Neutron Star Observations and Their Implications for the Nuclear Equation of State

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1 Neutron Star Observations and Their Implications for the Nuclear Equation of State J. M. Lattimer Department of Physics & Astronomy Stony Brook University May 24, May, 2016, JINA-CEE International Symposium Neutron Stars in the Multi-Messenger Era: Prospects & Challenges Athens, Ohio

2 Outline The Dense Matter Equation of State and Neutron Star Structure General Causality, Maximum Mass and GR Limits Neutron Matter and the Nuclear Symmetry Energy Theoretical and Experimental Constraints on the Symmetry Energy Extrapolating to High Densities with Piecewise Polytropes Constraints on Neutron Star Radii Universal Relations Observational Constraints on Radii Photospheric Radius Expansion Bursts Thermal Emission from Quiescent Binary Sources Effects of Systematic Uncertainties Further Observations of Masses and Radii

3 Neutron Star Structure Tolman-Oppenheimer-Volkov equations dp dr dm dr = G (mc 2 + 4πpr 3 )(ε + p) c 4 r(r 2Gm/c 2 ) = 4π ε c r 2 2 p(ε) maximum mass small range of radii M(R) Equation of State Observations

4 Extremal Properties of Neutron Stars The most compact and massive configurations occur when the low-density equation of state is soft and the high-density equation of state is stiff (Koranda, Stergioulas & Friedman 1997). ε o is the only EOS parameter p = ε ε o causal limit The TOV solutions scale with ε o soft = p = 0 ε o = stiff w = ε/ε o y = p/ε o x = r Gε o /c 2 z = m G 3 ε o /c 2

5 Causality + GR Limits and the Maximum Mass A lower limit to the maximum mass sets a lower limit to the radius for a given mass. Similarly, a precise (M, R) measurement sets an upper limit to the maximum mass. 1.4M stars must have R > 8.15M if M max 2.01M. = M R curves for minimally compact EOS = quark stars p ε = c2 s c 2 = s

6 vankerkwijk 2010 Romani et al Although simple average mass of w.d. companions is 0.23 M larger, weighted average is 0.04 M smaller Demorest et al Fonseca et al Antoniadis et al Barr et al Champion et al. 2008

7 Mass-Radius Diagram and Theoretical Constraints GR: R > 2GM/c 2 P < : R > (9/4)GM/c 2 causality: R > 2.9GM/c 2 normal NS SQS R R = 1 2GM/Rc 2

8 The Radius Pressure Correlation 9.52 ± 0.49 (km fm 3/4 MeV 1/4 ) 7.06 ± ± 0.14 Lattimer & Prakash (2001) Lattimer & Lim (2013)

9 Nuclear Symmetry Energy and the Pressure The symmetry energy is the difference between the energies of pure neutron matter (x = 0) and symmetric (x = 1/2) nuclear matter: S(ρ) = E(ρ, x = 0) E(ρ, x = 1/2) Usually approximated as an expansion around the saturation density (ρ s ) and isopin symmetry (x = 1/2): E(ρ, x) = E(ρ, 1/2)+(1 2x) 2 S 2 (ρ)+... S 2 (ρ) = S v + L ρ ρ s ρ s S v 31 MeV, L 50 MeV C. Fuchs, H.H. Wolter, EPJA 30(2006) 5 symmetry energy Extrapolated to pure neutron matter: E(ρ s, 0) S v + E(ρ s, 1/2) S v B, p(ρ s, 0) = Lρ s /3 Neutron star matter (beta equilibrium) is nearly neutron matter: (E + E e ) x = 0, p(ρ s, x β ) Lρ s 3 [ 1 ( 4Sv c ) ] 3 4 3S v /L 3π 2 ρ s

10 Theoretical Neutron Matter Calculations Auxiliary Field Diffusion Monte Carlo Calculations (Gandolfi & Carlson) Hamiltonians of two- and three-nucleon interactions for pure neutron matter Two-body interactions are obtained by fitting nucleon-nucleon scattering data up to laboratory energies of 350 MeV, which also fit elastic phase shifts up to much higher energies. (At saturation density, two neutrons on the Fermi surface with momenta ±k f have a relative laboratory-frame energy of about 240 MeV.) Three-body interactions are fitted to low energy properties of light nuclei. Four- and higher-body forces have been found to be negligible.

11 2-body only Gandolfi et al. (2015)

12 Chiral Lagrangian Approach (Drischler, Hebeler & Schwenk) Expand the effective interaction in powers of the nucleon momenta. The form of the operators that can occur is determined by the imposition of chiral symmetry. Short-range correlations are included through the effective low energy interactions; in many cases, the many-body calculations become perturbative. As in the QMC approach, the interaction parameters are determined by fitting both scattering data and properties of light nuclei.

13 Drischler, Hebeler & Schwenk (2015)

14 Nuclear Experimental Constraints The liquid droplet model is a useful frame of reference. Its symmetry parameters S v and S s are related to S v and L: S s al [ 1 + L K ] sym S v r o S v 6S v 12L Binding Energy (linear correlation between S v and L) [ E sym S v I S ] 1 s. S v A 1/3 Giant Dipole Resonance (dipole polarizability, also linear) α D AR ( ) S s. 20S v 3 S v A 1/3 Neutron Skin Thickness (approximately, L constant) ( 3 2r o I S s r np 1 + S ) 1 ( s ) S s. 5 3 S v S v A 1/3 3 S v A 1/3

15 Theoretical and Experimental Constraints H Chiral Lagrangian G: Quantum Monte Carlo S v L constraints from Hebeler et al. (2012) Neutron matter constraints are compatible with experimental constraints.

16 Neutron Star Crusts The evidence is overwhelming that neutron stars have crusts. Neutron Star Cooling, both long term (ages up to millions of years) and transient (days to years), supports the existence of km thick crusts with mass M. Pulsar glitches are best explained by crustal n 1 S 0 superfluidity. Crustal I /I The crust EOS is very well understood.

17 Piecewise Polytropes We know the crust EOS for n < 0.4n s. What about higher densities? Read, Lackey, Owen & Friedman (2009) found high-density EOS can be modeled as piecewise polytropes with 3 segments. The break points n n s and n 2 3.7n s optimized fits to the entire family of modeled EOSs. For n < n 0 use crust EOS. For n 0 < n < n 1 use neutron matter EOS. Arbitrarily choose n 3 = 7.4n s. 0 < Γ 2 < Γ 2c or p 1 < p 2 < p 2c. Γ 2 < Γ 3 < Γ 3c or p 2 < p 3 < p 3c. nm n 0, p 0 n1, p 1 n2, p 2 n 3, p 3

18 Maximum Mass and Causality Constraints

19 Maximum Mass and Causality Constraints

20 Radius - p 1 Correlation

21 Mass-Radius Constraints from Causality

22 Upper Limits to Maximum Mass

23 Universal Relations With the assumptions Known crust EOS Bounded neutron matter EOS (p min < p 1 < p max ) Two piecewise polytropes for p > p 1 Causality is not violated M max is limited from below tight correlations among the compactness, moment of inertia, binding energy and tidal deformability result.

24 Moment of Inertia - Compactness Correlations

25 Moment of Inertia - Compactness Correlations

26 Binding Energy - Compactness Correlations

27 Binding Energy - Compactness Correlations

28 Tidal Deformatibility - Moment of Inertia

29 Tidal Deformability - Moment of Inertia

30 Moment of Inertia - Radius Constraints

31 Simultaneous Mass/Radius Measurements Measurements of flux F = (R /D) 2 σteff 4 and color temperature T c λ 1 max yield an apparent angular size (pseudo-bb): R /D = (R/D)/ 1 2GM/Rc 2 Observational uncertainties include distance D, interstellar absorption N H, atmospheric composition Best chances are: Isolated neutron stars with parallax (atmosphere??) Quiescent low-mass X-ray binaries (QLMXBs) in globular clusters (reliable distances, low B H-atmosperes) Bursting sources (XRBs) with peak fluxes close to Eddington limit (gravity balances radiation pressure) F Edd = cgm κd 2 1 2GM/Rc 2

32 Photospheric Radius Expansion X-Ray Bursts F Edd = GMc 1 2GM κd 2 R ph c 2 F Edd EXO Galloway, Muno, Hartman, Psaltis & Chakrabarty (2006) A = f 4 c (R /D) 2 A = f 4 c (R /D) 2

33 PRE Burst Models Observational measurements: F Edd, = GMc 1 2β, κd Determine parameters: A = F σt 4 α = F Edd, A γ = Af c 4 c 3 = R κf Edd, α, Solution: = f 4 c ( ) 2 R D GM β = Rc 2 κd = β(1 2β) fc 4 c 3 β = 1 1 8α 4 ±, α independent of D for real solutions.

34 PRE M R Estimates α min ± Ozel & Freire (2016)

35 4U A PRE Example

36 QLMXB M R Estimates Ozel & Freire (2016)

37 Combined R fits Assume all sources have the same radius with the same mass distribution as measured from pulsar timing (taken to be Gaussian with M = 1.46 ± 0.21M ). Ozel & Freire (2015)

38 Piecewise-Polytrope Radii Distributions

39 Role of Systematic Uncertainties Apparent tension between nuclear physics expectations and astronomical observations for the value of the mean radius. Possible explanations: Non-uniform temperature distributions R 2 1 R 2 T 4 = R1 2 T1 4 + R 2 T2 4 ( R 2 3 hν ) 0 0 kt ( 3 hν 0 kt 2 ( 3 hν ) 0 e hν 0/kT 1 + R2 2 kt 1 ) e hν 0/kT 2 0 R 1 = R 2 = Ry/ 2, x = T 2 /T 1, z = T /T 1 z 4 = y 2 (1 + x 4 )/2, For 0 < x < 1, z < 1 and y > 1, so that z = 1 + (1 z/x)e 3z/x e 3z R R 2 2 = y 2 R 2 > R 2

40 Interstellar absorption Atmospheric composition: In quiescent sources, He or C atmospheres can produce about 50% larger radii. Non-spherical geometries: In bursting sources, improper to use spherically-symmetric Eddington flux formula. Disc shadowing: In burst sources, leads to underprediction of A = fc 4 (R /D) 2, overprediction of α 1/ A, and underprediction of R α.

41 Folding Observations with Piecewise Polytropes

42 Bayesian Analyses Ozel et al. (2015) Steiner et al. (2012)

43 Observations of Gravitational Radiation R = 11 km R = 15 km Takami, Rezzolla and Baiotti (2014) M 1 = M 2 Chirp mass and tidal deformability measurable during inspiral. Frequency peaks may be tightly correlated with compactness. Maximum mass determinations from black hole formation. R-mode instabilities in rotating neutron stars.

44 Additional Proposed Radius and Mass Constraints Pulse profiles Hot or cold regions on rotating neutron stars alter pulse shapes: NICER and LOFT will enable timing and spectroscopy of thermal and non-thermal emissions. Light curve modeling M/R; phase-resolved spectroscopy R. Moment of inertia Spin-orbit coupling of ultra- relativistic binary pulsars (e.g., PSR ) vary i and contribute to ω: I MR 2. Supernova neutrinos Millions of neutrinos detected from a Galactic supernova will measure BE= m B N M, < E ν >, τ ν. QPOs from accreting sources ISCO and crustal oscillations Neutron star Interior Composition ExploreR NASA Large Observatory For x-ray Timing ESA/NASA

45 Conclusions Neutron matter calculations and nuclear experiments are consistent with each other and set reasonably tight constraints on symmetry energy behavior near the nuclear saturation density. These constraints, together with assumptions that neutron stars have hadronic crusts and are causal, predict neutron star radii R 1.4 in the range 12.0 ± 1.2 km. Astronomical observations of photospheric radius expansion X-ray bursts and quiescent sources in globular clusters suggest R ± 1 km, unless maximum mass and EOS priors are implemented. Should observations require smaller or larger neutron star radii, a strong phase transition in extremely neutron-rich matter just above the nuclear saturation density is suggested. Or should GR be modified?

46 Consistency with Neutron Matter and Heavy-Ion Collisions