Potential Model Approaches to the Equation of State

Transcription

1 Potential Model Approaches to the Equation of State C. Constantinou IKP, FZ Jülich Wednesday, December 3, 214 PALS: M. Prakash, B. Muccioli & J.M. Lattimer Workshop on NEOS for Compact Stars and Supernovae FIAS, Goethe University, Frankfurt, 3-5 December, 214

2 Outline Equation of State (EOS) Potential model of Akmal, Pandharipande, and Ravenhall (APR) Potential model with momentum-dependent interactions (MDI)

3 Equation of State From microscopic interactions to macroscopic state variables. Z = Tr exp( βĥ) For SN/NS: Nucleon-nucleon interaction in many-body setting. Necessary in hydrodynamic simulations of supernova explosions µ T µν =, T µν = pg µν + wu µ u ν + τ µν and neutron star structure calculations m(r) = 4π r ɛ(r )r 2 dr dp dr = (ɛ + p)[gm(r) + 4πGr 3 p] r[r 2Gm(r)] (interior mass) (hydrostatic equilibrium)

4 EOS Laboratory Constraints At saturation density (n =.16 ±.1 fm 3 ): Compression modulus, K = 23 ± 3 MeV (giant monopole resonances) Energy per particle, E/A = 16 ± 1 MeV (fits to masses of atomic nuclei) Symmetry energy, S 2 = 3 35 MeV (fits to masses of atomic nuclei, GDR) Slope of S 2, L = 4 7 MeV (variety of experiments) Effective mass, M /M =.8 ± 1 (neutron evaporation spectra)

5 EOS Constraints from Neutron Stars Largest observed mass, (binaries) M = 2.1 ±.4 M Largest observed frequency, Ω = 114 rad/s (pulsars) Inferred radius range, 1 km R 12.5 km (photospheric emission, thermal spectra)... Solution of Tolman-Oppenheimer-Volkoff (TOV) equations and EOS predicts M max, R max, I max, Ω max, etc.

6 NR Potential Model:APR Due to Akmal & Pandharipande (Phys. Rev. C. 56, 2261 (1997)): where v 18,ij = v NN = v 18,ij + V IX,ijk + δv(p ij ) p=1,18 V IX,ijk = V 2π ijk v p (r ij )O p ij + v em + V R ijk (Argonne) (Urbana) δv(p) = P2 8m u [P.r P., u] + 2 8m2 8m [(σ 2 i σ j ) P., u] (relativistic boost correction)

7 NR Potential Model:APR Parametric fit by Akmal, Pandharipande, and Ravenhall (Phys. Rev. C. 58, 184 (1998)). Hamiltonian density: [ 2 H APR = 2m + (p 3 + (1 x)p 5 )ne p4n [ 2 + 2m + (p 3 + (1 x)p 5 )ne p4n ] τ n ] τ n +g 1 (n)(1 (1 2x) 2 ) + g 2 (n)(1 2x) 2 τ n(p) - neutron(proton) kinetic energy density p i - fit parameters Skyrme-like Landau effective mass: m i = ( ε ki k i ) 1 k Fi = kfi [ 1 m + 2 ] 1 2 (p 3 + Y i p 5 )ne p4n

8 T= Thermodynamics Energy per particle E(n, x) = H/n E(n, 1/2) + S 2 (n)(1 2x) Pressure Chemical Potentials Incompressibility P(n, x) = n 2 E n n2 [ K 9n ( n n 1) + (1 2x) 2 L 3n ] +... µ i = E n i nj K = 9n 2 H n 2 Symmetry energy S 2 = E x=1/2 x 2 Slope of S 2 L = 3n ds2 dn ( ) 1 Susceptibilities χ ij = µi n j Sound Speed ( cs ) 2 c = dp dɛ

9 APR (Ska*) Predictions At saturation density, n =.16 fm 3 : K = 266 MeV (263) E = 16 MeV ( 16) S v = 32.6 MeV (32.9) m /m =.7 (.61) L v = 58.5 MeV (74.6) E (MeV) T = MeV 1 8 Ska 6 4 APR 2 x = n (fm -3 ) [1] C. Constantinou, B. Muccioli, M. Prakash, & J. M. Lattimer, Phys. Rev. C 89, 6582 (214) [2] H. S. Kohler, Nucl. Phys. A258, 31 (1976)

10 NS properties 2.5 Y p = 2 Ska M max = 2.19M (2.22) R max = 1.2 km (11.7) R 1.4 = 11.8 km (13.8) M G /M O APR R (km)

11 Effective Mass m*/m 1 x =.1.9 APR n.4.3 p n (fm -3 ) 1 x =.1.9 Ska n.4.3 p n (fm -3 )

12 Finite T Single-particle energy spectrum: ε i = k 2 i n i = 1 H + H ε ki + V i τ i n i ( ) 2m 3/2 2π 2 i T 2 F1/2i τ i = F αi = ( ) 1 2m 5/2 i T 2π 2 2 F 3/2i x α i e ψ i e x i + 1 dx i x i = ε ki T, ψ i = µ i V i T = ν i T

13 Finite T Non-Relativistic Johns-Ellis-Lattimer s (JEL) Scheme (ApJ, 473 (12),1996) Fermi-Dirac integrals as algebraic functions of the degeneracy parameter ψ only: F 3/2 = F α 1 = 1 F α α ψ ψ = 2 3f (1 + f )1/4 M 2 2 M p m f m m= ( 1 + f a ) 1/2 + ln [ (1 + f /a) 1/2 ] 1 (1 + f /a) 1/2 + 1

14 Finite T Rest of state variables : Energy density Chemical potentials Entropy density Pressure Free energy density ε = 2 2m n τ n + 2 2m τ p p +g 1(n) [ 1 (1 2x) 2] + g 2(n)(1 2x) 2 µ i = T ψ i + V i ( ) s i = T 3 τ + n i (V i µ i ) 2m i P = T (s n + s p ) + µ n n n + µ p n p ε F = ε Ts

15 Degenerate Limit Landau Fermi Liquid Theory Interaction switched-on adiabatically Entropy density and number density maintain their free Fermi-gas forms: s i = 1 [f ki ln f ki + (1 f ki ) ln(1 f ki )] V k i n i = 1 f ki (T ) V dε δs δt s i = 2a i n i T k a i = π2 2 m i k 2 Fi level density parameter

16 Degenerate Limit Other thermodynamics via Maxwell s relations: Energy density dε ds = T ε(n, T ) = ε(n, ) + T 2 n i a in i Pressure Chemical potentials Free energy density dp dt = n2 d(s/n) dn p(n, T ) = p(n, ) + [ i a i n i n d(a i n i ) dn ] T 2 dµ dt = ds dn [ µ(n, T ) = µ(n, ) T 2 ai 3 + ] n j a j dm j j mj dn i df dt = s F(n, T ) = F(n, ) T 2 n i a in i

17 Non-Degenerate Limit ( ) z 1 1. F α Γ(α + 1) z z α+1 2. Invert F 1/2 to get z : ( ) 2, ( z = nλ3 γ + 1 nλ 3 2 3/2 γ λ = 3. Plug z in F α s : F 3/2 = 3π1/2 4 F 1/2 = π1/2 2 2π 2 m T nλ 3 γ nλ 3 γ 1/2 nλ3 F 1/2 = π γ ) 1/2 ( quantum concentration [ nλ 3 ] 2 5/2 γ [ 1 1 nλ 3 ] 2 3/2 γ )

18 Thermal Pressure Exact Degenerate Non-Degenerate P Thermal [MeV/fm 3 ] T = 2 MeV x = n B [fm -3 ] Exact calculations agree with analytical limits where expected. Note flattening at high densities.

19 Thermal Susceptibilities dµ n,th /dn n (MeV fm 3 ) x =.1 T = 2 MeV APR & Ska.1.1 x =.1 T = 5 MeV APR & Ska.1.1 n (fm -3 ) dµ n,th /dn p (MeV fm 3 ) T = 2 MeV T = 5 MeV x =.1 x =.1 Ska APR Ska APR n (fm -3 ) In ND Limit, to leading order, χ ii T n i. However, χ ij T m i m i n j.

20 Thermal Symmetry Energy T = 2 MeV APR T = 5 MeV APR 8 6 Non- Degenerate 6 S 2 th (MeV) 4 2 Exact Non- Degenerate 4 2 Exact Degerate Degerate n (fm -3 ) -2 Negative at high densities.

21 Symmetry Energy 9 4 Symmetry Energies (MeV) APR S 2 S 2 +S 4 E S 2 +S 4 +S Thermal Symmetry Energy (MeV) S 2,th T = 2 MeV APR F th = F th (n,y p =,T) -F th (n,y p =,T) S 2,th +...+S 12,th S 2,th +...+S 3,th S 2,th T = 2 MeV S 2,th +...+S 3,th APR F th = F th (n,y p =.2,T) -F th (n,y p =,T) S 2,th +...+S 12,th n (fm -3 ) n (fm -3 ) S 2 = 1 2 E 8 x 2 = 1 2 F x=1/2,s 8 x 2 E pnm E snm F pnm F snm x=1/2,t 1.5 1

22 MDI: Motivation Single particle potential, U NR (n, p) p 2 U RMFT (n, ε) ε; inconsistent with optical model fits to nucleon-nucleus scattering data. Microscopic calculations (RBHF, variational calculations, etc.) show distinctly different behaviors in their momentum dependence, consistent with optical model fits. The above features were found necessary to account for heavy-ion data on transverse momentum and energy flow (in conjuction with K 23 MeV).

23 MDI: Hamiltonian Density H = 1 2m (τ n + τ p ) + A 1 (n n + n p ) 2 + A 2 (n n n p ) 2 2n 2n B (n n + n p ) σ+1 + [1 σ + 1 n σ y (n n n p ) 2 (n n + n p ) 2 + C l d 3 p i d 3 p f i ( r i, p i )f i ( r i, p i i ) ) n 2 + C u n i i 1 + ( pi p i Λ d 3 p i d 3 p j f i ( r i, p i )f j ( r j, p j ) ) 2 ; i j. 1 + ( pi p j Λ [1] G. M. Welke, et al., Phys. Rev. C 38, 211 (1988) [2] C. B. Das, S. Das Gupta, C. Gale & Bao-An Li, Phys. Rev. C 67, (23) ]

24 MDI vs. SkO * 1 8 T = MeV E (MeV) 6 MDI(A) 4 SkO' x = n (fm -3 ) P (MeV fm -3 ) T = MeV x = n (fm -3 ) SkO' MDI(A) µ n (MeV) (a) T = MeV MDI(A) SkO' x =.1.3 * P. G. Reinhard et al., Phys. Rev. C 6, (1999) n (fm -3 )

25 Predictions Property MDI SkO Experiment n (fm 3 ) ±.2 E (MeV) ±1 K (MeV) ±2 m /m ±.1 S v (MeV) L v (MeV) K v (MeV) ±2 M max (M ) ±.4 R max (km) ± 1. n c (fm 3 ) R 1.4 (km) ±.7 n c (fm 3 ) NS maximum masses in excess of 2 M can also be obtained from a reparametrization of the MDI model, but at the expense of losing close similarity with the results of the SkO model.

26 Single-Particle Potentials U n (MeV) U i (n i, n j, p i ) = U i (n i, n j ) + R i (n i, n j, p i ) R i (n i, n j, p i ) = 2C l 2 n (2π ) 3 d 3 p i (a) n =.17 fm -3 T = MeV x = MDYI N SkO UV14-TNI BHF-TBF (b) n =.34 fm C u 2 n (2π ) 3 (c) n = fm d 3 p j f p i 1 + ( pi p i Λ 1 + ) 2 f pj ( ) 2 pi p j Λ BHF-TBF: W. Zuo et al. PRC 74, (26) UV14-TNI: R. B. Wiringa PRC 38, 2967 (1988) k (fm -1 )

27 Effective Mass m i = m 1 + m p Fi U i p i pfi 1.9 neutron neutron proton.8 m * /m x = x =.1.4 MDI(A) proton (a) n (fm -3 ) SkO'.4 (b) n (fm -3 ) Isospin Splitting: mn > mp C l > C u (MDI) t 1 (1 + 2x 1 ) > t 2 (1 + x 2 ) (Skyrme)

28 Finite Temperature: Exact Self-consistency problem: R(n, p) requires R(n, p ) for all p. Iterative numerical procedure adopted similar Hartree-Fock theory: ε = H 1. Initial guess: R o = R(T = ). Then get ɛ o = p2 2m + U(n) + Ro (n, p). 2. Use ɛ o in n = 2 d 3 p 3. Use ɛ o, µ o to get R e ɛ µ T to get µ o. 4. Repeat until sufficient convergence is achieved. s = 2 d 3 p[f p ln f p + (1 f p ) ln(1 f p )] P = ε + Ts + µn

29 Finite Temperature: Limits Degenerate Limit : Use FLT Non-Degenerate Limit Assume R (n, p ) = in the calculation of R(n, p). Expand thermodynamic integrals about z =. Then use steepest descent method to evaluate resulting exponential integrals: g(x)e f (x) dx = 2πg(x f (xo) o)e f (x o ) { [ f (3) (x o ) f (x o ) 3/2 + 1 g (x o ) 2g(x o ) f (x o ) 1 2g(x o ) ] 2 1 f (4) (x o ) 8 f (x o ) 2 g (x o )f (3) (x o ) f (x o ) 2 }

30 Results: Entropy per particle S (k B ) S (k B ) Deg. (b) Non- MDI(A) Deg. T = 5 MeV (a) Deg. MDI(A) T = 2 MeV Non-Deg. Exact x = Exact x = n (fm -3 ) (c) (d) SkO' SkO' T = 2 MeV T = 5 MeV Deg. Deg. Non- Deg. Non- Deg. x = Exact Exact x = n (fm -3 ) Exact calculations agree with analytical limits where expected. MDI enters degenerate regime at lower densities.

31 Results: Specific Heat at Constant Volume (a) T = 2 MeV (b) T = 5 MeV C v (k B ) 1 x = SkO' MDI(A) SkO' 1.5 MDI(A) x = n (fm -3 ) C v controls the density at which the core rebounds The ratio C p /C v features in adiabatic sound wave propagation. Due to T -dependence of spectrum (via R).

32 Summary m controls thermal effects. S 2 F in the presence of a phase transition and at finite-t. MDI is well able to support NS with M 2 M solar. The T -dependence of the spectrum of MDI leads to a more robust C v which may be relevant to supernovae.