Constraining Hadronic Superfluidity with Neutron Star Precession
a r X i v :a s t r o -p h /0302441v 2 28 A u g 2003Constraining Hadronic Super?uidity with Neutron Star
Precession
Bennett Link
Montana State University,Department of Physics,Bozeman MT 59717?
(Dated:February 2,2008)
Abstract I show that the standard picture of the neutron star core containing coexisting neutron and proton super?uids,with the proton component forming a type II superconductor threaded by ?ux tubes,is inconsistent with observations of long-period (～1yr)precession in iso-lated pulsars.I conclude that either the two super?uids coexist nowhere in the stellar core,or the core is a type I superconductor rather than type II.Either possibility would have interesting implications for neutron star cooling and theories of spin jumps (glitches).Physical Review Letters,in press
Neutron super?uidity and proton superconductivity in neutron stars could have a number of interesting consequences for observed spin behavior and thermal evolution.Interaction of super?uid vorticity with the nuclei of the inner crust or superconducting?ux tubes in the core could lead to the jumps in spin rate,glitches,seen in many neutron stars[1].The speci?c heat of the stellar interior is determined by the state of the matter,while neutrino emission processes which cool a young neutron star are strongly suppressed in the presence of hadronic super?uids(see,e.g.,[2]).The properties of condensed hadronic systems are also of interest in studies of heavy nuclei near the neutron drip line[3]and light halo nuclei [4].The properties of hadronic systems in beta equilibrium is therefore a central problem in both nuclear astrophysics and nuclear physics.
Reliable predictions of the pairing states of the neutron star core are not yet possible as they require extrapolation of nucleon-nucleon potentials well above nuclear saturation density,ρs≡2.8×1014g cm?3.The current picture of the neutron star interior posits
that the outer core consists of mostly3P2or3P2-3F2super?uid neutrons,with about5% of the mass in type II1S0superconducting protons,normal electrons and fewer muons [5,6,7,8,9,10,11].Above a density?1.7ρs,the pairing situation is essentially unknown [11].
The compelling evidence for precession in isolated pulsars[12,13,14]provides new probes of the state of a neutron star’s exotic interior.The periodic timing behavior of PSR B1828-11and correlated changes in beam pro?le have been interpreted as due to precession with a period of～1yr and an amplitude of?3?[15,16,17].The measured precession period implies a fractional distortion of the star(in addition to its rotational distortion)of??10?8. This deformation could be sustained by magnetic stresses[18,19],crust stresses[20],or a combination of the two.
The picture of the outer core I will consider is as follows.The neutron?uid rotates by es-tablishing a triangular array of quantized vortex lines,parallel to the axis of the angular mo-mentum of the super?uid and with an areal density of n v=2m n?n/πˉh?104P(s)?1cm?2, where m n is the neutron mass,?n is the angular velocity of the super?uid and P is the spin period.The average vortex spacing is l v≡n?1/2v?10?2P(s)cm.If magnetic?ux penetrates the superconducting core,it is organized into quantized?ux tubes,with an areal density nΦ=B/Φ0～1019B12cm?2,where B12≡1012B,B is the average core?eld in Gauss and Φ0≡πˉh c/e=2×10?7G cm?2is the?ux quantum.The average spacing between?ux tubes
is lΦ≡n?1/2Φ?3000B?1/212fm.The magnetic?eld in the core of a?ux tube is approximately the lower critical?eld for the superconducting transition,H c1(?1015G).Unlike the vortex array,which is expected to be nearly rectilinear,the?ux tube array is likely to have a very complicated and twisted structure[21].Hence,the vortices are entangled in the far more numerous?ux tubes.In this Letter I show that this entanglement restricts the precession to be of very high frequency and low amplitude,in con?ict with observations.
A?ux tube has a core of normal protons;the radius of this region is of order the proton coherence lengthξp?30fm.Outside the?ux tube,the magnetic?eld falls o?exponentially over a distance equal to the London length:Λp?80fm.(Type II superconductivity occurs √
when
ωcv=(H c1B/4πρp)1/2k,where k is the excitation wavenumber[25].Taking k=π/R,where R is the stellar radius,ρp=1.5×1013g cm?3and B=5×1012G,gives the characteristic frequency at which magnetic stresses are communicated through the core,ωcv,0?10rad
s?1.The frequencyωcv,0represents an approximate upper limit to the precession frequency of the star as a whole.
To calculate the precession dynamics,I assume that pressure gradients and magnetic stresses force the crust and the charged?uid to move together,and refer to the charged core ?uid plus crust,whose spin rate we observe,as the“body”,even though it contains<～10% of the star’s mass.By the arguments given above,the core neutron?uid,which accounts for >～90%of the stellar mass,has its angular momentum?xed to this body,through pinning of the vortices to the?ux tubes.I assume that the1S0vortices of the inner crust are not pinned to nuclei,and nearly follow the body’s rotation axis so that they have a negligible e?ect on the precession dynamics[26].With these idealizations,let us write the the inertia tensor of the charged?uid as the sum of a spherical piece,a centrifugal bulge that follows the instantaneous angular velocity and an oblate,biaxial deformation bulge aligned with the body’s principal axis:
I c=I0,cδ+?I? n?n??13δ .(1) Here I0,c is the moment of inertia of the charged?uid(plus any components tightly coupled to it)when non-rotating and spherical,δis the unit tensor,n?is a unit vector along the body’s angular velocity?c,n d is a unit vector along the principal axis of inertia,?I?is the increase in oblateness about?c due to rotation and?I d is the portion of the body’s deformation that is frozen in the body.Let the neutron?uid’s angular momentum vector L n be perfectly tied to the core?ux tube array,so that L n is?xed with respect to the body. The total angular momentum is L=L n+L c,where L c is the total angular momentum of the body.The Euler equations in the body frame are I c·˙?c+?c×(L c+L n)=0.De?ne principal axes in the body(x1,x2,x3),where x3is along the major principal axis(?x3=n d). The principal moments of inertia are I1=I0+2?I?/3??I d/3=I2=I3(1+?)?1,where ?≡?I d/I1>0.Let the angle between x3and L n beα.If L c,L n and?c are all aligned,the star is in a state of minimum energy for a given angular momentum and does not precess.
A likely precessional state is one in which L c and L n are perturbed slightly about this stationary point.To de?ne angles,let x3,L n,L c and?c all lie in a plane at t=0,withθ
the angle between x 3and L ,and θ′the angle between ?c and L (see Fig.1).Linearizing Euler’s equations in ?c 1,?c 2and α,gives the solutions:
?c 1(t )=A cos ?p t ?θ0?c ,?c 2(t )=A sin ?p t,(2)
where ?p ≡??c +L n /I 1is the body-frame precession frequency,and,
A
I 1 1?p ?θ 1+?p I 1?p .(3)
The motion of ?c is a circle of angular radius |A |/?c ,about an axis that takes an angle θ0with respect to x 3in the x 1?x 3plane,completing one revolution in a time 2π/?p .Removing the pinned component (L n =0)gives the familiar result of ?p =??c ,A/?c ?θ.Restoring the pinned neutron ?uid,and taking L n ?I n ?c gives ?p ?(I n /I 1)?c ?10?c ,independent of αand θ.When ?p exceeds ωcv,0,as for PSR B1828-11,the precession frequency is likely to be closer to ωcv,0?10rad s ?1,still very high.As long as L n is pinned to the body,the star precesses at high frequency for any ?nite αor θ.
First consider a state in which L n and L c are both aligned at t =0,but ?c is not along L .In this case,α=θ,giving
A
?p ,θ0=θ 1???c
?c =?θ 1+?p
of the neutron vorticity with magnitude h/2m n[27].For simplicity,takeκ=κ?x3and ?n=?n?x3and?n=?c≡?.At t=0,when?n,?c and L all lie in the x1?x3
plane,the angular velocity of the body is?c=?(?sinβ?x1+cosβ?x3).The instantaneous Magnus force per unit length of vortex as a function of position in the star is,for small angles,f m=??x1ρnκ?βx3.If f m exceeds f p,the pinning force per unit length on a typical vortex,the vortices will cut through the?ux tubes that are in their way.This condition gives|x3|>f p/ρnκ?β.For?=16rad s?1(PSR B1828-11),the inferredβof3?and a densityρs=3×1014g cm?3;|f m|exceeds f p=1016dyne cm?1for|x3|>2×10?2R,that is,the Magnus force will force the vortices through the?ux tubes almost everywhere in the star.This process is highly dissipative.
As a vortex is forced through a?ux tube,quantized vortex waves,kelvons,are excited, which propagate along the vortex and eventually dissipate as heat[28].In the rest frame of a straight vortex along the?z axis,suppose a straight?ux tube in the y?z plane approaches at speed v.SinceΛn<Λp,the?nite(magnetic)radius of the vortex can be ignored.Let the vortex and?ux tube overlap at t=0.The vector separation between a point at the center of the?ux tube which will coincide with the vortex at t=0is s(t)=vt?x.As a simple model of the interaction force,consider f int(s(t))=F p(s/Λp)exp[(1?s2/Λ2p)/2]δ(z)?x, whereδ(z),the Dirac-delta function,gives the distribution of the interaction force along the vortex(justi?ed below).
The relative velocity between vortices and?ux tubes is v?R?cβin the initial stage that ?ux tubes cut through vortices;takingβcomparable to the observed wobble angle gives v=106cm s?1for PSR B1828-11.As a?ux tube passes through a vortex,it excites kelvons of characteristic frequencyω0=v/Λp.Kelvons on a free vortex are circularly polarized waves.The frequency of a kelvon is related to its wavenumber byωk=ˉh k2/2μwhereμis the e?ective mass of a kelvon,given byμ=m n/πΛ.The dimensionless parameterΛis?0.116?ln(kξn)for wavenumbers in the range l?1v< The total energy transferred to a vortex per scattering in a potential is given in?rst-order perturbation theory by[28] ˉh ?E= Here f+(z,t)≡f int(z,t)·λ?whereλ?≡(?x+i?y)/√ dt >～4F2p R3?n B ˉh 1/2,(7) a lower limit,since the excitation of?ux tubes,which is also dissipative,was ignored. Di?erent choices for the dependence of the interaction force on s give the same scaling on the parameters appearing in eq.(7),with slightly di?erent numerical factors.The use of eq.(6)assumes that cuttings of the vortex at di?erent locations can be treated as separate events,with the excitations due to di?erent cuttings adding incoherently.This will be the case as long as k0lΦ?1.For v=106cm s?1and B12=1,k0lΦis～2, so the approximation of kelvons as distinct wavepackets is a somewhat crude one,but the lower limit in eq.(7)should be a reasonable estimate for the velocities of interest.Taking v=106cm s?1,μ=0.06m n,R=10kmΛp=80fm and F p=0.1MeV fm?1gives a dissipation rate of dE/dt～1041erg s?1.Now consider the excess rotational energy of the precessing star.The energy in the body E rot is related to the energy in the inertial frame E0by E rot=E0?L·?.Most of the angular momentum is in the neutrons,so L?L n. The excess rotational energy is thus?E rot?I n?2nβ2/2?2×1044erg.The characteristic damping time isτd≡?E rot(dE/dt)?1<～1hr.Over this short timescale,the precession damps to small amplitude.Whenβis<～0.06?,the Magnus force cannot drive the vortices though the?ux tubes anywhere in the star;L n is now?xed in the body,and therefore cannot follow the total angular momentum,so the star precesses at frequency?p?ωcv,0?10rad s?1.In general,then,long-period precession is not possible. To summarize,these estimates show that a neutron star core containing coexisting neutron vortices and proton?ux tubes cannot precess with a period of～1yr.Since ?p=??c+L n/I1,the fraction of the neutron component’s moment of inertia that is pinned against?ux tubes must be???10?8.Hence,observations require that neutron vortices and proton?ux tubes coexist nowhere in the star.Either the star’s magnetic?eld does not penetrate any part of the core that is a type II superconductor,which seems highly unlikely, or at least one of the hadronic?uids is not super?uid.This latter possibility appears unlikely in the face of pairing calculations which predict coexisting neutron and proton super?uids in the outer core[5,6,7,8,9,10,11]. If the core is a type I superconductor,at least in those regions containing vortices,the magnetic?ux could exist in macroscopic normal regions that surround superconducting re-gions that carry no?ux.In this case,the magnetic?eld would not represent the impediment to the motion of vortices that?ux tubes do,and the star could precess with a long period. Perhaps PSR B1828-11and other precession candidates are giving us the?rst clue that neu-tron stars contain a type I superconductor.Another,strange possibility,is that“neutron stars”are in fact composed of strange quark matter[30]. 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