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Tkachenko modes as sources of quasiperiodic pulsar spin variations Jorge Noronha∗ Frankfurt Institute for Advanced Studies, J. W. Goethe-Universität, D-60054 Frankfurt am Main, Germany Armen Sedrakian† Institute for Theoretical Physics, J. W. Goethe-Universität, D-60054 Frankfurt am Main, Germany We study the long wavelength shear modes (Tkachenko waves) of triangular lattices of singly quantized vortices in neutron star interiors taking into account the mutual friction between the superfluid and the normal fluid as well as the shear viscosity of the normal fluid. The set of Tkachenko modes that propagate in the plane orthogonal to the spin vector are weakly damped if the coupling between the superfluid and the normal fluid is small. In strong coupling, their 8 oscillation frequencies are lower and are undamped for small and moderate shear viscosities. The 0 periodsofthesemodesareconsistentwiththeobserved∼100−1000dayvariationsofspinforPSR 0 1828-11. 2 n I. INTRODUCTION calvariations ofthe density of the vortexlines as well as a periodic variations in the rotation and spin-down speeds J of the star. 4 Duringthelastdecademountingevidencehasemerged Ruderman [20] pointed out that the frequency of the 2 for the existence of long-period oscillations in a handful Tkachenkomodes is of the orderofseveralhundreddays ] of pulsars [1, 2, 3, 4, 5]. An outstanding example of and these modes could be responsible for the quasiperi- h this kind of phenomenonis observedin PSR 1828-11[3]. odic timing residuals observed in the Crab pulsar. Lit- p Its timing residuals are modulated with periods of 256 tle attention has been paid to the role played by the - o and 511 days, while a 1009-day periodicity is inferred Tkachenko modes in neutron stars since Ruderman’s r with lower confidence. These timing residuals coincide 1970 paper. Here we reinvestigate the propagation of t s with periodic modulations of the pulse shape, which is Tkachenko modes in neutron star superfluids and calcu- a a strong indication for the precessional motion(s) of the latehowthesemodesaredampedbymutualfrictionand [ pulsar with the periods quoted above [3]. In fact, var- the shear viscosity of normal matter. These factors are 2 ious models of precessing neutron stars [6, 7, 8, 9, 10] clearlyimportantforthecontinuouspropagationofthese v fit the timing data fairly well. However, it is known modes, which would certainly lead to observable effects. 6 that free precession is incompatible with the existence Provided that the observed variations are indeed caused 7 of a superfluid in the pulsar’s interior if the superfluid by the shear modes of the lattice we can ask the ques- 8 2 is strongly coupled to the normal fluid [11, 12, 13, 14]. tion: whatdothequasi-sinusoidalvariationstellusabout . The mutual friction for the superfluids present in the the microscopic physics governingsuperfluids in neutron 8 crust and core of neutron stars, which are derived us- stars? 0 7 ing microscopic calculations, cover a broad range of val- This question can be answered by describing the 0 ues. Areliablecalculationnecessarilyinvolvesthesuper- physics present at micro- and mesoscopic scales (the up- : conducting and superfluid properties of the fluid(s) and per limit on these scales is set by the size of the neutron v the non-superconducting material, which resists to vor- vortex) in terms of the few parameters that enter the i X tex motion. While precessionremains a viable model for equations of superfluid hydrodynamics. These parame- r the quasiperiodic oscillations observed in pulsar timing ters include the kinetic coefficients and other local char- a data, here we follow a different route [15] by exploring acteristics (e. g., density, baryon and lepton fractions, the propagation of Tkachenko modes in pulsars as the etc). Thepredictionsobtainedwithinthishydrodynamic source of long term variations. description can be presented in a form that is indepen- Chargeneutralsuperfluidsinneutronstarinteriorsro- dentofthedetailsofphysicsatthemicro-andmesoscopic scales. tate by forming an array of singly quantized vortices. In their lowest energy state the vortices form a two- In this paper we pursue this “model independent” dimensional triangular lattice. The lattice supports col- approach, although some comments about the physics lectiveelasticmodes,Tkachenkowaves,inwhichthevor- present at intermediate scales are in order. A key prob- tices are displaced parallel to each other [16, 17, 18, 19]. lem concerning the crust of neutron stars is the question Theirundamped propagationwouldleadtovariationsof ofwhetherneutronvorticesarepinnedto thecrustalnu- the angular momentum of the superfluid due to the lo- clei both under static and dynamical conditions. The static problem alwayshas pinning solutions for infinites- imally attractive interactions between a neutron vortex and a nucleus (for repulsive interactions the vortex is ∗Electronicaddress: noronha@fias.uni-frankfurt.de localized in between the lattice sites) [21, 22]. The dy- †Electronicaddress: [email protected] namical problem of repinning under the action of exter- 2 nal superflow [23] allows for repinning solutions only for ditional dynamical variable,the local deformationof the strong pinning potentials and current estimates for the vortex lattice ǫ(r), which can be incorporated into the pinning energy do not favor repinning solutions. How- Bekarevich-Khalatnikov superfluid hydrodynamics [46]. ever, if the axisymmetry of the problem is lost, as in the A hydrodynamic description of superfluids that includes caseofprecessingneutronstars,thevorticesmaynotpin lattice deformations has been studied by a number of at all [24]. authors [47, 48, 49, 50] and in this work we use the Another issue concerning neutron star crusts is that Baym-Chandlerversionofsuperfluidhydrodynamics[49] they may often undergo starquakes [25, 26], which drive tostudythe Tkachenkomodesandtheirdampingdueto thevortexlatticeoutofitsequilibriumstate. Infact,the mutual friction and shear viscosity. form of the lattice may change from a triangular shape Thefluidmotionsarenaturallyseparatedintothecen- into some other structure reflecting the symmetries of ter of mass motion (known as first sound) and second the underlying nuclear lattice. Moreover, other possible sound, whichcorrespondsto temperature dependent rel- scenarios include cases where the vortices may acquire ative oscillations between the superfluid and the normal multiple quanta of circulation or the lattice may contain component. Thesemotionsareconvenientlydescribedin impurities andvacancies. Thus,our workingassumption terms of the total mass current j = ρ v +ρ v and N N S S that the ground state of the lattice consists of a simple the relative velocity w = v v , where ρ (ρ ) and N S S N − triangular arrayof vortices may not be applicable to the v (v )arethesuperfluid(normalfluid)densityandve- S N crusts of frequently quaking neutron stars. locity, respectively. In addition, one needs an equation Regarding the cores of neutron stars, a major ques- for the time variations of the lattice deformation ǫ(r). tion is whether the proton fluid is a type-I [27, 28, 29] We note in passing that ρ = ρ +ρ is the total mass N S or type-II superconductor [30, 31, 32, 33, 34, 35, 36, 37, density and the tiny effects arising due to the inertia of 38, 39, 40] (it may also become unpaired at high densi- the vortex lines are neglected. ties). The friction in the case of type-II superconductors The linearized version of the fundamental superfluid is large [38], which implies that precession is impossible. hydrodynamic equations written for the net mass cur- If, however, the proton fluid is a type-I superconductor rent,the relativevelocity,andthe superfluidvelocityare the friction is found to be small and compatible with precession [15]. ∂j +(2Ω j)+C+σ+∇P +ρ∇φ = 0, (1) Finally, we would like to point our that the superflu- ∂t × ids may develop quantum turbulence when the velocity ∂w σ +(2Ω w) f = 0, (2) difference between the superfluid and the normal fluid ∂t × − ρ − S exceeds a certain critical value. Turbulent states in su- ∂v ∂ǫ ∇P perfluid He may be generated in many ways, e. g., by S + 2Ω + +∇φ = 0, (3) ∂t (cid:18) × ∂t(cid:19) ρ driving the normal fluid along the vortex lines through a temperature gradient. Neutron stars superfluids may where Ω = (0,0,Ω) is the spin vector (Ω is the pulsar also develop turbulent states [41, 42, 43], which could rotation frequency), P = P0 ρ(Ω r)2/2, P0 is the be caused either by unstable precession [44] or tectonic pressure in the fluid at rest, a−nd σ is×the vortex elastic activity in the crusts [45]. force density defined as This paper is organized as follows. In Sec. II we recapitulate the equations of superfluid hydrodynamics ∂2ǫ that include the combined effects of vortex tension, mu- σ =µS 2∇⊥·(∇⊥·ǫ)−∇2⊥ǫ −2Ωλ∂z2, (4) tual friction, and shear viscosity. Sec. III is devoted (cid:2) (cid:3) to the derivation of the characteristic equation for the where ∇ is the gradient in the x y plane and µ = ⊥ S Tkachenkoandinertialmodesandtheirnumericalstudy. ρ ~Ω/8m istheshearmodulusoft−hetriangularvortex S N Our conclusions are summarized in Sec. IV. lattice calculated by Tkachenko [16, 17, 18, 19]. More- over,m isthebareneutronmassandthevortextension N is given by II. SUPERFLUID HYDRODYNAMICS WITH VORTEX TENSION ~ρ b S λ= ln , (5) 8m (cid:18)a(cid:19) N It is convenientat this pointto discuss the three main length scales that appear in our study in more detail. where a is the coherence length and b = (π~/√3m Ω) N The relevant length scales are the vortex core radius is the vortex radius of the triangular lattice. The New- 10−12 cm, the intervortex spacing 10−3 cm, and tonian gravitationalpotential φ satisfies the equation ∼the size of the superfluid phase 105∼cm (we refer to ∼ 2 2 these three different scales as micro, meso, and macro- φ= (φ +φ )=4πG(ρ +ρ ), (6) S N S N ∇ ∇ scopic scales). A hydrodynamic description requires av- eraging over the mesoscopic scales. In order to describe whereGistheNewton’sconstantandφ andφ arethe S N the deformations of the vortex lattice, one needs an ad- gravitationalpotentials of the superfluid and the normal 3 fluid,respectively. TheforcedensityCisdefinedasC = ofsuperfluiddynamicsarebeyondthescopeofthis work i τ , where τ is the viscous stress tensor (see Refs. [41, 42, 43]). k ik ik ∇ 2 τ = η v + v δ ∇ v , (7) ik − (cid:18)∇i Nk ∇k Ni− 3 ik · N(cid:19) III. OSCILLATION MODES whereasηistheshearviscosity. Notethatwedonottake We consider plane wave perturbations with respect to into account the effects from bulk viscosity and thermal the equilibrium, which corresponds to uniform rotation. conductivity. Finally, the mutual friction force is We use a Cartesian system of coordinates where the z- ∂ǫ axisisdirectedalongthespinvectorω. Thevectorsjand f = βρ n ω v w can be decomposed into transverse and longitudinal S N (cid:20) ×(cid:20) ×(cid:18)∂t − (cid:19)(cid:21)(cid:21) parts, i.e., j = j +j and w = w +w . The transverse t l t l +β′ρ ω ∂ǫ v , (8) parts we are interested in satisfy the condition S N (cid:20) ×(cid:18)∂t − (cid:19)(cid:21) ∇ j =∇ w =0. (9) t t where ω = ∇ v is the quantum circulation vector, · · S n ω/ω,andβ×andβ′ arethephenomenologicalmutual Theperturbationequationforthetransversecomponents ≡ friction coefficients. of the vectors j and w derived from Eqs. (1)-(3) are The dissipative terms in the hydrodynamic equations (hereafter the subscript t is suppressed) defined above (such as the mutual friction forces) are ∂j the most general expressions that can be used in the i +(2ǫ Ω j +σ +k τ )P = 0, (10) lmn m n l m lm il description of a rectilinear vortex lattice in equilibrium ∂t that are still compatible with the conservation laws and ∂wi σl +(2ǫ Ω w f )P = 0, (11) the assumption that the dissipative function is a posi- ∂t lmn m n− ρ − l il S tive quadratic form of the perturbations (higher order ∂v ∂ǫ i n terms are neglected). Furthermore, we would like to re- +2ǫlmnΩm Pil = 0. (12) ∂t ∂t markthatBaym-Chandlerhydrodynamicsonlydescribes In the equations above we used the projector P = il the linear order corrections in the lattice displacements, δ k k /k2 where k is the wave vector. Our coordi- il i l − which are considered to be small. The neutron vortex natesystemissuchthatthe wavevectorliesonthez x − lattice in neutron stars may, however, become unstable plane, i.e., k=(ksin θ, 0, kcos θ) where θ is the angle towards forming a tangle with turbulent superfluid flow, formed by the vectors Ω and k. which wouldrequire a thoroughrevisionofthe hydrody- Writing the time perturbations as j (t) j e2Ωpt i i ∼ namic equations. In particular, the form of the mutual (a similar definition is used for the other vectors) we frictionforce (8) shouldbe changedto the onesuggested obtain, after some algebra, the characteristic equation by Groter and Mellink [51] where f w. These aspects det K =0 where ij ∝ || || p η˜αd (γ h 1) η˜γ αd γ γ h S S S N − − − −  d+γSg p η˜ γSγNg η˜γS  Kij = βˆg β−ˆ∗h p+βˆ(d+γ g) βˆ∗(−1 γ h) . N N  − − − −   βˆ∗g βˆh βˆ∗(d+γ g) p+βˆ(1 γ h)  − N − N  (13) We used the following shorthand notations in the defini- the density because µ ρ and λ ρ . The den- S S S tion of the matrix K : γ = ρ /ρ, d1/2 = cos θ, sity appears only through∼the normaliza∼tionof the shear ij N/S N/S βˆ∗ = 1 βˆ′, η˜ = ηk2/(2Ωρ), α = (4 d)/3, βˆ = γ−1β, viscosity η˜ ρ−1. The eigenmodes of the matrix in and βˆ′ =−γ−1β′. Moreover,we defined− N Eq.(13)prov∼idetheoscillatorymodesinthegeneralcase N where the shear viscosity of normal matter and the mu- k2 tual friction are included. In the non-dissipative limit g = 4Ω2ρ [µS −d(µS −2Ωλ)], (14) (β = β′ = η = 0) the modes separate into two indepen- S dentsetsthatdescribetheinertialandTkachenkomodes, k2 respectively. The (real) eigenfrequencies of these modes h = [µ d(µ +2Ωλ)]. (15) 4Ω2ρS S − S in units of 2Ω are Note that the coefficients g and h are independent of p = id1/2, p = i[(d+g)(1 h)]1/2, (16) I T ± ± − 4 256 days, should be identified with the fundamental os- Cos θ = 0 cillation mode. Oscillations with larger periods should 2 Cos θ = 10−6 thenbeidentifiedwiththehigher-orderharmonicsofthis Cos θ = 10−3 mode. A period of 256 days can be obtained by adopt- ing R = 7.7 km, which translates into k = 8.16 10−6 c cm−1. This value is close to the upper limit for t×he size s] y 0 of any superfluid region inside a medium-heavy neutron a g P [d sstuapre.rflItuiids irmegpioonrtawnhtetroe rTekmaachrkentkhoatwiafvtehsecwanidbthe ofofutnhde lo isatleastoneorderofmagnitude(ormore)smallerthan the values ofR quotedabovethe modes will be too fast -2 c to account for the long-period oscillations observed in PSR 1828-11 (as long as the renormalization effects due to mutual friction discussed below are neglected). We now consider the effects of the shear viscosity and -4-6 -4 -2 0 the mutual friction on the propagation of Tkachenko log k [cm-1] modes. It is convenient to use the drag-to-lift ratios ζ and ζ′ instead of β and β′ to describe mutual friction. These ratios are related by the following equations FIG.1: (Coloronline)DependenceoftheperiodP =2π/|pT| ζ β(1 ζ′) of the Tkachenko modes on the wave vector for d = 0 (solid β = , β′ =1 − . (17) black line), d = 10−12 (dashed red line), and d = 10−6 [(1 ζ′)2+ζ2] − ζ − (dashed-dotted blue line). For large wavelengths the periods are on the order of 100 days. In particular, for Ω⋆ = 15.51 Microscopic calculations indicate that ζ′ ≃ 0. The limit Hz we obtain that P(kmin,d=0)=331 days if the core size ζ 0 corresponds to weak coupling between the vor- → is Rc =10 km and 256 days when Rc =7.7 km. tices and the normal fluid, while ζ implies strong → ∞ coupling. Figure 2 shows the dependence of the modes derived fromEq.(13)onthedrag-to-liftratioζ forseveralvalues wheretheindicesI andT refertoinertialandTkachenko of the shear viscosity and d=0. The value of η˜is deter- modes, respectively. minedassumingaconstantdensityof3 1014gcm−3. In If the Tkachenko modes are generated within super- × the limit where ζ and η vanish we recoverthe results for fluid shells with the width of R 10 km their corre- c ∼ the non-dissipative case discussed above. For η = 0 the s6p.2o8ndin1g0−w6avcemv,ecwthoircsharseetosftthheeolordweerrolfimkmitino=n t2hπe/Rwcav∼e realpartofthe Tkachenkomode, whichisdoublydegen- × erate, vanishes only in a narrow window of values of ζ. vector. Sincethehydrodynamicdescriptionbreaksdown at length scales 10b (b 10−3 cm is the intervortex For larger values of ζ, which corresponds to the strongly ∼ ∼ coupledregion,therealpartreachesanasymptoticvalue distance) the wave vector is bounded from above by the value kmax = 73.3 cm−1. We are interested in the small thatisabout25%smallerthanitsvalueintheundamped limit. Note that in our plots only the regions where the wave vector limit k kmin that describes vortex density ∼ modes change significantly are shown. waves across the entire superfluid shell. Assuming that the normal fluid is inviscid, the results The parameters g and h are of the order of s = inFig.2implythatthereareoscillationswithevenlonger (~2km2in/2mn)(8~Ω)−1. For instance,whenk =kmin and periods in the strongly coupled limit. The Tkachenko Ω = 15.51 Hz (the rotation frequency of PSR 1828-11) ⋆ modesaresignificantlydampedbymutualfrictioninthe weobtainthats=10−14. Therefore,fors 1theeigen- regionwheretherealpartvanishes. Therethenumberof ≪ modes correspondingto Tkachenkowavesin the dissipa- imaginary roots of the characteristic equation increases tionless limit are given by pT = i[(d+g)]1/2. In the by one. Moreover, one of the imaginary roots is given ± limit d g where the wave vectors are highly collinear by Im ω = iβΩ, which continues beyond the figure’s y ≪ to the spin vector we obtain that pT = i√g and, in scale. This reflects the damping of the differential rota- ± the opposite limit d g, the Tkachenko modes become tion between the superfluid and the normal fluid caused ≫ identical to the inertial modes, i.e., pT =pI. by mutual friction. This damping has no effect on the Fig. 1 displays the period of the Tkachenko modes Tkachenko modes in strong coupling. For moderate val- without dissipation as a function of their wave vector. uesofviscosity(η =5 1017dynscm−2)therealpartof × Only the long wavelengthperturbations have periods on the Tkachenko mode is reduced in the strongly coupled the order of 100 days, which are then relevant for obser- region. In this case, its imaginary part is smaller than vations. Inthislimittheperiodsrapidlydecreaseforper- the real part and, therefore, the oscillations are weakly turbationswithfinited. TheperiodP(kmin,d=0)=331 damped. Forlargevaluesoftheshearviscosity(η 1019 days for Ω =15.51 Hz suggests that the shortest of the dyn s cm−2) the real part of the Tkachenko mod∼e van- ⋆ periods observed in PSR 1828-11, which corresponds to ishes in the strongly coupled limit. Finally, note that 5 -14 d=10 1 d = 0 η=0 1 0.5 η=0 0.5 0 1 0 19 η=10 1 17 ΩT0.5 η=5 10 ω/ ΩT0.5 0 ω/ 1 0 19 1 0.5 η=5 10 19 η=10 0.5 0 -10 -9 -8 -7 -6 log ζ 0 -10 -9 -8 -7 -6 log ζ FIG. 3: (Color online) The same as in Fig. 2 in case of d = 10−14 andη=0(upperpanel),η=1019 (middlepanel),and η = 5×1019 (lower panel) in dyn s cm−2 units. Note that FIG. 2: (Color online) Dependence of the real (solid black both theTkachenkoand inertial modes are displayed here. line) andimaginary (dashedredline) partsoftheTkachenko modes (ω = ip) on the drag-to-lift ratio ζ for η = 0 (upper panel),η=7.5×1017 (middlepanel),andη=1×1019 (lower larger because η T−2. panel)indynscm−2 units. Allmodesarenormalized bythe ∼ non-dissipative value of the Tkachenko mode ΩT = 2π/P. The modes are computed taking d = 0, which means that IV. CONCLUSIONS thereare noinertial modes. OurresultsindicatethatTkachenkomodesarebroadly there are no inertial modes when d=0. consistent with the weakly coupled theories between the Themodeswhend=10−14areshowninFig.3. Asdis- superfluidandthenormalfluid,independentofthevalue cussedabove,in the non-dissipativelimit the Tkachenko the shear viscosity. The subclass with d = cos2θ = 0 and inertial modes coincide for sufficiently large d. For has periods that are consistent with the lowest observed η = 0 the modes can be distinguished in the strongly periodicity in PSR 1828-11 of 256 days. coupled limit because the Tkachenko mode vanishes for TheexistenceofTkachenkomodesinthestronglycou- sufficiently large values of ζ. When larger viscosities are pledregiondependsontheshearviscosityofnormalmat- considered(η >1019 dyns cm−2)the differencebetween ter. For low viscosities the Tkachenko modes are (in the real parts of the inertial and Tkachenko modes can strong coupling) renormalized to values that are a few be clearly resolved. If we increase η even further we see times smaller than their non-dissipative limits. This im- that the real part of the inertial mode decreases and the plies that in strong coupling the Tkachenko oscillations imaginary part, which increases with η, becomes rele- have periods that are larger than their non-dissipative vant. Finally, the real part of the inertial mode vanishes counterparts. In fact, the damping caused by mutual at η 5 1019 dyn s cm−2. friction is not always strong enough to preclude an os- ≃ × The outer cores of neutron stars are mainly composed cillatory behavior. Therefore, we conclude that the long of light baryons, which pair in the isospin triplet states, term variation in the spin of PSR 1828-11 can in princi- and leptons. For densities of 2 3 1014 g cm−3 and plebeexplainedintermsofTkachenkooscillationswithin temperatures of T 108 K the−she×ar viscosity of the superfluidshellsforabroadrangeofvaluesofthemutual ∼ electron fluid was determined to be in the interval be- friction and the normal fluid shear viscosity. Finally, we tween 8 40 1017 dyn s cm−2 [52]. This value of the note that larger wave vector oscillations corresponding − × temperature is a realistic upper bound on the tempera- to periodic motions on shorter length scales may lead to ture in the core of neutron stars except for very young phenomena that are responsible for the observed timing objects such as the Vela and Crab pulsars. For colder noise in pulsars. stars the viscosity could be a few orders of magnitude Ourmodelnecessarilyinvolvescertainapproximations. 6 For instance, we have adopted the two-fluid superfluid ACKNOWLEDGMENTS hydrodynamics, which should be modified in order to account for the multiple fluids in the neutron star’s core [33, 34, 35, 36, 37, 38, 39]. Furthermore, the cylin- dricalsymmetry ofour setupandthe assumptionofuni- form density need to be relaxed in more realistic treat- ments of spherical superfluid shells with density gradi- J.N. acknowledges support by the Frankfurt Interna- ents. tional Graduate School for Science (FIGSS). [1] J. M. Cordes, in Planets around Pulsars, ed. J. A. [28] M. Alford, G. Good and S. Reddy, Phys. Rev. C 72, Phillips, S. E. Thorsett and S. R. Kulkarni, (ASP, San 055801 (2005) [arXiv:nucl-th/0505025]. Francisco, 1993), pg. 43. [29] D. M. Sedrakian, A. D. Sedrakian and G. F. Zharkov, [2] F. D’Alessandro, Astrophys. and Space Science 246, 73 Mon. Not. Roy. Astron. Soc. 290, 203 (1997) (1997). [arXiv:astro-ph/9710280]. [3] I.H. Stairs, A. G. Lyne,and S. L. Shemar, Nature 406, [30] G. Baym, C. J. Pethick, and D. Pines, Nature 224, 673 484 (2000). (1969). [4] G. Hobbs, The Discovery and Timing of Radio Pulsars, [31] G. Baym andC. Pethick,Ann.Rev.Nucl.Part.Sci. 25, Ph.D thesis, Universityof Manchester, 2002. 27 (1975). [5] G. Hobbs, A. Lyne and M. Kramer, Chin. J. Astron. [32] P. Muzikar and C. J. Pethick, Phys. Rev. B 24, 2533 Astrophys.6, 169 (2006). (1981). [6] T. Akgun, B. Link and I. Wasserman, Mon. Not. Roy. [33] G.A.VardanyanandD.M.Sedrakyan,Sov.Phys.JETP Astron.Soc. 365, 653 (2006) [arXiv:astro-ph/0506606]. 54, 919 (1981). [7] I. Wasserman, Mon. Not. Roy. Astron. Soc. 341, 1020 [34] G. Mendell, Astrophys.J. 380, 515 (1991); ibid,pg 530. (2003) [arXiv:astro-ph/0208378]. [35] G. Mendell and L. Lindblom, Ann. Phys. 205, 110 [8] C. Cutler, G. Ushomirsky and B. Link, Astrophys. J. (1991). 588, 975 (2003) [arXiv:astro-ph/0210175]. [36] G.Mendell,Mon.Not.Roy.Astron.Soc.296,903(1998) [9] B.LinkandR.I.Epstein,Astrophys.J.556,392L(2001) [37] A.D.SedrakianandD.M.Sedrakian,Astrophys.J.447, [arXiv:astro-ph/0101434]. 305 (1995). [10] D. I. Jones and N. Andersson, Mon. Not. Roy. Astron. [38] A. Sedrakian and J. M. Cordes, Mon. Not. Roy.Astron. Soc. 324, 811 (2001) [arXiv:astro-ph/0011063]. Soc. 307, 365 (1999). [arXiv:astro-ph/9806042]. [11] J. Shaham, Astrophys.J. 214, 251 (1977). [39] N. Andersson and G. Comer, Classical and Quantum [12] A. Sedrakian, I. Wasserman and J. M. Cordes, Astro- Gravity 23, 5505 (2006). phys.J. 524, 341 (1999) [arXiv:astro-ph/9801188]. [40] T. Akgun and I. Wasserman, Mon. Not. Roy. Astron. [13] B. Link,Phys. Rev.Lett. 91, 101101 (2003). Soc. 383, 1551 (2008). [14] B. Link,Astron. and Astrophys.458, 881L (2006); [41] C. Peralta, A. Melatos, M. Giacobello and A. Ooi, As- B. Link,Astrophys. SpaceSci. 308, 435 (2007). trophys. J. 635, 1224 (2005) [arXiv:astro-ph/0509416]. [15] A.Sedrakian, Phys.Rev. D 71, 083003 (2005). [42] C. Peralta, A. Melatos, M. Giacobello and A. Ooi, As- [16] V.K.Tkachenko,Sov.Phys.JETP 22,1282 (1966) [Zh. trophys. J. 651, 1079 (2006) [arXiv:astro-ph/0607161]. Eksp.Teor. Fiz. 49, 1875]. [43] N. Andersson, T. Sidery and G. L. Comer, [17] V.K.Tkachenko,Sov.Phys.JETP 23,1049 (1966) [Zh. Mon. Not. Roy. Astron. Soc. 381, 747 (2007) Eksp.Teor. Fiz. 50, 1573]. [arXiv:astro-ph/0703257]. [18] V. K. Tkachenko, Sov. Phys. JETP 29, 945 (1969) [Zh. [44] K. Glampedakis, N. Andersson and D. I. Jones, Eksp.Teor. Fiz. 56, 1763]. arXiv:0708.2693 [astro-ph]. [19] V.K.Tkachenko,1973,Zh.Eksp.Teor.Fiz.Pis’maRed., [45] A. Melatos and C. Peralta, arXiv:0710.2455 [astro-ph]. 17, 617. [46] I. M.Khalatnikov, An Introduction to the Theory of Su- [20] M. Ruderman,Nature225, 619 (1970). perfluidity, (WestviewPress, New York,2000). [21] P. Donati and P. M. Pizzochero, Phys. Lett. B 640, 74 [47] G. E. Volovik andV.S. Dotsenko,Sov.Phys.JETP 51, (2006). 65 (1980) [Zh. Eksp.Teor. Fiz. 78, 132]. [22] P.Avogadro, F.Barranco, R.A.Broglia andE.Vigezzi, [48] G. Baym and E. Chandler, J. Low Temp. Phys. 50, 57 Phys.Rev.C 75, 012805(R) (2007). (1983). [23] A. Sedrakian, Mon. Not. Roy. Astron. Soc. 277, 225 [49] E. Chandler and G. Baym, J. Low Temp. Phys. 62, 119 (1995). (1986). [24] B.LinkandC.Cutler,Mon.Not.Roy.Astron.Soc.336, [50] Baym-Chandlerhydrodynamicshasalso beenapplied to 211 (2002) [arXiv:astro-ph/0108281]. the problem of the spin-up of superfluid He II confined [25] L. M. Franco, B. Link and R. I. Epstein, Astrophys. J. between two parallel plates in A. Reisenegger, J. Low 543, 987 (2000) arXiv:astro-ph/9911105. Temp. Phys. 92, 77 (1993). [26] J. Middleditch, F. E. Marshall, Q. D. Wang, E. V. Got- [51] C. J. Gorter and J. H. Mellink, Physica 15, 285 (1949). thelf, and W. Zhang, Astrophys.J. 652, 1531 (2006) [52] E. Flowers and N. Itoh,Astrophys.J. 230, 847 (1979). [27] J. Charbonneau and A. Zhitnitsky, Phys. Rev. C 76, 015801 (2007) [arXiv:astro-ph/0701308].