Evolution of Neutron Stars and Observational Constraints Evolution of Neutron Stars and Observational Constraints James Lattimer Stony Brook University, USA Abstract The structure and evolution of neu[.]
7 03001 (2010) EPJ Web of Conferences 7, DOI:10.1051/epjconf/20100703001 © Owned by the authors, published by EDP Sciences, 2010 Evolution of Neutron Stars and Observational Constraints James Lattimer Stony Brook University, USA Abstract The structure and evolution of neutron stars is discussed with a view towards constraining the properties of high density matter through observations The structure of neutron stars is illuminated through the use of several analytical solutions of Einstein’s equations which, together with the maximally compact equation of state, establish extreme limits for neutron stars and approximations for binding energies, moments of inertia and crustal properties as a function of compactness The role of the nuclear symmetry energy is highlighted and constraints from laboratory experiments such as nuclear masses and heavy ion collisions are presented Observed neutron star masses and radius limits from several techniques, such as thermal emissions, X-ray bursts, gammaray flares, pulsar spins and glitches, spin-orbit coupling in binary pulsars, and neutron star cooling, are discussed The lectures conclude with a discusson of proto-neutron stars and their neutrino signatures The participation at this summer school was partially supported by the HISS Dubna program of the Helmholtz association and by the US Department of Energy Grant DE-FG02-ER40317 References Lattimer, J.M & Prakash, M., Science 304 (2004) 536-542 Lattimer, J.M & Prakash, M., Phy Rep 442 (2007) 109-165 Steiner, A.W., Prakash, M., Lattimer, J.M & Ellis, P., Phy Rep 411 (2005) 325-375 This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20100703001 EPJ Web of Conferences Neutron Star Structure James M Lattimer lattimer@astro.sunysb.edu Department of Physics & Astronomy Stony Brook University J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.1/64 J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.2/64 Credit: Dany Page, UNAM 03001-p.2 Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008) Neutron Star Structure Tolman-Oppenheimer-Volkov equations of relativistic hydrostatic equilibrium: dp G (m + 4πpr )( + p) = − dr c r(r − 2Gm/c2 ) dm = 4π r2 dr c p is pressure, is mass-energy density Useful analytic solutions exist: • Uniform density = constant = c [1 − (r/R)2 ] √ = pp∗ − 5p • Tolman VII • Buchdahl J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.3/64 Spherically Symmetric General Relativity ds2 = eλ(r) dr2 + r2 dθ2 + sin2 θdφ2 − eν(r) dt2 Static metric: Einstein’s equations: 8π(r) = 8πp(r) = p (r) = r m(r) = 4π Mass: −λ(r) −λ(r) λ (r) , − e + e r2 r (r) ν , − − e−λ(r) + e−λ(r) r r p(r) + (r) ν (r) − (r )r2 dr , Boundaries: r=0 e−λ(r) = − 2m(r)/r m(0) = p (0) = (0) = 0, r=R m(R) = M, p(R) = 0, eν(R) = e−λ(R) = − 2M/R Thermodynamics: d(ln n) = mn(r) = N = d d d de =− dν, h= , = n(m + e), p = n2 +p dp dn dn p = : n = n , e = e0 ((r) + p(r))e(ν(r)−ν(R))/2 − n0 e0 ; R 4πr2 eλ(r)/2 n(r)dr; BE = N m − M J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.4/64 03001-p.3 EPJ Web of Conferences Uniform Density Fluid m(r) = e−λ(r) = ν(r) = p(r) = (r) BE M = e = 4π M r , β≡ R − 2β(r/R)2 , 3 1 − 2β − − 2β(r/R)2 , 2 √ − 2β(r/R)2 − − 2β √ , − 2β − − 2β(r/R)2 constant; n(r) = constant √ sin−1 2β 3β √ − − 2β + β + ··· 4β 2β 14 pc < ∞ =⇒ β < 4/9 c2s = ∞ J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.5/64 Tolman VII (r) = c − (r/R)2 ≡ c [1 − x] e−λ(r) = − βx(5 − 3x) eν(r) = p(r) = n(r) = (1 − 5β/3) cos2 φ, β −λ(r) tan φ(r) − (5 − 3x) , 3βe 4πR2 (r) + p(r) cos φ(r) mb cos φ1 φ(r) = w(r) = β w1 − w(r) −1 + φ1 , , φ1 = φ(x = 1) = tan 3(1 − 2β) ⎡ ⎤ e−λ(r) ⎦ − 2β ln ⎣ x − + , w1 = w(x = 1) = ln + 3β 3β tan φc (P/)c = 15 − , β c2s,c = tan φc tan φc + β BE 11 7187 β+ β + ··· M 21 18018 π pc < ∞ =⇒ φc < , β < 0.3862 c2s,c < =⇒ β < 0.2698 J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.6/64 03001-p.4 Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008) Buchdahl = √ p∗ p − 5p eν(r) = (1 − 2β)(1 − β − u(r))(1 − β + u(r))−1 , λ(r) e 8πp(r) 8π(r) = = = mb n(r) = u(r) = r = = (1 − 2β)(1 − β + u(r))(1 − β − u(r))−1 (1 − β + β cos Ar )−2 , A2 u(r)2 (1 − 2β)(1 − β + u(r))−2 , 2A2 u(r)(1 − 2β)(1 − β − 3u(r)/2)(1 − β + u(r))−2 , 3/2 −1 p(r) p∗ p∗ p(r) − , c2s (r) = −5 p∗ p(r) −1 p∗ β sin Ar = (1 − β) , −1 Ar p(r) r(1 − 2β)(1 − β + u(r))−1 , π 2πp∗ (1 − 2β)−1 , R = (1 − β) 2p∗ (1 − 2β) A pc = p∗ β , c = p∗ β(1 − β), 2 nc mb = p∗ β(1 − 2β)3/2 BE β β2 = (1 − β)(1 − 2β)−1/2 (1 − β)−1 + + ··· M 2 c2s,c < =⇒ β < 1/6 J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.7/64 Maximally Compact Equation of State Koranda, Stergioulas & Friedman (1997) p() = 0, p() = − o , ≤ ≥ o o This EOS has a parameter 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 1/2 y = mo dy dx dq dx 1/2 , x = ro , = 4πx2 (1 + q) = − q = p−1 o (y + 4πqx3 )(1 + 2q) x(x − 2y) The solution with the maximum central pressure and mass and the minimum radius: ymax = 0.0851, xmin /ymax = 2.825 qmax = 2.026, 1/2 o s GMmax MeV fm−3 , Mmax = 4.2 pmax = 307 M , Rmin = 2.825 s o c2 Moreover, the scaling the axially-symmetric case, yielding extendsto 1/2 Mmax 1/2 s −1/2 ∝ o , Pmin = 0.82 ms Pmin ∝ Rmin o J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.8/64 03001-p.5 EPJ Web of Conferences Maximum Mass, Minimum Period Theoretical limits from GR and causality • Mmax = 4.2(s /f )1/2 M Rhoades & Rufni (1974), Hartle (1978) • Rmin = 2.9GM/c2 = 4.3(M/M ) km Lindblom (1984), Glendenning (1992), Koranda, Stergioulas & Friedman (1997) ã c < 4.5 ì 1015 (M /Mlargest )2 g cm−3 Lattimer & Prakash (2005) • Pmin (0.74 ± 0.03)(M /Msph )1/2 (Rsph /10 km)3/2 ms Koranda, Stergioulas & Friedman (1997) • Pmin 0.96(M /Msph )1/2 (Rsph /10 km)3/2 ms (empirical) Lattimer & Prakash (2004) • c > 0.91 × 1015 (1 ms/Pmin )2 g cm−3 • cJ/GM 0.5 (empirical) (empirical, neutron star) J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.9/64 Another Kind of Star – Self-Bound Finite, large surface energy density Lake’s solution eν(r) = eλ(r) = 4πpR2 = 4πR2 = m = c2s = surf c = (1 − 52 β(1 − 15 x)2 , (1 − β) (1 − β(1 − 35 x))2/3 , (1 − 52 β(1 − 35 x))2/3 − 2(1 − β)2/3 βx (1 − 52 β(1 − x)) β 2/3 , − (1 − β) − 52 β(1 − 15 x) (1 − 52 β(1 − 35 x))2/3 3(1 − β)2/3 β β) − 52 β(1 − 13 x) (1 − 52 β(1 − 35 x))5/3 , M x3/2 (1 − (1 − β(1 − 35 x))2/3 (2 − 5β + 3βx) (2 − 5β + 3βx)5/3 2 + (2 − 5β) − 5β x , 5(2 − 5β + βx)3 22/3 (1 − β)2/3 2/3 5 1− β (1 − β)−5/3 1− β 0.30 < c2s,c < 0.44, 0.265 < surf 2ns ), xDU = 2+(1+21/3 )3 0.148 If x < xDU , bystander nucleons needed: modied Urca process is then dominant (n, p) + n → (n, p) + p + e− + νe , (n, p) + p → (n, p) + n + e+ + ν¯e Neutrino emissivities: ˙M U RCA Beta equilibrium composition: xβ (3π n)−1 T μn 4Esym c 2 3 ˙DU RCA 0.04 n ns 0.5−2 J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.32/64 03001-p.17 EPJ Web of Conferences Direct Urca Threshold Klähn et al., Phys Rev C74 (2006) 035802 J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.33/64 Neutron Star Cooling Page, Steiner, Prakash & Lattimer (2004) J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.34/64 03001-p.18 Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008) Direct Urca Constraint? Apparently, some/most neutron stars don’t have accelerated cooling If direct Urca doesn’t occur for these stars, the direct Urca density threshold is above − 3ns , ruling out too-rapid density-dependence for Esym (n) Also, hyperon threshold density is high However, suppression of accelerated cooling by superuidity could invalidate this Klähn et al., Phys Rev C74 (2006) 035802 J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.35/64 Radiation Radius • Combination of ux and temperature measurements yields apparent angular diameter (pseudo-BB): R∞ R = d d − 2GM/Rc2 • 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 J010043.1-721134 in the SMC: R∞ ≥ 10.8 km (Esposito & Mereghetti 2008) J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.36/64 03001-p.19 EPJ Web of Conferences RX J1856-3754 J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.37/64 Radiation Radius: Nearby Neutron Star RX J1856-3754: 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 atmosphe R∞ ≈ 17 km Ho et al 2007 J.M Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.38/64 03001-p.20 ... Dubna, 18 July 2008 – p.24/64 03001-p.13 EPJ Web of Conferences The Pressure of Neutron Star Matter Expansion of cold nucleonic matter energy near ns and isospin symmetry x = 1/2: E(n, x) P (n,... Droplet extension: consider the neutron/ proton asymmetry of the nuclear surface E(A, Z) = (−av + Sv δ )(A − Ns ) + as A2/3 + aC Z /A1/3 + μn Ns Ns is the number of excess neutrons associated with the... Heavy Ion Collisions and Astrophysics (DM2008) Direct Urca Constraint? Apparently, some/most neutron stars don’t have accelerated cooling If direct Urca doesn’t occur for these stars, the direct