Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Hadronic Matter and the Equation of State Jurgen¨ Schaffner-Bielich Institut fur¨ Theoretische Physik NewCompstar School 2017 – Neutron stars: theory, observations and gravitational wave emission University of Sofia, Bulgaria, September 11-15, 2017 Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Content 1 Introduction: Compact Objects 2 Matter in White Dwarfs 3 Equation of State of Degenerate Gases 4 Mass-Radius Relations 5 Maximum Masses of Compact Stars 6 Scaling Solutions for Compact Stars 7 Modelling Compact Stars 8 Quarks in Neutron Stars! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Compact Objects compact objects: endpoint of stellar evolution, white dwarfs, neutron stars, and black holes differences to ordinary stars: do not burn nuclear fuel, not supported by thermal pressure against gravity white dwarfs: supported by degeneracy pressure of electrons neutron stars: supported by interaction pressure of nucleons black holes: completely collapsed, singularity! all are essentially static (except mini black holes) over the lifetime of the universe and (astronomically) small in size Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Comparison of compact objects mass radius density (g/cm3) GM=R −6 Sun M⊙ R⊙ 1.4 10 7 −4 white dwarf ≤ 1:44M⊙ ∼ 0:01R⊙ ≤ 10 ∼ 10 −5 15 −1 neutron star 0:1 ::: 2 − 3M⊙ 10 R⊙ ≤ 10 ∼ 10 2 3 black hole MPlanck ::: Muniverse 2GM=c ∼ M=R ∼ 1 33 10 (M⊙ = 1:989 · 10 g, R⊙ = 6:9599 · 10 cm) huge range of densities, involves all four fundamental forces (strong, weak, electromagnetic, gravity) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Etymology of compact objects white dwarfs: high effective temperatures, much ’whiter’ than ordinary stars neutron stars: dominantly neutrons in the interior, − inverse β-decay of e + p ! n + νe, 57 giant nuclei (ρ ≈ ρnuclei) with 10 baryons! black holes: no light (matter) can escape, appears ’black’ to observer Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Observations white dwarfs: by optical telescopes (Sirius B!) neutron stars: optical (ESO, Hubble Space Telescope), x-ray satellites, radio (pulsating source of radiation: pulsars!) black holes: indirectly by mass-radius constraint, infalling objects (x-rays), supermassive ones by observing active galactic nuclei (AGN) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Stellar Evolution (Credit: NASA/CXC/M.Weiss) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Formation of Compact Objects endpoints of stellar evolution: M < 0:08M⊙: no stable nuclear burning ! brown dwarfs M < 0:4M⊙: hydrogen burning ! H-He white dwarfs M < 8M⊙: helium burning ! C-O white dwarfs (our Sun!) 8 < M < 10M⊙: carbon burning, degenerate Ne-O-Mg core, core collapse supernova ! neutron star M > 10M⊙: Ne-O-Si burning, degenerate Fe core, core collapse supernova ! neutron star M & 25M⊙: core collapse supernova ! black hole M ∼ 100M⊙: upper mass limit for stars (formation of black holes also by supermassive stars (first stars), primordial black holes (’mini black holes’ with M > 1015g) or collision of compact stars in binary systems) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Matter in White Dwarfs relation between pressure and energy density ! the equation of state (EoS) global properties: large-scale dynamical response to matter to gravity, rotation, electromagnetic fields local properties: internal stress, dissipation, emissivity Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Thermodynamics I first law of thermodynamics: ( ) ( ) ϵ 1 X d = −P · d + T · ds + µ · dY (1) n n i i i for different species with a concentration Yi = ni =n n: number density of baryons (B is conserved!) ni : number density of species i (not conserved!) ϵ: energy density ϵ/n: energy per baryon (E=A) s: entropy density µi : chemical potential for species i Yi : particle fraction of species i (Yi = ni =n) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Thermodynamics II Thermodynamic relations for ϵ = ϵ(n; s; Yi ): @(ϵ/n) @(ϵ/n) P = − = n2 (2) @(1=n) @n @(ϵ/n) @(ϵ/n) @ϵ T = ; µi = = (3) @s @Yi @ni in equilibrium reactions between particles are in detailed balance, i.e. each reaction is balanced by backreaction and ni is constant ! Yi is not an independent quantity! system close to equilibrium, thermally isolated (δQ = T ds = 0), constant volume (d(1=n) = 0), constant energy (no work, d(ϵ/n) = 0), hence X µi · dYi = 0 (4) i Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Weak Reactions look at weak reaction − e + p ! n + νe (5) − − then dYe = dYp = dYn = dYνe and µe + µp = µn + µνe (6) composition determined by chemical potentials at fixed values of n and ϵ number of independent quantities required: consider mixture of baryons (neutrons, protons, . ) and − − leptons (e , µ , νe, νµ,...) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Independent quantities all reactions conserve baryon number (nB), lepton number (nL) and charge (nQ) choose nB, nL, and nQ, then all chemical potentials are fixed in terms of the corresponding chemical potentials µB, µL and µQ in general for a species i: µi = Bi · µB + Li · µL + Qi · µQ (7) thermodynamic quantities are a function of µi and T only astrophysical matter is globally charge neutral: nQ = 0 fixes µQ no trapped neutrinos for cold matter: µνe = µνµ = 0 cold degenerate matter (the temperature is much smaller than the Fermi energy T ≪ EF ): T = 0 =) everything given in terms of µB or nB! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Beta-equilibrium I ideal cold gas of electrons, protons, and neutrons treat neutrons and protons as free particles first (in white dwarfs they are bound in nuclei) − e + p ! n + νe (8) reaction proceeds, when e− has enough energy to overcome 2 the mass difference of (mn − mp)c = 1:29 MeV inverse reaction − n −! e + p +ν ¯e (9) is blocked, if the density of e− is high enough, so that all available energy levels are occupied assume equilibrium (T = 0, µν = 0), then µe + µp = µn (10) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Beta-equilibrium II the Fermi energy equals the chemical potential, hence q i 2 2 µi = EF = pF;i + mi (11) with the Fermi momenta pF;i and the equilibrium condition implies q q q 2 2 2 2 2 2 pF;e + me + pF;p + mp = pF;n + mn (12) number densities are given by Z pF;i dp3 1 = · = 3 ni gi 3 gi 2 pF;i (13) 0 (2π) 6π with the degeneracy factor gi = 2si + 1. Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Beta-equilibrium III Charge neutrality implies −! · 3 · 3 −! ne = np ge pF;e = gp pF;p pF;e = pF;p (14) determines everything in terms of µB or nB. Thermodynamic quantities: P = Pe+Pp+Pn ϵ = ϵe+ϵp+ϵn nB = np+nn nQ = np−ne = 0 (15) neutrons appear at pF;n = 0, then eq. (12) reads q q 2 2 2 2 pF;e + me + pF;p + mp = mn (16) or for the nonrelativistic limit for protons pF;p ≪ mp: q 2 2 − pF;e + me = mn mp (17) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Beta-equilibrium IV critical electron density (which is equal to the proton density) ( ) 1 3=2 n = (m − m )2 − m2 = 7:4 · 1030 cm−3 (18) c 3π2 n p e amounts to a mass density of 7 −3 ρc = mp · nc = 1:2 · 10 g cm (19) the neutron fraction increases drastically then so that nn ≫ np.