’s information criterion applied to the glitch size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B: Error estimation of the glitch activity . . . . . . . . . . . . . . . . . .. iii iv 1 8 8 10 12 20 24 24 26 34 37 39 41 44.

(5) 1 Chapter 1. INTRODUCTION In this introduction, I will give a brief background on neutron stars and the glitch phenomenon. The latter will be focused on how glitches can be produced and the importance of their study, which motivated this work. Neutron stars are the remnants of supernova explosions of massive stars. Once the core of the progenitor star has completely burned to iron, there are no more sources of energy to prevent the gravitational collapse. As the core rapidly collapses due to gravity, electrons and protons “squeeze” together forming neutrons and neutrinos. The neutrinos easily escape from the contracting core but the neutrons pack closer together until their density is similar to that of an atomic nucleus. At this point, the neutrons occupy the smallest space possible and become degenerate. A quantum pressure arises that is able to compete against gravity, thereby preventing the gravitational collapse. Therefore, a remnant core supported against its own gravity by the degeneracy pressure of neutrons (hereafter a neutron star) is born. These characteristics make neutron stars one of the densest objects in the Universe and provide a unique environment to study the physics of extremely dense matter. Moving inside from the neutron star surface, one can distinguish qualitatively different layers of matter: 1. The outer crust 106 g cm−3 . ρ . 4 × 1011 g cm−3 , corresponding to solid region in which a Coulomb lattice of heavy nuclei coexists with a relativistic degenerate electron gas. 11 −3 14 −3 2. The inner crust 4 × 10 g cm . ρ . 2 × 10 g cm , which consists of a lattice of neutron-rich nuclei together with a fluid of electrons and neutrons, the latter in a superfluid state. 3. The outer core 2 × 1014 g cm−3 . ρ . 1015 g cm−3 , composed mostly of superfluid neutrons and a few % of superconducting protons, relativistic electrons, and muons..

(6) 2 . . 4. The inner core ρ & 1015 g cm−3 , which is largely unknown, but likely a liquid containing exotic particles, such as mesons, hyperons and free quarks. For more details on the physics of neutron stars, see, for instance Shapiro & Teukolsky (1983). Under certain conditions, neutron stars can be directly observed. For instance, several neutron stars have been found at the centers of supernova remnants emitting X-rays. In binary systems, neutron stars can be found accreting material from their companions, emitting X-rays powered by the gravitational energy of the accreting material. More often, neutron stars are found rotating very fast with extreme magnetic fields, which accelerate jets of particles along the magnetic poles producing very powerful beams of electromagnetic radiation. If the magnetic field axis is not aligned with the rotation axis, those jets of particles and light are swept around as the star rotates. When the beam crosses our line of sight, we observe a pulse. In other words, we see neutron stars turn on and off as their beams sweep over Earth in each rotation period. This kind of neutron stars are known as pulsars (Figure 1.1).. Figure 1.1: Chart record of individual pulses from one of the first pulsars discovered, PSR B0329+54. The pulses occur at regular intervals of about 0.714 s. [From Pulsars by Richard N. Manchester and Joseph H. Taylor, W. H. Freeman and Company. Copyright © 1977.]. The pulses can be tracked with high precision, allowing to measure the rotation period of these stars, P = 2π/Ω, where Ω is the angular velocity, and their time derivative, Ṗ very precisely. The spin-down process is usually modeled in terms of a magnetic dipole rotating in vacuum, which loses rotation energy through electromagnetic radiation according to.

(7) 3 the relation IΩΩ̇ ∝ −µ2 Ω4 ,. (1.1). where dots indicate time derivatives, I is the moment of inertia, and µ is the magnetic moment of the star. This allows to estimate several properties of neutron stars. For instance, we can estimate the magnetic field B ∝ (P Ṗ) 1/2 at the surface of the star. For all the neutron stars observed as pulsars, the magnetic field is B ∼ 108 − 1015 G. Also, we can estimate the spin-down time, τc = P/2 Ṗ, as a rough estimate of the stellar age (by assuming that µ = constant and the initial period was much faster than the present one), which roughly agrees (for neutron stars with τc . 10 kyr) with historical records of the supernova explosion. For example, the Crab pulsar has P = 0.033 s and Ṗ = 4.2 × 10−13 , which implies τc = 1260 yr. The historical record of the supernova explosion was in A. D. 1054 (Ho & Andersson 2012), corresponding to 964 yr. Most pulsars (the so-called “classical pulsars”) have rotation periods P ∼ 0.01−8 s, and inferred magnetic field strength B ∼ 1010 − 1013 G. The oldest, but most rapidly spinning, have periods P ∼ 10−3 − 10−2 s, and magnetic field strengths as low as 108 − 109 G, and are called “millisecond pulsars”. In general, classical and millisecond pulsars do not exhibit strong radiative events, and their observed luminosities are in agreement with the rotational energy loss rate given by Eq. (1.1). This would make these objects be “rotation-powered pulsars”. On the other hand, there is a population of strongly magnetized (B ∼ 1014 − 1015 G), but slowly rotating (P ∼ 2 − 12 s) pulsars. Those are known as soft gamma-ray repeaters (SGRs) and anomalous X-ray pulsars (AXPs). These objects are distinguishable from classical and millisecond pulsars, because they exhibit strong radiative events (such as burst and flares) and their observed X-ray luminosities are much larger than the rotational energy loss rate defined by Eq. (1.1). Therefore, an additional source of energy is required, the most likely being the decay of their magnetic field (Thompson & Duncan 1995, 1996). This would make these objects be magnetically powered stars, or “magnetars”. The rotation of neutron stars is very stable, and, as mentioned earlier, it shows a spindown trend that often can be described, over many years, by a simple model. However, several of these stars show “glitches”, sudden increases in the rotation frequency, usually accompanied by a change in the spin-down rate (Figure 1.2). The first glitch was observed in the Vela pulsar (PSR B0833−45) in March 1969,.

(8) 4 Rotation history of PSR B0355+54 6.39468. 6.39465. Frequency ν (Hz). 6.39462. 6.39460. 6.39457. 6.39455. 6.39452. 6.39450 42000. 44000. 46000. 48000. 50000. MJD (days). 52000. 54000. Figure 1.2: Rotation frequency as a function of time for the PSR B0533+54. The data comes from the Jodrell Bank Observatory (JBO) and was provided by Cristóbal Espinoza.. corresponding to a fractional change in the frequency ∆ν/ν ∼ 2.2 × 10−6 (Radhakrishnan & Manchester 1969). Furthermore, the glitch was also accompanied by a change in the spin-down rate ∆ ν̇/ν̇ ∼ 10−2 (Reichley & Downs 1969). The first explanation for this glitch was a “starquake” (Ruderman 1969; Baym et al. 1969). This scenario assumes that neutron stars are born with a high angular velocity which makes their shape oblate, but as the star spins down, centrifugal forces on the crust decrease, and gravity pulls it towards a less oblate shape. However, the rigidity of the crust resists the change in the shape of the star, thereby stressing it until the maximum strain that it can support is reached. Beyond this point, the crust cracks, the stress is relieved, and the excess oblateness due to the crust rigidity, is reduced. By conservation of angular momentum, a decrease ∆I in the moment of inertia of the star produces an increase ∆ν in the star’s rotation frequency, giving origin to a glitch. From this model, a simple calculation gives ∆I ∆ν = , I ν. (1.2). where I and ν, are the moment of inertia and the rotation frequency before the starquake,.

(9) 5 respectively. For a canonical neutron star (M ∼ 1.4M and R ∼ 106 cm), the sudden decrease in the moment of inertia responsible for the glitch observed in the Vela pulsar would correspond to an effective decrease of ∆R ∼ 1 cm in the equatorial radius of the star. Further, if the external magnetic torque remains constant during the glitch, the change in the spin-down rate should be ∆I ∆ ν̇ = , I ν̇. (1.3). where ν̇ is the spin-down rate before the starquake. The model predicts ∆ν/ν ' ∆ ν̇/ν̇. Since the observed ∆ ν̇/ν̇ ∆ν/ν, the starquake model cannot explain the glitch observed in the Vela pulsar. Another prediction of the starquake model was that the crust needs from hundred to thousand of years to accumulate the necessary stresses to cause another starquake. However, 2.5 years after the first glitch observed in the Vela pulsar, Reichley & Downs (1971) announced the discovery of a new glitch with ∆ν/ν ∼ 2 × 10−6 . This detection ruled out the starquake model as the origin of glitches, and alternative models were needed to explain these phenomena. The alternative model (and nowadays, the standard scenario of glitches) relies on the prediction that neutrons in the interior of a neutron star are in a superfluid state. In particular, glitches are believed to be caused by a rapid transfer of angular momentum from a superfluid of neutrons inside the star, to the external solid crust and all the components coupled to it (Anderson & Itoh 1975; Pines & Alpar 1985). In this picture, as a neutron star’s crust spins down under magnetic torque, a fraction of the superfluid of neutrons in the interior keeps rotating faster, weakly affected by the external magnetic torque. As a result, a differential rotation develops between the outer crust and a fraction of the neutron superfluid. The more rapidly rotating component acts as a reservoir of angular momentum that is occasionally depleted to accelerate the rest of the star, when a critical state of the velocity lag is reached, giving origin to a glitch. The study of glitches is a direct way to test for the presence of superfluids inside neutron stars and investigate their properties. Further, glitches can be used to constrain the structural properties of neutron stars and the equation of state of dense matter. For example, Link et al. (1999) calculated that for the Vela pulsar, at least 1.4% of the star’s moment of inertia is necessary to produce the observed glitches. Glitches are not periodic events and only a few objects exhibit more than one glitch per.

(10) 6 year. This fact complicates a deep study of glitches, both theoretically and statistically, in individual pulsars. However, since the start of long-term observing programs of pulsars, the number of glitches detected has increased, revealing statistical trends in the glitch behavior of a population of pulsars. For instance, Lyne et al. (2000) showed that the rate of spin-ups due to glitches, ν̇g (glitch activity), increases with | ν̇| in pulsars with | ν̇| between 10−14 and 10−11 Hz s−1 . Espinoza et al. (2011) confirmed this result by using a larger sample, and they found that the glitch activity peaks for pulsars with a characteristic age τc ∼ 10 kyr, and decreases for longer values of τc . They also found that ν̇g is very small in the youngest pulsars (τc ∼ 1 kyr). Despite the small number of glitches detected in individual pulsars, different efforts have been made to study the distributions of glitches and the correlations between sizes and inter-glitch times in individual pulsars. For the latter, a remarkable result is the strong correlation between the glitch size and the waiting time until the next glitch, exhibited by the PSR J0537−6910 (Middleditch et al. 2006; Ferdman et al. 2018; Antonopoulou et al. 2018). On the other hand, Melatos et al. (2008) obtained that the glitch size distribution in individual pulsars is consistent with a power law, and the waiting time distribution an exponential. The only exceptions are PSR J0537−6910 and PSR B0833−45 (the Vela pulsar), which exhibit large glitches of similar sizes in more regular intervals of time. However, these distributions are strongly dependent on the completeness of the glitch sample, which is limited by detection capabilities. Therefore, the distributions and the best models obtained by Melatos et al. (2008) might not be representative of the real distributions and must be considered only as reference distributions of the actual data. Espinoza et al. (2014), using 29 years of daily observations of the Crab pulsar, showed that its glitch size distribution is consistent with either a power law or a log-normal distribution. Further, they found a decrease below 0.05 µHz in the glitch size distribution, which suggests that glitches in neutron stars could have a minimum size..

(11) 7 This thesis is structured in the following way: Chapter 2 reproduces the paper by Fuentes et al. (2017), presenting a statistical study of the glitch activity in a population of radio pulsars and magnetars. In chapter 3, we focus on the glitch behavior of individual pulsars. In particular, we study the sequence of glitches as a time series in the seven pulsars known with at least 10 glitches detected. Finally, chapter 4 shows the main Conclusions of both chapters..

(12) 8 Chapter 2. THE GLITCH ACTIVITY OF NEUTRON STARS The work presented in this chapter (and published as Fuentes et al. 2017) is an extension of the previous study by Espinoza et al. (2011). It presents the building and analysis of a sample of glitches. The events were taken from published systematic records of timing observations of hundreds of pulsars. We focus on the frequency step sizes and their rate (the glitch activity) and study how they depend on long-term spin properties (spin frequency, spin-down rate, and combinations of these, such as energy-loss rate, magnetic field, and spin-down age). This chapter is organized as follows: Section 2.1 describes the new database and how neutron stars and glitch detections were selected to avoid bias in our sample. In §2.2 we analyze the glitch size distribution and classify glitches according to their sizes. Section 2.3 presents a study of the cumulative effect of glitches on the rotation of neutron stars and a discussion of the relation between the glitch activity and the spin-down rate. Finally, §2.4 shows a Discussion of the main results obtained in this chapter. 2.1. THE NEW DATABASE. Today, more than 700 pulsars are regularly monitored at the Jodrell Bank Observatory (JBO), some of them from as early as 1978 (Hobbs et al. 2004). These long-term observations are essential to finding glitches and studying their properties. In order to build a sample that is as unbiased as possible, we included all pulsars that have been regularly monitored for glitches in clearly defined time spans, regardless of whether glitches were found or not. Selecting only those pulsars for which glitches have been detected would bias the sample towards the presence of glitches. According to this scheme, we included 778 pulsars monitored at JBO, containing 296 glitches in the rotation of 111 pulsars. These glitches are the JBO events in (Espinoza et al. 2011) plus 69 newer glitches measured until 2015 and published in the JBO online glitch catalog 1. In order to expand the sample, we also included observations of 118 pulsars, performed 1 http://www.jb.man.ac.uk/pulsar/glitches.html.

(13) 9 by the Parkes observatory, which include 73 gliches in the rotation of 23 pulsars, as reported by Yu et al. (2013). In case of overlap between the observations spans of the JBO and Parkes pulsars, we considered the earliest and the latest epoch between the two to define the start and end of the searched time spans. In order to improve the statistics for pulsars with small characteristic ages (τc = P/2 Ṗ = ν/2| ν̇|), we also added the two X-ray pulsars PSRs J1846−0258 and B0540−69, which have been monitored for about 15 years each and have been searched for glitches (Livingstone et al. 2011; Ferdman et al. 2015). With these two additions, the database contains all the rotation-powered pulsars known with τc < 2 kyr. It is not possible to obtain a complete sample for pulsars of larger characteristic ages because many of them have either not been regularly monitored or not been searched for glitches. Finally, to compare the glitch activity between rotation-powered pulsars and magnetars, we included the observations of five magnetars. They have been observed continuously for 16 years on average, and Dib & Kaspi (2014) reported a set of 11 glitches in the whole of their timing dataset. We constructed a database containing rotational information (ν, ν̇) with ν̇ corrected for the Shklovskii effect (Shklovskii 1970; Camilo et al. 1994), glitch measurements ∆ν, and the observation spans over which glitch searches have been performed (on average, 17.5 years for each pulsar). Here, ∆ν corresponds to the frequency increase due to the glitch. We did not take ∆ ν̇ steps into account because not all glitches have these parameters measured in a consistent way. Altogether, our sample contains the rotational information of 903 neutron stars, as shown in Figure 2.1, with a total of 384 glitches in 141 of them. The sample does not have a well-defined selection criteria, being mostly determined by having pulsars bright enough that they could be regularly monitored without an extreme commitment of observing time. In addition, the observing time spans are not uniform, as additional pulsars were added as they were discovered. On the other hand, it is important to note that the sample includes nearly a third of all pulsars known to date, with representatives across the P − Ṗ diagram (see Fig. 2.1), and none of the pulsars were selected directly because of their glitch properties (presence or absence of glitches, their frequency, or size). Thus, it should be close to the best possible available sample for the study performed in the present work, and the biases present should not affect our conclusions..

(14) 10 −9 −10 −11 −12. 10. Hz. s −1. −13. = Hz. s −1. ν̇. −15. 10 − 1. 2. −16. Hz. s −1. ν̇. −17. =. −. log Ṗ. −. 10 −. −14. −. 10 −. 14. −18 =. Known NSs No glitches Only small glitches Large glitches Magnetar’s glitches. ν̇. −19 −20 −21 −22. −3. −2. −1. 0. 1. 2. log P (s). Figure 2.1: Rotation period versus its time derivative ("P- Ṗ diagram") for all known neutron stars. Lines of constant spin-down rate ν̇ are shown and labeled. The small dark blue and medium light amber dots denote the known neutron stars not in our database, and the neutron stars in our database with no detected glitches, respectively. The large orange and turquoise dots represent those pulsars in our database with large glitches, and only small glitches detected, respectively. The turquoise triangles correspond to the magnetars in our database, which only have small glitches detected. P and Ṗ for neutron stars not in our database were taken from the ATNF pulsar catalog 2(Manchester et al. 2005).. 2.2. THE GLITCH SIZE DISTRIBUTION. The distribution of the glitch magnitude ∆ν of all glitches in our database is shown in Fig. 2.2, and is in agreement with the bimodal shape reported by Espinoza et al. (2011) 2 http://www.atnf.csiro.au/research/pulsar/psrcat.

(15) 11 and Ashton et al. (2017). There is a broad distribution of small glitches and a very narrow distribution of large glitches which peaks around 20 µHz. It is worth mentioning that this peak contains 70 large glitches detected in 38 different pulsars, where the main contributor is PSR J1420−6048, with 5 large glitches. This confirms that this peak of large glitches is by no means the effect of only a handful of pulsars. 60 Magnetars. Counts. 50 40 30 20. Normalized counts. Normalized counts. 10 0.5. One gaussian fit Two gaussians fit. 0.4 0.3 0.2 0.1 0.5. Three gaussians fit. 0.4 0.3 0.2 0.1 0.0 −5. −4. −3. −2. −1. 0. 1. 2. log ∆ν (µ Hz). Figure 2.2: Histogram of the glitch size ∆ν of all glitches in our database. In the upper panel the error bars correspond to the square root of the number of events per bin. The middle and lower panels show the best fits with one, two, and three Gaussians. Magnetar glitches were not included in the latter panels nor in the fits. The solid and dashed lines represent the best fits and their components, respectively. The shaded region indicates that glitches of sizes smaller than 0.01 µHz may be missing due to detectability issues.. As noted by (Espinoza et al. 2011), the left edge of the distribution is unconstrained.

(16) 12 since detections of very small glitches are strongly limited by the cadence of the observations, their sensitivity, and the intrinsic rotational noise of the pulsar. Given that these properties vary from pulsar to pulsar, it is not straightforward to set a universal lower detectability limit for the glitch size distribution of all pulsars (Fig. 2.2). However, based on the comparison of glitch sizes obtained by different authors, Espinoza et al. (2011) argued that for sizes below ∆ν/ν ∼ 10−8 the sample is likely to start being incomplete. Using the detectability limits for individual sources proposed by (Espinoza et al. 2014), it is possible to calculate average detection limits for all glitches. For an observing cadence of 30 days and a rotational noise (or minimum sensitivity) of 0.01 rotational phases, glitch detection is severely compromised below sizes ∆ν ∼ 10−2 µHz. This is a conservative limit because many pulsars are observed more frequently and exhibit lower noise levels. In the following, we model the glitch size distribution as a sum of Gaussian distributions. We note, however, that the detection issues discussed above must be considered when using such functions to describe the population of small glitches. Models composed of one and up to four Gaussians were tested against the data. The best fits obtained are shown in Fig. 2.2 (middle and lower panels; see Table A.1 for the parameters of the fits). Using Akaike’s information criterion (hereafter AIC, see Appendix A for a brief description of the test and the results obtained), we conclude that among the whole set of candidate models, a mixture of three Gaussians gives the best description of the glitch size distribution. Interestingly, all fits performed with more than one Gaussian give a common component (with nearly identical parameters) that contains all the glitches with magnitudes ∆ν ≥ 10 µHz. Because of this and the multimodal nature of the distribution, we classify glitches as large and small using ∆ν = 10µHz as the dividing line. We also made fits for the ∆νν distribution, and the AIC suggests, as for ∆ν, a non-unimodal behavior, although with two broader peaks of similar height. This further supports the multimodal interpretation of the glitch size distribution. 2.3. GLITCH ACTIVITY. A practical way to quantify the cumulative effects of a collection of spin-ups due to glitches on a pulsar’s rotation is through the glitch activity parameter. This parameter is defined as the time-averaged change of the rotation frequency due to glitches. Because of the short length of the observation spans available, it is not possible to detect enough.

(17) 13 glitches for a robust estimation for each pulsar. To avoid this problem, we studied the combined glitch activity for groups of pulsars sharing a common property. Following Lyne et al. (2000) and Espinoza et al. (2011), the average glitch activity for each group is PP i j ∆νi j ν̇g = P , (2.1) i Ti where the double sum runs over every change in frecuency ∆νi j due to glitch j of the pulsar i, and Ti is the time over which pulsar i has been observed and searched for glitches. This analysis also includes those pulsars that have been searched, but not found to glitch so far. Since the errors in the measurements of the glitch sizes are smaller than the Poisson fluctuations in the number ofqglitches detected fue to finite observation spans, the errors PP 2 P for ν̇g are estimated as δ ν̇g = i j ∆νi j / i Ti (for more details see Appendix A). Unlike those presented in Lyne et al. (2000) and Espinoza et al. (2011), these errors account for the presence of glitches of differernt sizes. However, our formula still does not take into account the possible contribution of rare, large glitches that were not detected because of the finite monitoring times. This implies that the error bars for ν̇g are likely underestimated. 2.3.1 DEPENDENCE ON SPIN PARAMETERS We grouped the pulsars in bins of width equal to 0.5 in logarithmic scale according to different properties (spin frequency ν, absolute value of the spin-down rate | ν̇|, and various p combinations of these, such as energy-loss srate Ėrot ∝ ν ν̇, magnetic field B ∝ ν̇/ν 3 , and spin-down age τc ∝ ν/ν̇). Figure 2.3 shows how ν̇g depends on these variables. Considering only rotation-powered pulsars, we observe that | ν̇|, Ėrot , and τc appear to give good correlations, whereas there are no clear correlations with B and ν. It is also apparent in Fig. 2.3 that the glitch activity of the magnetars with the smallest characteristic ages is lower than that of the rotation-powered pulsars with similar characteristic ages. However, their activity is larger than that of rotation-powered pulsars of equal spin-down power. The only parameter for which the glitch activity of magnetars appears to follow the same relation as for rotation-powered pulsars is the spin-down rate. Because there is almost no overlap between the spin-frequencies and magnetic fields of magnetars, and those of the rotation-powered pulsars, the comparison is not possible for these parameters. Interestingly, however, it seems that the glitch activity of rotation-powered pulsars and magnetars does not appear to depend directly on their dipolar.

(18) 14. −13. log ν̇g (Hz s−1 ). log ν̇g (Hz s−1 ). −13 −15 −17 −19 −21. −16. −14. log |ν̇| (Hz. −12. s−1 ). −17 −19 −21. −10. 0. −1. 1. 2. log ν (Hz). 3. −13. log ν̇g (Hz s−1 ). −13. log ν̇g (Hz s−1 ). −15. −15 −17 −19 −21. 8. 10. 12. 14. log B (G). −15 −17 −19 −21. 4. 6. log τc (yr). 8. 10. log ν̇g (Hz s−1 ). −13 −15 −17 −19 −21. 30. 32. 34. log Ė (erg. 36. 38. s−1 ). Figure 2.3: Glitch activity as a function of various pulsar parameters (all of them combinations of frequency ν and its time-derivative ν̇). Black dots are the ν̇g values calculated according to Eq (2.1), for rotationpowered pulsars grouped in bins of width 0.5 in the logarithm (base 10) of the variable on the horizontal axis. The crosses denote bins (groups of pulsars) with no detected glitches, and the gray squares represent individual magnetars..

(19) 15 magnetic field strength. On the other hand, in the case of magnetars, some glitches are contemporaneous with X-ray bursts, which are thought to be powered by the decay of their strong magnetic fields (Dib & Kaspi 2014; Kaspi & Beloborodov 2017). Similarly, the largest glitches in the high magnetic field rotation powered pulsars PSRs J1119−6127 and J1846−0258 were accompanied by changes in their emission properties (Livingstone et al. 2010; Weltevrede et al. 2011; Archibald et al. 2016). This possible connection between glitches and magnetospheric processes has lead to the idea that some glitches in high magnetic field neutron stars could have a different origin. Our results show, however, that the spin-down rate might be what determines the rate and size of these glitches, just as it does for ordinary classical pulsars. We choose | ν̇| as the best parameter to study ν̇g , because we can interpret our results in terms of simple physical concepts, and because this is the only parameter for which magnetars follow a similar tendency to rotation-powered pulsars. It is worth mentioning that we tested a broad range of combinations of the form ν a | ν̇| with different values of a, finding that none of them give a substantially better correlation with ν̇g than | ν̇|. Figure 2.4 confirms the relation ν̇g ∝ | ν̇| already reported by Lyne et al. (2000) and Espinoza et al. (2011) for pulsars with −14 < log | ν̇| < −10.5 (we always take the units of | ν̇| as Hz s−1 ). The mean value of the ratio ν̇g/| ν̇| for this range is 0.012 ± 0.001, corresponding to the horizontal line in panel (b) of Fig. 2.4. This represents the fraction of the spin-down “recovered” by glitches, which can be interpreted as the minimum fraction of the star’s moment of inertia in a decoupled internal component (Link et al. 1999). However, the linear trend between the glitch activity and the spin-down rate seems to fail towards the extremes, which we will explore further in Sect. 2.3.2. Table 2.1 shows additional information related to each bin in Fig 2.4..

(20) 16 −12. No glitches detected Individual pulsars Grouped pulsars. −13. log ν̇g (Hz s−1 ). −14 −15. (a). −16 −17 −18 −19 −20 −17. 0. −16. −15. −14. −13. −12 −1 s ). −11. −10. −9. −15. −14. −13. −12. −11. −10. −9. log |ν̇| (Hz. (b). log ν̇g /|ν̇|. −2. −4. −6. −17. −16. log |ν̇| (Hz s−1 ). Figure 2.4: Panel (a) shows log ν̇g versus log | ν̇|. Panel (b) shows log ( ν̇g /| ν̇|) versus log | ν̇|. The horizontal line corresponds to the average ratio ν̇g /| ν̇| = 0.012 ± 0.001, calculated over the bins with −14 < log | ν̇| < −10.5. In both panels the crosses correspond to pulsars that have no glitches detected, whereas large black dots and small gray dots represent the bins (groups) and individual pulsars, respectively.. 2.3.2 A COMMON TREND IN THE ACTIVITY OF NEARLY ALL PULSARS Motivated by the results in Sect. 2.2 showing that the largest glitches separate from.

(21) 17 Table 2.1: Statistics of glitches for pulsars binned by their spin-down rate. The first column is the bin number. The second and third columns correspond to log | ν̇| for the group of pulsars in each bin (the central value of each logarithmic interval), and the sum of the observation time of all pulsars in that bin. The next two columns contain the number of large glitches and the total number of glitches, respectively. The last two columns correspond to the number of pulsars with glitches, and the total number of pulsars in each bin, respectively.. # bin. log | ν̇| (Hz s−1 ). P. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16. −16.75 −16.25 −15.75 −15.25 −14.75 −14.25 −13.75 −13.25 −12.75 −12.25 −11.75 −11.25 −10.75 −10.25 −9.75 −9.25. 117 0 0 430 0 0 1233 0 0 2478 0 3 2675 0 11 1973 0 25 2083 0 35 1706 1 29 1312 3 26 745 4 38 493 8 74 357 37 78 66 13 19 44 4 8 16 0 2 46 0 25. Ti (yr). N` Nt Npg 0 0 0 3 8 16 20 18 14 15 15 18 5 2 1 1. Np 7 25 70 139 142 105 113 105 81 48 33 20 5 3 1 1. the rest, we re-computed the glitch activity separately for large glitches (∆ν ≥ 10 µHz) and for small glitches (the remainder), showing the results in Fig. 2.5. We observe that large glitches determine the linear relation between ν̇g and | ν̇| in the range −13.5 < log | ν̇| < −10.5, and calculate the ratio ν̇g/| ν̇| = 0.010 ± 0.001. This value is consistent with the value obtained in the previous section, when considering all glitches and including bin 7, which has no large glitches detected. Accordingly, we use the linear relation ν̇g = 0.01| ν̇| as a reference throughout the paper. The activity due to small glitches follows a roughly similar, though noisier, increasing trend with | ν̇|. There are no large glitches detected in pulsars with the smallest and the largest spindown rates (log | ν̇| < −13.5 and log | ν̇| > −10). The 14th bin (log ν̇ = −10.25, Table 2.1) fails to follow the linear trend, despite having four large glitches with average size of 37 µHz. For the accumulated observing time in this bin, approximately 30 additional large.

(22) 18 glitches of 20 µHz are required to reach the activity value predicted by the linear relation. On the other side, the 7th bin (log ν̇ = −13.75) follows the linear trend even though it has no large glitches detected. According to the linear relation and the accumulated observing time, only one large glitch of 10 µHz is necessary to obtain the value predicted by the linear trend, which is very close to the current activity value. However, instead of one large glitch, there are smaller glitches that account for 9.78 µHz, making the activity consistent with the linear relationship. −12. No glitches detected Large glitches Small glitches. −13. −15. (a). log Ṅ` (yr−1 ). log ν̇g (Hz s−1 ). −14. No large glitches detected Large glitches. −1. −16 −17. −2. (b). −3. −18 −4. −19 −20 −21 −17. −5 −16. −15. −14. −13. −12. log |ν̇| (Hz s−1 ). −11. −10. −9. −17. −16. −15. −14. −13. −12. log |ν̇| (Hz s−1 ). −11. −10. −9. Figure 2.5: Panel (a) shows log ν̇g versus log | ν̇|. The black squares and gray diamonds represent the glitch activity separately for large and small glitches, respectively. In both cases, ν̇g was calculated for the same bins (groups) of pulsars, respectively. The straight line shows the linear relation ν̇g = 0.01| ν̇|. The crosses denote bins with no detected glitches. Panel (b) shows log Ṅ` versus log | ν̇|. In panel (b) the black squares and the crosses represent the bins or groups of pulsars with large glitches and no large glitches detected, respectively.. Since the proportionality between the glitch activity and the spin-down rate is dominated by large glitches, and these have a very narrow size distribution, we expect that the rate of large glitches, Ṅ` , will also be proportional to | ν̇| (Fig. 2.5 panel b). Because the number of large glitches is expected to follow a Poisson distribution, the expected dispersion in the rate of large glitches can be estimated in a more reliable way than that in the glitch activity. This allows us to test in a statistically meaningful way whether or not the identified trend applies to all pulsars. Figure 2.6 confirms that Ṅ` /| ν̇| is approximately constant and its mean value is (4.2 ± 0.5) × 102 Hz−1 (which we calculated considering only the | ν̇| bins for pulsars with −13.5 <.

(23) 19 log | ν̇| < −10.5). We observe that except for the three bins with the largest spin-down rate, all others are consistent with this trend. The non-detection of large glitches in the region of small | ν̇| is consistent with the small expected rate and the finite monitoring time, as illustrated by the shaded area in Fig. 2.6. Based on this relation, the expected number exp of large glitches, N` , for the three bins with the highest | ν̇| (log | ν̇| > −10.5) is (taking the integers values) 30 ± 5, 36 ± 6, and 325 ± 18, respectively. This strongly contradicts the only four large glitches detected in bin 14 and the absence of large glitches in bins 15 and 16 (which contain only PSR B0540−69 and the Crab pulsar, respectively; Table 2.1). Thus, we can confidently rule out the linear relation between ν̇g and | ν̇| for the largest values of the latter variable, but it remains consistent consistent for all | ν̇| < 10−10.5 Hz s−1 .. log Ṅ` /|ν̇| (Hz−1 ). 5. 4. 3. 2. 1 −17. −16. −15. −14. −13. −12. log |ν̇| (Hz s−1 ). −11. −10. −9. Figure 2.6: log Ṅ` /| ν̇| versus log | ν̇|. The horizontal line corresponds to the logarithm of the mean value h Ṅ` /| ν̇|i = (4.2 ± 0.5) × 102 Hz−1 , calculated over the bins with −13.5 < log | ν̇| < −10.5. The shaded region indicates the expected dispersion around this average value, based on a Poisson distribution of the number of large glitches and the available observing time spans. The black squares represent the observed values of the ratio Ṅ` /| ν̇| for bins (groups) of pulsars. The crosses denote bins with no large glitches detected.. Next, we test whether the individual pulsars within each bin are also consistent with this trend. Since the number of large glitches for each pulsar is small (in most cases zero), the usual χ2 test is not applicable. Instead, we use Fisher’s test (Fisher 1925), based on the statistic k X 2 X2k = −2 ln pi , (2.2) i=1.

(24) 20 with two-tailed p-values for each pulsar calculated as pi = min{P(x ≤ Nòbs ), P(Nòbs ≥ x)},. (2.3). where P(x ≤ Nòbs ) is the (Poisson) probability of obtaining a value x smaller or equal to the actual observed value Nòbs , based on the fixed ratio Ṅ` /| ν̇| = (4.2 ± 0.5) × 102 Hz−1 calculated above, and the observation time of the pulsar (and analogously for P[Nòbs ≥ x]). If the null hypothesis is true, pi is uniformly distributed between 0 and 1, and therefore the fisher statistic will follow a χ2 distribution with 2k degrees of freedom. On the other 2 will be large, leading to the hand, when the individual p-values pi are very small, X2k rejection of the global null hypothesis. Table 2.2 shows the results of the test, where the global p-value for each bin was calculated as 2 2 pbin = P χ2 ≥ X2k |X2k ∼ χ22k ,. (2.4). 2 , given that is, the probability of obtaining a χ2 value at least as large as our observed X2k 2 ∼ χ 2 . Based on the values obtained for p , the null hypothesis the null hypothesis X2k bin 2k can be strongly ruled out for the last three bins, whereas we cannot rule out that all pulsars with | ν̇| < 10−10.5 Hz s−1 , even those without glitches, follow the identified trend.. 2.4. DISCUSSION. We have shown that all pulsars with | ν̇| < 10−10.5 Hz s−1 and magnetars are consistent with a single trend, dominated by large glitches, in which the glitch activity ν̇g is equal to 0.01| ν̇|. The large collection of pulsars with no detected glitches is also consistent with this trend. For instance, the predicted rate of large glitches for pulsars with log | ν̇| = −14.25 (bin 6) is one large glitch every ∼ 104 yr, whereas the accumulated observing time in this bin is only 1973 yr (Table 2.1), and this mismatch becomes even more extreme for the bins with lower | ν̇|. This means that there are no reasons to reject the idea that every neutron star will eventually experience a large glitch and will, in the long-term, follow the above relationship. On the other hand, we can not rule out the possibility that, for example, the glitch mechanism could fail to produce large glitches in the pulsars with the.

(25) 21 Table 2.2: Results of Fisher’s method. The first column denotes the bin number, whereas the second and third columns contain the Fisher statistic given by Eq. (2.2) and the p-value defined in Eq.(2.4). 2 / dof # bin X2k. pbin. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16. 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.99 0.49 0.07 2 × 10−10 3 × 10−18 3 × 10−104. 0.00005 / 12 0.0007 / 50 0.006 / 140 0.04 / 276 0.12 / 286 0.30 / 210 0.96 / 226 13.29 / 210 23.73 / 162 25.23 / 96 32.13 / 66 40.42 / 40 18.09 / 10 57.24 / 6 79.30 / 2 475.21 / 2. smallest spin-down rates, or produce substantially more of them than predicted from the linear relation. A similar amount of superfluid neutrons inside all neutron stars could be responsible for the common glitch activity levels observed. However, despite this uniformity in the collective behavior, it appears that the glitch mechanism possesses an intrinsic bi-modality (Fig. 2.2). It could be possible that there are two or more different glitch mechanisms in action, giving rise to different size distributions. The cumulative distributions of glitch sizes for five individual pulsars, presented in Fig. 16 of Espinoza et al. (2011), suggests that this bi-modality could be extended to define at least two types of pulsars: those with large glitches and those with only small glitches; but we find no evidence for this in the glitch activity data. 2.4.1 THE GLITCH BEHAVIOR OF THE MOST RAPIDLY EVOLVING PULSARS The glitch activity for the last two bins was calculated using only one pulsar per bin.

(26) 22 (see Table 2.1) and the values obtained do not follow the tendency defined by the rest of the population. These are the pulsars with the largest spin-down rates and correspond to PSR B0540−69 (bin 15) and the Crab pulsar (PSR B0531+21, bin 16). PSR B0540−69 has been observed for ∼ 15.8 yr (Ferdman et al. 2015) and its activity is especially low, compared to all pulsars with high spin-down rates (see Fig. 2.4). In the same bin it is possible to include the X-ray pulsar PSR J0537−6910, which was not considered in our sample since it does not belong to any of the monitoring programs taken into account for our database, and including it would have artificially biased the sample towards the presence of glitches. It has been observed for 13 yr, and 45 glitches have been detected (Marshall et al. 2004; Middleditch et al. 2006; Antonopoulou et al. 2018; Ferdman et al. 2018), 30 of which are large according to our classification in §2.2, making it consistent with the linear trend identified above. If we included this pulsar in our dataset, the glitch activity in the associated bin would be only slightly lower than predicted by the relation ν̇g = 0.01| ν̇| (see Fig. 2.7).. PSR J0537-6910 −→. log ν̇g (Hz s−1 ). −12. −13. PSR B0531+21 −→. −14. −15. PSR B0540-69 −→. −16. −14. −13. −12. log |ν̇| (Hz. −11. s−1 ). −10. −9. Figure 2.7: A zoom-in to the zone ν̇g = 0.01| ν̇| plus the last three bins, with the highest | ν̇| values. The straight line shows the linear relation ν̇g = 0.01| ν̇|. The glitch activity of PSR J0537−6190 was plotted using a gray square to show that this young pulsar follows the general trend. The activity obtained when including this pulsar in the appropriate bin is represented by the gray dot.. Marshall et al. (2015) reported a large increase in the spin-down rate of PSR B0540−69 that remained for < 3 years. Based on their data, they placed a limit of ∆ν < 12 µHz for.

(27) 23 a hypothetical glitch responsible for this increase. With a glitch of the maximum allowed size, the activity of PSR B0540−69 would be 2(2) × 10−14 Hz s−1 , very similar to the activity of the Crab pulsar 1.1(5) × 10−14 Hz s−1 , but still low if the pulsar was to follow the relationship above. One possible explanation for the discrepancy in the glitch activity of these pulsars could be their age, as proposed by Alpar et al. (1996) to explain the differences between the Crab and Vela pulsars. Indeed, PSR B0540−69 and the Crab pulsar are among the youngest pulsars known, with supernova remnant ages equal to 1000+600 −240 and 962 yr, respectively (Park et al. 2010; Ho & Andersson 2012). However, other similarly young pulsars, such as PSRs J1119−6127 and J1846−0258 (in bins 13 and 14, respectively), but with lower spin-down rates, have experienced large glitches during their monitored rotations and exhibit glitch activities in closer agreement with the main trend. We conclude that it is possible that the glitch mechanism might not be able to operate normally when the spin-down rate is too high. Perhaps, once PSR B0540−69 and the Crab pulsar have evolved, and their spin-down rates have decreased, their glitches will have settled into the trend followed by the rest of the population..

(28) 24 Chapter 3. TIME SERIES OF NEUTRON STAR GLITCHES In this chapter, we study the sequence of glitches in pulsars with more than 10 detected glitches as a time series, in order to determine whether this sequence and the relation between glitch sizes and time intervals between them are random or follow some regularities. It is organised as follows. Section 3.1 presents the sample of pulsars studied in this work. In §3.2 we study the correlation between the glitch size and the time until the following glitch, in particular, we propose and test two hypotheses that could explain why the rest of the pulsars do not show a correlation as clear as PSR J0537−6910. In §3.3 we look for different correlations between the glitch sizes and time intervals between them, such as ∆νk vs ∆τk (size of the glitch versus the time since the preceding glitch), and ∆νk vs ∆νk−1 (size of the glitch versus the size of the previous glitch). 3.1. THE PULSARS WITH MORE GLITCHES DETECTED. To date, there are seven pulsars with at least 10 glitches detected (Figure 3.1). PSRs B0531+21 (the Crab pulsar), B1758−23, and J0631+1036 have been observed regularly by the Jodrell Bank observatory (JBO), whose pulsar timing programme provides the necessary cadence to follow systematically the rotation of pulsars, allowing the detection of glitches. PSRs J1341−6220 and B0833−45 (the Vela pulsar) have been observed by the Parkes telescope and others in Australia. PSR J0537−6910 is the only one not detected in the radio band and was observed for 13 years by the Rossi X-ray Timing Explorer (RXTE). For detailed information, such as glitch epochs and magnitudes of all the glitches in each pulsar, refer to the JBO online glitch catalog 1. Figure 3.2 shows the size ∆ν of each glitch as a function of its ocurrence time, for all the pulsars in the sample. PSRs B0833−45 (the Vela pulsar) and J0537−6910 seem to produce glitches of similar sizes, particularly large glitches, and in fairly regular time intervals, whereas glitches in the rest of pulsars have an irregular occurrence and cover a 1 http://www.jb.man.ac.uk/pulsar/glitches/gTable.html 2 http://www.atnf.csiro.au/research/pulsar/psrcat.

(29) 25 B=. log Ṗ. −10 −12 −14 −16 −4. B=. 10 12. 10 14. Known NSs B0531+21 J0631+1036 B1737−30 B1758−23 B0833−45 J1341−6220 J0537−6910. G. G. 3 yr 10 = τc 5 yr 10 = τc. −3. −2. −1 log P (s). 0. 1. 2. Figure 3.1: Upper part of the P − Ṗ diagram. The pulsars considered in our sample have at least 10 glitches detected and are labelled with different colors and symbols. Lines of constant spin-down age τc and magnetic field B are shown and labelled. P and Ṗ for neutron stars not in our sample were taken from the ATNF pulsar catalog.2.. wide distribution of sizes. The gray area in all panels corresponds to a period of time in which there were no observations. Because of this, for the Crab pulsar we only consider the sample of glitches from January 1984 until 2018, since in that time span it was observed every day. It is important to mention that the observation spans of all the pulsars in the sample are different from each other, and not all of them are monitored with the same cadence. This, added to the intrinsic noise of each pulsar, puts into question the completeness of the glitch dataset of each pulsar..

(30) 26 PSR J0537−6910. 2. Ng = 45. 0 −2 −4. PSR B1758−23. 2. Ng = 13. 0 −2 −4. PSR J0631+1036. 2. Ng = 15. log ∆ν (µHz). 0 −2 −4. PSR J1341−6220. 2. Ng = 27. 0 −2 −4. PSR B1737−30. 2. Ng = 35. 0 −2 −4. PSR B0833−45 (The Vela pulsar). 2. Ng = 20. 0 −2 −4. PSR B0531+21 (The Crab pulsar). 2. Ng = 26. 0 −2 −4. 40000. 42500. 45000. 47500. 50000. MJD (days). 52500. 55000. 57500. 60000. Figure 3.2: Glitch sizes ∆ν (in logarithmic scale) as a function of time for all pulsars in the sample. In each panel the gray area denotes a period of time in which there were no observations, and Ng is the number of glitches detected in the respective pulsar, until 13th June 2018. All panels share the same scales in the MJD and ∆ν axes.. 3.2. TIME SERIES CORRELATIONS: GLITCH SIZE AND TIME TO THE FOLLOWING GLITCH. Different studies have shown that for PSR J0537−6910 the magnitude of its glitches is strongly correlated with the waiting time until the following glitch (Middleditch et al. 2006; Ferdman et al. 2018; Antonopoulou et al. 2018, , see Fig. 3.3, left panel). This.

(31) 27 correlation can help to understand how the glitch mechanism works. For instance, if a glitch is produced when a finite reservoir of angular momentum reaches a critical state, and large glitches deplete more of the reservoir than small ones, then, after a large glitch, the time it takes for the reservoir again to reach the critical state and produce another glitch is larger than after a small glitch. Furthermore, for practical purposes, this relation might be useful to design observation campaigns for the detection of future glitches. For PSR J0537−6910, we compute the glitch size versus the waiting time until the following glitch for different sub-samples: using all the glitches, and only glitches equal or larger than a certain cutoff ∆νc , which varies from 1 to 10 µHz. Figure 3.3 shows the correlation for the most representative sub-samples. When considering all glitches, Pearson’s correlation coefficient is r p = 0.94 and the respective p-value is pp ∼ 10−22 . In addition, this correlation seems to be exceptional, since until ∆νc = 7 µHz it is still very strong and essentially unaffected by removing the smallest glitches, giving correlation coefficients as large as 0.90 and with a very low p-value ∼ 10−12 . However, by considering only very large glitches (∆νc = 10 µHz, according to our classification in Sect. 2.2), like the giant glitches observed in the Vela pulsar, the correlation gets much worse (r p = 0.54) although it remains statistically significant (pp = 0.005). The best correlation is obtained when considering all the glitches together, rather than different sub-samples separately. This suggests that the strong correlation observed is not an effect of only the largest glitches, and it indicates that all glitches use a common reservoir of angular momentum..

(32) 28. ∆τk+1 (yr). PSR J0537−6910. 0.8. 0.8. 0.8. 0.6. 0.6. 0.6. 0.4. 0.4. 0.4. 0.2. 0.2. 0.2 ∆ν ≥ 7 µHz. All glitches 0.0. 0. 10. 20 30 ∆νk (µHz). 40. 0.0. 0. 10. 20 30 ∆νk (µHz). 40. ∆ν ≥ 10 µHz 0.0. 0. 10. 20 30 ∆νk (µHz). 40. Figure 3.3: ∆νk as a function of ∆τ k+1 . From left to right, the sub-samples consider: all glitches, and only the ones larger than ∆νc = 7, and 10 µHz, respectively.. We test whether this correlation is also present in the other pulsars of our sample, and show the results in Fig. 3.4 and Table 3.1. None of them exhibits a correlation as clear as PSR J0537−6910. However, for PSRs B1758−23, J0631+1036, and J1341−6220, the Pearson’s correlation coefficients are larger than 0.5 and the p-values are very low ∼ 10−3 . Therefore, at 5% of significance (p-values < 0.05), we can reject the null hypothesis that ∆νk and ∆τk+1 are uncorrelated. We also compute the Spearman’s rank correlation coefficient to avoid making conclusions affected by outliers, which could be the case for PSR J0631+1036, whose high Pearson’s correlation coefficient is clearly dominated by the last data point. We only obtain different results for PSRs J0631+1036 and the Crab pulsar, both having very small Spearman’s rank correlation coefficients, and negative in the case of the Crab pulsar. Although the correlation coefficient in PSR J0631+1036 is strongly dependent on the last point, it is interesting that this point corresponds to the largest glitch and largest waiting time until the following glitch, which is expected if the correlation exists. Further, it is interesting that for all the pulsars in the sample, the Pearson’s correlation coefficients are positive (for the Spearman’s rank correlation coefficient, only the Crab pulsar shows a negative coefficient). Given the positive correlation by PSR J0537−6910, the probability of obtaining the same positive sign in the correlation coefficients of all the other pulsars, only by chance, is very low, (1/2) 6 = 0.016, and therefore it suggests that a positive correlation between ∆νk and ∆τk+1 could exist in all pulsars..

(33) 29 PSR B1758−23. PSR J0631+1036. PSR J1341−6220 1.75. 3.0. 5. 1.50. 2.5 ∆τk+1 (yr). 4. 1.25. 2.0. 1.00. 3. 1.5. 2. 1.0. 0.75 0.50. 0.5. 1 0.0. 0.5. 1.0 ∆νk (µHz). 1.5. 0.25 0. PSR B1737−30. 4.0. 2. ∆νk (µHz). 4. 6. 0. PSR B0833−45 (The Vela pulsar). 2. 4 ∆νk (µHz). 6. 8. PSR B0531+21 (The Crab pulsar) 6. 3.5. 4. 5. 3.0 ∆τk+1 (yr). 0.00. 3. 2.5. 4. 2.0. 3. 2. 1.5. 2. 1.0. 1. 1. 0.5 0.0 0. 1. 2 3 ∆νk (µHz). 4. 0. 0 0. 10. 20 ∆νk (µHz). 30. 0. 2. 4 ∆νk (µHz). 6. Figure 3.4: Glitch magnitude ∆νk as a function of the waiting time until the next glitch ∆τ k+1 , for the rest of pulsars in the sample. Table 3.1: Correlation coefficients between ∆νk and ∆τ k+1 . The first and second columns contain the names of the pulsars considered in the sample, and the respective number of glitches detected, respectively. The third and fourth columns correspond to Pearson’s linear correlation coefficient r p and the respective p-value pp . The last two columns are the same as the third and fourth but for the Spearman’s correlation.. PSR Name. Ng. rp. J0537−6910 B1758−23... J0631+1036 J1341−6220 B1737−30... B0833−45... B0531+21.... 45 13 15 27 35 20 26. 0.94 0.76 0.70 0.56 0.29 0.24 0.04. pp. rs. ps. 10−22 0.95 10−23 0.003 0.80 0.001 0.005 0.14 0.62 0.002 0.63 0.0004 0.08 0.26 0.12 0.31 0.30 0.21 0.85 −0.01 0.94.

(34) 30 PSRs J0631+1036, B1737−30, and the Crab have mainly small glitches. The weak correlations observed for these pulsars could be due to either of the following effects: (1) Glitches below some cutoff size are not detected, thus increasing the intervals of time between the detected glitches by random amounts, worsening the correlation. (2) There are two classes of glitches happening in all of these pulsars: larger glitches (including all or most of those in PSR J0537-6910) that use a certain angular momentum reservoir and thus follow the correlation, and smaller glitches caused by a different mechanism and happening at random times. In the following, we test the first hypothesis, leaving the second for future work. 3.2.1 INCOMPLETENESS AT SMALL GLITCH SIZES AND ITS EFFECT ON THE CORRELATION In order to test the first hypothesis, we simulate a generic pulsar with 100 glitches with a perfect correlation between ∆νk and ∆τk+1 , and then we remove the smallest events to understand the effect of their absence on the correlation. The procedure is the following: (1) We model the glitch size distribution from a power law distribution in which each decade of the glitch size x = ∆ν contributes to the glitch rate as dN dN ∼x , d(log x) dx. (3.1). and its contribution to the glitch activity is x. dN dN ∼ x2 , d(log x) dx. (3.2). with dN/dx = x −α . We consider several different choices for the power-law index 1 ≤ α ≤ 3, but we only show the results for α = 1, 1.2, and 1.4, which generate more realistic distributions, as we show in the results. We note that for α = 1, each decade of glitch size contributes equally to the glitch rate, and the glitch activity (see Eq. 2.1 and Sec. 2.3) is strongly dominated by the large glitches. In order to ensure that both the glitch rate and the activity are finite, we normalize the distribution by.

(35) 31 considering the maximum glitch detected so far among all pulsars in our sample, which corresponds to x max = 42 µHz, in PSR J0537−6910. The lower limit x min is defined in such a way that, after reducing the sample of glitches as we explain in step (4), the resulting sample covers the observed range of glitch sizes. We adopt those simple distributions because the sample of small glitches is incomplete and the bimodal distribution that we show in Fig. 2.2 does not represent the real distribution of the glitch sizes in individual pulsars (see Fig. 3.5).. (2) We generate randomly the magnitude of a glitch ∆νk from the distribution described in (1). (3) We compute the time to the next glitch ∆τk+1 by using the size of the current glitch ∆νk ∆τk+1 ∝ ∆νk . (3.3) We do not include any dispersion in ∆τk+1 . (4) We repeat the steps (2) and (3) until we complete a sequence of 100 glitches. Then, the 80 smallest are removed, leaving a reduced sample of 20 to be analyzed (a reasonable number of glitches for real samples), covering roughly the real observed range of glitch sizes. Then, we calculate the time interval between each pair of successive glitches in the reduced sample and the linear correlation coefficient between ∆νk and ∆τk+1 ..

(36) 32. dN d log x. PSR J0537−6910. PSR B1758−23. PSR J0631+1036. 30. 6. 6. 25. 5. 5. 20. 4. 4. 15. 3. 3. 10. 2. 2. 5. 1. 1. 0. −3. −2. −1 0 log x (µHz). 1. 2. 0. −3. −2. PSR J1341−6220. −1 0 log x (µHz). 1. 2. 12. −2. 12. 14. 1. 2. 1. 2. 12 10. 8. 8. −1 0 log x (µHz) PSR B0833−45. 16. 10. 10 dN d log x. −3. PSR B1737−30 14. 14. 8 6. 6 4. 4. 2. 2. 0. 0. −3. −2. −1 0 log x (µHz). 1. 2. 1. 2. 0. 6 4 2 −3. −2. −1 0 log x (µHz). 1. 2. 0. −3. −2. −1 0 log x (µHz). PSR B0531+21 14 12. dN d log x. 10 8 6 4 2 0. −3. −2. −1 0 log x (µHz). Figure 3.5: Glitch size distributions dN/d log x for all pulsars in our database.. Figure 3.6 shows the distributions and correlations for the reduced sample of the simulated glitches. We observe that for α = 1, 1.2, and 1.4, the effect of missing many small glitches is not strong on the correlation. In more than 90 % of the realizations, the correlation coefficients (both Pearson’s and Spearman’s) are larger than 0.95 (the average over all the simulations is hri = 0.99). For α larger than 1.4, the distribution becomes.

(37) 33 α =1. α = 1.2 7. 12. 5. 6. 10. 5. 4 dN d log x. α = 1.4. 6. 8. 4 3. 6 3. 2 1 0. 2. 1 −3. −1 0 log x (µHz). −2. 1. 2. 8. 6 ∆τk+1 (A. U.). 4. 2. 4. 0. −3. −2. −1 0 log x (µHz). 1. 2. 0. 6. 6. 5. 5. 4. 4. 3. 3. 2. 2. 1. 1. −3. −2. −1 0 log x (µHz). 0.0. 0.5. 1.0 1.5 ∆νk (µHz). 1. 2. 2. 0. 0 0. 10. 20 ∆νk (µHz). 30. 0 0.0. 2.5. 5.0 7.5 ∆νk (µHz). 10.0. 12.5. 2.0. Figure 3.6: In each column, from left to right, the upper panels show the glitch size distributions dN/d log x ∝ x 1−α , with alpha 1, 1.2, and 1.4, respectively, and the lower panels show the respective ∆νk versus ∆τ k+1 , for the reduced sample of glitches. The colors represents two different realizations which are consistent with the average value of the correlation coefficient over all the realizations.. narrow towards the inferior limit x min , and therefore, since a large fraction of the simulated glitches are of very similar sizes, after removing the 80 smallest glitches, the correlation gets significantly worse (with correlation coefficients between 0.9 and 0.4). Although in these cases the correlation coefficients are similar to those exhibited by the real data, the distributions of glitch sizes differ strongly from the real data. From these simulations, we conclude that it is unlikely that the non-detection of ∼ 80% of the glitches in the observation spans is the explanation for the bad correlations observed..

(38) 34 3.3. OTHER CORRELATIONS. We also look for other correlations between the glitch sizes and time intervals between them, specifically for ∆νk vs ∆τk (size of the glitch versus the time since the preceding glitch), and ∆νk vs ∆νk−1 (size of the glitch versus the size of the previous glitch), but we did not find a clear correlation (Figure 3.7 and 3.8, and Table 3.2). Table 3.2: Correlation coefficients for the pairs of variables (∆νk , ∆τ k ), and (∆νk , ∆νk−1 ). The first and second columns contain the names of the pulsars considered in the sample, and the respective number of glitches detected, respectively. r ij and pij correspond to a correlation coefficient and its p-value. The index i is equal to a for ∆νk vs ∆τk , and b for ∆νk vs ∆νk−1 . The index j is p for Pearson’s correlation, and s for Spearman’s rank correlation.. PSR Name. Ng. J0537−6910 B1758−23... J0631+1036 J1341−6220 B1737−30... B0833−45... B0531+21.... 45 13 15 27 35 20 26. r pa. ppa. −0.08 0.60 −0.02 0.94 −0.09 0.75 −0.29 0.14 −0.06 0.71 0.55 0.01 0.68 0.005. r sa. psa. r pb. −0.12 −0.04 −0.12 −0.16 −0.17 0.27 0.47. 0.41 0.89 0.65 0.40 0.31 0.24 0.02. −0.13 −0.02 −0.09 −0.33 −0.11 −0.08 0.02. ppb. r sb. psb. 0.38 −0.16 0.29 0.92 −0.04 0.89 0.73 0.23 0.40 0.09 −0.23 0.24 0.50 −0.008 0.96 0.71 −0.12 0.59 0.89 −0.14 0.52. For ∆νk vs ∆νk−1 , in most of the cases, the correlation coefficients are close to zero, and the p-values are larger than 0.2. However, despite that individual pulsars do not show a significant correlation, it could be meaningful for the whole sample because almost all the pulsars have negative correlation coefficients, with the exception of the Crab pulsar for Pearson’s coefficient, and PSR J0631+1036 for Spearman’s coefficient. The probability of getting this result (for 6 of 7 pulsars the same sign), just by chance, regardless of whether the sign of the correlation coefficient is positive or negative, is equal to 2 × pbinom (7, 6) = 0.1, n where pbinom (n, k) = k (1/2) n is the probability of getting exactly k successes in n trials described by a Binomial distribution in which the probability of success is 0.5. This probability is not low enough to draw a strong conclusion, but if we are able to confirm this with more data available in the future, it could establish an interesting constraint into the glitch mechanism: The size of a glitch is regulated by the size of the previous glitch..

(39) 35 PSR J0537−6910 40. 5. 1.2 4. 1.0. 3. 0.8. 20. 0.6. 2. 0.4. 10. 1. 0.2 0.0. 0 0. 10. 20 ∆νk (µHz). 30. 0 0.0. 0.5. PSR J1341−6220. 1.0 ∆νk (µHz). 1.5. 0.0. PSR B1737−30. 2.5. 5.0 7.5 ∆νk (µHz). 10.0. PSR B0833−45 35. 8. 4. 6 ∆νk−1 (µHz). PSR J0631+1036. 6. 1.4. 30 ∆νk−1 (µHz). PSR B1758−23. 1.6. 30 25. 3. 20 4. 2. 2. 1. 15 10 5. 0. 0 0. 2. 4 ∆νk (µHz). 6. 8. 0 0. 1. 2 3 ∆νk (µHz). 4. 0. 10. 20 ∆νk (µHz). 30. PSR B0531+21 6. ∆νk−1 (µHz). 5 4 3 2 1 0 0. 5. ∆νk (µHz). 10. Figure 3.7: Glitch magnitude ∆νk versus the magnitude of the preceding glitch ∆νk−1 , for all pulsars in the sample..

(40) 36 In the case of ∆νk vs ∆τk , for all pulsars except Vela and the Crab, the correlation coefficients are negative but at 5% of significance we cannot reject that both variables are uncorrelated. The case of the Vela and Crab pulsars is different. Both pulsars show positive correlation coefficients (either Pearson or Spearman), and the p-values are smaller than 0.05. However, for the case of Vela this result is dominated by outliers, as the Spearman’s coefficient shows. PSR J0537−6910. PSR B1758−23. PSR J0631+1036. 0.8. 2.5. 0.6 ∆τk (yr). 3.0. 5. 0.7. 4. 2.0. 0.5 0.4. 3. 1.5. 2. 1.0. 0.3 0.2 0.1 0.0. 0. 10. 20 ∆νk (µHz). 30. 0.0. PSR J1341−6220. ∆τk (yr). 0.5. 1.0 ∆νk (µHz). 1.5. 3.0. 1.25. 2.5. 1.00. 2.0. 0.75. 1.5. 0.50. 1.0. 0.25. 0.5 2. 4 ∆νk (µHz). 6. 8. 5.0 7.5 ∆νk (µHz). 10.0. 4. 3. 2. 1. 0.0 0. 2.5. PSR B0833−45. 3.5. 1.50. 0.0. PSR B1737−30. 4.0. 1.75. 0.00. 0.5. 1. 0. 1. 2 3 ∆νk (µHz). 4. 0. 0. 10. 20 ∆νk (µHz). 30. PSR B0531+21 6 5. ∆τk (yr). 4 3 2 1 0 0. 5. ∆νk (µHz). 10. Figure 3.8: Glitch magnitude ∆νk versus the waiting time since the preceding glitch ∆τ k , for all pulsars in the sample..

(41) 37 Chapter 4. SUMMARY AND CONCLUSIONS. In chapter 2, we presented a statistical study of glitches in pulsars and magnetars, using a large database of pulsars whose selection was independent of their glitch properties. The main conclusions are the following: (2.1) The glitch size distribution is at least bimodal, with two well-defined classes: a broad distribution of small glitches and a narrow one with large glitches peaked at around 20 µ Hz. (2.2) For pulsars and magnetars with 10−13.5 < | ν̇| < 10−10.5 Hz s−1 , the glitch activity ν̇g is directly proportional to the spin-down rate | ν̇|, following ν̇g = (0.010 ± 0.001) | ν̇|. This relationship is dominated by large glitches. For all pulsars in this interval, the rate of large glitches is consistent with the proportionality relation Ṅ` = (4.2 ± 0.5) × 102 Hz−1 | ν̇|. This is also consistent for the pulsars with | ν̇| < 10−13.5 Hz s−1 , which have not been observed long enough to detect large glitches. Thus, we showed that the glitch activity of every pulsar with | ν̇| < 10−10.5 Hz s−1 , including those with no glitches, is statistically consistent with the above relationships. (2.3) The activity due to small glitches also increases with | ν̇|. We did not present an explicit relation between the activity of small glitches and | ν̇| because it depends strongly on the choice of the cutoff value between small and large glitches, and on detectability issues. (2.4) Pulsars with | ν̇| > 10−10.5 Hz s−1 present an intrinsically different behavior, with a much lower rate of large glitches..

(42) 38 Chapter 3 presented a statistical study of glitches as a time series in the seven pulsars with more than 10 glitches detected so far. The main conclusions are the following: (3.1) Among the pulsars studied, only the PSRs J0537−6910, B1758−23, and J1341−6220 show a clear correlation between the glitch size, ∆νk , and the interval of time until the following glitch, ∆τk+1 . For those objects, the values of r and the p-values are larger than 0.5, and ∼ 10−3 , respectively. (3.2) All the pulsars studied have positive correlation coefficients between ∆νk , and ∆τk+1 , but not as strong as PSR J0537−6910. However, although for the rest of pulsars the correlation seems not to be strong, it could be meaningful for all pulsars together since the probability of having all the same sign of the correlation coefficient is low. (3.3) Furthermore, simulations of correlated glitches show that the non-detection of several small glitches is not the reason for the weak correlation between ∆νk and ∆τk+1 , observed in the rest of pulsars. (3.4) Although the negative correlation between ∆νk and ∆νk−1 is not strong individually for each pulsar, it could be meaningful for all pulsars together since the probability of having all the same sign of the correlation coefficient is low. (3.5) We only found a significant and positive correlation between ∆νk and ∆τk in the Crab pulsar. The rest of pulsars show negative correlation coefficients and the pvalues associated do not allow to reject the null hypothesis that both variables are uncorrelated. Future studies, based on the analysis of glitches in individual pulsars, should provide information helping to clarify whether there are two or more glitch mechanisms giving rise to the observed bi-modal glitch size distribution. For this, a consistent analysis that allows to determine the smallest detectable size of a glitch for a pulsar is needed. Furthermore, the detection of new glitches in those objects can help clarify whether the correlations obtained in individual pulsars are a real phenomenon..

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