Neutron Star Mass and Radius Constraints on the Dense Matter Equation o

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1 Neutron Star Mass and Radius Constraints on the Dense Matter Equation of State Department of Physics & Astronomy Stony Brook University 20 June 2011 Collaborators: E. Brown (MSU), K. Hebeler (OSU), D. Page (UNAM), C.J. Pethick (NORDITA), M. Prakash (Ohio U.), A. Schwenk (TU Darmstadt), A. Steiner (MSU) MICRA 2011 Perimeter Institute

2 Outline Neutron Star Structure Neutron Star Limits from General Relativity and Causality Mass Measurements 2 M Neutron Stars? Limits to the Extent of Quark Matter Neutron Star Radii Relation to the Nuclear Symmetry Energy Thermal Emission from Cooling Neutron Stars Photospheric Radius Expansion X-Ray Bursters The Universal Mass-Radius Relation and the Neutron Star EOS Consistency with Neutron Matter Expectations Implications for Other Laboratory Constraints Time Permitting: The Evolution of Neutron Stars Neutron Star Cooling and the Direct Urca Process Cas A: A Direct Detection of Core Superfluidity?

3 Neutron Star Structure Tolman-Oppenheimer-Volkov equations p(ε) dp dr dm dr = G (m + 4πpr 3 )(ε + p) c 2 r(r 2Gm/c 2 ) = 4π ε c 2 r 2 maximum mass M(R)

4 Extreme 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). p = ε ε o causal limit ε 0 is the only EOS parameter The TOV solutions scale with ε 0 soft = p = 0 = stiff w = ε/ε 0 y = p/ε 0 x = r Gε 0 /c 2 z = m G 3 ε 0 /c 2 ɛ o

5 Extreme Properties of Neutron Stars M max = 4.1 (ε s /ε 0 ) 1/2 M (Rhoades & Ruffini 1974) M B,max = 5.41 (m B c 2 /µ o )(ε s /ε 0 ) 1/2 M R min = 2.82 GM/c 2 = 4.3 (M/M ) km µ B,max = 2.09 GeV ε c,max = ε 0 51 (M /M largest ) 2 ε s p c,max = ε 0 34 (M /M largest ) 2 ε s n B,max 38 (M /M largest ) 2 n s BE max = 0.34 M P min = 0.74 (M /M sph ) 1/2 (R sph /10 km) 3/2 ms = 0.20 (M sph,max /M ) ms

6 Maximum Energy Density in Neutron Stars p = s(ε ε 0 )

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 = contours 1 2GM/Rc 2

8 Black hole? Firm lower mass limit? M > 1.68 M { 95% confidence Freire et al { Although simple average mass of w.d. companions is 0.27 M larger, weighted average is 0.08 M smaller } w.d. companion? statistics? Demorest et al Champion et al. 2008

9 PSR J ms pulsar in 8.69d orbit with 0.5 M white dwarf companion. Shapiro delay tightly confines the edge-on inclination: sin i = Pulsar mass is 1.97 ± 0.04 M Distance > 1 kpc, B G t(µs) Demorest et al. 2010

10 Black Widow Pulsar PSR B ms pulsar in circular 9.17h orbit with a M c 0.03 M companion. Pulsar is eclipsed for minutes each orbit; eclipsing object has a volume much larger than the companion or its Roche lobe. It is believed the companion is ablated by the pulsar leading to mass loss and an eclipsing plasma cloud. Companion nearly fills its Roche lobe. Ablation by pulsar leads to eventual disappearance of companion. The optical light curve does not represent the center of mass of the companion, but the motion of its irradiated hot spot. pulsar radial velocity eclipse NASA/CXC/M.Weiss

11 Implications of Maximum Masses M max > 2 M Upper limits to energy density, pressure and baryon density: ε < 13.1εs p < 8.8εs nb < 9.8n s Lower limit to spin period: P > 0.56 ms Lower limit to neutron star radius: R > 8.5 km Upper limits to energy density, pressure and baryon density in the case of a quark matter core (s = 1/3): ε < 7.7εs p < 2.0εs nb < 6.9n s M max > 2.4 M Upper limits to energy density, pressure and baryon density: ε < 8.9εs p < 5.9εs nb < 6.6n s Lower limit to spin period: P > 0.68 ms Lower limit to neutron star radius: R > 10.4 km Upper limits to energy density, pressure and baryon density in the case of a quark matter core (s = 1/3): ε < 5.2εs p < 1.4εs nb < 4.6n s

12 Low-Mass Neutron Stars Theoretial limit for minimum mass of neutron stars is about 0.09 M. Practical limit, on the basis that neutron stars form from lepton-rich proto-neutron stars, can t exceed of 1 M and could be close to 1.2 M. Recent refined mass determinations of X-ray pulsar binaries (Rawls, Orosz, McClintock and Torres (2011): Vela X-1: 1.77 ± 0.08 M 4U : 0.87 ± 0.07 M (eccentric orbit), 1.00 ± 0.10 M (circular orbit) SMC X-1: 1.04 ± 0.09 M Y l = 0.4, s in = 1, s out = 4 5 Y l = 0.4, s = 1 2 Y ν = 0, s = 1 2 T = 0 Strobel, Schaab & Weigel (1999) LMC X-4: 1.29 ± 0.05 M Cen X-3: 1.49 ± 0.08 M Her X-1: 1.07 ± 0.36 M

13 Neutron Star Matter Pressure and the Radius p Kn γ γ = d ln p/d ln n 2 R K 1/(3γ 4) M (γ 2)/(3γ 4) R p 1/2 f n 1 f M 0 (1 < n f /n s < 2) Wide variation: 1.2 < p(ns ) MeV fm 3 < 7 n s

14 The Radius Pressure Correlation R p 1/4 Lattimer & Prakash (2001)

15 The Pressure of Neutron Star Matter Expansion of cold nucleonic matter energy near n s and isospin symmetry x = 1/2: E(n, x) E(n, 1/2) + E sym (n)(1 2x) c 4 x(3π2 nx) 1/3, ] P(n, x) n 2 [ de(n, 1/2) dn + de sym (1 2x)2 dn µ e = c(3π 2 nx) 1/3, E(n, 1/2) B + K ( ) 18 Beta Equilibrium: E = µ p µ n + µ e = 0. x n ( ) 3 x β (3π 2 n) 1 4Esym, c ( ) P β Kn2 n 1 9n 0 n s E sym (n s ) S v 30 MeV, c 200 MeV/fm, + c 4 nx(3π2 nx) 1/3, ( 1 n ) 2. n s + n 2 (1 2x β ) 2 de sym dn + E symnx β (1 2x β ) n n s = x β 0.04, P β n 2 s (de sym /dn) ns.

16 The Uncertain E sym (n) C. Fuchs, H.H. Wolter, EPJA 30(2006) 5

17 Nuclear Structure Considerations Information about E sym (n) can be extracted from nuclear binding energies and models for nuclei. For example, consider the schematic liquid droplet model (Myers & Swiatecki): S s S v R E(A, Z) a v A + a s A 2/3 + 0 ( ) Sv n E sym (n) 1 d 3 r Fitting binding energies results in a strong correlation between S v and S s, but not definite values. Another observable: neutron skin thickness δr S s /S v. Blue: E < 0.01 MeV/b Green: E < 0.02 MeV/b Gray: E < 0.03 MeV/b Circle: Moeller et al. (1995) Crosses: Best fits Dashed: Danielewicz (2004) Solid: Steiner et al. (2005) S v 1 + (S s /S v )A 1/3 A + a C Z 2 A 1/3

18 Radiation Radius The measurement of flux and temperature yields an apparent angular size (pseudo-bb): R d = R d 1 1 2GM/Rc 2 Observational uncertainties include distance, interstellar H absorption (hard UV and X-rays), atmospheric composition Best chances for accurate radii: Nearby isolated neutron stars (parallax measurable) Quiescent X-ray binaries in globular clusters (reliable distances, low B H-atmosperes)

19 Inferred M-R Probability Estimates from Thermal Sources Steiner, Lattimer & Brown 2010

20 Photospheric Radius Expansion X-Ray Bursts F Edd = GMc κd 1 2GM 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

21 Systematics with R ph = R F Edd = GMc κd GM R ph c 2 = GMc 1 2β κd 2 κ 0.2(1 + X ) cm 2 g 1 A = F σt 4 = f 4 c ( ) 2 R D α = F Edd κd A c 3 fc 2 = β(1 2β) γ = Ac3 fc 4 F Edd κ = R β(1 2β) 3/2 β = 1 4 ± α R = αγ If R ph >> R, α < 1/

22 M R Probability Estimates from PRE Bursts EXO α = 0.14 ± 0.01 R ph = R 4U α = 0.18 ± 0.02 EXO α = 0.14 ± 0.01 R ph > R 4U α = 0.18 ± U α = 0.26 ± 0.10 Özel et al. 2009, 2010, 2011 α = 0.21 ± U Steiner, Lattimer & Brown 2010, 2011 α = 0.26 ± 0.10 α = 0.21 ± 0.06

23 Bayesian TOV Inversion ε < 0.5ε 0 : Known crustal EOS 0.5ε 0 < ε < ε 1 : EOS parametrized by K, K, S v, γ ε 1 < ε < ε 2 : n 1 ; ε > ε 2 : Polytropic EOS with n 2 inferred p(ε) EOS parameters (K, K, S v, γ, ε 1, n 1, ε 2, n 2 ) uniformly distributed M and R probability distributions for 7 neutron stars treated equally. Steiner, Lattimer & Brown 2010 inferred M(R)

24 Inferred Model EOS Parameters K K Steiner, Lattimer & Brown 2010 S v γ

25 Consistency with Neutron Matter and Heavy-Ion Collisions

26 With More Extreme Assumptions M max 2.4M Smaller f c Skyrme forces with too large symmetry energy Fiducial SLB 2010 Strange Quark Stars

27 Radius and Maximum Mass Limits Hebeler et al preliminary! M max implied by R of 1.4 M star.

28 Neutron Matter and the Symmetry Energy Fits to nuclear binding energies result in a strong, nearly linear, correlation between volume and surface symmetry energy coefficients of the liquid droplet model. This correlation is dependent on the nature of the liquid droplet model and how it treats the interaction between the Coulomb effects on the nuclear surface, and does not translate directly into a correlation between S v and L = 3(dS v /d ln n) ns. Finite nucleus models, such as Thomas-Fermi and Hartree or Hartree-Fock, for a particular nuclear interaction, can be fit to binding energies to obtain the correlation between S v and L. Neutron matter studies (Hebeler & Schwenk; Carlson et al.) indicate that E sym and de sym /d ln n) ns are also correlated. Comparing these correlations could constrain the properties of the symmetry energy. It could be dependent on the nature of the nuclear interaction model, but this has not been thoroughly explored.

29 Neutron Matter and Mass Fit Symmetry Correlations

30 Neutron Star Cooling Cas A J Page, Steiner, Prakash & Lattimer (2004)

31 Transitory Rapid Cooling MU emissivity: ε MU T 8 PBF emissivity (f 10): ε PBF F (T ) T 7 T 8 f ε MU Specific heat: C V T Neutrino dominated cooling: C V dt /dt = L ν core temperature No p superconductivity With p superconductivity = T (t/τ) 1/6 τ PBF = τ MU /f (d ln T /d ln t) transitory (1 10)(d ln T /d ln t) MU (1 25)(d ln T /d ln t) MU (p SC) Very sensitive to n 3 P 2 critical temperature (T C ) and existence of proton superconductivity surface temperature Page et al T C

32 Cas A Remnant of Type IIb (gravitational collapse, no H envelope) SN in 1680 (Flamsteed). 3.4 kpc distance 3.1 pc diameter Strongest radio source outside solar system, discovered in X-ray source detected (Aerobee flight, 1965) X-ray point source detected (Chandra, 1999) 1 of 2 known CO-rich SNR (massive progenitor and neutron star?) Spitzer, Hubble, Chandra

33 Cas A Superfluidity X-ray spectrum indicates thin C atmosphere, T e K (Ho & Heinke 2009) Page et al years of X-ray data show cooling at the rate d ln T e = 1.23 ± 0.14 d ln t (Heinke & Ho 2010) Modified ( Urca: d ln Te ) d ln t MU 0.08 We infer that T C 5 ± K T C (t C L/C V ) 1/6

34 Conclusions Maximum neutron star mass may be above 2 M based on pulsar mass measurements and inferred from astrophysical observations of photospheric radius expansion bursts and thermal emissions. Quark stars or quark cores in neutron stars are nearly impossible if M max > 2.4 M. In spite of large estimated errors for individual stars, collective M R data predict a remarkably tight pressure-density relation. Astrophysical observations are consistent with predictions of neutron matter calculations and heavy-ion experiments. Neutrons are superfluid and protons superconducting in neutron star interiors. Predictions: A relatively soft nuclear symmetry energy, γ = 0.3 ± 0.1. Smallish radii of 1.4 M neutron stars, 11.3 ± 0.3 km. Small neutron skin thickness of 208 Pb, δr 0.15 ± 0.02 fm. A small n 3 P 2 gap, T C K. A p 1 S 0 gap with T C > K. If Mmax 2.4 M, radius estimates increase by 1 km.