The Complete APR Equation of State

Transcription

1 The Complete Equation of State C. Constantinou IKP, FZ Jülich 4 May 6, Athens JINA-CEE International Symposium on Neutron Stars in Multi-Messenger Era: Prospects and Challenges Collaborators: B. Muccioli, M. Prakash & J.M. Lattimer C. Constantinou The Complete Equation of State

2 Matter in Astrophysical Phenomena Core-collapse Proto-neutron Mergers of compact supernovae stars binary stars Baryon Density(n ) Temperature(MeV) 3 5 Entropy(k B ).5 Proton Fraction n.6 fm 3 (equilibrium density of symmetric nuclear matter) Objective: Construction of EOS based on the best nuclear physics input for use in hydrodynamic simulations of core-collapse supernova explosions and binary mergers as well as the thermal evolution of proto-neutron stars. Status: Nearly there. C. Constantinou The Complete Equation of State

3 : Construction Due to Akmal & Pandharipande, Phys. Rev. C. 56, 6 (997) (AP): where v 8,ij = v NN = v 8,ij + V IX,ijk + δv(p ij ) p=,8 V IX,ijk = V π ijk v p (r ij )O p ij + v em + V R ijk (Argonne) (Urbana) δv(p) = P 8m u + [P.r P., u] + 8m 8m [(σ i σ j ) P., u] (relativistic boost correction) C. Constantinou The Complete Equation of State

4 π Condensation.5 SNM η(q) η(q) ρ=.6 ρ=.36 LDP ρ=.36 HDP PNM ρ=.6 ρ=. LDP ρ=. HDP Figure taken from AP. Distribution of excess pions η(q) = ξ(q) m π +q where q - momentum of exchanged pions ξ - pion-exchange nucleon-nucleon interaction contribution, exhibits sharp enhancement (especially in the case of SNM) q [fm ] C. Constantinou The Complete Equation of State

5 : Hamiltonian Density Parametric fit by Akmal, Pandharipande, and Ravenhall (Phys. Rev. C. 58, 84 (998)). Hamiltonian density: H = [ ] m + (p3 + ( x)p5)ne p 4n τ n [ ] + m + (p3 + xp5)ne p 4n τ p +g (n)( ( x) ) + g (n)( x) τ n(p) - neutron(proton) kinetic energy density p i - fit parameters Skyrme-like Landau effective mass: ( ) [ mi ε ki = k i k Fi = kfi m + ] (p3 + Y ip 5)ne p 4n x - proton fraction, Y p = x, Y n = x C. Constantinou The Complete Equation of State

6 T= Thermodynamics Energy per particle E(n, x) = H/n E(n, /) + S (n)( x) +... Pressure P(n, x) = n E n n [ K 9n ( n n ) + ( x) L 3n ] +... Chemical Potentials µ i = E n i nj Incompressibility K = 9n H n Symmetry energy S = 8 Slope of S L = 3n ds dn ( ) Susceptibilities χ ij = µi n j Sound Speed ( cs c ) = dp dɛ E x=/, S x 4 = E x 4 x=/,... C. Constantinou The Complete Equation of State

7 () Predictions At saturation density, n =.6 fm 3 : K = 66 MeV (63) 4 T = MeV E = 6 MeV ( 6) S = 3.6 MeV (3.9) M /M =.7 (.6) L = 58.5 MeV (74.6) NS properties: M max =.9M (.) R max =. km (.7) R.4 =.8 km (3.8) n du =.79/.99 fm 3 (.38/.54) E (MeV) M G /M O x = Y p = R (km) C. Constantinou The Complete Equation of State

8 Finite T Single-particle energy spectrum: ε i = k i H τ i + H ε ki + V i n i ( ) n i = m i T 3/ π F/i τ i = F αi = ( ) m 5/ i T F π 3/i x α i e ψ i e x i + dx i x i = ε ki T, ψ i = µ i V i T = ν i T C. Constantinou The Complete Equation of State

9 Finite T Non-Relativistic Johns-Ellis-Lattimer s (JEL) Scheme (ApJ, 473 (),996) Fermi-Dirac integrals as algebraic functions of the degeneracy parameter ψ only: F 3/ = F α = F α α ψ ψ = 3f ( + f )/4 M M p mf m m= ( + f a ) / + ln [ ] ( + f /a) / ( + f /a) / + C. Constantinou The Complete Equation of State

10 Finite T Rest of state variables : Energy density ε = m n τ n + τ mp p [ +g (n) ( x) ] + g (n)( x) Chemical potentials µ i = T ψ i + V i ( ) Entropy density s i = 5 τ T 3 +n i (V i µ i ) Pressure P = T (s n + s p) + µ nn n + µ pn p ε Free energy density F = ε Ts m i C. Constantinou The Complete Equation of State

11 Consistency Checks To infer thermal contributions, eliminate terms that depend only on density : X th = X (n, x, T ) X (n, x, ) Compare graphically the exact X th with its degenerate and non-degenerate limits C. Constantinou The Complete Equation of State

12 Degenerate Limit Landau Fermi Liquid Theory Interaction switched-on adiabatically Entropy density and number density maintain their free Fermi-gas forms: s i = [f ki ln f ki + ( f ki ) ln( f ki )] V k i n i = f ki (T ) V k dε δs δt s i = a i n i T a i = π m i k Fi level density parameter C. Constantinou The Complete Equation of State

13 Degenerate Limit Other thermodynamics via Maxwell s relations: Energy density dε ds = T ε(n, T ) = ε(n, ) + T n i a in i Pressure dp dt d(s/n) = n dn p(n, T ) = p(n, ) + i [ a i n i n d(a i n i ) dn ] T Chemical potentials dµ dt = ds dn µ(n, T ) = µ(n, ) T [ ai 3 + j n j a j m j ] dmj dn i Free energy density df dt = s F(n, T ) = F(n, ) T n i a in i C. Constantinou The Complete Equation of State

14 Non-Degenerate Limit ( ) z. F α Γ(α + ) z z +... α+. Invert F / to get z : ( ) z = nλ3 + nλ, ( 3 γ 3/ γ λ = 3. Plug z in F α s : π m T F 3/ = 3π/ 4 F / = π/ ) / ( quantum concentration nλ 3 γ nλ 3 γ F / = π / nλ 3 γ [ + ] nλ 3 5/ γ [ ] nλ 3 3/ γ ) C. Constantinou The Complete Equation of State

15 Effective Mass x =..3.5 x = m*/m n.5 n p.3. p C. Constantinou The Complete Equation of State

16 Thermal Energy E th (MeV) x =.5. (a) T = MeV x =.5. (b) T = 5 MeV E th (MeV) Degenerate x =. (a) Exact T = MeV.5 (b) Degenerate Non- Degenerate x =. Non- Degenerate Exact T = 5 MeV Exact calculations agree with analytical limits where expected. Around nuclear saturation density exact results needed. Differences between models more pronounced at high densities. C. Constantinou The Complete Equation of State

17 Entropy (a) T = MeV (b) T = 5 MeV 5 5 (a) T = MeV (b) T = 5 MeV 5 S (k B ) 3 x =.5 x = S (k B ) 4 3 Degenerate x =..5 Exact Non- Degenerate Degenerate x =..5 Exact Non- Deg Exact calculations agree with analytical limits where expected. Differences between models more pronounced at high densities. C. Constantinou The Complete Equation of State

18 Thermal Pressure P th (MeV fm -3 ) (a) (b).5.5 x =. x =. T = MeV T = 5 MeV P th (MeV fm -3 ) 4 3 (a).5 x =. Non-Deg. Exact Deg. T = MeV (b) Non-Deg..5 x =. Deg. Exact T = 5 MeV Differences between models more pronounced at high densities. Note flattening at high densities. C. Constantinou The Complete Equation of State

19 Thermal Chemical Potential µ p,th (MeV) x =. x =. T = MeV T = 5 MeV µ p,th (MeV) Non-Deg. (c).5 x =. Exact Deg. T = MeV Non-Deg. Deg..5 x =. (d) Exact T = 5 MeV Differences at intermediate densities. Not always <. C. Constantinou The Complete Equation of State

20 Thermal Symmetry Energy 8 T = MeV T = 5 MeV 8 6 Non- Degenerate 6 S th (MeV) 4 Exact Non- Degenerate 4 Exact Degerate Degerate Significant thermal contributions at low densities. Negative at high densities. C. Constantinou The Complete Equation of State

21 Thermal Susceptibilities dµ n,th /dn n (MeV fm 3 ) x =..5 T = MeV &.. x =..5 T = 5 MeV & dµ n,th /dn p (MeV fm 3 ) T = MeV T = 5 MeV x =. x = In ND Limit, to leading order, χ ii T n i. However, χ ij T m i m i n j. C. Constantinou The Complete Equation of State

22 Thermal Index-General Considerations The thermal properties of the EOS are relevant in a merger at the onset of matter transfer between the two compact objects which leads to shock heating. These can be characterized by the thermal index, Γ th. Γ th = + P th ε th Degenerate Limit Nonrelativistic Γ th = + 3 Q 4 5π a nt dl F dn Q = + 3 n m dm dn n 5 3 Relativistic ( Γ th = + Q π a T ( Q) ( ) Q = + M ( ) 3n M dm dn E F L F 5 3 pf 4 E F 4 ) n 4 3 CC, B. Muccioli, M. Prakash & J.M. Lattimer, Phys. Rev. C 9, 58. C. Constantinou The Complete Equation of State

23 Γ th.6. x =. =.3 =.5 npeγ np (a) MDI(A) T=5 MeV x =. =.3 =.5 npeγ np (b).6 T=5 MeV. Nuc. Nuc.+e+γ x =. =.3 =.5 MFT T = 5 MeV (c).6. Γ th.6 np np.6 Nuc..6 Γ th. npeγ T= MeV npeγ T= MeV. Nuc.+e+γ T = MeV..6 np np.6 Nuc..6. npeγ T= MeV npeγ T= MeV. Nuc.+e+γ T = MeV Weak x and T dependence but n dependence cannot be ignored. Significant contributions from electrons. C. Constantinou The Complete Equation of State

24 .... Adiabatic Index Γ S Exact Deg Exact Deg S = 3 (a) MDI(A) Y e =.5 S = (c) SkO' Y e =.5 Exact ND Exact ND S = 3 (b) MDI(A) Y e =.5 S = (d) SkO' Y e = Relevant during the early inspiralling phase of a merger when the two objects interact only gravitationally. Γ S (n, S) = ln P ln n S = n P Γ S (n, T ) = C P n P C V P n T Relation to sound speed, c s: P n S ( cs ) P c = Γ S h+mn Only nucleons mechanical instability With leptons, nuclear matter is stable n X such that P dp dn = P th P th n S (indep. of S in Deg. Limit).... C. Constantinou The Complete Equation of State

25 LDP - HDP transition 6 Mechanical stability requires dp dn But, in the transition from LDP to HDP, pressure decreases (negative incompressibility). Boundaries of the phase coexistence region defined by P L (n L ) = P H (n H ) µ L (n L ) = µ H (n H ) Across the region, s is discontinuous l = T [s H (n H ) s L (n L )] first-order transition P (MeV fm -3 ) T = MeV x = T = MeV.5 n H n t 5 MeV x n L C. Constantinou The Complete Equation of State

26 Numerical Implementation of Maxwell Construction Define the average chemical potential µ = i=n,p,e Y iµ i and the function E = n tµ P. Expand these in Taylor series about n t: µ(n) = µ(n t) + (n n t) dµ dn nt dµ E(n) = E(n t) + (n nt ) dn nt Equilibrium across the coexistence region implies: n L = n t + n H = n t + µ H (n t ) µ L (n t ) µ L (nt )/ [µ L (nt )/ +µ H (nt )/ ] µ L (n t ) µ H (n t ) µ H (nt )/ [µ L (nt )/ +µ H (nt )/ ] µ (MeV) P (MeV fm -3 ) T = MeV x =.4. (a) T = 5 MeV x =.4. (b) T = MeV x =.4. (c) T = 5 MeV x =.4. (d) C. Constantinou The Complete Equation of State

27 Liquid-Gas Phase Transition The critical temperature and density of the transition for a given composition are obtained by the condition dp = d P =. dn dn nc,t c nc,t c P (MeV fm -3 ) x =.5 T c = 7.9 MeV (a) T = MeV (n c, T c) is the termination point of the phase boundary separating the homogeneous and the inhomogeneous phases of nuclear matter. T c / T c (x=.5).8 (a) (b)...4 x x n c / n c (x=.5) C. Constantinou The Complete Equation of State

28 Subnuclear Regime C. Constantinou The Complete Equation of State

29 in Subnuclear Regime Lattimer-Swesty approach (Nucl. Phys. A 535, 33 (99)): F total = F N + F α + F bulk + F e + F γ Single representative species of heavy nucleus; described by the compressible liquid-drop model: F N = F bulk,in + F s + F C + F tr. F S - Semi-infinite matter in TF approximation; curvature and neutron skin ignored. F C = 4 5 πe xin n inr N D(u); D(u) = 3 u/3 + u (point + lattice + finite size) { [ ] } F tr = u( u)n in h(t )T ln A u( u)n in n Q A 5/ (massive non-interacting point particles w/ modifications) α-particles represent light nuclei; treated as non-interacting Boltzmann gas having hard sphere interactions with nuclei: F α = ( u)n αf α Nucleons have hard-sphere interactions with α s and nuclei: F bulk = ( u)( v αn α)n outf bulk Pasta : Use smooth generalized functions which modify F C and F S appropriately. C. Constantinou The Complete Equation of State

30 in Subnuclear Regime To find tbe equilibrium state of the system, F total must be minimized with respect to nuclear radius r, proton fraction x in, nucleon density n in, volume occupied by nuclei u, and density of α-particles n α: F r = (nuclear size optimization) F x in = (chemical equilibrium) un in n in u F n in F = u (pressure equilibrium) F n α = (chemical equil. of α s with nucleons) Constraints: n = un in + ( u)[4n α + (n n,out + n p,out)( n αv α)] (baryon # conservation) ny e = ux in n in + ( u)[n α + n p,out( n αv α)] (charge conservation) C. Constantinou The Complete Equation of State

31 in Subnuclear Regime Advantages: Issues: Thermodynamic consistency. Consistent treatment of matter inside and outside nuclei. Separation of bulk and surface effects. Relatively low demand for computational resources. Collective motion and related thermal effects in nuclear dissociation regime. Processes sensitive to the full ensemble of nuclei present in the inhomogeneous phase. C. Constantinou The Complete Equation of State

32 Results Free energy density Ftotal (MeV fm-3) Ftotal (MeV fm-3) --.5 T (MeV) --3 Ye= T (MeV) Ye= MeV fm n (fm-3) MeV fm n (fm-3) Volume fraction occupied by heavy nuclei u (none) u (none) T (MeV) T (MeV) - Ye=.5 - Ye= n (fm ) -3 n (fm ) C. Constantinou The Complete Equation of State -

33 Results Pressure Ptotal (MeV fm-3) Ptotal (MeV fm-3) T (MeV) -4-5 Ye= n (fm-3) MeV fm T (MeV) - MeV fm-3-5 Ye= n (fm-3) - - Nuclear size A (none) A (none) 4 Ye=.5 Ye= T (MeV) T (MeV) n (fm ) -3 n (fm ) C. Constantinou The Complete Equation of State -

34 Conclusions EOS at finite T with consistent treatment of the subnuclear regime and of the transition to a pion-condensed phase (almost) ready; to serve as a benchmark for other nonrelativistic calculations. The nucleon effective mass m is crucial in the determination of thermal effects in the bulk homogeneous phase. The thermal and the adiabatic indices Γ th and Γ S depend weakly on composition and temperature but their density dependence especially in the neighborhood of the saturation density of SNM cannot be ignored. Future: Apply LS to MFTs and models w/ finite-range interactions; comparison with the Shen EOS, Phys. Rev. C 83, 358 (). Better treatment of nuclear properties (e.g. HFB description of heavy nuclei, etc.). SN, PNS, BM simulations using (the correct). C. Constantinou The Complete Equation of State