Unified description of neutron-star interiors Nicolas Chamel Institute of Astronomy and Astrophysics Université Libre de Bruxelles, Belgium in collaboration with S. Goriely (ULB), J. M. Pearson (UMontréal), A. F . Fantina (ULB, GANIL) P . Haensel & J. L. Zdunik (CAMK) Y. D. Mutafchieva & Zh. Stoyanov (INRNE) A. Potekhin (Ioffe) CAMK, Warsaw, 28 March 2018
Prelude Haensel, Potekhin, Yakovlev, “Neutron Stars - 1. Equation of State and Structure” (Springer, 2007) The interior of a neutron star exhibits very different phases (gas, liquid, solid, superfluid, etc.) over a very wide range of densities with possibly exotic particles (hyperons, quarks) in the inner core. Blaschke&Chamel, contribution to the White Book of the COST Action MP1304, arXiv:1803.01836
Main motivation Ad hoc matching of different models can lead to significant errors on the neutron-star structure & dynamics. 2.5 Unified 3.0 n m =0 . 01 n m = n c n m = n t 2.0 n m = n 0 2.5 n m =0 . 5 n 0 − n 0 n m =0 . 1 n 0 − n t 2.0 1.5 M ( M ⊙ ) l cr (km) 1.5 1.0 1.0 0.5 0.5 12 13 14 15 1.0 1.5 2.0 2.5 R (km) M ( M ⊙ ) Fortin et al., Phys.Rev.C94, 035804 (2016) Combining different microscopic inputs lead to multiple interpretations of astrophysical phenomena (degeneracy). This calls for a unified description of neutron-star interiors.
Outline Internal constitution of a neutron star 1 ⊲ Main assumptions on dense stellar matter ⊲ Constraints from laboratory experiments ⊲ Predictions from the nuclear-energy density theory ⊲ Phase transitions in the inner core ? Description of specific neutron-star classes 2 ⊲ Highly-magnetized neutron stars ⊲ Accreting neutron stars Conclusions & perspectives 3
Neutron-star surface The surface of a neutron star is expected to be made of iron, the end product of stellar nucleosynthesis (identification of broad Fe K emission lines from accretion disk around neutron stars in LMXB). Compressed iron can be studied with nuclear explosions and laser-driven shock-wave experiments ... But at pressures corresponding to about 0 . 1 0 . 1 0 . 1 mm below the surface (for a star with a mass of 1 . 4 M ⊙ and a radius of 12 km) ! Stixrude, Phys.Rev.Lett. 108, 055505 (2012) Ab initio calculations predict various structural phase transitions .
Crystal Coulomb plasma At a density ρ eip ≈ 2 × 10 4 g cm − 3 (about 22 cm below the surface), the interatomic spacing becomes comparable with the atomic radius. Ruderman, Scientific American 224, 24 (1971) At densities ρ ≫ ρ eip , atoms are crushed into a dense plasma of nuclei and free electrons . Nuclei become more neutron rich with increasing pressure.
Description of the outer crust of a neutron star Main assumptions: cold “catalyzed” matter (full thermodynamic equilibrium) Harrison, Wakano and Wheeler, Onzième Conseil de Physique Solvay (Stoops, Brussels, Belgium, 1958) pp 124-146 the crust is stratified into pure layers made of nuclei A Z X electrons are ∼ uniformly distributed and are highly degenerate T < T F ≈ 5 . 93 × 10 9 ( γ r − 1 ) K � 1 / 3 p F � ρ 6 Z � 1 + x 2 γ r ≡ r , x r ≡ m e c ≈ 1 . 00884 A nuclei are arranged on a perfect body-centered cubic lattice T < T m ≈ 1 . 3 × 10 5 Z 2 � ρ 6 � 1 / 3 ρ 6 ≡ ρ/ 10 6 g cm − 3 K A Pearson et al.,Phys.Rev.C83, 065810 (2011) Chamel&Fantina,Phys.Rev.D93,063001 (2016)
Experimental “determination” of the outer crust The composition of the crust is completely determined by experimental atomic masses down to about 200m for a 1 . 4 M ⊙ neutron star with a 10 km radius The physics governing the structure of atomic nuclei (magicity) leaves its imprint on the composition. Due to β equilibrium and electric charge neutrality, Z is more tightly constrained than N : only a few layers with Z = 28. Kreim, Hempel, Lunney, Schaffner-Bielich, Int.J.M.Spec.349-350,63(2013)
Plumbing neutron stars to new depths Precision mass measurements of 82 Zn by the ISOLTRAP collaboration at CERN’s ISOLDE radioactive-beam facility in 2013 allowed to "drill" deeper. Wolf et al.,Phys.Rev.Lett.110,041101(2013) The composition is very sensitive to uncertainties in nuclear masses. Errors of a few keV / c 2 can change the results. Deeper in the star, recourse must be made to theoretical models.
Theoretical challenge Models of dense matter required to compute all the necessary inputs to astrophysical simulations of neutron stars should be: versatile : applicable to compute all properties under various conditions thermodynamically consistent : avoid spurious instabilities as microscopic as possible : make reliable extrapolations numerically tractable : systematic calculations over a wide range of temperatures, pressures, composition, magnetic field The nuclear energy density functional theory is the most suitable (quantum) approach. Nucleons are treated as independent quasiparticles in a self-consistent potential field (Hartree-Fock-Bogolyubov method). Dobaczewski&Nazarewicz, in ”50 years of Nuclear BCS” (World Scientific Publishing, 2013), pp.40-60; Chamel,Goriely,Pearson, ibid., pp.284-296 In principle, this theory describes the many-body system exactly provided the exact functional is known (Hohenberg-Kohn theorem)...
Nuclear-matter uncertainties Because the exact functional is unknown, phenomenological functionals are employed. How to quantify nuclear-matter uncertainties ? The energy per nucleon of nuclear matter at T = 0 around saturation density n 0 and for asymmetry η = ( n n − n p ) / n , is usually written as e ( n , η ) = e 0 ( n ) + S ( n ) η 2 + o � η 4 � where e 0 ( n ) = a v + K v 18 ǫ 2 − K ′ 162 ǫ 3 + o � ǫ 4 � with ǫ = ( n − n 0 ) / n 0 S ( n ) = J + L 3 ǫ + K sym 18 ǫ 2 + o � ǫ 3 � is the symmetry energy The lack of knowledge is embedded in a v , K v , K ′ , etc. In order to make meaningful comparisons, energy-density functionals corresponding to different values of these parameters should be fitted using the same protocole .
Brussels-Montreal Skyrme functionals (BSk) For application to extreme astrophysical environments, functionals should reproduce global properties of both finite nuclei and infinite homogeneous nuclear matter. Experimental data/constraints : nuclear masses (rms ∼ 0 . 5 − 0 . 6 MeV / c 2 ) nuclear charge radii (rms ∼ 0 . 03 fm) symmetry energy 29 ≤ J ≤ 32 MeV incompressibility K v = 240 ± 10 MeV Many-body calculations using realistic interactions : equation of state of pure neutron matter 1 S 0 pairing gaps in nuclear matter effective masses in nuclear matter stability against spin and spin-isospin fluctuations Chamel et al., Acta Phys. Pol. B46, 349(2015)
Neutron-matter stiffness 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, 035804 (2010).
Neutron-matter constraints at low densities All three functionals are consistent with ab initio calculations at densities relevant for the inner crust and outer core of neutron stars:
Symmetry-energy constraint The functionals BSk22-26 were also fitted to realistic neutron-matter equations of state but with different values for J = 29 − 32 MeV: Goriely, Chamel, Pearson, Phys.Rev.C 88, 024308 (2013).
Theoretical predictions of the outer crust a 0m t 56 Fe 56 Fe Predictions from HFB-21 of odd nuclei 79 Cu, 121 Y. 62 62 outer Ni Ni 64 64 crust Composition dominated by 66 66 86 Kr 86 Kr inner nuclei with N close to magic 100m crust number N = 82. 84 Se 84 Se N ≈ 50 Sr isotopes are the most 82 Ge abundant nuclides ( ∼ 40%). 82 Ge 10km 200m 80 Zn 80 Zn core 78 Ni 79 Cu 82 Zn 80 Ni 78 Ni 80 126 Ru 124 Mo 124 Mo 300m 122 Zr 122 Zr 124 N ≈ 82 121 Y 120 120 122 Sr Sr 122 124 124 126 HFB-19 HFB-21 only 0.04% in Earth’s crust Pearson et al.,Phys.Rev.C83,065810(2011) Wolf et al.,Phys.Rev.Lett.110,041101(2013)
Role of the symmetry energy The composition of the outer crust is only slightly influenced by the density dependence of the symmetry energy S ( n ) . 2 − ( 12 π 2 ( � c ) 3 P ) 1 / 4 The proton fraction varies roughly as Y p = Z A ∼ 1 8 S 0.4 0.35 Y p HFB-22 HFB-24 HFB-25 0.3 HFB-26 -5 -4 10 10 -3 ] n [fm Pearson et al., Eur. Phys. J.A50,43(2014); Pearson et al. in prep.
Equation of state of the outer crust The pressure, determined by electrons, is almost independent of the composition. Analytical fits : http://www.ioffe.ru/astro/NSG/BSk/
Stratification and equation of state Transitions between adjacent crustal layers are accompanied by density discontinuities. P 2 + P 1 2 1+2 1 n n max n min 1 2 Mixed solid phases cannot exist in a neutron star crust because P has to increase strictly monotonically with ¯ n (hydrostatic equilibrium).
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