Neutron Stars. James M. Lattimer. Department of Physics & Astronomy.

Transcription

1 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 1/75 Neutron Stars James M. Lattimer Department of Physics & Astronomy

2 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 2/75 Outline Structure Dense Matter Equation of State Observational Constraints Formation and Evolution

3 Neutron Stars: The Early History 1920 Rutherford predicts the neutron 1931 Landau anticipates single-atom stars but does not predict 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 Prediction that neutron stars have intense magnetic fields Hewish and Okoye discover an intense radio source in the Crab nebula Colgate and White perform simulations of supernovae leading to neutron stars Wheeler predicts Crab nebula powered by a rotating neutron star Pacini makes first pulsar model C. Schisler discovers a dozen pulsing radio sources, including the Crab pulsar, using secret military radar in Alaska Schlovsky proposes accretion on neutron star as source of X-rays from Sco 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 T. Gold identifies pulsars with magnetized, rotating neutron stars The term pulsar first appears in print, in the Daily Telegraph. J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 3/75

4 1969 Glitches observed and proposed to be evidence for superfluidity in neutron star 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. It shows the orbital decay due to gravitational radiation predicted by Einstein s General Theory of Relativity Chart recording of PSR used as album cover for Unknown Pleasures by Joy Division (#19/100 greatest British albums ever) First millisecond pulsar, PSR B , discovered by Backer, Kulkarni, et al. at Arecibo Discovery of first extra-solar planetary system orbiting PSR B by Wolszczan and Frail Prediction of magnetars (Duncan & Thompson) Hulse and Taylor receive Nobel Prize Kouveletiou et al. discover SGR , first magnetar. sleevage.com/joy-division-unknown-pleasures/ 2004 Largest burst of energy seen in Galaxy since SN 1604 from magnetar SGR ; brighter than full moon in gamma rays; more energy emitted than Sun over 100,000 years Hessels et al. discover PSR J ad; fastest known rotation rate, 716 Hz Hessels et al. discover PSR J , first binary with two pulsars. J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 4/75

5 sleevage.com/joy-division-unknown-pleasures/ J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 5/75

6 Pulsars: What are they? Extreme regularity of pulsing implies source is a massive object. Small period, 1 ms, can only be explained by compact objects from orbital motion, vibrations or from spin. A binary with total mass M = 1 M and period P = s, according to Kepler s Law, has a separation a ( ) M 1/3 ( ) P 2/3 a = = AU = 93 miles. 1 year M Smaller than any solar mass object except a neutron star or black hole. Vibrational frequency is of order 2πν G ρ; 30 Hz implies ρ g cm 3. The mass-shedding rotational limit is of order ( ) 2π 2 R < GM P R 2 ρ > 3 ( ) 2πP g cm 3. 4π G Only a neutron star or black hole works. There is no known mechanism for black holes to emit pulses of energy. The discovery that the Crab pulsar is slowing down is only consistent with a spinning object and rules out binaries or pulsating stars. A binary that is losing energy goes into a tighter orbit with a smaller period. A vibrating object that is losing energy develops a higher frequency of oscillation. J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 6/75

7 Credit: Dany Page, UNAM J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 7/75

8 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, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 8/75

9 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 9/75 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

10 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 10/75 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

11 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 11/75 Uniform Density Fluid m(r) = 4π 3 ǫr3 Mx 3/2, β M R e λ(r) = 1 2βx, [ ] 3 e ν(r) 1 2 = 1 2β 1 2βx, 2 2 [ ] 1 2βx 1 2β p(r) = ǫ 3 1 2β, 1 2βx ǫ(r) = constant; n(r) = constant ( BE M = 3 sin 1 2β ) 1 2β 3β 4β 2β β2 + c 2 s = p c < = β < 4/9

12 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 12/75 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 = 2 tan φ c 15 3 β 1 3, c2 s,c = tan φ c ( BE M β β2 + ) 1 β 5 tan φ c + 3 p,c 2 s < = φ c < π/2 = β <

13 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 13/75 Buchdahl s Solution: Relativistic n=1 Polytrope 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 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 < = β < 1/5 ) 1

14 0.30 < c 2 s,c < 0.44, < ǫ surf ǫ c < 1 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 14/75 Tolman IV Variation: A Self-Bound Star Finite, large surface energy density, as in a strange quark star K. Lake s solution e ν(r) = e λ(r) = 4πpR 2 = [ β ( x)] 2 4πǫR 2 = 3(1 β) 2/3 β m = c 2 s = ǫ surf ǫ c =, (1 2β) [ β ( x)] 2/3 [ β (, x)] 2/3 2(1 β) 2/3 βx β β ( x) (1 β) 2/3 1 5 β(1 x) 2 [ β ( x)] 2/ β(1 1 3 x) [ β ( x)] 5/3, (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 2/3 (1 β) 2/3 + (2 5β) 2 5β 2 x (1 53 ) ( β 1 5 ) 2/3 2 β (1 β) 5/3., ],

15 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 15/75 GR: R > 2GM/c 2 Mass-Radius Diagram P < : R > (9/4)GM/c 2 causality: R > 2.9GM/c2 normal NS SQS R = R 1 2GM/Rc 2 contours

16 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 16/75 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).

17 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 17/75 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

18 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 ( ) Mmax P min Rmin 3 ǫo 1/2 R 3/2 ( ) 1/2 M, P min = 0.76 ms 10 km M J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 18/75 ǫ o

19 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 19/75 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

20 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 20/75 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) a/m = cj/gm (empirical)

21 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 21/75 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 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.

22 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 22/75 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.

23 PSR , Kramer et al. (2007) J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 23/75

24 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 24/75 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

25 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 25/75 Median of probability for pulsar mass is 2.74 M 0.1% chance M < 1.5 M PSR J B in NGC 6440, Freire et al. (2008)

26 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 26/75 Constraints from Pulsar Masses and Spins PSR J B XTE J ν = 1122 Hz Kaaret et al Not confirmed to be a rotation rate PSR PSR J ad ν = 716 Hz Hessels et al. 2006

27 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 27/75 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 M0 (1 < n f /n s < 2) Wide variation: 1.2 < p(n s) MeV fm 3 < 7 n s

28 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 28/75 The Radius Pressure Correlation R p 1/4 Lattimer & Prakash (2001)

29 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 29/75 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

30 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 30/75 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, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 31/75 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 [ ǫ(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 P = n 2 (ǫ/n) [ ( ) ] = n2 K n 1 + S v (1 2x) 2 + 2an ( ns ) 2/3 T 2 n n s 9 n s 3 n µ n = ǫ n x ǫ n x = B + K ) ) (1 (1 nns 3 nns n + 2S v (1 4x 2 ) a ( ns ) 2/3 T 2 18 n s 3 n 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

32 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 32/75 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 4x 2 ) 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

33 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 33/75

34 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 34/75 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) = Skewness C. Fuchs, H.H. Wolter, EPJA 30(2006) 5

35 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 35/75 The Uncertain E sym (n) C. Fuchs, H.H. Wolter, EPJA 30(2006) 5

36 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 36/75 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, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 37/75 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, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 38/75 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, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 39/75 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, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 40/75 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

41 Cold Catalyzed Matter J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 41/75

42 Cold Catalyzed Matter J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 42/75

43 Supernova Matter J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 43/75

44 Proto-Neutron Stars J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 44/75

45 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 45/75 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 ]

46 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 46/75 BE(M, R) Lattimer & Prakash (2001) BE (0.60 ± 0.05) GM Rc 2 [ 1 GM 2Rc 2 ] 1

47 Model Simulations J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 47/75

48 Model Simulations J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 48/75

49 Model Signal J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 49/75

50 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 50/75 Neutron Star Cooling 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 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 )

51 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 51/75 Direct Urca Threshold Klähn et al., Phys. Rev. C74 (2006)

52 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 52/75 Neutron Star Cooling Page, Steiner, Prakash & Lattimer (2004)

53 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 53/75 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) Redshifts from Pulse Shape Modulation Gravitational Radiation from Neutron Star Mergers (Masses, Radii from tidal Love numbers) Significant dependence on symmetry energy

54 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 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 Crustal Moment of Inertia I 8π R 6 p t I 3 IMc 2 Binding Energy β B.E. (0.60 ± 0.05) 1 β/2 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 54/75

55 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 55/75 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 and temperature distribution Typical optical excess (factor f 6) above Rayleigh-Jeans tail of X-ray blackbody distribution (T X 2T O ). Non-uniform temperature: R R,X 1 + ftx /T O 4R,X. 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)

56 RX J J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 56/75

57 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 57/75 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

58 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 58/75 Radiation Radius: Globular Cluster Sources Webb & Barret 2007 M Tuc ω Cen

59 EXO J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 59/75 Cooling Following An X-Ray Burst Galloway, Muno, Hartman, Psaltis & Chakrabarty (2006) F EDD F EDD Af 4 = R /D Af 4 = R /D

60 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 60/75 EXO Özel, Güver & Psaltis (2008)

61 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 61/75 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) R 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

62 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 62/75 Giant Flares in Soft Gamma-Ray Repeaters (SGRs) Quasi-periodic oscillations observed following giant flares in three soft gamma-ray repeaters (Israel et al. 2005; Strohmayer & Watts 2005, 6; Watts & Strohmayer 2006) which are believed to be highly magnetized neutron stars (magnetars). Fields decay and twist, becoming periodically unstable. Eventually, the field lines snap and shift, launching starquakes and bursts of gamma-rays. Torsional shear modes are much easier to excite than radial modes. Watts & Reddy (2006)

63 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 63/75 Observations Typical frequencies observed: Hz, Hz, 625 Hz (SGRs , , ) Frequencies of fundamental mode (28-29 Hz) agree well with expected torsional mode of neutron star crust (Duncan 1998) Watts & Strohmayer (2006)

64 Analysis Samuelsson & Andersson Include crust elasticity in a relativistic context of axial oscillations; calculate both fundamental and overtone frequencies using the Cowling approximation: neglect dynamical nature of spacetime (i.e., perturbations of gravitational field). Ignores magnetic fields and superfluidity and is thus insensitive to core structure. F + A F + BF = 0 e A = r 4 e (ν λ)/2 (ǫ + p)vr 2 B = e λ vr 2 [e ν ω 2 vt 2 (l 1)(l + 2)r 2 ] F (R) = F (R c ) = 0 F : amplitude of fluid oscillations ν, λ: metric functions: at surface e ν e λ 1 2β v r, v t : radial/transverse shear velocities Analytical approximation: ω >> v t /r; v r v t v F (R) = F (R t ) = 0 R R t Bdr = nπ ω 2 nπvω n = 0 : n > 0 : e(ν λ)/2 v2 (l 1)(l+2) ω 0,l e ν/2 (l 1)(l+2) v 2RR c 2RR t e ν = 0 v/r ω n (1 2β)nπv 1 Mv R 2 1 H 1 Watts & Strohmayer (2006) J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 64/75

65 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 65/75 Neutron Star Seismology f n=0 v t /R f n>0 v r (1 2β)/ v r M/[R 2 (H 1)] Strohmayer & Watts (2005) Samuelsson & Andersson (2006) Lattimer & Prakash (2006)

66 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 66/ 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

67 Constraints From X-Ray Pulse Shapes Pulse profiles for accretion-powered pulsations from XTE J1814 from June 5-27, 2003 (excluding bursts). Stellar spin frequency is 314 Hz. Analysis yields two χ 2 minima, the smaller being excluded by its unphysical nature. Leahy, Morsink, Chung & Chou 2008) SAX J Leahy, Morsink & Cadeau 2007) J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 67/75

68 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 68/75 Other Observable Quantities Redshift z = (1 2GM/Rc 2 ) 1/2 1 Possible lines from active X-ray bursters XTE J z < 0.38, 4U < z < 0.30, EXO z (possibly Fe XXV and XXVI Cottam, Paerels & Mendez 2002). Gravitational light-bending analysis of emissions from Her X-1: < z < (1.29 < M/M < < R/km < 13.1) Permits determination of M and R if both z and R known. R = R (1 + z) 1, M = R c 2 (1 + z) 1 [1 (1 + z) 2 ]/2G QPOs Accreting matter in low mass X-ray binaries emits X-rays due to conversion of gravitational energy into thermal energy, kt GM/R. Fourier analysis of the X-rays often show quasi-periodic frequency peaks, including a high-frequency peak around 1300 Hz. It is thought that this peak could be associated with the orbital frequency ω = GM/RA 3 of the inner radius of accretion disk (Lamb & Miller 2000). In GR, the innermost circular stable orbit about a non-rotating star has a radius R ISCO = 6GM/c 2. From R < R A < R ISCO, M < c 3 6 3/2 Gω = 2.2 ( ) 1000 Hz ν M, R < c 6 1/2 ω = 19.5 ( ) 1000 Hz ν km.

69 Main figure courtesy D. Page; inset R. van der Klies 2000 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 69/75

70 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 70/75

71 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 71/75 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

72 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 72/75 Moments of Inertia and Precession Spin-orbit coupling: SA = L = G(4M A +3M B ) 2M A a 3 c 2 L SA

73 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 72/75 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

74 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 72/75 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

75 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 72/75 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

76 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 72/75 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

77 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 72/75 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

78 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 73/75 I(M, R) Lattimer & Prakash (2001) [ ( ) ] 4 I (0.237 ± 0.008)MR M km + 90 M km R M R M

79 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 74/75 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)

80 J.M. Lattimer, Connecting Quarks With the Cosmos Summer School, Seattle, July 2009 p. 75/75 EOS Constraint Lattimer & Schutz (2005) Bejger & Haensel (2003)