Vortex unpinning due to crustquake initiated neutron excitation and pulsar glitches
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed September 18, 2020 (MN L A TEX style file v2.2)
Vortex unpinning due to crustquake initiated neutronexcitation and pulsar glitches
Biswanath Layek & Pradeepkumar R. Yadav Department of Physics, Birla Institute of Technology and Science, Pilani Campus, Pilani, Jhunjhunu 333031, Rajasthan, India
September 18, 2020
ABSTRACT
Pulsars undergoing crustquake release strain energy, which can be absorbed in a smallregion inside the inner crust of the star and excite the free superfluid neutrons therein.The scattering of these neutrons with the surrounding pinned vortices may unpin alarge number of vortices and effectively reduce the pinning force on vortex lines. Suchunpinning by neutron scattering can produce glitches for Crab like pulsars and Velapulsar of size in the range ∼ − − − , and ∼ − − − , respectively. Althoughwe discuss here the crustquake initiated excitation, the proposal is very generic andequally applicable for any other sources, which can excite the free superfluid neutrons,or can be responsible for superfluid - normal phase transition of neutron superfluid inthe inner crust of a pulsar. Key words: stars: neutron, pulsars: general, scattering, gravitational waves.
Pulsars are known to be excellent time-keepers. However, asignificant number of pulsars show sporadic spin-up events,namely glitches. A total of 555 glitches in 190 pulsars havebeen catalogued and reported to date (Espinoza et al.2011). The size of glitches lie in the range ∼ − − − ,with a typical interglitch time of a few years. Although, themodels associated with pinning-unpinning of superfluid vor-tices (Anderson & Itoh 1975) are considered to be the mostpopular models for explaining glitches, the crustquake (Ru-derman 1969) model finds its place in the literature quiteregularly in the study of glitches or otherwise (see Haskell& Melatos (2015) for a detailed review). There have beendiscussions in the literature suggesting the involvement ofcrustquake in neutron star physics, such as an explanationfor the giant magnetic flare activities observed in magne-tars (Thompson & Duncan 1995; Lander et al. 2015), as apossible source of gravitational waves (GW) from isolatedpulsars (Keer & Jones 2015; Layek & Yadav 2020). On theother hand, though the basic picture of the superfluid modelis well accepted, it has a few issues which are not understoodwith certainty. For instance, the value of a very importantquantity used in this model, namely, the pinning strengthis not calculated from first principles. Similarly, the precisemechanism which triggers vortex unpinning is not knownwith certainty. In fact, there are suggestions (Melatos et al. E = B ∆ (cid:15) ) due to crustquake (Baym & Pines 1971)excites the unbound superfluid neutrons that exist in theinner crust. These excited neutrons should share theirenergy with the pinned vortices through scattering. As aresult, a large number of vortices ( ∼ ) existing in the c (cid:13) a r X i v : . [ a s t r o - ph . H E ] S e p Layek, Yadav neighbourhood of the quake site can be released, causingthe star to spin-up. Here, the size of the glitch depends onthe number of vortices released due to the excitations. ForCrab like pulsars and Vela, the size of the glitches will beshown to lie in the range ∼ − − − and ∼ − − − ,respectively.The paper is organized in the following manner. Insection 2, we briefly review the relevant features of thecrustquake model. We present our work in the subsequentsections. In section 3, we determine the number of vorticesthat can be unpinned through neutron-vortex scattering.Here, we will provide the expression for glitch size. Themechanism for unpinning of vortices and the time of oc-currence of glitches will be discussed in sections 4 and 5,respectively. We will present our results in section 6. Here,we will also make a brief comment on the future scope ofthis study. Finally, we conclude our work in section 7. The crustquake in a neutron star is caused due to theexistence of a solid elastic deformed crust of thicknessabout 1 km (Ruderman 1969; Smoluchowski & Welch1970). The deformation of the crust is characterised by itsellipticity/oblateness (cid:15) = I zz − I xx I , where I zz , I xx and I arethe moment of inertia about z-axis (rotation axis), x-axisand of the spherical star, respectively (Baym & Pines 1971).At an early stage of formation, the crust solidified withinitial oblateness (cid:15) (unstrained value) at a much higherrotational frequency of the star. As the star slows down,the oblateness (cid:15) ( t ) decreases, leading to the development ofstrain in the crust. Once the critical stress is reached, crustcracks followed by a sudden change of oblateness ∆ (cid:15) . Asa result, the moment of inertia (MI) of the star decreases,which leads to an increase in its rotational frequency. In thecrustquake model, the glitch size is related to ∆ (cid:15) through ∆ΩΩ = − ∆ II = ∆ (cid:15) , and it is completely determined by theextent to which ∆ (cid:15) is changed due to crustquake. Theinterglitch time being proportional to ∆ (cid:15) , is also determinedby the change of oblateness. Note that for crustquake tobe a successful model for glitches, it should be a regularevent for a pulsar, with a frequency of once in a few years.This requires the crustal strain to be released partially in acrustquake event. Equivalently, the fractional strain release, η = ∆ (cid:15)(cid:15) should always be less than unity. In fact, smallerthe value of this fraction, larger the number of crustquakeevents that are likely to occur during the rotational historyof a pulsar. In our model, we will take a fixed value of∆ (cid:15) = 10 − motivated by the crustquake model of glitchesfor Crab like pulsars. By ‘Crab like’, we mean pulsars withthe characteristic age (Andersson et al. 2012) and glitchsize similar to that of Crab pulsar. We will show belowthat the condition η < (cid:15) , which is determined by thecritical (breaking) strain u cr that a star can sustain withoutbreaking.We will now present the relationship between the criti-cal strain u cr and the ellipticity of the star (cid:15) by mentioninga few relevant parameters (see Baym & Pines (1971); Jones (2002) for details). The total energy of the deformed pulsaris given by (Baym & Pines 1971), E = E + L I + A(cid:15) + B ( (cid:15) − (cid:15) ) . (1)Where E is the contribution of gravitational potential en-ergy of the spherical pulsar. L and I are the angular mo-mentum and the moment of inertia of the deformed pulsar,respectively. The coefficient A ( (cid:39) erg) arises as a correc-tion of gravitational energy due to deviation from sphericity.The coefficient B ( (cid:39) erg) is related to the modulus ofrigidity of the star’s crust (Baym & Pines 1971; Jones &Andersson 2001). Within an approximation B << A , theupper bound on star’s ellipticity can be written in terms ofthe critical strain u cr as (Baym & Pines 1971; Jones 2002;Layek & Yadav 2020), (cid:15) < BA u cr (cid:39) − u cr . (2)Using the above equation, the possible upper bound on (cid:15) can be obtained from the values of u cr , as estimated theo-retical in several works (Horowitz & Kadau 2009; Chugunov& Horowitz 2010; Baiko & Chugunov 2018). Horowitz& Kadau (2009) have done detailed molecular dynamicssimulations to estimate the magnitude of crustal breakingstrain of neutron star. Simulations were performed throughmodelling the crust as monocrystalline and polycrystallinematerials and they obtained the critical strain u cr = 0 . (cid:15) < − . Recently,Baiko & Chugunov (2018) have followed a semi-analyticalapproach to study the crustal properties of a neutronstar, including the analysis on crustal breaking strain. Forpolycrystalline materials, they have obtained the value u cr = 0 .
04. For this value, the upper bound of ellipticity isgiven by (cid:15) = 0 . × − . From the observational perspective,there were several attempts (Abadie et al. 2011; Aasi et al.2013, 2014; Abbott et al. 2020) to constrain star’s ellipticityby observations. Any asymmetric mass distribution ofpulsar relative to its rotation axis, such as triaxiality (Jones2002), mountains (Haskell et al. 2006; Bhattacharyya 2020)that can be characterised by ellipticity parameter are thesource of continuous gravitational waves. As the strainamplitude of such gravitational waves is proportional to (cid:15) ,their non-observation naturally puts an upper limit on theellipticity of the star. Here, we mention recent results byAbbott et al. (2020), that are based on the analysis of LIGOand VIRGO data obtained from the searches of continuousgravitational waves from a few selected isolated pulsars.The results were presented for three recycled pulsars, alongwith two relatively young pulsars Crab and Vela. We willquote the results for Crab and Vela that are relevant in thecontext of our present model. As per the analysis in Abbottet al. (2020), the upper limits of (cid:15) were constrained at 10 − and 10 − for Crab and Vela, respectively.The typical fractional strain released η = ∆ (cid:15)(cid:15) cannow be obtained for a fixed value of ∆ (cid:15) = 10 − andputting the values of (cid:15) as quoted above. Firstly, within thetheoretical uncertainties in the estimate of u cr , the valuesof (cid:15) in the range (1 . − . × − provides η in the range ∼ . − .
02. For the values of (cid:15) as constrained by theobservations, η will be even smaller. For Crab and Vela, thevalues are given by η = 10 − and η = 10 − , respectively. c (cid:13) , 000–000 rustquake initiated neutron excitation & glitches As we see from above, the set of values of η satisfy thecondition η < (cid:15) = 10 − throughout this work to be consistent with theconditions that are required in the crustquake model, i.e.,the glitch size, interglitch time and the fractional strainreleased. The corresponding value of strain energy is thengiven by ∆ E = B ∆ (cid:15) (cid:39) erg. We assume that thereleased energy is absorbed in the inner crust and excitesthe neutrons in the bulk neutron superfluid. We will showthat the excited neutrons, in turn, can unpin a large numberof vortices through neutron-vortex scattering from a localregion in the equatorial plane. We calculate the number ofunpinned vortices and estimate the glitch size. Note, withthe fixed value of ∆ (cid:15) = 10 − , the interglitch time is alwaysfixed to be about one year, irrespective of the glitch sizeproduced by the local unpinning in our model. We willshow that for Crab like pulsars and Vela, the glitch size stillcan vary in the range ∼ − − − , without affecting theinterglitch time. We assume that superfluid vortices are pinned at t = 0,and a fraction of these vortices get unpinned by neutronexcitations caused by crustquake at t = t p . The interglitchtime t p is expected to be of the same order as the timeduration of successive crustquake events. We take thepicture that crustquake occurs in a local region aroundthe equatorial plane in the outer crust of the star (seeFig. 1). We choose the quake site to be in the equatorialplane motivated by the picture proposed by Baym & Pines(1971) in their work on the crustquake model for glitches.There was also a detailed study (Franco et al. 2000) on thedevelopment of the crustal strain, which arises due to theslowing down of the star. By including the effects of themagnetic field, the authors have calculated the strain angleand found out that the strain angle is maximum in theequatorial plane, making it most likely place for the quakesite. The absorption of strain energy ∆ E = B ∆ (cid:15) should ex-cite the free superfluid neutrons that exist outside the nu-clei surrounding the pinned vortices. The sharing of energythrough scattering by these excited neutrons with the vor-tex core neutrons should result in the unpinning of vorticescausing the glitch event. Assume Ω p is the angular velocityof the pinned vortices that remains fixed during t = 0 to t = t p . Ω c ( t ) is the angular velocity of the corotating crust-core coupled system with Ω c (0) = Ω p . The development ofdifferential angular velocity δ Ω = Ω p − Ω c ( t ) between vor-tices and the rest follows the time evolution of the star andcan be written as (at t = t p )Ω p − Ω c ( t p )Ω c ( t p ) ≡ (cid:16) δ ΩΩ (cid:17) t p (cid:39) t p τ , (3)where τ = − Ω2 ˙Ω is the characteristic age of pulsar and weassume t p << τ . For ease of notation, from now onwardwe assume, ( δ ΩΩ ) t p ≡ δ ΩΩ . The glitch size can be estimated applying the model of superfluid vortices,∆ΩΩ = (cid:16) I p I c (cid:17)(cid:16) δ ΩΩ (cid:17)(cid:16) N v N vt (cid:17) , (4)where I p I c is the MI ratio of bulk neutron superfluid in theinner crust to the rest of the star (Ruderman 1976). N vt isthe total number of pinned vortices in the equatorial planein the inner crust. The ratio N v N vt is incorporated since onlya fraction of vortices is expected to be affected by the ex-cited neutrons. In standard superfluid model (Anderson &Itoh 1975), the above ratio is almost unity and δ Ω shouldbe replaced by its critical value δ Ω cr . Where δ Ω cr is themaximum value of δ Ω at which the magnus force balancesthe pinning force. The magnus force per unit length on avortex line is given by f m = ρκRδ Ω. Equating this with thepinning force per unit length f p = E p bξ , we get (Alpar et al.1984), δ Ω cr = E p ρκRbξ , (5)where, E p and ρ are the pinning energy per pinning site (tobe estimated later) and the local mass density, respectively. κ = h m n is the quantum vorticity with m n being the neu-tron mass. ξ (cid:39)
10 fm is the coherence length of the bulksuperfluid, and the nucleus-nucleus distance is denoted by b ( (cid:39)
100 fm). R ( (cid:39)
10 km) is the distance of the inner crustfrom the centre of the star. The numerical value of δ Ω cr will be estimated in the next section. We now estimate thenumber of vortices N v that are expected to be affected bythe neutron excitation. First, we take a region of volume V p within which the free neutrons should be excited. Therelevant volume can be estimated by energy balance as B ∆ (cid:15) = N e ∆ f = ∆ f E f n f V p (6)or equivalently, V p = B ∆ (cid:15) E f n f ∆ f , (7)where the free neutron number density in the superfluidstate is denoted by n f , and ∆ f denotes the superfluid freeenergy gap of the neutrons. N e is the number of excitedneutrons. The mass density ρ of the inner crust lies in therange ∼ (10 − ) gm-cm − . The Fermi momentumof the free neutrons at ρ (cid:39) × gm-cm − has beencalculated by several authors (Pastore et al. 2011; Chamel &Haensel 2008; Sinha & Sedrakian 2015) and found to be oforder k f (cid:39) .
20 fm − . The corresponding value of neutrondensity is given by, n f = k f π = 2 . × − fm − , whichincreases as one goes deeper in the crust. For our case, therelevant region of interest is the outer part of the innercrust, and we will take the above value for the estimateof V p . The superfluid gap parameter ∆ f = 0 .
06 MeV for k f (cid:39) .
20 fm − (Pastore et al. 2011; Chamel & Haensel2008; Sinha & Sedrakian 2015). Putting the values of vari-ous quantities in Eq. (7), we get V p = 5 . × m . Notethat the calculation of V p assumes the isotropic distributionof energy from the quake site, and the geometry is taken tobe a cubical shape as shown in Fig. 1.The isotropic distribution of energy from the quakesite is an assumption in estimating the volume of the c (cid:13)000
20 fm − (Pastore et al. 2011; Chamel & Haensel2008; Sinha & Sedrakian 2015). Putting the values of vari-ous quantities in Eq. (7), we get V p = 5 . × m . Notethat the calculation of V p assumes the isotropic distributionof energy from the quake site, and the geometry is taken tobe a cubical shape as shown in Fig. 1.The isotropic distribution of energy from the quakesite is an assumption in estimating the volume of the c (cid:13)000 , 000–000 Layek, Yadav
Ω ∆ l ∆ R R rustquake site l vortex linepinning sites Figure 1.
Schematic representation (cross-sectional view, not toscale) of quake site and the nearby volume element V p , where localunpinning may occur by absorbing energy due to crustquake. R ( (cid:39)
10 km) is the distance of the inner crust from the centre ofthe star and ∆ R (cid:39) affected region. In principle, the volume of the affectedregion should depend on the Fermi energy ( E f ) and pairingenergy (∆ f ) of the neutron superfluid in that region.Although, the distances to which the energy transportsin the azimuthal direction (with respect to the rotationaxis) and along the altitude of the star are expected tobe the same, the distance across the inner crust should bedifferent from the other two directions. In this work, we willnot take into account such anisotropy and assume cubicalgeometry only. Here, we mention the works of Lander et al.(2015) and Akbal & Alpar (2018), where the authors haveconsidered a cubical geometry in their respective studies.Lander et al. (2015) have studied the crustquake eventdue to the development of magnetic strain as a result ofinternal magnetic field evolution of the star. In their study,the authors have considered a cubical crustquake geometryto calculate the required magnetic field strength for crustbreaking. Similarly, Akbal & Alpar (2018) have studiedvortex unpinning (Although, the unpinning mechanism iscompletely different from ours.) due to the crustal platemovement triggered by crustquake. In their work, the sizeof the broken plate and the number of unpinned vorticesare calculated by modelling cubical shape as one of thequake site geometries. The number of unpinned vorticesestimated by the authors turned out to be of a similarorder, irrespective of the assumed geometries in their work.In this spirit and for simplifying calculations, we considera cubical geometry to test our model by estimating thevolume V p of the affected region, number of unpinnedvortices N v and the glitch size ∆ΩΩ .Now denoting ∆ l as the length of each side of the cube,we now express the volume V p in terms of number of vortices N v as V p = (∆ l ) = N v ∆ ln v , (8)where n v = m n Ω π (cid:126) = 10 cm − ( Ωs − ) is the areal density ofvortices. Number of vortices in the equatorial plane, whichare expected to be unpinned due to neutron excitation, can be estimated using Eq. (7) and Eq. (8) as N v = B ∆ (cid:15) n v E f ∆ l n f ∆ f = 3 . × (cid:16) Ωs − (cid:17) . (9)In the above equation, the numerical factor has been cal-culated by taking ∆ l = ( V p ) / = 172 m and the values ofother parameters are taken as mentioned earlier. Note thatfor Crab/Vela Ω (cid:39) , there are about N v = 10 vorticeswhich can be released from the volume V p . Finally, usingEq. (3) and Eq. (4), the glitch size is obtained as∆ΩΩ = (cid:16) . × N vt (cid:17)(cid:16) Ωs − (cid:17)(cid:16) I p I c (cid:17)(cid:16) t p τ (cid:17) , (10)where the total number of vortices in the crust is given by N vt (cid:39) (2 πR ∆ R ) n v = (cid:16) . × Ωs − (cid:17) . (11)Here, ∆ R (cid:39) N vt in Eq. (10), we get∆ΩΩ (cid:39) − (cid:16) I p I c (cid:17)(cid:16) t p τ (cid:17) . (12)By taking t p of the same order as typically observed inter-glitch time of pulsars, we estimate the glitch size using Eq.(12) and results are discussed in section 6. Of course, theglitch will arise provided N v (cid:39) vortices are unpinnedby neutron excitation from the region of our interest. Wenow discuss the mechanism which ensures the unpinning ofthe vortices. We assumed that the crustquake occurs in the outer crust inthe vicinity of the equatorial plane and the energy releasedby this event is distributed isotropically from the quakesite. The energy ∆ E (cid:39) erg absorbed in the volume V p = 5 . × m should excite a N e (= ∆ E f n f V p (cid:39) )number of neutrons from the bulk neutron superfluid.Ignoring small finite temperature ( kT (cid:39) .
01 MeV) cor-rection, each of these excited neutrons has approximately E f = 0 .
83 MeV amount of energy. If the total energy ofthe excited neutrons is more than the pinning energy ofall vortices enclosed in volume V p (see Fig. 1), then theinelastic scattering of these neutrons with the vortex coreneutrons should unpin the vortices. Note that the pinningenergy E p acts as the binding energy of the vortex-nucleussystem, and it arises due to the interaction of the vortexwith the nucleus. Thus the sharing of energy by the excitedneutrons with the vortex core neutrons increases the kineticenergy of the later. In fact, the energy of the excitedneutrons equivalently can be treated as the activationenergy, which helps to overcome the pinning barrier. Wewill show below that the excited neutrons have the requiredenergy to overcome the barrier. The inelastic collision canbe represented as, excited neutron ( ∼ E f ) + pinned vortex ( − E P ) → de-excited neutron ( E f − E p ) + free vortex. The quantities in bracket denote the energy of variousobjects. The negative sign in front of E p ( >
0) signifies thebinding energy of the pinned vortex. It should be noted that c (cid:13) , 000–000 rustquake initiated neutron excitation & glitches following unpinning; the vortex is free to move (i.e., freevortex) outward with radial velocity v r . Here we make afew comments on the possibility of repinning of the outwardmoving vortices. Since the study of repinning initiatedby Sedrakian (1995), the mechanism and consequencesof repinning have been discussed often in the literature,whether in the context of creep theory (Alpar et al. 1984),or in the standard theory of superfluid vortices (Anderson& Itoh 1975). One of the important consequences ofrepinning, namely, the acoustic radiation is quite relevantto our present model. As per the studies in Warszawskiet al. (2012); Warszawski & Melatos (2012b), the acousticradiation caused by repinning (so-called ‘acoustic knock-on’ as per the terminology used in Warszawski et al. (2012))is believed to play an important role in the process ofvortex avalanche. The avalanche can be a viable process forour model to produce large size glitches through acousticknock-on (and proximity knock-on) caused by repinning ofoutgoing vortices. In this present work, we skip the studiesof vortex avalanche (which will be explored in future)except making a few comments in the results & discussionsection.We now compare the pinning energy with the energy ofexcited neutrons. The pinning mechanism as per discussionin Ref. Alpar et al. (1984) depends on the density ρ . It wassuggested that for ρ > gm-cm − , the vortex lines arepinned to lattice nuclei with pinning energy per site (Alparet al. 1984; Link & Epstein 1991), E p = 38 γ ∆ E f n f V (cid:39) . × − MeV . (13)Where V = πξ is the overlap volume between the vortexand the nucleus. The size of vortex core ξ ( (cid:39)
10 fm) is of thesame order as the nuclear radius. The numerical value of γ is of order unity (Alpar et al. 1984, 1989). For the density ρ < gm-cm − , the lines are preferably pinned in betweennuclei (interstitial pinning) and the pinning energy E p hasbeen calculated to be of the order 1 KeV per site (Link &Epstein 1991), which is approximately the same as obtainedin Eq. (13). The above two pictures though qualitatively dif-ferent, provide approximately the same value of E p for thepinning region of our interest and both pictures are consis-tent with our unpinning mechanism. The total amount ofpinning energy associated with all the vortices within vol-ume V p can be estimated as E tp = (cid:16) E p d v b (cid:17) V p = 3 . × MeV . (14)Where d v (= 0 .
01 cm) and b (= 100 fm) are inter vortexand inter-nuclear distance, respectively. The value of E p istaken from Eq. (13). The energy as estimated in Eq. (14)should now be compared with the total kinetic energy ofthe excited neutrons in the same volume V p , E te (cid:39) N e E f = (cid:16) ∆ E f n f (cid:17) E f V p = ∆ n f V p (cid:39) MeV . (15)Thus, we see from Eq. (14) and (15) that the excitedneutrons have the required energy to unpin the vorticesthat are contained in the volume V p .For the sake of numbers, with a typical inter-nuclear distance of about a 100 fm at baryon density ρ (cid:39) gm-cm − , a single vortex line passes through ∆ lb (cid:39) number of nuclei within the cube of height ∆ l (cid:39)
172 m andabout 10 nuclear sites which host pinned vortices in a sin-gle vortex line will be affected due to the neutron-vortexscattering. Since all the vortices that lie inside the volume V p are unpinned, the pinning force on the whole vortex linespassing through V p is reduced. The fractional decrease inpinning force per unit length is given by∆ f l f l = ∆ ll (cid:39) . , (16)where f l is the pinning force per unit length (Anderson& Itoh 1975; Pizzochero 2011) of vortex lines and ∆ f l isthe decrease of pinning force per unit length due to theeffects as mentioned above. The above numerical value inEq. (16) has been calculated by taking the length of a vor-tex line (assuming straight), threaded in the inner crust(see Fig. 1) as, l (cid:39) √ R ∆ l (cid:39) δ Ω cr . The numerical value of δ Ω cr can be estimated fromEq. (5). Taking E p = 5 . × − MeV from Eq. (13) and ρ = 5 × gm-cm − , we get δ Ω cr = 0 .
09 rad-s − . Thefractional decrease of the above quantity should be of thesame order as ∆ f l f l , i.e.,∆( δ Ω cr ) δ Ω cr (cid:39) . . (17)If the build-up differential angular frequency δ Ω at t p doesnot differ much from its critical value δ Ω cr , then the magnusforce effectively should be able to move the vortex lines fromthe pinning site. The relative difference can be estimated as δ Ω cr − δ Ω δ Ω cr = 1 − (cid:16) t p τ (cid:17)(cid:16) Ω0 . − (cid:17) ≡ − x. (18)The numerical value of (1 − x ) can be determined bytaking the typical values for Crab/Vela, t p = 1 year, τ = (10 − ) years and Ω (cid:39) s − . For these setof values, the numerical value of (1 − x ) turns out to beof a similar order, such that Eq. (17) is satisfied. Note,the purpose of the above exercise is to check whetherthe pinning force is decreased enough for the vortex linespassing through the volume V p to move under magnusforce, even when δ Ω at t = t p is less than the critical value δ Ω cr . It is indeed true, as suggested by the above set ofarguments. We conclude this part by saying that crustquakeinitiated neutron excitations can unpin a large number ofvortex lines which pass through the volume V p and henceproduces glitch through the local unpinning. Now we estimate the time of occurrence of glitches t g following the crustquake event. The time t g is the sum ofthe time taken for unpinning followed by the time for thevortex to move outward and share their excess angularmomentum to the crust. First, we estimate the time forunpinning, which is determined by the relaxation timescale ( τ nn ) of neutron-vortex scattering. The calculation c (cid:13)000
09 rad-s − . Thefractional decrease of the above quantity should be of thesame order as ∆ f l f l , i.e.,∆( δ Ω cr ) δ Ω cr (cid:39) . . (17)If the build-up differential angular frequency δ Ω at t p doesnot differ much from its critical value δ Ω cr , then the magnusforce effectively should be able to move the vortex lines fromthe pinning site. The relative difference can be estimated as δ Ω cr − δ Ω δ Ω cr = 1 − (cid:16) t p τ (cid:17)(cid:16) Ω0 . − (cid:17) ≡ − x. (18)The numerical value of (1 − x ) can be determined bytaking the typical values for Crab/Vela, t p = 1 year, τ = (10 − ) years and Ω (cid:39) s − . For these setof values, the numerical value of (1 − x ) turns out to beof a similar order, such that Eq. (17) is satisfied. Note,the purpose of the above exercise is to check whetherthe pinning force is decreased enough for the vortex linespassing through the volume V p to move under magnusforce, even when δ Ω at t = t p is less than the critical value δ Ω cr . It is indeed true, as suggested by the above set ofarguments. We conclude this part by saying that crustquakeinitiated neutron excitations can unpin a large number ofvortex lines which pass through the volume V p and henceproduces glitch through the local unpinning. Now we estimate the time of occurrence of glitches t g following the crustquake event. The time t g is the sum ofthe time taken for unpinning followed by the time for thevortex to move outward and share their excess angularmomentum to the crust. First, we estimate the time forunpinning, which is determined by the relaxation timescale ( τ nn ) of neutron-vortex scattering. The calculation c (cid:13)000 , 000–000 Layek, Yadav of τ nn deserves a separate work, and we will provide herean order of magnitude estimate following the approach ofFeibelman (1971). To understand postglitch behaviour ofpulsars, the author (Feibelman 1971) has worked out adetailed calculation of relaxation time scale τ en of electronsby considering the scattering of thermal electrons with thevortex core neutrons. The contribution of neutron-vortexscattering was not considered in their calculation. It ismainly due to the presence of very few thermally excitedneutrons in the bulk superfluid at a lower temperature( kT (cid:39) .
01 MeV) of the star. Note that the probability ofexcited neutrons in the bulk superfluid is suppressed by thefactor e − ∆ fkT . Hence, the thermal neutron-vortex scatteringis expected to be negligible compared to electron-neutronscattering. However, the presence of excited neutrons inour case is not of thermal origin. The neutrons are excitedby the absorption of energy B ∆ (cid:15) , and the fraction ofexcited neutrons relative to superfluid neutrons is given by ∆ f E f (cid:39) .
07. Therefore, the scattering of these neutrons withthe vortex core neutrons should not be suppressed, andit should naturally provide a finite relaxation time scale τ nn .We will now estimate τ nn following the expression of τ en (see Equation (45) of (Feibelman 1971)), τ nn = (cid:16) Ω c Ω (cid:17)(cid:16) π g n (cid:17)(cid:16) E f E fv (cid:17) (cid:16) E fv ∆ v (cid:17)(cid:16) E fv m n c (cid:17) / × (cid:16) (cid:126) ∆ v (cid:17)(cid:32) exp (cid:16) √ π (cid:17) K (cid:16) √ π (cid:17) (cid:33) exp (cid:16) π ∆ v f E fv (cid:17) s . (19)Here, Ω c Ω is the ratio of upper critical angular speed ofneutron fluid (Ω c = 10 ) to the angular speed of thestar. For Crab/Vela the ratio is of order 10 . The couplingstrength associated with neutron-neutron interactions canbe described by the neutron g factor, g n = − . K ( √ π )is the zero order Bessel function. E fv and ∆ v denote theFermi energy and superfluid gap parameter associated withthe neutron vortex core, respectively. The factor β = kT in the calculation of τ en (Feibelman 1971) is due to the fi-nite temperature probability distribution of electrons. In ourcase, β should be replaced by bulk superfluid energy gap ∆ f .For the case of nuclear-vortex pinning, we take the approx-imation E fv (cid:39) E f = 0 .
83 MeV and ∆ fv (cid:39) ∆ f = 0 .
06 MeV.For interstitial pinning, the approximation will be replacedby equality. Substituting these quantities in Eq. (19), we get τ nn = 3 . × − s . (20)Thus, as we see from Eq. (20) that the unpinning of vorticesare almost instantaneous. Note, the relaxation time scale forelectron τ en has been estimated (Feibelman 1971) to be onthe order of days to years. The shorter time scale of τ nn ascompared to τ en is expected due to the difference of couplingstrength in electron-neutron and neutron-neutron interac-tions. The former is dipole-magnetic moment interactionand hence the strength of the interaction is proportional to αg n , where α = e π = . The later interaction is solely dueto magnetic moment of the neutrons with the interactionstrength proportional to g n . We should mention that adetailed calculation is required for the precise estimate of τ nn . From the perspective of the occurrence of glitches following crustquake, as we see below that even a feworders of magnitude change in the value of τ nn will havenegligible contribution to t g . Next, we determine the time( t c ) taken by the unpinned vortices to move toward theouter crust and share their excess angular momentum tothe crust. The radial velocity v r of the unpinned vortices is v r (cid:39) R ( δ Ω) = ( t p τ ) Ω R (cid:39) (10 − ) cm-s − , where wehave used Eq. (3) and τ (cid:39) (10 − ) years are the age ofCrab and Vela, respectively. Thus the time t c (cid:39) v r ∆ l lies inthe range ∼ (0 . − .
7) s. We see that the glitch due tovortex unpinning occurs at t g = τ nn + t c (cid:39) (0 . − .
7) safter the crustquake. The implication of the time of occur-rence t g of pulsar glitches in our model will be discussed inthe next section.Note that the change of oblateness of the star due tocrustquake is also expected to produce a glitch (as per thecrustquake model) of order 10 − . The glitch rise time ∆ t isapproximately determined (Ruderman 1991; Haskell et al.2015) by the speed of shear wave v = (cid:113) µρ = 3 × cm s − .Where µ = 10 dynes-cm is the shear modulus of the crustand ρ (cid:39) gm-cm − is the average crust density. Thus,the time for the shear wave to propagate along the stellarsurface of the radius ( R ) is given by ∆ t (cid:39) πR/v = 0 .
01 s(Baym & Pines 1971). From the perspective of distinguisha-bility of the glitches produced by two different sources, theglitch rise time ∆ t in the crustquake model needs to be com-pared with the time of occurrence of glitch t g produced bylocal unpinning. We will discuss this issue in the next sec-tion. We have discussed (in section 4) our novel mechanism oflocal unpinning of vortices in a given region caused by shar-ing of energy by the excited neutrons with the vortex coreneutrons. We will now estimate the glitch size ∆ΩΩ caused bythe local unpinning for Crab like pulsars and Vela pulsarsusing the Eq. (12). The glitch size depends on the numberof vortices N v released due to local unpinning. For the fixedinput energy B ∆ (cid:15) , this number depends on the propertiesof bulk neutron superfluid via E f and ∆ f . The values of E f and ∆ f are taken from the literatures as noted in sec-tion 3. The energy input is provided by the strain energyreleased B ∆ (cid:15) due to the crustquake. The value of this en-ergy ( (cid:39) erg) is set based on the arguments providedin section 2. The interglitch time t p is set by the frequencyof occurrence of crustquake events and is proportional tothe change of oblateness ∆ (cid:15) due to crustquake. We choose∆ (cid:15) = 10 − in accordance with the interglitch time of ap-proximately one year. The ratio of moment of inertia of su-perfluid component in the inner crust to the rest of the staris of order I p I c (cid:39) − as per the evidences through severalstudies (see, for example, Ref. Ruderman (1976)). Puttingthese values in Eq. (12) and by taking the typical charac-teristic age of Crab and Vela in the range τ (cid:39) (10 − )years, we get the glitch size ( t p (cid:39) (cid:16) ∆ΩΩ (cid:17) (cid:39) − − − , (21) c (cid:13) , 000–000 rustquake initiated neutron excitation & glitches where the relatively larger (smaller) value of glitch size isfor Crab (Vela) pulsar.The above estimate of glitch size corresponds to unpin-ning of N v (cid:39) vortices (Eq. (9)), out of total N vt ∼ vortices (Eq. (11)) present in the inner crust. Note that theMI ratio I p I c is taken as 10 − for the estimate of glitch size.It was suggested (Ruderman 1976) that this ratio takesdifferent values depending on the presence or absence ofnormal neutron fluid (called as ‘transition region’) betweenthe inner crust and the interior neutron superfluid. Inthe presence of a normal layer, the unpinned vortices arerequired to share their excess angular momentum to arelatively larger part of the corotating system and hence, I p I c (cid:39) − is relatively smaller. In the absence of such layer,the above ratio is increased and approximately is given by I p I c = 0 .
1. In fact, Piekarewicz et al. (2014) suggested thatwithin theoretical uncertainties in the equation of state,the neutron star can have I p I c = 0 .
1. For this value, therewill be about one order of magnitude enhancement in theglitch size. For Crab like pulsars and Vela, the glitch sizecan be of order 10 − and 10 − , respectively. Interestingly,such glitches have been observed for Crab (Basu et al.2020; Ggercinolu & Alpar 2020) and Vela pulsar (Cordeset al. 1988; Jankowski et al. 2015). Note, the interglitchtime remains same in our model due to the fixed value of∆ (cid:15) = 10 − , even though the glitch size varies. Thus, thecorrelation between the glitch size and the waiting timedoes not exist in our model, which is consistent with thestatistical study for most of the pulsar glitches (Warszawski& Melatos 2012b).From the perspective of the time of occurrence ofglitches, we mentioned that the time interval betweencrustquake and the glitch produced by unpinned vorticesshould be decided by the relaxation time scale ( τ nn ) ofexcited neutrons, followed by the time t c taken by the vor-tex to reach the outer crust and share their excess angularmomentum. In section 5, we have attempted to provide anorder of magnitude estimate of τ nn ( (cid:39) − s) following thework of (Feibelman 1971). The total time duration turnsout to be t g = τ nn + t c (cid:39) (0 . − .
7) s, which is the time ofoccurrence of glitch by vortex unpinning after crustquake.We point out here that too small value of neutron relaxationtime scale τ nn compared to electron-neutron relaxationtime scale τ en is quite significant from the perspective ofobservation of glitches following crustquake. The longertime scale (few days to few months) of τ en is crucial inexplaining postglitch behaviour of the star. In contrast, τ nn and t c set the time of occurrence of glitch immediatelyafter the crustquake event. As crustquake itself producesglitch (because of rearrangement of the shape of thepulsar), the time scale of τ nn of similar order as τ en couldhave been noticeable through the recurrence of anotherglitch within a few days/months. The non-observation ofanother glitch within such interval follows from the factthat τ nn ( (cid:39) − s) is too small. Also for the same reason,the time t g ( (cid:39) t c ) can now be identified as the glitch risetime in our model, where the beginning of glitch coincideswith the simultaneous unpinning of all the vortices. Notethat t c (cid:39) (0 . − t g with the glitch rise time∆ t (cid:39) .
01 s in the crustquake model, we see that theglitch produced through vortex unpinning lags behindthe glitch produced due to crustquake. The conclusion isof course within the uncertainty in the estimation of t g and ∆ t . This distinguishing feature should be reflectedin the pulse profile for glitches, provided the subsecondresolution in pulsar timing is achieved. As of now, thebest resolved time observed for spin-up of Vela pulsar hasbeen reported (McCulloch et al. 1990) to be ≈ . vortices may act as atrigger mechanism to unpin the nearby vortices, which lie inthe equatorial plane. Among various suggestions on vortexavalanches (Melatos et al. 2008; Warszawski et al. 2012;Warszawski & Melatos 2012b; Akbal & Alpar 2018), we findthat the knock-on pictures (Melatos et al. 2008; Warszawski& Melatos 2012b) fit quite well in our model. In proximityknock-on, presence of the azimuthal component of thevortex velocity can make an individual vortex knock-on theother vortices present in the equatorial plane and nearbythe cubical volume V p . Note that for completely outwardmotion, the vortices should not encounter vortices along itstrajectory. In our future work, we would like to determinethe trajectory of unpinned vortices, estimate the numberof vortices that can be released through this process. Theacoustic knock-on caused by repinning of vortices can alsobe a viable process in our model. In future, we would like toimplement these mechanisms to study the avalanche process.Other than glitches, another interesting phenomenon ofinhomogeneous vortex line movement could be the genera-tions of gravitational waves from an isolated pulsar (Jones2002; Bagchi et al. 2015; Layek & Yadav 2020). We wouldlike to explore this possibility in future following the ap-proach of Warszawski & Melatos (2012a), and estimate thestrain amplitude associated with the gravitational waves. We proposed a novel mechanism for the unpinning ofsuperfluid vortices in the inner crust of a pulsar. It occursthrough the scattering of excited neutrons with the vortexcore neutrons. The excitation of neutrons are caused by the c (cid:13)000
01 s in the crustquake model, we see that theglitch produced through vortex unpinning lags behindthe glitch produced due to crustquake. The conclusion isof course within the uncertainty in the estimation of t g and ∆ t . This distinguishing feature should be reflectedin the pulse profile for glitches, provided the subsecondresolution in pulsar timing is achieved. As of now, thebest resolved time observed for spin-up of Vela pulsar hasbeen reported (McCulloch et al. 1990) to be ≈ . vortices may act as atrigger mechanism to unpin the nearby vortices, which lie inthe equatorial plane. Among various suggestions on vortexavalanches (Melatos et al. 2008; Warszawski et al. 2012;Warszawski & Melatos 2012b; Akbal & Alpar 2018), we findthat the knock-on pictures (Melatos et al. 2008; Warszawski& Melatos 2012b) fit quite well in our model. In proximityknock-on, presence of the azimuthal component of thevortex velocity can make an individual vortex knock-on theother vortices present in the equatorial plane and nearbythe cubical volume V p . Note that for completely outwardmotion, the vortices should not encounter vortices along itstrajectory. In our future work, we would like to determinethe trajectory of unpinned vortices, estimate the numberof vortices that can be released through this process. Theacoustic knock-on caused by repinning of vortices can alsobe a viable process in our model. In future, we would like toimplement these mechanisms to study the avalanche process.Other than glitches, another interesting phenomenon ofinhomogeneous vortex line movement could be the genera-tions of gravitational waves from an isolated pulsar (Jones2002; Bagchi et al. 2015; Layek & Yadav 2020). We wouldlike to explore this possibility in future following the ap-proach of Warszawski & Melatos (2012a), and estimate thestrain amplitude associated with the gravitational waves. We proposed a novel mechanism for the unpinning ofsuperfluid vortices in the inner crust of a pulsar. It occursthrough the scattering of excited neutrons with the vortexcore neutrons. The excitation of neutrons are caused by the c (cid:13)000 , 000–000 Layek, Yadav absorption of strain energy released due to the crustquakeevent. We take a cubical shape region near the mostprobable quake site around the star’s equatorial plane anddetermine the volume ( ∼ m ) where a fraction ( ∆ f E f )of bulk superfluid neutrons are excited. The scatteringof these exited neutrons with the vortex core neutronsresults in the unpinning of vortices from the above volume.The Crab and Vela pulsar with Ω (cid:39) s − can releaseabout 10 vortices as a result of local unpinning. Thesize of the glitches have been estimated to lie in the range ∼ − − − , and ∼ − − − for Crab and Velapulsar, respectively. The glitches, though vary in size, havethe same frequency of occurrence of about once in a year.We estimated the relaxation time scale of excitedneutrons through neutron-vortex scattering and the valueof τ nn (cid:39) − s justifies the absence of multiple glitcheswithin the time interval of a few days or months. The glitchrise time t c ∼ (0 . −
2) s in our model also turns out tobe consistent with the typical feature of the glitch profile(sudden spin-up event). At the same time, this commonfeature (i.e., small glitch rise time) of all crustquake initi-ated glitch models makes it difficult to choose one amongvarious models (Ruderman 1991; Link & Epstein 1996;Akbal & Alpar 2018).The model for unpinning proposed here has the poten-tial to explore further by implementing the knock-on pictureto study vortex avalanches. Also, sudden release of a largenumber of vortices can have consequences on the emissionof gravitational radiation. We would like to explore these inour future work. Finally, though we have discussed the ex-citation of neutron superfluid initiated by crustquake, thisproposal is very generic. The approach used here should beapplicable for any other sources, which have the potentialto excite the superfluid neutrons, or can make superfluid -normal phase transition in the inner crust of a pulsar. It willbe interesting to look for such sources.
We would like to thank Partha Bagchi and Arpan Das foruseful discussions. We thank the anonymous reviewer forcritical comments and constructive suggestions on the pre-vious version of this manuscript.
No new data were generated or analysed in support of thisresearch.
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