Mass, Radius and Equation of State of Neutron Stars

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1 Mass, Radius and Equation of State of Neutron Stars J. M. Lattimer Yukawa Institute of Theoretical Physics Kyoto University and Department of Physics & Astronomy Stony Brook University February 13 and 15, 2014 International School on Neutron Star Matter YITP, Kyoto, Japan, 4 7 March 2014

2 Outline Structure Dense Matter Equation of State Formation and Evolution Observational and Experimental Constraints

3 Dany Page, UNAM

4 Amazing Facts About Neutron Stars Densest objects this side of an event horizon: g cm 3 Four teaspoons, on the Earth, would weigh as much as the Moon. Largest surface gravity: cm s 2 This is 100 billion times the Earth s gravity. Fastest spinning objects known: ν = 716 Hz This spin rate was measured for PSR J ad, in the globular cluster Terzan 5 located 9 kpc away. (33 pulsars have been found in this cluster.) If the radius is about 15 km, the velocity at the equator is one fourth the speed of light. Largest known magnetic field strengths: B = G Highest temperature superconductor: T c = 10 billion K The highest known superconductor on the Earth is mercury thallium barium calcium copper oxide (Hg 12 T l3 Ba 30 Ca 30 Cu 45 O 125 ), at 138 K. Highest temperature, at birth, anywhere in the Universe since the Big Bang: T = 700 billion K PSR B has fastest measured stellar velocity in the Galaxy: 1083 km/s = c/300 The only place in the universe except for the Big Bang where neutrinos become trapped.

5 Neutron Stars: History 1920 Rutherford predicts the neutron 1931 Landau anticipates single-nucleus stars but not neutron stars 1932 Chadwick discovers the neutron W. Baade and F. Zwicky predict existence of neutron stars as end products of supernovae Oppenheimer and Volkoff predict upper mass limit of neutron star Hoyle, Narlikar and Wheeler predict neutron stars rapidly rotate Hewish and Okoye discover an intense radio source in the Crab nebula Colgate and White perform simulations of supernovae leading to neutron stars C. Schisler discovers a dozen pulsing radio sources, including the Crab pulsar, using secret military radar in Alaska. X Hewish, Bell, Pilkington, Scott and Collins discover first PSR , Aug The Crab Nebula pulsar is discovered, found to be slowing down (ruling out binary and vibrational models), and clinched the connection to supernovae The term pulsar first appears in print, in the Daily Telegraph.

6 1969 Glitches observed; evidence for superfluidity in neutron star crust Accretion powered X-ray pulsar discovered by Uhuru (not the Lt.) Hewish awarded Nobel Prize (but Bell and Okoye were not) Binary pulsar PSR discovered by Hulse and Taylor., orbital decay due to GR gravitational radiation 1979 Chart recording of PSR used as album cover for Unknown Pleasures by Joy Division (#19/100 greatest British album) First millisecond pulsar, PSR B , discovered by Backer et al. at Arecibo Discovery of first extra-solar planets orbiting PSR B by Wolszczan and Frail Hulse and Taylor receive Nobel Prize 1998 Kouveletiou et al. discover first magnetar 2004 SGR flares: largest burst of energy seen in Galaxy since SN 1604, brighter than full moon in γ rays, more energy emitted than Sun in 100,000 years Hessels et al. discover PSR J ad; fastest rotation rate, 716 Hz Hessels et al. discover PSR J , first two-pulsar binary 2013 Antoniadis et al. find most-massive PSR J , 2.01 M 2013 Ransom et al. find first pulsar in triple system (two white dwarfs) sleevage.com/joy-division-unknown-pleasures/

7 sleevage.com/joy-division-unknown-pleasures/

8 Neutron Star Structure Tolman-Oppenheimer-Volkov equations of relativistic hydrostatic equilibrium: dp dr dm dr = G (m + 4πr 3 p/c 2 )(ɛ + p) c 2 r(r 2Gm/c 2 ) = 4π ɛ c 2 r 2 p is pressure, ɛ is mass-energy density Useful analytic solutions exist: Uniform density ɛ = constant Tolman VII ɛ = ɛ c [1 (r/r) 2 ] Buchdahl ɛ = pp 5p Tolman IV e ν = α + β(r/r) 2

9 Uniform Density Fluid m(r) = 4π ( r ) 2 3 ɛr 3 = Mx 3/2 M, x, β R R e λ(r) = 1 2βx, [ ] 3 e ν(r) 1 2 = 1 2β 1 2βx, [ 2 2 ] 1 2βx 1 2β p(r) = ɛ 3 1 2β, 1 2βx ɛ(r) = constant; n(r) = constant BE M = 3 ( sin 1 2β ) 1 2β 3β 4β 2β β2 +, cs 2 = p c < = β < 4/9

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

11 Buchdahl s Solution: Relativistic n=1 Polytrope 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 + cs 2 < = β < 1/5 ) 1

12 Tolman IV Variation: A Self-Bound Star (Quark Star) e ν(r) = e λ(r) = 4πpR 2 = [ β ( x)] 2, (1 2β) [ β ( x)] 2/3 [ β (, x)] 2/3 2(1 β) 2/3 [ βx ] β β ( x) 1 (1 β) 2/ β(1 x) [ β (, x)] 2/3 4πɛR 2 = 3(1 β)2/3 β [ β(1 1 3 [ x)] β ( x)] 5/3 c 2 s = ɛ surf ɛ c =, m = (1 β)2/3 Mx 3/2 [ β ( x)] 2/3 [ ] (2 5β + 3βx) (2 5β + 3βx) 5/3 5(2 5β + βx) 3 + (2 5β) 2 5β 2 x, 2 2/3 (1 β) (1 2/3 53 ) ( β 1 5 ) 2/3 2 β (1 β) 5/ < c 2 s,c < 0.44, < ɛ surf ɛ c < 1

13 Important Questions How Does the Structure of Neutron Stars Depend On the Nucleon-Nucleon Interaction? The Neutron Star Maximum Mass and Causality The Neutron Star Radius and the Nuclear Symmetry Energy Does Exotic Matter (Hyperons, Kaons/Pions, Deconfined Quarks) Exist in Neutron Star Interiors? How Do Nuclear Experiments Constrain the Nuclear Symmetry Energy and Neutron Star Radii? Binding Energies Heavy ion Collisions Neutron Skin Thicknesses Dipole Polarizabilities Giant (and Pygmy) Dipole Resonances Pure Neutron Matter What Astrophysical Constraints Exist? Mass Measurements of Pulsars and X-ray Binaries Photospheric Radius Expansion Bursts Thermal Emission from Isolated and Quiescent Binary Sources Pulse Modeling of X-ray Bursts, QPOs, Gravitational Radiation, etc.

14 Neutron Star Structure Tolman-Oppenheimer-Volkov equations dp dr dm dr = G (m + 4πpr 3 /c 2 )(ε + p) c 2 r(r 2Gm/c 2 ) = 4π ε c r 2 2 p(ε) mass maximum R p1/4 M(R) Equation of State Observations

15 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

16 The Radius Pressure Correlation R p 1/4 Red points satisfy M max > 2M Lattimer & Prakash (2001)

17 Neutron Star Structure Newtonian Gravity: dp dr = Gmρ r ; dm 2 dr = 4πρr 2 ; ρc 2 = ε Newtonian Polytropes: p = Kρ γ ; M K 1/(2 γ) R (4 3γ)/(2 γ) ρ < ρ s : γ 4; ρ > ρ 3 s: γ 2 maximum mass p(ε) p = Kρ 4 2 R K 1/2 M p = Kρ 4/3 ρ/ρ s = 1.0 M K 3/2 R 0

18 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). 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

19 Extremal Properties of Neutron Stars The maximum mass configuration is achieved when x R = , w c = 3.034, y c = 2.034, z R = 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

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

21 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. 1.4M strange quark matter stars (and likely hybrid quark/hadron stars) must have R > 11 km. M R curves for maximally compact EOS = = quark stars p ε = c2 s c 2 = s

22 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

23 Roche Model for Stellar Rotation (c.f., Shapiro & Teukolsky 1983) ρ 1 P = µ = (Φ G + Φ c ) Evaluate at equator: Ω 2 R 3 eq 2GM = R eq R p 1 Mass-shedding limit Ω 2 shed = GM/R3 eq : R eq /R p = 3 2 GR: Cook, Shapiro & Teukolsky (1994): ( ) 3/2 2 GM Ω shed = 3 R 3 P shed ms ( ) 3/2 ( ) 1/2 R M km M Φ G GM/r, Φ c = 1 2 Ω2 r 2 sin 2 θ Shape: Ω 2 R 3 sin 2 θ = R 1 Bernoulli integral: 2GM R p H = µ + Φ G + Φ c = GM/R p µ = ( [ p R p 0 ρ 1 dp = µ n µ n0 R(θ) = 1 ( ) ]) cos 1 Ω sin θ 3 cos Limit: R p R(θ) = sin(θ) 3 sin(θ/3). Ω shed

24 The Equation of State in Neutron Stars The EOS of an ideal Fermi gas Thermodynamic relations Degeneracy Limiting cases Interactions Properties of bulk nucleonic matter Phase coexistence Nuclear symmetry energy Finite nuclei the liquid droplet model Nuclei in dense matter

25 EOS of an Ideal Fermion Gas Single particle energy Occupation index E 2 = m 2 c 4 + p 2 c 2 [ ( E µ f = 1 + exp T )] 1

26 Thermodynamics Number and energy densities n = g h 3 fd 3 p; 0 Landau quasi-particle entropy formula: Pressure P = g 3h 3 0 ε = g h 3 Efd 3 p; µ = ns = g h 3 [f ln f + (1 f ) ln(1 f )] d 3 p 0 p E p fd 3 p = n 2 n = Degeneracy parameters φ = µ T = ψ + mc2 T ; 0 ( ) (F /n) n T ( ) P ; ns = µ T ( ) P = n(s + φ); T φ ( ) F n T = µn ε + nts = µn F ( ) P T µ ( ) P = n(s + ψ) ψ T

27 Pressure and Degeneracy Parameters n c = g ( mc ) 3 2π 2

28 Limits Non-relativistic p << mc x = p 2 /(2mT ), ψ = (µ mc 2 )/T n = g(2mt )3/2 4π 2 3 Relativistic p >> mc 0 ε = nmc 2 + P = 2 3 (ε nmc2 ), x 1/2 1 + e x ψ dx g(2mt )3/2 = 4π 2 3 F 1/2 (ψ) gt (2mT )3/2 4π 2 3 F 3/2 (ψ) n = g ( ) 3 T 2π 2 F 2 (φ) c ε = gt ( ) 3 T 2π 2 F 3 (φ) c P = ε 3, s = 5F 3/2(ψ) 3F 1/2 (ψ) ψ s = 4F 3(φ) 3F 2 (φ) φ

29 Limits ( ) [ 3 n = g φt 6π 2 c P = ε 3 = nφt 4 n = g 6π 2 ( 2mψT P [ 1 + ( ) ] 2 π 1 + φ + ( ) ] 2 + n 4/3 π φ 2 ) 3/2 [ = 2 3 (ε nmc2 ) = 2nψT 5 ( π ψ [ π ψ, s = π2 φ + EDER ) ] 2 +, s = π2 2ψ + ( ) ] 2 EDNR + n 5/3 { ( n = g T ) 3 } π 2 c e φ, s = 4 φ P = ε 3 = nt NDER { n = g ( ) } mt 3/2 2π e ψ ; s = ψ P = 2 3 (ε NDNR nmc2 ) = nt

30 Limit of Relativistic Gas With Pairs in Thermal Equilibrium µ = µ + = µ = φt, [ n = n + n = g ( µ ) ( ) ] 3 2 πt 6π 2 1 +, c µ [ P = P + + P = ε 3 = gµ ) ( 3 πt s = s + + s = gt ( µ 6n c ( µ 24π 2 c [ ) ( πt µ µ ) ] 2 Cubic Solution: [ (q µ = r q/r, r = 3 + t 2) ] 1/2 1/3 + t, t = 3π2 g n( c)3, q = (πt )2. 3. ) ( ) ] 4 πt, 15 µ

31 Non-Relativistic Fermi Gases With Interactions ε = t Single particle energy, t = n, p 2 2m t ɛ t = m t c 2 + = δε δτ t, 2 2m t τ t + U(n n, n p ) p 2 2m t (n n, n p ) + V t(n n, n p, τ n, τ p ) V t = δε = 2 δn t 2 s τ s (m s ) 1 n s + U n t Number and kinetic densities n t = 1 ( ) 2m 3/2 t T 2π F 1/2(η t ); τ t = 1 ( ) 2m 5/2 t T 2π F 3/2(η t ) η t = µ t m t c 2 [ ( )] 1 V t ɛt µ t ; f t = 1 + exp T T P = [ ] n t V t + 2 τ t 3m U, ns = [ 5 2 ] τ t t t 6m t t T n tη t

32 Bulk Matter Energy and Pressure

33 Schematic Free Energy Density n: number density; x: proton fraction; T : temperature n s 0.16 ± 0.01 fm 3 : nuclear saturation density B 16 ± 1 MeV: saturation binding energy K 240 ± 20 MeV: incompressibility parameter S v 30 ± 6 MeV: bulk symmetry parameter L 60 ± 60 MeV: symmetry stiffness parameter a ± MeV 1 : bulk level density parameter [ F (n, x, T ) = n B + K ( 1 n ) ] 2 n ( + S v (1 2x) 2 ns ) 2/3 a T 2 18 n s n s n [ ( ) ] 2 (F /n) P = n = n2 K n 1 + S v (1 2x) 2 + 2an n 9 n s 3 µ n = F n x n F x n s (1 nns ) (1 3 nns ) + 2S v n n s (1 2x) a 3 = B + K 18 ˆµ = 1 F n x = µ n n µ p = 4S v (1 2x) n s s = 1 F ( n T = 2a ns ) 2/3 T ; ε = F + nts n ( ns ) 2/3 T 2 n ( ns ) 2/3 T 2 n

34 Phase Coexistence Negative pressure: matter is unstable to separating into two phases of different densities (and possibly proton fractions). Physically, this represents coexistence of nuclei and vapor. Neglecting finite-size effect, bulk coexistence approximates the EOS at subnuclear densities. Free Energy Minimization With Two Phases F = ɛ nts = uf I + (1 u)f II, n = un I + (1 u)n II, ny e = ux I n I + (1 u)x II n II. n II = n un I 1 u, x II = ny e un I x I n un I F = u F I + (1 u) F ( ) II u = u(µ n,i µ n,ii ) n I n I n II 1 u F = u F I + (1 u) F ( ) II uni = un I (ˆµ I ˆµ II ) x I x I x [ II n ( un I ) F FII nii n I = F I F II + (1 u) + F ( )] II uni u n II 1 u x II n un I = µ ni = µ nii, µ pi = µ pii, P I = P II

35 Critical Point (Y e = 0.5) ( ) P = n T ( 2 ) P n 2 = 0 T n c = 5 12 n s ( ) 1/3 ( 5 5K T c = 12 32a ( ) 1/3 ( 12 5Ka s c = 5 8 ) 1/2 ) 1/2

36

37 Nuclear Symmetry Energy Defined as the difference between energies of pure neutron matter (x = 0) and symmetric (x = 1/2) nuclear matter. Expanding around saturation density (ρ s ) and symmetric matter (x = 1/2) E(ρ, x) = E(ρ, 1/2)+(1 2x) 2 S 2 (ρ)+... S(ρ) = E(ρ, x = 0) E(ρ, x = 1/2) C. Fuchs, H.H. Wolter, EPJA 30(2006) 5 S 2 (ρ) = S v + L 3 ρ ρ s ρ s +... S v 31 MeV, L 50 MeV symmetry energy Connections to 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 (in beta equilibrium): [ (E + E e ) = 0, p(ρ s, x β ) Lρ s x 3 1 ( LSv c ) ] 3 4 3S v /L 3π 2 ρ s

38 Nuclear Mass Formula Bethe-Weizsäcker (neglecting pairing and shell effects) E(A, Z) = a v A + a s A 2/3 + a C Z 2 /A 1/3 + S v AI 2. Myers & Swiatecki introduced the surface asymmetry term: E(A, Z) = a v A + a s A 2/3 + a C Z 2 /A 1/3 + S v AI 2 S s I 2 A 2/3. a v = B, a s 18 MeV, a C 0.75 MeV, S s 40 MeV, I = (N Z)/A Optimum Nucleus E/A A E/A I = a s S s I 2 A 4/3 + a C 3 6 A 1/3 (1 I ) 2 = 0 = 2S v I 2S s A 1/3 I a C 2 A2/3 (1 I ) = 0 A = 2(a s S s I 2 ) 45 a C (1 I ) 2 (1 I ) 2 60 a C I = 1 4S v A 2/3 4S s /A + a C 8 Valley of Beta Stability: Z A 2 a C A 5/3 8(S v S s A 1/3 )

39 Liquid Droplet Model H = H B (n, α) + n p V C /2 + (Q/n) (n ) 2, H B (n, α) = Bn + n(k/18)(n/n s 1) 2 + S 2 (n)α 2 /n. ( n = n n + n p, α = n n n p ; V C = Ze2 3 R 2 r 2 ) 2R 2 Extremize total energy for fixed A = nd 3 r, N Z = αd 3 r: [ Q δ δ [H µn] = 0, δn ] (Q/n) [H µα] = 0. δα 2 d dr n n (n ) 2 = [H B + n p V c /2] µ, n n µ 3Ze 2 /(8R) B, Q(n ) 2 n 2 (n/n s 1) 2, n n s / ( 1 + e x y ), r = ax, a = 18Q/K, A 4πn s (ay) 3 /3. 0 = [H B + n p V C /2] µ = 2S 2 α/n V C /2 µ, α α = n [ ( Sv I + Ze2 H2 r 2 )] S 2 H 0 8R H 0 R 2, H i = Sv S 2 ( r R ) i nd 3 r A.

40 Liquid Droplet Model Continued Parameters fixed by experiment: K 240 MeV 0.9 dn t = a 0.1 n = 4a ln fm, which gives a = fm, Q = (K/18)[t /(4 ln 3)] MeV fm 2. As a check, surface tension σ o = Qn s a (n ) 2 Ze2 [H µn]dz = 2Q dz ndz n 8R Ze2 n s a ( 8R Z 1/3) MeV fm 2. Then surface energy parameter is a s = 4πr 2 o σ MeV, which matches expectations from liquid droplet fits to nuclear masses. If symmetry energy can be expanded S 2 (n) = S v [ i b i(n/n s ) i ] 1, H i = 4πn s(ay) 3+i [ ] [ (3 + i)t 1 (3 + i)ar i 1 + = 3 y 3 + i T ] y +, T = b b b b 4 +, b i = 1. i

41 The total symmetry and Coulomb energies are α 2 E sym + E C = S 2 n d 3 r + 1 n p V C d 3 r 2 = AI 2 S v + 3Z 2 e 2 [ 1 + AI ( 1 5H )] 2 H 0 5R 8Z 3H 0 AI 2 S v = + 3Z 2 e 2 [ 1 AI S s 1 + S s A 1/3 /S v 5R 12Z S v A 1/3 + S s if we identify 3T /y = 3aT /R = S s A 1/3 /S v. Static dipole polarizability is α D = zα d d 3 r/(2ɛ), where α d is the value of α found by extremizing δ ( Hd 3 r ɛ zαd 3 r ) /δα = 0. α d = n ɛz + V C /2, α D = H 2AR 2 AR2 2S 2 12S v 20S v Mean square radii rn,p 2 1 = (n ± α)r 2 d 3 r = R2 2(N, Z) 1 ± I r np = 2r o(1 I 2 ) 1/2 [ 3 I S s 3(1 + S s A 1/3 /S v ) 5 S v ( [ 3 5 ± I H 2 ± Ze2 H 0 8RS v ( 3Ze2 140S v r o ] S s A 1/3 ) = 4m 1. S v ( )] H 2 2 H 4, H 0 S s A 1/3 S v )].

42 Correlations from Experiments The three relations for E sym, α D, and r np can be combined with experimental data to give information about S s and S v and about L and S v if we can relate S s to L and S v. This is done using T. S s T = r o = b a S v 2 b b 3+, S v S 2 = i b i ( n n s ) i, If S 2 = S v + (L/3)(n/n s 1), then S s La/r o. If S 2 = S v + (L/3)(n/n s 1) + (K sym /18)(n/n s 1) 2, and K sym = 6L 18S v so that S 2 (0) = 0, then S s S v 3a 2r o [ 1 + L 3S v + ( L 3S v ) 2 ] b i = 1. Experiments can thus give a specific combination of values for S v and L, i.e., a correlation in S v L space. Masses: d(s s /S v )/ds v 0.26; dl/ds v 19 Dipole polarizability: d(s s /S v )/ds v 0.17; dl/ds v 14 Neutron skin thickness: d(s s /S v )/ds v 0.08; dl/ds v 6. i

43 Nuclear Binding Energies E sym (N, Z) = I 2 (S v A S s A 2/3 ) χ 2 = i (E ex,i E sym,i ) 2 /N χ vv = 2 N i I i 4 A 2 i χ ss = 2 N i I i 4 A 4/3 i χ vs = 2 N i I i 4 A 5/3 i σ Sv = 2χss χ vv χ ss χ 2 sv σ Ss = 2χvv χ vv χ ss χ 2 sv α = 1 2 tan 1 r vs = χvs χvv χ ss 2χvs χ vv χ ss Liquid Droplet Model S E sym (N, Z) = v AI 2 1+(S s/s v )A 1/3

44 Neutron Skin Thickness r np r o 4 S s I 15 S v +S s A 1/3

45 Heavy Ion Collisions Reaction Mechanisms in Heavy Ion Collisions peripheral Coulomb barrier to Fermi energies central Proton and neutron currents j j n p E sym Esym ( ρ) ( ρ) I + I ρ ρ Diffusion Drift Isospin migration Isospin fractionation, multifragm Sensitive to Sym. Energy and slope depending on observable Wolter, NuSYM11

46 Giant Dipole Resonances E 1 S v 1+ 5S s 3Sv A 1/ MeV< S 2 (0.1 fm 3 ) <24.9 MeV

47 Dipole Polarizability The linear response, or dynamic polarizability, of a nuclear system excited from its ground state to an excited state, due to an external oscillating dipolar field. Data from Tamii et al. (2011)

48 Nuclear Experimental Constraints Isobaric Analog States

49 Theoretical Neutron Matter Calculations H&S: Chiral Lagrangian GC&R: Quantum Monte Carlo

50 Supernovae

51 P Ṗ Diagram τ c = P 2Ṗ B 3 10 PṖ 19 G E Ṗ P 3 erg/s P in seconds log [ Period (s) ]

52 P-pdot

53 P-pdot

54 MSPs

55 P-pdor

56 Magnetars

57 Magnetar

58 Clock

59 Timing

60 Gravity waves

61 Binary Mass Measurements Mass function f (M 1 ) = P(v 2 sin i) 3 2πG = (M 1 sin i) 3 (M 1 +M 2 ) 2 > M 1 f (M 2 ) = P(v 1 sin i) 3 2πG = (M 2 sin i) 3 (M 1 +M 2 ) 2 > M 2 In an X-ray binary, v optical has the largest uncertainties. In some cases sin i 1 if eclipses are observed. If no eclipses observed, limits to i can be made based on the estimated radius of the optical star. X-ray timing Optical spectroscopy

62 Pulsar Mass Measurements Mass function for pulsar precisely obtained. It is also possible in some cases to obtain the rate of periastron advance 17 yr 1 in PSR 0737 and the Einstein gravitational redshift + time dilation term: ( ω = 3 2π ) 5/3 ( GM ) 2/3 1 e 2 P c 2 ) 1/3 em 2 (2M 2 + M 1 ) ( G M 2 c 2 ) 2/3 ( P γ = 2π Gravitational radiation leads to orbit decay: ( Ṗ = 192π 2πG ) 5/3(1 5c 5 P e 2 ) ( 7/ In some cases, can constrain Shapiro time delay, r is magnitude and s = sin i is shape parameter. 24 e M 1/2 96 e4) M 1 M 2

63 PSR J PSR J Kramer et al. (2007)

64 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

65 Black Widow Pulsar PSR B ms pulsar in circular 9.17h orbit with a 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

66 Pulsars Around Sgr A*

67 J

68 J0337

69 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 Antoniadis et al Champion et al. 2008

70 Radiation Radius The measurement of flux and temperature yields an apparent angular size (pseudo-bb): R d = R 1 d 1 2GM/Rc 2 Observational uncertainties include distance, interstellar H absorption (hard UV and X-rays), atmospheric composition Nearby isolated neutron stars (parallax measurable) Quiescent X-ray binaries in globular clusters (reliable distances, low B H-atmosperes) Bursting sources in which Eddington flux is measured F Edd = GMc κd 2 1 2GM R ph c 2

71 RX J

72 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 Walter, Eisenbeiβ, Lattimer, Kim, Hambaryan & Neuhäuser (2010) determined D 115 ± 8 pc based on HST Advanced Camera for Surveys observations (double the resolution) A magnetic hydrogen atmosphere model (Ho et al. 2007) suggests M 1.29 M and R 11.6 km Redshift or gravity measurements, which would allow simultaneous M and R determinations, are not yet possible. pixel size HST parallactic ellipse

73 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

74 PRE Burst Models Ozel et al. z ph = z β = GM/Rc 2 Steiner et al. z ph << z F Edd = GMc 1 2β κd A = F σt 4 = fc 4 κd α = F Edd A γ = Af c 4 c 3 = R κf Edd α ( ) 2 R D Fc 2 = β(1 2β) c3 β = 1 4 ± α α 1 8 required. F Edd = GMc κd α = β 1 2β θ = cos 1 ( [ 1 54α 2) β = ( ) θ 3 sin 6 3 ( ) ] θ cos 3 1 α required. 27 α EXO U U KS SAX J ± ± ± ± ± 0.036

75 M R PRE Burst Estimates F Edd,, (R /D) 2 f 4 c, D, f c from Ozel et al. z ph = z Lattimer & Steiner (2013)

76 M R PRE Burst Estimates F Edd,, (R /D) 2 fc 4, D from Ozel et al. z ph = 0 Altered uncertainties for f c, D Lattimer & Steiner (2013)

77 Quiescent Sources in Globulars Hot neutron stars in globular clusters Globular clusters evolve: more massive stars, including binaries, sink to center via long-range stellar encounters. Close binaries formed in encounters. Episodes of accretion in close binaries heats neutron stars: they are reborn. Following accretion, they become quiescent, low-mass X-ray sources. Accretion suppresses surface B fields. Atmospheric composition is H. 47 Tuc Reiko Kuromizu

78 M R QLMXB Estimates M (M ) ω Cen M M28 Absorption (N H ) determined self-consistently from spectra R (km) Guillot et al. (2013)

79 M R QLMXB Estimates M (M ) ω Cen M M28 P(M, R) from H atmosphere models of Guillot et al. (2013), adjusted for alternate N H values of Dickey & Lockman (1990) R (km) Lattimer & Steiner (2013)

80 Bayesian TOV Inversion ) 3 P (MeV/fm ε < 0.5ε 0 : Known crustal EOS 0.5ε 0 < ε < ε 1 : EOS parametrized by K, K, S v, γ Polytropic EOS: ε 1 < ε < ε 2 : n 1 ; ε > ε 2 : n inferred p(ε) Fid. EOS, PREs+QLMXBs w/adj. N H ε (MeV/fm ) M (M ) EOS parameters K, K, S v, γ, ε 1, n 1, ε 2, n 2 uniformly distributed M max 1.97 M, causality enforced All 10 stars equally weighted Fiducial EOS, PREs, QLMXBs w/adjusted N H Lattimer & Steiner 2013 inferred M(R) RX J1856 Ho et al. (2007) R (km)

81 Astronomy vs. Astronomy vs. Physics Ozel et al., PRE bursts z ph = z: R = 9.74 ± 0.50 km. Suleimanov et al., long PRE bursts: R 1.4 > 13.9 km Guillot et al. (2013), all stars have the same radius, self N H : R = km. Lattimer & Steiner (2013), TOV, crust EOS, causality, maximum mass > 2M, z ph = z, alt N H. Lattimer & Lim (2013), nuclear experiments: 29 MeV < S v < 33 MeV, 40 MeV < L < 65 MeV, R 1.4 = 12.0 ± 1.4 km. M (M ) Fiducial EOS, PREs, QLMXBs w/adjusted N H R (km)

82 Can Hyperons Appear in Abundance in Neutron Stars? X H < 0.2 Jiang, Li & Chen (2012) Miyatsu, Yamamumo & Nakazato (2013) Bednarek et al. (2011) Weissenborn, Chatterjee & Schaffner-Bielich (2012)

83 Hyperon Stars with Small Radii Have Few Hyperons 0.1 fi 0.01 Jiang, Li & Chen (2012)

84 More Hyperon Stars Whittenbury et al. (2013) Schramm et al. (2013) Bednarek, Manka & Pienkos (2013) Colucci & Sedrakian (2014)

85 Still More Hyperon Stars Lopes & Menezes (2013)

86 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 ultrarelativistic 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 Large Observatory For x-ray Timing ESA/NASA NASA

87 Constraints from Observations of Gravitational Radiation Mergers: Chirp mass M = (M 1 M 2 ) 3/5 M 1/5 and tidal deformability λ R 5 (Love number) are potentially measurable during inspiral. λ λm 5 is related to Ī IM 3 by an EOS-independent relation (Yagi & Yunes 2013). Both λ and Ī are also related to M/R in a relatively EOS-independent way (Lattimer & Lim 2013). Neutron star - neutron star: M crit for prompt black hole formation, f peak depends on R. Black hole - neutron star: f tidal disruption depends on R, a, M BH. Disc mass depends on a/m BH and on M NS M BH R 2. Rotating neutron stars: r-modes

88 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 )

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

90 Minimal Cooling 3 Neutron P F gap: "a" 2 max cn (T = 10 K) LEE HEE A B 8 C D b d e 11 f c a LEE HEE 9 Neutron HEE 3 P F gap: "b" LEE 2 1 A 2 5 B C 6 b a 3 4 D c de f (T cn max = 3x10 9 K) 7 LEE HEE 13 In minimal cooling, it is supposed there is no rapid neutrino cooling due to direct Urca processes among nucleons, hyperons, Bose condensates or quarks. Envelope and atmospheric variations lie within the shaded regions. Stellar mass and radii have little effect. Any objects failing to match cooing curves are candidates for rapid cooling (possibly due to high stellar masses).

91 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

92 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

93 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

94 Conclusions Nuclear experiments set reasonably tight constraints on symmetry energy parameters and the symmetry energy behavior near the nuclear saturation density. Theoretical calculations of pure neutron matter predict very similar symmetry constraints. These constraints predict neutron star radii R 1.4 in the range 12.0 ± 1.4 km. Combined astronomical observations of photospheric radius expansion X-ray bursts and quiescent sources in globular clusters suggest R ± 0.6 km. The nearby isolated neutron star RX J appears to have a radius near 12 km, assuming a solid surface with thin H atmosphere (Ho et al. 2007). The observation of a 1.97 M neutron star, together with the radius constraints, implies the EOS above the saturation density is relatively stiff; abundance of hyperons or any phase transition must be small.

95 What Was Not Included Neutron Star Mergers Gravitational Radiation from Mergers Nucleosynthsis from Mergers Quadrupole Polarizability