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12.4 Pulsar glitches

Since the discovery by Jocelyn Bell and Anthony Hewish in 1967 of highly-periodic radio sources soon identified with rotating neutron stars (Hewish was awarded the Nobel Prize in Physics in 1974 [305]), more than 1700 pulsars have been found at the time of writing (pulsar timing data are available online at [26]). Pulsars are the most precise clocks with rotation periods ranging from about 1.396 milliseconds for the recently discovered pulsar J1748–2446ad [195] up to several seconds. The periodicity of arrival time of pulses is extremely stable. The slight delays associated with the spin-down of the star are at most of a few tens of microseconds per year. Nevertheless, longterm monitoring of pulsars has revealed irregularities in their rotational frequencies.
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Figure 74: Glitch Δ Ω∕Ω ∼ 9 × 10− 9 observed in the Crab pulsar by Wong et al. [425]

The first kind of irregularity, called timing noise, is random fluctuations of pulse arrival times and is present mainly in young pulsars, such as the Crab, for which the slow down rate is larger than for older pulsars. Indeed, correlations have been found between the spin-down rate and the noise amplitude [275]. Timing noise might result from irregular transfers of angular momentum between the crust and the liquid interior of neutron stars. A second kind of irregularity is the sudden jumps or “glitches” of the rotational frequency, which have been observed in radio pulsars and more recently in anomalous X-ray pulsars [234232106233296]. An example of a glitch is shown in Figure 74View Image. Evidence of glitches have also been reported in accreting neutron stars [153]. These glitches, whose amplitude vary from Δ Ω ∕Ω ∼ 10−9 up to Δ Ω∕Ω ∼ 16 × 10− 6 for PSR J1806–2125 [199Jump To The Next Citation Point], as shown in Figure 75View Image, are followed by a relaxation over days to years and are sometimes accompanied by a sudden change of the spin-down rate from −6 −5 |Δ Ω˙∕˙Ω | ∼ 10 – 10 to − 3 −2 |Δ Ω˙∕˙Ω | ∼ 10 –10. By the time of this writing, 171 glitches have been observed in 50 pulsars. Their characteristics and the references can be found at [24]. The time between two successive glitches is usually a few years. One of the most active pulsars is PSR J1341–6220, for which 12 glitches have been detected during 8.2 years of observation [416].

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Figure 75: Amplitudes of 97 pulsar glitches, including the very large glitch − 6 Δ Ω∕Ω ∼ 16 × 10 observed in PSR J1806–2125 [199].

Very soon after the observations of the first glitches in the Crab and Vela pulsars, superfluidity in the interior of neutron stars was invoked to explain the long relaxation times [41]. The possibility that dense nuclear matter becomes superfluid at low temperatures was suggested theoretically much earlier, even before the discovery of the first pulsars (see Section 8.2). Following the first observations, several scenarios were proposed to explain the origin of these glitches, such as magnetospheric instabilities, pulsar disturbance by a planet, hydrodynamic instabilities or collisions of infalling massive objects (for a review of these early models, see Ruderman [350]). Most of these models had serious problems. The most convincing interpretation was that of starquakes, as briefly reviewed in Section 12.4.1. However, large amplitude glitches remained difficult to explain. The possible role of superfluidity in pulsar glitches was first envisioned by Packard in 1972 [317]. Soon after, Anderson and Itoh proposed a model of glitches based on the motion of neutron superfluid vortices in the crust [10Jump To The Next Citation Point]. Laboratory experiments were carried out to study similar phenomena in superfluid helium [68408409]. It is now widely accepted that neutron superfluidity plays a major role in pulsar glitches. As discussed in Sections 12.4.2 and 12.4.3, the glitch phenomenon seems to involve at least two components inside neutron stars: the crust and the neutron superfluid. Section 12.4.4 shows how the observations of pulsar glitches can put constraints on the structure of neutron stars.

12.4.1 Starquake model

Soon after the observations of the first glitches in Vela and in the Crab pulsars, Ruderman [349] suggested that these events could be the manifestations of starquakes (see also [184] and references therein). As a result of centrifugal forces, rotating neutron stars are not spherical but are slightly deformed, as can be seen in Figure 40View Image. If the star were purely fluid, a deceleration of its rotation would entail a readjustment of the stellar shape to a more spherical configuration. However, a solid crust prevents such readjustment and consequently the star remains more oblate. The spin-down of the star thus builds up stress in the crust. When this stress reaches a critical level, the crust cracks and the star readjusts its shape to reduce its deformation. Assuming that the angular momentum is conserved during a starquake, the decrease ΔI < 0 of the moment of inertia I is therefore accompanied by an increase Δ Ω > 0 of the rotational frequency Ω according to

Δ-Ω- ΔI-- Ω = − I . (278 )
For a purely fluid rotating star, the moment of inertia can be written as
I = I0(1 + ε), (279 )
where I0 is the moment of inertia of a nonrotating spherical star. The parameter ε is proportional to Ω2 and is typically very small. For instance, for the Crab and Vela pulsars, ε ∼ 10 −4 and 10− 5, respectively. Since the decrease of the moment of inertia of a pulsar can be at most equal to that of a purely fluid star, this model predicts that the glitch amplitude is
Δ Ω δt ---- < ε-- , (280 ) Ω τ
where τ = ˙Ω ∕2Ω and δt is the time between two successive starquakes (on the order of years for the Vela and Crab pulsars). This model is consistent with the glitches observed in the Crab pulsar and explains the weak glitch activity of young pulsars by the fact that the internal temperatures are still too high for the crust to store a large stress. However, for the Vela pulsar, with 4 τ ∼ 10 years and −5 ε ∼ 10, this model predicts glitch amplitude of Δ Ω∕Ω ∼ 10− 9, about three orders of magnitude smaller than those observed. The starquake model fails to explain all the observations of pulsar glitches. Therefore, other mechanisms have to be invoked.

12.4.2 Two-component models

Due to the interior magnetic field, the plasmas of electrically charged particles inside neutron stars are strongly coupled and co-rotate with the crust on very long timescales on the order of the pulsar age [132], thus following the long-term spin-down of the star caused by the electromagnetic radiation. Besides, the crust and charged particles are rotating at the observed angular velocity of the pulsar due to coupling with the magnetosphere. In contrast, neutrons are electrically neutral and superfluid. As a consequence, they can rotate at a different rate by forming quantized vortex lines (Section 8.3.2). This naturally leads to the consideration of the stellar interior as a two-fluid mixture. A model of this kind was first suggested by Baym et al. [40] for interpreting pulsar glitches as a transfer of angular momentum between the two components. Following a sudden spin-up of the star after a glitch event, the plasma of charged particles readjusts to a new rotational frequency within a few seconds [133]. Moreover, as discussed in Section 8.3.7, neutron superfluid vortices carry magnetic flux giving rise to an effective mutual friction force acting on the superfluid. As a result, the neutron superfluid in the core is dynamically coupled to the crust and to the charged particles, on a time scale much shorter than the post-glitch relaxation time of months to years observed in pulsars like Vela, suggesting that glitches are associated with the neutron superfluid in the crust. This conclusion assumes that the distribution of proton flux tubes in the liquid core is uniform. Nevertheless, one model predicts that every neutron vortex line is surrounded by a cluster of proton flux tubes [369370]. In this vortex-cluster model, the coupling time between the core superfluid and the crust could be much longer than the previous estimates and could be comparable to the postglitch relaxation times.

The origin of pulsar glitches relies on a sudden release of stresses accumulated in the crust, similar to the starquake model. However, the transfer of angular momentum from the rapidly-rotating neutron superfluid to the magnetically-braked solid crust and charged constituents during a glitch allows much larger spin-up than that due solely to the readjustment of the stellar shape. Neutron superfluid is weakly coupled to a normal charged component by mutual friction forces and thus follows the spin-down of the crust via a radial motion of the vortices away from the rotation axis unless the vortices are pinned to the crust. In the latter case, the lag between the superfluid and the crust induces a Magnus force, acting on the vortices producing a crustal stress. When the lag exceeds a critical threshold, the vortices are suddenly unpinned. Vortex motion could also be initiated by a temperature perturbation, for instance the heat released after a starquake [267]. As a result, the superfluid spins down and, by the conservation of the total angular momentum, the crust spins up leading to a glitch [10]. If the pinning is strong enough, the crust could crack before the vortices become unpinned, as suggested by Ruderman [351Jump To The Next Citation Point352Jump To The Next Citation Point356Jump To The Next Citation Point353Jump To The Next Citation Point]. These two scenarios lead to different predictions for the internal heat released after a glitch event. It has been argued that observations of the thermal X-ray emission of glitching pulsars could thus put constraints on the glitch mechanism [251].

In the vortex creep model [9] a postglitch relaxation is interpreted as a motion of vortices due to thermal fluctuations. Even at zero temperature, vortices can become unpinned by quantum tunneling [268]. The vortex current increases with the growth of temperature and can prevent the accumulation of large crustal stress in young pulsars, thus explaining the low glitch activity of these pulsars. In the model of Alpar et al. [67] a neutron star is analogous to an electric circuit with a capacitor and a resistor, the vortices playing the role of the electric charge carriers. The star is, thus, assumed to be formed of resistive regions, containing a continuous vortex current, and capacitive regions devoid of vortices. A glitch can then be viewed as a vortex “discharge” between resistive regions through capacitive regions. The permanent change in the spin-down rate observed in some pulsars is interpreted as a reduction of the moment of inertia due to the formation of new capacitive regions. A major difficulty of this model is to describe the unpinning and repinning of vortices.

Ruderman developed an alternative view based on the interactions between neutron vortices and proton flux tubes in the core, assuming that the protons form a type II superconductor [354355]. Unlike the vortex lines, which are essentially parallel to the rotation axis, the configuration of the flux tubes depends on the magnetic field and may be quite complicated. Recalling that the number of flux tubes per vortex is about 1013 (see Sections 8.3.3 and 8.3.4), it is therefore likely that neutron vortices and flux tubes are strongly entangled. As superfluid spins down, the vortices move radially outward dragging along the flux tubes. The motion of the flux tubes results in the build up of stress in the crust. If vortices are strongly pinned to the crust, the stress is released by starquakes fracturing the crust into plates like the breaking of a concrete slab reinforced by steel rods when pulling on the rods. These plates and the pinned vortices will move toward the equator thus spinning down superfluid and causing a glitch. Since the magnetic flux is frozen into the crust due to very high electrical conductivity, the motion of the plates will affect the configuration of the magnetic field. This mechanism naturally explains the increase of the spin-down rate after a glitch observed in some pulsars like the Crab, by an increase of the electromagnetic torque acting on the pulsar due to the increase of the angle between the magnetic axis and the rotation axis.

12.4.3 Recent theoretical developments

Other scenarios have recently been proposed for explaining pulsar glitches, such as transitions from a configuration of straight neutron vortices to a vortex tangle [324], and more exotic mechanisms invoking the possibility of crystalline color superconductivity of quark matter in a neutron star core [4]. These models, and those briefly reviewed in Section 12.4.2, rely on rather poorly known physics. The strength of the vortex pinning forces and the type of superconductivity in the core are controversial issues (for a recent review, see, for instance, [367] and references therein). Besides, it is usually implicitly assumed that superfluid vortices extend throughout the star (or at least throughout the inner crust). However, microscopic calculations show that the superfluidity of nuclear matter strongly depends on density (see Section 8.2). It should be remarked that even in the inner crust, the outermost and innermost layers may be nonsuperfluid, as discussed in Section 8.2.2. It is not clear how superfluid vortices arrange themselves if some regions of the star are nonsuperfluid. The same question also arises for magnetic flux tubes if protons form a type II superconductor.

Andersson and collaborators [16] have suggested that pulsar glitches might be explained by a Kelvin–Helmholtz instability between neutron superfluid and the conglomeration of charged particles, provided the coupling through entrainment (see Section 8.3.7) is sufficiently strong. It remains to be confirmed whether such large entrainment effects can occur. Carter and collaborators [81] pointed out a few years ago that a mere deviation from the mechanical and chemical equilibrium induced by the lack of centrifugal buoyancy is a source of crustal stress. This mechanism is always effective, independently of the vortex motion and proton superconductivity. In particular, even if the neutron vortices are not pinned to the crust, this model leads to crustal stress of similar magnitudes than those obtained in the pinned case. Chamel & Carter [94Jump To The Next Citation Point] have recently demonstrated that the magnitude of the stress is independent of the interactions between neutron superfluid and normal crust giving rise to entrainment effects. But they have shown that stratification induces additional crustal stress. In this picture, the stress builds up until the lag between neutron superfluid and the crust reaches a critical value, at which point the crust cracks, triggering a glitch. The increase of the spin-down rate observed in some pulsars like the Crab can be explained by the crustal plate tectonics of Ruderman [351352356353], assuming that neutron superfluid vortices remain pinned to the crust. Even in the absence of vortex pinning, Franco et al. [149] have shown that, as a result of starquakes, the star will oscillate and precess before relaxing to a new equilibrium state, followed by an increase of the angle between the magnetic and rotation axis (thus increasing the spin-down rate).

12.4.4 Pulsar glitch constraints on neutron star structure

Basing their work on the two-component model of pulsar glitches, Link et al. [269Jump To The Next Citation Point] derived a constraint on the ratio If∕I of the moment of inertia If of the free superfluid neutrons in the crust to the total moment of inertia I of the Vela pulsar, from which they inferred an inequality involving the mass and radius of the pulsar. However, they neglected entrainment effects (see Sections 8.3.6 and 8.3.7), which can be very strong in the crust, as shown by Chamel [90Jump To The Next Citation Point91Jump To The Next Citation Point]. We will demonstrate here how the constraint is changed by including these effects, following the analysis of Chamel & Carter [94Jump To The Next Citation Point].

The total angular momentum J of a rotating neutron star is the sum of the angular momentum f J of free superfluid neutrons in the crust and of the angular momentum c J of the “crust” (this includes not only the solid crust but also the liquid core, as discussed in Section 12.4.2). As reviewed in Sections 10.2 and 10.3, momentum and velocity of each component are not aligned due to (nondissipative) entrainment effects. Likewise, it can be shown that the angular momentum of each component is a superposition of both angular velocities Ωf and Ωc [94Jump To The Next Citation Point];

f ff fc J = I Ωf + I Ωc, (281 )
J c = IcfΩf + IccΩc , (282 )
where ff I, fc cf I = I and cc I are partial moments of inertia, which determine f ff fc I = I + I and Ic = Icf + Icc. As discussed in [94Jump To The Next Citation Point], Ifc = Icf is expected to be positive in the core and negative in the crust.
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Figure 76: Schematic picture showing the variations Δ Ω and δΩ of the pulsar angular frequency Ω, during a glitch and in the interglitch period, respectively.

Let us denote (discontinuous) variations of some quantity Q during a glitch by ΔQ and (continuous) variations of this quantity during the interglitch period by δQ, as illustrated in Figure 76View Image. The total angular momentum J = J f + J c can be assumed to be conserved during a glitch, therefore,

f c ΔJ = − ΔJ . (283 )
If no torque were acting on the neutron superfluid in the interglitch period, its angular momentum Jf would be conserved and we would have f δJ = 0. However, neutron superfluid is weakly coupled to the magnetically-braked crust via friction forces induced by the dissipative motion of quantized vortex lines, as discussed in Section 8.3.5. Consequently, Jf does not remain exactly constant but decreases δJ f ≤ 0,
Ifc δΩf ≤ − -ff δΩc . (284 ) I
Friction effects prevent a long term build up of too large a deviation of the superfluid angular velocity Ωf from the externally observable value Ω = Ωc. This means that the average over many glitches (denoted by ⟨...⟩) of the change of relative angular velocity Ωf − Ω should be approximately zero
⟨Δ Ωf + δΩf ⟩ ≃ ⟨Δ Ω + δΩ ⟩. (285 )
Combining Equations (283View Equation), (284View Equation) and (285View Equation), it can be shown that the partial moments of inertia are constrained by the following relation obtained by Chamel & Carter [94Jump To The Next Citation Point]
(If)2 IIff ≥ 𝒢 . (286 )
The dimensionless coupling parameter 𝒢 [269Jump To The Next Citation Point] depends only on observable quantities and is defined by
𝒢 = Ag -Ω---, (287 ) |Ω˙av |
where Ag is the glitch activity [286]
1 ∑ Δ Ωi Ag = -- -----, (288 ) t i Ω
where the sum is over the glitches occurring during a time t and Ω˙ av is the average spin-down rate. According to the statistical analysis of 32 glitches in 15 pulsars by Lyne et al. [276], the parameter 𝒢 in Vela-like pulsars is 𝒢 ≃ 0.017. The coupling parameters for selected pulsars are shown in Figure 77View Image. In particular, an analysis of the Vela pulsar shows that 𝒢 ≃ 0.014.
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Figure 77: Coupling parameter 𝒢 for selected glitching pulsars. From [269Jump To The Next Citation Point].

Microscopic calculations [90Jump To The Next Citation Point91Jump To The Next Citation Point] show that the ratio f ff I ∕I is smaller than unity [94] (assuming that only neutron superfluid in the crust participates in the glitch phenomenon). We, thus, have

f f2 I- (I-)- I ≥ II ff ≥ 0. (289 )
Adopting this upper bound and substituting in Equation (286View Equation), Link et al.[269] derived the following constraint on the mass M and radius R of the Vela pulsar
M R ≥ 3.6 + 3.9--- km . (290 ) M ⊙
Let us emphasize that this constraint has been obtained by neglecting entrainment effects between neutron superfluid and crust. However, these effects are known to be very strong [90Jump To The Next Citation Point], so that the inequality (290View Equation) should be taken with a grain of salt.
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