Nuclear symmetry energy and Neutron star cooling

Transcription

1 Nuclear symmetry energy and Neutron star cooling Yeunhwan Lim 1 1 Daegu University. July 26, 2013 In Collaboration with J.M. Lattimer (SBU), C.H. Hyun (Daegu), C-H Lee (PNU), and T-S Park (SKKU) NuSYM13 1 / 29

2 Table of contents Nuclear Symmetry energy Nuclear Force Models Nuclei in Neutron star Crust Thermal evolution of neutron stars Nuclear equation of state Neutron star cooling Nuclear superfluidity Conclusion 2 / 29

3 Nuclear Symmetry energy Large uncertainties in nuclear physics at high density (ρ > ρ 0 ) Energy per baryon e(ρ, x) = e(ρ, 1/2)+S 2 (ρ)(1 2x) 2 + The symmetry energy parameter S v = S 2 (ρ 0 ), L = 3ρ 0 (ds 2 /dρ) ρ0, K sym = 9ρ 2 0 (d2 S 2 /dρ 2 ) ρ0, Q sym = 27ρ 3 0 (d3 S 2 /dρ 3 ) ρ0 The density dependence of the symmetry energy in nuclear astrophysics - The neutronization of matter in core-collapse supernovae - The radii and crust thinckness of neutron stars - The cooling rate of neutron stars - The r-process nucleosynthesis 3 / 29

4 We can estimate the range of symmetry energy from experiments and theories. Nuclear Mass fitting Liquid droplet model, Microscopic nuclear force model (Skyrme force model) Neutron Skin Thickness 208 Pb, Sn with RMF and Skyrme Dipole Polarizabilities Heavy Ion Collisions Neutron Matter Theory Quantum Monte-Calro, Chiral Lagrangian Astrophysical phenomenon Neutron stars mass and radius 4 / 29

5 The optimal points for S v and L S v and L from nuclear mass constraints Figure: The allowed region of Sv, L from experiment and observation from J.M.Lattimer and Y. Lim, ApJ. 771, 51 (2013) 5 / 29

6 Nuclear Force Model - Potential Model Skyrme Force Model (EDF) - Non relativistic potential model - Two body and many body effects (contact force) v ij =t 0 (1+x 0 P σ)δ(r i r j )+ 1 2 t 1(1+x 1 P 1 [ σ) p 2 2 ij δ(r i r j )+δ(r i r j )p 2 ] ij + t 2 (1 + x 2 P σ) 1 2 p ij δ(r i r j )p ij t 3(1+x 3 P σ)ρ(r) ǫ δ(r i r j ) (1) + i 2 W 0p ij δ(r i r j )(σ i +σ j ) p ij, - 10 parameters can be fitted with nuclear mass data 6 / 29

7 Uniform nuclear matter (x 0, x 1, x 2 = 1, x 3, t 0, t 1, t 2, t 3,ǫ) (m s, m n, B,ρ 0, K, S v, L,ω 0 ) Binding energy : B = 16 MeV Saturation density : n 0 = 0.16 fm 3 Incompressibility : K = MeV Effective mass of nucleons in symmetric nuclear matter : ms = 0.7M Effective mass of nucleons in pure nuclear matter : mn = 0.6M Symmetry energy : S v = MeV Density derivative of symmetry energy : L = MeV Nuclear surface tension or t : ω 0 = 1.15MeV/fm 2, t = fm. 7 / 29

8 ( ) E A = Cn2/3 (1+βn 0 0 )+ 3t 0 8 n 0 + t 2/ nǫ+1, C = 3 2 3π 2 (2) 0 10M 2 Ms M = 1, β = M 1+βn 0 P(n 0 ) 8 2(3t 1 + t 2 ) Mn M = 1, θ = M 1+θn 0 4 2(t 1 t 1 x 1 ) (3) ) t 0n 0 + t 3 16 (ǫ+1)nǫ+1 (4) 0 = 0 = 2 ( n 0 3 Cn βn 0 K = 9n 2 2 E/A n 2 = 2Cn 2/ Cβn 5/ t 3 n=n0 16 ǫ(ǫ+1)nǫ+1 0 (5) S v = n2 2 E/A 2 α 2 = 5 α=0 9 Cn2/3 0 5CM 36 2(t 2 + 3t 1 x 1 )n 5/3 0 t ( ) x 3 n ǫ+1 0 (6) )n 0 L v = 3n 0 S v n = 10 n=n0 9 Cn2/3 25CM (t 2 + 3t 1 x 1 )n 5/3 0 t ( ) (1+ǫ) 2 + x 3 n ǫ+1 3t ω 0 = n 0 T 0 t I ω/i t, t = t 0 4 ( x 0 0 ( 1 )n (7) 2 + x 0 (Q nn + Q np)n 0 2T 0 I t. (8) 8 / 29

9 Nuclei in Neutron Star outer crust LDM, TF, and HF for NS outer crust - Easy to deal with because of no neutrons outside nuclei Electron contribution and Lattice structure (BCC) Up to ρ /fm 3 (µ n < 0) Beta equilibrium state for LDM and TF, semi β for HF Atomic and mass number are different Pressure and energy density are almost same 9 / 29

10 SLy4 EOS SLy4 + LDM SLy4 + TF SLy4 + HF Z e-10 1e-09 1e-08 1e-07 1e-06 1e ρ (fm -3 ) Figure: Atomic number of nuclei in the outer crust of neutron star from SLy4 + LDM, TF, and HF 10 / 29

11 e-05 SLy4 EOS SLy4 + LDM SLy4 + TF SLy4 + HF 1e-06 P (MeV/fm 3 ) 1e-07 1e-08 1e-09 1e-10 1e-11 1e-12 1e-13 1e-10 1e-09 1e-08 1e-07 1e-06 1e ρ (fm -3 ) Figure: Pressure in the outer crust of neutron star from SLy4 + LDM,, LDM, and HF 11 / 29

12 1 0.1 SLy4 EOS SLy4 + LDM SLy4 + TF SLy4 + HF 0.01 e (MeV/fm 3 ) e-05 1e-06 1e-07 1e-08 1e-10 1e-09 1e-08 1e-07 1e-06 1e ρ (fm -3 ) Figure: Energy density in the outer crust of neutron star from SLy4 + LDM,, LDM, and HF 12 / 29

13 Nuclei in Neutron Star inner crust LDM, TF, and HF for NS inner crust Electron and neutron contribution and Lattice structure (BCC) Up to ρ 0.5ρ 0 (µ n > 0) Beta equilibrium state for LDM and TF, semi β for HF Nuclear pasta phase might exists - Coulomb energy favors geometry - easy to deal with in LDM, difficult in TF and HF where f surf = udσ r N, f Coul = 4π 5 (ρ i x or N e) 2 D d (u) (9) [ 5 D d (u) = d 2 1 d 4 2 u1 2/d + d 2 ] u. 2 (10) ( d 2 ) 1/3 f surf + f Coul = βd, D = u D d (u) 9 (11) 13 / 29

14 Nuclear density profile at the inner crust ρ = 2.0x10-2 /fm 3 SLy4 + HF, neutron, N=476 SLy4 + HF, proton, Z=20 SLy4 + TF, neutron, N= SLy4 + TF, proton, Z= SLy4 + LDM, neutron N= SLy4 + LDM, proton Z=34.57 ρ (fm -3 ) r (fm) Figure: Nuclear density profile in the inner crust of neutron star at ρ = /fm / 29

15 Crust thickness Transition density from energy difference Crust thickness L (MeV) Sv (MeV) Figure: Contour plot of crust thickness of 1.4M for S v and L 15 / 29

16 Crust thickness Transition density from thermodynamic instability Crust thickness L (MeV) Sv (MeV) Figure: Contour plot of crust thickness of 1.4M for S v and L 16 / 29

17 The relativistic equations of thermal evolution Diffusion equation 1 4πr 2 e 2Φ 1 2Gm c 2 r r (e2φ L r) = Q ν Cv T e Φ t L r 4πr 2 = κ 1 2Gm c 2 r e Φ r (TeΦ ) Q ν : Neutrino emission rate : Q ν = Q ν(t,ρ n,ρ p) C v : Heat capacity (specific heat) : C v = C v(t,ρ n,ρ p) κ : Thermal conductivity : κ = κ(t,ρ n,ρ p) e Φ : General relativistic metric function : e Φ = L (T ) is defined on even (odd) grid : L 2i, T 2i+1 Two boundary conditions : L 0 = 0, T s = T s(t b ) Henyey method is used to find new temperature 1 2GM rc 2 (12a) (12b) T n+1 i = T n i + t dt i n dt T n+1 i n+1 = Ti n dti + t dt 17 / 29

18 Equation of state Lots of nuclear force models are available EOS based on RMF : TM1, FSU Gold, NL3,... EOS based on variational principle : APR EOS based on EDF : SLy4 (0,, 10) Phenomenological model : PAL EOS should explain the maximum mass of neutron stars greater than 2.02M ρ 0 (fm 3 ) B (MeV) K (MeV) S v (MeV) L (MeV) SLy APR TMA TM NL EDF EDF EDF EDF EDF / 29

19 Neutron stars mass and radii M-R relation M-R relation SLy4 APR TM1 TMA NL3 Steiner et al PSR J S v = MeV, L = MeV S v = 29.0 MeV, L = 40.5 MeV S v = MeV, L = 30.0 MeV S v = 32.7 MeV, L = 61.9 MeV S v = 29.0 MeV, L = 61.9 MeV Steiner et al PSR J M ( M O ) 1.5 M ( M O ) R ( km ) R ( km ) Figure: Mass and radius relation for neutron stars 19 / 29

20 Physics from EOS Nuclear matter in the core of neutron stars Easy to calculate (Relativistic, Non-relativistic) Three approaches for crust : Liquid droplet, Thomas Fermi, Hartree-Fock Among the many EOSs, APR, SLy series are probably the best Composition of constituents : Protons, neutrons, electrons On and off direct URCA process (proton fraction) Boundaries of inner crust and out crust, (neutron drip) Boundaries of URCA process Atomic number in case of crust Need to calculate Q ν, C v, and κ Effective masses for proton and neutron Effective masses are involved in formulae for Q ν, C v, and κ Volume fraction of heavy nuclei 20 / 29

21 Cas A Neutron Star Casiopea A supernova remnant First light : Chandra observations (1999) Distance : 3.4 kpc Age : 330± 10 years 21 / 29

22 Temperature profile of CAS A neutron star Cas A neutron star T s and best fit Cas A Best Fit 6.18 T s (K) Log(t/year) d ln T s d ln t = (best fit) vs (standard cooling) 22 / 29

23 NS Cooling (Standard) Standard cooling : DU, MU, NB, No superfluidity, No bose condensation Neutron star cooling curve EDF Standard Cooling APR SLy4 TM1 TMA PAL Cas A S v =32.36 MeV, L=42.23 MeV, 1.4 M S v =29.0 MeV, L=30 MeV, 1.4 M S v =32.7 MeV, L=30 MeV, 1.4 M S v =32.7 MeV, L=61.9 MeV, 1.4 M S v =29.0 MeV, L=61.9 MeV, 1.4 M T s (K) 6 T s (K) Log(t/year) Log(t/year) Direct URCA process turned on in case of TM1, TMA and PAL Standard cooling process cannot explain the fast cooling in Cas A S v and L effect on thermal relaxation time 23 / 29

24 Standard cooling with EDF EDF Standard Cooling S v =32.36 MeV, L=40 MeV, 1.4 M L=50 MeV, 1.4 M L=60 MeV, 1.4 M L=70 MeV, 1.4 M L=80 MeV, 1.4 M L=90 MeV, 1.4 M L=100 MeV, 1.4 M T s (K) Log(t/year) 24 / 29

25 Nuclear Superfluidity Nuclear ground state can be achived with Cooper pairing ex) Even-Even nuclei is more stable than even-odd, odd-odd Nuclear superfluidity is uncertain at high densities (ρ ρ 0 ). 3 P 2 N, 1 S 0 P for higher densities (Neutron star core) 1 S 0 N for lower densities (Neutron star crust) Nuclear superfluidity gives reduction factor for neutrino emission, heat capacity, and thermal conductivity. Q DU Q DU R DU, Q Mod,U Q DU R Mod,U, Q B Q B R B C n,p C n,pr n,p, C crust n Cn crust R n, κ core n κ core n R κ,n It opens neutrino emission process from Cooper Pair Breaking and Formation (PBF). Q PBF = m n p F (n n) m n m T 9 7 NνR(T/Tc,n) erg cm 3 s 1 nc (cf. Q mod,n = ( m n m n ) 3 ( m )( p np m p n 0 ) 1/3 T 8 9 αnβnr mod,nerg cm 3 s 1 ) 25 / 29

26 NS Cooling (Enhanced) Enhanced cooling : Pair Breaking and Formation (PBF) NS ( 1.4M O ) Cooling Curve with APR EOS and PBF Scale factor, s=0 s=0.5 s=1.0 s=1.5 s=2.0 SLy4 1.6M DH 1.5M Cas A obs. 6 T s APR 1.4M s=1.0 SLy4 1.6M DH 1.5M Log(t/year) Observation of Cas A NS indicates the existence of nuclear superfluidity Different EOSs (APR, SLy4) may give same slope but different critical temperature for nuclear matter EOSs good for Cas A cooling are passing SLB criteria area. 26 / 29

27 NS Cooling (Enhanced) Enhanced cooling with EDF EDF Enhanced Cooling L=40 MeV, 1.4 M L=50 MeV, 1.4 M L=60 MeV, 1.4 M L=70 MeV, 1.4 M L=80 MeV, 1.4 M L=90 MeV, 1.4 M L=100 MeV, 1.4 M T s (K) Large L s give low T c s Log(t/year) L = 40 MeV L = 50 MeV L = 60 MeV T c ( 3 P 2 ) K K K 27 / 29

28 Critical Temperature Fast cooling in Cas A NS can be explained with nuclear superfluidity and PBF T c for 3P 2 is K. The range of S v and L might be confirmed from Cas A cooling rate. 6e+09 5e+09 Critical Temperatures of 3P2 N, 1S0 P, 1S0 N T c 3P2 N T c 1S0 P T c 1S0 N 4e+09 T c (K) 3e+09 2e+09 1e e-06 1e ρ (fm -3 ) 28 / 29

29 Conclusion It s not likely that DURCA happened in CAS A NS mass of CAS A NS > 1.4M L < 70 MeV Superfluidity gives the critical temperature of 3 P 2 Need to use flexible RMF (eg, FSU-GOLD) model to compare with EDF 29 / 29