Neutron-star properties with unified equations of state

Transcription

1 Neutron-star properties with unified equations of state Nicolas Chamel in collaboration with J. M. Pearson, S. Goriely, A. F. Fantina Institute of Astronomy and Astrophysics Université Libre de Bruxelles, Belgium ECT* - July 11, 2013

2 Punch line The interior of a neutron star contains very different phases of matter. A unified and consistent description is very challenging but of utmost importance for hydrodynamical simulations. Chamel&Haensel, Living Reviews in Relativity 11 (2008), 10

3 Outline 1 Brussels-Montreal nuclear energy density functionals 2 Unified equations of state of neutron stars 3 Superfluidity in neutron-star crusts

4 Brussels-Montreal nuclear energy density functionals

5 Brussels-Montreal Skyrme functionals in a nut shell Fitting strategy Fit to both experimental data (all atomic masses with Z, N 8 from AME) and N-body calculations using realistic forces. For this purpose, it was necessary to generalize the standard Skyrme functional. Main features of the latest functionals: generalized pairing functionals fitted to realistic 1 S 0 pairing gaps in symmetric and neutron matter (BSk16-17) generalized Skyrme to remove spurious instabilities in infinite homogeneous matter (BSk18) different degrees of neutron-matter stiffness and effective mass splitting (BSk19-21) fit of the 2012 AME with different symmetry energy coefficients (BSk22-26)

6 Neutron-matter equation of state at high densities The functionals BSk19, BSk20 and BSk21 were fitted to realistic neutron-matter equations of state with different of degrees of stiffness: Goriely, Chamel, Pearson, Phys. Rev. C 82, (2010).

7 Neutron-matter equation of state at low densities All three functionals yield similar equations of state at subsaturation densities consistent with microscopic calculations using realistic NN interactions:

8 Nuclear-matter equation of state Our functionals are also in good agreement with BHF calculations not only in neutron matter but also in symmetric nuclear matter (not fitted).

9 Effective masses Our functionals yield a qualitatively correct splitting of effective masses (M n > M p in neutron-rich matter) in agreement with giant resonances and many-body calculations using realistic forces. M M q = 2ρ q ρ M M s ( + 1 2ρ q ρ ) M M v BSk19 BSk20 BSk21 EBHF Ms/M Mv/M BSk21 is in good quantitative agreement with Extended Brueckner Hartree-Fock (EBHF) calculation from Cao et al.,phys.rev.c73,014313(2006).

10 HFB-19,HFB-20 and HFB-21 mass tables All three mass models fit the 2149 measured masses with Z, N 8 from the 2003 AME with a rms error of 0.58 MeV. How well do these models extrapolate? data set model ǫ(m) [MeV] σ(m) [MeV] 2011 AME new HFB (154) HFB HFB Chamel, Fantina, Pearson, Goriely, Phys.Rev.C84,062802(R)(2011).

11 Latest Brussels-Montreal Skyrme functionals Our latest functionals were fitted to the 2012 AME with different symmetry energy coefficients: HFB-22 HFB-23 HFB-24 HFB-25 HFB-26 J [MeV] NeuM BHF BHF BHF BHF APR σ(m) [MeV] ǫ(m) [MeV] σ(m nr ) [MeV] ǫ(m nr ) [MeV] Mv/M Goriely, Chamel, Pearson, to appear in Phys.Rev.C (2013).

12 Unified equations of state of neutron stars

13 Description of neutron star crust below neutron drip Cold catalyzed matter hypothesis The interior of a neutron is supposed to be in full thermodynamic equilibrium at zero temperature. We have determined the equilibrium structure of the outer crust of a neutron star for ρ 10 4 g cm 3 using the BPS model: fully ionized atoms arranged in a bcc lattice uniform relativistic electron gas. The only microscopic inputs are nuclear masses. We have made use of the experimental data from the 2011 Atomic Mass Evaluation complemented with our HFB mass tables. Pearson,Goriely,Chamel,Phys.Rev.C83,065810(2011).

14 Composition of the outer crust of a neutron star Sequence of equilibrium nuclides with increasing depth: HFB-19 HFB Fe 56 Fe 62 Ni 62 Ni 64 Ni 64 Ni 66 Ni 66 Ni 86 Kr 86 Kr 84 Se 84 Se 82 Ge 82 Ge 80 Zn 80 Zn - 79 Cu 78 Ni 78 Ni 80 Ni 80 Ni 126 Ru Mo 124 Mo Zr 122 Zr 124 Zr Y 120 Sr 120 Sr 122 Sr 122 Sr 124 Sr 124 Sr 126 Sr - The first 8 nuclides are determined by experimental masses. Predominance of N 50 and 82 nuclei. Deeper (below 200 m for a 1.4M neutron star with a 10 km radius) the composition is more model-dependent. Measurements of neutron-rich nuclei are crucially needed. Pearson,Goriely,Chamel,Phys.Rev.C83, Wolf et al.,prl110,

15 Description of neutron star crust beyond neutron drip We have determined the equilibrium structure of the inner crust of a neutron star for ρ g cm 3 using the Extended Thomas-Fermi+Strutinsky Integral method (ETFSI): spherical neutron-proton clusters coexisting with a neutron liquid using parametrized density distributions (Wigner-Seitz approximation for the Coulomb energy) uniform relativistic electron gas. Pearson,Chamel,Goriely,Ducoin,Phys.Rev.C85,065803(2012). Advantages of ETFSI method very fast approximation to the full Hartree-Fock method avoids the difficulties related to boundary conditions but include proton shell effects (neutron shell effects are much smaller and are therefore omitted) Chamel et al.,phys.rev.c75(2007),

16 Structure of the inner crust of a neutron star (I) nucleon distributions With increasing density, the clusters keep essentially the same size but become more and more dilute: Crust-core transition properties n cc (fm 3 ) P cc (MeV fm 3 ) BSk BSk BSk SLy BSk21 n = n = n = n = n = n = r [fm] n n (r), n p (r) [fm -3 ] The crust-core transition is very smooth: the crust dissolves continuously into a uniform mixture of nucleons and electrons.

17 Structure of the inner crust of a neutron star (II) composition With SLy4, BSk19, BSk20 and BSk21, only Z = 40 is found. Comparison of inner- and outer-crust codes at drip point; results for latter code in parentheses. e is the internal energy per nucleon, and P the pressure. n drip (fm 3 ) Z N e (MeV) P (10 4 MeV fm 3 ) BSk (38) 96 (88) (-1.87) 5.1 (4.9) BSk (38) 95 (88) (-1.87) 5.1 (4.9) BSk (38) 94 (86) (-1.90) 5.0 (4.9) SLy (38) 93 (82) (-1.96) 4.7 (4.8) The few % discrepancies can be attributed to the neglect of pairing, the neglect of neutron shell effects, the parametrized density distributions, the neglect of rotational and vibrational corrections.

18 Structure of the inner crust of a neutron star (II) composition The ordinary nuclear shell structure seems to be preserved apart from Z = 40 (quenched spin-orbit?). The energy differences between different configurations become very small as the density increases: SLy4 n= fm -3 n=0.06 fm Z BSk19 n= fm n=0.06 fm Z In a real neutron star, a large range of values of Z can be expected due to thermal effects.

19 Structure of the inner crust of a neutron star (III) composition Impact of proton pairing (BCS approximation) - preliminary results with BSk n= fm -3 n=0.055 fm -3 with pairing without pairing Z Even though pairing smoothes out shell effects, it does not change appreciably the composition.

20 Unified equations of state of neutron stars The same functionals used in the crust can be also used to compute the equation of state of the core, assuming it is made of n,p e and µ. 2.5 M [Solar masses] BSk19 BSk20 BSk21 PSR J PSR J ad R [km] M max/m BSk (1.84) BSk (2.20) BSk (2.3) Masses for the microscopic EoSs are indicated in parenthesis. Mass measurements of PSR J and more recently J rule out the softest of our EoS. Chamel et al.,phys.rev.c84,062802(r)(2011).

21 Unified equation of state of hybrid stars However, these observations do not necessarily rule out the BSk19 functional itself. M (M sun ) BSk19 BSk20 BSk21 PSR J PSR J ad R (km) Assuming a phase transition occurs at a baryon density above 0.2 fm 3 provided it lowers the Gibbs free energy for a given pressure: M max/m BSk BSk BSk Massive neutron stars could still be compatible with the BSk19 functional if their core contains exotic matter with a stiff enough EoS. Chamel et al.,a& A553, A22 (2013)

22 Neutron-star composition and direct Urca Depending on their composition, neutron stars may cool very rapidly via the direct Urca process. 0.5 Y p SLy4 BSk19 BSk21 BSk20 direct Urca n [fm -3 ] The direct Urca process n p + e + ν e and p + e n+ν e is allowed if the proton fraction Y p exceeds a critical threshold. The low luminosity from CTA1, SAX J , 1H and several young supernova remnants suggest that the direct Urca has occurred. This would rule out the SLy4, BSk19 and BSk20 EoSs. Chamel et al.,phys.rev.c84,062802(r)(2011).

23 Composition and symmetry energy The composition is mainly determined by the symmetry energy : consider n,p,e matter neglect any finize-size effects and assume e(n,η) e 0 (n)+s(n)η 2 with η = 1 2Y p assume electric charge neutrality n e = n p (no muons) assume β equilibrium µ e = µ n µ p µ e (n)y 1/3 p 4S(n)(1 2Y p ) Remark: depending on its definition, the symmetry energy may differ by a few MeV at saturation and much more at higher densities.

24 Composition and symmetry energy Full calculations of the neutron-star composition for BSk22 (J = 32, BHF), BSk24 (J = 30, BHF) and BSk26 (J = 30, APR): S 2 (n) [MeV] BSk22 BSk24 BSk n [fm -3 ] Y p BSk22 BSk24 BSk26 outer crust inner crust core n [fm -3 ] Symmetry energy defined here as S(n) = e(n,η = 1) e(n,η = 0). Same hierarchy between the forces if S(n) defined as (1/2)( 2 e/ η 2 ) η=0.

25 Astrophysical constraints on the symmetry energy Direct Urca constraint: it should not accur in stars with mass M = 1 1.5M but should be allowed in massive stars Klähn et al.,phys.rev.c74, Critical masses for direct Urca: Mass-radius constraint: Observations of 3 X-ray bursters and 3 transient low-mass X-ray binaries in globular clusters Steiner et al., ApJ.722, M durca/m BSk BSk BSk26 - M [M sun ] BSk22 BSk24 BSk R [km] The BSk22 EoS (J = 32) seems to be ruled out by observations.

26 Superfluidity of neutron-star crusts

27 Superfluidity in neutron-star crusts with the Wigner-Seitz approximation Pairing properties have been already studied using the HFB method and the Wigner-Seitz (W-S) approximation. However, this approach is not well suited for treating the deep region of the crust Chamel et al., Phys.Rev.C75(2007) k F =1.1 fm -1 n, MeV Z=26 Z=20 Spurious shell effects 1/R 2 can be very large at the crust bottom and are enhanced by the self-consistency. Baldo et al., Eur.Phys.J. A 32, 97(2007) r, fm

28 Nuclear band theory These limitations can be overcome by using the band theory of solids Chamel, Nucl.Phys.A747(2005)109. Floquet-Bloch theorem: ϕ αk (r) = e i k r u αk (r) u αk (r +l) = u αk (r) for any lattice translation vector l. α rotational symmetry around lattice sites (clusters) k translational symmetry (unbound neutrons)

29 Example of neutron band structure Body-centered cubic crystal of zirconium like clusters with N = 160 (70 unbound) and ρ = g.cm 3 W-S approximation band theory E (MeV) p 1/2 2g 9/2,2g 7/2 4p 2h 3/2,4s 11/2,2h 1/2 9/2 3d 5/2,3d 3/2 3p 1/2 2f 7/2,2f 5/2 Energy (MeV) 0,6 0,5 0,4 0,3 0,2 0,1 0 3p 3/2 Chamel et al, Phys.Rev.C75 (2007), N Σ Γ Λ P F H G N

30 Pairing in the deep region of the inner crust The HFB equations are approximated by the anisotropic multi-band BCS equations: αk = 1 2 β k v pair βk αkα kβk k β k k E k βk k k tanh Eβk k 2k B T v pair = αkα kβk k β k k d 3 r v π [ρ n (r),ρ p (r)] ϕ αk (r) 2 ϕ βk k (r) 2 E αk = (ε αk µ) αk ε αk, µ and ϕ αk (r) are obtained from band structure calculations using the ETFSI mean fields Chamel et al., Phys.Rev.C81, (2010).

31 Analogy with terrestrial multi-band superconductors Multi-band superconductors were first studied by Suhl et al. in 1959 but clear evidence were found only in 2001 with the discovery of MgB 2 (two-band superconductor) In neutron-star crusts, the number of bands can be huge up to a thousand! both intra- and inter-band couplings must be taken into account

32 Neutron pairing gaps Results obtained for BSk16 nn f is the density of unbound neutrons u is the gap in neutron matter at density nn f u is the gap in neutron matter at density n n n [fm 3 ] Z A nn f [fm 3 ] F [MeV] u [MeV] u [MeV] αk (T)/ αk (0) is a universal function of T The critical temperature is approximately given by the usual BCS relation T c F the nuclear clusters lower the gap by 10 20%

33 Pairing in the shallow region of the inner crust 3D HFB calculations with Bloch boundary conditions using the ETFSI mean fields. Preliminary results for 185 Sn at n = fm 3 with BSk16: 0.3 [MeV] density weighted average of (r) Spectral gap Spatial average of (r) T [MeV] T cu Pairing is enhanced for T > T c! This agrees with recent calculations from Margueron& Khan, Phys.Rev.C86,065801(2012).

34 Entrainment Despite the absence of viscous drag, the crust can still resist the flow of the neutron superfluid due to non-local and non-dissipative entrainment effects. Carter, Chamel and Haensel, Nucl.Phys.A748,675(2005). Bragg diffraction Neutrons that are coherently scattered by the lattice do not propagate. Neutron conduction Only neutrons in the conduction band can move throughout the crust.

35 How free are neutrons in neutron-star crusts? On average most neutrons are actually entrained by the crust! Chamel,Phys.Rev.C85,035801(2012). n (fm 3 ) nn/n f n (%) nn/n c n f (%) The density n c n = n f n/m n of conduction neutrons can be much smaller than the density n f n of free neutrons! Open issues: pastas? quantum and thermal fluctuations? impurities and defects? Entrainment can impact the interpretation of several astrophysical phenomena.

36 Pulsar glitches and superfluidity Sudden pulsar spin-ups have been generally attributed to sudden transfers of angular momentum between the superfluid in the crust and the rest of the star. However, due to entrainment the superfluid does not carry enough angular momentum to explain Vela glitches! Chamel, PRL 110, (2013).

37 Neutron-star cooling Due to entrainment, the Bogoliubov-Anderson bosons are strongly mixed with longitudinal lattice phonons: the specific heat is enhanced while the thermal conductivity is reduced. Speeds [c] superfluid bosons longitudinal lattice phonons lowest mixed mode highest mixed mode log 10 C V [fm -3 ] n lph sph tph - e ρ [g cm -3 ] ρ [g cm -3 ] Entrainment may therefore affect the thermal relaxation of neutron-star crusts. Chamel,Page,Reddy,Phys.Rev.C87,035803(2013).

38 Summary We have developed a set of generalized Skyrme functionals fitted to experimental atomic masses and N-body calculations. We have computed a set of unified and consistent EoS for all regions of a neutron star, spanning different degrees of neutron-matter stiffness and symmetry energies. Neutron-star observations rule out some of these EoS. With these functionals, we have studied superfluidity in the crust, and we have found that entrainment effects may impact the interpretation of various observations. Take home message: Our functionals can be applied to compute consistently the microscopic ingredients needed for modeling neuton stars.