Draft version October 2, 2020 Typeset using LATEX twocolumn style in AASTeX63
A Quake Quenching the Vela Pulsar
Ashley Bransgrove,1 Andrei M. Beloborodov,1, 2 and Yuri Levin1, 3
1Physics Department and Columbia Astrophysics Laboratory, Columbia University, 538 West 120th Street, New York, NY 10027 2Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85741, Garching, Germany 3Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 6th floor, New York, NY 10010
(Accepted May 14, 2020) Submitted to ApJ
ABSTRACT The remarkable null pulse coincident with the 2016 glitch in Vela rotation indicates a dynamical event involving the crust and the magnetosphere of the neutron star. We propose that a crustal quake associated with the glitch strongly disturbed the Vela magnetosphere and thus interrupted its radio emission. We present the first global numerical simulations of a neutron starquake. Our code resolves the elasto-dynamics of the entire crust and follows the evolution of Alfv´enwaves excited in the magnetosphere. We observe Rayleigh surface waves propagating away from the epicenter of the quake, around the circumference of the crust — an instance of the so-called whispering gallery modes. The Rayleigh waves set the initial spatial scale of the magnetospheric disturbance. Once launched, the Alfv´enwaves bounce in the closed magnetosphere, become de-phased, and generate strong electric currents, capable of igniting electric discharge. Most likely, the discharge floods the magnetosphere with electron-positron plasma, quenching the radio emission. We find that the observed ∼ 0.2 s disturbance is consistent with the damping time of the crustal waves if the crust is magnetically coupled to the superconducting core of the neutron star. The quake is expected to produce a weak X-ray burst of short duration.
Keywords: magnetic fields — pulsars: general — pulsars: (PSR J0835-4510)
5 1. INTRODUCTION ∆ν/ν ≈ 10− (Espinoza et al. 2011). The so-called 1.1. Glitches ‘Crab-like’ pulsars feature strong jumps in spin-down with ∆ν/ ˙ ν˙ ∆ν/ν, power-law glitch-size distribu- Pulsars are highly stable rotators, which slowly spin tions, and exponential wait-time distributions (Melatos down. However, they show two types of irregularity et al. 2008). The so-called ‘Vela-like’ pulsars glitch dubbed timing noise and glitches. Timing noise is the quasi-periodically, with consistently large magnitude slow stochastic deviation from regular spin-down, most (Espinoza et al. 2011). prominent in young pulsars [Hobbs et al.(2010), Lyne The standard theoretical picture of a pulsar glitch in- et al.(2010)]. A glitch is a sudden increase in the spin volves a sudden transfer of angular momentum to the frequency ν, sometimes accompanied by a change in the arXiv:2001.08658v2 [astro-ph.HE] 30 Sep 2020 crust due to the catastrophic unpinning of superfluid spin-down rateν ˙. vorticity (Anderson & Itoh 1975). In this picture, the The first pulsar glitch was observed in the Vela pul- crust (ion lattice) spins down due to external torques sar (Radhakrishnan & Manchester 1969), and by now while the rotation of the crustal neutron superfluid re- there are more than 520 recorded glitches in 180 pul- mains unchanged as long as its vorticity (quantized vor- sars (Manchester 2018) with glitch magnitude (rela- tices) is pinned to the lattice. When the rotation mis- 12 tive frequency change) ranging from ∆ν/ν ≈ 10− to match builds up to some threshold, many vortices are unpinned simultaneously and migrate away from the Corresponding author: Ashley Bransgrove axis of rotation, spinning down the superfluid and spin- [email protected] ning up the crust, thus bringing the two components closer to corotation. 2
Quakes have been proposed in the past as a possible shear waves. In this paper, we do not provide an ar- mechanism for triggering the glitch [Ruderman(1976), gument for why a large stress should build up in Vela’s Alpar et al.(1994), Link & Epstein(1996), Larson & crust. However, we argue that a quake is able to connect Link(2002), Eichler & Shaisultanov(2010)]. Quakes the 2016 glitch with the observed major magnetospheric are expected to occur when the crust is stressed beyond disturbance coincident with the glitch. a critical strain ∼ 0.1, leading to its mechanical fail- The quake mechanism of exciting the magnetosphere ure (Horowitz & Kadau 2009). However, there exists no of a neutron star was previously studied in several works compelling reason why such large stresses should ever [Blaes et al.(1989); Thompson & Duncan(1995), Tim- build up in the crusts of typical pulsars, which are rel- okhin et al.(2000), Timokhin(2007)]. The wave trans- atively weakly magnetized and slowly spinning. There- mission coefficient at the crust-magnetosphere interface fore, other ideas for the glitch trigger were explored (see was calculated by Blaes et al.(1989), who considered Haskell & Melatos(2015) for a review). Nevertheless, in quakes as possible triggers of gamma-ray bursts (GRBs). this paper we argue that the 2016 glitch in the Vela pul- We consider much less energetic events, and thus we do sar, and the accompanying major magnetospheric tran- not expect a bright GRB to accompany a glitch. Other sient observed by Palfreyman et al.(2018), were trig- key differences are that our model is two-dimensional gered by a quake. (2D), time-dependent, and includes the self-consistent magnetic coupling to both the magnetosphere and the 1.2. The 2016 December Vela Glitch liquid core. These advances are essential for our model of the 2016 December event. We also include a liquid On 2016 December 12 a glitch of magnitude ∆ = ν/ν ocean, which was absent in the study of Blaes et al. 1 431 × 10 6 was observed in the Vela pulsar (PSR . − (1989), but find that it has little effect on the phenom- J0835-4510) with the 26 m telescope at Mount Pleas- ena that we study. ant, Tasmania, and the 30 m telescope at Ceduna, South We find that the quake shear waves spread sideways Australia (Palfreyman et al. 2018). For the first time, and fill the whole crust. Therefore, seismic crustal oscil- each single radio pulse was recorded during the glitch, lations populate the entire magnetosphere with Alfv´en and the pulse shape was seen to change dramatically. waves. The Alfv´enwaves bounce in the closed magneto- First, a broad pulse was detected, followed by a single sphere, become de-phased, and generate strong electric null (missing) pulse. The following two pulses showed an currents. De-phasing, in concert with growing wave am- unusually low linear polarization. Ashton et al.(2019) plitude in the outer magnetosphere leads to charge star- constrained the rise time of the glitch to be less than 12.6 vation, and e discharge. The discharge can flood the s. Additionally they found evidence for a slow-down of ± magnetosphere with plasma, interrupting the observed the pulsar immediately before the spin-up glitch. radio emission. We also find that excitation of Alfv´en Detection of the radiative feature accompanying the waves in the liquid core efficiently drains energy from the 2016 Vela glitch was challenging because of its very crustal oscillations, and thus limits the quake duration. short duration (two pulses, ∼0.2 s) and no subsequent Assuming the mean magnetic field at the crust-core in- long-term change in the pulse shape. This is differ- terface is comparable to the surface dipole field, and ent from the known behavior of high-B pulsars, such that the field in the core is bunched into flux tubes or as PSR J1119-6127 which showed persistent abnormal domains (as is expected for type-II and type-I supercon- radio pulsations in the months following its 2007 glitch ductors, respectively), we find that the quake amplitude (Weltevrede et al. 2011). Note also that no significant is exponentially reduced on the timescale ∼ 0.2 s, fast radiative change had been associated with a glitch in a enough to cause a single null. canonical radio pulsar until the dedicated observation of The paper is organized as follows. In Section2 Vela in 2016 by Palfreyman et al.(2018). we present the relevant parameters of Vela, and other This observation shows for the first time that the mag- physics input required by our model. In Sections3 and netosphere can be affected by a glitch – an event consid- 4 we provide an analytic description of the proposed ered to originate from the interior of the neutron star. picture of the 2016 event. Section5 outlines the formal- We see no plausible mechanism for the coupling between ism and numerical method for the full three-dimensional the pulsar interior and the magnetosphere other than (3D) problem, although we only present results in 2D seismic motions of the crust (a quake). Excitation of axisymmetry in this work. In Section6 we show four seismic motions requires a sudden change of elastic stress sample numerical models, and the results are further on the timescale 1 ms (the wave crossing time of the discussed in Section7. crust thickness). The quake is possible if the crust is stressed beyond its critical strain and “fails”, launching A Quake Quenching the Vela Pulsar 3
2. VELA MODEL 2.1. Observed Parameters of the Vela pulsar
OCEAN
˙ 11 2 Its spin-down rate Ω = −9.8432 × 10− rad s− gives a measurement of the magnetic dipole moment of the star q 3 3 30 3 µdip = 3c IΩ˙ /(2Ω ) ≈ 3.4 × 10 G cm , assuming I ≈ 1045 g cm2 for the star’s moment of inertia (Manch- ester et al. 2005). The corresponding dipole magnetic 3 12 3 field is Bd ≡ µdip/r? = 3.4 × 10 (r?/10 km)− G, where r? is the neutron star radius. The spin-down power of Vela is given by
36 1 Lsd = IΩΩ˙ ≈ 7 × 10 erg s− . (2)
The pulsed radio emission at frequencies around 1.4 GHz Figure 1. Schematic picture of the neutron star and its has a much smaller luminosity (Manchester et al. 2005), magnetosphere, indicating relevant length scales and char- acteristic densities. The gray shaded region represents the 28 1 LGHz ≈ 10 erg s− . (3) closed magnetosphere. The observed bolometric luminosity of the pulsar is 1/3 dominated by GeV gamma-rays from the outer mag- where a = (4πni/3)− is the mean inter-ion spacing netosphere (Abdo 2009), (Potekhin & Chabrier 2000). This defines the crystal- lization density ≈ 8 × 1034 erg s 1 (4) LGeV − . 3 6 7 T Z − A 3 ρ = 8 × 10 g cm− , (6) The apparent surface temperature of Vela (as measured crys 108 26 56 by a distant observer) is = (7 85 ± 0 25) × 105 K Ts∞ . . where A and Z are the ion mass and charge numbers. (Page et al. 1996). It is related to the actual surface We adopt the value ρ = 108 g cm 3 for all of our p 2 crys − temperature Ts by Ts∞ = Ts 1 − 2GM/r?c (Thorne numerical simulations. 6 1977). We will use the approximate Ts ≈ 10 K. The density profile of the neutron star ( ) (where 2.2. Magnetosphere, Ocean, Crust, and Core ρ r r is the radial coordinate) is obtained by integrating the In the magnetosphere, the plasma mass density ρ sat- equation of general relativistic hydrostatic equilibrium, 2 2 isfies ρc B /4π, and so Alfv´en waves propagate with using the SLy equation of state (Douchin & Haensel almost the speed of light. This changes in the ocean 15 3 2001), with a central density ρ = 10 g cm− . We use 2 2 where density ρ > ρB ≡ B /4πc , the OPAL equation of state for the ocean with temper- 8 2 ature T = 10 K(Rogers et al. 1996). We also make use 3 B 3 of the analytical fitting formula in Haensel & Potekhin ρB = 10 12 g cm− . (5) 3.4 × 10 G (2004) for the crust and the ocean. This gives a neutron star with mass M = 1.4M and radius r? = 11.69 km. The ocean is an excellent thermal conductor, and is ef- fectively isothermal in the deeper layers. According to For the SLy equation of state, there is a phase tran- the temperature profiles of Potekhin et al.(2016) the sition at the bottom of the crust that occurs at fixed 32 3 6 pressure P = 5.37 × 10 erg cm− . In our model, the ocean of a Vela-like pulsar with Ts = 10 K has uni- 8 6 crust-core boundary is located at r = 10.8 km, with form temperature T ∼ 10 K for densities ρ & 10 c g cm 3, which is in agreement with the analytic for- density − 14 3 mula of Gudmundsson et al.(1983). The solid-liquid ρc = 1.27 × 10 g cm− . (7) phase transition, which defines the top of the crust, is The neutron star structure is summarized in Figure1. 2 2 set by the Coulomb parameter Γ = Z e /akBT ≈ 175, The crust-ocean boundary is located at radius rcrys = 4
11.66 km, and the thickness of the crust is H ≈ 860 m. area A0. The energy of the quake is 2 The mass of the crust is Mc = 1.6×10− M . The ocean 2 2 µ0 39 0 A0 is ∼ 30 m deep. EQ ∼ ∆`A0 ∼ 10 3 11 2 erg. 2 10− 10 cm The speed of crustal shear waves is controlled by (8) the shear modulus of the crustal lattice µ. At densi- The wave propagates toward the stellar surface with ties far above the crystallization density, µ is propor- 1/2 8 1 speed vs = (µ/ρ) ≈ 10 cm s− and crosses the crust tional to the Coulomb energy density of the lattice and 5 2 thickness H ∼ 10 cm on the timescale is approximately given by µ ≈ 0.12 ni(Ze) /a where 1/3 H a ∼ ni− is the separation of the ion lattice with den- τ ∼ ∼ 1 ms. (9) vs sity ni (Strohmayer et al. 1991). At densities ρ below 11 3 The thickness of the shear layer sets the characteristic the neutron drip density, ρdrip ≈ 4 × 10 g cm− , it gives µ ∝ ρ4/3. In the deeper crust µ scales almost lin- frequency of the generated waves. As a concrete exam- early with ρ. The shear modulus has a sharp cutoff at ple, consider the smooth deformation √ density ρcrys, so that µ = 0 in the ocean. " # ξ0 2(z − z ) The star’s magnetic field is frozen in its core, crust, ξ(z) = erf Q , (10) 2 ∆` and ocean. In our axisymmetric numerical models, we assume that the magnetic field in the magnetosphere where z < 0 is the distance below the stellar surface. has a dipolar configuration aligned with the axis of ro- It corresponds to a shear layer of thickness ∆` at depth tation. We also need to include magnetic stresses in- zQ. The characteristic length scale of the deformation is 1 p side the crust, when computing the transmission of the `0 ≡ ξ(dξ/dz)− = π/8∆`. The characteristic angular seismic waves into the magnetosphere. For computa- frequency is tional simplicity we assume that the field inside the crust is that of a monopole, chosen so that the field 1 12 vs 4 ∆` − 1 at the surface equals 3 × 10 G. The spherical symme- ω ∼ ≈ 2 × 10 rad s− . (11) ` 104 cm try of the background configuration dramatically speeds 0 up the computation of crustal oscillations, because the The quake can excite a broad spectrum of waves ex- vibrational eigenfunctions used in our spectral code are tending to frequencies well above this characteristic fre- easily computed through the separation of angular and quency. radial variables (see Section 5.2 for details).1 An im- portant feature of our model is that the magnetic field 3.1. One-dimensional Model of Waves lines connecting the rotating star with the light cylinder Much insight about the transmission of seismic waves are assumed to be open, and their footprints on the star into the magnetosphere and the core can be obtained form the two “polar caps.” In the simplest case of a from studying the propagation and transmission of ra- nearly aligned rotator, the angular size of the polar cap dially directed seismic waves. A classic one-dimensional 1/2 is θp ≈ (r?/RLC) ≈ 0.05. (1D) model of this type was developed by Blaes et al. (1989). Following their approach, we approximated the crust as a 1D slab with the normal along the z-axis (which would be in the radial direction for a spherical crust). The shear displacement ξ(z) is in they ˆ-direction. For the timescales of interest, the star is an ideal con- 3. QUAKE EXCITATION OF SHEAR WAVES ductor, so the magnetic field is perturbed by the dis- We model the quake as a sudden change in shear stress placement along the y-axis, By = Bz∂ξ/∂z, as required in the deep crust, which launches an elastic wave with by the flux-freezing condition. As a first approximation, an initial strain amplitude 0. The quake is triggered in the magnetosphere is also described by ideal MHD. 4 a region of vertical thickness ∆` ∼ 10 cm (comparable The magneto-elastic wave equation is given by to the hydrostatic pressure scale height) and horizontal ∂2ξ ∂ ∂ξ ρ˜ = µ˜ , (12) ∂2t ∂z ∂z 1 Replacing the dipole field with monopole below the stellar sur- face only slightly changes the crust dynamics and the calculated whereρ ˜ andµ ˜ are the effective mass density and shear displacements of the magnetospheric footpoints. In the magne- modulus, respectively, such that tosphere itself, the waves are followed in the correct dipole back- 2 2 ground. Had we kept the dipole field throughout, we would get B B ρ˜ = ρ + z , µ˜ = µ + z . (13) similar results with a much greater computational effort. 4πc2 4π A Quake Quenching the Vela Pulsar 5
v˜s of elasticity, which is applicable in the limit of 1. In 10 vA particular we assume that nowhere in the solid crust vs does the strain exceed the critical value crit ∼ 0.1 9 (Horowitz & Kadau 2009). This condition is satisfied for a quake with a typical strain in the deep trigger re- 3 8 gion 0 < 2 × 10− . Our numerical models in Section6 11 2
v (cm/s) have the starquake area A0 ∼ 3 × 10 cm , which gives
10 38 the quake energy EQ ∼ 10 erg (Equation8). log 7 For waves excited on scales comparable to the hydro- static scale height of the crust (as assumed in our quake 6 scenario), the WKB approximation is not accurate, and the exact solution should be obtained numerically. More ρB ρcrys ρrefl ρdrip ρc 5 importantly, the 1D model is insufficient, as the quake 4 6 8 10 12 14 waves propagate at different angles, and after reflection 3 log10ρ (g/cm ) from the surface layers, they tend to spread sideways to
Figure 2. Wave speedv ˜s(ρ) in the magnetosphere, ocean, fill the entire crust. The numerical simulations of this crust, and core (thick black line). The dashed line shows the process are presented in Sections5 and6 below. Here we Alf´enspeed vA(ρ), and the dotted line shows the elastic wave estimate the transmission coefficients analytically using speed vs(ρ). the simple 1D model.
1/2 The wave speed is given byv ˜s = (˜µ/ρ˜) and shown in 3.2. One-dimensional Wave Transmission into the 8 1 Figure2. It equals vs ≈ 10 cm s− in the deep crust and Magnetosphere grows to the speed of light in the magnetosphere. The The wave reflection occurs in the upper crust, which wave speed in the liquid core equals the Alfv´enspeed, is defined by ρdrip < ρ < ρcrys. For a vertically prop- which depends on B and the density of matter coupled agating wave, the transmission coefficient (the ratio of to the Alfv´enwave, as discussed in Section 3.3 below. transmitted to incident energy flux) is given by iωt For a harmonic time dependence ξ ∝ e− with 4 1 4Z Z 4Z ω & 10 rad s− the wave propagation may be described crust mag mag Tm = 2 ≈ , (18) in the WKB approximation. Then an upward propagat- (Zcrust + Zmag) Zcrust ing wave and its reflection from the low-density surface where the impedance Z =v ˜sρ˜ is evaluated in the layers are given by Blaes et al.(1989) upper crust at the transmission layer ρrefl, Zcrust ≈ ρrefl v˜s(ρrefl), and in the magnetosphere, Zmag ≈ ρBc 1 h i(u+ωt) i(u ωt)i ξ ∝ √ e− + A e − , (14) R Zcrust. In the relevant region,ρ ˜ ≈ ρ ρB, and the ρv˜s shear wave speed may be approximated as where Z z 2 2 ω 2 15 1/3 b cm B u ≡ − dz0 . (15) v˜s ≈ 10 7ρ9 + 2 , b = 12 , v˜s ρ9 s 3.4 × 10 G (19) The first term in brackets in Equation (14) is the upward where we normalized to the characteristic dipole field propagating wave, and the second term with the com- B of the Vela pulsar, and ρ = ρ/(109 g cm 3). Note plex amplitude A is the reflected wave. The scaling 9 − R thatv ˜ (ρ) is non-monotonic (see Figure2). The wave of the overall amplitude ξ ∝ (ρv˜ ) 1/2 comes from the s s − speed first decreases from v ≈ 2 × 108 cm s 1 in the conservation of energy flux in the wave F ∼ ρv ω2ξ2. In s − s deep crust to 9 × 107 cm s 1 at ρ = 109 g cm 3. This particular, usingv ˜ ∝ ρ1/6 in the upper crust, one finds − − s decrease shortens the wavelength by a factor of ∼ 2, 7/12 so that it remains comparable to or shorter than the ξ ∝ ρ− (ρ < ρdrip), (16) hydrostatic scale height. However, as ρ further decreases 9 3 and the strain in the shear wave is below 10 g cm− , the wave speed steeply grows, and the length scale of this change soon becomes shorter than ∂ξ ξω 1 3/4 ≡ = ∝ ∝ ρ− . (17) the wavelength. Therefore, reflection mainly occurs at ∂z v˜ 1/2 3/2 s ρ v˜s 9 3 ρrefl just below 10 g cm− . The reflection condition The strain can become large in the low-density regions may be written as (Blaes et al. 1989) and cause a secondary failure of the crust. However, in d 2 this work we choose to remain within the linear theory v˜ ∼ ωv˜s. (20) dz s 6
Pressure in the upper crust is dominated by relativistic 3.3. Wave Transmission into the Core degenerate electrons, and the hydrostatic balance gives The bottom of the crust is magnetically coupled to 3 1/3 the relation |z| ≈ 6 × 10 ρ9 cm, where z < 0 is the the liquid core. The core supports a multitude of MHD depth below the stellar surface. Using this relation and modes, which get excited while draining elastic wave Equation (19), we obtain the equation for ρrefl, energy from the crust (Levin 2006). The Alfv´encross-
1 2 ing time of the core τA ∼ r?/vA ∼ 1 s is longer than 3b2 ω b2 / 1 − ∼ √4 1/3 + (21) the characteristic lifetime of crustal waves (estimated 4/3 ρ9 . 7ρ9 7ρ9 2 below). Effectively, the waves escape into the core as if it were an infinite reservoir. Under such conditions, At high frequencies one can keep only the second terms the transmission coefficient for a vertically propagating on both sides of the equation, which gives shear wave at the crust-core interface can be estimated 6/5 as 9 b g 4ZcrustZcore ρrefl ≈ 2 × 10 3 (ω > ωeva) . (22) Tc = 2 , (24) ω4 cm (Zcrust + Zcore) where Z and Z are the impedances of the crust One can show that the reflection condition (20) does crust core 4 1 and the outer core, not apply when ω < ωeva ≈ 2 × 10 rad s− . In that case, the reflection occurs deeper in the crust due to the Zcrust = ρ>v˜s,Zcore = ρ
2 ii) Decoupling of protons from other species in the core The fundamental frequency of the liquid ocean is ωocean = 5 −1 reduces the effective mass density participating in the v˜s/Ho ∼ 3 × 10 rad s where Ho ∼ 30 m is the scale height 4 of the ocean. For the characteristic frequency of the crustal os- oscillation to the proton density, ρ< = ρp. This reduc- cillations ω ωocean, the ocean can be viewed as attached to tion of ρ< (by a factor of ∼ 10) decreases the transmis- the moving crust. Effectively, the waves are transmitted directly from the solid crust to the extended magnetosphere above the sion coefficient by a factor ∼ 3. ocean. 3 4/7 3/7 Blaes et al.(1989) obtained a different result Tm ∝ B ω , because they considered neutron stars with lower B = 1011 G. In 4 Even in the presence of strong vortex-flux-tube interactions, a that case, ρrefl is much lower, and the hydrostatic stratification negligible fraction of the neutron mass couples to the oscillations is different because the degenerate electrons are sub-relativistic. we are considering [see van Hoven & Levin(2008)] A Quake Quenching the Vela Pulsar 7
The net effect is an enhancement of the transmission for waves propagating to radii r r? in the outer mag- 1/2 1/2 coefficient Tc, by a factor of ∼ 6. netosphere as δB/B ∝ F /B ∝ B− . In particular, 3 The Alfv´enspeed in the outer core is for a dipole magnetosphere, B ∝ r− , and so
1/2 3/2 BBc 6 1 δB 4 1/2 r vA = ∼ 5 × 10 cm s− , (27) ≈ 10− F?,26 . (33) 4πρp B r? and the resulting transmission coefficient is The emitted Alfv´enwaves bounce in the closed mag- netosphere on the light-crossing timescale tb and can 2 Tc ∼ 2 × 10− . (28) accumulate energy and δB during the quake. This ac- cumulation occurs on field lines that do not extend too The transmitted waves are lost for the quake. Since Tc far from the star, so that their tb is shorter than the for the superconducting core is ∼ 5 times greater than quake duration. Tm, the lifetime of crustal waves is controlled by their Alfv´enwaves can be thought of as the propagating leakage to the core rather than to the magnetosphere. shear of the magnetic field lines. They require electric The characteristic lifetime is given by current j along B as long as the wavevector k has a k component perpendicular to B, k 6= 0. This com- 2τ ⊥ τcore = ∼ 100 ms. (29) ponent is inevitably present, since the field lines are Tc curved. The waves develop different phases on differ- ent field lines, and thus amplify the gradients of δB in 4. MAGNETOSPHERIC WAVES AND ELECTRIC the direction perpendicular to the field lines. 5 DISCHARGE The electric current j may be estimated as k 4.1. Electric Current of Alfv´enWaves c c δB j ∼ k δB ∼ , (34) The magnetospheric disturbance may be described as k 4π ⊥ 4π ` ⊥ ideal MHD Alfv´enwaves as long as there is enough 1 where ` ∼ k− is the spatial scale of the wave variation plasma in the magnetosphere to support electric cur- ⊥ perpendicular⊥ to B. The length scale ` is initially de- rents. The energy flux of the Alfv´enwaves into the ⊥ magnetosphere is approximately given by termined by the elasto-dynamics of the crust. But once Alfv´en waves on neighbouring field lines accumulate EQTm 26 EQ,38 erg a difference in path length similar to the wavelength, F? ∼ ∼ 4 × 10 2 , (30) τA A12 s cm they are effectively de-phased. Therefore, ` decreases, ⊥ and so j grows as the Alfv´enwaves keep bouncing in where is the area through which the crustal wave k A the closed magnetosphere. The growth of j may be energy is leaking into the magnetosphere. Initially, at estimated as follows. k times comparable to τ = H/vs ∼ 1 ms, the waves emerge from the quake area ≈ . Later, grows as the A A0 A Let us consider a dipole magnetosphere and let θ be waves spread horizontally through the crust. the polar angle measured from the dipole axis. It is con- The Alfv´enwaves are ducted along the magnetic field venient to label the field lines by the poloidal magnetic lines, and their flux F changes proportionally to the flux function, local magnetic field B, µ sin2 θ ψ = dip , (35) B r F = F? . (31) B? which is constant along a field line. In the axisymmetric magnetosphere, ψ =const on each flux surface formed This fact follows from F dS =const where dS = dψ/B by a field line rotated about the axis of symmetry. A is the cross-sectional area of a field-line bundle carrying closed field line with footprints on the star at θ? and infinitesimal magnetic flux dψ. The flux F determines 2 π−θ? extends to radius rmax = r?/ sin θ?, and its length the wave amplitude , δB is ∼ 3rmax. The bounce cycle of Alfv´enwaves along a closed field line takes time ∝ ∝ 1, so two field 1/2 tb rmax ψ− 8πF 8 δB ≈ ∼ 3 × 10 F26 G. (32) c 5 In particular, in axisymmetry, δBB is azimuthal, and its gradient is The relative perturbation of the magnetic field is small in the poloidal plane. This gradient has a component perpendic- 4 1/2 ular to the background dipole field B and generates ∇ ×δB k B. near the star, δB?/B? ≈ 10− F?,26. However, it grows 8 lines separated by a small ∆ψ have different tb, One can see that the Alfv´enwaves generated by the quake can become charge-starved, especially when one ∆tb ∆ψ ≈ − . (36) takes into account the growth of δB due to the accu- t ψ ? b mulation of waves trapped in the closed magnetosphere. After time t, the accumulated phase mismatch between waves on flux surfaces separated by ∆ψ is Once charge starvation is reached, a parallel electric ∆φ ∆ψ field will be induced to support ∇ × B. The resulting ≈ − . (37) ωt ψ parallel voltage may be estimated as
De-phasing on a given scale ∆ψde occurs when |∆φ| ∼ 4πj 2 Φ ∼ k ` ∼ δB ` . (43) π, and so ∆ψde(t) ∼ πψ/ωt. At a radius r > r?, the c ⊥ ⊥ distance ` between the poloidal field lines separated ⊥ The voltage is maximum for the largest ` at which by ∆ψ is ⊥ de starvation occurs. This scale ` is given by the condition ∆ψ πr tan θ ⊥ ` (t) ≈ r de ∼ . (38) ⊥ 2 δB ∂ψ/∂θ ωt ∼ 4πMρGJ, (44) ` This gives the current density (Equation 34) ⊥ which yields c δB ( ) ∼ (39) 2 j t 2 ωt. c(δB) 4πF k 2π r tan θ Φ ∼ = . (45) 2MΩB MΩB Note that F/B =const (Equation 31), so the generated 4.2. e± Discharge voltage is approximately the same at all r along the field In the canonical pulsar picture, the rotating closed line and can be estimated with F = F? and B = B?. magnetosphere is filled with plasma that sustains the This gives corotation electric field E = −v ×BB/c (here v = Ω ×r). Φ This implies the characteristic minimum plasma density e 9 1 2 ∼ 3 × 10 M− F?,26. (46) (Goldreich & Julian 1969), mec
|∇ · E| |Ω · B| This voltage exceeds the threshold for e± discharge, as nGJ = ≈ . (40) 6 7 4πe 2πce particle acceleration to γ ∼ 10 − 10 is sufficient to ig- nite creation by emitting high-energy curvature pho- The actual plasma density may be higher by a multi- e± tons (Ruderman & Sutherland 1975). This process will plicity factor M, n = Mn . This factor is believed GJ flood the magnetosphere and the open field-line bundle to be large in the open field-line bundle, in some cases with plasma. Therefore, the quake should be capable exceeding 103, because the open field lines are twisted e± of interrupting the normal radio pulsations of Vela. and sustain continual e± discharge. The value of M in the closed magnetosphere is unknown and likely much lower, because this zone is not active and generates no 5. SETUP OF THE NUMERICAL SIMULATION discharge. It may, however contain e± pairs created by In this section, we outline the formalism and the setup gamma-rays entering from the open field lines (Chen & of our numerical simulations. We are able to simulate Beloborodov 2014). the elasto-dynamics of the crust in 3D; however we are The existing plasma in the closed zone can sustain currently limited to the 2D axisymmetric simulations Alfv´enwaves with the maximal current of the magnetosphere. Since the two computations are M |Ω · B| coupled, we are restricting ourselves to the 2D axisym- jmax = ceMnGJ = . (41) 2π metric simulations of the whole system.
When j exceeds jmax, the waves become charge starved, and thek ideal MHD approximation must break (Blaes 5.1. Dynamics of the Crust et al. 1989). From Equations (33), (39), and (41), we We use the linearized equations of motion (see, e.g. find McDermott et al.(1988), Blaes et al.(1989)). For sim- j c (δB/B) ωt plicity, the background state of the crust is assumed to k ∼ 2 jmax 4π M Ω r tan θ have a potential magnetic field, ∇ × B = 0 and j = 0. 1/2 The background is static and has E = 0. A displacement ω4 δB?/B? r t ˙ ∼ 10 4 (42). ξ(t,rr) creates motion with velocity ξ = dξξξ/dt ≈ ∂ξξξ/∂t in M tan θ 10− r? 0.1 s A Quake Quenching the Vela Pulsar 9 the linear order. The momentum and continuity equa- 5.2. Spectral Method tions are In order to numerically solve Equation (53), we pre- 1 ρξ¨ = ∇ · σ + δjj × B + g δρ − ∇δp, (47) fer to use a spectral method for superior stability and c accuracy over a large range of densities. Our formal- ism follows closely that of van Hoven & Levin(2012). δρ = −∇ · (ρξξ), (48) Equation (53) is written in the form where σ is the elastic stress tensor of the crustal 2 Coulomb lattice, g is the gravitational acceleration, and ∂ ξ = Lˆ(ξ) = Lˆel(ξ) + Lˆmag(ξ), (54) p is the pressure; perturbations are denoted by δ. The ∂t2 quake waves involve a fraction of the Coulomb energy ˆ ˆ density of the lattice, which is much smaller than the where the linear differential operators Lel and Lmag give hydrostatic pressure. Therefore, compressive motions the acceleration due to elastic and magnetic forces, re- and radial displacements are negligible, and hereafter spectively. The elastic acceleration is we consider only solenoidal deformations (∇·ξ = 0) and ˆ 1 2 set = 0. In this model, = 0, = 0, and the Lel(ξ) = (∇µ · ∇)ξ − (ξ · ∇)∇µ + µ∇ ξ . (55) ξr δρ δp ρ density of the crust is spherically symmetric. The stress tensor for an isotropic and incompressible The operator Lˆmag is greatly simplified by approxi- solid is (Landau & Lifshitz 1970), mating the crustal magnetic field as purely radial (a 2 ∂ξi ∂ξj monopole) with Br = B0(r?/r) , where B0 is the typ- σij = µ + , (49) ical magnetic field strength in the crust. In reality ∂xj ∂xi B varies over the crust. We use the fiducial value of where µ is the crustal shear modulus. The linear theory r B = 3 × 1012 G. The magnetic acceleration is then of elasticity is applicable in the limit of small strain. 0 For the short timescales considered in this problem, 2 ˆ 1 µB ∂ ξ the crust is effectively an ideal conductor. In the con- Lmag = [∇ × ∇ × (ξ × B)] × B = r 2 , ˙ 4πρ ρ ∂r r ductor rest frame, which is moving with velocity ξ, the (56) electric field must vanish, 2 where µB ≡ Br /4π depends only on r. We use spherical ξ˙ × B coordinates r, θ, and φ. δEE + = 0. (50) c We separate variables t, r, θ, and φ in Equation (54), and define magneto-elastic modes ξ as the eigenfunc- Then the induction equation ∂BBB/∂t = −c∇ × E gives nlm tions of the operator Lˆ with the boundary conditions of δBB = ∇ × (ξ × B). (51) zero stress at the boundaries (free oscillations of the sys- The excited electric current δjj is related to δBB and δEE tem), ˆ 2 by the Maxwell equation, L(ξnlm) = −ωnlmξnlm. (57) 4π 1 ∂δEE Here is the eigenfrequency of the mode with ra- δjj = ∇ × δBB − ωnlm c c ∂t dial, polar, and azimuthal numbers n, l, and m, respec- 1 = ∇ × ∇ × (ξ × B) + ξ¨ × BB. (52) tively. The modes ξnlm(r) form an orthogonal basis for c2 a Hilbert space with the inner product Substitution of Equations (49) and (52) into Equation Z (47) gives the elasto-dynamic wave equation, hηη,ββi = ρηη · β d3rr, (58) ¨ ¨ 2 ρξ + ρB ξ =(∇µ · ∇)ξ − (ξ · ∇)∇µ + µ∇ ξ V ⊥ 1 (53) where η and β are arbitrary vector functions defined + [∇ × ∇ × (ξ × B)] × BB, 4π over the volume of the crust V. Therefore, an arbitrary 2 2 solenoidal displacement field of the crust ξ(rr, t) may be where ρB = B /4πc and ξ is the displacement per- ⊥ decomposed as pendicular to B. In the crust, Equation (53) describes oscillations of the magnetized solid. In the liquid ocean, X ξ(rr, t) = a (t)ξ (r), (59) µ −→ 0 and Equation (53) describes pure Alfv´enwaves. nlm nlm n,l,m The dynamics of the crust and the ocean of interest oc- 3 3 curs at densities ρ ρB ∼ 10 g cm− where the term where ¨ ρB ξ can be neglected. hξ(rr, t),ξξnlmi ⊥ anlm(t) = . (60) hξnlm,ξξnlmi 10
Effectively, the spectral method replaces the crust with For simplicity, we approximate the background mag- many oscillators. Equation (54) describes free oscil- netic field B as purely radial so that B = Br. Since lations, with no external forces, and is reduced to the core is effectively an infinite reservoir on the quake 2 a¨nlm(t) + ωnlmanlm(t) = 0. In the presence of magnetic timescale (Section 3.3), there are only inward propagat- coupling to the magnetosphere/core, external forces ing waves with the displacement of the form ξ(t+r/vA), f mag and f core appear at the upper/lower boundaries where vA is the Alfv´enspeed in the core. The magnetic of the crust, field of the emitted waves is related to the displacement
f ext = f mag + f core. (61) ξ by the flux-freezing condition,
Then each oscillator is driven by the projection of the 1 Br ˙ δBB< = ∇ × (ξ × B) = ∂r(Brr ξ) ≈ Br∂rξ = ξξ. external force on the eigenmode, r vA (65) 2 hf ext(rr, t),ξξnlmi Here subscript “<” stands for the core region immedi- a¨nlm(t) + ωnlmanlm(t) = . (62) ˙ hξnlm,ξξnlmi ately below the crust, and ξ is the time derivative of the displacement at the interface. The initial conditions anlm(t = 0) are determined by The presence of δBB< implies that the core applies the initial displacement ξ0 and Equation (60). We then Maxwell stress to the bottom of the crust. The extracted evolve the spectral coefficients anlm, our effective dy- momentum flux is namical variables, using Equation (62). B δB σ = − r h,< , (66) rh 4π 5.3. Basis Functions where h = θ, φ labels the horizontal component. Since For the class of solenoidal displacements we are con- the crustal modes are calculated with the stress-free sidering, and the above operators, the natural choice of boundary condition δBB = 0, the external stress must basis functions is be included as a driving term in the oscillation Equa- tion (62). The external force appearing in this equa- ξ = ζ (r)r × ∇Y , (63) nlm nl lm tion is applied to the bottom layer of the crust of some thickness ∆r and density ρ (just above the interface), where ζnl contains the radial part of the eigenfunction, > so that f ρ ≈ σ /∆r. Approximating the layer as and r × ∇Ylm is the third vector spherical harmonic. ext > rh Substitution of Equation (63) into Equation (54) results infinitesimally thin, the external force at the crust-core in the following Sturm-Liouville problem: interface becomes B δBB dµ˜ dζ ζ µ˜ d dζ f = − r < δ(r − r ). (67) 2 nl nl 2 nl core 4 c −ωnlρζnl = − + 2 r πρ> dr dr r r dr dr (64) ζ Substituting the core Afv´enspeed v = B /(4πρ )1/2, − [l(l + 1)µ + 2µ ] nl , A r < B r2 we obtain ρ< ˙ The radial eigenfunctions ζnl(r) and eigenvalues ωnlm = f core = − vA ξ δ(r − rc), (68) ρ> ωnl do not depend on the azimuthal mode number m due to the spherical symmetry of ˜. Note that in the µ where ρ< is the mass density of the core infinitesimally limit µB → 0 Equation (64) is the same as Equation (23) below the crust-core interface. One can see that coupling in McDermott et al.(1988). to the core is equivalent to adding a damping force ∝ ξ˙. We use a high-order Sturm-Liouville solver to numeri- The projection of f core onto each basis function is cally find the eigenfunctions and eigenvalues of Equation computed once at the beginning of the simulation and (64). The details are given in AppendixA. stored in an array (see AppendixB).
5.4. Coupling to the Core 5.5. Coupling to the Magnetosphere The magnetic field is frozen in the crust and the liq- In this work, we model the pulsar magnetosphere as uid core, and so crustal oscillations deform the magnetic dipole, and treat the magnetospheric waves as linear per- field lines and launch Alfv´enwaves into the core. The turbations, using the framework of force-free electrody- feedback of these waves on the crust dynamics is incor- namics. In force-free electrodynamics the inertia of the porated in our simulations as follows. plasma is negligible compared to the inertia of the mag- netic field, and the equation of motion is replaced by the A Quake Quenching the Vela Pulsar 11 condition Between the solid crust and the force-free magneto- j × B ρeE + = 0. (69) sphere there is the liquid ocean. The ocean dynam- c ics can be calculated by extending the magnetosphere It implies E · B = 0 and E · j = 0, so there is no dis- model so that each 1D string includes a heavy part sipation. This approximation is valid if there is enough at the footprint where the string mass density is in- plasma to sustain electric currents excited in the per- creased and the shear wave is decelerated below c as turbed magnetosphere. For linear perturbations about 1/2 vA/c = (ρ/ρB + 1)− . The technical motivation for a stationary background state with E = 0 (in the coro- treating the ocean motions as part of the magneto- tating frame) and ∇ × B = 0 the force-free condition spheric dynamics is that it is liquid and hence “force- becomes δjj × B = 0. Substitution of δjj from Equa- free” — it does not sustain any shear forces. Note tion (52) then gives however that the ocean depth is small compared with the crust thickness, and at wave frequencies of inter- ¨ 1 ρB ξ = [∇ × ∇ × (ξ × B)] × BB. (70) est, it moves together with the crust at the footprints of ⊥ 4π the magnetospheric field lines. Effectively, the magneto- Note that only the perpendicular displacement ξ = ξ ⊥ sphere is attached to the solid crust, and in the numer- enters the force-free wave equation. ical models presented in Section6 the presence of the The wave equation gives the dispersion relation for ocean will be neglected. We also performed more de- eigen modes ξ ∝ exp(−iωt + k · r), tailed simulations with ocean dynamics included, which 2 support this approximation for Vela. ω 2 2 ξ = k ξ + k (k · ξ), (71) Solving the magnetospheric field-line dynamics re- c k ⊥ quires two boundary conditions. For closed field lines, where k and k are the components of k parallel and k ⊥ the boundary conditions are applied at the two foot- perpendicular to B, respectively. The eigen modes in- points where the field line intersects with the surface clude shear Alfv´enwaves (k · ξ = 0) with dispersion re- of the neutron star. The field line is attached to the lation ω = k c, and compressive (called “fast”) modes. k star and its footprint displacement equals the instanta- The perturbations are generated by the shear motions neous displacement of the uppermost layer of the crust, of the crust at the footprints of the magnetospheric field ξ(t, r?), which is determined by Equation (62). lines, and these motions should launch Alfv´enwaves. For open field lines, only one end is attached to the Their conversion to fast modes in the magnetosphere star, giving one boundary condition ξ(r?). The other is a second-order effect, which is negligible as long as end is at the outer boundary of the computational do- δB/B 1. main. At this end, we apply the condition of free escape, The group speed of Alfv´enwaves is parallel to B, so which means that there are only outgoing Alfv´enwaves. they are ducted along the magnetic field lines. For the Outgoing waves are functions of t − χ/c and satisfy the linear dynamics of Equation (70), each poloidal field line condition behaves like an independent string, with no coupling to ∂ξξ 1 ∂ξξ = − , (73) other field lines. Then effectively we need to solve a 1D ∂χ c ∂t wave equation along each poloidal field line. χend χend In axisymmetry, ∂/∂φ = 0, the Alfv´enwaves have the In our simulations, the magnetosphere is sampled with ˆ displacement in the φ-direction, ξ = ξφ φ. It is con- 275 closed and 50 open flux surfaces. The outer bound- 7 venient to work in the so-called magnetic flux coordi- ary of the open field lines is set at rmax = 10 cm, and 8 nates (ψ, χ, φ). The coordinate ψ represents surfaces the last closed field line extends to RLC = 4.2 × 10 of constant poloidal flux (for a dipole magnetosphere it cm — the light cylinder radius of Vela. We follow the is given by Equation (35)), and χ is the length along dynamics of each field line by solving the string Equa- poloidal field lines in the φ = const plane (Goedbloed tion (72) with the boundary condition ξ(r?) at the foot- et al. 2010). Equation (70) can be written in the flux prints and Equation (73) at the outer boundary. The coordinates as magnetospheric dynamics is coupled to the crustal oscil- 2 2 lations at r?, so the crust and the magnetosphere evolve ∂ ξφ(ψ, χ) c ∂ 2 ∂ ξφ(ψ, χ) 2 = r B , (72) together as a coupled system. The coupled differential ∂t r B ∂χ ⊥ ∂χ r ⊥ ⊥ Equations (62) and (72) are integrated numerically us- where r = r sin θ is the cylindrical radius. Each flux ing the fourth-order Runge-Kutta scheme, as described surface⊥ in the magnetosphere is effectively a 1D string in AppendicesB andC. (with mass density and tension both proportional to The feedback of the emitted magnetospheric waves on Br ) supporting shear wave propagation with speed c. the crust oscillations is implemented similarly to the ⊥ 12 crust-core interaction described in Section 5.4. In the the beginning, we observe shear waves propagating to- axisymmetric model, both the displacement and the ward the surface and launching Alfv´enwaves into the perturbed magnetic field are in the φ-direction. Let magnetosphere directly above the quake region (which δB = Bφ,> be the perturbed field immediately above the is at the north polar cap in model A1). Due to the stellar surface. The magnetospheric stress BrBφ,>/4π large impedance mismatch at both the crust-core and is communicated directly to the solid crust at the bot- the crust-magnetosphere interfaces, most of the quake tom of the ocean, where density ρ = ρcrys. To extract energy remains trapped inside the crust, and the waves the required momentum flux σrφ = −BrBφ,>/4π from bounce many times between the two interfaces. Some ˆ the crust, we apply force fmag = −(σrφ/ρcryst∆r) to the waves are launched in the θ-direction with a large sur- upper layer of the solid material with a small thickness face amplitude and cross the circumference of the crust ∆r, in a time πr?/v˜s ∼ 30 ms. These surface waves are the Br δBB> so-called “whispering gallery modes” (Rayleigh 1894). f mag ≈ δ(r − rcrys). (74) 4πρcrys However, most of the shear wave energy remains con- centrated at the north pole for a longer time, and grad- The magnetospheric perturbation B is related to the φ ually spreads toward the south pole after many small displacement ξ (ψ, χ) by the flux-freezing condition, φ angle reflections at the interfaces. As the centroid of ∂ ξφ the shear wave energy passes the magnetic equator the δBφ = Br . (75) ⊥ ∂χ r luminosity of Alfv´enwaves into the magnetosphere, LA, ⊥ drops because Br is small. After ∼ 200 ms the wave This allows one to express f mag in the form energy has spread throughout the entire crust, and the same luminosity of Alfv´enwaves is measured from the ρ(r ) ∂ ξ φ > 2 φ north and south poles. The evolution of LA is shown in fmag = vA(r>) cos α r δ(r − r?), ρcrys ⊥ ∂χ r Figure8. ⊥ r> (76) The magnetospheric Alfv´enwaves are initially coher- where α is the angle between the magnetic flux surface ent when launched from the surface (Figure4, top right), and the radial direction. In the model where the mag- with the perpendicular length scale determined by the netosphere is directly attached to the solid crust (ne- length of the elastic waves in the crust. After a light- glecting the thin ocean), vA(r>) = c and ρ(r>) = ρB. crossing time (∼ 45 ms for the last closed field-line) all This approximation is used in the simulations presented of the Alfv´enwaves become de-phased (Figure5, top below. A more detailed model of magnetospheric waves right). The regions where |j /cρGJ| > 1 are mapped in k with the ocean at the footprints would have vA(r>) ≈ Figures4 and5. We find that avoiding charge starvation 1/2 2 B/(4πρcrys) ≈ 10− c and ρ(r>) = ρcrys. It would and the ignition of e± discharge requires the magneto- 3 explicitly follow the wave acceleration to c as it crosses spheric plasma to have a high multiplicity M & 10 , the ocean. in agreement with the estimates in Section 4.2. After three rotations of Vela, LA has dropped by a factor of 6. SAMPLE MODELS ∼ 2 − 3. Less than 3% of the quake energy EQ has been transferred to the magnetosphere (Figure9). We have calculated four sample models: A1, B1, and The dynamics in model B1 is the same except that the A2, B2. Their parameters are given in Table1, and elastic waves spread from a different quake region, now the initial displacement of the disturbed crust is shown located at latitude θ ∼ π/4 instead of the north pole Figure3. In all the models, the quake has energy = EQ (Figures6 and7). The energy budget and the timescale 1038 erg. for injecting the Alfv´enwaves into the magnetosphere Models A1 and B1 have no crust-core coupling, rep- are similar to those in model A1. At first, Alfv´enwaves resenting a pulsar with a magnetic field confined to are only launched into the closed field-lines (Figure6), the crust and not penetrating the core. Models A2 but after ∼ 20 ms the crustal shear waves have spread to and B2 have strong crust-core coupling; they assume the north polar cap, and Alfv´en waves are launched into a superconducting core, and the poloidal component the north open field-line bundle, and the entire closed of the magnetic field at the crust-core interface B ≈ magnetosphere. Their luminosity L remains quite con- 3 4×1012 G, similar to the measured surface dipole field A . stant for the remainder of the simulation. After 3 rota- of Vela. tions of Vela, ∼ 3% of the initial elastic energy has been The dynamical picture of quake development is quite transmitted into the magnetosphere. similar in all four models. As an example, the snap- shots of model A1 are shown in Figures4 and5. At A Quake Quenching the Vela Pulsar 13
Table 1. Sample models.
Model Quake Location Core Core vA ρ 0 A0 EQ A1 Polar cap Decoupled — — 4.4 × 10−4 3 × 1011 cm2 1038 erg A2 Polar cap Superconducting 5 × 106 cm s−1 0.1 4.4 × 10−4 3 × 1011 cm2 1038 erg B1 θ = π/4 Decoupled — — 1.3 × 10−4 1 × 1012 cm2 1038 erg B2 θ = π/4 Superconducting 5 × 106 cm s−1 0.1 1.3 × 10−4 1 × 1012 cm2 1038 erg
x (106 cm) Models A2 and B2, which include the crust-core cou- 0.0 0.2 0.4 0.6 0.8 1.0 1.2 pling, show a significant difference from models A1 and 1.2 B1: the lifetime of crustal waves is significantly reduced, because the wave energy is drained into the core. This 1.0 draining occurs exponentially, because it results from ˙ the damping force fcore ∝ ξ (Equation 68). The evolu- tion of the crustal wave energy is well approximated by 0.8
t cm) Ecrust ≈ EQ exp − , (77) 6 0.6 10
τcore ( z with τcore ≈ 86 ms in both models A2 and B2 (Figure9). The luminosity of Alfv´enwaves into the magnetosphere 0.4 LA decays on the same characteristic timescale. After three rotations of Vela, ∼ 1% of the initial elastic en- 0.2 ergy is in the magnetosphere, and ∼ 95% of the initial A1, A2 energy has been transmitted into the liquid core. The 0.0 luminosity LA has decreased by a factor of ∼ 20. The evolution of and the wave energy in all four models 0.06 0.03 0.00 0.03 0.06 LA − − is summarized in Figures8 and9. ξφ (m) x (106 cm) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 7. DISCUSSION 1.2 Glitches give deep insight into the exotic dynamics of quantum fluids that likely exist in pulsar interiors. 1.0 One of the unsolved theoretical issues is the cause of the nearly simultaneous unpinning of billions of super- fluid vortices over a macroscopic 10 − 103 m length that 0.8 must take place during a glitch. The catastrophic un- cm) pinning is required to explain the glitches’ magnitudes, 6 0.6 10 especially the giant glitches with the relative spin-up of ( 5 z ∼ 10− observed in Vela. Crustal quakes have been sug- gested as one of the candidates for the glitch trigger, but 0.4 not considered promising for Vela. Indeed, what could deform the crust so dramatically that it would have a 0.2 mechanical failure? Vela’s external magnetic field is two B1, B2 orders of magnitude smaller than that of magnetars, and 0.0 thus the magnetic stresses are not obviously sufficient to break the crust. Furthermore, Vela is spinning at 1% of 0.03 0.02 0.01 0.00 0.01 0.02 0.03 − − − the break-up angular velocity, and thus its relative rota- ξφ (m) 4 tional deformation is ∼ 10− , which is smaller than the Figure 3. Top: initial conditions used for models A1 and critical strain of the crust. Therefore, rotational defor- A2. Bottom: initial conditions used for models B1 and B2. mation is also unlikely to lead to a quake. Color shows the amplitude of the azimuthal displacement ξφ. Nonetheless, the remarkable observations of the 2016 The amplitude is scaled so that each initial condition has the 38 glitch by Palfreyman et al.(2018) force one to seriously initial energy E = 10 erg. The gray dashed lines show the consider a quake as a trigger. The change in the magne- boundaries of the crust. tospheric activity indicates its strong disturbance by the 14
Figure 4. Model A1 at t = 2 ms. Top left: displacement ξφ of the crust near the epicenter of the quake. The dashed lines show the boundaries of the crust. Top right: toroidal perturbation of the magnetic field Bφ/B (left), and the ratio |jk/cρGJ | (right). The green curves show the poloidal magnetic field. The two field lines closest to the axis of symmetry are the boundary of the open field-line bundle. The gray dashed circle is the surface of the neutron star. Bottom: displacement ξφ(r, θ) in the entire crust, plotted on the r-θ plane.
Figure 5. Same as Figure4 but at time t = 50 ms. A Quake Quenching the Vela Pulsar 15
Figure 6. Model B1 at t = 2 ms. Top left: displacement ξφ of the crust near the epicenter of the quake. The epicenter is at θ = 45 ◦, and we have rotated the figure by −45 ◦ (˜x = x − z andz ˜ = x + z). The dashed lines show the boundaries of the crust. Top right: toroidal perturbation of the magnetic field Bφ/B (left), and the ratio |jk/cρGJ | (right). The green curves show the poloidal magnetic field. The two field lines closest to the axis of symmetry are the boundary of the open field-line bundle. The gray dashed circle is the surface of the neutron star. Bottom: displacement ξφ(r, θ) in the entire crust, plotted on the r-θ plane.
Figure 7. Same as Figure6 but at time t = 50 ms. 16
Polar cap quake Off-axis quake 1037 1037
1036 1036
1035 1035 ) )
s 34 s 34 / 10 / 10 s s g g r r
e 33 e 33 ( (
10 10 A A
L A1: Total luminosity L B1: Total luminosity 1032 A1: North polar cap 1032 B1: North polar cap A1: South polar cap B1: South polar cap 1031 A2: Total luminosity 1031 B2: Total luminosity A2: North polar cap B2: North polar cap 1030 A2: South polar cap 1030 B2: South polar cap 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (s) t (s)
Figure 8. Luminosity of Alfv´enwaves emitted into the magnetosphere, LA. The luminosity has been averaged into 20 ms bins to remove the noise from fast oscillations. Left: models A1 and A2 (initial quake under the polar cap). Right: models B1 and B2 (initial quake at θ ∼ π/4). Red is used for models with no crust-core coupling (A1 and B1), and blue for models with strong crust-core coupling (A2 and B2). For each model, we show LA from the entire stellar surface (solid curve), and the contributions from the north (dashed) and south (dotted) polar caps.
Model A1 Model A2
1038 1038
1037 1037
E (ergs) 1036 E (ergs) 1036 crust core crust magnetosphere 35 35 10 magnetosphere 10 total t/τcore total EQe−
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (s) t (s)
Model B1 Model B2
1038 1038
1037 1037
E (ergs) 1036 E (ergs) 1036 crust core crust magnetosphere 35 35 10 magnetosphere 10 total t/τcore total EQe−
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t (s) t (s)
Figure 9. Evolution of the quake energy. The four panels show the results for models A1, A2, B1 and B2. The energy retained by the crustal oscillations Ecrust (dashed curve) is reduced by the transmission into the magnetosphere (dotted) and (in models A2, B2) transmission into the core (dotted-dashed). As required by energy conservation, the sum of the retained 38 and transmitted energies remains equal to EQ = 10 erg (horizontal solid line). The blue dashed line shows the analytical approximation to Ecrust(t) (Equation 77) with τcore = 86 ms. A Quake Quenching the Vela Pulsar 17 glitch on a timescale shorter than 0.1 s. The only plau- nitude greater than the external dipole component re- sible way for such a disturbance to be delivered from the sponsible for the pulsar spin-down. The existence of star’s interior is through a shear wave that reaches the ultra-strong internal fields would not require the as- interface between the crust and the magnetosphere. The sumption of superconductivity to explain the short life- high-frequency elastic wave can shake vortex pinning time of the quake. In addition, it would indicate that the sites in the crust. The resulting magnus force can unpin Vela glitches are due to the crustal superfluid, contrary vortices in a macroscopic region causing a glitch (Eich- to models that invoke the core superfluid [e.g. Ruder- ler & Shaisultanov 2010). Alternatively, plastic failure man et al.(1998), Sidery & Alpar(2009)]. The theoret- of the crust could generate sufficient heat to thermally ical challenges pertaining to pulsar exteriors would also unpin many vortices (Link & Epstein 1996). be considerable: the damping of the strong magneto- In this paper, we studied an important ingredient spheric waves and their impact on pair production and of such a scenario — the seismic motion in the crust pulsar radio emission will need to be understood. and its coupling to the magnetosphere and the core. The methodology developed in this paper is not lim- We have shown that the seismic activity, once created, ited to studies of quakes in pulsars, but can also be spreads through the crust and engages the whole magne- used for studies of magnetars, where superstrong crustal tosphere in Alfv´en-type oscillations. Even for a modest- quakes were proposed as triggers of giant X-ray flares amplitude quake, we find that the magnetospheric dis- (Thompson & Duncan 1996). turbance can cause an electric discharge that produces Finally, we note that the quake we invoked for the gamma-rays and e± pairs. We are unable to make spe- Vela glitch is capable of producing a weak X-ray burst. cific predictions for the quake effect on the radio lumi- We found the Alfv´enwave energy deposited in the mag- 2 36 nosity LGHz, because the mechanism of pulsar emission netosphere EA ∼ 10− EQ ∼ 10 erg. This energy is is poorly understood. However, it is reasonable to ex- dissipated through the discharge, and a large fraction of pect that the appearance of a new powerful e± source EA should be emitted in the X-ray band. In particular, changes LGHz for the duration of the quake, and could X-rays are emitted by e± created near the star in ex- shut down the radio pulsations as observed in the Vela cited Landau states, and cascading down to the ground glitch in 2016 December. The seismic motion in the state. The duration of the X-ray burst is comparable to crust is damped on a short timescale through emission the dissipation timescale for the magnetospheric Alfv´en of Alfv´enwaves into the liquid core. This process is waves. The burst is much brighter than than the normal sped up by the enhanced magnetic tension due to the pulsating X-ray luminosity of Vela; however, its detec- bunching of the magnetic field into flux tubes in the su- tion is challenging because of the short duration and the perconducting core of Vela. As a result, the damping modest fluence. timescale for the crustal oscillations is as short as ∼ 0.2 s, comparable to the duration of the observed pulse dis- turbance. New detailed observations would help confirm the 8. ACKNOWLEDGMENTS presence of magnetospheric disturbances during glitches. We thank Mal Ruderman, Andrei Gruzinov, and If such disturbances turn out to be common, they will re- Xinyu Li for useful discussions. A.M.B. is supported quire a paradigm shift that should include crustal quakes by NASA grant NNX17AK37G, a Simons Investigator as a common phenomenon in young pulsars. This could Award (grant No. 446228), and the Humboldt Founda- indicate internal magnetic fields that are orders of mag- tion.
APPENDIX
A. ELASTIC MODES
The elastic modes ζnl(r) and corresponding frequencies ωnl are found by solving the eigenvalue equation [Equation (64)], dµ˜ dζ ζ µ˜ d dζ ζ −ω2 ρζ = nl − nl + r2 nl − [l(l + 1)µ + 2µ ] nl . (A1) nl nl dr dr r r2 dr dr B r2 Following McDermott et al.(1988) Equation (64) is reduced to two first-order ordinary diffrential equations by intro- ducing the dimensionless variables ζnl S1 ≡ , (A2) r 18
µr˜ ? dζnl ζnl S2 ≡ 2 − , (A3) ω M? dr r where S1 refers to a dimensionless amplitude, and S2 is a dimensionless stress. In terms of these variables, the equation for ζnl becomes 2 dS1 ω M? r = S2, (A4) dr µ˜ r?
2 2 dS2 µr? ω ρr r = 2 l(l + 1) − 2 − S1 − 3S2. (A5) dr ω M? µ
In the limit µB −→ 0 Equations (A4) and (A5) reduce to Equations 25(a) and (b) of McDermott et al.(1988). The mag mag appropriate boundary conditions for these unforced modes are zero magnetic stress σrθ = σrφ = 0 and zero elastic el el stress σrθ = σrφ = 0 at the boundaries. These conditions are expressed through the single equation
dζ ζ µ˜ nl − nl = 0, (A6) dr r or in terms of the variable S2, S2(ri) = 0, (A7) where ri is either the radius of the crust-core interface (rc), or the surface of the crust (r?). The amplitude of the displacement is arbitrary, as the problem is linear. We set the amplitude at the crust-core interface
S1(rc) = 1. (A8)
Equations (A4) and (A5), together with the boundary conditions Equations (A7) and (A8), constitute a well posed Sturm-Liouville problem. The Sturm-Lioville problem is solved by ‘shooting’ (integrating) from the crust-core interface and varying the eigen- value until the boundary condition Equation (A7) is satisfied at the surface of the crust. We have implemented a fourth-order Runge-Kutta integrator, and used it in two modes: i) Scanning: for each value of l the eigenvalue is varied coarsely through all possible values up to some maximum frequency. The frequencies for which S2(r?) is min- imized are recorded as estimates of the eigenvalues, together with the corresponding value of n. ii) Root finding: for each (n, l) Newton-Raphson method is used to converge on the eigenvalue ωnl for which |S2(r?)| < ? (typically we set 12 ? = 10− ). The frequencies from the scanning mode are used as first guesses for the Newton-Raphson iterations. When finding modes we use a uniform radial grid of 50, 000 points. As a test we check the orthogonality of our modes. We typically find Z r? 2 9 ρr ζnlζn0ldr = δnn0 ± 10− . (A9) rc We also studied the time-dependent propagation of a radial l = 0 wave using our elastic modes. This was compared to the same wave propagation using a 1D finite difference solver. The two methods produced the same time-dependent solution. To test the convergence, we found one set of modes on a grid of 20, 000 points, and another on a grid of 50, 000 points. We ran simulations of 2D axisymmetric elastic waves with both sets of modes, using the same initial conditions. The time-dependent solutions were indistinguishable, indicating that our elastic modes and frequencies are converged to a sufficient accuracy for our dynamical simulations. The obtained normalized modes ζnl and their frequencies ωnl are stored and used for the dynamical simulations described below.
B. CRUST DYNAMICS: NUMERICAL METHOD
The spectral method follows the dynamics of the crust through the coefficients anlm(t). Since we are only consid- ering axisymmetric dynamics in this work, the index m is set to zero, and ξφ is the only nonzero component of the displacement. The displacement is written as a sum over basis functions (orthogonal eigenmodes),
nmax lmax ˆ X X ξ(t, r, θ) = ξφ(t, r, θ)φ = anl(t)ξnl(r, θ), (B10) n=0 l=1 A Quake Quenching the Vela Pulsar 19 where finite nmax and lmax are chosen to truncate the infinite series. The product nmax × lmax is the total number of the eigenmodes in our simulations. The basis functions are
dY (θ) ξ = ξφ φˆ = ζ (r) l0 φφ,ˆ (B11) nl nl nl dθ where Yl0 = Pl(cos θ) are the Legendre polynomials and the radial eigenfunctions ζnl(r) are found as described in AppendixA. The initial conditions are set by projecting ξ(t = 0) on to the basis functions ξnl for each (n, l),
Z r? Z π 2 φ anl(t = 0) = hξ(rr, t = 0),ξξnli = dr dθ r sin θ ρ ξφ(t = 0) ξnl, (B12) rc 0 where we have used that the modes are orthonormal. The integration is done numerically on a uniform (r, θ) grid of Nr ×Nθ = 1000×600 points using the fifth-order accurate Simpsons rule. The Legendre polynomials Pl and derivatives are computed once at the beginning of the simulation and stored. The time evolution of anl is given by the equation of motion 2 a¨nl(t) + ωnlanl(t) = hf ext(rr, t),ξξnli, (B13) where f ext = f core + f mag is the force on the crust due to the core and magnetosphere, and hf ext(rr, t),ξξnli is a matrix containing the projection of f ext onto the basis functions. The force of the core on the crust (Equation (68)) is written as nmax lmax ρ< ˙ ρ< X X f core = −vA δ(r − rc)ξ = −vA δ(r − rc) a˙ nl(t)ξnl, (B14) ρ> ρ> n=0 l=1 ˙ where we have used Equation (B10) to express the ξ in terms of the coefficientsa ˙ nl. Then the projection of f core onto the basis functions is given by n l Xmax Xmax hf core(rr, t),ξξnli = a˙ n0l0 (t)Cn0l0nl, (B15) n0=0 l0=1 where Z r? Z π 2 ρ< φ φ 2 Cn0l0nl = − dr dθr sin θρvA δ(r − rc)ξn0l0 ξnl = −rc vAρ
The components of the coupling matrix Cn0l0nl = 0 for l0 6= l; therefore, it is not necessary to sum over l0 in Equation (B15). The matrix Cn0l0nl is calculated once at the beginning of each simulation and stored. The force of the magnetosphere on the crust is φ ρB 2 ∂ ξφ fmag = c cos α δ(r − r?) r . (B17) ρcrys ⊥ ∂χ r ⊥ r> φ As ξφ is evolved self-consistently in the magnetosphere (AppendixC), the force fmag is calculated at each time step and used to evaluate
Z r? Z π Z π 2 φ φ 2 φ ∂Yl0 hf mag(rr, t),ξξnli = dr dθr sin θ ρ fmag(t, r, θ) ξnl = r?ρ(r?)ζnl(r?) dθ sin θfmag(t, r?, θ) , (B18) rc 0 0 ∂θ where the integral is evaluated numerically at each time step on a uniform grid of Nθ points using the fifth-order Simpsons rule. Equation (B13) is integrated in time together with using the fourth-order Runge-Kutta integration, with a constant time step ∆t = min{∆tcrust, ∆tmag}, where ∆tcrust is the largest stable time step for the crust, and ∆tcore is the largest stable time step for the magnetosphere (see AppendixC). We use ∆ tcrust = kc/max{ωnl} with kc ≤ 0.1, where max{ωnl} is the highest frequency of all of the modes we are using. We have found that for a free crust (without external forcing terms), our code conserves energy to one part per million. If the external forcing terms are included, some additional error is introduced, and energy is usually conserved to one part in 105. We use (nmax, lmax) = (300, 200), a total of 60,000 modes. More radial modes are needed (nmax > lmax) to properly resolve the wave transmission through the upper layers of the crust where the scale height is very small. The only relevant scale in the θ-direction is introduced by the initial conditions. We have tried independently increasing nmax to 600, and lmax to 400, and we observe the same results. 20
C. MAGNETOSPHERE DYNAMICS: NUMERICAL METHOD
In the magnetosphere, we calculate the small azimuthal displacement ξφ, using the so-called magnetic flux coordinates (ψ, χ, φ), where ψ =const defines surfaces of constant poloidal flux, and χ is the length along poloidal the field lines in the φ = const plane. The dependence of the Cartesian position vector x on the coordinates ψ and χ is found by integrating the equation dxx(ψ, χ) B = . (C19) dχ |B| The footpoints of the field lines are chosen to coincide with the grid points used in the projection Equation (B18). We chose the grid spacing along the field lines so that the light-crossing time of each grid cell is the same. When we include the liquid ocean, the grid spacing remains large in the magnetosphere, but becomes very small in the ocean where the density increases. By using this grid spacing, we are not limited to a prohibitively small time step by the Courant condition. The time evolution of ξφ(ψ, χ) is given by the wave equation
2 ∂ ξφ(ψ, χ) B ∂ 2 ∂ ξφ(ψ, χ) 2 = r B . (C20) ∂t 4πr ρB ∂χ ⊥ ∂χ r ⊥ ⊥ We are effectively solving a 1D wave equation for each flux surface ψ. The right-hand side of Equation (C20) is evaluated using the second-order finite difference formulas given by Bowen & Smith(2005). The first derivatives use a three-point stencil, and the second derivatives use a four-point stencil, so that second-order accuracy is preserved when the grid spacing is nonuniform. We integrate Equation (C20) in time, together with Equation (B13) for the crust using the fourth-order Runge-Kutta integration. The crust provides the boundary condition for ξφ(ψ, χ) at the surface in the magnetosphere, and the magnetosphere communicates to the crust through the force Equation (B17). The stable time step for the magnetosphere is ∆tmag = kcdtχ, where dtχ is the light-crossing time of a grid cell, and kc < 0.5. We set the time step for the simulation ∆t = min{∆tcrust, ∆tmag}, where ∆tcrust is the largest stable time step for the crust (see AppendixB). We find that ∼ 600 grid points are required for the projection Equation (B18), which results in ∼ 50 open flux surfaces (∼ 25 at each pole), and ∼ 275 closed flux surfaces.
D. A TEST FOR WAVE TRANSMISSION
102 100
1 3/5 0 101 10− ω ω ∝ ∝
2 10− 100 =0
m 10 3 n,l − T 7 1 ω E 10− ∝
4 10− 2 10−
5 10− Analytic 3 10− Code 6 10− 104 105 104 105 ω (rad/s) ω (rad/s)
Figure 10. Left: initial energy spectrum of waves in this test problem. Right: transmission coefficient of waves into the magnetosphere Tm(ω). The thick black line shows the transmission coefficient found by solving the analytic reflection conditions (Section3), and the gray dots show the numerical transmission coefficient measured using our code for an l = 0 radial wave. The dotted lines show the powe-law scalings in each frequency range.
In order to test the implementation of the crust-magnetosphere coupling in our numerical model, we have measured the frequency-dependent transmission coefficient Tm(ω) using our code. We initialize the simulation by launching a A Quake Quenching the Vela Pulsar 21 purely radial l = 0 wave in the crust. The magnetosphere is chosen to be a radial monopole with outflow boundary conditions on all flux surfaces, so that no Alfv´enwaves return to the crust. The setup is effectively 1D, so that we should recover the transmission coefficient calculated in Section3 for a Cartesian slab crust. The energy spectrum of the initial condition is shown in Figure 10 (left panel). The initial displacement is a smoothed step function, similar to the 2D initial conditions used in Section6. It corresponds to a strain layer of thickness ∆` ∼ 104 cm, similar to the pressure scale height in the deep crust. The energy spectrum peaks around 4 ω ∼ v˜s/`0 ∼ 2 × 10 rad/s. We measure the transmission coefficient by calculating the exponential decay time of the energy in each mode τm. The transmission coefficient for a given elastic mode is then calculated as Tm = 2τ/τm, where τ ≈ 1 ms is the elastic wave crossing time of the crust. This gives the effective transmission coefficient as a function of the mode frequency, Tm(ω), which we compare with the analytically calculated Tm (Figure 10). The two lowest-frequency modes deviate from the analytical result, because they are reflected deep in the crust near neutron drip, where the exact density profile used in the code deviates from the approximation ρ ∝ |z|3 used in the analytical model. There are few data points at low frequencies in Figure 10 because there are few elastic modes in that frequency range. We performed similar simulations with different initial conditions and found nearly the same Tm(ω).
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