Mass-Radius Relation of Compact Stars with Quark Matter INT Summer Program "QCD and Dense Matter: From Lattices to Stars", June 2004, INT, Seattle, Washington, USA JurgenÄ Schaffner{Bielich Institut furÄ Theoretische Physik/Astrophysik J. W. Goethe UniversitÄat Frankfurt am Main { p.1 Phase Diagram of QCD T early universe RHIC Tc Quark-Gluon Plasma Hadrons Color Superconductivity nuclei neutron stars µ mN / 3 c µ ² Early universe at zero density and high temperature ² neutron star matter at zero temperature and high density ² lattice gauge simulations at ¹ = 0: phase transition at Tc ¼ 170 MeV { p.2 Neutron Stars ² produced in supernova explosions (type II) ² compact, massive objects: radius ¼ 10 km, mass 1 ¡ 2M¯ 14 3 ² extreme densities, several times nuclear density: n À n0 = 3 ¢ 10 g/cm { p.3 Masses of Pulsars (Thorsett and Chakrabarty (1999)) ² more than 1200 pulsars known ² best determined mass: M = (1:4411 § 0:00035)M¯ (Hulse-Taylor-Pulsar) ² shortest rotation period: 1.557 ms (PSR 1937+21) { p.4 ² n = 104 ¡ 4 ¢ 1011 g/cm3: outer crust or envelope (free e¡, lattice of nuclei) ² n = 4 ¢ 1011 ¡ 1014 g/cm3: Inner crust (lattice of nuclei with free neutrons and e¡) Structure of Neutron Stars — the Crust (Dany Page) ² n · 104 g/cm3: atmosphere (atoms) { p.5 ² n = 4 ¢ 1011 ¡ 1014 g/cm3: Inner crust (lattice of nuclei with free neutrons and e¡) Structure of Neutron Stars — the Crust (Dany Page) ² n · 104 g/cm3: atmosphere (atoms) ² n = 104 ¡ 4 ¢ 1011 g/cm3: outer crust or envelope (free e¡, lattice of nuclei) { p.5 Structure of Neutron Stars — the Crust (Dany Page) ² n · 104 g/cm3: atmosphere (atoms) ² n = 104 ¡ 4 ¢ 1011 g/cm3: outer crust or envelope (free e¡, lattice of nuclei) ² n = 4 ¢ 1011 ¡ 1014 g/cm3: Inner crust (lattice of nuclei with free neutrons and e¡) { p.5 Structure of a Neutron Star — the Core (Fridolin Weber) quark−hybrid traditional neutron star star N+e N+e+n n,p,e, µ n superfluid hyperon c o n d u c t star e r i n neutron star with p g pion condensate u n,p,e, s p µ r o u,d,s t quarks o − n π s H crust Σ,Λ,Ξ,∆ Fe − absolutely stable K 6 3 strange quark 10 g/cm matter 11 10 g/cm 3 3 µ u d s 10 14 g/cm m s strange star nucleon star R ~ 10 km M ~ 1.4 M { p.6 Neutron Star Matter for a Free Gas Hadron p,n §¡ ¤ others appears at: ¿ n0 4n0 8n0 > 20n0 (Ambartsumyan and Saakyan, 1960) but the corresponding equation of state results in a maximum mass of only Mmax ¼ 0:7M¯ < 1:44M¯ (Oppenheimer and Volkoff, 1939) =) effects from strong interactions are essential to describe neutron stars! { p.7 4 N§: §He hypernucleus bound by isospin forces §¡ atoms: potential is repulsive N¥: attractive ! 7 ¥ hypernuclear events U¥ = ¡28 MeV at n = n0 quasi-free production of ¥: U¥ = ¡18 MeV ¤¤: attractive ! 5 ¤¤ hypernuclear measurements more attractive than N¤? YY: Y= ¤; §; ¥, unknown! Baryon–Baryon Interactions N¤: attractive ! ¤-hypernuclei for A = 3 ¡ 209 U¤ = ¡30 MeV at n = n0 { p.8 N¥: attractive ! 7 ¥ hypernuclear events U¥ = ¡28 MeV at n = n0 quasi-free production of ¥: U¥ = ¡18 MeV ¤¤: attractive ! 5 ¤¤ hypernuclear measurements more attractive than N¤? YY: Y= ¤; §; ¥, unknown! Baryon–Baryon Interactions N¤: attractive ! ¤-hypernuclei for A = 3 ¡ 209 U¤ = ¡30 MeV at n = n0 4 N§: §He hypernucleus bound by isospin forces §¡ atoms: potential is repulsive { p.8 ¤¤: attractive ! 5 ¤¤ hypernuclear measurements more attractive than N¤? YY: Y= ¤; §; ¥, unknown! Baryon–Baryon Interactions N¤: attractive ! ¤-hypernuclei for A = 3 ¡ 209 U¤ = ¡30 MeV at n = n0 4 N§: §He hypernucleus bound by isospin forces §¡ atoms: potential is repulsive N¥: attractive ! 7 ¥ hypernuclear events U¥ = ¡28 MeV at n = n0 quasi-free production of ¥: U¥ = ¡18 MeV { p.8 YY: Y= ¤; §; ¥, unknown! Baryon–Baryon Interactions N¤: attractive ! ¤-hypernuclei for A = 3 ¡ 209 U¤ = ¡30 MeV at n = n0 4 N§: §He hypernucleus bound by isospin forces §¡ atoms: potential is repulsive N¥: attractive ! 7 ¥ hypernuclear events U¥ = ¡28 MeV at n = n0 quasi-free production of ¥: U¥ = ¡18 MeV ¤¤: attractive ! 5 ¤¤ hypernuclear measurements more attractive than N¤? { p.8 Baryon–Baryon Interactions N¤: attractive ! ¤-hypernuclei for A = 3 ¡ 209 U¤ = ¡30 MeV at n = n0 4 N§: §He hypernucleus bound by isospin forces §¡ atoms: potential is repulsive N¥: attractive ! 7 ¥ hypernuclear events U¥ = ¡28 MeV at n = n0 quasi-free production of ¥: U¥ = ¡18 MeV ¤¤: attractive ! 5 ¤¤ hypernuclear measurements more attractive than N¤? YY: Y= ¤; §; ¥, unknown! { p.8 Neutron Star Matter and Hyperons Hyperons appear at n ¼ 2n0! ² nonrelativistic potential model (Balberg and Gal, 1997) ² quark-meson coupling model (Pal et al., 1999) ² relativistic mean–field models (Glendenning, 1985; Knorren, Prakash, Ellis, 1995; JS and Mishustin, 1996) ² relativistic Hartree–Fock (Huber, Weber, Weigel, Schaab, 1998) ² Brueckner–Hartree–Fock (Baldo, Burgio, Schulze, 2000; Vidana et al., 2000) ² chiral effective Lagrangian’s (Hanauske et al., 2000) { p.9 Composition of Neutron Star Matter 100 n GM1 p 10−1 e UΣ=+30 MeV 10−2 µ UΞ=−18 MeV 0 Baryon fraction Ξ −3 10 Λ Ξ− 10−4 0.0 0.3 0.6 0.9 1.2 1.5 Density (fm−3) ² ¤s are present around n = 2n0 ² repulsive potential for §s: § hyperons do not appear at all! ¡ ² ¥ present in matter before n = 3n0 { p.10 Phase Transition to Hypermatter 100 n p 10−1 e 10−2 µ 0 Ξ GM1(0.7) Baryon fraction −3 Λ 10 Ξ− Σ+ 10−4 0.0 0.3 0.6 0.9 1.2 1.5 Density (fm−3) ² first order phase transition ² mixed phase for a wide range of densities ² all hyperons (¤, ¥0, ¥¡) appear at the start of the mixed phase { p.11 Hypercompact Neutron Stars ² new stable solution in the mass–radius diagram! ² neutron star twins: Mhyp » Mn but Rhyp < Rn ² selfbound compact stars for strong attraction with R = 7 ¡ 8 km (JSB, Hanauske, Stöcker, Greiner, PRL 89, 171101 (2002)) { p.12 , p.13 { stars neutron and arfs dw stars white ¤ ¦ ¥ © neutron besides ansition! tr than Stars ¤ ¥ ¦ phase ¤ equations ¥ V compact O © order (2000)) T first more the Compact le! y Thoma to , ¨ an ¤ of ¤ stab stars ¦ © JSB or © , f is ¥ §¨ © le ates §¨ © solution ¥ ¦ ¤ ¥ ¦ amily ¤ Greiner , F possib gener solution third tler ¢ £ ² ² ² ¡ ¡ (Scher Third A Model For Cold And Dense QCD hadrons / massless massive quarks quarks µ µ min χ µ Two possibilities for first-order chiral phase transition: ² A weakly first-order chiral transition (or no true phase transition), =) one type of compact star (neutron star) ² A strongly first-order chiral transition =) two types of compact stars: a new stable solution with smaller masses and radii { p.14 A Simple Model of Dense QCD ² star made of a gas of u, d and s quarks ² interaction taken into account perturbatively up to 2 2 ®s ; ®s = g =4¼ ² ®s runs according to the renormalization group equation ² No bag constant is introduced! ² star temperature ¿ typical chemical potentials ¡! zero temperature ² ms = 100 MeV ¿ ¹min = mN =3 ¡! three flavor massless quarks ² charge neutrality and ¯ equilibrium: ¹s = ¹d = ¹u ´ ¹ { p.15 Equation of State in pQCD 1.0 ) 3 free gas 0.5 Λ=3µ Λ=2µ Pressure (GeV/fm 0.0 0 1 2 3 Energy density (GeV/fm3) Nearly linear behavior of the pressure with the energy density ) approximation with an effective nonideal bag model: N ­(¹) = ¡ f a ¹4 + B 4¼2 e® e® 1=4 case 2: Be® = 199 MeV, ae® = 0:628 (· 4% ) 1=4 { p.16 case 3: Be® = 140 MeV, ae® = 0:626 (· 2%) Mass-radius and maximum density of pure quark stars 2 B1/4=145 MeV Λ=3µ 1.5 B1/4=200 MeV sun 1 Λ=2µ M/M 0.5 0 0 5 10 15 Radius (km) ² case 2: Mmax = 1:05 M¯, Rmax = 5:8 km, nmax = 15 n0 ² case 3: Mmax = 2:14 M¯, Rmax = 12 km, nmax = 5:1 n0 { p.17 Heavy Quark Stars? (Rüster and Rischke (2004)) 3 φ=0 φ>0, 50% φ>0, 100% 2.5 φ>0, 150% φ>0, 200% 2 ] ⋅ ° M 1.5 star mass [ 1 0.5 0 0 2 4 6 8 10 12 14 16 star radius [km] ² quark star with color–superconducting quarks ² uses NJL model for pairing quarks ² increased interactions gives heavy quark stars ² heavy quark stars also for HDL calculation (Strickland and Andersen (2002)) { p.18 The two possible scenarios 0.6 free 0.4 massive baryons/quarks massless quarks 0.2 Pressure p/p 0 µ µ strong µ weak min χ χ ² Weak: phase transition is weakly first order or a crossover ! pressure in massive phase rises strongly ² Strong: transition is strongly first order ! pressure rises slowly with ¹ ² asymmetric matter up to » 2n0: [Akmal,Pandharipande,Ravenhall (1998)] 2 4 n mq E=A » 15 MeV (n=n0) ! pB » 0:04 ¢ pfree µn0 ¶ µ ¹ ¶ { p.19 Quark star twins? 2 Λ=3µ ρ ρ c=3 0 1.5 ρ 2.5 0 ρ 2 0 sun 1 Λ=2µ M/M 0.5 hadronic EoS 0 0 5 10 15 Radius (km) ² Weak: ordinary neutron star with quark core (hybrid star) ² Strong: third class of compact stars possible with maximum masses M » 1 M¯ and radii R » 6 km ² Quark phase dominates (n » 15 n0 at the center), small hadronic mantle { p.20 Constraints on the Mass–Radius Relation (Lattimer and Prakash (2004)) ² spin rate from PSR B1937+21 of 641 Hz: R < 15:5 km for M = 1:4M¯ ² observed giant glitch from Vela pulsar: moment of inertia changes by 1.4% ² implies a mass-radius relation for glitches from the crust ² problematic, constraint not fulfilled by most EoS! { p.21 How To Measure Masses and Radii of Compact Stars ² mass from binary systems (pulsar with a companion star) ² radius and mass from thermal emission, for a blackbody: 2 L1 4 R1 F1 = = σ T 4¼d2 SB e®;1 µ d ¶ with Te®;1 = Te® =(1 + z) and R1 = R=(1 + z) ² redshift: 2GM ¡1=2 1 + z = 1 ¡ µ R ¶ ² need to know distance and effective temperature to get R1 ² radius measured depends on true mass and radius of the star ² additional constraint from redshift measurement from e.g.