The high density equation of state for neutron stars and (CC)-supernovae

Transcription

1 The high density equation of state for neutron stars and (CC)-supernovae Part I: Equations of state for neutron star matter Micaela Oertel Laboratoire Univers et Théories (LUTH) CNRS / Observatoire de Paris/ Université Paris Diderot NewComsptar School, Barcelona, September,22-26,24 Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 1 / 52

2 Outline 1 Introduction 2 Constructing an equation of state 3 Nuclear interaction models Nucleon-Nucleon interaction Models for heavier nuclei Constraints on the EoS Models for bulk matter 4 Exotic hadronic matter Hyperons Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 2 / 52

3 Outline 1 Introduction 2 Constructing an equation of state 3 Nuclear interaction models Nucleon-Nucleon interaction Models for heavier nuclei Constraints on the EoS Models for bulk matter 4 Exotic hadronic matter Hyperons Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 2 / 52

4 Outline 1 Introduction 2 Constructing an equation of state 3 Nuclear interaction models Nucleon-Nucleon interaction Models for heavier nuclei Constraints on the EoS Models for bulk matter 4 Exotic hadronic matter Hyperons Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 2 / 52

5 Outline 1 Introduction 2 Constructing an equation of state 3 Nuclear interaction models Nucleon-Nucleon interaction Models for heavier nuclei Constraints on the EoS Models for bulk matter 4 Exotic hadronic matter Hyperons Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 2 / 52

6 Nowadays Crab pulsar Vela pulsar Almost 2000 neutron stars have been observed as pulsars, among others Crab, Vela, Geminga, Hulse-Taylor double pulsar,... Hubble (blue, optic), Chandra (red, X) Chandra (X) RX J Some ( the magnificant seven ) have been observed only by their thermal emission Magnetars have extremely high magnetic fields ( G at the surface) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 3 / 52

7 Why a neutron star? Theoretical argument : high density electrically neutral matter in β-equilibrium high neutron fraction (later...) Observational argument : Neutron stars are formed in supernova events. For SN1987A 24 antineutrinos have been observed, 11 Kamiokande-II 8 IMB 5 Baksan This is generally interpreted as a proof for the classical scenario with two main sources of (anti-)neutrino production, the neutronization reaction p + e n + ν e and the inverse process n + e + p + ν e SN1987A Superkamiokande Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 4 / 52

8 Physical ingredients to model these systems First step, building equilibrium models : give the gravitational law (self-gravitating body), here General Relativity described by Einstein s equations ; assume that matter can be treated as a perfect fluid, specified by the energy-momentum tensor of the form T αβ = (ε + p ) c 2 u α u β + p g αβ. write the equilibrium conditions following from conservation of energy and momentum µ T µν = 0 ; assume thermal equilibrium ; give a law for the pressure as a function of the system s thermodynamical state equation of state (EoS). The EoS specifies the nuclear matter properties and, in particular the strength of the interaction between particles, which is to equilibrate gravity. Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 5 / 52

9 Physical ingredients to model these systems First step, building equilibrium models : give the gravitational law (self-gravitating body), here General Relativity described by Einstein s equations ; assume that matter can be treated as a perfect fluid, specified by the energy-momentum tensor of the form T αβ = (ε + p ) c 2 u α u β + p g αβ. write the equilibrium conditions following from conservation of energy and momentum µ T µν = 0 ; assume thermal equilibrium ; give a law for the pressure as a function of the system s thermodynamical state equation of state (EoS). The EoS specifies the nuclear matter properties and, in particular the strength of the interaction between particles, which is to equilibrate gravity. Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 5 / 52

10 Physical ingredients to model these systems First step, building equilibrium models : give the gravitational law (self-gravitating body), here General Relativity described by Einstein s equations ; assume that matter can be treated as a perfect fluid, specified by the energy-momentum tensor of the form T αβ = (ε + p ) c 2 u α u β + p g αβ. write the equilibrium conditions following from conservation of energy and momentum µ T µν = 0 ; assume thermal equilibrium ; give a law for the pressure as a function of the system s thermodynamical state equation of state (EoS). The EoS specifies the nuclear matter properties and, in particular the strength of the interaction between particles, which is to equilibrate gravity. Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 5 / 52

11 R [km] R ~ 3000 Fe R [km] formation ~ 10 R [km] R s ~ 200 R g ~ 100 R ν ~ 50 Fe, Ni νe Si M(r) [M ] Si burning shell Si ν e Fe, Ni M(r) [M ] nuclear matter ( > nuclei Si burning shell δ PNS δ < free n, p νe 1.3 gain layer 1.5 cooling layer p n Si M(r) [M ] R [km] ~ 100 R [km] R s ~ 100 km position of shock formation R [km] R ns ~ 10 R ν PNS 0.5 heavy nuclei α,n n, p c Fe, Ni Fe00 11 free n, p 11 Ni α r process? α,n, 12 C, seed Si M(r) [M ] Si burning shell Si M(r) [M ] Si burning shell Ni Si He O 3 M(r) [M ] Matter composition and equation of state Matter composition changes dramatically through the core collapse Starting point : onion like structure with iron/nickel core+ degenerate electrons Upon compression (+deleptonisation) : heavier and more neutron rich nuclei For n B > n 0 /2 : nuclei disappear in favor of free nucleons Composition of matter above n 0 and at T > 10 MeV relatively unknown All this has to be covered by the EoS! Different stages of a supernova R Fe shock radius of Initial Phase of Collapse (t ~ 0) ο ) δ ~ M Ch Bounce and Shock Formation (t ~ 0.11s, c 2 o) δ Shock Stagnation and ν Heating, Explosion (t ~ 0.2s) ν e ν e ν e ν e,µ,τ,ν e,µ,τ ν e ν e ν e,µ,τ,ν e,µ,τ R Fe R Fe R ν Neutrino Trapping M hc Shock Propagation and ν e Burst (t ~ 0.12s) nuclear matter nuclei (Janka et al. 07) (t ~ 0.1s, ~10¹² g/cm³) δ Neutrino Cooling and Neutrino Driven Wind (t ~ 10s) 9 Be, ν e ν e ν e ν e ν e ν e ν e ν e ν e,µ,τ,ν e,µ,τ ν e,µ,τ,ν e,µ,τ Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 6 / 52

12 Matter composition and equation of state Matter composition changes dramatically through the core collapse Starting point : onion like structure with iron/nickel core+ degenerate electrons Upon compression (+deleptonisation) : heavier and more neutron rich nuclei For n B > n 0 /2 : nuclei disappear in favor of free nucleons Composition of matter above n 0 and at T > 10 MeV relatively unknown All this has to be covered by the EoS! Z (Proton Number) Z (Proton Number) Nuclear abundancies at different stages T= 9. GK, ρ= 6.80e+09 g/cm 3, Y e = Log (Mass Fraction) N (Neutron Number) T= GK, ρ= 3.39e+11 g/cm 3, Y e = Log (Mass Fraction) N (Neutron Number) (Janka et al. 07) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 7 / 52

13 Matter composition and the equation of state Matter composition changes dramatically through the core collapse Starting point : onion like structure with iron/nickel core+ degenerate electrons Upon compression (+deleptonisation) : heavier and more neutron rich nuclei For n B > n 0 /2 : nuclei disappear in favor of free nucleons Composition of matter above n 0 and at T > 10 MeV relatively unknown All this has to be covered by the EoS! Phase diagram of bulk nuclear matter (RMF/TMA) T [MeV] Y p =0.5 Y p =0.3 Y p = n B [fm -3 ] (Hempel & Schaffner-Bielich 09) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 8 / 52

14 How to construct an equation of state? An equation of state is a relation between two or more functions describing the thermodynamic state of matter, e.g. temperature, density, pressure, energy, i.e. it describes the state of matter under a given set of physical conditions. It assumes thermal equilibrium. Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, 24 9 / 52

15 Reminder of thermodynamic identities There are different thermodynamic potentials depending on the temperature (T )/the entropy (S), the volume (V )/the pressure (p), the particle number (N)/the chemical potential (µ), or the corresponding densities (s, n) the energy density ε(s, n i ) the free energy density f(t, n i ) = ε T s the grand canonical potential density ω(t, µ i ) = ε T s i µ i n i the conjugate variables are related via derivatives, e.g. n i = ω µ i There is a chemical potential associated with each conserved quantity (charge, baryon number, lepton number) and the individual chemical potentials are linear combinations of µ q, µ B, µ l, e.g. µ proton = µ B + µ q. At zero temperature the entropy vanishes. Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

16 Part I : Neutron star EoS Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

17 An equation of state for neutron stars Start with homogeneous neutron star matter : f(t, n B, n q, n l ) would be a pertinent (and convenient) equation of state soon after their birth in supernovae ( minutes), neutron stars are sufficiently cool for temperature effects on the EoS to be neglected in general strong, electromagnetic, and weak reactions are at equilibrium (this includes in particular β-equilibrium) global charge neutrality should be fulfilled (n q = 0) all state variables are functions of only one parameter, chosen to be e.g. the baryon number density n B. Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

18 An equation of state for neutron stars Start with homogeneous neutron star matter : f(t, n B, n q, n l ) would be a pertinent (and convenient) equation of state soon after their birth in supernovae ( minutes), neutron stars are sufficiently cool for temperature effects on the EoS to be neglected in general strong, electromagnetic, and weak reactions are at equilibrium (this includes in particular β-equilibrium) global charge neutrality should be fulfilled (n q = 0) all state variables are functions of only one parameter, chosen to be e.g. the baryon number density n B. Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

19 More precisely... Homogeneous (bulk) matter in the core of neutron stars should be charge neutral : q i n i = 0. For matter composed of neutrons, i protons, and electrons this means just n p = n e. should be in β-equilibrium. This means the reactions p + e n + ν e and n p + e + ν e should be in equilibrium. In bulk matter this can be achieved by the following condition on the chemical potentials : µ p + µ e = µ n + µ νe. Remind that the chemical potential corresponds to the energy needed to add one particle to the Fermi sea. Since neutrinos can freely leave the (cold) neutron star, their chemical potential (= µ l ) is zero and the β-equilibrium condition reduces to µ p + µ e = µ n. Attention : in the literature, the chemical potentials of the nucleons are sometimes defined without the particle mass. If this is the case, the masses should be added to the above relation explicitly. Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

20 More precisely... Homogeneous (bulk) matter in the core of neutron stars should be charge neutral : q i n i = 0. For matter composed of neutrons, i protons, and electrons this means just n p = n e. should be in β-equilibrium. This means the reactions p + e n + ν e and n p + e + ν e should be in equilibrium. In bulk matter this can be achieved by the following condition on the chemical potentials : µ p + µ e = µ n + µ νe. Remind that the chemical potential corresponds to the energy needed to add one particle to the Fermi sea. Since neutrinos can freely leave the (cold) neutron star, their chemical potential (= µ l ) is zero and the β-equilibrium condition reduces to µ p + µ e = µ n. Attention : in the literature, the chemical potentials of the nucleons are sometimes defined without the particle mass. If this is the case, the masses should be added to the above relation explicitly. Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

21 The free Fermi gas The most simple equation of state would be a free Fermi gas of n, p, e. n i = p3 F 3π 2 with pi F = µ 2 i m2 i giving µ i = (n i 3π 2 ) 2/3 + m 2 i charge neutrality (n p = n e ) and the β-equilibrium condition then allow to determine n e as a function of n B = n n + n p n i = n i (n B ) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

22 The free Fermi gas Y e n B [fm -3 ] Y e = n e /n B charge neutral matter in β-equilibrium becomes neutron rich Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

23 The free Fermi gas The most simple equation of state would be a free Fermi gas of n, p, e. µ 2 i m2 i giving µ i = n i = p3 F 3π 2 with pi F = (n i 3π 2 ) 2/3 + m 2 i charge neutrality (n p = n e ) and the β-equilibrium condition then allow to determine n e as a function of n B = n n + n p n i = n i (n B ) energy density ε = 2 i=n,p,e P i F 0 = (m i n i d 3 p (2π) 3 p 2 + m 2 i (3π 2 ) 2/3 n 5/3 i ) + 3 2m i 4 (3π2 ) 1/3 n 4/3 e i=n,p (Here I have used the non-rel. approximation for nucleons and neglected me) pressure p = ε + i=n,p,e µ i n i (note that i=n,p,e µ i n i = µ B n B ) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

24 NS models : Tolman-Oppenheimer-Volkov system Consider hydrostatic solutions in spherical symmetry Einstein and hydrostatic equations then reduce to : dm dr dφ dr dp dr = 4πr 2 ε ( = 1 2Gm ) 1 ( Gm rc 2 r 2 ( = ε + p ) dφ c 2 dr + 4πG p ) c 2 r In the newtonian limit m(r) describes the enclosed mass and Φ(r) the gravitational potential. This system of partial differential equations is called the Tolman-Oppenheimer-Volkov (TOV) system Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

25 NS models : Tolman-Oppenheimer-Volkov system Consider hydrostatic solutions in spherical symmetry Einstein and hydrostatic equations then reduce to : dm dr dφ dr dp dr = 4πr 2 ε ( = 1 2Gm ) 1 ( Gm rc 2 r 2 ( = ε + p ) dφ c 2 dr + 4πG p ) c 2 r In the newtonian limit m(r) describes the enclosed mass and Φ(r) the gravitational potential. This system of partial differential equations is called the Tolman-Oppenheimer-Volkov (TOV) system Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

26 From a book of statistical mechanics : The process which has been discussed leads to the formation of a degenerate gas of neutrons. It is mainly the quantum pressure of this gas which is stabilising the star. Free gas a good EoS? Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

27 The free Fermi gas But life is not as simple! 0.8 M g [M sun ] R [km] Historical note : A neutron Fermi gas EoS led Oppenheimer and Volkov (Oppenheimer& Volkov, Phys. Rev. 55 (1939) 374) to predict maximum NS masses of 0.7M But : Maximum neutron star mass < 1M in contradiction with observations! Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

28 The nuclear interaction comes into play... Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

29 Some basic principles of the nucleon-nucleon interaction Mainly two experimental sources of information : deuteron properties since the deuteron is the only bound two-nucleon state (di-neutron and di-proton are unbound) NN diffusion experiments The main conclusions are : the NN interaction is charge independent introduction of isospin (Heisenberg, 1932) Proton and neutron have isospin 1/2 with different orientations Small isospin violating contributions only from electromagnetic corrections and differences in mass. the NN interaction is short-range ( fm) with an attractive part and strong repulsion below roughly 0.5 fm the NN interaction is spin-dependent and there are important pairing correlations Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

30 NN interaction Nucleon-nucleon (NN) interaction relatively well known Potential described in terms of meson exchange (π, ) Paris potential, Bonn potential, Argonne potential... : very precise description of phase shift data, deuteron Problem with three-body force Recently : modern potentials based on chiral perturbation theory (effective field theory EFT of low-energy QCD) big advantage : consistent three-body forces (success for neutron-rich nuclei) Ab-initio calculations up to A 12 Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

31 The nuclear landscape limits of the chart : neutron and proton driplines inclination due to electromagnetic forces existence of magic numbers with strongly bound nuclei (shell closures) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

32 Models for heavier nuclei 1. The liquid drop model Principal idea : density of nuclei approximately constant (liquid drops) Binding energy determined from volume, surface,asymmetry, Coulomb and pairing terms : B = a V A + a S A 2/3 (N Z) + a 2 Z A A a 2 (mod(n,2)+mod(z,2) 1) C + a A 1/3 P A 1/2 Nuclear binding energies can be very precisely reproduced 2. The shell model Original idea : nucleons move in a (harmonic oscillator) potential and fill up the shells Inspired by atomic shell model, but LS interaction important (magic numbers)! Residual interaction very precisely fitted to experimental data Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

33 Models for heavier nuclei : mean field models Selfconsistent approach (variational principle) : minimizing of φ H φ with product ansatz (Slater determinant) for the wave function φ Take effective interaction to take correlations into account Modern interpretation in terms of Kohn-Sham energy density functional theory Good description of the nuclear chart up to very heavy nuclei 3. The non-relativistic Skyrme and Gogny interaction Zero range force (Skyrme), finite range (Gogny) Spin-orbit interaction added by hand 4. The relativistic mean field (RMF) Relativistic (special relativity) treatment of the nucleons Interaction described via effective (!) meson exchange Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

34 Models for bulk matter Ab-initio calculations up to A 12 (not adequate for the description of nuclear matter!) for nuclear matter there are two types of models 1 phenomenological models with effective interactions liquid drop mean field 2 ab initio microscopic calculations starting from the basic NN interaction Brueckner-Hartree-Fock (BHF) Self-consisten Green s function Variational techniques V lowk... Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

35 Constraints on the EoS 1. Symmetric nuclear matter For symmetric nuclear matter (i.e. n n = n p ), the energy per baryon as a function of density has a minimum at nuclear matter saturation density, n 0 = 0.16fm 3. The binding energy at this point is E b = 16MeV. Several coefficients of an expansion of the energy per baryon around saturation are measured in nuclear experiments (x = n B /n 0 1) : E(n B, β) = E b Kx K x 3 + β 2 (J Lx K symx 2 ) incompressibility K 240MeV skewness coefficient K symmetry energy J 32MeV symmetry energy slope coefficient L 50 MeV symmetry incompressibility K sym, -500 MeV < K sym < 100 MeV A free Fermi gas cannot reproduce these features! E/A [MeV] n B [fm -3 ] Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

36 Constraints on the EoS 2. Other nuclear experiments data from transversal matter flow in heavy ion collisions is sensitive to the EoS of symmetric matter caveats : matter not really in thermal equilibrium transport calculations are reinterpreted in terms of EoS in principle only valid for the model employed within the transport at high densities many new degrees of freedom (resonances, mesons,... ) relation to the nuclear EoS? Flow constraint from heavy ion collisions P [MeV fm -3 ] UB LB extrapolated NLρ, NLρδ DBHF (Bonn A) DD D 3 C KVR KVOR DD-F n [fm -3 ] (Klähn et al. PRC 2006) meson production in heavy-ion collisions, in particular subthreshold kaon production soft EoS in the range 2 3n 0 Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

37 Constraints on the EoS 3. Main present constraint from astrophysical observations : neutron star masses Observed masses in binary systems (NS-NS, NS-WD, X-ray binaries) with most precise measurements from double neutron star systems. All NS-NS systems give masses close to 1.4M Recently two precise mass measurements in NS-WD binaries M = 1.97 ± 0.04M (PSR J ) (Demorest et al, Nature 20) M = 2. ± 0.04M (PSR J ) (Antoniadis et al, Science 23) (Lattimer & Prakash, ) 2 M probably not the end of the story (e.g. indications for a 2.4 M NS in B (van Kerkwijk et al., ApJ 21)) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

38 Constraints on the EoS 4. NS radius determinations Measurements of rotational frequency f = 716 Hz (PSR J ad) (Hessels et al, Science 2006) f = 1122 Hz not confirmed! (XTE J ) (Kaaret et al. ApJ 2007) Theory : Kepler frequency f K = 1008 Hz (M/M ) 1/2 (R/10km) 3/2 (Haensel et al. A&A 2009) A measured frequency of 1.4 khz would constrain R 1.4 < 9.5 km! Bayesian inversion of TOV equations with observed M G R relations (Steiner et al. ApJ 20) Radius determinations so far model dependent (assumptions on atmosphere, distance,...) (e.g. Heinke et al ) and theoretical uncertainties (rotation,... ) M (M ) FPS SQM3 BGN1H1 GMGS APR DH SQM1 BM165 (M. Fortin, CAMK) r ph >>R R (km) (Steiner et al. ApJ 20) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

39 Constraints on the EoS 5. Direct URCA threshold the direct URCA process n p + e + ν and p + e n + ν, a very efficient cooling process, is suppressed in the cold neutron star due to energy and momentum conservation. It can reappear in the very dense core if the proton fraction exceeds 11-14%. in contradiction with observed surface temperatures if Y p > 11 14% for too low densities main caveat : masses of NS with observed surface temperature not known Why is direct URCA suppressed? Consider the reaction n p + e + ν. Energy conservation E n = E p + E e + E ν, where n, p, e have E i E Fi ± T, thus the neutrino has E ν T. β equilibrium µ n = µ p + µ e and charge neutrality give p F,e = p F,p p 2 F,n/(2m n ) p F,n Momentum conservation p n = p p + p e + p ν with p ν T can then no longer be fulfilled, since p F,n < p F,p + p F,e. Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

40 Constraints on the EoS 6. Baryon mass versus gravitational mass low mass NS stars in binary NS systems could be formed in an electron capture SN of a O-Ne-Mg core with very little mass loss two candidate systems M = ± 0.0M (PSR J /B) (Kramer et al, astro-ph/ ) M = ± 0.007M (PSR J companion) (Ferdman et al, MNRAS 24) constraints on the baryon mass of the low-mass object by evolution arguments (Podsiadlowski et al. MNRAS 2005), assuming no mass loss and a numerical simulation (Kitaura et al. A&A 2006) caveats mass loss assumption (simulation gives little mass loss) uncertainties on evolutionary scenario does the contraint obtained from the simulation only reflect the properties of the (two) EoS employed? multi-d effects, magnetic field,... M B versus M G -range for different EoS M [M sol ] NLρ NLρδ DBHF (Bonn A) DD D 3 C KVR KVOR DD-F M N [M sol ] (Klähn et al. PRC 2006) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

41 Models for bulk matter Ab-initio calculations up to A 12 (not adequate for the description of nuclear matter!) for nuclear matter there are two types of models 1 phenomenological models with effective interactions liquid drop mean field 2 ab initio microscopic calculations starting from the basic NN interaction Brueckner-Hartree-Fock (BHF) Self-consistent Green s function Variational techniques V lowk... Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

42 The non-relativistic Skyrme mean field model Starting point is a functional for the energy density ε(n n, n p ) Zero range interaction Spin-orbit interaction added by hand (very important for magic numbers of nuclei, but absent for bulk matter) The interaction part in general contains the following terms a central term n 2 B, (n n n p) 2 a density dependent term n α B(n 2 B, (n n n p) 2 ) an effective mass term τnb, τ in i (τ is the kinetic energy density, same form as Fermi gas) a spin-orbit term, a gradient term and a tensor term, which do not contribute to nuclear matter calculations How to proceed to obtain the pressure? How to proceed to get the EoS for neutron star matter? E/A [MeV] SLy4, Y p = 0.1 SLy4, Y p = 0.3 SLy4, Y p = n B [fm -3 ] Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

43 A Skyrme EoS 1. Pressure Thermodynamic relations ε = µ i and n i X P = ε + µ in i i 2. Neutron star EoS Add leptons, i.e. electrons and possibly muons (as free Fermi gas) Charge neutrality n q = 0 β-equilibrium µ n = µ p + µ e (and muons?) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

44 A Skyrme EoS The proton fraction in neutron star matter for different parametrisations Y p SLy4 + µ SLy4 Sk0p Gs SV n B [ fm -3 ] Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

45 A Skyrme EoS P [MeV/fm 3 ] SLy4 + µ SLy4 Sk0p Gs SV ε [MeV/fm 3 ] Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

46 The relativistic mean field (RMF) Neutron matter Relativistic (special relativity) treatment of the nucleons Interaction described via effective (!) meson exchange : L RMF = ψ (iγ µ µ M g σ σ +... ) ψ binding energy per nucleon E/A [MeV] NL3 DD D 3 C density ρ [fm -3 ] µσ µ σ 1 2 m2 σσ symmetric nuclear matter 80 In the original version (Walecka model) only σ, ω In order to reproduce data, many refinements : isovector channel (ρ, δ), nonlinear σ-interaction, density dependent couplings,... binding energy per nucleon E/A [MeV] NL3 DD D 3 C density ρ [fm -3 ] S. Typel, nucl-th/0556 Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

47 (Dirac)-Brueckner-Hartree-Fock calculations Starting point is the bare NN-interaction V Construction of the Brueckner G-matrix : G(ω) ij,kl = V ij,kl + a,b V ij,ab Q ab ω e a e b G(ω) ab,kl The Pauli operator Q prevents the baryons (they are fermions!) to be scattered to states below their respective Fermi momenta The single-particle energies are determined self-consistently e i ( k) = M i + k 2 + ij G(ω) ij,ij ij 2M i j Three-body forces necessary to reproduce empirical behaviour of E/A(n B ). E/A (MeV) density (fm -3 ) M. Baldo, G.F. Burgio, arxiv : Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

48 Constraints on the EoS 7. Ab-initio calculations of neutron (and symmetric) matter any of the many-body techniques mentioned before, i.e. (D)BHF, SCGF,... Monte-Carlo techniques recently also V lowk from chiral EFT forces + Hartree-Fock Vlowk is a renormalisation group improved interaction potential idea : start at an interaction scale Λ and evolve it to lower energies the effective interactions at a lower scale Λ then contain the higher energy dynamics (e.g. three-body forces are generated) Pressure for (low-density) neutron matter V lowk + HF vs MC P [MeV fm 3 ] NN+3N Gandolfi et al n [fm 3 ] (Hebeler et al. ApJ 23) remarks all these methods have problems (of different nature, e.g. finite size effects in Monte-Carlo) truncation of the initial interaction to N-body-forces the results, in particular for symmetric matter, do not all agree Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

49 0-10 Energy per baryon for symmetric matter, different approaches V 4 V E/A [MeV] V 8 V E/A [MeV] ~ V 6 (BHF) ρ [fm -3 ] BHF SCGF -20 FHNC AFDMC -25 BBG ρ [fm -3 ] (Baldo et al. PRC 22) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

50 60 Energy per baryon for neutron matter, different approaches V 4 V 6 60 E/A [MeV] E/A [MeV] V 8 BHF SCGF FHNC AFDMC BBG GFMC V ρ [fm -3 ] ρ [fm -3 ] (Baldo et al. PRC 22) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

51 Neutron, protons, electrons, is it all? Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

52 Let s have a look on the chemical potentials d 3 p Remind : We are at zero temperature n = 2 θ(µ E) (2π) 3 A state can be populated if the chemical potential exceeds its energy. µ B [MeV] SLy4 + µ SLy4 Sk0p Gs SV n B [ fm -3 ] µ q [MeV] SLy4 + µ SLy4-400 Sk0p Gs -500 SV n B [ fm -3 ] Example : At which density could muons (m µ = MeV), treated as free Fermi gas, typically appear? Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

53 Short reminder of hadronic states Hadrons : Hadrons are strongly interacting particles, composed of quarks and gluons. There are two different types of hadrons, baryons and mesons. Baryons : Fermionic states carrying baryon number, simplest example proton and neutron Hyperons (Y) are baryons containing net strangeness, but no charm, bottom or top Mesons : Bosonic states, for example π + Kaons have net strangeness Pions ( 140 MeV) by far lightest hadrons Baryon octet s = 0 n p Σ 0 s = 1 Σ Λ s = 2 Ξ Ξ 0 Pseudoscalar meson nonet Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52 Σ +

54 Can there be hyperons in the core? Hyperons (Y) are baryons containing strangeness, for example : S = -1 : Λ(mΛ = 1115MeV), Σ ±, Σ 0 (m Σ = 1190MeV) S = -2 : Ξ 0, Ξ (m Ξ = 1318MeV) They can appear in the core of neutron stars if their chemical potential becomes large enough to make the conversion of N into Y energetically favorable. Strangeness is conserved in strong reactions which would mean that a strangeness chemical potential has to be introduced. In neutron stars, time scales are in general long enough to suppose weak equilibrium which means that strangeness is not conserved. µ Λ = µ Σ 0 = µ Ξ 0 = µ B µ Σ ± = µ B ± µ q µ Ξ = µ B µ q Examples of (weak) reactions to produce hyperons (produced in laboratories) : n + n n + Λ p + e Λ + ν e n + n p + Σ Without interaction (E( p = 0) = m), in general Σ would appear first, then Λ. Experimental information on hyperonic interaction scarce Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

55 Hyperons in neutron stars In most models hyperons appear at n B 2 3n 0 The order in which the different hyperons appear depends on the interaction Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

56 Hyperons in neutron stars 10 0 n MC1-HF/NY abundances p e - µ Ξ Ξ 0 Λ n ρ[fm -3 ] MC1-H/NY FG abundances e - p µ Σ Λ (Massot, Margueron & Chanfray, 21) Σ 0 Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

57 Hyperons in neutron stars In most models hyperons appear at n B 2 3n 0 The order in which the different hyperons appear depends on the interaction Hyperons add d.o.f. particles can be distributed on more Fermi seas softening of the EOS lower maximum mass of neutron stars This argument is true without interaction! Short-range repulsion needed (e.g. from three-body forces) In microscopic models (BHF) this seems to be a problem (Vidaña et al., 20) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

58 Hyperons in neutron stars Gravitational Mass [M ] PSR J PSR J Hulse-Taylor Radius R [km] (Vidaña et al., 20) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

59 Hyperons in neutron stars In most models hyperons appear at n B 2 3n 0 The order in which the different hyperons appear depends on the interaction Hyperons add d.o.f. particles can be distributed on more Fermi seas softening of the EOS lower maximum mass of neutron stars This argument is true without interaction! Short-range repulsion needed (e.g. from three-body forces) In microscopic models (BHF) this seems to be a problem (Vidaña et al., 20) In phenomenological models less difficult (models adjusted to provide necessary repulsive interaction) (Haensel et al. 21,...) Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52

60 Hyperons in neutron stars Two RMF models with different hyperonic interactions DDHδ GM1 PSR J M/M sol R [km] Micaela Oertel (LUTH) EoS for NS and SN Barcelona, September 22-26, / 52