The Equation of State for Neutron Stars from Fermi Gas to Interacting Baryonic Matter. Laura Tolós

Transcription

1 The Equation of State for Neutron Stars from Fermi Gas to Interacting Baryonic Matter Laura Tolós

2 Outline

3 Outline

4 Neutron Star (I) first observations by the Chinese in 1054 A.D. and prediction by Landau after discovery of neutron by Chadwick in 1932 produced in core collapse supernova explosions usually refer to compact objects with M 1-2 M and R Km extreme densities up to 5-10 times nuclear density ρ 0 (ρ 0 =0.16 fm -3 => n 0 = g/cm 3 ) n Universe ~ g/cm 3 n Sun ~ 1.4 g/cm 3 n Earth ~ 5.5 g/cm 3

5 Neutron Star (II) usually observed as pulsars with rotational periods from milliseconds to seconds magnetic field : B ~ G temperature: T ~ K (1 ev=10 4 K)

6 Pulsars are magnetized rotating neutron stars emitting a highly focused beam of electromagnetic radiation oriented long the magnetic axis. The misalignment between the magnetic axis and the spin axis leads to a lighthouse effect: from Earth we see pulses Since 1967, 2500 pulsars have been discovered. Most of them have been detected as radio pulsars, but also some observed in X-rays and an increasingly large number detected in gamma rays. Their period P ranges from ms for PSRJ ad up to 8.5 s for PSR J

7 Magnetic fields

8 Bulk properties: Masses > 2000 pulsars known best determined masses: Hulse-Taylor pulsar M= ± M Hulse-Taylor Nobel Prize 94 figures by Weiseberg PSR J M=(1.97 ± 0.04) M ; PSR J M=(2.01 ± 0.04) M 1 Demorest et al 10; 2 Antoniadis et al 13 Lattimer 16

9 Bulk properties: Radius analysis of X-ray spectra from neutron star (NS) atmosphere: Fortin et al 15: Ø RP-MSP: Bodganov 13 Ø BNS-1: Nattila et al 16 Ø BNS-2: Guver & Ozel 13 Ø QXT-1: Guillot & Rutledge 14 Ø BNS+QXT: Steiner et al 13 RP-MSP: X-ray emission from radio millisecond pulsars BNS: X-burst from accreting NSs QXT: quiescent thermal emission of accreting NSs theory + pulsar observations: R 1.4M ~11-13 Km Lattimer and Prakash 16 Some conclusions: ü marginally consistent analyses, favored R < 13 Km (?) ü future X-ray telescopes (NICER, extp) with precision for M-R of ~ 5% ü GW signals from NS mergers with precision for R of ~1 km Bauswein and Janka 12; Lackey and Wade 15 adapted from Fortin s NewCompstar Annual Meeting 16

10 Internal structure and composition: the Core A. Watts et al. 15

11 Atmosphere few tens of cm, ρ 10 4 g/cm 3 made of atoms Outer crust or envelope few hundred m s, ρ= g/cm 3 made of free e - and lattice of nuclei Inner crust 1-2 km, ρ= g/cm 3 made of free e -, neutrons and neutron-rich atomic nuclei ~ρ 0 /2: uniform fluid of n,p,e - Outer core ρ 0 /2-2ρ 0 is a soup of n,e -,μ and possible neutron 3 P 2 superfluid or proton 1 S 0 superconductor Inner core (?) 2-10 ρ 0 with unknown interior made of hadronic, exotic or deconfined matter

12 The Core of a Neutron Star Fridolin Weber

13 Strange Quark Matter Hypothesis working hypothesis: strange quark matter constitutes the ground state of strong interaction rather than 56 Fe Mark Alford s lectures Pauli principle Bodmer,Witten & Terazawa

14 Equation of State of dense matter: free Fermi gas The Equation of State (EoS) is the relation between the mass/energy density and pressure: P(n), P(E) Extreme Conditions in the stellar interior! * Atoms are ionized * Particles can be degenerate and ultrarelativistic * Pressure due to radiation can be significant As a first approximation, let s consider the simplest thermodynamical system, the ideal gas. The ideal gas consists of a large number of particles occupying quantum states whose energy is not affected by the interaction between the particles. Depending on the conditions of the stellar interior the gas can be: Classical or Quantum (degenerate) and the energies of the particles can be: Non-relativistic or Relativistic

15 Ideal Gas Classical vs quantum (degenerate) Distribution of the particles in the quantum states, f(e p ) A gas is classical when the average occupation of any quantum state is small f(e p )= 1 e e p µ kt ± 1 ' e ep µ Relativistic vs non relativistic f(e p )= 1 e e p µ e 2 p = m 2 c 4 + p 2 c 2 kt ± 1 +1 Fermi-Dirac distribution - 1 Bose-Einstein distribution kt 1 μ decreases or T increases Non-relativistic: e p = mc 2 + p2 2m v p = p m Ultra-relativistic: e p = pc v p = c

16 General formulae Number of (wave vector) states with magnitude between k and k+dk g V s (2 ) 3 4 k2 dk Internal energy: Number of particles: Broglie relation (p= h/λ=hk), number of momentum states between p and p+dp: E = N = Z 1 Z0 1 0 _ e p f(e p )g(p)dp f(e p )g(p)dp g(p)dp = g s V h 3 4 p2 dp degeneracy factor Pressure: de=t ds- P dv + µ dn de p dv = de p dp P = 1 3V dp dv Z 1 0 de p dp = pc2 p / V e p = v p 1/3! dp dv = (definition) p 3V pv p f(e p )g(p)dp = N 3V hpv pi

17 P = 1 3V Z 1 0 pv p f(e p )g(p)dp = N 3V hpv pi Two kinematical limits: Non-relativistic: e p = mc 2 + p2 2m v p = p m P = 2N 3V h p2 2m i = 2 3 of kinetic energy density Ultra-relativistic: e p = pc v p = c P = N 3V hpci = 1 3 of kinetic energy density Pressure is smaller in ultrarelativistic gases!! Up to here, the expressions found are applicable to an ideal gas in its most general form: for a dilute classical gas or for a dense gas where quantum effects are important

18 Ideal Classical (Diluted) Gas f(e p )= P = kt V e µ kt Z 1 0 (integrating by parts) N =e µ kt Z 1 0 e Comparing expressions for P A more intuitive way of determining if a gas is classical (diluted) is by checking whether the interparticle separation is much larger that the typical de Broglie wave length λ=h/p, which is determined by non-relativistic: ultra-relativistic: e e p V kt gs h 3 4 p2 dp e p V kt gs h 3 4 p2 dp non-relativistic: ultra-relativistic: h p2 = h p h p mkt! = h p hc kt! kt hc 1 e e p µ kt ± 1 ' e ep µ kt 1 P = N V 2m i = 3 2 kt hpci =3kT mkt 3 kt = kt (average kinetic energy per particle) h 2 3/2 Gases formed by particles with mass, such as electrons and ions, can behave classically or quantum mechanically, depending on their density

19 Free Fermi Gas When the concentration of fermions becomes large, the interparticle separation may be comparable with the de Broglie wavelength. In particular, the high density requirement for a non-relativistic quantum gas 3/2 mkt can be viewed as a low temperature requirement Therefore a quantum gas is a cold gas but the coldness is set by the density. Temperatures of 10 6 K can be low if the densities are very high! Let s consider then a cold gas of fermions (electrons or neutrons at T=0) The ground state is referred to as the free Fermi sea The fermions have fallen into quantum states with the lowest possible energy Pauli principle! They are distributed so that each quantum state is occupied up to a certain energy (Fermi level, e F ) and quantum states above this level are empty. h 2 kt h2 2/3 m

20 This distribution is the T=0 limit of the Fermi-Dirac distribution (setting μ=e F ) 1 n(k) f(e k ) 1 k F e F e F (the energy of the most energetic particle) is called the Fermi energy and k F is the Fermi wave-number (p F = ħ k F is the Fermi momentum): e F =e(p F ) Relation between the Fermi momentum and the density: X Z N = g s (k F ~ V k ) =g s (2 ) 3 d 3 k (k F ~ V 4 k ) =g s (2 ) 3 3 k3 F ~ k! N V = = g s k 3 F 6 2 k F = 6 2 g s 1/3! F = 2 k F =2 Note that de Broglie wave-length of the most energetic particle is comparable with ρ -1/3, that is of the order of the average distance between fermions 1/3 gs 6 2

21 Equation of state for a non-relativistic free Fermi gas Non relativistic case: p F c = ħ k F c << m c /3 k F = mc ~! g s mc h g s Internal energy: h mc (neutrons) ~ m (electrons) ~10-12 m 3 neutrons are still not relativistic when electrons are! The Compton wavelength of a particle is equivalent to the wavelength of a photon whose energy is the same as the restmass energy of the particle e p = mc 2 + p2 Z 2m kf E = mc 2 + ~2 k 2 2m 0 g s V (2 ) 3 4 k2 dk = Nmc 2 + g s = N mc E = Ne() with e() =mc ~ 2 2m V 4 (2 ) 3 3 k3 F ~ 2 kf 2 2m 6 2 2/3 2/3 g s ~ 2 k 2 F 2m 3 5

22 From internal energy, we can obtain the pressure and chemical potential so we finally obtain for internal energy and pressure For ultrarelativistic free Fermi gas, the EoS becomes less stiff (do it!)

23 Baryonic (nucleonic) matter in the core A Fermi gas model for only neutrons inside neutron stars is unrealistic: 1. real neutron star consists not just of neutrons, but contains a small fraction of protons and electrons - to inhibit the neutrons from decaying into protons and electrons by their weak interactions! 2. the Fermi gas model ignores nuclear interactions, which give important contributions to the energy density 3. more exotic degrees of freedom are expected, in particular hyperons, due to the high value of density at the center and the small mass difference between nucleons and hyperons

24 Baryonic (nucleonic) matter in the core A Fermi gas model for only neutrons inside neutron stars is unrealistic: 1. real neutron star consists not just of neutrons, but contains a small fraction of protons and electrons - to inhibit the neutrons from decaying into protons and electrons by their weak interactions! 2. the Fermi gas model ignores nuclear interactions, which give important contributions to the energy density 3. more exotic degrees of freedom are expected, in particular hyperons, due to the high value of density at the center and the small mass difference between nucleons and hyperons

25 1. Neutrons, protons and electrons in β-equilibrium The composition of neutron star matter is found by demanding equilibrium against weak interaction processes (β-stability). Therefore, the reaction for the decay of a free neutron: (responsible for the free neutron lifetime of 15 minutes) is halted in neutron star matter by the presence of protons and electrons. Protons and neutrons in their lowest levels of their corresponding Fermi seas are occupied and the reaction is Pauli blocked. In this regime the decay reaction is equilibrated with the electron capture one: -This equilibrium can be expressed in terms of the chemical potentials. Since the mean free path of the ν e is >> 10 km, they freely escape - Charge neutrality is also ensured by demanding ρ p =ρ e, i.e. k Fp =k Fe ρ p = ρ e Note that baryon number is conserved too: ρ =ρn+ ρ p

26 2. Nuclear interactions the EoS can only fulfill known properties of nuclear matter and nuclei if nuclear interactions are considered - saturation density - binding energy per nucleon e() - nuclear compressibility K =9@P/@ - symmetry energy x p = p / 0 =0.16 ± 0.02 fm 3 E( e( 0 )= ) 15.6 ± 0.2 MeV K( 0 ) MeV S( 0 ) = 32.5 MeV - also causality imposes that speed of sound should be smaller than speed of light (v s /c) 2 = dp/d(e)

27 One may find in the literature many EoS s obtained with different approaches (phenomenologial, microscopic) using various types of interactions (nonrelativistic, relativistic, effective theories, meson-exchange) We employ the simple parametrization of the nucleonic energy per baryon from H. Heiselberg, M. Hjorth-Jensen, Phys. Reports 328 (2000) 237 S 0 u = / 0 x p = p / e 0 =15.8 MeV, S 0 =32 MeV, ϒ=0.6, δ=0.2 Symmetric nuclear matter (fitted to reproduce saturation density, binding energy and incompressibility modulus at saturation density) Pure neutron matter e e

28 β-stable matter with neutrons, protons and electrons ρ p = ρ e µ e =hc(3πρx - p ) 1/3 (ultrarelativistic) This particular model gives and explicit expression for S(ρ)=S 0 (ρ/ρ 0 ) ϒ This gives rise to a third degree equation that determines x p for each u=ρ/ρ 0. In fact, there is an analytical solution. (do it!) n, p, e - 1 n [MeV fm -3 ] P [MeV fm -3 ] Particle fractions p e n B [fm -3 ] n B [fm -3 ] n B [fm -3 ]

29 Next lecture: The Equation of State for Neutron Stars Hyperonic Matter and The Hyperon Puzzle