Mass, Radius and Equation of State of Neutron Stars

Home , Crab Pulsar

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

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Outline

I Structure

I Dense Matter Equation of State

I Formation and Evolution

I Observational and Experimental Constraints

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Dany Page, UNAM

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Amazing Facts About Neutron Stars 15 −3 I Densest objects this side of an event horizon: 10 g cm Four teaspoons, on the Earth, would weigh as much as the Moon. 14 −2 I Largest surface gravity: 10 cm s This is 100 billion times the Earth’s gravity. I Fastest spinning objects known: ν = 716 Hz This spin rate was measured for PSR J1748-2446ad, 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. 15 I Largest known magnetic ﬁeld strengths: B = 10 G I Highest temperature superconductor: Tc = 10 billion K The highest known superconductor on the Earth is mercury thallium barium calcium copper oxide (Hg12Tl3Ba30Ca30Cu45O125), at 138 K. I Highest temperature, at birth, anywhere in the Universe since the Big Bang: T = 700 billion K I PSR B1508+55 has fastest measured stellar velocity in the Galaxy: 1083 km/s = c/300 I The only place in the universe except for the Big Bang where neutrinos become trapped. J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Neutron Stars: History

1920 Rutherford predicts the neutron 1931 Landau anticipates single-nucleus stars but not neutron stars 1932 Chadwick discovers the neutron. 1934 W. Baade and F. Zwicky predict existence of neutron stars as end products of supernovae. 1939 Oppenheimer and Volkoff predict upper mass limit of neutron star. 1964 Hoyle, Narlikar and Wheeler predict neutron stars rapidly rotate. 1965 Hewish and Okoye discover an intense radio source in the Crab nebula. 1966 Colgate and White perform simulations of supernovae leading to neutron stars. 1967 C. Schisler discovers a dozen pulsing radio sources, including the Crab pulsar, using secret military radar in Alaska. X-1. 1967 Hewish, Bell, Pilkington, Scott and Collins discover “first” PSR 1919+21, Aug 6. 1968 The Crab Nebula pulsar is discovered, found to be slowing down (ruling out binary and vibrational models), and clinched the connection to supernovae. 1968 The term “pulsar” first appears in print, in the Daily Telegraph. J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars 1969 “Glitches” observed; evidence for superfluidity in neutron star crust. 1971 Accretion powered X-ray pulsar discovered by Uhuru (not the Lt.). 1974 Hewish awarded Nobel Prize (but Bell and Okoye were not). 1974 Binary pulsar PSR 1913+16 discovered by Hulse and Taylor., orbital decay due to GR gravitational radiation 1979 Chart recording of PSR 1919+21 used as album cover for Unknown Pleasures by Joy Division (#19/100 greatest British album). 1982 First millisecond pulsar, PSR B1937+21, discovered by Backer et al. at Arecibo. 1992 Discovery of first extra-solar planets orbiting PSR B1257+12 by Wolszczan and Frail. 1993 Hulse and Taylor receive Nobel Prize 1998 Kouveletiou et al. discover first magnetar 2004 SGR 1806-20 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. sleevage.com/joy-division-unknown-pleasures/ 2004 Hessels et al. discover PSR J1748-2446ad; fastest rotation rate, 716 Hz. 2005 Hessels et al. discover PSR J0737-3039, first two-pulsar binary 2013 Antoniadis et al. find most-massive PSR J0348+0432, 2.01 M 2013 Ransom et al. find first pulsar in triple system (two white dwarfs) J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars sleevage.com/joy-division-unknown-pleasures/

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Neutron Star Structure

Tolman-Oppenheimer-Volkov equations of relativistic hydrostatic equilibrium:

dp G (m + 4πr 3p/c2)( + p) = − dr c2 r(r − 2Gm/c2) dm = 4π r 2 dr c2

p is pressure, is mass-energy density Useful analytic solutions exist:

I Uniform density = constant 2 I Tolman VII = c [1 − (r/R) ] √ I Buchdahl = pp∗ − 5p ν 2 I Tolman IV e = α + β(r/R)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Uniform Density Fluid

4π r 2 M m(r) = r 3 = Mx 3/2, x ≡ , β ≡ 3 R R e−λ(r) = 1 − 2βx, 3 1 2 eν(r) = p1 − 2β − p1 − 2βx , 2 √ 2√ 1 − 2βx − 1 − 2β p(r) = √ √ , 3 1 − 2β − 1 − 2βx (r) = constant; √ n(r) = constant BE 3 sin−1 2β 3β 9 = √ − p1 − 2β ' + β2 + ··· , M 4β 2β 5 14 2 cs = ∞

pc < ∞ =⇒ β < 4/9

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Tolman VII

2 (r) = c 1 − (r/R) ≡ c [1 − x] e−λ(r) = 1 − βx(5 − 3x) eν(r) = (1 − 5β/3) cos2 φ, 1 p β p(r) = 3βe−λ(r) tan φ(r) − (5 − 3x) , 4πR2 2 (r) + p(r) cos φ(r) n(r) = mb cos φ1 s w − w(r) β φ(r) = 1 + φ , φ = φ(x = 1) = tan−1 , 2 1 1 3(1 − 2β)  s  " s # 5 e−λ(r) 1 1 − 2β w(r) = ln x − + , w = w(x = 1) = ln + .  6 3β  1 6 3β s r ! 2 tan φ 3 1 1 β (p/) = c − , c2 = tan φ tan φ + c 15 β 3 s,c c 5 c 3 BE 11 7187 ' β + β2 + ··· M 21 18018 2 p, cs < ∞ =⇒ φc < π/2 =⇒ β < 0.3862

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Buchdahl’s Solution: Relativistic n=1 Polytrope √ = p∗p − 5p eν(r) = (1 − 2β)(1 − β − u(r))(1 − β + u(r))−1, eλ(r) = (1 − 2β)(1 − β + u(r))(1 − β − u(r))−1(1 − β + β cos Ar 0)−2, 8πp(r) = A2u(r)2(1 − 2β)(1 − β + u(r))−2, 8π(r) = 2A2u(r)(1 − 2β)(1 − β − 3u(r)/2)(1 − β + u(r))−2, 3/2 s ! r −1 p p(r) 2 1 p∗ mbn(r) = p∗p(r) 1 − 4 , cs (r) = − 5 p∗ 2 p(r) β 1r p −1 u(r) = sin Ar 0 = (1 − β) ∗ − 1 , Ar 0 2 p(r) r 0 = r(1 − 2β)(1 − β + u(r))−1, r 2 −1 π A = 2πp∗(1 − 2β) , R = (1 − β) . 2p∗(1 − 2β) p p 5 p p = ∗ β2, = ∗ β(1 − β), n m = ∗ β(1 − 2β)3/2 c 4 c 2 2 c b 2 BE 3 β β2 = (1 − β)(1 − 2β)−1/2(1 − β)−1 ' + + ··· M 2 2 2 2 cs < ∞ =⇒ β < 1/5

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Tolman IV Variation: A Self-Bound Star (Quark Star)

2 1 − 5 β 1 − 1 x eν(r) = 2 5 , (1 − 2β) 2/3 1 − 5 β 1 − 3 x eλ(r) = 2 5 , 2/3 1 − 5 β 1 − 3 x − 2(1 − β)2/3βx 2 5 " # β 1 − 5 β(1 − x) 4πpR2 = 1 − (1 − β)2/3 2 , 1 − 5 β 1 − 1 x 5 3 2/3 2 5 1 − 2 β 1 − 5 x 3(1 − β)2/3β 1 − 5 β(1 − 1 x) (1 − β)2/3Mx 3/2 4πR2 = 2 3 , m = 5 3 5/3 5 3 2/3 1 − 2 β 1 − 5 x 1 − 2 β 1 − 5 x (2 − 5β + 3βx) (2 − 5β + 3βx)5/3 c2 = + (2 − 5β)2 − 5β2x , s 5(2 − 5β + βx)3 22/3(1 − β)2/3 5 5 2/3 surf = 1 − β 1 − β (1 − β)−5/3. c 3 2 2 surf 0.30 < cs,c < 0.44, 0.265 < < 1 c

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Important Questions

I How Does the Structure of Neutron Stars Depend On the Nucleon-Nucleon Interaction?

I The Neutron Star Maximum Mass and Causality

I The Neutron Star Radius and the Nuclear Symmetry Energy

I Does Exotic Matter (Hyperons, Kaons/Pions, Deconﬁned Quarks) Exist in Neutron Star Interiors?

I How Do Nuclear Experiments Constrain the Nuclear Symmetry Energy and Neutron Star Radii?

I Binding Energies

I Heavy ion Collisions

I Neutron Skin Thicknesses

I Dipole Polarizabilities

I Giant (and Pygmy) Dipole Resonances

I Pure Neutron Matter

I What Astrophysical Constraints Exist?

I Mass Measurements of Pulsars and X-ray Binaries

I Photospheric Radius Expansion Bursts

I Thermal Emission from Isolated and Quiescent Binary Sources

I Pulse Modeling of X-ray Bursts, QPOs, Gravitational Radiation, etc.

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Neutron Star Structure Tolman-Oppenheimer-Volkov equations dp G (m + 4πpr 3/c2)(ε + p) = − dr c2 r(r − 2Gm/c2) dm ε = 4π r 2 dr c2 -maximum mass p(ε) - R ∝ p1/4 -- M(R-)

b e v to sqainfState ObservationsEquationof J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Mass-Radius Diagram and Theoretical Constraints GR: R > 2GM/c2 P < ∞ : R > (9/4)GM/c2

causality: R ∼> 2.9GM/c2 — normal NS — SQS R — R∞ =√ 1−2GM/Rc2

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars The Radius – Pressure Correlation

R ∝ p1/4

Red points satisfy > Mmax ∼ 2M

Lattimer & Prakash (2001)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Neutron Star Structure Newtonian Gravity: dp Gmρ dm = − ; = 4πρr 2; ρc2 = ε dr r 2 dr Newtonian Polytropes: p = Kργ; M ∝ K 1/(2−γ)R(4−3γ)/(2−γ) 4 ρ < ρs : γ ' 3 ; ρ > ρs : γ ' 2 -maximum mass p(ε) - p = Kρ2 4 R ∝ K 1/2M0 2.5 - p = Kρ4/3 M ∝ K 3/2R0 ρ/ρs = 1.0 -

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Extremal Properties of Neutron Stars

I 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).

ε0 is the only EOS parameter ε o − ε The TOV = p solutions scale causal limit with ε0

w = ε/ε0 y = p√/ε0 2 x = r √Gε0/c 3 2 soft ⇒ z = m G ε0/c =

= stiﬀ ⇐ ◦ p = 0

o J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Extremal Properties of Neutron Stars

The maximum mass conﬁguration is achieved when xR = 0.2404, wc = 3.034, yc = 2.034, zR = 0.08513. 1/2 I Mmax = 4.1 (εs /ε0) M (Rhoades & Ruﬃni 1974) 2 1/2 I MB,max = 5.41 (mB c /µo )(εs /ε0) M 2 I Rmin = 2.82 GM/c = 4.3 (M/M ) km

I µb,max = 2.09 GeV 2 I εc,max = 3.034 ε0 ' 51 (M /Mlargest ) εs 2 I pc,max = 2.034 ε0 ' 34 (M /Mlargest ) εs 2 I nB,max ' 38 (M /Mlargest ) ns

I BEmax = 0.34 M 1/2 3/2 I Pmin = 0.74 (M /Msph) (Rsph/10 km) ms = 0.20 (Msph,max /M ) ms

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Maximum Energy Density in Neutron Stars

p = s(ε − ε0)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Causality + GR Limits and the Maximum Mass A lower limit to the maximum mass sets a M − R curves for= maximally compact EOS lower limit to the = ⇒ ⇒ radius for a given mass. Similarly, a precise (M, R) measurement sets an upper limit to the maximum mass. quarkss stars

1.4M stars must have R > 8.15M . 2 ∂p cs ∂ε = c2 = s 1.4M strange quark matter stars (and likely hybrid quark/hadron stars) must have R > 11 km.

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Mass-Radius Diagram and Theoretical Constraints GR: R > 2GM/c2 P < ∞ : R > (9/4)GM/c2

causality: R ∼> 2.9GM/c2 — normal NS — SQS R — R∞ =√ 1−2GM/Rc2

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Roche Model for Stellar Rotation

(c.f., Shapiro & Teukolsky 1983) 3/2 1/2 Pshed R M −1 ' 1.0 ρ ∇P = ∇µ = −∇(ΦG + Φc ) ms 10 km M 1 2 2 2 Φ ' −GM/r, Φ = − Ω r sin θ 2 G c 2 Shape: Ω2R3 sin θ R = − 1 Bernoulli integral: 2GM Rp " #! H = µ + ΦG + Φc = −GM/Rp R 1 2 1 Ω sin θ 2 R p −1 p = + cos cos−1 1 − 2 µ = 0 ρ dp = µn − µn0 R(θ) 3 3 3 Ωshed Evaluate at equator: R sin(θ) Limit: p = . Ω2R3 R R(θ) 3 sin(θ/3) eq = eq − 1 2GM Rp Mass-shedding limit 2 3 3 Ωshed = GM/Req : Req/Rp = 2 GR: Cook, Shapiro & Teukolsky (1994): 1.43–1.51 3/2 r 2 GM Ω = shed 3 R3

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars The Equation of State in Neutron Stars

I The EOS of an ideal Fermi gas

I Thermodynamic relations

I Degeneracy

I Limiting cases

I Interactions

I Properties of bulk nucleonic matter

I Phase coexistence

I Nuclear symmetry energy

I Finite nuclei – the liquid droplet model

I Nuclei in dense matter

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars EOS of an Ideal Fermion Gas

Single particle energy E 2 = m2c4 + p2c2 Occupation index E − µ−1 f = 1 + exp T

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Thermodynamics

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

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Pressure and Degeneracy Parameters

g mc 3 nc = 2 2π ~

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Limits

Non-relativistic p << mc x = p2/(2mT ), ψ = (µ − mc2)/T g(2mT )3/2 Z ∞ x 1/2 g(2mT )3/2 n = 2 3 x−ψ = 2 3 F1/2(ψ) 4π ~ 0 1 + e dx 4π ~ 3/2 2 gT (2mT ) ε = nmc + F3/2(ψ) 4π2~3 2 5F (ψ) P = (ε − nmc2), s = 3/2 − ψ 3 3F1/2(ψ) Relativistic p >> mc g T 3 n = F2(φ) 2π2 ~c gT T 3 ε = F3(φ) 2π2 ~c ε 4F (φ) P = , s = 3 − φ 3 3F2(φ)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Limits

 3 2  g φT π π2  n = 2 1 + + ··· , s = + ···   6π ~c φ φ  EDER 2  P = ε = nφT 1 + π + ··· ∝ n4/3   3 4 φ 

 3/2 2  g 2mψT 1 π π2  n = 2 2 1 + + ··· , s = + ···   6π ~ 8 ψ 2ψ  EDNR 2  P = 2 (ε − nmc2) = 2nψT 1 + 1 π + ··· ∝ n5/3   3 5 2 ψ 

g T 3 φ n = π2 c e , s = 4 − φ ε ~ NDER P = 3 = nT ( ) mT 3/2 ψ 5 n = g 2π 2 e ; s = 2 − ψ 2 ~ 2 NDNR P = 3 (ε − nmc ) = nT

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Limit of Relativistic Gas With Pairs in Thermal Equilibrium

µ = µ+ = −µ− = φT , " # g µ 3 πT 2 n = n+ − n− = 1 + , 6π2 ~c µ " # ε gµ µ 3 πT 2 7 πT 4 P = P+ + P− = = 1 + 2 + , 3 24π2 ~c µ 15 µ " # gT µ 2 7 πT 2 s = s+ + s− = 1 + . 6n ~c 15 µ

Cubic Solution:

h 1/2 i1/3 µ = r − q/r, r = q3 + t2 + t , 3π2 (πT )2 t = n( c)3, q = . g ~ 3

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Non-Relativistic Fermi Gases With Interactions

X 2 ε = ~ τ + U(n , n ) 2m∗ t n p t t Single particle energy, t = n, p 2 2 p t = mt c + ∗ + Vt (nn, np, τn, τp) 2mt (nn, np) 2 δε δε 2 X ∂(m∗)−1 ∂U ~ = , V = = ~ τ s + 2m∗ δτ t δn 2 s ∂n ∂n t t t s s t Number and kinetic densities ∗ 3/2 ∗ 5/2 1 2mt T 1 2mt T nt = F1/2(ηt ); τt = F3/2(ηt ) 2π2~2 ~2 2π2~2 ~2 µ − m c2 − V − µ −1 η = t t t ; f = 1 + exp t t t T t T 2 2 X τt X 5 τt P = n V + ~ − U, ns = ~ − n η t t 3m∗ 6m∗T t t t t t t

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Bulk Matter Energy and Pressure

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Schematic Free Energy Density

n: number density; x: proton fraction; T : temperature −3 ns ' 0.16 ± 0.01 fm : nuclear saturation density B ' 16 ± 1 MeV: saturation binding energy K ' 240 ± 20 MeV: incompressibility parameter Sv ' 30 ± 6 MeV: bulk symmetry parameter L ' 60 ± 60 MeV: symmetry stiﬀness parameter a ' 0.065 ± 0.010 MeV−1: bulk level density parameter " 2 2/3 # K n n 2 ns 2 F (n, x, T ) = n −B + 1 − + Sv (1 − 2x) − a T 18 ns ns n 2 2/3 2 ∂(F /n) n K n 2 2an ns 2 P = n = − 1 + Sv (1 − 2x) + T ∂n ns 9 ns 3 n ∂F x ∂F µ = − n ∂n n ∂x 2/3 K n n n a ns 2 = −B + 1 − 1 − 3 + 2Sv (1 − 2x) − T 18 ns ns ns 3 n 1 ∂F n µˆ = − = µn − µp = 4Sv (1 − 2x) n ∂x ns 1 ∂F n 2/3 s = − = 2a s T ; ε = F + nTs n ∂T n J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars 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 = − nTs = uFI + (1 − u)FII , n = unI + (1 − u)nII , nYe = uxI nI + (1 − u)xII nII .

n − unI nYe − unI xI nII = , xII = 1 − u n − unI ∂F ∂FI ∂FII −u = u + (1 − u) = u(µn,I − µn,II ) ∂nI ∂nI ∂nII 1 − u ∂F ∂FI ∂FII −unI = u + (1 − u) = unI (ˆµI − µˆII ) ∂xI ∂xI ∂xII n − unI ∂F ∂FII nII − nI ∂FII −unI = FI − FII + (1 − u) + ∂u ∂nII 1 − u ∂xII n − unI

=⇒ µnI = µnII , µpI = µpII , PI = PII

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Critical Point (Ye = 0.5)

∂P ∂2P = 2 = 0 ∂n T ∂n T

5 n = n c 12 s 5 1/3 5K 1/2 T = c 12 32a 121/3 5Ka1/2 s = c 5 8

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Nuclear Symmetry Energy

Deﬁned as the diﬀerence between energies of pure neutron matter (x = 0) and symmetric (x = 1/2) nuclear matter. S(ρ) = E(ρ, x = 0) − E(ρ, x = 1/2)

Expanding around saturation density C. Fuchs, H.H. Wolter, EPJA 30(2006) 5 (ρs ) and symmetric matter (x = 1/2)

2 E(ρ, x) = E(ρ, 1/2)+(1−2x) S2(ρ)+...

L ρ − ρs S2(ρ) = Sv + + ... 3 ρs 6 symmetry energy S ' 31 MeV, L ' 50 MeV v ?

Connections to neutron matter: E(ρs , 0) ≈ Sv + E(ρs , 1/2) = Sv − B, p(ρs , 0) = Lρs /3 Neutron star matter (in beta equilibrium): " 3 # ∂(E + Ee ) Lρs LSv 4 − 3Sv /L = 0, p(ρs , xβ) ' 1 − 2 ∂x 3 ~c 3π ρs J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Nuclear Mass Formula

Bethe-Weizs¨acker (neglecting pairing and shell eﬀects) 2/3 2 1/3 2 E(A, Z) = −av A + as A + aC Z /A + Sv AI . Myers & Swiatecki introduced the surface asymmetry term: 2/3 2 1/3 2 2 2/3 E(A, Z) = −av A + as A + aC Z /A + Sv AI − Ss I A .

av = B, as ' 18 MeV, aC ' 0.75 MeV, Ss ' 40 MeV, I = (N − Z)/A Optimum Nucleus ∂E/A a − S I 2 a = − s s A−4/3 + C A−1/3(1 − I )2 = 0 ∂A 3 6 ∂E/A a = 2S I − 2S A−1/3I − C A2/3(1 − I ) = 0 ∂I v s 2 2 2(as − Ss I ) 45 A = 2 ' 2 ' 60 aC (1 − I ) (1 − I ) a 1 I = C ' −2/3 4Sv A − 4Ss /A + aC 8 5/3 A aC A Valley of Beta Stability: Z ' − −1/3 2 8(Sv −Ss A )

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Liquid Droplet Model

0 2 H = HB (n, α) + npVC /2 + (Q/n)(n ) , 2 2 HB (n, α) = −Bn + n(K/18)(n/ns − 1) + S2(n)α /n. Ze2 3 r 2 n = n + n , α = n − n ; V = − n p n p C R 2 2R2 Extremize total energy for ﬁxed A = R nd 3r, N − Z = R αd 3r: δ δ [H − µn] = 0, [H − µα¯ ] = 0. δn δα d Q ∂(Q/n) ∂[H + n V /2] 2 n0 − (n0)2 = B p c − µ, dr n ∂n ∂n 2 0 2 2 2 x−y µ ' 3Ze /(8R) − B, Q(n ) ' n (n/ns − 1) , n ' ns / 1 + e , p 3 r = ax, a = 18Q/K, A ' 4πns (ay) /3.

∂[H + n V /2] 0 = B p C − µ¯ = 2S α/n − V /2 − µ,¯ ∂α 2 C 2 2 Z i 3 n Sv I Ze H2 r Sv r nd r α = + − 2 , Hi = . S2 H0 8R H0 R S2 R A

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Liquid Droplet Model Continued

Parameters ﬁxed by experiment: K ' 240 MeV

Z 0.9 dn t90−10 = a 0 = 4a ln 3 ' 2.3 fm, 0.1 n 2 2 which gives a = 0.523 fm, Q = (K/18)[t90−10/(4 ln 3)] ' 3.65 MeV fm . As a check, surface tension Z ∞ Z ∞ (n0)2 Ze2 Z ∞ σo = [H − µn]dz = 2Q dz − ndz −∞ −∞ n 8R −∞ Qn Ze2n a ' s − s ' 1.17 − 0.007Z 1/3 MeV fm−2. a 8R2 2 Then surface energy parameter is as = 4πro σ0 ' 19.2 MeV, which matches expectations from liquid droplet ﬁts to nuclear masses. P i −1 If symmetry energy can be expanded S2(n) = Sv [ i bi (n/ns ) ] , 4πn (ay)3+i (3 + i)T 1 T H = s 1 − + ··· = 3 − + ··· , i (3 + i)ARi y 3 + i y 3 11 25 X T = b + b + b + b + ··· , b = 1. 1 2 2 6 3 12 4 i i J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars The total symmetry and Coulomb energies are Z α2 1 Z E + E = S d 3r + n V d 3r sym C 2 n 2 p C S 3Z 2e2 AI 5H = AI 2 v + 1 + 1 − 2 H0 5R 8Z 3H0 AI 2S 3Z 2e2 AI S = v + 1 − s −1/3 1/3 1 + Ss A /Sv 5R 12Z Sv A + Ss

−1/3 if we identify 3T /y = 3aT /R = −Ss A /Sv . R 3 Static dipole polarizability is αD = zαd d r/(2), where αd is the value of α found by extremizing δ R Hd 3r − R zαd 3r /δα = 0.

2 2 −1/3 z + VC /2 H2AR AR 5 Ss A αd = n , αD = ' 1 + = 4m−1. 2S2 12Sv 20Sv 3 Sv Mean square radii Z 2 2 2 2 1 2 3 R 3 H2 Ze H2 rn,p = (n ± α)r d r = ± I ± − H4 , 2(N, Z) 1 ± I 5 H0 8RSv H0 2r (1 − I 2)−1/2 r3 S 3Ze2 10 S A−1/3 r = o I s − 1 + s . np −1/3 3(1 + Ss A /Sv ) 5 Sv 140Sv ro 3 Sv

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Correlations from Experiments

The three relations for Esym, αD , and rnp can be combined with experimental data to give information about Ss and Sv and about L and Sv if we can relate Ss to L and Sv . This is done using T . i ro Ss 3 11 Sv X n X T = − = b1+ b2+ b3+··· , = bi , bi = 1. 3a Sv 2 6 S2 ns i i

If S2 = Sv + (L/3)(n/ns − 1), then Ss ' La/ro . 2 If S2 = Sv + (L/3)(n/ns − 1) + (Ksym/18)(n/ns − 1) , and Ksym = 6L − 18Sv so that S2(0) = 0, then " # S 3a L L 2 s ' 1 + + . Sv 2ro 3Sv 3Sv

Experiments can thus give a speciﬁc combination of values for Sv and L, i.e., a correlation in Sv − L space. I Masses: d(Ss /Sv )/dSv ∼ 0.26; dL/dSv ∼ 19 I Dipole polarizability: d(Ss /Sv )/dSv ∼ 0.17; dL/dSv ∼ 14 I Neutron skin thickness: d(Ss /Sv )/dSv ∼ −0.08; dL/dSv ∼ −6

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Nuclear Binding Energies

2 2/3 Esym(N, Z) = I (Sv A − Ss A )

2 P 2 χ = i (Eex,i − Esym,i ) /N 2 P 4 2 χvv = N i Ii Ai 2 P 4 4/3 χss = N i Ii Ai 2 P 4 5/3 χvs = N i Ii Ai

q 2χss σSv = 2 χvv χss −χsv

q 2χvv σSs = 2 χvv χss −χsv α = 1 tan−1 2χvs 2 χvv −χss χvs rvs = − √ χvv χss Liquid Droplet Model

2 Sv AI Esym(N, Z) = −1/3 1+(Ss /Sv )A

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Neutron Skin Thickness

q rnp 4 SsI ' −1/3 ro 15 Sv +SsA

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Heavy Ion Collisions

Reaction Mechanisms in Heavy Ion Collisions

Coulomb barrier to Proton and neutron currents Fermi energies central peripheral ∂E (ρ) j − j ∝ E (ρ)∇I + sym I∇ρ n p sym ∂ρ

“Diffusion” “Drift”

Symmetry energy slope

Isospin Isospin fractionation, Asy-stiff migration multifragm

Asy-soft

Sensitive to Sym. Energy and slope depending on observable

Wolter, NuSYM11

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Giant Dipole Resonances

www.tunl.duke.edu r Sv E ∝ −1 1+ 5Ss A−1/3 3Sv

−3 23.3 MeV< S2(0.1 fm ) <24.9 MeV

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars 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)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Nuclear Experimental Constraints

Isobaric Analog States

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Theoretical Neutron Matter Calculations

H&S: Chiral Lagrangian

GC&R: Quantum Monte Carlo

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Supernovae

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars P − P˙ Diagram

τ = P c 2P˙

p B ' 3 · 1019 PP˙ G P˙ −E˙ ' 1047 erg/s P3 P in seconds

log [ Period (s) ] J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars P-pdot # G )$# &3 3B: : (

5 3 3 3 & 33 <3 ( "

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars P-pdot # G )$# &3 3B: : (

&B: 5 (

" >3 5 5 33 3 3 3 53 3 @3 !"+!""%

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars MSPs% ) - .

' % !,*? 5 3 53 34 3D !,*0 5 )B ; :A G

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars P-pdor # G )$# &3 3B: : (

&B: 5 ( = " & 5B:: : : 3 (

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars MagnetarsK3< %L 5 3 ; ) !"!0+!1 ' & !"""; 3 ( 5 : =

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars ' I+G)Magnetar % '!,""J!0

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Clock 4

!*+,-. "")""@N!*?"!!) H1 61601!,0?"?0/ J+" """"""""""""! @3 3!33 = @3 3!1""= @3 ; 53 5 F 3 =

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Timing % % % & ( &@)@ ( ! ? /

%+% H@

?""% ?

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Gravity' K2 5 3 waves @

9 4 78F 5 43 % 5 33 % % 3 1+!" !"" = 3 & " (:> &>@(: &@(

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Binary Mass Measurements X-ray timing Mass function 3 P(v2 sin i) f (M1) = 2πG 3 (M1 sin i) = 2 (M1+M2) > M1 3 P(v1 sin i) f (M2) = 2πG 3 (M2 sin i) = 2 (M1+M2) Optical spectroscopy > M2 In an X-ray binary, voptical 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. J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars 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: 3 2π 5/3 GM 2/3 ω˙ =1−e2 P c2 2/3 P 1/3 G γ = 2π eM2(2M2 + M1) M2c2 Gravitational radiation leads to orbit decay: 5/3 P˙ = −192π 2πG (1 − e2)−7/2 1 + 73 e2 + 37 e4 M1M2 5c5 P 24 96 M1/2 In some cases, can constrain Shapiro time delay, r is magnitude and s = sin i is shape parameter.

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars PSR J0737-3039

PSR J0737-3039 Kramer et al. (2007)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars PSR J1614-2230

3.15 ms pulsar in 8.69d orbit with 0.5 M white dwarf companion. Shapiro delay tightly conﬁnes the edge-on inclination: sin i = 0.99984 Pulsar mass is 1.97 ± 0.04 M Distance > 1 kpc, B ' 1.8 × 108 G ) s µ ( t

∆ Demorest et al. 2010

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Black Widow Pulsar PSR B1957+20

1.6ms pulsar in circular 9.17h orbit with a ∼ 0.03 M companion. Pulsar is eclipsed for 50-60 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 J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Pulsars Around. 6 %J@ Sgr A* • ) 8 D

• 5 .# 8 K @ > . L 0 ;:? • .# ,# 6

A- 4- ; J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars J0337+1715 !+

D ; 5

; . 1" 6 D ! D &% ' D! 5

& 6 D;

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars J0337 % )6 + > % %6? • 5 6- 5 6 • % 6 66 > 5: ?+ • " C ::; " • "P C FE " • "P C : " > 71I? • & 5 F E>:? 6 • , ) • % 6 6 >6 D G 5: I? • & )

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars vanKerkwijk 2010 Romani et al. 2012

Although simple average mass of w.d. companions is 0.23 M larger, weighted average is 0.04 M smaller

Demorest et al. 2010

Antoniadis et al. 2013 Champion et al. 2008

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Radiation Radius

I The measurement of ﬂux and temperature yields an apparent angular size (pseudo-BB): R R 1 ∞ = d d p1 − 2GM/Rc2 I Observational uncertainties include distance, interstellar H absorption (hard UV and X-rays), atmospheric composition I Nearby isolated neutron stars (parallax measurable) I Quiescent X-ray binaries in globular clusters (reliable distances, low B H-atmosperes) I Bursting sources in which Eddington ﬂux is measured s GMc 2GM FEdd = 2 1 − 2 κD Rphc

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars RX J1856-3754

→

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Astrometry of RXJ 1856-3754

I Redshift or gravity measurements, which would allow simultaneous M and R I Walter & Lattimer (2002) determined determinations, are not yet D = 117 ± 12 pc and v ' 190 km/s possible. from 1996-1999 HST Planetary Camera observations pixel size

I Star’s age is probably 0.5 million years I Walter, Eisenbeiβ, Lattimer, Kim, 1996-1999 HST Hambaryan & Neuh¨auser(2010) determined D ' 115 ± 8 pc based parallactic ellipse on 2002-2004 HST Advanced Camera for Surveys observations (double the resolution)

I A magnetic hydrogen atmosphere model (Ho et al. 2007) suggests M ' 1.29 M and R ' 11.6 km

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Photospheric Radius Expansion X-Ray Bursts

GMc q 2GM ⇐ FEdd = 2 1 − 2 κD Rphc ⇐ FEdd

EXO 1745-248

Galloway, Muno, Hartman, Psaltis & Chakrabarty (2006)

−4 2 A = f (R∞/D) −4 2 c A = fc (R∞/D)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars PRE Burst Models

2 Ozel et al. zph = z β = GM/Rc Steiner et al. zph << z

GMc GMc p F = 1 − 2β FEdd = Edd κD κD F R 2 p A = ∞ = f −4 ∞ α = β 1 − 2β 4 c −1 2 σT∞ D θ = cos 1 − 54α F κD " √Edd 1 √ θ α = 2 3 = β(1 − 2β) A Fc c β = 1 + 3 sin 4 3 6 3 Af c R∞ γ = c = # κFEdd α θ 1 1√ − cos β = ± 1 − 8α 3 4 4 r 1 1 α ≤ required. α ≤ ' 0.192 required. 8 27 α EXO 1745-248 4U 1608-522 4U 1820-30 KS 1731-260 SAX J1748.9-2021 0.188 ± 0.035 0.247 ± 0.058 0.235 ± 0.04 0.199 ± 0.032 0.177 ± 0.036

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars M − R PRE Burst Estimates

2 −4 FEdd,∞, (R∞/D) fc , D, fc from Ozel et al.

zph = z

Lattimer & Steiner (2013)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars M − R PRE Burst Estimates

2 −4 FEdd,∞, (R∞/D) fc , D from Ozel et al.

zph = 0 Altered uncertainties

for fc , D Lattimer & Steiner (2013)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Quiescent Sources in Globulars Hot neutron stars in globular clusters

I Globular clusters evolve: more massive stars, including binaries, sink to center 47 Tuc via long-range stellar encounters.

I Close binaries formed in encounters.

I Episodes of accretion in close binaries heats neutron stars: they are reborn.

I Following accretion, they become quiescent, low-mass X-ray sources.

I Accretion suppresses surface B ﬁelds.

I Atmospheric composition is H.

Reiko Kuromizu

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars M − R QLMXB Estimates

2.4 ω Cen M13 2.2 6304 6397 Absorption (NH ) 2 M28 determined self-consistently 1.8 from spectra ) 1.6

1.4 M (M

1.2

0.8

0.6 6 8 10 12 14 16 18 20 R (km) Guillot et al. (2013) J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars M − R QLMXB Estimates

2.4 ω Cen M13 2.2 6304 6397 2 M28 1.8 P(M, R) from H atmosphere

) models of Guillot et al. (2013), 1.6 adjusted for alternate NH values 1.4

M (M of Dickey & Lockman (1990)

1.2

0.8

0.6 6 8 10 12 14 16 18 20 R (km) Lattimer & Steiner (2013) J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Bayesian TOV Inversion

0 I ε < 0.5ε0: Known crustal EOS I EOS parameters K, K , Sv , γ, ε1, n , ε , n uniformly distributed I 0.5ε0 < ε < ε1: EOS 1 2 2 0 parametrized by K, K , Sv , γ I Mmax ≥ 1.97 M , causality enforced I Polytropic EOS: ε1 < ε < ε2: n1; ε > ε2: n2 I All 10 stars equally weighted 1 2.5 1 103 Fiducial EOS, PREs, QLMXBs w/adjusted N inferred p(ε) 0.9 Lattimer & Steiner 2013 H 0.9 0.8 0.8 2 102 0.7 0.7 ) 3 0.6 0.6 ) 1.5 0.5 RX J1856 0.5 10 Ho et al. (2007)

Fid. EOS, PREs+QLMXBs w/adj. N 0.10.5 inferred M(R) 0.1 H 10-1 0 0 200 400 600 800 1000 1200 1400 1600 6 8 10 12 14 16 18 ε (MeV/fm3) R (km) J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Astronomy vs. Astronomy vs. Physics

Ozel et al., PRE bursts zph = 2.5 1 z: R = 9.74 ± 0.50 km. Fiducial EOS, PREs, QLMXBs w/adjusted N H 0.9 Suleimanov et al., long > 0.8 PRE bursts: R1.4 ∼13.9 km 2 Guillot et al. (2013), all 0.7 stars have the same radius, 0.6 self N : R = 9.1+1.3 km. ) @ H −1.5 1.5 @ Lattimer & Steiner (2013), 0.5 TOV, crust EOS, causality, M (M 0.4 maximum mass > 2M , 1 zph = z, alt NH . 0.3 Lattimer & Lim (2013), 0.2 nuclear experiments: 0.5 0.1 29 MeV < Sv < 33 MeV, 40 MeV < L < 65 MeV, 0 R = 12.0 ± 1.4 km. 6 8 10 12 14 16 18 1.4 R (km)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Can Hyperons Appear in Abundance in Neutron Stars?

XH < 0.2

Miyatsu, Yamamumo & Nakazato (2013) Jiang, Li & Chen (2012)

Weissenborn, Chatterjee & Bednarek et al. (2011) Schaﬀner-Bielich (2012)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Hyperon Stars with Small Radii Have Few Hyperons

0.1 i f

0.01

Jiang, Li & Chen (2012)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars More Hyperon Stars

Whittenbury et al. (2013)

Schramm et al. (2013)

Colucci & Sedrakian (2014) Bednarek, Manka & Pienkos (2013)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Still More Hyperon Stars

Lopes & Menezes (2013)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Additional Proposed Radius and Mass Constraints Neutron star Interior Composition ExploreR I Pulse proﬁles Hot or cold regions on rotating neutron stars alter pulse shapes: NICER and LOFT will enable timing and spectroscopy of NASA thermal and non-thermal emissions. Light curve modeling → M/R; phase-resolved spectroscopy → R.

I Moment of inertia Spin-orbit coupling of ultra- relativistic binary pulsars Large Observatory For x-ray Timing (e.g., PSR 0737+3039) vary i and contribute toω ˙ : I ∝ MR2.

I Supernova neutrinos Millions of neutrinos detected from a Galactic supernova will measure BE= mB N − M, < Eν >, τν . ESA/NASA I QPOs from accreting sources ISCO and crustal oscillations J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Constraints from Observations of Gravitational Radiation

Mergers: 3/5 −1/5 Chirp mass M = (M1M2) M and tidal deformability λ ∝ R5 (Love number) are potentially measurable during inspiral. λ¯ ≡ λM−5 is related to I¯ ≡ IM−3 by an EOS-independent relation (Yagi & Yunes 2013). Both λ¯ and I¯ are also related to M/R in a relatively EOS-independent way (Lattimer & Lim 2013).

I Neutron star - neutron star: Mcrit for prompt black hole formation, fpeak depends on R.

I Black hole - neutron star: ftidal disruption depends on R, a, MBH. Disc mass depends on a/MBH and on −2 MNSMBHR . Rotating neutron stars: r-modes

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Urca Processes

If x < xDU , bystander nucleons Gamow & Sch¨onberg proposed the needed: modiﬁed Urca process. direct Urca process: nucleons at the − (n, p) + n → (n, p) + p + e + νe , top of the Fermi sea beta decay. (n, p) + p → (n, p) + n + e+ +ν ¯ − e n → p + e + νe , + p → n + e +ν ¯e Neutrino emissivities: 2 −6 ˙MU ' (T /µn) ˙DU ∼ 10 ˙DU . Energy conservation guaranteed by beta equilibrium Beta equilibrium composition: 2 −1 3 µn − µp = µe xβ ' (3π n) (4Esym/~c) ' 0.04 (n/n )0.5−2 . Momentum conservation requires s |kFn| ≤ |kFp| + |kFe |.

Charge neutrality requires kFp = kFe , therefore |kFp| ≥ 2|kFn|. 3 Degeneracy implies ni ∝ kFi , thus x ≥ xDU = 1/9. With muons 2 (n > 2ns ), xDU = 2+(1+21/3)3 ' 0.148

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Neutron Star Cooling

Cas A e J1119-6127 m

Page, Steiner, Prakash & Lattimer (2004)

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Minimal Cooling

3 max 9 3 max 9 Neutron P−F2 gap: "b" (T cn = 3x10 K) Neutron P−F2 gap: "a" (T cn = 10 K) LEE LEE A HEE A HEE 1 3 3 1 4 4 7 7 2 2 10 10 9 5 9 5 8 B 8 B 6 C 6 12 C 12 13 13 D D b 11 b e 11 de f d f c c a a LEE HEE LEE HEE

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).

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Transitory Rapid Cooling No p superconductivity With p superconductivity 8 MU emissivity:ε ˙MU ∝ T PBF emissivity (f ∼ 10): 7 8 ε˙PBF ∝ F (T ) T ∝ T ' f ε˙MU core temperature Speciﬁc heat: CV ∝ T

Neutrino dominated cooling: CV dT /dt = −Lν

=⇒ T ∝ (t/τ)−1/6 τPBF = τMU /f TC

(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)

1 Page et al. 2009

Very sensitive to n S0 critical surface temperature temperature (TC ) and existence of proton superconductivity

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars 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 1947. 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

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Cas A Superﬂuidity

X-ray spectrum indicates thin C Page et al. 2010 atmosphere, 8 Te ∼ 1.7 × 10 K (Ho & Heinke 2009) 10 years of X-ray data show cooling at the rate d ln Te d ln t = −1.23 ± 0.14 (Heinke & Ho 2010) Modiﬁed Urca: d ln Te ' −0.08 d ln t MU We infer that 8 TC ' 5 ± 1 × 10 K −1/6 TC ∝ (tC L/CV )

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars Conclusions

I Nuclear experiments set reasonably tight constraints on symmetry energy parameters and the symmetry energy behavior near the nuclear saturation density.

I Theoretical calculations of pure neutron matter predict very similar symmetry constraints.

I These constraints predict neutron star radii R1.4 in the range 12.0 ± 1.4 km.

I Combined astronomical observations of photospheric radius expansion X-ray bursts and quiescent sources in globular clusters suggest R1.4 ∼ 12.1 ± 0.6 km.

I The nearby isolated neutron star RX J1856-3754 appears to have a radius near 12 km, assuming a solid surface with thin H atmosphere (Ho et al. 2007).

I 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.

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars What Was Not Included

I Neutron Star Mergers

I Gravitational Radiation from Mergers

I Nucleosynthsis from Mergers

I Quadrupole Polarizability

J. M. Lattimer Mass, Radius and Equation of State of Neutron Stars