Proto - Neutron Stars and Neutron Stars

Transcription

1 Department of Physics & Astronomy Stony Brook University 6 7 May 2011 Ćompstar 2011: School Catania Gravitational Waves and Electromagnetic Radiation from Compact Stars

2 Outline The Evolution of Neutron Stars Protoneutron Stars Neutron Star Cooling and the Direct Urca Process Cas A: A Direct Detection of Core Superfluidity? 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 Dependence on Nuclear Symmetry Energy Thermal Emission from Cooling Neutron Stars Photospheric Radius Expansion X-Ray Bursters Consistency with Neutron Matter Expectations Other Astrophysical Constraints Laboratory Constraints

3 Proto-Neutron Stars

4 Proto-Neutron Star Evolution dp = G(M + 4πr 3 P)(ρ + P/c 2 ) dr r(r 2GM/c 2 ) dy ν dm = e φ (4πr 2 F ν e φ ) + S = 4πr 2 ν ρ dτ N dr dy e dn 4πr 2 n = S ν = dτ dr 1 2GM/rc 2 du = P d(1/n) e 2φ L νe 2φ dφ dp = 1 dτ dτ N P + ρc 2 In the diffusion approximation, fluxes are driven by density gradients: F ν = L ν = 0 0 c 3 ( n ν (E ν ) λ ν r 4πr 2 i cλ i E 3 n ν (E ν ) λ ν r ɛ i (E ν ) de ν. r ) de ν, λ ν, ν and λ i E are mean free paths for number and energy transport, respectively. n ν, ν (E ν ) is the number density of e-type and ɛ i (E ν ) iis the energy density of i = e, µ, τ-type neutrinos at energy E ν.

5 Proto-Neutron Star Evolution We examine the Newtonian case. Combine with the first law of thermodynamics to obtain the rate of change of the total lepton number and the entropy: n dy L dt nt ds dt = n dy e dt + dy ν dt = 1 r 2 r r 2 F ν, = 1 L ν 4πr 2 n dy i µ i r dt. n,p,e,ν There are two main sources of opacity: 1. ν-nucleon absorption. Affects only e types. 2. Neutrino-electron scattering. Inelastic scattering affects all types of neutrinos. Mean free paths for these processes are approximately: 1. λ ν λ ν 5 cm, λ ν E 2 ν ; 2. λ i E 100 cm, λi E T 1 E 2 ν.

6 Proto-Neutron Stars Analytic Analysis Neutrino fluid Eν 2 n ν (E ν ) = 2π 3 ( c) 3 f ν(e ν ), f ν (E ν ) = [1 ] + e (Eν µν)/t 1 Diffusion approximation F ν = c 3 0 L ν = 4πr 2 ɛ ν (E ν ) = E ν n ν [ ] n ν (E ν ) n ν (E ν ) λ ν λ ν de ν r r 0 i cλ i E 3 ɛ i (E ν ) de ν r Number transport dominated by degenerate electron neutrino absorption λ ν λ 0 (T 0 /T ) [1 + (E ν µ ν )/(πt )] 1, λ 0 50 cm, T 0 10 MeV Energy transport dominated by all-flavor neutrino scattering λ i E λ C (T0 3 /TEν 2 )(n s /n) 1/3, λ C 0.2 km For degenerate neutrinos f ν r f ν(1 f ν ) T µ ν r δ(e ν µ ν ) µ ν r

7 The Deleptonization of a Proto-Neutron Star Energy transport dominated by degenerate electron neutrinos propagating through degenerate matter. Number transport equation Y L Y ν ny L t ( ) YL Y ν 0 = cλ 0 18π 2 ( c) 3 1 r 2 r [ r 2 ( T0 T ) 2 µ 3 ν r 5, 6π 2 ny ν ( c) 3 = µ 3 ν, µ 3 ν = µ 3 ν,0ψ(x)φ(t) Dimensionless radius: x = x 1 r/r Eigenvalue equation, assuming T (r, t) = T c 20 MeV, R 20 km: τ D dφ φ dt = 1 [ x 2 x 2 ψ ] = 1 ψ x x Solutions: φ = exp( t/τ D ), ψ = sin(x)/x, ψ(x 1 = 0) = x 1 = π τ D = 3 ( ) 2 ( ) 2 ( ) R Tc YL 16 s, τ D M 4/3 cλ 0 x 1 T 0 Y ν ] 0

8 Neutrino Signal of Deleptonization Neutrino number flux: F ν (x, t) = cλ ( 0 µν,0 18π 2 c Outer boundary: x 1 ( ψ/ x) x1 = 1 Initial flux: F ν (R, t) = cλ 0 18π 2 R ( µν,0 c ) 3 ( T0 T c ) 2 x 1 R ψ(x) φ(t) x ) ( ) 3 2 T0 φ(t) = F ν (R, 0)φ(t) T c F ν (R, 0) = neutrinos cm 2 s 1 This ignores the large neutrino flux originating from the hot, shocked mantle, about cm 2 s 1.

9 Heating During Deleptonization Beta equilibrium: µ j dy j = ( µ n + µ p + µ e µ ν )dy e + µ ν dy L = µ ν dy L j Energy transport equation: nt s t = 1 L ν 4πr 2 n r j µ j Y j t nµ ν Y L t s at, a 0.05 MeV 1 ( ) s ds a dt µ Y L YL ν t Y ν 0 sf 2 si 2 3a ( ) YL 2 Y ν ( Yν Y ν,0 µ ν,0 Y ν,0 0 ) 1/3 Y ν µ ν,0 t s i 1, s f 2.5, T f 50 MeV; s f M 1/6

10 Core Cooling Following deleptonization, µ ν << T, 0 E νf ν de ν π 2 T 2 /12. Energy transport equation: L ν = 4πr 2 cλ C 6 nt s t = 1 L ν 4πr 2 n r j ( T0 c µ j dy j dt ) 3 T r 1 L ν 4πr 2 r T s dt T dt = cλ ( ) 4/3 ( ) 3 [ C ns T0 1 6n s n c r 2 r 2 T ] r r n, T are core density and temperature following deleptonization. Specific heat dominated by baryons: s a(n s /n ) 2/3 T, a 0.1 MeV 1. Assume separable solution: T = T ψ(x)φ(t) φ τ C t = 1 [ ψ 2 x 2 x 2 ψ ] = 1 x x

11 Core Cooling φ τ C t = 1 [ ψ 2 x 2 x 2 ψ ] = 1 x x ψ given by n = 2 Lane-Emden solution; φ(t) = 1 t/τ C : ψ(x 1 = 0) = x and x 1 ( ψ/ x) x=x1 0.55, τ C = 6an st R 2 cλ C x 2 1 ( ) 3 ( ) 2/3 c n M 4/3 T 0 n s Emergent Luminosity: L ν (R, t) = 4πRT cλ C 6 ( ) 3 ( T0 x ψ ) φ(t) = cf 3(0) c x x 1 2( c) 3 R2 T e (t) 4 T = 50 MeV, n = 4n s τ C 18 s, L ν (R, 0) 11 bethe s 1, T e (0) 4.0 MeV and < E ν >= [F 3 (0)/F 2 (0)]T e (0) 12 MeV. < E ν > obs [F 5 (0)/F 4 (0)]T e (0) 20 MeV

12 Model Simulations

13 Model Simulations

14 Model Signal

15 Urca Processes Gamow & Schönberg proposed the direct Urca process: nucleons at the top of the Fermi sea beta decay. n p + e + ν e, p n + e + + ν e Energy conservation guaranteed by beta equilibrium µ n µ p = µ e Momentum conservation requires k Fn k Fp + k Fe. Charge neutrality requires k Fp = k Fe, therefore k Fp 2 k Fn. Degeneracy implies n i k 3 Fi, thus x x DU = 1/9. With muons (n > 2n s ), x DU = 2 2+(1+2 1/3 ) If x < x DU, bystander nucleons needed: modified Urca process. (n, p) + n (n, p) + p + e + ν e, (n, p) + p (n, p) + n + e + + ν e Neutrino emissivities: ɛ MU (T /µ n ) 2 ɛ DU 10 6 ɛ DU. Beta equilibrium composition: x β (3π 2 n) 1 (4E sym / c) (n/n s )

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

17 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 1 S 0 critical temperature (T C ) and existence of proton superconductivity surface temperature Page et al T C

18 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

19 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

20 Credit: Dany Page, UNAM

21 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)

22 Tolman VII Solution [ ε(r) = ε c 1 (r/r) 2 ] ɛ c [1 x] λ(r) = ln [1 βx(5 3x)], ν(r) = ln [ (1 5β/3) cos 2 φ ], 1 p(r) = [tan 4πR 2 φ(r) 3βe λ(r) β2 ] (5 3x), ɛ(r) + p(r) cos φ(r) n(r) =, φ(r) = w 1 w(r) + tan 1 β m b cos φ 1 2 3(1 2β), [ ] w(r) = ln x e λ(r) 1, w 1 = w(x = 1) = ln 3β β. 3β ( ) ( P = 2 tan φ c 3 ε c 15 β 1 ) 3, 1 β c2 s,c = tan φ c 5 tan φ c + 3 BE M β β2 + p c < = φ c < π/2, β < c 2 s,c < 1 = β <

23 Buchdahl s Solution ɛ = p p 5p e ν(r) = (1 2β)(1 β u(r))(1 β + u(r)) 1, e λ(r) = (1 2β)(1 β + u(r))(1 β u(r)) 1 (1 β + β cos Ar ) 2, 8πp(r) = A 2 u(r) 2 (1 2β)(1 β + u(r)) 2, 8πɛ(r) = 2A 2 u(r)(1 2β)(1 β 3u(r)/2)(1 β + u(r)) 2, m b n(r) = ( p(r) p p(r) 1 4 p )3/2, c 2 s (r) = ( 1 2 u(r) = β ( ) 1 1 Ar sin Ar p = (1 β) 2 p(r) 1, r = r(1 2β)(1 β + u(r)) 1, A 2 = 2πp (1 2β) 1 π, R = (1 β) 2p (1 2β). p p(r) 5 p c = p 4 β2, ɛ c = p 2 β(1 5 2 β), n cm b = p β(1 2β)3/2 2 BE M = (1 3 2 β)(1 2β) 1/2 (1 β) 1 β 2 + β2 2 + c 2 s,c < 1 = β < 1/6 ) 1

24 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 soft = p = 0 = stiff The TOV solutions scale with ε 0 ɛ o

25 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 A phenomenological limit for hadronic matter (Lattimer & Prakash 2004) P min 1.00 (M /M sph ) 1/2 (R sph /10 km) 3/2 ms = 0.27 (M sph,max /M ) ms

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

27 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 contours

28 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

29 PSR J A 3.15 ms pulsar in an 8.69d orbit with an 0.5 M white dwarf companion. Shapiro delay yields edge-on inclination: sin i = Pulsar mass is 1.97 ± 0.04 M Distance > 1 kpc, B 10 8 G t(µs) Demorest et al. 2010

30 Black Widow Pulsar PSR B A 1.6ms pulsar in a circular 9.17h orbit with a low-mass ( 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. pulsar radial velocity eclipse NASA/CXC/M.Weiss

31 Black Widow Pulsar PSR B Ablation by pulsar leads to eventual disappearance of companion. Light curve shows enormous brightness variations. Its shape restricts inclination to 63 < i < 66. The optical light curve does not represent the center of mass of the companion, but the motion of its irradiated hot spot. Without correction for companion s finite size, M P > 1.6 M. Correcting for its size, 2.0 < M P /M < 2.6. optical radial velocity optical light curve 19 Magnitude = 100 = Reynolds 2007 Phase van Kerkwijk 2010

32 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: ε < 7.7εs p < 2.0εs nb < 6.9n s M max > 2.4 M Upper limits to energy density, pressure, 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, baryon density in the case of a quark matter core: ε < 5.2εs p < 1.4εs nb < 4.6n s

33 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

34 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

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

36 Polytropes Newtonian dimensional analysis: M n c R 3, p M2 R 4, R K 1/(3γ 4) M (γ 2)/(3γ 4) When γ 2: R K 1/2 M 0 p 1/2 f n 1 f M 0 General Relativistic analysis using Buchdahl s analytic solution: ε = pp 5p, R = (1 β) d ln R d ln p = 1 (1 β)(2 β) n,m 2 (1 3β + 3β 2 ) π 2p (1 2β), (1 10 p/p ) (1 + 2 p/p ). For M = 1.4M, R = 14 km, n = 1.5n s, ε = 1.5m b n s km 2 : β = 0.148, p = , p/p = d ln R d ln p n,m

37 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 (n), c ( ) P β = Kn2 n 1 9n 0 n s + c 4 nx(3π2 nx) 1/3, ( 1 n ) 2. n s + n 2 (1 2x β ) 2 de sym dn + E sym(n)nx β (1 2x β ) E sym (n s ) S v 30 MeV, c 200 MeV/fm, n n s = x β 0.04, P β ns 2 de sym dn. ns

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

39 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): E(A, Z) a v A + a s A 2/3 + Fitting binding energies results in a strong correlation between S v and S s, but not definite values. R [ ] S v S s S v n E sym (n) 1 d 3 r 0 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

40 Radiation Radius Combination of flux and temperature measurements yields apparent angular diameter (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)

41 RX J

42 Astrometry of RXJ Walter & Lattimer (2002) determined D = 117 ± 12 pc and v 190 km/s from HST Planetary Camera observations Star s age is probably 0.5 million years Kaplan, van Kerkwijk & Anderson (2002): D = 140 ± 40 pc using same data van Kerkwijk & Kaplan (2007, conference proceeding) revised this to D = 161 ± 16 pc based on High-Resolution Camera of the Advanced Camera for Surveys HST observations (double the resolution) Walter, Eisenbeiβ, Lattimer, Kim, Hambaryan & Neuhäuser (2010) determined D 115 ± 8 pc with HST data Two component black body gives minimum angular diameter: R > 13 km. But redshift or gravity measurement not yet possible. pixel size HST parallactic ellipse

43 Quiescent Sources in Globulars Hot neutron stars in globular clusters? Globular clusters evolve, more massive stars, including binaries, sink to middle. Close binaries formed by encounters. Episodes of accretion onto neutron stars in close binaries heats them: they are reborn. They become quiescent, low-mass X-ray sources emitting thermal X-rays. Accretion suppresses surface B fields. Atmospheric composition is H. 47 Tuc Reiko Kuromizu

44 M-R Probability Distributions Steiner, Lattimer & Brown 2010

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

46 Systematics Assuming 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 Real solutions require α < 1/8 β = 1 4 ± α R = αγ

47 PRE Burst M R Predictions EXO α = 0.14 ± 0.02 R ph = R 4U α = 0.18 ± 0.02 EXO α = 0.14 ± 0.02 R ph > R 4U α = 0.18 ± U α = 0.26 ± U Özel, Baym & Güver 2010 α = 0.26 ± 0.11 Steiner, Lattimer & Brown 2010

48 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)

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

50 Consistency with Neutron Matter Calculations

51 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 unlikely if M max 2.4 M. In spite of large estimated errors for individual stars, taken as an ensemble, they predict a remarkably tight pressure-density relation. Astrophysical observations are consistent with predictions of neutron matter calculations. 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.