Theory of Nuclear Matter for Neutron Stars and ... - Graduate Physics

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Appendix BJEL thermodynamic integrationTo compute the properties of finite-temperature matter, it is necessary to calculate the Fermiintegrals F 1/2 and F 3/2 . Directly calculating these integrals by usual integral methods is notadvisable forapplicationslike hydrodynamics, which require ahighdegreeofthermodynamicconsistency. The approach we adopt is due to Johns, Ellis & Lattimer[12] (hereafter referredtoasJEL),whichisamodificationofalessaccuratescheme originatedbyEggleton, Faulkner& Flannery[8]. These approaches involve polynomial interpolations for arbitrary degeneracyand relativity. However, in the finite-range model, nucleons are treated non-relativistically,so the interpolation method can be considerably simplified from the general case.The degeneracy parameter Ψ, and an associated variable f, are defined in terms of thechemical potential and temperature assuch thatΨ = µ−VT= 2√f ′ = dfdΨ =1+ f a +ln √1+f/a−1√1+f/a+1,f√1+f/a.(B.1)(B.2)In the above, a is a fitting parameter and is given in table B. The Fermi integrals areexpressed as polynomials in f with additional parameters p m , where m = 0...M, that arefixed by the requirements of yielding exact results for the pressure, energy and entropy inthe extremely degenerate and non-degenerate limits or by fitting intermediate results. Using116
M 3 2a 0.433 1p 0 (e 2 /a) √ π/32 = 5.34689 (e 2 /a) √ π/32 = 2.31520p 1 16.8441 a −1/4 [π 2 −8+56/(5a)]/3 = 44.35653p 2 a −1/4 [π 2 −8+88/(5a)]/3 = 17.4708 32a −5/4 /15 = 2.13333p 3 32a −5/4 /15 = 6.07364 —the results of JEL, one findsTable B.1: Non-relativistic fermion coefficientsF 3/2 (Ψ) = 3f(1+f)1/4−M2 3/2 M∑F 1/2 (Ψ) = f′ (1+f) 1/4−M√2F −1/2 (Ψ) =− f′f +a F 1/2(Ψ)m=0∑ Mm=0+ √ 2 f(1+f)1/4−M1+f/ap m f m ,p m f m [1+m−(M − 1 4 ) f1+fM∑p m f[(1+m) m 2 −m=0−(M − 1 4 ) f1+f (3+2m−(M + 3 4 ) f1+f ) ].],(B.3)The parameters are given in Table B for two cases, M = 2 and 3. With the 1 free parameterof the M = 2 scheme (ı.e., a), accuracy is better than 3%. With the 2 free parameters ofthe M = 3 scheme, i.e., a and p 1 , accuracy improves about 100-fold.For EOStable, we have a large rangeof temperature anddensity, so the electron can havefour different regimes such as, non-relativistic non-degenerate, non-relativistic degenerate,relativistic non-degenerate, andrelativistic degenerate. We treat electrons asnon-interactingfermions except for the Coulomb interaction. In this case, we add another parameter g toaccount for relativistic effects. Then the pressure from electrons can be written, aswherep = m (e me) 3 fg 5/3 ∑pmn f m g n ,π 2 (1+f) M (1+g) N−3/2g = T m e√1+f ,(B.4)(B.5)and the electron’s degeneracy parameter is given eq. (B.1). The coefficient p mn is providedin Ref.[12].Among the thermodynamic quantities in EOS table, we may calculate ∂p/∂n, ∂p/∂T, and117
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Theory of nuclear matter of neutron
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Abstract of the DissertationTheory
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To Jaenyeong, Jonghyun, and Jongbum
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3.3 Optimized Parameter Set . . . .
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List of Figures1.1 Mass and radius
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SymbolsNuclear Physics SymbolE ener
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ω δt 90−10QtT cTaylor expansion
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List of Tables1.1 Range of Tables .
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Chapter 1IntroductionStars give us
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Since the mass of neutron star is d
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shown in Fig. 1.2, LS220 is the onl
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density:p = ρ2ρ o[ K9µ n = −B
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nuclear energy density functional i
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2.3.2 Thermodynamic QuantitiesThe t
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point r and with momentum p isU n (
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Thus, in the case of zero temperatu
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For standard nuclear matter, u = 1
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a α L α U β L β U σ0.5882 1.11
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Figure 2.2: Bulk equilibrium of T =
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Figure 2.4: Coexistence curves for
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proton fraction x are irrelevant fo
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potential model analogue for the Ha
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with the convention that z II is th
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Figure 2.10: The surface tension of
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Figure 2.12: Contour plots of surfa
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2.6.2 Dense Matter and the Wigner-S
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Figure 2.15: Proton number per unit
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Chapter 3Modified model† The trun
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We choose the value ǫ = 1/3. One m
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where∆V t = T oρ o[( ρρ o) ǫ[
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Q 1 to a small number also results
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where p = (S v ,L) and M ij = 1 ∂
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3.4 Nuclear Surface TensionCompared
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1.41.2FRTF II ω 0 , T=0 MeVFRTF II
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For the finite-range term, we use a
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Landau’s quasi-particle formula g
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The symmetry energy in nuclear matt
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Figure 4.2: The left figure shows t
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Figure 4.4: This figure shows the b
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Figure 4.5: In a neutron star, heav
Page 81 and 82: The energy exchange from the gradie
Page 83 and 84: Figure 4.8: The left panel shows th
Page 85 and 86: 4.5 Astrophysical application4.5.1
Page 87 and 88: 0.550.5GaussianFRTF IIFRTF I0.450.4
Page 89 and 90: neutron matter respectively. Those
Page 91 and 92: Table 5.3: Nuclear properties of si
Page 93 and 94: which aref Rc = a 0 +a 1 x+a 2 x 2
Page 95 and 96: parametrized[ht]504540EDF, Z nucFit
Page 97 and 98: Chapter 6Nuclear Equation of State
Page 99 and 100: Coulomb, and, symmetry part [24], i
Page 101 and 102: We can eliminate N s , leading to t
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Page 105 and 106: 2.52GSRsSGISIVSkaSkI1SkI2SkI3SkI5St
Page 107 and 108: HNsn, p, α,eFigure 6.7: In the Wig
Page 109 and 110: Table 6.1: Surface tension analytic
Page 111 and 112: free energy density for the alpha p
Page 113 and 114: Here s ′ = ∂s(u)/∂u.The minim
Page 115 and 116: 6.5.2 Determination of Coulomb surf
Page 117 and 118: gives analytic derivatives of therm
Page 119 and 120: 200180160Atomic number, Y p =0.45,
Page 121 and 122: for Fermi integral. This fitting fu
Page 123 and 124: A.1 Taylor expansion integrationThe
Page 125 and 126: thenwhere0u − i+1 = fu− i +J
Page 127 and 128: We separate the two integrals as in
Page 129 and 130: Each w i can be obtained[ae −∆/
Page 131: By expanding e r 0/a −e −r 0/a
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Page 137 and 138: Appendix CPhase coexistenceBulk equ
Page 139 and 140: To summarize, we need to solve 5 eq
Page 141 and 142: D.1 Non-uniform electron density ap
Page 143 and 144: Appendix ENuclear Quantities in Non
Page 145 and 146: Table E.1: Non-relativistic Skyrme
Page 147 and 148: [20] C.J. Pethick and D.G. Ravenhal
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Theory of Nuclear Matter for Neutron Stars and ... - Graduate Physics

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