ANGULAR MOMENTUM TRANSFER IN VELA-LIKE PULSAR GLITCHES

Glitches are sudden spin-ups observed in the otherwise decreasing rotational frequency of a pulsar (Lyne et al. 2000). Their origin is still debated: the giant spin-ups observed in the 20 known Vela-like glitchers (Espinoza et al. 2011) could indicate the presence of bulk superfluidity inside these stars. In this scenario, giant glitches would represent the natural macroscopic outcome of the interaction between quantized neutron vortex lines, which carry the angular momentum of the rotating chargeless superfluid, and the Coulomb lattice of neutron-rich nuclear clusters, which coexists with the neutron superfluid in the inner crust (Negele & Vautherin 1973). Indeed, this interaction can pin vortices to the normal component of the star, thus freezing the superfluid vorticity and storing its angular momentum. Only when the hydrodynamical lift on a vortex (Magnus force), which increases as the pulsar slows down, equals the pinning force on a line, is the vortex unbound from the lattice; thus, free to move under the action of drag forces, it can transfer its angular momentum to the normal component of the star. According to Anderson & Itoh (1975), giant glitches are due to the sudden and simultaneous depinning of a large number of accumulated vortices, followed by the rapid transfer of their angular momentum to the observable normal crust (which consists of the outer crust plus all the other charged components in the star, electrons, protons, and nuclear clusters, strongly coupled together by the pulsar magnetic field). Such a storage and trigger mechanism would have a natural periodicity, as indeed observed in Vela (Dodson et al. 2007).

The vortex scenario for glitches was roughly compared to existing observations in a simple but instructive toy model, which assumed a cylindrical, uniform-density star with a cylindrical pinning shell (corresponding to the inner-crust) close to the surface (Pines et al. 1980; Alpar et al. 1981; Anderson et al. 1982). Although the naive treatment of the vortex–nucleus and vortex–lattice interactions gave pinning forces three orders of magnitude larger than that required to explain the average interval between glitches observed in Vela (Δt_gl ≈ 3 years), the model predicted the correct orders of magnitude for the typical glitch parameters known at the time, namely the jump in angular velocity, ΔΩ_gl ≈ 10⁻⁶Ω, and the jump in angular acceleration, $\Delta \dot{\Omega }_{\rm gl}\approx 10^{-2}\dot{\Omega }$ (the pre-glitch, steady-state parameters for Vela being Ω_Vela = 70 Hz and $\dot{\Omega }_{\rm Vela}=-9.8\times 10^{-11}$ Hz s⁻¹). In spite of these positive preliminary results, Pines et al. (1980) carefully pointed out that the effects of some crucial corrections had to be taken into account before drawing any conclusion: the spherical geometry of the star, the radial density profile required by gravitational equilibrium, the density dependence of the pinning interaction, and the presence of different superfluid phases along the star profile. To date, however, this has not been done in any coherent and consistent model; thus, the explanation of giant glitches in terms of vortices is not yet tested against observations, leaving the origin of these spin-ups an open question.

The post-glitch recovery of pulsars, on the other hand, has been successfully interpreted in terms of vortex motion under drag forces. Early on, the phenomenological model of Baym et al. (1969) explained the slow relaxation to steady state following a glitch as due to the weak interaction between a normal and a superfluid component, each rotating rigidly. Following a glitch, the response of the model is linear in the initial perturbation, and relaxes back to steady-state conditions exponentially, with a relaxation time which is inversely proportional to the strength of the interaction between the two components. The simple two-component model was then reformulated in terms of vortex motion to allow for differential rotation of the superfluid (Pines et al. 1980). Eventually, two scenarios were developed to describe vortex dynamics between glitches: thermally activated creep of strongly pinned vortices (Alpar et al. 1984a, 1984b; Link et al. 1993) or corotation of unpinned vortices under weak drag forces (Jones 1990, 1991, 1993). The vortex creep model was motivated by the large pinning forces obtained in early calculations (Alpar 1977; Epstein & Baym 1988). Later on, however, the microscopic vortex–nucleus interaction was shown to be one order of magnitude smaller than what found earlier (Donati & Pizzochero 2003, 2004, 2006). Moreover, Jones (1992) argued that the mesoscopic vortex–lattice interaction, necessary to calculate the macroscopic pinning force on a vortex line, is likely to be a factor α₁ ∼ 10⁻² smaller than that naively assumed in early calculations, due to the random orientation of the macro-crystals forming the inner crust, as well as to the rigidity of vortex lines on distances of order 10²–10³R_ws (with R_ws the radius of the Wigner–Seitz cells describing the nuclear lattice). Significantly smaller pinning forces favor the corotation model where unpinned vortex lines are weakly coupled to the normal crust by small drag forces; thus, in steady state they (nearly) corotate with the superfluid, while the response to perturbations is linear. Finally, the Christmas 1988 Vela glitch (Flanagan 1990) showed that the creep model can fit observations only in the linear response regime (Alpar et al. 1989) in which case it is equivalent to the corotation model, but with a different temperature dependence of the drag parameters. Observationally, the post-glitch recovery of Vela is well described by a sum of exponential terms with different amplitudes and relaxation times (Dodson et al. 2002), and this can be explained in terms of the linear response of regions of the superfluid characterized by different drag parameters. The dissipative force has been evaluated for several densities of interest, and the corresponding drag parameters yield relaxation times and glitch rise times compatible with observations (Jones 1990, 1992; Epstein & Baym 1992).

Three aspects of current post-glitch models are relevant here.

• 1.  
  Although simulations successfully reproduce the observed recovery of Vela (Larson & Link 2002), the glitch itself is always introduced by hand, as an ad hoc initial condition.
• 2.  
  Core and crust vortices are taken as physically disconnected, namely a layer of normal matter is assumed between the S-wave neutron superfluid found at subnuclear densities and the P-wave neutron superfluid found above nuclear saturation (Jones 1990). Recent microscopic calculations, however, do not show any discontinuity of this kind (Zhou et al. 2004), thus indicating continuous vortex lines throughout the star.
• 3.  
  In the core, the superconducting state of protons determines in part the mutual friction on neutron vortex lines. A type II superconductor corresponds to the strong-drag limit, with vortices entangled in a dense array of magnetic flux tubes (Link 2003); for a type I superconductor, instead, both the weak- and strong-drag limits have been suggested (Sedrakian 2005; Jones 2006). To date, the microscopic nature of the proton superconductor is far from settled; therefore, both scenarios of weak and strong mutual friction in the core should be taken into account in the study of glitches. Post-glitch models, however, are not affected by this theoretical uncertainty, since they involve mostly vortices in the equatorial regions, lying entirely in the inner crust.

A typical neutron star has total momentum of inertia I_tot ≈ 10⁴⁵ g cm², while its inner crust has I_ic ≈ 10⁻²I_tot. The accidental coincidence of the ratio I_ic/I_tot with the early observations of $\Delta \dot{\Omega }_{\rm gl}\approx 10^{-2}\dot{\Omega }$ and with the fact that about 1.7% of the Vela spin-down is reversed during a glitch led us to consider glitches as related to crust vorticity alone and thus to the assumption of disconnected neutron superfluids. This scenario, however, has direct implications on the glitch energetics. Indeed, since vortex lines in the core are strongly coupled to the normal component, being magnetized by entrainment effects (Alpar et al. 1984c), the normal crust comprises most of the star and I_c ≈ I_tot. This implies that the angular momentum transferred during the glitch is ΔL_gl = I_cΔΩ_gl ≈ 10⁴¹ erg s and the corresponding glitch energy is ΔE_gl = ΔL_glΩ ≈ 10⁴³ erg; both values appear too large. On the one hand, 10⁴¹ erg s corresponds to the difference in angular momentum of the entire inner crust between glitches, thus requiring some unlikely mechanism that freezes the vorticity everywhere in the crust for about 3 years, and then releases it simultaneously. On the other hand, observations of the power wind nebula surrounding Vela indicate an upper limit of ∼10⁴² erg to the glitch energy (Helfand et al. 2001).

In this Letter, we present the first realistic (with several approximations, but still preserving the essential physics) and consistent model to determine where in the star the vorticity is pinned, how much of it is pinned, and for how long. The model has been tested against observations using realistic equations of state (EoSs) for dense matter and implementing general relativistic hydrostatic equilibrium (P. M. Pizzochero et al., in preparation). Moreover, initial dynamical simulations based on the multifluid formalism of Andersson & Comer (2006) confirm the main assumptions and predictions of the model (Haskell et al. 2011). Here, however, we will discuss a fully analytical, Newtonian version of the model: it yields the correct orders of magnitude for all relevant variables, but provides deeper insight than any numerical treatment.

We now outline the main assumptions of the model and present the resulting equations; details and calculations, together with a parameter study of the solutions, will be given in a longer article (P. M. Pizzochero, in preparation; hereafter, Paper I).

We describe the core and inner crust as an n = 1 polytrope of mass M and radius R_s. The actual radius of the star will be larger because of the overlying outer crust; its presence, however, can be ignored here since it contributes negligibly to the mass and moment of inertia of the normal component. The polytropic relation P∝ρ² is a very soft EoS for dense matter; realistic soft EoSs yield R_s ≈ 10 km for M = 1.4 M_☉. The density profile is u = sin (πξ)/(πξ), with a dimensionless radius ξ = r/R_s and density u = ρ/λ, normalized to its central value λ = πM/(4R³_s). The radius of the core, R_c, corresponds to the density ρ_c = 0.6ρ₀ (in units of nuclear saturation density, ρ₀ = 2.8 × 10¹⁴ g cm⁻³), where nuclei merge into nuclear matter. The inner crust has ξ > x_c, where x_c = R_c/R_s > 0.9; for ξ > 0.9, the approximation u = 1 − ξ is sufficiently accurate (Figure 1, left). The total momentum of inertia is

$$\begin{equation} I_{\rm tot}=\frac{2(\pi ^2-6)}{3\pi ^2}MR_s^2=0.26MR_s^2, \end{equation} \tag{ 1 }$$

while the inner crust has M_ic = 4.9(1 − x_c)²M and I_ic = 12.6(1 − x_c)²I_tot.

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The standard superfluid fraction, Q, is introduced to describe the protons and nuclei of the normal crust, I_c = (1 − Q)I_tot, and the neutron superfluid component, I_s = QI_tot, of the star. Although the neutron fraction varies with density, a typical average value is Q ≈ 0.95 (Zuo et al. 2004). Regarding proton superconductivity in the core, here we choose the weak-drag limit. The model, however, also gives reasonable results for the jump parameters in the strong-drag limit; the decoupling of the core vorticity, pinned by flux-tubes, reduces both ΔL_gl and the moment of inertia responding at the glitch (cf. Paper I).

The density dependence of f_pin(ρ), the (mesoscopic) pinning force per unit length, is taken as in Figure 1 (right panel). This is a reasonable first approximation for a parameter study in terms of the maximum value, f_m; indeed, pinning goes to zero at ρ_c (no more nuclei) and at neutron drip ρ_d = 0.0015ρ₀ (no more neutrons), while it is expected to be maximum around densities where the pairing gap peaks. Moreover, we have performed a numerical simulation to evaluate f_pin(ρ) in a bcc lattice, with random crystal orientations and proper vortex rigidity. We obtain profiles compatible with Figure 1, with maximum values of order f_m ≈ 10¹⁵ dyn cm⁻¹ at densities ρ_m ≈ 0.2ρ₀. We also find, as already noted by Link (2009), that attractive and repulsive vortex–nucleus interactions are equivalent for pinning vortices to the lattice (F. Grill & P. M. Pizzochero, in preparation).

The geometry of the model is shown in Figure 2. The star spins around the z-axis, and continuous vortex lines are assumed through the core. Following the results of Ruderman & Sutherland (1974), we can reduce the problem to axial symmetry by integrating the density-dependent quantities along the vortex lines; these quantities will then depend only on the cylindrical radius x = R/R_s, with R being the distance from the rotational axis. We distinguish two cylindrical zones, separated by x_c = R_c/R_s: the "crust" (x > x_c), with vortices lying entirely in the inner crust, and the "core" (x < x_c), with vortices crossing the star core. In particular, we can integrate the pinning and Magnus forces to obtain an estimate of their total values on a vortex. If ω(x) = Ω_s(x) − Ω indicates the lag between the local superfluid angular velocity and that of the rigid normal crust, the critical lag for depinning, ω_cr(x), is obtained by equating these two forces

$$\begin{eqnarray} &&\int _v{d}z\,f_{\rm pin}[\rho (x,z)]=x\omega _{\rm cr}(x)\kappa \int _v{d}z\,\rho (x,z), \end{eqnarray} \tag{ 2 }$$

where κ = πℏ/m_N.

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In Figure 3, we show the resulting profile: ω_cr(x) presents a sharp peak in the "crust," with maximum ω_max located very close to x_m = 1 − u_m = 1 − ρ_m/λ; we will take ρ_m = 0.2ρ₀. In most of the "core," instead, ω_cr(x) has a roughly uniform value ω_min ≈ 10⁻²ω_max (as x → 0 it diverges; similar to the outer crust, however, this region can be neglected). We find (cf. Paper I)

$$\begin{eqnarray} &&\omega _{\rm max}=\omega _{\rm cr}(x_m)=\frac{4}{\kappa }\frac{R_s^2}{M}\frac{g_{\rm pin}(x_m)}{g_{\rm mag}(x_m)}f_m, \end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray} g_{\rm pin}(x)&=&\frac{1}{2(1-x)}\left[\sqrt{1-x^2}-x^2\ln \left(\frac{1+\sqrt{1-x^2}}{x}\right)\right]\nonumber\\ \end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray} g_{\rm mag}(x)&=&\pi x\left[\ln \left(\frac{1+\sqrt{1-x^2}}{x}\right)-\sqrt{1-x^2}\right]. \end{eqnarray} \tag{ 4b }$$

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In the "crust," this estimate of ω_cr(x) should be reasonable, since pinning is continuous along the vortices (Ruderman & Sutherland 1974; Jones 1990). In the "core," instead, pinning is discontinuous: vortex lines are attached to the lattice only at their extremities, while most of their length lies in a pinning-free region (having selected the weak-drag limit). We can expect individual string-like excitations of the pinned vortices, which could detach them from the lattice well before ω_min is reached. Indeed, the collective rigidity of vortex bundles in coherent motion, which explains the axial symmetry predicted by the Taylor–Proudman theorem (Ruderman & Sutherland 1974), is actually not observed in laboratory experiments with superfluid vortices attached to the rotating vessel only at their ends (Adams et al. 1985).

Although this issue requires and deserves further study, the crucial point is that vortices in the "core" are pinned very weakly. On the other hand, drag forces due to magnetization correspond to very short relaxation times and thus very small steady-state lags (Alpar et al. 1984c). Moreover, Link (2009) has shown that vortex repinning is dynamically possible if the lag falls below a critical value (smaller than the critical lag for depinning). These considerations and the profile in Figure 3 naturally suggest the following scenario: as the star slows down, vortices in the "core" are continuously depinned and then rapidly repinned; this dynamical creep allows a steady removal of the excess vorticity on short effective timescales τ_c. Although the value of τ_c is not relevant here, dynamical simulations of Vela glitches suggest τ_c ∼ 10⁰–10¹ s (Haskell et al. 2011), compatible with mutual friction dominated by vortex magnetization. On timescales Δt ≲ τ_c (e.g., during a glitch), the "core" vorticity is only partially coupled to the normal crust (the rest being pinned or responding on longer timescales), and only a detailed study of the dynamics can provide a direct estimate of the coupled fraction. On longer timescales Δt ≫ τ_c, however, the dynamical creep ensures full effective coupling of the two components with (average) lag of order $|\dot{\Omega }|\tau _c$; in steady state, this scenario is then equivalent to the corotation model.

The excess "core" vorticity will be repinned in the "crust," where pinning increases rapidly by orders of magnitude. At any time t after a glitch, the lag $\omega (t)=|\dot{\Omega }_{\infty }|t$ defines a radial distance x(t) as in Figure 3; here $\dot{\Omega }_{\infty }$ indicates the steady-state (pre-glitch) angular acceleration. We now assume that the excess vorticity, corresponding to the entire region x < x(t) and to the lag ω(t), is accumulated in a thin vortex sheet at x(t); as the star slows down and ω(t) increases, the sheet is pushed outward by the increasing Magnus force and moves with x(t). When ω(t) reaches the value ω_max, the sheet is at the pinning peak, x_m, and the vorticity accumulated in the sheet is finally released simultaneously, causing the glitch. This picture reminds one of a snowplow, pushing accumulated snow up an incline and eventually reaching its top edge. The interval between glitches is

$$\begin{eqnarray} &&\Delta t_{\rm gl}=\frac{\omega _{\rm max}}{|\dot{\Omega }_{\infty }|}. \end{eqnarray} \tag{ 5 }$$

This scenario is compatible with post-glitch relaxation; indeed, after a glitch, the unpinned vortices in the "crust" are under the same conditions as those considered in current post-glitch models.

Although the "snowplow" model is quite schematic, it contains a plausible mechanism for storing and releasing vorticity, as actually confirmed by parallel dynamical simulations (Haskell et al. 2011). In particular, the model allows us to calculate directly the angular momentum L_v(x) of the vortex sheet at x. In Figure 4, we show the reduction of angular momentum, ℓ_v(x) = L_v(x)/L_v(0), when uniformly distributed vorticity contained within x is accumulated in a sheet at x; at the peak, x_m, the reduction is of order 10⁻³. For comparison, we also show the significantly different results for a uniform-density, cylindrical or spherical star; we see how spherical symmetry and a realistic density profile are both crucial for obtaining the correct order of magnitude of ℓ_v(x).

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The angular momentum stored during Δt_gl and released at the glitch, ΔL_gl, can be calculated from the number of vortices removed from the interior and accumulated at x_m, namely, ΔN_v(x_m) = 2πR²_sx²_mω_max/κ. We find (cf. Paper I)

$$\begin{eqnarray} &&\Delta L_{\rm gl}=I_{v}(x_m)\omega _{\rm max}, \end{eqnarray} \tag{ 6 }$$

with an effective moment of inertia

$$\begin{eqnarray} &&I_{v}(x)=\frac{3\pi ^4}{2(\pi ^2-6)}g_{v}(x)QI_{\rm tot}=\pi ^2g_{v}(x)QMR_s^2, \end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray} &&g_{v}(x)=\frac{x^2}{6}\left[\sqrt{1-x^2}\left(1+2x^2\right)-3x^2\ln \left(\frac{1+\sqrt{1-x^2}}{x}\right)\right].\nonumber\\ \end{eqnarray} \tag{ 8 }$$

The glitch rise time is very short, τ_gl < 40 s (Dodson et al. 2002); we introduce a new parameter, Y_gl, which globally describes the fraction of vorticity coupled to the normal crust on timescales of order τ_gl (the steady-state coupled fraction, corresponding to long timescales and to pre-glitch conditions, is Y_∞ = 1). The value of Y_gl depends on the detailed short-time dynamics of the "core" vorticity; in order to get an estimate of the observables, only this quantity is needed. From angular momentum conservation and variation of the crust equation of motion we find the glitch jump parameters

$$\begin{eqnarray} \Delta \Omega _{\rm gl}&=&\frac{\Delta L_{\rm gl}}{I_{\rm tot}[1-Q(1-Y_{\rm gl})]} \end{eqnarray} \tag{ 9a }$$

$$\begin{eqnarray} \frac{\Delta \dot{\Omega }_{\rm gl}}{\dot{\Omega }_{\infty }}&=&\frac{Q(1-Y_{\rm gl})}{1-Q(1-Y_{\rm gl})}. \end{eqnarray} \tag{ 9b }$$

After fixing the basic stellar parameters M, R_s, and Q (more generally, M and an EoS), the model has two free parameters, f_m and Y_gl. It must predict three observables: the interval between glitches, the jump in angular velocity, and the jump in angular acceleration during a glitch. In the case of Vela, the average observed values are Δt_gl ≈ 3 years and ΔΩ_gl = 1.2 × 10⁻⁴ Hz (Lyne et al. 2000); we already mentioned that early observations gave $\Delta \dot{\Omega }_{\rm gl}/\dot{\Omega }_{\infty }\approx 10^{-2}$. More recent data, however, indicate much larger values; in particular, the year 2000 glitch (Dodson et al. 2002) added to the already known short-, middle-, and long-time relaxation components (τ_i ≈ 10⁴, 10⁵, 10⁶ s, with $\Delta \dot{\Omega }_i/\dot{\Omega }_{\infty }\approx 0.44,0.044,0.009$ for i = 1, 2, 3), a fourth and very short one, with τ₄ = 1.2 ± 0.2 minutes and $\Delta \dot{\Omega }_4/\dot{\Omega }_{\infty }=18\,\pm\, 6$ (1σ errors). In the 2004 glitch, however, such a component was observed only barely above noise and no firm conclusion could be drawn from the weak data (Dodson et al. 2007). Waiting for future observations, there is nonetheless evidence that right after a glitch $\Delta \dot{\Omega }_{\rm gl}/\dot{\Omega }_{\infty }$ is larger than unity.

In order to test the model against observations, we consider a standard neutron star with M = 1.4 M_☉, R_s = 10 km, and Q = 0.95. If we take f_m = 1.1 × 10¹⁵ dyn cm⁻¹, from Equations (3)–(8) we find that ω_max = 0.01 Hz, and thus Δt_gl = 3.1 years and ΔL_gl = 9.5 × 10³⁹ erg s (also ΔE_gl = 6.7 × 10⁴¹ erg). If we then take Y_gl = 0.05, from Equation (9) we obtain ΔΩ_gl = 1.3 × 10⁻⁴ Hz and $\Delta \dot{\Omega }_{\rm gl}/\dot{\Omega }_{\infty }=9.3$, in good general agreement with observations. In Paper I, we analyze the parameter dependence of these results; we find that the model is quite robust under physically meaningful variations of all the basic parameters (M, R_s, Q, ρ_c, ρ_m, ρ_d).

In conclusion, assuming continuous vortices throughout the star, we find that maximum pinning forces of order f_m ≈ 10¹⁵ dyn cm⁻¹ can accumulate ≈10¹³ vortices in the inner crust of a standard neutron star, and release them every ≈3 years, transferring an angular momentum ΔL_gl ≈ 10⁴⁰ erg s. This is one order of magnitude smaller than that inferred from the (microscopically inconsistent) assumption of disconnected vortices, yet it yields the observed glitch parameters, provided one assumes a small coupled fraction Y_gl < 10%. The model is compatible with post-glitch recovery and with the presently known microphysics; the numerical results follow from implementing both spherical geometry and a realistic density profile, and they are robust.

This work was supported by CompStar, a Research Networking Programme of the European Science Foundation (http://www.compstar-esf.org/).