Thermal Properties of Dense Matter The Homogeneous Phase C. Constantinou IKP, FZ J¨ ulich 17 August 2015, Stockholm Workshop on Microphysics In Computational Relativistic Astrophysics Collaborators: M. Prakash, B. Muccioli & J.M. Lattimer C. Constantinou Thermal Properties of Dense Matter
The Models 1 1 0.8 0.8 Landau 0.6 0.6 m*/m Dirac 0.4 0.4 M*/m 0.2 0.2 MDI(A) SkO' MFT 0.2 dlog(m*)/dlog(n) 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 SNM Dirac/3 PNM 0 0 Landau -0.2 -0.2 -0.4 -0.4 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 n (fm -3 ) ◮ m ∗ = E ∗ ◮ m ∗ = F = ( p 2 F + M ∗ 2 ) 1 / 2 ◮ For n large, m 1+ β ( x ) n dm ∗ dn ≃ 0 ◮ Minimum at n s.t. M ∗ + dM ∗ p F dp F = 0 C. Constantinou Thermal Properties of Dense Matter
Degenerate Limit Thermodynamics ◮ Interaction switched-on adiabatically ◮ Entropy density and number density maintain their free Fermi-gas forms: 1 � s = [ f p ln f p + (1 − f p ) ln(1 − f p )] V p 1 � = f p ( T ) n V p d ε δ s ◮ � ⇒ s = 2 anT δ T a = π 2 m ∗ level density parameter p 2 2 F C. Constantinou Thermal Properties of Dense Matter
Degenerate Limit Thermodynamics Rest of thermodynamics via Maxwell’s relations or other identities: ◮ Energy density d ε ds = T ε th = anT 2 dT = − n 2 d ( s / n ) ◮ Pressure dP dn P th = 2 ; Q = 1 − 3 m ∗ dm ∗ n 3 nQT 2 2 dn d µ ◮ Chemical potential dT = − ds dn 1 − 2 Q T 2 � � µ ( n , T ) = − a 3 � C V = T d ( s / n ) ◮ Specific Heats n = 2 aT � dT � � C P = T d ( s / n ) P = 2 aT � dT � C. Constantinou Thermal Properties of Dense Matter
Degenerate Limit Thermodynamics Beyond Leading Order Degenerate limit implications: ◮ η = µ − ǫ ( p =0) ≫ 1 ⇒ Sommerfeld T ◮ ǫ = p 2 2 m + U ( n , p ; T ) → p 2 2 m + U ( n , p ; 0) For a general U ( n , p ), define an effective mass function � − 1 � � 1 + m ∂ U ( n , p ) � M ( n , p ) = m . � p ∂ p � n Relation to Landau m ∗ : M ( n , p = p F ) = m ∗ Applying the Sommerfeld expansion to the integral of the entropy density gives s = 2 anT − 16 5 π 2 a 3 nT 3 (1 − L F ) M ′ 2 M ′′ M ′ � L F = 7 m ∗ 2 + 7 m ∗ + 3 F ≡ ∂ M ( n , p ) 12 p 2 F 12 p 2 F F ; M ′ � 4 p F F F � m ∗ ∂ p � p = p F C. Constantinou Thermal Properties of Dense Matter
Degenerate Limit Thermodynamics Beyond Leading Order ◮ Thermal Energy: E th = aT 2 + 12 5 π 2 a 3 T 4 (1 − L F ) ◮ Thermal Pressure: P th = 2 8 � � 1 − L F + n dL F 3 anQT 2 − 5 π 2 a 3 nQT 4 2 Q dn ◮ Thermal Chemical Potential: � 1 − 2 Q � T 2 + 4 � (1 − L F )(1 − 2 Q ) − ndL F � 5 π 2 a 3 T 4 µ th = − a 3 dn ◮ Specific Heat at constant volume: C V = 2 aT + 48 5 π 2 a 3 T 3 (1 − L F ) � ∂ Pth � 2 � � ◮ Specific Heat at constant pressure: C P = C V + T ∂ T � n n 2 ∂ P ∂ n | T C. Constantinou Thermal Properties of Dense Matter
Results: S and E th 25 MDI(A) SkO' MFT 25 2 MDI(A) SkO' MFT 2 SNM 20 SNM 20 1.5 1.5 15 T = 20 MeV 15 1 1 10 10 0.5 5 0.5 5 T = 20 MeV E th (MeV) 0 0 0 0 S (k B ) 25 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 FLT FLT 25 2 PNM 2 FLT+NLO FLT+NLO 20 PNM 20 Exact Exact 1.5 1.5 15 15 1 10 1 10 0.5 5 0.5 5 0 0 0 0 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 n (fm -3 ) n (fm -3 ) ◮ The three models produce quantitatively similar results. ◮ Agreement with exact results extended to n ≃ 0 . 1 fm − 3 . ◮ Better agreement for PNM than for SNM. C. Constantinou Thermal Properties of Dense Matter
Results: P th and µ th 0 SNM 0 4 SNM 4 -2 -2 3 3 -4 -4 2 T = 20 MeV 2 -6 T = 20 MeV -6 1 MDI(A) SkO' MFT -8 MDI(A) SkO' MFT 1 P th (MeV fm -3 ) -8 µ n,th (MeV) -10 0 -10 0 FLT 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0 PNM 4 PNM 0 4 FLT+NLO -2 Exact -2 3 3 -4 -4 FLT 2 FLT+NLO 2 -6 -6 Exact 1 -8 1 -8 -10 0 -10 0 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 n (fm -3 ) n (fm -3 ) ◮ Model dependence is evident- due to dm ∗ dn . ◮ Agreement with exact results extended to n ≃ 0 . 1 fm − 3 . ◮ Better agreement for PNM than for SNM. C. Constantinou Thermal Properties of Dense Matter
Results: Specific Heats 7 7 6 6 MDI(A) SkO' MFT MDI(A) SkO' MFT 1.5 5 1.5 SNM 5 SNM 4 4 1 1 3 3 2 T = 20 MeV 2 0.5 0.5 1 T = 20 MeV 1 3 0 0 C P (k B ) C V (k B ) 3 0 0 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 2.5 FLT 0 0.2 0.4 0.6 0.8 0.2 0.4 FLT 0.6 0.8 0.2 0.4 0.6 0.8 2.5 PNM PNM FLT+NLO 1.5 FLT+NLO 1.5 2 Exact 2 Exact 1.5 1 1.5 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 n (fm -3 ) n (fm -3 ) ◮ The MDI and MFT C V exceed the classical value of 1.5 in the nondegenerate limit. In this regime the T -dependence of the spectrum becomes important. ◮ The peaks in C P are due to the proximity to the nuclear liquid-gas phase transition. C. Constantinou Thermal Properties of Dense Matter
Binary Mergers and the EOS ◮ The EOS is necessary for a complete description of the dynamics of a merger. ◮ The EOS is relevant to: ◮ GW frequency ◮ Size, type, and lifetime of remnant ◮ EOSs used in simulations: ◮ Realistic ◮ Polytropic, P = κ n Γ S ◮ Ideal Fluid, P th = (Γ th − 1) ε th C. Constantinou Thermal Properties of Dense Matter
Γ th -General Considerations ◮ Γ th = 1 + P th ε th ◮ Degenerate Limit ◮ Nonrelativistic n → 0 Γ th = 1 + 2 5 π 2 a 2 nT 2 dL F 4 5 3 Q − − → dn 3 Q = 1 + 3 m ∗ dm ∗ n 2 dn ◮ Relativistic � p 4 � n →∞ Γ th = 1 + Q 15 π 2 a 2 T 2 (1 − Q ) 8 L F − 5 4 3 + − → F 3 E ∗ 4 3 F � 2 � � � M ∗ 1 − 3 n M ∗ dM ∗ Q = 1 + E ∗ dn F ◮ CC, B. Muccioli, M. Prakash & J.M. Lattimer, arXiv:1504.03982 C. Constantinou Thermal Properties of Dense Matter
Γ th -SkO’ 2.2 Y p = 0.1 2 = 0.3 1.8 = 0.5 1.6 No Leptons SkO' 1.4 Deg. on top of Exact T = 50 MeV 1.2 ◮ No T dependence: 2.2 For Skyrme models, 0.0001 0.001 0.01 0.1 1 2 1.8 P th ( n , T ) = P id th ( n , T ; m ∗ ) Q 1.6 1.4 ε th ( n , T ) = ε id th ( n , T ; m ∗ ) T = 20 MeV 1.2 Γ th P id 2.2 th = 2 th 0.0001 0.001 0.01 0.1 1 2 ε id 3 1.8 Γ th = 8 3 − m ∗ 1.6 1.4 m T = 10 MeV 1.2 ◮ Weak x dependence 2.2 0.0001 0.001 0.01 0.1 1 2 1.8 1.6 ◮ Sharp rise in homogeneous phase 1.4 T = 5 MeV 1.2 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 n (fm -3 ) C. Constantinou Thermal Properties of Dense Matter
Γ th -MDI Y p = 0.5 2 (a) Deg = 0.3 = 0.1 1.6 Exact MDI(A) ◮ Weak T dependence T = 50 MeV 1.2 0.0001 0.001 0.01 0.1 1 ◮ Weak x dependence 2 Deg 1.6 Exact ◮ Maximum around n ∼ n 0 : T = 20 MeV 1.2 Γ th d Γ th dn = 0 ⇒ 0.0001 0.001 0.01 0.1 1 2 Deg + n d 2 m ∗ dm ∗ m ∗ dm ∗ n � � 1 − dn 2 = 0 dn dn 1.6 Exact T = 10 MeV 1.2 0.0001 0.001 0.01 0.1 1 2 1.6 Exact = Deg. T = 5 MeV 1.2 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 C. Constantinou Thermal Properties of Dense Matter
Γ th -MFT x = 0.1 (c) 2 = 0.3 Deg. = 0.5 1.6 Exact MFT T = 50 MeV ◮ Weak T dependence 1.2 0.0001 0.001 0.01 0.1 1 2 Deg. ◮ Weak x dependence 1.6 Exact T = 20 MeV 1.2 ◮ Maximum around 2 n 0 Γ th 0.0001 0.001 0.01 0.1 1 2 Exact = Deg. 1.6 T = 10 MeV 1.2 0.0001 0.001 0.01 0.1 1 2 Exact = Deg. 1.6 T = 5 MeV 1.2 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 C. Constantinou Thermal Properties of Dense Matter
Γ th -Leptons and photons included Y e = 0.5 Y e = 0.1 Y e = 0.1 (b) 2 (a) (c) 2 2 = 0.3 = 0.3 = 0.3 Deg Deg = 0.5 Deg = 0.5 = 0.1 1.6 1.6 1.6 MDI(A) SkO' MFT Exact T = 50 MeV Exact T = 50 MeV Exact T = 50 MeV 1.2 1.2 1.2 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 2 2 2 Deg Deg Deg 1.6 1.6 1.6 Exact T = 20 MeV Exact T = 20 MeV Exact T = 20 MeV 1.2 1.2 1.2 Γ th th 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 2 2 2 Deg Deg Deg 1.6 1.6 1.6 Exact Exact Exact T = 10 MeV T = 10 MeV T = 10 MeV 1.2 1.2 1.2 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 2 2 2 Deg Deg Deg 1.6 1.6 1.6 Exact Exact Exact T = 5 MeV T = 5 MeV T = 5 MeV 1.2 1.2 1.2 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 n (fm -3 ) n → 0 ◮ Γ th → 4 − 3 ◮ Maximum even for Skyrme C. Constantinou Thermal Properties of Dense Matter
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