The Nuclear Equation of State and Neutron Star Structure

Transcription

1 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 1/68 The Nuclear Equation of State and Neutron Star Structure James M. Lattimer Department of Physics & Astronomy Stony Brook University

2 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 2/68 Outline Structure Dense Matter Equation of State Formation and Evolution Observational Constraints

3 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 3/68 PSR sleevage.com/joy-division-unknown-pleasures/

4 Credit: Dany Page, UNAM J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 4/68

5 Amazing Facts About Neutron Stars Densest objects this side of an event horizon: g cm 3 Four teaspoons, on the Earth s surface, 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, which is located in the globular cluster Terzan 5 located 28,000 light years 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 The Sun s field strength is about 1 G. Highest temperature superconductor: Tc = 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 Fastest measured velocity of a massive object in the Galaxy: 1083 km/s This velocity was measured for PSR B , and is 1/300 the speed of light. It is actually an underestimate, since only the transverse velocity can be measured. The only place in the universe except for the Big Bang where neutrinos become trapped. J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 5/68

6 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 6/68 Spherically Symmetric General Relativity Static metric: ds 2 = e λ(r) dr 2 + r 2 ( dθ 2 + sin 2 θdφ 2) e ν(r) dt 2 Einstein s equations: Mass: Boundaries: 8πǫ(r) = 1 ( r 2 1 e λ(r)) + e λ(r) λ (r), r 8πp(r) = 1 ( r 2 1 e λ(r)) + e λ(r) ν (r) r p p(r) + ǫ(r) (r) = ν (r). 2 m(r) = 4π r 0 ǫ(r )r 2 dr, r = 0 m(0) = p (0) = ǫ (0) = 0, e λ(r) = 1 2m(r)/r r = R m(r) = M, p(r) = 0, e ν(r) = e λ(r) = 1 2M/R, Thermodynamics: d(ln n) = dǫ ǫ + p = 1 2 dǫ dp dν, h = dǫ de, ǫ = n(m + e), p = n2 dn dn mn(r) = (ǫ(r) + p(r))e (ν(r) ν(r))/2 n 0 e 0 ; p = 0 : n = n 0, e = e 0 N = R 0 4πr 2 e λ(r)/2 n(r)dr; BE = Nm M

7 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 7/68 Neutron Star Structure Tolman-Oppenheimer-Volkov equations of relativistic hydrostatic equilibrium: dp dr dm dr = G (m + 4πpr 3 )(ǫ + p) c 2 r(r 2Gm/c 2 ) = 4π ǫ c 2r2 p is pressure, ǫ is mass-energy density Useful analytic solutions exist: Uniform density ǫ = constant Tolman VII ǫ = ǫ c [1 (r/r) 2 ] Buchdahl ǫ = pp 5p

8 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 8/68 Uniform Density Fluid m(r) = 4π 3 ǫr3, β M R e λ(r) = 1 2β(r/R) 2, [ 3 e ν(r) ] 1 2 = 1 2β 1 2β(r/R) 2, 2 2 [ 1 2β(r/R) p(r) = ǫ 2 ] 1 2β 3 1 2β, 1 2β(r/R) 2 ǫ(r) = constant; n(r) = constant ( BE M = 3 sin 1 2β ) 1 2β 4β 2β 3β β2 + p c < = β < 4/9 c 2 s =

9 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 9/68 Tolman VII ǫ(r) = ǫ c [ 1 (r/r) 2 ] ǫ c [1 x] e λ(r) = 1 βx(5 3x) e ν(r) = (1 5β/3) cos 2 φ, 1 p(r) = [ 3βe λ(r) 4πR 2 tan φ(r) β2 ] (5 3x), ǫ(r) + p(r) cos φ(r) n(r) = m b cos φ 1 φ(r) = w 1 w(r) + φ 1, φ 1 = φ(x = 1) = tan 1 β 2 3(1 2β), [ ] w(r) = ln x e λ(r) 1, w 1 = w(x = 1) = ln 3β β. 3β (P/ǫ) c = 2tan φ c 15 3 β 1 3, c2 s,c = tan φ c BE M β β2 + ( p c < = φ c < π 2, β < ) 1 β 5 tan φ c + 3 c 2 s,c < 1 = β <

10 Buchdahl ǫ = 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, 8πǫ(r) = 2A 2 u(r)(1 2β)(1 β 3u(r)/2)(1 β + u(r)) 2, m b n(r) = ( ) 3/2 ( p(r) 1 p p(r) 1 4, c 2 s (r) = 2 u(r) = p β Ar sin Ar = (1 β) ( 1 2 ) 1 p 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 ) 1 p c = p 4 β2, ǫ 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 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 10/68

11 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 11/68 Tolman IV Variation: A Self-Bound Star Finite, large surface energy density, e.g., strange quark star Lake s solution e ν(r) = [1 5 2 β(1 1 5 x)]2, (1 2β) e λ(r) (1 5 2 = β(1 3 5 x))2/3 (1 5 2 β(1 3 5 x))2/3 2(1 β) 2/3 βx, [ 4πpR 2 β = β(1 1 5 x) 1 (1 β) 2/3 (1 5 ] β(1 x)) 2 (1 5 2 β(1 3, 5 x))2/3 4πǫR 2 = 3(1 β) 2/3 β m = c 2 s = ǫ surf ǫ c = (1 β) 2 3 Mx 3/ β(1 1 3 x) (1 5 2 β(1 3 5 x))5/3, (1 5 2 β(1 3 5 x))2/3 [ ] (2 5β + 3βx) (2 5β + 3βx) 5/3 5(2 5β + βx) 3 2 2/3 (1 β) 2/3 + (2 5β) 2 5β 2 x, (1 53 ) ( β 1 5 ) 2/3 2 β (1 β) 5/ < c 2 s,c < 0.44, < ǫ surf ǫ c < 1

12 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 12/68 Mass-Radius Diagram and Theoretical Constraints GR: R > 2GM/c 2 P < : R > (9/4)GM/c 2 causality: R > 2.9GM/c2 normal NS SQS R = R 1 2GM/Rc 2 contours

13 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 13/68 Mass Measurements In X-Ray Binaries Mass function f(m 1 ) = P(v 2 sin i) 3 2πG = (M 1 sin i) 3 (M 1 +M 2 ) 2 > M 1 X-ray timing 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, sini 1 if eclipses are observed. If eclipses are not observed, limits to i can be made based on the estimated radius of the optical star. Optical spectroscopy

14 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 14/68 Pulsar Mass Measurements Mass function for pulsar precisely obtained. It is also possible in some cases to obtain the rate of periastron advance and the Einstein gravitational redshift + time dilation term: ω = 3(2π/P) 5/3 (GM/c 2 ) 2/3 /(1 e 2 ) γ = (P/2π) 1/3 em 2 (2M 2 + M 1 )(G/M 2 c 2 ) 2/3 Gravitational radiation leads to orbit decay: ( P = 192π 2πG ) 5/3 5c 5 P (1 e 2 ) ( 7/ e2 + 37e4) M 1 M 2 96 M 1/2 In some cases, can constrain Shapiro time delay, r is magnitude and s = sin i is shape parameter. 17 yr 1 in PSR 0737

15 PSR , Kramer et al. (2007) J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 15/68

16 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 16/68 Black hole? Firm lower mass limit? M < 1.17 M (95%) 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? Champion et al. 2008

17 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 17/68 Roche Model for Maximal Rotation (c.f., Shapiro & Teukolsky 1983) ρ 1 P = h = (Φ G + Φ c ), Φ G GM/r, Φ c = 1 2 Ω2 r 2 sin 2 θ Bernoulli integral: H = h + Φ G + Φ c = GM/R p Enthalpy h = p 0 ρ 1 dp = µ n (ρ) µ n (0) in beta equilibrium Numerical calculations show R p is nearly constant for arbitrary rotation Evaluate at equator: Ω2 R 3 eq 2GM = R eq R 1 Mass-shedding limit Ω 2 shed = GM R 3 : R eq eq R = 3 2 GR: Cook, Shapiro & Teukolsky (1994): Ω shed = ( ) 2 3/2 GM 3 R 3 P shed = 1.0 (R/10 km) 3/2 (M /M) 1/2 ms GR: Lattimer & Prakash (2005): 0.96 ± 3% Shape: Ω2 R 3 sin 2 θ = R 1 2GM R p R p R(θ) = cos[1 3 cos 1 (1 2( Ω sin θ ) Ω 2 )] shed R Limit: p R(θ) = cos[1 3 cos 1 (1 2sin 2 θ)] = sin(θ) 3 sin(θ/3).

18 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 18/68 Extreme Properties of Neutron Stars The most compact configurations occur when the low-density equation of state is "soft" and the high-density equation of state is "stiff". soft = p = 0 ǫ o = stiff p = ǫ ǫ o causal limit ǫ 0 is the only EOS parameter The TOV solutions scale with ǫ 0

19 Maximally Compact Equation of State Koranda, Stergioulas & Friedman (1997) p(ǫ) = 0, ǫ ǫ o p(ǫ) = ǫ ǫ o, ǫ ǫ o This EOS has a parameter ǫ o, which corresponds to the surface energy density. The structure equations then contain only this one parameter, and can be rendered into dimensionless form using y = mǫ 1/2 o, x = rǫ 1/2 o, q = pǫ 1 o. dy dx dq dx = 4πx 2 (1 + q) = (y + 4πqx3 )(1 + 2q) x(x 2y) The solution with the maximum central pressure and mass and the minimum radius: q max = 2.026, y max = , x min /y max = ( ) ( ) 1/2 ǫo p max = 307 MeV fm 3 ǫs, M max = 4.1 M, R min = GM max c 2. ǫ s Moreover, the scaling extends to the axially-symmetric case, yielding ( ) 1/2 ( ) 1/2 ( ) Mmax P min Rmin 3 ǫo 1/2 ǫs R 3/2 ( ) 1/2 M, P min = 0.82 ms = 0.76 ms ǫ o 10 km M J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 19/68 ǫ o

20 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 20/68 Maximum Possible Density in Stars The scaling from the maximally compact EOS yields ( ) ( M ǫ c,max = ǫ s M M max M max ) 2 g cm 3. A virtually identical result arises from combining the maximum compactness constraint (R min 2.9GM/c 2 ) with the Tolman VII relation ǫ c,v II = 15 8π M R 3 = 15 8π ( c 2 2.9G ) 3 1 M 2 Maximum possible density: 2.2 M ǫ max < g cm 3

21 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 21/68 Maximum Mass, Minimum Period Theoretical limits from GR and causality M max = 4.2(ǫ s /ǫ f ) 1/2 M Rhoades & Ruffini (1974), Hartle (1978) R min = 2.9GM/c 2 = 4.3(M/M ) km Lindblom (1984), Glendenning (1992), Koranda, Stergioulas & Friedman (1997) ǫ c < (M /M largest ) 2 g cm 3 Lattimer & Prakash (2005) P min 0.74(M /M sph ) 1/2 (R sph /10 km) 3/2 ms Koranda, Stergioulas & Friedman (1997) P min 0.96 ± 0.03(M /M sph ) 1/2 (R sph /10 km) 3/2 ms (empirical) Lattimer & Prakash (2004) ǫ c > (1 ms/p min ) 2 g cm 3 (empirical) cj/gm (empirical, neutron star)

22 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 22/68 Constraints from Pulsar Spins XTE J ν = 1122 Hz Kaaret et al Not confirmed to be a rotation rate PSR J ad ν = 716 Hz Hessels et al. 2006

23 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 23/68 BE(M, R) Lattimer & Prakash (2001) BE (0.60 ± 0.05) GM Rc 2 [ 1 GM 2Rc 2 ] 1

24 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 24/68 Moment of Inertia I = 8π 3c 4 = 2c2 3G R 0 R 0 r 4 [ǫ(r) + p(r)] e (λ(r) ν(r))/2 ω(r)dr r 3 ω(r) dj(r) dr dr, where d dr j(r) = e (λ(r)+ν(r))/2 ; [ r 4 j(r) dω(r) ] = 4r 3 ω(r) dj(r) dr dr ; j(r) = 1, ω(r) = 1 2GI R 3 c 2, dω(0) dr = 0. Combining these: With φ = dln ω/dln r, φ(0) = 0, I = c2 dω(r) R4 6G dr dφ = φ j (3 + φ) (4 + φ)dln dr r dr, I = φ Rc 2 6G R3 ω R = φ ( R R 3 c 2 ) 6 G 2I = R3 φ R c 2 G(6 + 2φ R ).

25 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 25/68 I(M, R) Lattimer & Prakash (2001) [ ( ) ] 4 I (0.237 ± 0.008)MR M km + 90 M km R M R M

26 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 26/68 The Uncertain Nuclear Force The density dependence of E sym (n) is crucial but poorly constrained. Although the second density derivative, the incompressibility K, for symmetric matter is known well, the third density derivative, the skewness, is not. == E sym (n) K 220 MeV = Skewness C. Fuchs, H.H. Wolter, EPJA 30(2006) 5

27 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 27/68 Schematic 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 220 ± 15 MeV: incompressibility parameter S v 30 ± 6 MeV: bulk symmetry parameter a ± MeV 1 : bulk level density parameter [ ( 1 n n s ) 2 + S v n ( ns ǫ(n, x, T) = n B + K (1 2x) 2 + a 18 n s n P = n 2 (ǫ/n) [ ( ) ] = n2 K n 1 + S v (1 2x) 2 n n s 9 n s µ n = ǫ n x ǫ n x = B + K ) ) (1 (1 nns 3 nns n + 2S v (1 2x) a 18 n s 3 ˆµ = 1 ǫ n x = µ n n µ p = 4S v (1 2x) n s s = 1 ǫ ( n T = 2a ns ) 2/3 T n ) 2/3 T 2 + 2an 3 ] ( ns ( ns n n ) 2/3 T 2 ) 2/3 T 2

28 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 28/68 Phase Coexistence Schematic energy density [ ǫ = n B + K ( 1 n ) ] 2 n ( + S v (1 2x) 2 ns ) 2/3 + a T 2 18 n s n s n [ ( ) ] P = n2 K n 1 + S v (1 2x) 2 + 2an ( ns ) 2/3 T 2 n s 9 n s 3 n µ n = B + K ) ) (1 (1 nns 3 nns n + 2S v (1 2x) a ( ns ) 2/3 T 2 18 n s 3 n n ( ns ) 2/3 ˆµ = µ n µ p = 4S v (1 2x), s = 2a T n s n Free Energy Minimization With Two Phases F = ǫ 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 F n I = 0, F x I = 0, F u = 0 = µ ni = µ nii, µ pi = µ pii, P I = P II Critical Point (Y e = 0.5) ( P ) ( 2 ) P = n T n 2 = 0 T n c = 5 ( ) 5 1/3 ( ) 5K 1/2 ( ) 12 1/3 ( 5Ka 12 n s, T c =, s c = 12 32a 5 8 ) 1/2

29 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 29/68

30 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 30/68 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, [ de(n,1/2) P(n, x) n 2 + de ] sym (1 2x)2 + c dn dn 4 nx(3π2 nx) 1/3, µ e = c(3π 2 nx) 1/3, E(n,1/2) B + K ( 1 n ) n s Beta Equilibrium: ( ) E x n = µ p µ n + µ e = 0. ( ) 3 x β (3π 2 n) 1 4Esym, c ( ) P β = Kn2 n 1 + n 2 (1 2x β ) 2 de sym 9n 0 n s dn + E sym 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 n 2 s de sym dn. ns

31 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 31/68 The Uncertain E sym (n) C. Fuchs, H.H. Wolter, EPJA 30(2006) 5

32 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 32/68 Neutron Star Matter Pressure p Kn γ γ = d ln p/d ln n 2 Wide variation: 1.2 < p(n s) MeV fm 3 < 7 n s

33 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 33/68 Polytropes Polytropic Equation of State: p = Kn γ n is number density, γ is polytropic exponent. Hydrostatic Equilibrium in Newtonian Gravity: Dimensional analysis: dp(r) dr = Gm(r)n(r) r 2, dm(r) dr = 4πnr 2 M n c R 3, p M2 R 4, R K1/(3γ 4) M (γ 2)/(3γ 4) When γ 2: R K 1/2 M 0 p 1/2 f n 1 f M0 General Relativistic analysis using Buchdahl s solution: π R = (1 β) 2p (1 2β), dln R dln p = 1 (1 β)(2 β) n,m 2 (1 3β + 3β 2 ) 1 10 p/p p/p. For M = 1.4M, R = 14 km, n = 1.5n s, ǫ = 1.5m b n s km 2 : β = 0.148, p = , p/p = dln R dln p n,m

34 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 34/68 The Radius Pressure Correlation R p 1/4 Lattimer & Prakash (2001)

35 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 35/68 Nuclear Symmetry Energy The density dependence of E sym (n) is crucial. Some information is available from nuclei (for n < n s ). Heavy ion collisions have potential for constraining it for n > n s. It is common to expand E sym (n) as E sym (n) J + L 3 ( ) n 1 n s + K sym 18 ( ) n n s J = E sym (n s ), L = 3n s ( Esym n ) n s, K sym = 9n 2 s ( 2 ) E sym n 2 n s Almost no information is available for K sym. What information can constrain E sym (n) from the laboratory?

36 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 36/68 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 (N Z) 2 /A 4/3. Droplet extension: consider the neutron/proton asymmetry of the nuclear surface. E(A, Z) = ( a v + S v δ 2 )(A N s ) + a s A 2/3 + a C Z 2 /A 1/3 + µ n N s. N s is the number of excess neutrons associated with the surface, I = (N Z)/(N + Z), δ = 1 2x = (A N s 2Z)/(A N s ) is the asymmetry of the nuclear bulk fluid, and µ n is the neutron chemical potential. From thermodynamics, N s = a sa 2/3 µ n = S s δ = I S v ( 1 + S s S v A 1/3 δ 1 δ = AI δ 1 δ, ) 1, E(A, Z) = a v A + a s A 2/3 + a C Z 2 /A 1/3 + S v AI 2 ( 1 + S s S v A 1/3 ) 1.

37 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 37/68 Nuclear Structure Considerations Information about E sym 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 S v (S s /S v )A 1/3 A + a cz 2 A 1/3 Fitting binding energies results in a strong correlation between S v and S s, but not definite values. 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)

38 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 38/68 Schematic Dependence Nuclear Hamiltonian: [ H = H B + Q 2 n 2, H B n B + K ) ] 2 (1 nns 18 + E sym (1 2x) 2 Lagrangian minimization of energy with respect to n (symmetric matter): H B µ 0 n = Q 2 n 2 = K 18 n (1 n n s ) 2, µ 0 = a v Liquid Droplet surface parameters: a s = 4πr 2 0 σ 0, σ 0 = t = + σ δ = S v Q 2 [H µ 0 n]dz = 0.9ns dn Qns 0.1n s n = 3 K ns 0 = S vt n s 3 n 1 ( Sv 0 1 u ( ) n p E sym S v = n s ns S s = 4πr 2 0 σ δ (H B µ 0 n) dn n = 4 45 du Qns 9 u(1 u) K ) 1 (H B µ 0 n) 1/2 dn E sym ( ) u Sv 1 du E sym QKn 3 s 0.28 (p = 1 2 ), 0.93 (p = 2 ), 2.0 (p = 1) 3 E sym S v + L 3 ( n n s ) = 2 3Sv L 1tan 1 ( 1 + S ) 1 v 1 + L 3L 3S v

39 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 39/68 Schematic Dependence S s S v t r = For Pb 208 : δr = 3 5 (R n R p ) t A 2Z Z(1 Z/A) = PREX experiment (E06002) at Jefferson Lab to measure the neutron radius of lead to about 1% accuracy (current accuracy is about 5%) using the parity violating asymmetry in elastic scattering due to the weak neutral interaction. Requires corrections for Coulomb distortions (Horowitz).

40 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 40/ K = From T. Klähn

41 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 41/68 Flow Constraint From Heavy Ions K = 0 MeV K = 100 MeV K = 200 MeV K = 300 MeV Consistent with 0 < K /MeV < 300 Danielewicz, Lacey & Lynch 2002

42 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 42/68 Nuclei in Dense Matter Liquid Droplet Model, Simplified F = u(f I + f LD /V N ) + (1 u)f II, f LD = f S + f C + f T f C = 3 Z 2 e 2 ( R N 2 u1/3 + u ) = 3 Z 2 e 2 D(u) 2 5 R N ( ) u f T = T ln n Q V N A 3/2 T = µ T T, n Q = f S = 4πR 2 N σ(µ s) ( mt 2π 2 ) 3/2 n = un I + (1 u)n II, Free Energy Minimization ny e = un I x I + (1 u)n II x II + u N s V N F z i = 0, z i = (n I, x I, R N, u, ν s, µ s ) µ n,ii = µ n,i + µ T A, P II = P I + 3σ 2R N ˆµ 3σ II = ˆµ I = µ s, N s = 4πRN 2 R N n I x i ) ( ) 1/3 (1 + ud 15σ, R N = D 8πn 2 I x2 I e2 D σ µ s

43 Cold Catalyzed Matter J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 43/68

44 Cold Catalyzed Matter J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 44/68

45 Supernova Matter J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 45/68

46 Proto-Neutron Stars J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 46/68

47 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 47/68 Proto-Neutron Star Evolution n dy L dt nt ds dt = n dy e dt + dy ν dt = 1 r 2 r r2 F ν = 1 4πr 2 L ν r n n,p,e,ν µ i dy i dt. In the diffusion approximation, fluxes are driven by density gradients: ( ) c n ν (E ν ) n ν (E ν ) F ν = λ ν λ ν de ν, 0 3 r r L ν = 4πr 2 cλ i E ǫ i (E ν ) de ν. 0 3 r i λ ν and λ i E s are mean free paths for number and energy transport, respectively. n ν(e ν ) and ǫ i (E ν ) are the number and energy density of species i = e, µ at neutrino energy 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.

48 Model Simulations J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 48/68

49 Model Simulations J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 49/68

50 Model Signal J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 50/68

51 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 51/68 M G [ M sun ] Effective Minimum Masses Strobel, Schaab & Weigel (1999) CNS HNS s1, HNS s2 LPNS s1 YL04, LPNSs2 YL04 EPNS s4s1 YL04 EPNS s5s1 YL04 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 M = 1.18 M R inf [ km ]

52 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 52/68 Neutron Star Cooling Gamow & Schönberg proposed the direct Urca process: nucleons at the top of the Fermi sea beta decay. c 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 kfi 3, thus x x DU = 1/9. 2 With muons (n > 2n s ), x DU = (1+2 1/3 ) 3 If x < x DU, bystander nucleons needed: modified Urca process is then dominant. (n, p) + n (n, p) + p + e + ν e, (n, p) + p (n, p) + n + e + + ν e Neutrino emissivities: ǫ MURCA ( T µ n ) 2 ǫ DURCA. Beta equilibrium composition: x β (3π 2 n) 1 ( 4Esym c ) 3 ( n 0.04 n s )

53 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 53/68 Direct Urca Threshold Klähn et al., Phys. Rev. C74 (2006)

54 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 54/68 Neutron Star Cooling Page, Steiner, Prakash & Lattimer (2004)

55 Light element envelopes 3 Light element 2 envelopes Neutron P gap = 0 3 Neutron P gap = 0 2 Heavy element envelopes Heavy element envelopes 3 Neutron P gap = "a" 3 Neutron P 2 gap = "b" 2 3 Neutron P gap = "a" 3 Neutron P gap = "b" 2 2 Minimal Cooling Paradigm Page, Lattimer, Prakash & Steiner (2009) J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 55/68

56 A possible constraint on E sym (n), i.e., it s not supersoft? J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 56/68 Minimal Cooling Paradigm If some observations are inconsistent with the MCP, then according to Sherlock Holmes, rapid cooling must occur for these exceptions. All sources are consistent with the MCP only IF tight conditions are placed on the magnitude and density dependence of the neutron 3 P 2 gap, AND some neutron stars have heavy Z envelopes and others have light Z envelopes, AND ALL core-collapse supernova remnants with no observable thermal emission contain black holes. Highly suggestive that rapid cooling occurs in some neutron stars (of higher masses?)

57 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 57/68 Possible Kinds of Observations Maximum and Minimum Mass (binary pulsars) Minimum Rotational Period Radiation Radii or Redshifts from X-ray Thermal Emission Crustal Cooling Timescale from X-ray Transients X-ray Bursts from Accreting Neutron Stars Seismology from Giant Flares in SGR s Neutron Star Thermal Evolution (URCA or not) Moments of Inertia from Spin-Orbit Coupling Neutrinos from Proto-Neutron Stars (Binding Energies, Neutrino Opacities, Radii) Pulse Shape Modulations Gravitational Radiation from Neutron Star Mergers (Masses, Radii from tidal Love numbers) Significant dependence on symmetry energy

58 Potentially Observable Quantities Apparent angular diameter from flux and temperature measurements β GM/Rc 2 R D = R 1 F 1 = D 1 2β σ f T 2 2 Redshift z = (1 2β) 1/2 1 Eddington flux Crust thickness mbc 2 F EDD = GMc κc 2 (1 2β)1/2 D2 Moment of Inertia 2 R R R t R Crustal Moment of Inertia Binding Energy ln H h t = pt 0 (H 1)(1 2β) H + 2β 1 dp n = µ n,t µ n,t (p = 0) (H 1) ( ) 1 2β 1. I (0.237 ± 0.008)MR 2 ( β β 4 ) M km 2 I I 8π 3 R 6 p t IMc 2 β B.E. (0.60 ± 0.05) 1 β/2 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 58/68

59 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 59/68 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 are from Nearby isolated neutron stars (parallax measurable) Quiescent X-ray binaries in globular clusters (reliable distances, low B H-atmosperes) X-ray pulsars in systems of known distance CXOU J in the SMC: R 10.8 km (Esposito & Mereghetti 2008)

60 RX J J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 60/68

61 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 61/68 Radiation Radius: Nearby Neutron Star RX J : Walter & Lattimer 2002 Braje & Romani 2002 Truemper 2005 D=120 pc BUT D=140 pc Kaplan, van Kerkwijk & Anderson 2002 Raises R limit to 19.5 km Magnetic H atmospher R 17 km Ho et al. 2007

62 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 62/68 Radiation Radius: Globular Cluster Sources Webb & Barret 2007 M Tuc ω Cen

63 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 63/68 The Neutron Star Crust Hydrostatic equilibrium in the crust: dp(r) m b n = dµ GM = m b r 2 2GMr/c 2 dr. µ t µ 0 m b c 2 = 1 [ 2 ln rt (R 2GM/c 2 ] ) R(r t 2GM/c 2. ) Defining ln H = 2(µ t µ 0 )/m b c 2, R R R t R = (H 1)(1 2β) H + 2β 1 (H 1)(1 2β) R2 2M. Crustal moment of inertia I = 8π 3c 4 R R r 4 (ǫ + p)e λ jωdr 8πω(R) 3Mc 2 0 p(r ) r 6 dp Approximately, 0 p(r ) r6 dp R 6 p t e 4.8 /R. p t < 0.65 MeV fm 3. I I 8πR4 p t 3M 2 c 2 ( MR 2 I ) 2β e 4.8 /R

64 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 64/ Pulsar Glitches Pulsars occasionally undergo glitches, when the spin rate "hiccups". Each glitch changes the angular momentum of the star by J = I liquid Ω. The glitches are stochastic, but total angular momentum transfer is regular. J(t) = I liquid Ω Ω Ω, J = I Ωd( Ω/Ω) liquid. dt A leading model is that as the crust slows due to pulsar s dipole radiation, the interior acquires an excess differential rotation. The acquired excess is limited: J ΩI crust. Crab Glitch time (days) I crust I liquid = spin rate I crust I I crust Ω Ω glitch instability time d ( Ω/Ω) dt liquid crust cumulative angular momentum 2X10 5 1X Vela Link, Epstein & Lattimer (1999) MJD

65 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 65/68 Moment of Inertia Spin-orbit coupling of same magnitude as post-post-newtonian effects (Barker & O Connell 1975, Damour & Schaeffer 1988) Precession alters inclination angle and periastron advance More EOS sensitive than R: I MR 2 Requires extremely relativistic system to extract Double pulsar PSR J is a marginal candidate Even more relativistic systems should be found, based on dimness and nearness of PSR J

66 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 66/68 Moments of Inertia and Precession Spin-orbit coupling: SA = L = G(4M A +3M B ) 2M A a 3 c 2 L SA

67 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 66/68 Moments of Inertia and Precession Spin-orbit coupling: SA = G(4M L = A +3M B ) 2M A a 3 c L 2 SA Precession Period: P p = 2(M A + M B )ac 2 GM B (4M A + 3M B ) P(1 e2 ) 74.9 years

68 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 66/68 Moments of Inertia and Precession Spin-orbit coupling: SA = G(4M L = A +3M B ) 2M A a 3 c L 2 SA Precession Period: Precession Amplitude SA L: δ i = S A L P p = 2(M A + M B )ac 2 GM B (4M A + 3M B ) P(1 e2 ) 74.9 years sin θ I A(M A + M B ) a 2 M A M B P P A sin θ ( ) sin θ 10 5

69 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 66/68 Moments of Inertia and Precession Spin-orbit coupling: SA = G(4M L = A +3M B ) 2M A a 3 c L 2 SA Precession Period: Precession Amplitude SA L: δ i = S A L P p = 2(M A + M B )ac 2 GM B (4M A + 3M B ) P(1 e2 ) 74.9 years sin θ I A(M A + M B ) a 2 M A M B Delay in Time-of-Arrival: ( M B t = M A + M B P P A sin θ ( ) sin θ 10 5 ) a c δ i cos i sin θ µs

70 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 66/68 Moments of Inertia and Precession Spin-orbit coupling: SA = G(4M L = A +3M B ) 2M A a 3 c L 2 SA Precession Period: Precession Amplitude SA L: δ i = S A L P p = 2(M A + M B )ac 2 GM B (4M A + 3M B ) P(1 e2 ) 74.9 years sin θ I A(M A + M B ) a 2 M A M B Delay in Time-of-Arrival: ( M B t = M A + M B P P A sin θ ( ) sin θ 10 5 ) a c δ i cos i sin θ µs Periastron Advance SA L: A P /A P N = ( ) 2πI A 2 + 3M B /M A MA + M B P A 3MA 2 + 3M2 B + 2M AM B Ga cos θ ( ) 10 4 cos θ

71 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 66/68 Moments of Inertia and Precession Spin-orbit coupling: SA = G(4M L = A +3M B ) 2M A a 3 c L 2 SA Precession Period: Precession Amplitude SA L: δ i = S A L P p = 2(M A + M B )ac 2 GM B (4M A + 3M B ) P(1 e2 ) 74.9 years sin θ I A(M A + M B ) a 2 M A M B Delay in Time-of-Arrival: ( M B t = M A + M B P P A sin θ ( ) sin θ 10 5 ) a c δ i cos i sin θ µs Periastron Advance SA L: A P /A P N = ( ) 2πI A 2 + 3M B /M A MA + M B P A 3MA 2 + 3M2 B + 2M AM B Ga cos θ ( ) 10 4 cos θ Moment of Inertia Mass Radius [ I (0.237 ± 0.008)MR M km ( ) ] M km R M R M

72 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 67/68 Comparison of Binary Pulsars PSR PSR PSR References a, b, c d e, f a/c (s) P (h) e M A (M ) ± ± ± M B (M ) ± ± ± T GW (M yr) i 87.9 ± P A (ms) θ A 13 ± ± 3.8 φ A 246 ± ± 20 P pa (yr) δt a /I A,80 (µs) 0.7 ± ± 1.1 A pa /(A 1PN I A,80 ) A 2PN /A 1PN a: Lyne et al. (2004); b: Solution 1, Jenet & Ransom (2004); c: Coles et al. (2004) d: Weisberg & Taylor (2002, 2004); e: Stairs et al. (2002, 2004); f: Bogdanov et al. (2002)

73 J.M. Lattimer, CompSchool2009, NBIA, August 2009 p. 68/68 EOS Constraint Lattimer & Schutz (2005) Bejger & Haensel (2003)