Neutron-star matter within the energy-density functional theory and neutron-star structure

Transcription

1 Institut d Astronomie et d Astrophysique Université Libre de Bruxelles Neutron-star matter within the energy-density functional theory and neutron-star structure Anthea F. Fantina (afantina@ulb.ac.be) N. Chamel, S. Goriely (IAA, Université Libre de Bruxelles) J. M. Pearson (Université de Montréal) Lecture at CSSP14, July 2014, Sinaia (Romania) 1

2 Outline v Motivation and astrophysical framework v Energy-density functional (EDF): - What is an EDF? - Example: Brussels-Montreal EDF - What about constraints from nuclear physics? v Equation of state (EoS): - How to construct an EoS? à neutron-star outer crust (BPS model) à neutron-star inner crust (ETFSI model) - Example: Brussels-Montreal EoSs v Neutron-star (NS) structure: - Build a NS - Example: NS structure with Brussels-Montreal EoSs - What about constraints from astrophysics? v Conclusions & Outlook 2

3 Astrophysical framework: SN & NS Credits: 3

4 Astrophysical framework: SN & NS Credits: 4

5 How compact is a NS? SUN M = kg R = km 1.4 g cm 3 2GM Rc dense matter! need of General Relativity! NEUTRON STAR M 1 2 M R 10 km g cm 3 2GM Rc

6 NS mass measurements to probe dense matter 2 M sun NS measurements: Demorest et al., Nature 467, 1081 (2010) Antoniadis et al., Science 340, (2013) Lattimer, Ann. Rev. Nucl. Part. Sci. 62, 485 (2012) 6

7 EoS for NS: the challenge Contrarily to a normal star, in a NS: ü matter is highly degenerate! ( T = 0 approximation ) ü very high density! composition uncertain à despite their name, NSs are not only made by neutrons! different states of matter : inhomogeneous, homogeneous, exotic particles? (see also other talks! e.g. Raduta, Sagert, Vidaña) UNIQUE LABORATORIES! (e.g. superfluidity, strongly deformed nuclei,... ) 7

8 NS crust structure Chamel & Haensel, Living Rev. Relativ. 11, 10 (2008) Neutron star crust : 1% mass, 10% radius 8

9 Which theoretical framework? Modelling nuclear systems under extreme conditions! Ab-initio Configur. Interact. EDF à very n-rich nuclei! Up to now, no experimental or observational information to probe all those regions in ρ, Y e à theory! Energy-density functional theory Bertsch, et al., SciDAD Rev. 6 (2007) 9

10 Outline v Motivation and astrophysical framework v Energy-density functional (EDF): - What is an EDF? - Example: Brussels-Montreal EDF - What about constraints from nuclear physics? v Equation of state (EoS): - How to construct an EoS? à neutron-star outer crust (BPS model) à neutron-star inner crust (ETFSI model) - Example: Brussels-Montreal EoSs v Neutron-star (NS) structure: - Build a NS - Example: NS structure with Brussels-Montreal EoSs - What about constraints from astrophysics? v Conclusions & Outlook 10

11 Energy-density functional (EDF) theory in a nutshell ² Functional: F[f(x)] = y : it is any rule associating a function to a number: input : function à output : number ² Density functional: describe an interacting system via its density ² The total energy of the system is given by (kinetic + interaction part): E = Z E(r)d 3 r EDF: algebraic function of nucleon densities, kinetic energy densities, spin-orbit densities à one can obtain ground state by E minimisation procedure (mean-field HF, or HFB / HF+BCS if pairing is added) à a functional exists which is minimised by the ground state density, but we do not know the functional! à functional deduced by effective (phenomenological) forces chosen to reproduce some global properties of nuclei and nuclear matter à e.g. Skyrme, or Gogny. Here: non-relativistic EDFs of Skyrme type Hohenberg & Kohn, Phys. Rev. B 136, 864 (1964); Kohn & Sham, Phys. Rev. A 140, 1133 (1965); Skyrme, Phil. Mag. 1, 1043 (1956); Déchargé & Gogny, PRC 21, 1568 (1980) 11

12 Energy-density functional (EDF) theory in a nutshell example: Skyrme EDF (e.g. Skyrme, Phil. Mag. 1, 1043 (1956); Chamel et al., PRC 80, (2009)) τ q, ρ q, J q : functions of the occupied wave functions x i, t i fitted on properties of finite nuclei (experim. mass, charge radii) and of nuclear matter (saturation) 12

13 Some properties of nuclear matter Energy around saturation (in a liquid drop model approach): E(n, x = Z/A) =E(n 0,x=1/2) K 1 n 3n 0 n0 + J(1 2x) 2 In SN & NS à n-rich matter à symmetry energy important (see Lattimer s talk): " n E sym = J + L 3n 0 n K sym n 3n 0 n0 2 # (1 2x) 2 related to NS crust-core boundary (e.g. Vidaña et al., PRC 80, (2009)) à Need of experimental information on properties of nuclear matter and nuclei: v incompressibility (K ) v symmetry energy (J, L, K sym ), v nuclear masses,... 13

14 Unified EoS within the EDF theory Ø Unified EoS based on the same nuclear model from EDF theory valid in all regions of NS interior (and SN cores) outer / inner crust and crust / core transition described consistently Ø EoS both at T = 0 à cold non-accreting NS and at finite T à SN cores, accreting NS Ø Satisfying: - constraints from nuclear physics experiments - astrophysical observations Ø Direct applicable for astrophysical applications 14

15 Nuclear models for the EoS: BSk Family of unified EoSs: microscopic mass models based on HFB method with Skyrme type functionals BSk19 BSk20 BSk21 and macroscopically deduced pairing force fit 2010 nuclear experimental mass data (2149 masses, rms = MeV) different degrees of nuclear-matter stiffness (BSk19 softer à BSk21 stiff) constrained to microscopic neutron matter EoS at T = 0 (FP, APR, LS) all have J = 30 MeV Goriely et al., PRC 82, (2010) BSk22 BSk23 BSk24 BSk25 BSk26 fit 2012 nuclear experimental mass data (2353 masses, rms= MeV) constrained to microscopic neutron matter EoS at T = 0 (LS, APR for BSk26) different symmetry energy coefficient (J = 32, 31, 30, 29, 30 MeV) Goriely et al., PRC 88, (2013) BSk** suitable to describe all the regions of NS 15

16 Constraints from nuclear physics: theoretical calculations (neutron matter) N. Chamel, talk ECT* (2013) Goriely et al., PRC 88, (2013) BSk** fitted to realistic neutron-matter EoSs with different stiffness 16

17 Constraints from nuclear physics: experiments Potekhin et al., Astron. Astrophysi. 560, A48 (2013) J,L consistent with experimental constraints for experim. constraints see also: Tsang et al., PRC 86, (2012); Lattimer and Lim, ApJ 771, 51 (2013) 17

18 Constraints from nuclear physics: experiments J,L consistent with experimental constraints Courtesy of N. Chamel for experim. constraints see also: Tsang et al., PRC 86, (2012); Lattimer and Lim, ApJ 771, 51 (2013) 18

19 Outline v Motivation and astrophysical framework v Energy-density functional: - What is an energy-density functional (EDF)? - Example: Brussels-Montreal functionals - What about constraints from nuclear physics? v Equation of state (EoS): - How to construct an EoS? à neutron-star outer crust (BPS model) à neutron-star inner crust (ETFSI model) - Example: Brussels-Montreal EoSs v Neutron-star (NS) structure: - Build a NS - Example: NS structure with Brussels-Montreal EoSs - What about constraints from astrophysics? v Conclusions & Outlook 19

20 NS EoS: outer crust Ground state of matter below neutron drip (isolated NSs): cold catalysed matter hypothesis à T = 0, matter in full thermodynamical equilibrium charge neutrality + β equilibrium structure made by perfect crystal with ONE nuclear species at lattice sites (e.g. bcc) (A,Z), + electrons (no free neutrons!). BPS model (Baym, Pethick, Sutherland, ApJ 170, 299 (1971)) minimise the Gibbs energy per nucleon: where: n is the baryon number density, P is the pressure: E is the energy density: P = P e + P L g = E + P n (electrons + lattice) E = n N M(Z, A)c 2 + E e + E L at constant P density of nuclei mass of nucleus e - energy lattice energy n N = n/a (exp or theory) (electrostatic contribution) For ref. other calculations of the EoS have been made, e.g.: Harrison & Wheeler (1958), Salpeter, ApJ 134, 669 (1961) 20

21 NS EoS: outer crust v Electron contribution: uniform relativistic electron gas & 10 6 g cm 3 e m ec 2 for : electrons are relativistic! P e = E e = where: m e c 2 (2 ) 2 m e c 2 (2 ) 2 3 e 3 e x = }c2 (3 2 n e ) 1/3 m e c 2 }c & n 1/3 e hx(1 + x 2 ) 1/2 (2x 2 /3 1) + ln x +(1+x 2 ) 1/2i hx(1 + x 2 ) 1/2 (1 + 2x 2 ) ln x +(1+x 2 ) 1/2i the relativistic parameter N.B.: there are corrections to this formula, e.g.: - electron correlation energy - electron screening (proportional to Z 4/3 ) - exchange energy (independent on Z) For ref, e.g.: Shapiro & Teukolsky, Black Holes, White Dwarfs, and Neutron Stars (1983) Haensel, Potekhin, Yakovlev, Neutron Stars 1 (2007) Salpeter, ApJ 134, 669 (1961) 21

22 NS EoS: outer crust v Lattice contribution (fully ionised atoms): Electrostatic corrections Wigner-Seitz approximation Space divided into cells, each one charge-neutral with ion in the centre and electrons uniformly distributed E L = E e e + E i e = 9 10 Z 2 e 2 r cell n e Z = /3 4 3 Z 2/3 e 2 n 4/3 e P L = 1 3 E L = ( for bcc) N.B.: corrections to this formula: - finite nuclear size - zero-point corrections For ref, e.g.: Shapiro & Teukolsky, Black Holes, White Dwarfs, and Neutron Stars (1983) Haensel, Potekhin, Yakovlev, Neutron Stars 1 (2007) Salpeter, ApJ 134, 669 (1961) 22

23 NS EoS: outer crust E = n N M(Z, A)c 2 + E e + E L outer crust structure determined by (measured) masses of n-rich nuclei Wolf et al., PRL 110, (2013) Kreim et al., arxiv1303:1343v1 (2013) 23

24 NS EoS: outer crust with BSk models Mass models: HFB (no approximations!) 200 m n n (Z/A) Z/A 2.0% 2.3% J. M. Pearson et al., PRC83, (2011) N.B.: Pressure continous function of star radius à density jump if transition from one stable nuclide to the other 24

25 NS crust structure Chamel & Haensel, Living Rev. Relativ. 11, 10 (2008) Neutron star crust : 1% mass, 10% radius 25

26 NS EoS: inner crust Beyond n drip (ρ g cm -3 ) à n-rich nuclei + e - + free n! i.e. when: µ n = g(t = 0) = M(A, Z)+Zµ e + Zµ L A m n c 2 E(Z,A) has to be extrapolated à which approach to use? 1. Compressible liquid-drop model (e.g. Baym, Bethe & Pethick, Nucl. Phys. A (1971); Douchin & Haensel Astron. Astrohys. 380, 151 (2001) ) - nuclei have sharp surface - nuclear energy given by sum of contribution (bulk, surface, Coulomb) - separation of nuclear matter inside and outside nuclei into two homogeneous phases 2. (Extended) Thomas-Fermi (e.g. Onsi et al., PRC 77, (2008) and refs. therein) - nuclei have smooth surface (smooth density profiles) - nuclear energy as a functional of the density (and density gradients) of species - consistent treatment nucleons inside and outside nuclei 3. Hartree-Fock / Hartree-Fock Bogoliubov (e.g. Negele & Vautherin, Nucl. Phys. A 207, 298 (1973); Grill et al., PRC 84, (2011) ) - quantum calculations: indipendent particle/quasiparticle à shell / pairing effects - also 3D calculations! e.g. Magierski & Heenen, PRC 65, (2002); Newton & Stone, PRC 79, (2009) 26

27 NS EoS: inner crust: ETFSI v ETF: Extended (4th order in ħ) Thomas-Fermi (for the T=0 case!) Onsi et al., PRC 77, (2008); PRC 55, 3139 (1997); PRC 50, 460 (1994) and Refs. therein - Energy density and energy per nucleon of a unit Wigner-Seitz cell is given by: E = E nuc + E e + E Coul +( n m n + p m p + n e m e )c 2 e nuc = 1 Z e = 1 Z E(r)d 3 r ESky ETF (r)d 3 r + A A - spherical neutron-proton clusters + n liquid + uniform relativistic e - gas in Wigner-Seitz cells, using parameterised density distributions: n q (r) =n B,q + n,q f q (r) f q (r) = cell apple 2 Cq R 1 + exp cell r R cell 1 exp 1 Ø Minimisation done directly with respect to density instead of wave functions à minimisation of the energy per nucleon at constant density n wrt cell parameters à semi-classical model à fast approximation to HF equations! cell r a q Cq First applied by Buchler & Barkat (e.g. Buchler & Barkat, PRL 27, 48 (1971)) Many further applications, e.g. Ogasawara & Sato, Prog. Theor. Phys. 68, 222 (1982); Oyamatsu, NPA 561, 431 (1993); Centelles et al., Nucl. Phys. A 537, 486 (1992); Cheng et al., PRC 55, 2092 (1997): relativistic ETF 27

28 NS EoS: inner crust: ETFSI E Sky is function of n q (r), τ q (r), J q (r) ETF : expansion of τ q (r), J q (r) as function of parametrised n q (r) (ρ q (r)) up to the 4th order in ħ e.g.: } 2 2m q f q q = }2 2m q m q m q q = }2 2m q 0q (T = 0) Onsi et al., PRC 55,3139 (1997) and Refs. therein à minimization wrt geometrical parameters of the cell, Z, and N 28 à q (r), J q (r) à approximation to the HF values

29 NS EoS: inner crust: ETFSI Ø but: ETF has no shell effects à added in perturbation (after minimisation) via the Strutinski Integral (SI) theorem e nuc = 1 Z ESky ETF (r)d 3 r + e p sh A cell v SI: proton quantum shell correction via Strutinsky-Integral theorem (perturbative corrections) e p sh = X Z } n i i,p d 3 2 r 2 m i p +ñ p Ũ p + J p Wp where: n i are the occupancy of states, ε i,p are the eigenvalues of the Schrodinger eq. the integral goes over the volume of the WS cell, the sum over the occupied single-particle proton states NB: With respect to outer crust model, some approximations have been made: ² neglect of pairing ² neglect of neutron shell effects ² parametrised density distributions ² neglect of vibrational and rotational correction to the energy Onsi et al., PRC 77, (2008) and Refs. therein; Pearson et al., PRC 85, (2012); Weiss et al., Cox and Giuli s Principle of Stellar Structure (2004) for e - part 29

30 NS EoS: inner crust / core with BSk models Pearson et al., PRC85, (2012) 30

31 NS EoS: inner crust / core with BSk models Pearson et al., PRC85, (2012) very smooth crust-core transition: crust dissolves continously into a uniform mixture of nucleons and electrons 31

32 NS EoS: summary of the model Ø OUTER CRUST (up to neutron drip) (J. M. Pearson et al., PRC83, (2011)) one nucleus (bcc lattice) + e -, in charge neutrality and β equilibrium minimization of the Gibbs energy per nucleon: BPS model Only microscopic inputs are nuclear masses à Experimental or microscopic mass models HFB19-26 Ø INNER CRUST (Pearson et al., PRC85, (2012) ) one cluster (spherical) + n, e - semi-classical model: Extended Thomas Fermi (4th order in ħ) + proton shell corrections (Strutinski Integral theorem) Ø CORE (Goriely et al., PRC 82, (2010), Goriely et al., PRC 88, (2013)) homogeneous matter: n, p, e -, muons in β equilibrium * same nuclear model to treat the interacting nucleons * here we do not consider possible phase transition! 32 transition to exotic matter in Chamel, Fantina, Pearson, Goriely, Astron. Astrophys. 553, A22 (2013)

33 Outline v Motivation and astrophysical framework v Energy-density functional (EDF): - What is an energy-density functional (EDF)? - Example: Brussels-Montreal functionals - What about constraints from nuclear physics? v Equation of state (EoS): - How to construct an EoS? à neutron-star outer crust (BPS model) à neutron-star inner crust (ETFSI model) - Example: Brussels-Montreal EoSs v Neutron-star (NS) structure: - Build a NS - Example: NS structure with Brussels-Montreal EoSs - What about constraints from astrophysics? v Conclusions & Outlook 33

34 Computing the NS structure Ø Nuclear models: BSk & BSk microscopic mass models that fit: ² available nuclear experimental mass data ² nuclear-matter properties from microscopic calculations Ø Build the NS: ² non-rotating NS à solve Tolman-Oppenheimer-Volkoff (TOV) equations: dp dr = G M r 2 1+ P Pr3 2GM c 2 Mc 2 1 rc 2 dm dr =4 r2 M EoS P(ρ) to close the system ² rigidly rotating NSs Method: solve Einstein eqs. in GR for stationary axi-symmetric configurations. Code: LORENE library ( developed at Observatoire de Paris-Meudon Refs on LORENE: Gourgoulhon, arxiv: (lectures given at 2010 CompStar school) Gourgoulhon et al., A&A 349, 851 (1999) Granclément & Novak, Liv. Rev. Relativ. 12, 1 (2009) 34

35 NS properties: M-central density relation BSk compatible with observations (for BSk : Fantina et al., A&A 559, A128 (2013)) 35

36 NS properties: M-R relation with rotation Fantina et al., Astron. Astrophys. 559, A128 (2013) BSk20, BSk21 compatible with observations, BSk19 too soft, (but if we consider a possible phase transition to exotic phase ) 36

37 NS properties: M-R relation with rotation BSk compatible with observations 37

38 NS properties: keplerian velocity The rotational frequency of a stable NS is limited by the keplerian frequency above which the NS will be disrupted as a result of mass shedding R [km] Fantina et al., Astron. Astrophys. 559, A128 (2013) only fast rotation increases considerably maximum mass ( 17-20%) but: rotation can affect structure of low-mass NSs 38

39 NS properties: M-R relation dark (light) shaded area: 1(2)-σ contour from Steiner et al Fantina et al., Astron. Astrophys. 559, A128 (2013) BSk20, SLy4 compatible with observations BSk21 marginally compatible 39

40 NS properties: M-R relation J = 30 MeV J = 32 MeV light (dark) shaded area: 1(2)-σ contour from Steiner et al Pearson, Chamel, Fantina, Goriely, Eur. Phys. J. A 50 (2014) astrophysical observations agree with J = 30 MeV 40

41 Outline v Motivation and astrophysical framework v Energy-density functional and Equation of state (EoS): - What is an energy-density functional (EDF)? - Example: Brussels-Montreal functionals - What about constraints from nuclear physics? v Equation of state (EoS): - How to construct an EoS? à neutron-star outer crust (BPS model) à neutron-star inner crust (ETFSI model) - Example: Brussels-Montreal EoSs v Neutron-star (NS) structure: - Build a NS - Example: NS structure with Brussels-Montreal EoSs - What about constraints from astrophysics? v Conclusions & Outlook 41

42 Conclusions on the unified EoSs v Unified EoSs for NS matter same nuclear model to describe all regions of NS v Nuclear models fitted on experimental nuclear data and nuclear-matter properties v Astrophysical observations can put constraints on EoSs of dense matter! v EoSs based on BSk consistent with astrophysical observations! BSk21-24 functional favoured by mass measurements Chamel et al., PRC84, (2011) Both mass measurements and astro observations favours J 30 MeV The softest EoS BSk19 seems to be ruled out by astrophysical observations, but BSk19 functional favoured by the analysis of HIC experiments. à discrepancy could be resolved by considering the occurrence of a transition to an exotic phase in neutron star cores (Chamel et al., A&A 553, A22 (2013)) v Some EoSs (BSk ) publicly available as: Ø tables at CDS VizieR (Fantina et al., A&A 559, A128 (2013), doi: / / ) Ø fit at: (Potekhin et al., A&A 560, A48 (2013)) 42

43 Outlooks on the EoSs Ø BSk EoSs for NS (T=0) and SN cores (finite T) T = 0: EoS : AVAILABLE! - table - analytical fit (easy to implement!) T 0: work in progress à generate tables for SN cores à implement in hydro codes Ø Application to accreting NS 43

44 Study of compact stars: NS & SN interdisciplinary! Microphysics nuclear physics v equation of state v weak processes v neutrino interactions Macrophysics hydrodynamical models v multi-d models v General Relativity v neutrino transport Nuclear physics experiments v nuclear structure (exotic nuclei, HIC) v β decays Gamow-Teller transitions v neutrino cross-sections Astrophysical observations v neutrino signal v light curves, e-m signals v gravitational waves v NS cooling 44

45 Thank you