COOLING OF NEUTRON STARS WITH COLOR SUPERCONDUCTING QUARK CORES

Transcription

1 COOLING OF NEUTRON STARS WITH COLOR SUPERCONDUCTING QUARK CORES David Blaschke Universität Bielefeld & JINR Dubna Collaboration: D. Aguilera, H. Grigorian, D. Voskresensky EoS and QCD Phase Transition Stability of Hybrid Stars Cooling of Hadronic Stars Hybrid Star Cooling: 2SC + X Quark Matter Perspectives - a Conjecture Picture taken from

2 INTRODUCTION Tc ~ 170 MeV T early universe ALICE crossover n = 0 B vacuum QCD Phase Diagram <ψψ> > 0 quark-gluon plasma RHIC hadronic fluid µ o n > 0 B SPS nuclear matter 922 MeV µ <ψψ> 0 quark matter crossover superfluid/superconducting phases? 2SC <ψψ> > 0 CFL neutron star cores T = 0, µ = 0 ψψ 0 hadron phase chiral symmetry is broken. T, µ ψψ = 0 symmetry phase all symmetries are restored. Moderate T and µ? heavy-ion collisions / compact stars interiors * S. Hands, The phase diagram of the QCD physics/ (2001)

3 MOTIVATION Even at weak coupling, any attractive quark-quark interaction condensate of Cooper pairs Color superconductivity To explore dense matter EoS (T, µ) with diquark condensates? with strangeness? To build a stable realistic neutron star with quark core - Quark matter EoS with hadron shell - Nuclear matter EoS

4 QUARK MATTER EOS MODEL Mass gap φ f and diquark gap f are order parameters for χsb and CSC β-equilibrium for the two flavor and three color case (N f = 2 u, d, N c = 3 r, b, g) requires different chemical potentials: µ q = (µ u + µ d )/2 and µ I = (µ u µ d )/2. Ω(φ, ; µ q, µ I, T ) = φ dqq 2 (N 4G 1 4G 2 2π 2 c 2){2E φ [ ( 0 +T ln 1 + exp E )] [ ( φ µ q µ I + T ln 1 + exp E )] φ µ q + µ I T T [ ( +T ln 1 + exp E )] [ ( φ + µ q µ I + T ln 1 + exp E )] φ + µ q + µ I } T T 4 dqq 2 {E 2π E [ 0 ( +T ln 1 + exp E φ µ )] [ ( I + T ln 1 + exp E φ + µ )] I T T [ ( )] [ ( )] +T ln 1 + exp E+ φ µ I T + T ln 1 + exp E+ φ + µ I T } Ω vac The dispersion relations: E φ 2 = q 2 + (m + g(q)φ) 2 for quarks of unpaired color E ± φ 2 = (E φ ± µ) 2 + g( q) 2 for quarks of paired colors

5 FORM FACTORS The following cases are considered: Gaussian (g G ),Lorentzian (g L ) and NJL cutoff (g NJL ). Gaussian gg(q) 2 = exp ( q 2 /Λ 2 G) Lorentzian gl 2 (q) = 1 Λ 2 + (q/λ L ) 4 NJL gnjl 2 (q) = Θ(1 q/λ NJL) Λ[GeV] G 1 Λ 2 m[mev] g G g L g NJL The parameters Λ, G 1, G 2 and m are fixed at T = µ = 0 by m π = 140 MeV, f π = 93 MeV, m q (0) = 330 MeV and by choosing G 1 /G 2 = 4/3. S. Schmidt, D. Blaschke, Y. Kalinovsky, Phys. Rev. C 50 (1994) 435. QUARK MATTER IN COMPACT STAR CORES The quark matter core of compact stars consists of u, d, e,, under the conditions: β-equilibrium d u + e + µ d = µ u + µ e + µ νe µ e + µ νe = 2µ I Charge neutrality 2 3 n u 1 3 n d n e = 0 n q + 3n I 6n e = 0

6 THERMODYNAMICAL POTENTIAL, GAP EQUATIONS The extrema of Ω correspond to Gap equations for the order parameters φ 0, 0 : Ω φ φ=φ0 = Ω ; = 0 φ=φ0 = 0 ; = 0 As T or µ changes, Ω can have several local minima in the φ, plane. The lowest minimum describes the lowest free energy state and is preferred. As an example, Ω(T, µ) at T = 0, µ = GeV is plotted. Two degenerate minima can coexist at the values: φ = 0.4 GeV, = 0 and φ = 0, = GeV. This corresponds to a first order transition at which two phases with equal pressure can coexist Ω φ = 0 [GeV] = 0 µ=0 µ=0.292 µ= φ Ω(φ 0, 0 ; µ q, µ I, T ) = ɛ T s µ q n q µ I n I = P, n i = Ω µ i T,φ=φ0, = 0

7 EFFECT OF FINITE TEMPERATURE GAP EQUATION SOLUTIONS AND µ I IN β-equilibrium µ I,, δ [GeV] φ, Τ = 0, T = 0 µ Ι, Τ = 0 φ, T = 20 MeV, T = 20 MeV µ I, T= 20 MeV φ, T = 40 MeV, T = 40 MeV µ I, T= 40 MeV µ q [GeV] m q = 0 (left) and m q = 2.41 MeV, (right) The diquark condensate does not feel the difference in partial or complete restoration of the chiral symmetry. At low temperatures color superconductivity at µ immediately above the first order phase transition. Color superconductivity persists up to Tc = 45 MeV. The onset of the diquark condensation is found at higher µ q if the temperature is increased. BCS theory predicts that T c = 0.57F (µ) 2 0 (T = 0)

8 EFFECT OF FORM FACTORS GAP EQUATION SOLUTIONS AND µ I IN β-equilibrium The maximum of the diquark condensate ( 150 MeV) roughly form factor independent. The critical values of the phase transition temperature and chemical potential depend on smoothness of form factors. [GeV] φ F G F L α=2 F L α=10 F L α=50 F NJL At T = 0 the critical chemical potentials are: µ G qc = 320 MeV, µ L qc = 345 MeV and µ qc NJL = 360 MeV. D.B., Fredriksson, Grigorian, Öztas, NPA 736 (2004) µ Ι µ q [GeV]

9 EOS FOR QUARK STAR MATTER TEMPERATURE AND ANTINEUTRINO EFFECTS P + ɛ = st + µ q n q + µ I n I, n q = n u + n d, n I = n u n d Temperature effect Antineutrino effect T = 0 T = 40 MeV T = 50 MeV T = 20 MeV 1000 P [MeV fm -3 ] P 10 ε P [MeV/fm 3 ] µ νe = 0 µ νe = 100 MeV µ νe = 200 MeV ε[mev fm -3 ] ε [MeV/fm 3 ]

10 CONFIGURATION OF QUARK STARS WITHOUT SHELL CONDENSATE AND TEMPERATURE EFFECT 2 Gaussian Tolman-Oppenheimer-Volkoff equations M [M sun ] n B [n 0 ] T = 0, = 0 T = 0 T = 40 MeV T = 60 MeV R [km] dp dr = [ɛ(r) + P (r)][m(r) + 4πr3 P (r)] r[r 2m(r)] m = 4π r 0 r 2 ɛ(r)dr The integral parameters M and R feel the existence of the diquark condensation: the EoS becomes softer. At higher temperature both M max and R max decrease approximately 10% compared to T = 0 and = 0 cases.

11 MASS DEFECT - ESTIMATION OF ENERGY RELEASE M [M sun ] N B [N sun ] Μ(Α) = 0.06 Μ sun M i (A) = 1.51 M sun M i (B) = 1.51 M sun M f (B) = 1.44 M sun M f (B) = 1.42 M sun Μ(Β) = 0.09 Μ sun Conserved N B (A) = 1.68 N sun, N B (B) = 1.57 N sun T = 0 T = 0, = 0 T = 40 MeV, = 0 T = 40 MeV Gaussian n B [n 0 ] During the cooling evolution the star must loose an amount of energy: the energy release will lead to an equivalent mass defect. We estimated it to about 0.1 M. It accidentally corresponds to the simple estimate (Hong et al., PLB 516 (2001) 362) ( /µ) 2 M erg

12 PNS EVOLUTION WITH NEUTRINO TRAPPING T = 40 MeV QM ν e νe ν µ = 72 MeV e T > 1 MeV QM 2SC µ = 72 MeV T = 0 µ = 0 2SC M [M sun ] M sun 0.07 M sun initial state A final state initial state B N B [N Bsun ] µ νe = 0 µ νe = 72 MeV µ νe = 150 MeV n q [n 0 ] Mass defect M M erg Effect due to Un-Trapping of (Anti-)Neutrinos Explosion possible: Phase transition 1 st Order GRB with time delay Cannon Ball Model Aguilera, D.B., Grigorian, Astron. & Astrophys. 416 (2004) 991.

13 PHASE TRANSITION: NUCLEAR MATTER QUARK MATTER Walecka model (p,n,e)+(σ,ω) Bag model (u,d,e) Maxwell construction: P Had = P Quark µ Had B 1000 EoS at T = 0 B 1/4 [MeV] 1800 = µ Quark B Walecka Model Bag Model, B 1/4 =190 MeV P[ MeV / fm 3 ] ε[ MeV / fm 3 ] Nuclear Matter B 1/4 = 220 B 1/4 =200 B 1/4 =190 B 1/4 =180 ε[ MeV / fm 3 ] µ = MeV µ B [ MeV]

14 STABILITY OF HADRONIC AND HYBRID STARS Gaussian model Lorentzian model

15 STABILITY OF ROTATING HYBRID STARS Mass - Radius relation HHJ - Gaussian formfactor model Phase diagram for rotating stars Springer LNP 578 Phys. of NS Int. M [ M sun ] R e [km] _ Ω = 0 Ω = Ω max with SC normal hadronic ε(0) [MeV/fm 3 ] Ω [khz] RSt line (normal) RSt line (2SC) phtc line (normal) phtc line (2SC) BH line (normal) BH line (2SC) Hadronic Configurations No Stable Rotation Quark Core Configurations M [M sun ] Black Holes

16 Normal Cooling for HJ EoS our crust, without medium effects COOLING log(t s [K]) PSR J in 3C58 Crab RX J E RX J log(t[yr]) Vela PSR Geminga PSR RX J Evolution of the surface temperature T s of a hadronic star Data: Yakovlev et al., A & A 389 (2002) L24; Calculation: D.B., Grigorian, Voskresensky, astro-ph/

17 THE NUCLEAR URCA PROCESS n p + e + Neutrinos leave the star (λ ν R): µ ν = 0 β-equilibrium: µ n = µ p + µ e Momentum conservation: p F,n = p F,p + p F,e p F,n p F,p + p F,e Charge neutrality: n p n e = 0 p3 F,p = p3 F,e 3π 2 3π 2 Triangle inequality: p F,n 2p F,p n n 8 n p n p n p +n n = x p 1 9 Luminosity: L ν = (2π) 4 d 3 p n (2π) 3 2E n... d 3 p ν (2π) 3 2E ν δ 3 ( p i )δ(e i ) M fi 2 f n (1 f p )(1 f e ) Emissivity: ɛ ν = L ν V 1027 ( m n m p m 2 N ) ( ) 1/3 n e ( T ) 6 erg n K cm 3 s

18 HADRON MATTER: DU CRITICAL DENSITIES AND CRITICAL MASSES OF STARS DU critical densities n c = 2.7 n 0 non linear Walecka (NLW) n c = 5.0 n 0 AV 18 + δv (HHJ) DU critical masses M c = 1.25 M - NLW M c = 1.84 M - HHJ 0.18 Charge Particles for WNL(250) & HJ(240) EoS Densities of protons electrons and muons n p,e,µ [fm-3 ] p - HHJ e - - HHJ µ - HHJ p - NLW e - - NLW µ NLW M [M sun ] HHJ, 240 MeV NLW 250 MeV n [fm -3 ] R [km] n(0) [fm -3 ] Heiselberg, Hjorth-Jensen, astro-ph/

19 GENERAL RELATIVISTIC COOLING EQUATIONS The energy flux per unit time l(r) through a spherical slice at distance r from the center is: l(r) = 4πr 2 k(r) (T eφ ) e 1 Φ 2M r r. The factor e Φ 1 2M r corresponds to relativistic corrections of time and distance scales. The equations for energy balance and thermal energy transport are: N B (le 2Φ ) = 1 n (ɛ νe 2Φ + c V t (T eφ )) (T e 2Φ ) = 1 le Φ N B k 16π 2 r 4 n where n = n(r) is the baryon number density, N B = N B (r) is the total baryon number in the sphere with radius r and N B r = 4πr2 n(1 2M r ) 1/2 D.B., Grigorian, Voskresensky, Astron. & Astrophys. 368 (2001) 561.

20 COOLING CURVES: HADRONS (NORMAL) STARS Our crust (Ts Tin) log(t[yr]) RX J Vela Geminga log(t s [K]) 1E RX J PSR PSR PSR J in 3C58 Crab RX J π -condensate and MMU Normal Cooling for HJ EoS our crust, without medium effects log(t[yr]) RX J Vela Geminga log(t s [K]) PSR PSR J in 3C58 1E RX J PSR Crab RX J Normal Cooling for HJ EoS our crust, with medium effects, π cond D.B., Grigorian, Voskresensky, astro-ph/

21 HADRON MATTER: PAIRING GAPS AND CRUST OF STARS Neutron and Proton pairing gaps Crust model 1.5 n 1S 0 Thick lines Yak. et. al Thin lines AV18 10 n 3P 2 p 1S [MeV] log 10 (T in [K]) our fit Tsuruta law Yakovlev (η = ) Yakovlev (η = 4.0*10-8 ) n / n log 10 (T s [K]) Yakovlev, Gnedin, Kaminker, Levenfish, Potekhin, astro-ph/ ; Yakovlev, Levenfish, Potekhin, Gnedin, Chabrier, A & A 417 (2004) 169; Takatsuka, Tamagaki, nucl-th/

22 COOLING CURVES: HADRONIC (SUPERCONDUCTING) STARS with AV18 gaps, π -condensate, MMU (3P 2 - suppressed) log(t s [K]) Cooling for HJ EoS (AV18) Our crust, with medium effects, 3P 2 *0.1 π cond PSR J in 3C58 Crab RX J E RX J Vela PSR Geminga PSR RX J with Gaps from Yakovlev at al log(t s [K]) Cooling for HJ EoS our crust, with medium effects, π cond. 3P 2 *0.1 PSR J in 3C58 Crab RX J E RX J log(t[yr]) Vela PSR Geminga PSR RX J log(t[yr]) D. Blaschke, H. Grigorian, and D.N. Voskresensky, (2004). arxiv:astro-ph/

23 COOLING CURVES: HADRONIC (SUPERCONDUCTING - ANOMALOUS) STARS with AV18 gaps, π -condensate, MMU (3P 2 - is NOT suppressed) log(t s [K]) Cooling for HJ EoS (AV18) Our crust, with medium effects PSR J in 3C58 Crab RX J E RX J Vela PSR Geminga PSR RX J with Gaps from Yakovlev at al log(t s [K]) Cooling for HJ EoS our crust, with medium effects, π cond. PSR J in 3C58 Crab RX J E RX J log(t[yr]) Vela PSR Geminga PSR RX J log(t[yr]) D. Blaschke, H. Grigorian, and D.N. Voskresensky, (2004). arxiv:astro-ph/

24 BAG MODEL FIT AND HYBRID STAR CONFIGURATION Mass - Radius relation HHJ - Gaussian formfactor model Bag model Fit for Gaussian SM EoS H. Grigorian et al., PRC 69 (2004) M [ M o ] HHJ -SM with 2SC HHJ -SM without 2SC HHJ R [km]

25 NEUTRINO PROCESSES IN QUARK MATTER: EMISSIVITIES Quark direct Urca (QDU) the most efficient processes d u + e + ν and u + e d + ν ɛ QDU ν α s uye 1/3 ζ QDU T9 6 erg cm 3 s 1, Compression u = n/n 0 2, strong coupling α s 1 Quark Modified Urca (QMU) and Quark Bremsstrahlung (QB) d + q u + q + e + ν and q 1 + q 2 q 1 + q 2 + ν + ν ɛ QMU ν ɛ QB ν ζ QMU T9 8 erg cm 3 s 1. Suppression due to the pairing QDU : ζ QDU exp( q /T ) QMU and QB : ζ QMU exp( 2 q /T ) for T < T crit,q 0.4 q e+ e e+ e+ ν+ ν ɛ ee ν = Ye 1/3 u 1/3 T9 8 erg cm 3 s 1, becomes important for q /T >> 1 BLASCHKE, GRIGORIAN, VOSKRESENSKY, ASTRON. & ASTROPHYS. 368 (2001) 561 d d u - ν e- u - ν e - u u FLOWERS, ITOH, APJ 250 (1981) 750; SCHAAB, VOSKRESENSKY, SEDRAKIAN, WEBER, WEIGEL, A & A 321 (1997)591 YAKOVLEV, LEVENFISH, SHIBANOV, PHYS. USP. 169 (1999) 825; BAIKO, HAENSEL, ACTA PHYS. POLON. B 30 (1999) 1097

26 EVOLUTION OF THE TEMPERATURE PROFILE = 50 MeV = 0.1 MeV = 0 Temperature profiles in MeV t = 1 sec t = 1 min t = 5 min t = 2 hours t = 5 hours t = 10 hours t = 1.5 days t = 0.1 year t = 1 year t = 3 years t = 4 years t = 6 years Radius in km

27 COOLING CURVES: HYBRID STARS WITH 2SC QUARK MATTER 2SC phase - 1 color is unpaired (mixed superconductivity) log(t s [K]) Cooling of Hybrid stars with 2SC Quark core HJ (Y - 3P 2 *0.1) with K = 240 MeV with Med. effects, our crust, Gaussian FF PSR J in 3C58 Crab RX J E RX J Vela PSR Geminga PSR RX J critical SC + X, X = 1 MeV log(t s [K]) 6,4 6,2 6 5,8 Cooling of Hybrid stars with 2SC Quark core HJ (Y - 3P 2 *0.1) with K = 240 MeV with Med. effects, our crust, Gaussian FF PSR J in 3C58 Crab 5, RX J E RX J log(t[yr]) Vela PSR Geminga PSR RX J log(t[yr]) D.B., Grigorian, Voskresensky, astro-ph/

28 COOLING CURVES: HYBRID STARS WITH 2SC QUARK MATTER 2SC + X phase, X = 100 kev log(t s [K]) Cooling of Hybrid stars with 2SC Quark core HJ (Y - 3P 2 *0.1) with K = 240 MeV with Med. effects, our crust, Gaussian FF PSR J in 3C58 Crab RX J E RX J Vela PSR Geminga PSR RX J critical SC + X phase, X = 30 kev log(t s [K]) Cooling of Hybrid stars with 2SC Quark core HJ (Y - 3P 2 *0.1) with K = 240 MeV with Med. effects, our crust, Gaussian FF PSR J in 3C58 Crab RX J E RX J log(t[yr]) Vela PSR Geminga PSR RX J critical log(t[yr]) D.B., Voskresensky, Grigorian, [arxiv:astro-ph/ ], and in preparation

29 CONJECTURE ABOUT MAGNETIZED QUARK STARS: NEUTRINO BEAMING AND GRB Magnetic vortex structure Characteristic length scales: ξ, λ, λ D N vort Φ q = B in,sπr 2, Φ q 6Φ 0, B λ Channeling of neutrino propagation Berdermann, D.B., Grigorian, Hong, Voskresensky in prep. (2004) star surface λ ν ν,ν + e ν ν e conversion ξ R G G ν,ν λ D G ξ < λ < λ D vortex superconductor θ ν

30 CONJECTURE ABOUT MAGNETIZED QUARK STARS: NEUTRINO BEAMING AND GRB Beaming angle as function of T and B θ ν arcsin λ ν R, λν = λ ν ( V N V o V V o ) α Evolution of luminosity, temperature, θ ν Berdermann, D.B., Grigorian, Hong, Voskresensky, in prep. (2004) 10 L [erg/s] 1e+53 1e+52 1e+51 1e+50 B = G B = G B = G B = G B = G θ ν [grad] T [MeV] B = G B = G T [MeV] θ ν [grad] t [s]

31 PERSPECTIVES THEORY OF EOS WITH CONSTRAINTS Nuclear Matter - 2SC Quark Matter Phase transition depends on details of the modeling, e.g. Coupling Constants, Form Factors Improve by using DSE Models 2SC Quark Matter Core is possible for NLW - Gaussian separable model Rôle of CFL phase at higher densities? EXPERIMENTAL OBSERVABLES Late time Cooling: Onset of Enhanced Cooling depends on Size of Quark Core, i.e. Mass/Spin of the Star Coupling of Rotational & Cooling evolution (in progress) Early Evolution (PNS): Explosive release of Binding energy is possible in antineutrino untrapping Second pulse of ν from supernovae Anisotropic Emission (magnetic vortices) may explain GRB - beaming