ANTI-GLITCHES WITHIN THE STANDARD SCENARIO OF PULSAR GLITCHES

The generally accepted scenario of neutron star (NS) glitches, proposed by Anderson & Itoh (1975), assumes that a sudden unpinning of a group of vortices from their pinning centers results in an abrupt increase of the observed NS rotation frequency—a glitch. Here we present a toy model that demonstrates that, under certain conditions (in particular, for a neutron critical temperature in the outer core that is not too high; see Sections 4 and 5 for details), an opposite effect—an anti-glitch—can take place due to avalanchelike unpinning of vortices. Our model takes into account that superfluid density depends on the relative velocities between the normal and superfluid NS components (hereafter the ΔV effect; see Section 2). When a group of vortices leaves the superfluid region, the velocity lag between the normal component and pinned superfluid component decreases and, due to the ΔV effect, the mass of the superfluid fraction increases. Such a redistribution of mass between the normal and superfluid liquid components, which has been ignored so far, can naturally lead to an anti-glitch for certain model parameters. Although there are a number of models describing anti-glitches (see, e.g., Parfrey et al. 2012, 2013; Duncan 2013; Lyutikov 2013; Tong 2014; García & Ranea-Sandoval 2014; Huang & Geng 2014; Katz 2014), the proposed effect allows one to explain the "sudden" spin down ³ of the magnetar 1E2259+586 (Archibald et al. 2013) within the generally accepted scenario of NS glitches.

Here we introduce the notion of superfluid density ρ_s and discuss how it can be affected by the relative motion of superfluid and normal currents induced in the system (see below for the definition of superfluid and normal currents). Below we set ℏ = k_B = 1.

Let us consider a non-relativistic degenerate Fermi liquid composed of identical particles. Assume that at a temperature T less than some critical temperature T_c they start to form Cooper pairs (become superfluid). In what follows, for simplicity, we assume that the particles pair in the spin-singlet ¹S₀ state and ignore the Fermi liquid effects (in particular, we assume that the particle effective mass coincides with its bare mass m).

Generally, at T < T_c it is instructive to think of the superfluid liquid as consisting of two components, the superfluid and normal ones. The superfluid component, which has a density of ρ_s, is associated with the Cooper-pair condensate, while the normal component with the density ρ_q = ρ − ρ_s (ρ is the total liquid density) is associated with the thermal (Bogoliubov) excitations (unpaired particles). A notable property of any superfluid system is that these components can flow, without friction, with two distinct velocities (e.g., Landau et al. 1980; Khalatnikov 2000). As a consequence, the total mass-current density ${\pmb {J}}$ (or the momentum density ${\pmb {P}}$) of the liquid can be presented as a sum of two terms,

$$\begin{equation} {\pmb {J}}={\pmb {P}}=\rho _{s} {\pmb {V}}_{s}+\rho _{q} {\pmb {V}}_{q}, \end{equation} \tag{ 1 }$$

where ${\pmb {V}}_{s}$ and ${\pmb {V}}_{q}$ are the velocities of the superfluid and normal components, respectively. Note that the velocity ${\pmb {V}}_{s}$ is related to the momentum $2 {\pmb {Q}}$ of a Cooper pair by the expression ${\pmb {V}}_{s}={\pmb {Q}}/m$. In the reference frame in which ${\pmb {V}}_{s}=0$, we have (see, e.g., Landau et al. 1980; Gusakov & Haensel 2005)

$$\begin{equation} {\pmb {J}}|_{{\pmb {V}}_{s}=0} = {\pmb {P}}|_{{\pmb {V}}_{s}=0} = \sum _{{\pmb {p}}, \sigma } \, {\pmb {p}} \, f(E_{\pmb {p}}+{\pmb {p}} \, \Delta {\pmb {V}}), \end{equation} \tag{ 2 }$$

where the summation goes over the Fermi momenta ${\pmb {p}}$ and spin states σ of the Bogoliubov thermal excitations, $\Delta \pmb {V} \equiv {\pmb {V}}_{s}-\pmb {V}_{q}$, f(x) = 1/(e^x/T + 1) is the Fermi–Dirac distribution function for thermal excitations, $E_{\pmb {p}} = \sqrt{\vphantom{A^A}\smash{{{v_{\rm F}^2 (|{\pmb {p}}|-p_{\rm F})^2 + \Delta ^2}}}}$ is their energy in the reference frame in which ${\pmb {V}}_{s}=0$, and p_F and v_F = p_F/m are the particle's Fermi momentum and Fermi velocity, respectively.

Finally, Δ is the superfluid energy gap, which generally depends on both T and $|\Delta {\pmb {V}}|\equiv \Delta V$: Δ = Δ(T, ΔV). The fact that sufficiently large ΔV can affect the gap was emphasized by Bardeen (1962) (see also Gusakov & Kantor 2013; Glampedakis & Jones 2014, where this effect was discussed in application to NSs). In the absence of currents ($\Delta {\pmb {V}}=0$), a gap can be approximated as (Yakovlev et al. 1999)

$$\begin{equation} \Delta (T,\, 0)=T \, \sqrt{1-\tau } (1.456-0.157/\sqrt{\tau }+1.764/\tau) \end{equation} \tag{ 3 }$$

(τ ≡ T/T_c) and decreases from the value Δ₀ = 1.764 T_c at T = 0 to 0 at T = T_c. The dependence of Δ on ΔV at T = 0 is similar: it decreases from Δ₀ at ΔV = 0 to 0 at ΔV ≡ ΔV₀ = eΔ₀/(2p_F) (see Gusakov & Kantor 2013 for details). In an arbitrary frame, Equation (2) can be rewritten as

$$\begin{equation} {\pmb {J}} = {\pmb {P}} = \rho {\pmb {V}}_{s} + \sum _{{\pmb {p}}, \sigma } \, {\pmb {p}} \, f(E_{\pmb {p}}+{\pmb {p}} \, \Delta {\pmb {V}}). \end{equation} \tag{ 4 }$$

Equations (1) and (4) allow us to find an expression for the normal density ρ_q and hence for the superfluid density ρ_s = ρ − ρ_q,

$$\begin{equation} \rho _{q}= - \sum _{{\pmb {p}}, \sigma } \frac{{\pmb {p}}\Delta {\pmb {V}}}{\Delta V^2} \, f(E_{\pmb {p}}+{\pmb {p}} \, \Delta {\pmb {V}}). \end{equation} \tag{ 5 }$$

As follows from this formula, ρ_q and ρ_s depend on not only T but also ΔV (this is the ΔV effect introduced in the beginning of the section); the latter dependence is especially pronounced at p_F ΔV ∼ Δ(T, 0). Usually, however, one considers a situation in which p_F ΔV ≪ Δ(T, 0). Then two simplifications can be made. First, one can neglect the dependence of the gap on ΔV, Δ(T, ΔV) ≈ Δ(T, 0) (Gusakov & Kantor 2013). Second, one can expand the function $f(E_{\pmb {p}}+{\pmb {p}} \Delta {\pmb {V}})$ in Equation (5) in a Taylor series, retaining only the linear term in $\Delta {\pmb {V}}$. The resulting expression for ρ_q then reduces to the standard one (e.g., Landau et al. 1980) and is independent of ΔV,

$$\begin{equation} \rho _{q}= - \sum _{{\pmb {p}}, \sigma } \frac{({\pmb {p}}\Delta {\pmb {V}})^2}{\Delta V^2} \, \frac{d f(E_{\pmb {p}})}{dE_{\pmb {p}}}. \end{equation} \tag{ 6 }$$

As will be clear from the subsequent consideration, the condition p_F ΔV ≪ Δ is not necessarily satisfied in NSs. Hence, in what follows, we will make use of the more general formula (5) rather than the standard Equation (6).

Figure 1 illustrates the dependence of ρ_s = ρ − ρ_q on ΔV for seven stellar temperatures: T/T_c = 0, 0.2, 0.4, 0.6, 0.75, 0.85, and 0.95. At T/T_c ≪ 1, when ΔV is small, all particles are paired and ρ_s ≈ ρ. As the velocity lag ΔV becomes larger, the superfluid fraction decreases and eventually disappears (ρ_s = 0) at ΔV = ΔV₀ = eΔ₀/(2p_F).

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The standard glitch scenario teaches us that when the crust slows down, a pinned superfluid continues to rotate with a higher frequency Ω_s because vortices, which determine Ω_s, cannot freely escape from the pinning region. (Note that by the "crust" we mean not only the solid crust itself but also all the NS components rigidly coupled to it. These include non-superfluid and charged particles, neutron thermal Bogoliubov excitations, as well as the unpinned superfluid neutron component.) At a given moment, a group of vortices unpins (the actual unpinning mechanism is unknown but is not important for us here) and transfers angular momentum to the crust. In the standard scenario, this event leads to a decrease of Ω_s and, due to angular momentum conservation, to an increase of Ω_c. However, the standard consideration ignores the ΔV effect. How will Ω_c change if we account for it? The answer follows from the angular momentum conservation,

$$\begin{equation} I_{c0} \Omega _{c0}+I_{s0} \Omega _{s0}= I_{c1} \Omega _{c1}+I_{s1} \Omega _{s1}, \end{equation} \tag{ 7 }$$

where the subscripts 0 and 1 refer to the corresponding quantities before and after the glitch, respectively. In Equation (7), I_s is the moment of inertia of the pinned superfluid component, I_c = I − I_s is the moment of inertia of the remaining components, and I is the total stellar moment of inertia. In the non-relativistic limit, I_s is given by the integral over the pinning region,

$$\begin{equation} I_{s}=\int _{\rm pinning\, region} \rho _s\left(T,\,\Delta \Omega \, r\,{\rm sin}\theta \right)r^2\,{\rm sin}^2 \theta \,\,{\rm d}V, \end{equation} \tag{ 8 }$$

and depends on the rotation lag ΔΩ. Here ρ_s is the neutron superfluid density, r is the radial coordinate, θ is the polar angle, and dV is the volume element.

Taking into account that the variation δΔΩ ≡ ΔΩ₁ − ΔΩ₀ = (Ω_s1 − Ω_c1) − (Ω_s0 − Ω_c0) of the rotation lag in the glitch event is much smaller than ΔΩ₀, one can expand I_s(ΔΩ) and I_c(ΔΩ) in a Taylor series near the point where ΔΩ = ΔΩ₀ = Ω_s0 − Ω_c0 and rewrite Equation (7) in the form

$$\begin{eqnarray} &&I_{c0} \Omega _{c0}+I_{s0} \Omega _{s0} = [I_{c0}+I_{c}^{\prime } (\delta \Omega _{s}-\delta \Omega _{c})] (\Omega _{c0}+\delta \Omega _{c}) \nonumber\\ &&\quad +[I_{s0}+I_{s}^{\prime } (\delta \Omega _{s}-\delta \Omega _{c})] (\Omega _{s0}+\delta \Omega _{s}), \end{eqnarray} \tag{ 9 }$$

where $I_{c}^{\prime }=-I_{s}^{\prime }=d I_{c}/d(\Delta \Omega)$, δΩ_s = Ω_s1 − Ω_s0, and δΩ_c = Ω_c1 − Ω_c0. Equation (9) yields

$$\begin{equation} \delta \Omega _{c}=-\frac{I_{s0}-I_{c}^{\prime }\Delta \Omega _0}{I_{c0}+I_{c}^{\prime }\Delta \Omega _0}\delta \Omega _{s}. \end{equation} \tag{ 10 }$$

This formula reduces to the standard result, δΩ_c = −I_s0/I_c0 δΩ_s, if one sets $I_{c}^{\prime }=0$. Because $I_{c}^{\prime } =-I_{s}^{\prime }>0$ (superfluid density ρ_s decreases with increasing ΔΩ), ΔΩ₀ = Ω_s0 − Ω_c0 > 0, and δΩ_s = Ω_s1 − Ω_s0 < 0, it follows from Equation (10) that we shall observe an abrupt pulsar spin down (an anti-glitch; δΩ_c < 0) if $I_{c}^{\prime } \Delta \Omega _0 >I_{s0}$ or, equivalently,

$$\begin{equation} |I_{s}^{\prime }| \Delta \Omega _0 >I_{s0}. \end{equation} \tag{ 11 }$$

Thus, when an anti-glitch occurs, both the pinned superfluid and the rest of the star decelerate (δΩ_c, δΩ_s < 0), while the moment of inertia redistributes (via formation of additional Cooper pairs) to satisfy the angular momentum conservation.

As follows from Equation (11), for an anti-glitch to occur, it is necessary to have a relatively large $|I^{\prime }_{s}|$. In other words, the superfluid neutron density ρ_s(T, ΔΩ r sinθ) in the pinning region should be quite sensitive to a variation of ΔΩ (see Equation (8)). As discussed in Section 2, it is necessary to have p_F ΔV ∼ Δ(T, 0) or p_F ΔΩ r sinθ ∼ Δ(T, 0) for that (here and below the quantities p_F, Δ(T, 0), ΔV, T_c etc. refer to neutrons). From this estimate, one obtains a typical value ΔΩ_typ of the rotation lag ΔΩ, at which one can expect to have an anti-glitch,

$$\begin{equation} \Delta \Omega _{\rm typ} \sim 1 \left[ \frac{\Delta (T,\, 0)}{10^8 \, {\rm K}} \right] \left(\frac{n_0}{n} \right)^{1/3} \left(\frac{10^6 \, {\rm cm}}{r} \right) \,\,\, {\rm rad} \,\, {\rm s}^{-1}, \end{equation} \tag{ 12 }$$

where n₀ = 0.16 fm⁻³ is the nucleon number density in atomic nuclei and $n=p_{\rm F}^3/(3 \pi ^2)$ is the neutron number density. Generally, ΔΩ_typ increases with T_c (see Equation (3)).

The condition (12) can hardly be satisfied in the crust, because T_c there is larger than 10⁹ K everywhere except for the narrow regions at the slopes of the neutron critical temperature profile T_c(ρ). The corresponding values of ΔΩ_typ are much higher than the possible frequency lags sustained by the vortex pinning (see Link 2014; Seveso et al. 2014). However, this condition is very likely to be met in the core.

Recently, Chamel (2012) demonstrated that entrainment in the crust can be strong. If correct, this result indicates that the crust is probably not enough to explain pulsar glitches and the core superfluid may be responsible for glitches as well (Andersson et al. 2012; Chamel 2013; see, however, Steiner et al. 2014; Piekarewicz et al. 2014). It is generally accepted (e.g., Baym et al. 1969) that protons in the core form a type II superconductor that harbors a magnetic field confined to flux tubes (for a discussion of other possibilities, see Link 2003; Jones 2006; Charbonneau & Zhitnitsky 2007; Alford & Good 2008, and references therein). Neutron vortices pin to magnetic flux tubes in the same way that they pin to nuclei in the crust. A purely poloidal configuration of the magnetic field cannot immobilize vortices so that they freely escape from the core as NS slows down. Opposite to this, the toroidal component efficiently prevents vortices from moving outward (Sidery & Alpar 2009; Gügercinoğlu & Alpar 2014). A number of numerical simulations in non-superfluid NSs (Braithwaite 2009; Ciolfi & Rezzolla 2013) and in superconducting NSs (Lander et al. 2012; Lander 2013, 2014) show that the toroidal component of the magnetic field, localized in the outer layers of the NS core, is necessary for stability of the magnetic field configuration. These simulations demonstrate that the toroidal field, which can be noticeably higher than the surface magnetic field, is confined within an equatorial belt of width ∼0.1 R (R is the stellar radius) at a distance r ∼ 0.8 R from the center (Lander et al. 2012; Lander 2013, 2014; Ciolfi & Rezzolla 2013). In our numerical calculations, we assume that the pinning region, which coincides with the localization region of the toroidal field, has the form of a torus with coordinates r ∈ [0.75 R, 0.85 R], θ ∈ [π/2 − π/12, π/2 + π/12], ϕ ∈ [0, 2π], where ϕ is the azimuthal angle. The moment of inertia pinned to such a region is about I_s ∼ 0.1 I for ΔV = 0 and T = 0 (see Figure 2 and a similar estimate of Gügercinoğlu & Alpar 2014).

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The rotation lag ΔΩ₀, at which a glitch/anti-glitch occurs, is uncertain. Clearly, it cannot exceed the maximum value of the critical lag ΔΩ_cr in the pinning region, required to unpin vortices from flux tubes by the Magnus force. The typical energy of vortex pinning to flux tubes is of the order of 100 MeV (see Link 2014 and the formula (A13) of Ruderman et al. 1998). It corresponds to $\Delta \Omega _{\rm cr}\sim 0.1\, B_{12}^{1/2}\,\rm rad\, s^{-1}$ (a more refined estimate can be found in Link 2014). In the case of magnetars, we obtain ΔΩ_cr ∼ (1–3) rad s⁻¹ for the magnetic field in the core B ∼ 10¹⁴–10¹⁵ G. Below ΔΩ₀ will be treated as a free parameter of our toy model.

The value of T_c in the core is also uncertain and varies from 10⁷ K (Schwenk & Friman 2004) to 10⁹–10¹⁰ K (e.g., Baldo et al. 1998). Note that, to calculate the gap, the latter authors used bare neutron–neutron interactions and ignored medium polarization effects, which can substantially overestimate T_c (Gezerlis et al. 2014). For illustration, in our calculations, we choose T_c = 1.5 × 10⁸ K in the pinning region (T_c in the inner core can be larger). This value does not contradict the results of microscopic calculations and, e.g., is close to T_c reported by Dong et al. (2013). It also agrees with the predictions of a minimal cooling scenario (see, e.g., model a2 in Figure 12 of Page et al. 2013).

Using input parameters from Section 4, we analyzed whether anti-glitches are possible if we allow for the ΔV effect. Our results are shown in Figures 2 and 3. To plot the figures, we employed an APR equation of state (Akmal et al. 1998) and considered an NS with the mass M = 1.4 M_☉.

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Figure 2 presents the normalized moment of inertia of the pinned superfluid I_s0, calculated with Equation (8), as a function of the rotation lag, ΔΩ₀. The function I_s0(ΔΩ₀) is plotted for four typical (Kaminker et al. 2014; Viganò et al. 2013) values of magnetar temperature, T = 1.2 × 10⁸ K (black lines online), T = 10⁸ K (red lines online), T = 8 × 10⁷ K (blue lines online), and T = 5 × 10⁶ K (magenta lines online). At small rotation lags, $|I_{s}^{\prime }|$ is too small to meet the condition (11) (see thin lines in the figure), but it increases with ΔΩ₀. Then, at some value of ΔΩ₀ (marked with filled circles) $|I_{s}^{\prime }| \Delta \Omega _0$ and I_s0 become equal to one another and at higher ΔΩ₀, the inequality (11) always holds (then a vortex avalanche leads to an anti-glitch; thick lines in the figure).

Figure 3 presents the ratio δΩ_c/δΩ_s (see Equation (10)) versus ΔΩ₀. We remind the reader that δΩ_c is the observed rotation frequency jump, while δΩ_s is the frequency jump of the pinned superfluid component. In the standard glitch scenario, δΩ_s is negative and depends on the number of unpinned vortices and on the place in the star where they repin. It is thus a poorly constrained parameter.

The curves are plotted for the same set of temperatures as in Figure 2. When ΔΩ₀ is small, the ΔV effect is negligible and solid lines almost coincide with the corresponding dashed lines, which are calculated by ignoring this effect. As ΔΩ₀ increases, δΩ_c/δΩ_s also increases and eventually reaches 0 (this moment is shown by filled circles in the figure). At larger rotation lags, δΩ_c becomes negative (δΩ_c/δΩ_s is positive); then a vortex avalanche leads to an anti-glitch, which generally has a size |δΩ_c| similar to that of glitches at the same δΩ_s. Note that for higher temperatures, Δ(T, 0) is smaller and a lower rotation lag is required to produce an anti-glitch (see estimate (12)).

Further increase of ΔΩ₀ leads to a gradual shrinking of the pinning region (due to transformation of the superfluid matter to normal matter with growing ΔΩ₀). This leads to a rapid decrease of $|I_{s}^{\prime }|$ (see Figure 2) and hence to a sharp decrease of δΩ_c/δΩ_s. Finally, when the superfluidity is completely destroyed in the whole pinning region, neither glitches nor anti-glitches are possible (δΩ_c/δΩ_s = 0).

As follows from Figure 3, anti-glitches can be produced for ΔΩ₀ ≳ (1–2) rad s⁻¹, in agreement with the estimate (12). Such values of ΔΩ₀ are comparable to the critical rotation lag ΔΩ_cr, estimated in Section 4 (recall that ΔΩ₀ cannot exceed ΔΩ_cr). Thus, the standard glitch scenario can account for anti-glitches. Note, however, that this conclusion is sensitive to the assumed value of T_c in the pinning region. For example, for T_c ∼ 10⁹ K (e.g., Baldo et al. 1998) one has ΔΩ₀ ∼ ΔΩ_typ ∼ 10 rad s⁻¹; for such T_c, anti-glitches will hardly occur in our model unless ΔΩ_cr is substantially larger for some reason.

Here we propose a toy model that allows us to describe an anti-glitch within the standard scenario of pulsar glitches formulated by Anderson & Itoh (1975). The main feature of our model accounts for the so-called ΔV effect—the dependence of the superfluid density on the relative velocity of normal and superfluid components (see Section 2).

We predict that magnetars are the most promising anti-glitching objects. The high magnetic field of magnetars provides strong pinning of vortices to flux tubes in the outer core, which leads to a large rotation lag between the normal and superfluid components. As we showed, such a rotation lag may be sufficient to transform a glitch to an anti-glitch.

Could a similar mechanism produce anti-glitches in the crust of a magnetar or an ordinary pulsar? Most probably not, because the critical rotation lag ΔΩ_cr (see Section 4) seems to be noticeably smaller (Link 2014; Seveso et al. 2014) than the typical rotation lag ΔΩ_typ (see Equation (12)), which is needed to affect ρ_s. There are two reasons for this. First, the pinning force per unit length is weaker for pinning to crust nuclei than for pinning to flux tubes in the core of a magnetar. Second, the rotation lag (∼ΔΩ_typ), which can lead to an anti-glitch in the crust, is severalfold larger than in the core because the neutron critical temperature T_c in the crust is higher. Due to these two factors, it seems implausible that vortex avalanches in the crust can lead to a substantial redistribution of the stellar moment of inertia and thus to an anti-glitch. However, if for some reason vortex unpinning occurs exclusively in the narrow region where T_c is substantially lower (i.e., on the slopes of the critical temperature profile), then anti-glitches could, in principle, be produced.

We summarize by concluding that (1) the same magnetar can exhibit both glitches (due to vortex unpinning in the crust) and, under certain conditions, anti-glitches (due to vortex unpinning in the core) and (2) it is not very likely that ordinary pulsars can demonstrate any anti-glitching activity.

This is an extended version of the contribution presented at the conference "Physics of Neutron Stars—2014" (2014 July 28–August 1, St. Petersburg, Russia). We are very grateful to its participants, especially, to Crist$\acute{{\rm o}}$bal Espinoza, Bennett Link, and Andreas Reisenegger for discussions and thoughtful comments. This study was supported by the Russian Science Foundation (grant number 14-12-00316).