Superfluid Density of Neutrons in the Inner Crust of Neutron Stars:

Transcription

1 PACIFIC 2018 (Feb. 14, 2018) Superfluid Density of Neutrons in the Inner Crust of Neutron Stars: New Life for Pulsar Glitch Models GW & C. J. Pethick, PRL 119, (2017). Gentaro Watanabe (Zhejiang Univ.) 1

2 Contents I. Matter in neutron star crusts II. Pulsar glitches & glitch models based on superfluidity III. Superfluid density of neutrons in the inner crust of neutron stars IV. Summary & conclusion 2

3 Preliminaries Neutron stars (NS) are extremely dense & neutron rich. rich High density: 1-2 Msolar inside a sphere of 10km rad. 1 Msolar ~ 1030 kg ~ 105 Mearth Density goes beyond ρ0 (normal nuclear density) Neutron higher densities: Neutronization escape stable nuclei on earth neutron star matter (NSM) x ~ 0.5 x < 0.1 x: proton fraction NSs: NSs:gigantic giganticneutron-rich neutron-richnuclei. nuclei. 3

4 Structure of neutron stars Solid state of n-rich heavy nuclei Neutronization: pn + e- nn + νe (by Dany Page)

5 Structure of neutron star matter low density neutron density proton density n-rich nucleus dripped neutrons Oyamatsu (1993) (gas phase) high density Negele & Vautherin, NPA 207, 298 (1973) 5

6 Importance of NS crusts Neutron star crust Thickness: ~ 1 km (cf. radius of NS: ~10km) Total mass: ~ O(0.01) Msolar (cf. mass of NS: ~1.4Msolar) Negligible? Depending on phenomena. Outer parts have direct consequences on observations. NS cooling Torsional oscillations of crusts Pulsar glitches 6

7 Pulsar glitches & glitch models based on superfluidity New Scientist 7

8 Pulsar glitches Magnetic dipole model μ : magnetic mom. α : angle btwn. B & Ω I : mom. of inertia Radhakrishnan & Manchester, Nature (1969) Glitches: Sudden (but small) increase of pulsar spinning rate Followed by a gradual relaxation of the spinning rate. 8

9 Glitch model based on n-superfluidity 2 fluids in NS crust dripped neutrons: superfluid nuclei (form a lattice): quantized vortices pinning center of vortices normal fluid charged coupled to a EM field decelerated by pulsar emission Relative velocity btwn n-superfluid & lattice of nuclei increases. Pinning can sustain n-sf is a large ang. mom. reservoir. 9

10 Glitch model based on n-superfluidity Magnus force Collective unpinning Vortex density nv ang. mom. transfer 10

11 Crisis of the glitch models Chamel, PRC 85, (2012) Band calculation without pairing. (HF with nuclear interaction of Skyrme type) Good approx. if the effect of pairing is weak. Superfluid density: (inverse of m*) energy density Reduction by a factor of 10! mom. / particle of bulk flow Insufficient superfluid density to explain glitches! Mom. of inertial of n-superfluid is too small. Andersson et al., PRL (2012); Chamel, PRL (2013); Delsate et al., PRD (2016). 11

12 Superfluid density Landau's two-fluid description of supefluids: normal component: superfluid component: Superfluid density ns : Density which contributes to SF flow. Response of current to the phase twist of SF order parameter. energy density mom. / particle of bulk flow 12

13 Band calculation by Chamel Interacting particles Free particles Chamel, PRC (2012) 13

14 Band calculation by Chamel Interacting particles Free particles Chamel, PRC (2012) dripped n / nucleus is large! 14

15 SF density of neutrons in the inner crust of NSs: New life for pulsar glitch models GW & Pethick, PRL 119, (2017). 15

16 Difficulty of the problem ξbcs ~ R (nuclear radius) Hydrodyn. theory is invalid. Martin & Urban, PRC (2016) Need to look at the band structure in detail. # of neutrons / nucleus >> 1 neutrons occupy ~500 bands Chamel, PRC (2012) 16

17 Take-home messages Both gaps are important! (band gap) vs (pairing gap) Pairing drastically reduces the effects of band gap when Δ (lattice potential height) matters even though Superfluid density may be large enough to account for glithces. New life for glitch models! GW & Pethick, PRL 119, (2017). 17

18 Poor man's analysis Scattering of quasiparticles by spin-indep. pot.: fermion quasiparticle coh. factor On the Fermi surface (k = [2mμ]1/2), uk = vk = 1/2. No net scattering on Fermi surface. Int. potential for particles and holes are equal and opposite. 18

19 Simple analysis by 2-band model particle (p) hole (p) particle (p-k) hole (p-k) E p : quasimom. of a quasiparticle (in units of pf) p K : reciprocal lattice vector (in units of pf) V : strength of the lattice pot. (in units of 2EF) Δ : pairing gap (in units of 2EF) Nested case: K = 2 Eigenvalues: (doubly degenerate) Pairing effect is important even if 19

20 Bogoliubov-de Gennes approach Basic equation (BdG eq.): Q : quasimom. per particle of superflow 1D sinusoidal pot. Calculate 20

21 Effects of the pairing gap (1) Pairing gap increasing. s Reduction Reduction of of nns due due to to band band gap gap isis suppressed suppressed by by paring paring gap. gap. 21

22 Effects of the pairing gap (2) Suppression of band structure by pairing Approximate fit: suppression factor 22

23 Application to NS crusts Obstacles: Lattice pot. in NS crusts has many Fourier components. Neutrons occupy ~ 500 bands. 3D lattice: average over the orientation of lattice is needed. Direct BdG approach is formidable. Take a shortcut! 23

24 Superfluid density in NS crusts Assumption: pairs of RLVs {Ki, -Ki} contribute to ns independently. longitudinal transverse effect of lattice pot. K-dep. effect of pairing gap approximate fit form factor of lattice pot. (VK -dep.) 24

25 Superfluid density in NS crusts Focus on the case where the reduction of ns is largest. In Chamel (2012): Avr. density n = 0.03 fm-3 No pairing limit (Δ = 0): (cf. Chamel's result ~ 0.1) Δ = 1MeV: Δ = 1.5MeV: Only 29% reduction! Superfluid density is large enough. Glitch models based on superfluidity are still tenable! 25

26 Summary & conclusion Study of neutron superfluid density in neutron star crusts Both pairing gap and band gap are imporant. matters rather than Effects of the band gap is suppressed in NS crusts. No pairing Pairing included Pulsar glitch models get new life! GW & Pethick, PRL 119, (2017). 26

27 27

28 Band calculation by Chamel Interacting particles Free particles Chamel, PRC (2012) 28

29 Application to NS crusts Focus on the case where the reduction of ns is largest. In Chamel (2012) Avr. density n = 0.03 fm-3 neutron Fermi energy EFo = 16.4 MeV ns / nno ~ 0.1 Free-space n-n int. gives Effect of induced int.: reduction by a factor of 2-3 [Gor'kov & Melik-Barkhudarov, JETP (1961)] 29

30 K and VK dependence in normal limit Normal limit: Δ = 0; for sinusoidal pot. Little effect of lattice for K > 2kF Scattering with mom. transfer > 2kF is kinematically forbiden. Almost linear wrt K/2kF Approximate fit: 30

31 Form factor of lattice pot. in NS crusts Fourier transform of MF pot. in Chamel's calculation. Reciprocal lattice vectors (RLVs) bcc lattice fcc in reciprocal space Min: 2nd: 3rd: etc. (12 RLVs) etc. (6 RLVs) etc. (24 RLVs) VK decreases rapidly with K. 31

32 Superfluid density in NS crusts (1) Assumption: pairs of RLVs {Ki, -Ki} contribute to ns independently. longitudinal transverse avr. over orientation for cubic symmetry contribution from many RLVs in crusts Only one of Ki & -Ki is included. sum integral density of nuclei 32

33 Comparison btwn n-matter & cold atomic gases Neutron matter in NS crust Particle separation Scattering length ~ 1 fm Cold Fermi gas at unitarity ~ 100 nm fm ~ 19 >> 1 very large! Temperature ~ 100 kev ~ 100 nk Degeneracy temp. ~ 100 MeV ~ 1 μk ~ 10-3 ~

34 BdG equations BEC Fermi superfluids Gross-Pitaevskii eq. Bogoliubov-de Gennes (BdG) eq. energy of st. with k BdG eq. Amp. of k & -k are empty Amp. of k & -k are occupied Number of particles: Pairing gap: Δ(r) depends on uk & vk Nonlinearity! Nonlinearity! 34

35 Standard regularization Take contact potential: Replace g by the low energy limit of T-matrix. 35