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Mon.Not.R.Astron.Soc.000,1–7() Printed2February2008 (MNLATEXstylefilev2.2) Damping of neutron star shear modes by superfluid friction P. B. Jones ⋆ University Of Oxford, Department of Physics, DenysWilkinson Building, Keble Road, Oxford, OX13RH 3 0 0 2 ABSTRACT n The forcedmotion ofsuperfluid vorticesin shearoscillations ofrotating solidneutron a star matter produces damping of the mode. A simple model of the unpinning and J repinningprocessesisdescribed,withnumericalcalculationsoftheconsequentenergy 7 decay times. These are of the order of 1 s for typical anomalous X-ray pulsars with rotationalangularvelocityΩ≈1rads−1 butbecomeveryshortforradiopulsarswith 1 Ω ≈ 10−100 rad s−1. The superfluid friction processes considered here may also be v significant for the damping of r-modes in rapidly rotating neutron stars. 2 1 Key words: stars: neutron-pulsars: general. 1 1 0 3 0 1 INTRODUCTION movementsofthesolidcrust,causedbyMaxwellstressesor / otherwise, may then lead to magnetospheric excitation and h Althoughtherehasbeenmuchrecentinterestinshearmodes possiblyobservablephenomena.Inthiscontext,thepresent p of neutron star oscillation (Blaes et al 1989; Thompson & - paper examines some processes, specific to neutron stars, o Duncan 1995; Duncan 1998), relatively little attention has whicharemoreaccessibletocalculationthanstress-induced r beenpaidtotheassociateddampingmechanismsapartfrom dislocationmovementandwhichmightbecapableofdamp- t couplingwithmagnetosphericAlfvénwaves.Blaesetalcon- s ingshearwavesenoughtoinvalidatethisassumption.Shear a sider that stress-induced motion of dislocations may be an wavesarereflectedmanytimesfrom theboundarywith the : important process, as is believed to be the case for shear v magnetosphereandpossiblyfromtheinnerboundaryofthe i waves in the Earth. But nothing is known about the dis- crustwith theliquidcore ofthestar wherethematterden- X tribution of linear and planar defects in the lattice of nu- sity is ρ 2 1014 g cm−3. The number of reflections ar cρblnieldiity=a.tA4m.t3ad×tteen1r0si1dt1ieengssicitnmiet−sh3ρeocbrreualosbtwouattbhotevheenierρusttr,roenist-sd-iisrnipdliuktechleyedshtmhoaoldt- tmhuestnebuetcraotn≈lesatas×tr roafdtihuesoarnddercsof≈cs1τ0s8/Rcm≈s−1012i,swthheereshRearis- nd wave propagation velocity. Reflection from each boundary neutron star matter is an amorphous heterogeneous solid, is complicated because it couples shear waves with longi- analogous with a disordered many-component alloy (Jones tudinal sound (see, for example, Landau & Lifschitz 1970, 1999),andagain,nothingisknownaboutitslikelyresponse p.103)andatρ ,mayevendependonthedimensionalityof to periodic stress. Thus there is no basis for even a rough c thesolid. estimate of the contribution to shear-mode Q-values from this mechanism. Itisknownthatlongitudinalsoundisstronglyabsorbed Blaes et al assume that coupling with magnetospheric in the core of a neutron star owing to bulk viscosity pro- Alfvénwavesisthedominantmechanismofdamping.These duced by the strangeness-changing four-baryon weak inter- authors show that, for a typical frequency of 104 Hz, shear action. The magnitude of the coefficient is very dependent waves are efficiently reflected from the boundary with the on the assumptions which must be made, principally about magnetosphere and, in the absence of other damping pro- theequationofstateatcoredensities,buttheabsorptionis cesses, would have a lifetime τ 1 s at a reference mag- alwaysstrongenough,withincertain temperatureintervals, s neticfluxdensityof1011 G.TheB≈-dependenceofτ isgiven to besignificant for shear-wave damping. s inanalyticalexpressionsvalidasymptoticallyatfrequencies The shear-wave velocity depends on nuclear parame- asmt alollwanfrdeqlaerngceiecsoamnpdarτedwiBth−14/074 Hatz.hTighhesferqaureenτcsie∝s.BT−he2 ters according to cs ∝ZA−2/3ρ1N/6, where Z and A are the s ∝ nuclear charge and mass number, and ρN is the spatially- couplingisthereforeanefficientmechanismforenergytrans- averaged matter density derived from the nuclei alone and ferfrom crust tomagnetosphere atthehighmagneticfields excluding the superfluid neutron continuum external to assumedinmagnetars(Thompson&Duncan1995).Sudden them. This elementary expression assumes a nuclear effec- tive mass equal to the free mass and, in conjunction with thenuclearparameterslistedbyNegele&Vautherin(1973), ⋆ E-mail:[email protected] gives a rapid increase of cs with depth at ρ < ρnd but no 2 P. B. Jones morethanaveryslow increaseatρ>ρ .(Thelattervari- lengths and times, a force f(u) per unit length of vortex, nd ationseemstobeindisagreementwithBlaesetalwhostate andthatfluctuationsfrom itarenotsolargeastomakeits that c is a decreasing function of depth at ρ > ρ .) As a usemeaningless.Thecomponentofthisforceparallelwithu s nd consequence of this variation, shear waves will be refracted is the dissipative force f (u). Its properties were considered k radially outward in all regions, as are longitudinal sound brieflyinapreviouspaper(Jones2002),butitseemsappro- waves,butonlyweaklysoatρ>ρ .Butinatypical1.4M priate to give here a more detailed account of what can be nd ⊙ neutronstar,thelayersofstrongrefraction atρ<ρ have deducedaboutitfromtheinteractionofavortex,comoving nd depthonly of thesame orderas the wavelength at therele- with superfluid of velocity v , and a single point defect in n vant frequency of 104 Hz. Even assuming, with Blaes et al, thesolidstructure.Thisleadstotheexcitationofphononsin that the Maxwell-stress generated events producing shear thesolidand,atfinitetemperatures,ofvortex-corequasipar- waves occur predominantly within this latter region, it fol- ticles (Jones 1990a,b), but Kelvin wave excitation is easily lows from this small depth and from the multiplicity of re- the most important process (Epstein & Baym 1992, Jones flections that shear-wave propagation in the neutron drip 1998).Inthecasethattheinteractionpotentialisattractive, region atρ>ρ mustbeconsideredasasignificantsource the coupling always becomes strong at sufficiently small v nd n of damping. Thus most of the present paper will describe so that the energy transfer to Kelvin waves must be calcu- an investigation in thiscontextof phenomenainvolvingthe lated non-perturbatively.In the strong-coupling region, the rapidforcedmotionofvorticeswhich,forwantofanybetter vortex is trapped in the potential for a time t v−2 and name, we shall refer to as superfluid friction. The relevant the energy transfer is E v−1. We refer tomJo∝nesn(1998) a ∝ n results for longitudinal sound absorption in the liquid core for details andfor a diagram showing atypical vortexorbit are considered in Section 3. within the potential. (For weak coupling, realized only at values of v larger than those considered here, the energy n transferchangestoE v−1/2 andthevortexorbitislittle a n ∝ perturbedbythepotential.) Fig. 1 shows schematically the 2 DESCRIPTION OF SUPERFLUID FRICTION effectofasequenceofsuchinteractions,withadilutedistri- An elementary description of superfluid friction is given by butionofpointdefects,onthevortexpath.Eachinteraction consideringaregionofsolidinwhichtheneutronsuperfluid produces a lateral displacement and introduces a trapping has rotational angular velocity Ωn. The superfluid velocity time tm. It is found that, averaged over impact parameter, vn is defined (see Sonin 1987) as the average over the unit thecomponents of the computed impulse on thevortex are cell of the local vortex lattice. All velocities are referred to approximately equal, cylindricalpolarcoordinatesrotatingwithconstantangular ∞ ∞ velocity Ω, coincident at some fiducial time with the an- F1(t)dt F2(t)dt , (1) gular velocity of the charged components of the star. The (cid:12)(cid:12)Z−∞ (cid:12)(cid:12)≈(cid:12)(cid:12)Z−∞ (cid:12)(cid:12) difference Ωn−Ω is always small compared with Ω. To the w(cid:12)(cid:12)heretheaxe(cid:12)(cid:12)sar(cid:12)(cid:12)easlabelled i(cid:12)(cid:12)ntheFig.1.Theconnection extent that steady spin-down of the superfluid occurs, it is between F, defined for interaction with an isolated point maintained by radial outward expansion of its vortex lat- defect, and the force per unit length f, which is an aver- tice; the Magnus force fM acting on an individual vortex is age overmacroscopic lengths and times for motion through balanced by the force per unit length f generated through a dense system of point defects, is a complex problem not its motion with velocity u, perpendicular to its local axis consideredhere.Onmicroscopic lengthandtime-scales, the − and relative tothesolid. Thequantitiesf and uhaveto be vortex has fluctuating motion and, as shown in Fig. 1, treated as averages over macroscopic lengths and times, an the macroscopically-averaged vortex velocity u = uuˆ is assumption which is satisfactory provided vortex curvature smaller in magnitude than the superfluid velo−city v−n with onthesesamescales isnegligible. Ashearwaveofwavevec- which the vortex would comove in the absence of interac- tor q is represented by a solid matter displacement vector tion, and has a component perpendicular to it. For general ξ=ξoexpi(q·r−ωt),withξ·q=0.Thevelocityrelativeto valuesofvn andforadensesystemofpointdefects,itmust thelocal solid isthen u=vL v⊥,wheretheshear-wave bepresumedthatasaconsequenceofequation(1)theforce displacementvelocityi−sv=ξ˙a−ndv⊥ isitscomponentper- f has,inadditiontothedissipativecomponentfk,acompo- pendiculartothevortexaxis.Thevelocityofapointonthe nentf parallelwithκˆ uˆ,wherethecirculationisκ=κκˆ ⊥ vortex axis is vL. Energy dissipation is obviously changed andκisitsquantum.In×othersuperfluidproblems,theexis- bythedisplacementvelocity,specifically,bythecomponent tenceof this non-dissipativecomponent has been viewed as v⊥. Energy is transferred irreversibly from the shear wave controversial.Thouless, Ao&Niu(1996) haveclaimed that itself and, for most wave directions and polarizations, from itdoesnotexistforneutralsuperfluidsinauniformnormal the rotational kinetic energy of the superfluid measured in background (see, however, Hall & Hook 1998, Sonin 1998, the rotating coordinate system. It is to be expected that andtherepliestherein),butasystemofpointdefectsisnot calculation of shear-wave energy-loss rates depends princi- ofthisclass.However,inthespecialcaseofaninfinitelinear pally on the assumptions made about the dissipative force, array of point defects, parallel with κ, the Green function andsoweareobliged toconsidertheseinasmuchdetailas used by Jones (1998) allows one to see that thecondition possible. t F2(t′)dt′ 0, (2) Z → −∞ 2.1 The dissipative force which is necessary for the onset of pinning, can always be The assumptions, made here for almost rectilinear vortices, satisfied at sufficiently small v . n are that it is possible to define, averaged over macroscopic The qualitative form inferred for the dissipative force Damping of neutron star shear modes by superfluid friction 3 velocity Ω are: ρ κ (v v )+f (u)uˆ+f (u)κˆ uˆ =0, (4) n × L− n k ⊥ × ∂v 1 n +2Ω v = P, (5) ∂t × L −ρn∇ where P is the pressure. Comparison with the equations given by Sonin (1987; equations 4.36 - 4.38) shows that the Tkachenko force, derived from the vortex lattice structure, and the force dependent on vortex curvature have been ex- cluded from equation (4). Both are quite negligible in the present context. These equations are macroscopic in that equation (5) contains the vortex density 2Ω/κ, although equation(4),withnoTkachenkoforce,isvalidforanisolated vortex.Atthelowvortexdensities(103−5 cm−2)ofrotating neutronstars,v doesnotchangerapidlywithtimeand,for n the purpose of understanding vortex unpinning and repin- ning at a qualitative level, can be treated as a constant in equation (4). Remembering that u = v v , and that all L − the velocity components with which we are concerned are perpendiculartoκˆ,thecomponentsofequation(4)parallel with uˆ and κˆ uˆ givean equation for u, × 2 2 (v v )2 = u f⊥ + fk . (6) − n (cid:18) − ρnκ(cid:19) (cid:18)ρnκ(cid:19) The roots of this equation are determined by the u- dependenceofthefunctionsf andf .Forconsistencywith k ⊥ equation (2), thelatter must satisfy the limit f (u) 0 as ⊥ → u 0. In general, there can be either one or three roots → in each of the sectors v v < v and v v > v . n m n m | − | | − | A simple example is the instance in which the solid is stationary in the rotating coordinates so that u = v , with L − forcecomponentsf =0,andf givenbyequation(3).The ⊥ k Figure 1. A vortex moves through, andinteracts with, adilute solutions are shown in Fig. 2 for a set of four values of v . n system of attractive point defects, shown schematically. In the In case (1), the single root corresponds with an unpinned absenceofinteraction,itwouldcomovewiththesuperfluidveloc- vortexmovingwithavelocityv v .Ifv weredecreased ity vn. However weak the attractive potential, the orbit within L ≈ n n to case (2), there would be three roots including a pinned the potential field, at sufficiently small vn, is always character- ized by a mean lateral displacement and by trapping for finite solutionatverysmallu.Furtherreductiontocase(3)leads times in regions close to each origin inside the small rectangu- to a critical value u = uc (uc = 0.44vm for b = 0.3vm) lar boxes. Averaged over macroscopic time and length-scales, a and to a transition to the pinned solution. At the smaller vortextypicallyhasvelocityvL,asshown. vn of case (4), only the pinned solution exists. If equation (6) has only one root, there is, of course, no discontinuity anduvariescontinuouslywithv .Obviously,thisdiscussion n isbasedonourdefinitionanduseofmacroscopicaveraging, per unit length of vortex, f (u), is shown in Fig. 2. Finite butgivessomeideaoftheunpinningandrepinningprocesses k temperatures and quantum-mechanical tunnelling lead to foranisolated vortex.Theverylowvortexdensitiesinneu- vortex creep and therefore a large and rapidly varying gra- tron matter suggest that these, ratherthan multiple-vortex dient close to u=0. However, those properties of f which processes involving vortex-vortex interactions, are the sig- k are important here can be expressed in thesimple function nificant consideration. f (u)=f 1 e−u/a 1 e−b/u (3) k m − − 2.2 The hydrodynamic equations (cid:0) (cid:1)(cid:0) (cid:1) where a,b and f are constants. We assume that b is of m Although the non-dissipative force component f is ex- the order of v = f /ρ κ, where ρ is the density of the ⊥ m m n n pected to be finite for general values of u, any attempt to superfluid continuum outside the nuclei. It is the only nat- give it an explicit form would bemuch less well-based than urally occurring temperature-independent scale of velocity equation (3). Thus the assumption f = 0 adopted here in theproblem. The precise value of the parameter a is not ⊥ is the only practical basis for obtaining numerical solutions important here except in connection with rotational noise of the hydrodynamic equations. Resolved into components generationasmentionedinSection3;itisverymanyorders parallel with the unit vectors ˆr and θˆ in cylindrical polar of magnitude smaller than b. coordinates, equations (4) and (5) are, The linearized equations for the macroscopic hydrody- namicsofanincompressiblesuperfluidrotatingwithangular f ρ κ(v v )=0, (7) kr− n Lθ− nθ 4 P. B. Jones sheardisplacementisnegligiblysmallanditisthereforeap- propriatetoadopttheinitialconditions P = P =0.It r θ ∇ ∇ will be convenient to eliminate v from equations (7)-(10) L infavourofu.Wealsoneglectanysmalldifferencesbetween the local κˆ and the zâxis. The solid velocity v = v sinωt o is resolved into components parallel and perpendicular to the zˆ axis, only the latter, v being present in equations ⊥ (7)-(10).Thevelocityvariablesarethenu ,u ,v andv , r θ nr nθ withinitialconditionsofv (0)=0andaspecifiedvalueof nr v (0). The initial values, u (0), obtained from equations nθ r,θ (7)and (8)are forapinned-vortexsolution which isalways assumedastheinitialcondition.Algebraicsolutionofequa- tions (7) - (10) is possible through linearization, which is validforshearwavesofsuchsmallamplitudethattheycan- not cause vortex unpinning. It gives a dissipation rate per unit length of vortex of the order of −1 ∂f (ρ κv )2 k . n o⊥ (cid:18)∂u(cid:19) Foranyphysicallyreasonable valuesoftheparameterscon- cerned,this is many orders of magnitudetoo small to beof interest in thepresent context. Therateofenergydissipationperunitlengthofvortex is f u, which satisfies · f u=f v f v . L · · − · Because f +f = 0 and f (v v ) = 0, this can be M M L n · − rewritten as f u=f v f v . (11) n · · − · Thismerelystatesthatthedissipationrateisthesumofthe energy loss rates for the shear mode (f v) and for the su- · perfluid.Forthechoseninitialconditions,equations(7)-(10) Figure2.Theforceperunitlengthfk (inunitsoffm)isshown havebeensolvednumerically,withattentiontotheinstabil- versus u (in units of vm). The function is that of equation (3) ities discussed in Section 2.1, for several complete cycles of except that the value of the parameter a has been greatly exag- theshearmodeinordertoobtainvaluesofitstime-averaged geratedforeaseofrepresentation.Forthesamereason,thesecond energylossrate.Theserateshavebeencalculatedforthein- parameterisb=0.3vm,avalueintermediatebetweenthosecon- terval104 <v <106 cms−1,wherev isthecomponent sideredinFigs.3and4.Possiblesolutionsofequation(6),forthe o⊥ o⊥ ofthedisplacementvelocityvperpendiculartoκ,andfora case v = f⊥ = 0, are given by the intersection of this function withthecirclesofradiusvn/vm.Forvn>vm(case1),thevortex distribution of values of its r,θ components. This allows us approximately comoves with the superfluid. With decreasing vn to obtain, for a given value of vo and for each of the shear (cases 2 and 3), the critical value uc is approached, leading to modepolarization states,theenergylossrateaveragedover vortexrepinning(case4) anisotropicdistributionofitswavevectorq.(Themultiple reflection of shear waves makes this average the most ap- propriatewayofpresentingthedata.)Theenergyloss-rates f +ρ κ(v v )=0, (8) kθ n Lr− nr perunitlengthofvortex,dividedbyρnκandaveragedover both polarization states, are defined as D and shown in 1 av v˙ =2Ωv P, (9) Fig. 3 as functions of v for pairs of values of f /ρ κ and nr Lθ− ρn∇r of b. They are quite insoensitive to ω, Ω and a.mThenrefore, we have assumed ω = 104 rad s−1, Ω = 10 rad s−1 and 1 v˙nθ =−2ΩvLr− ρn∇θP. (10) ath=e u1n0p−i4nncmings−th1rtehshroouldg,hloousts.-rEatxecsepdtofonrotvedleopceitniedssctlroosnegtlyo Withregardtoboundaryconditions,theonlypracticalway on the initial pinned-vortex velocity. This is demonstrated ofobtainingnumericalsolutionsistoassumeaninfinitesys- bythebrokencurvein Fig. 3which isfor u (0)=0, anini- r tem. Thus the boundary condition that vn should have no tialstateofanexactlycorotatingsuperfluid,vnr =vnθ =0. perpendicularcomponentonthesphericalsurfaceofthesu- Awayfromthethreshold,itisalmostindistinguishablefrom perfluidatρndisneglected.Solutionsarethereforevalidonly the curve for ur(0) = a and so we have assumed this lat- for wavenumbersq > 10−4 cm−1 large enough tobe consis- ter initial condition for all other curves shown. In general, ∼ tentwiththeinfinitesystemassumption.Equations(7)-(10) energy loss-rates become large at values of v sufficient to o alsoneglectfluctuationsinpressureandgravitationalpoten- producevortexunpinningandincreasetoasymptoticvalues tial, a reasonable approximation at the wavenumbers con- ρ κD f b. The mode decay time τ at a given ampli- n av m sidered here. Any pressure fluctuation caused by the local tudev ca≈nbefoundfromitsenergydensity 1ρ v2 andthe o 2 N o Damping of neutron star shear modes by superfluid friction 5 Figure 3. The shear-mode energy loss-rate per unit length of Figure4.Theverticalaxisgivestheright-handsideofequation vortex, averaged over polarization and for an isotropic distribu- (12),whichisproportionaltoΩτ averagedoverpolarizationand tionofwavevectors, isshownasafunctionofvelocityamplitude for an isotropic distribution of wavevectors. The parameter sets vo for given vm and b. The broken curve, for b= vm = 105 cm are those of Fig. 3. Damping is strongest at values of vo in the s−1,isfortheinitialconditionvnr =vnθ =0. vicinityofvm. loss-rate per unit volume, equal to the product of ρnκDav s−1 mustexistinsomeintervalsofmatterdensityifangular and the vortex numberdensity 2Ω/κ. Thefunction momentumtransferfromcrustsuperfluidistheoriginof,or v2 ρ a substantial contributor to, pulsar glitches. Much smaller o =4Ωτ n (12) values may exist in other regions, and the limit vm 0 D ρ → av N must be approached near ρ and ρ . In Section 2.1, it was nd c is shown in Fig.4. Assuming the lattice parameters calcu- argued that b should be of the order of v , but in Figs. m latedbyNegele&Vautherin(1973),thedensityratioρ /ρ 3 and 4, results are also given for values an order of mag- n N appearing in our, perhaps naive, model of the shear wave nitude smaller. From Fig. 3, we can see that the asymp- varies from zero at the threshold ρ to approximately 8 in totic value D v b gives a very approximate estimate, nd av m theinterval1013 <ρ<1014 gcm−3 which includesmost of 2Ωρ v bθ(v v≈ ), for the energy-loss rate per unit vol- n m o m − theneutron-dripvolume.TheproductΩτ islessthanunity ume, where θ is here the unit step function. The order of for a substantial intervalof shear-wave amplitude. magnitudeof theenergy decay time is then ρ v2 τ = N o , (13) 4ρ Ωv2 θ(v v ) n m o m − 3 DISCUSSION AND CONCLUSIONS for b = v . It is a function only of v /v , so that even a m o m Themotivation for expressing theforce parameter f asin small-amplitudeshearwaveisstronglydampedinregionsof m Figs. 3and4 isthatv =f /ρ κisan upperlimit for the weakvortex-pinning.Itisalsotemperature-independent,be- m m n superfluidvelocity v relative to coordinates rotating with low the superfluid critical temperature, because it depends nθ angularvelocityΩ.Thusvaluesoftheorderof104 106 cm principally on f at values of u many orders of magnitude − k 6 P. B. Jones greaterthana.Theconventionallydefinedshearviscosityη shear waves. But the superfluid friction calculated in the for the solid (see Landau & Lifshitz 1970, p. 155) is related present paper damps such modes, based on shear defor- with the decay time by η = µ/τω2, where µ is the shear mations, directly. The damping times found here are much modulus. shorterthanthoseconsideredbyDuncanandwouldruleout This process has some similarity with stress-induced theexistence of other than small mode-amplitudes. motion of dislocations, considered briefly by Blaes et al. The presence of Ω in equations (12) and (13) repre- Both involve dissipative motion of a topological defect, ei- sents no more than the mean vortex density of the super- therin thesuperfluidorder parameter or in thelattice, but fluid.Therefore,superfluidfrictionwouldproduceextremely it isnot possibletoobtain evenan orderofmagnitudeesti- strongdampinginmorerapidlyrotatingneutronstars,such mate of the importance of the latter process. The force per as the typical radio pulsar with Ω 101−2 rad s−1. This unitlength onalinedislocation isoftheorderof˜bµv /c , may also be of relevance to the A≈XP and SGR sources, o⊥ s where˜b is the Burgers vector (see Landau & Lifshitz 1970, of which all examples so far observed have long periods. p.133);itistypicallysomeordersofmagnitudesmallerthan Energy transfer through coupling between shear waves and ρ κD .Thedislocation densityisnotknown,butcouldbe magnetospheric Alfvén waves, as in the magnetar model of n av many orders of magnitude greater than the vortex density, Thompson & Duncan (1995), is unlikely to be efficient at 2Ω/κ 103−5 cm−2. But the time-averaged velocity of an short periods owing to thevery strong damping. ≈ individual dislocation under stress is probably very many Superfluid friction is also likely to be relevant to other orders of magnitudesmaller than vm. current problems such as the possibility of vortex unpin- ThecouplingwithlongitudinalsoundreferredtoinSec- ning in precessing neutron stars (Link & Cutler 2002), and tion 1 introduces additional damping. The most significant the damping of r-modes in rapidly rotating neutron stars source is the bulk viscosity produced by the strangeness- (Lindblom,Owen & Morsink 1998; Andersson, Kokkotas & changing four-baryon weak interaction. The amplitude de- Schutz 1999). Many papers on the latter problem have as- cay time is sumed that the solid crust does not participate in the displacement amplitude of the mode, is stationary in rotating 2ρc2(0) τ = ℓ , (14) coordinates, and therefore acts only as a boundary condi- nl ζω2 tion on a thin viscous or turbulent dissipative boundary atmatterdensityρandangularfrequencyω,wherec (0)is layer. But more recent work (Levin & Ushomirsky 2001; ℓ the longitudinal sound velocity in the zero-frequency limit. Yoshida & Lee 2001) has recognized that the mode ampli- Thebulkviscositycoefficientζ isafunctionbothoftemper- tude must penetrate the crust significantly at high values atureandangularfrequency.Italsodependsontheequation of Ω. (The radial component of the r-mode displacement, of state at core densities. For recent evaluations, based on though small, must penetrate the crust. Also, an element differing assumptions, we refer to Jones (2001), Haensel et of solid crust moving with velocity v in the rotating coor- al (2002) and Lindblom & Owen (2002). For ω = 104 rad dinates experiences a Coriolis force which, for Ω 102 rad s−1,atemperatureof109Kandζ =3 1028 gcm−1s−1 (see s−1, is of the same order of magnitude as the sh≈ear stress Jones2001;equation45),thedecaytim×eisτ =3 10−2 s, integrated over its surface.) A mode with amplitude large nl × which would increase by several orders of magnitude if the enough forits velocity ofdisplacement tobeof theorderof temperature were to decrease to 108K. The evaluation of vm willbesubjecttostrongdissipationinthecrust,theen- Lindblom&Owengivesalarger ζ andthusanevenshorter ergyloss-ratetherebeingoftheorderof2Ωρnvmbθ(v vm) − decay time. perunitvolume,asforshearwaves.Unlikekinetictheoryor Fig.3ofBlaesetalshowsthecomputedAlfvénwavelu- weak-interaction mechanisms of viscous dissipation (Jones minosity as a function of time following generation of shear 2001; Haensel et al 2002; Lindblom & Owen 2002), this is waves by a starquake, assuming a magnetic flux density notcutoffbyexponentialfactorsinvolvingsuperfluidenergy equal to their reference value of 1011 G. Energy begins to gaps but is approximately temperature-independent below enter the magnetosphere about 10−3s after the event and the critical temperatures. However,the detailed calculation the luminosity persists for times just exceeding 6 10−1s. required to obtain the r-mode lifetime is beyond the scope × AnomalousX-raypulsars(AXP)andsoft gamma repeaters of thepresent paper. (SGR)areassociated with longrotation periods, Ω 1rad Thesecondtermontheright-handsideofequation(11) s−1and,inthemagnetarmodel,withsurfacemagnet≈icfields istheenergy-lossratefortheneutronsuperfluid.Forallthe exceeding1014 G.Theseverylargefieldsindicate,assuming cases in Fig. 3, except for that with the initial condition the B-dependence of τs given by Blaes et al and referred ur(0)=0,thesuperfluidloseskineticenergyintherotating to in Section. 1, that superfluid friction damping times are frame of reference and is gradually brought into corotation probably rather longer than the time τ for energy trans- with thecharged componentsof thestar. Theinitial rate is s fer by coupling of the shear modes with magnetospheric of thesame order as for theshear mode. Alfvén waves. In magnetars, superfluid friction is unlikely Superfluid spin-down in the presence of shear waves is to limit the efficiency of this process. A similar conclusion, governed by the form of fk(u) at u < um, the position um though depending on temperature, holds for the bulk vis- ofitsmaximum beingu √abforthespecificform given m ≈ cosity damping. by equation (3). In the absence of shear waves, the solid Duncan (1998) has examined the excitation and de- matter velocity is v = 0 and steady-state superfluid spin- tectability of global seismic modes, particularly low-order down is possible, in principle, if the required vortex-creep toroidalmodes, bystarquakeevents.Heassumesthatthese velocities, v = rΩ˙/2Ω at radius r, are smaller than u . L m − modesaredampedthroughtheirinteractionwithfaultlines Theactualformoff inthisregionisunknown,becauseitis k inthesolidandtheconsequentgenerationofhigh-frequency dependent on thermal activation and quantum-mechanical Damping of neutron star shear modes by superfluid friction 7 tunnelling,butitwouldnotbeunreasonabletoassumethat Haensel P., Levenfish K. P., Yakovlev D. G., 2002, A&A, it allows this condition to be satisfied given the very small 381, 1080 values, v <10−5 cm s−1 required. Underthis assumption, Hall H. E., Hook J. R., 1998, Phys.Rev.Lett., 80, 4356 L some perturbing movement of the solid is needed to induce Jones P.B., 1990a, MNRAS,243, 257 vortexunpinning.Itsmagnitudedependsontheactualform Jones P.B., 1990b, MNRAS,244, 675 of f (u)near itsmaximumat u ,butit is likelythat quite Jones P.B., 1998, Phys.Rev.Lett., 81, 4560 k m small values would be sufficient to satisfy the condition Jones P.B., 1999, Phys.Rev.Lett., 83, 3589 | v v > v in one or more, possibly small, regions of Jones P.B., 2001, Phys.Rev.D, 64, 084003 n m − | superfluid. Following our discussion at the end of Section Jones P.B., 2002, MNRAS,335, 733 2.1, we can see that vortices in these regions would remain Landau L. D., Lifshitz E. M., 1970, Theory of Elasticity, unpinnedwithrapidsuperfluidspin-down,eveniftheinitial Pergamon, Oxford disturbanceweretodecaytonegligibleamplitudes,untilthe Levin Y., Ushomirsky G., 2001, MNRAS,324, 917 critical value u=u is reached. Thus a significant fraction, LindblomL.,OwenB.J.,MorsinkS.M.,1998,Phys.Rev. c 1 (u /v )2+(f (u )/f )2, of the available superfluid Lett., 80, 4843 − c m k c m anguplar momentum in these regions is transferred to the Lindblom L., Owen B. J., 2002, Phys. Rev.D,65, 063006 charged components of the star in discontinuous glitch or Link B., Cutler C., 2002, MNRAS,336, 211 noise events. Lyne A. G., Shemar S. L., Graham-Smith F., 2000, MN- It is worth comparing the crust movements considered RAS,315, 534 here with those assumed in the glitch model of Ruderman, Negele J. W., Vautherin D., 1973, Nucl. Phys., A207, 298 Zhu & Chen (1998). The poloidal velocities of the latter Ruderman M., Zhu T., Chen K., 1998, ApJ, 492, 267 model arenot necessarily large enough to generate asignif- Sonin E. B., 1987, Rev.Mod. Phys., 59, 87 icant shear-wave energy density;it is possible that they are Sonin E. B., 1998, Phys.Rev.Lett., 81, 4276 no more than episodes of plastic flow occurring in times Thompson C., Duncan R.C., 1995, MNRAS,275, 255 smaller than the observed upper limit ( 102s) for pul- Thouless D. J., Ao P., Niu Q., 1996, Phys. Rev. Lett., 76, ≈ sar spin-up, thus with velocities many orders of magnitude 3758 smaller than theshear-wavevelocity c .Inthiscase, super- Yoshida S., LeeU., 2001, ApJ,546, 1121 s fluid is spun up or down by the inward or outward radial displacement of the solid with pinned vortices (see Jones ThispaperhasbeentypesetfromaTEX/LATEXfileprepared 2002). Vortices are not necessarily unpinned although this bythe author. could occur in any region where the superfluid velocity v n reaches v . But crust movement with poloidal velocities of m theorderofc isnotexcluded,inwhichcaseglitcheswould s beinitiatedbythe(v=0)velocityperturbationsconsidered 6 here. It is possible that both kinds of movement contribute to theobserved glitch population. This would not be inconsistent with the value of the glitchactivityparameterdefinedbyLyneetal(2000),which isbasedontheobservationthatforallexceptveryyoungor old pulsars, the summed glitch amplitudes δΩ divided by gi thetotal observation time is T−1 δΩ 1.7 10−2Ω˙. i gi ≈− × For this group of pulsars, angularPmomentum of the same order as that of the total crust superfluid is transferred by glitchesratherthanbythesteady-statesuperfluidspin-down describedearlierinthisparagraph.Thusallglitchesinradio pulsarsmay beinitiated bycrust movements,with poloidal velocitiesoftheorderofc ormuchsmaller.Suchmovements s occur less frequently,if at all, in very young or old neutron stars for which steady-state superfluid spin-down is more usual. 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