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

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to explain the existence of quark matter but it’s still an open question. MIT bag modeldescribes the quarks are freely moving in a bag. If the distance between quarks is increases,the force increases so strong that they cannot get out of the bag. To maintain the bag,negative pressure is introduced, which is known as the bag constant (B). Since we cannot doany experiment to reveal the existence of quark matter at such high densities on the earthyet, the theoretical prediction of quark matter depends on the parameters, especially thebag constant in quark matter.One thing is for sure is that the densities of neutron stars are extremely high. It is believedthat the central density of the core in neutron stars is 4ρ 0 ∼ 10ρ 0 † , where a ρ 0 is the nuclearsaturation density, 3×10 14 g/cm 3 .The typical radius of neutron star is around 10km. The mass of neutron is about 1.2M ⊙ to2.0 M ⊙ . Since neutron stars are so compact, the surface gravity of neutron stars is extremelyhigh, about 2×10 11 times earth gravity. In these conditions, general relativity is essential tofind out the mass-radius relation of neutron stars. The Tolman-Oppenhimer-Volkov (TOV)equation solves the internal structure of spherically symmetric neutron stars. Thus, thenuclear physics and the TOV equation play together to give mass-relation of neutron stars.The nuclear physics beyond ρ 0 is not known well and the central density is much higher thanρ 0 , various nuclear force models give different mass-radius relation of neutron stars. Fig.Figure 1.1: Mass and radius relation of cold neutron stars [19]. Each nuclear force modelshows different mass-radius relation of neutron stars.1.1 shows the mass-radius relation of cold neutron stars using various nuclear force models.† ρ 0 ≃ 0.16fm −3 . ρ 0 denotes nuclear saturation density which is central density of heavy nuclei. But, ingeneral, it is a nuclear parameter to be determined from standard nuclear matter properties.2
Since the mass of neutron star is dominated by its core and the density of the core is muchbeyond the nuclear saturation density, the nuclear force models should be investigated morethan the current level of understanding.In contrast to neutron stars, supernovae explosions and neutron star mergers result in extremelyhigh temperature. Thus, we need to know thermodynamic properties of nuclearmatter for a wide range of densities (10 −9 ∼ 1.6/fm 3 ) , temperature (0 ∼ 30 MeV), andproton compositions (0 ∼ 0.56). This thesis concerns nuclear physics under such extremeconditions, which are far beyond those achievable by nuclear physics experiments on earth.Hence, a theoretical extrapolation is needed for high density, high temperature, and low protonfraction. For zero temperature, the extrapolation of nuclear properties at high densitydensity ( 10ρ 0 ) can be checked by the mass-radius relation of neutron stars [53]. Unfortunately,the extrapolation to high temperature cannot be checked with anything fromexperiment or astrophysical observations at this time.The nuclear thermodynamic information is called ‘Nuclear Equation Of State’ (EOS) andprovided as a tabulated form because of memory constraints. The table should contain freeenergy density, pressure, entropy as a function of baryon number density, proton fraction,and temperature.For the simulations of supernovae explosions, only a few EOS tables available now. The mostfamous one is Lattimer & Swesty [22] (LS) EOS in which they combined non-relativistic potentialmodel with liquid droplet approach. In their EOS they considered phase transitionsfrom three dimensional nuclei to three dimensional bubble. It is difficult in their code toarbitrarily vary nuclear parameters.H. Shen et al. [59] (STOS) built a table using relativistic mean field model (RMF) andThomas Fermi approximation. To performthe Thomas Fermi approximation, they employedthe parametrized density profile method, in which density profile follows a mathematicalpolynomial (see chapter 5) so to avoid the numerical difficulty of differential equations. Anew version [60] is available now and it contains hyperon interactions.G. Shen et al. [61] (SHT) provided a few version of tables using RMF parameter sets. Theyemployed the Hartree approximation to find the nuclear density profile. It is difficult andvery time consuming to develop another table using their code. Table 1.1 show the range ofTable 1.1: Range of TablesLS 220 STOS SHTρ(fm −3 ) 10 −6 ∼ 1 (121) 7.58×10 −11 ∼ 6.022 (110) 10 −8 ∼ 1.496 (328)Y p 0.01 ∼ 0.5 (50) 0 ∼ 0.65 (66) 0 ∼ 0.56 (57)T (MeV) 0.3 ∼ 30 (50) 0.1 ∼ 398.1 (90) 0 ∼ 75.0 (109)the independent variables and the number of grid points (number in the parenthesis). STOStables deals wide range of densities and temperature. The number of grid points, however,is small so that the interpolations from that table might not be suitable for the simulation,which needs high thermodynamic consistencies. That is, except for the LS EOS, the pressureand entropy densities from STOS and SHT are obtained from the numerical derivatives(P = −ρ 2∆F,S = ∆ρ −∆F ). But this numerical formalism might not give enough accuracy for∆T3
Page 1 and 2: Theory of nuclear matter of neutron
Page 3 and 4: Abstract of the DissertationTheory
Page 5 and 6: To Jaenyeong, Jonghyun, and Jongbum
Page 7 and 8: 3.3 Optimized Parameter Set . . . .
Page 9 and 10: List of Figures1.1 Mass and radius
Page 11 and 12: SymbolsNuclear Physics SymbolE ener
Page 13 and 14: ω δt 90−10QtT cTaylor expansion
Page 15 and 16: List of Tables1.1 Range of Tables .
Page 17: Chapter 1IntroductionStars give us
Page 21 and 22: shown in Fig. 1.2, LS220 is the onl
Page 23 and 24: density:p = ρ2ρ o[ K9µ n = −B
Page 25 and 26: nuclear energy density functional i
Page 27 and 28: 2.3.2 Thermodynamic QuantitiesThe t
Page 29 and 30: point r and with momentum p isU n (
Page 31 and 32: Thus, in the case of zero temperatu
Page 33 and 34: For standard nuclear matter, u = 1
Page 35 and 36: a α L α U β L β U σ0.5882 1.11
Page 37 and 38: Figure 2.2: Bulk equilibrium of T =
Page 39 and 40: Figure 2.4: Coexistence curves for
Page 41 and 42: proton fraction x are irrelevant fo
Page 43 and 44: potential model analogue for the Ha
Page 45 and 46: with the convention that z II is th
Page 47 and 48: Figure 2.10: The surface tension of
Page 49 and 50: Figure 2.12: Contour plots of surfa
Page 51 and 52: 2.6.2 Dense Matter and the Wigner-S
Page 53 and 54: Figure 2.15: Proton number per unit
Page 55 and 56: Chapter 3Modified model† The trun
Page 57 and 58: We choose the value ǫ = 1/3. One m
Page 59 and 60: where∆V t = T oρ o[( ρρ o) ǫ[
Page 61 and 62: Q 1 to a small number also results
Page 63 and 64: where p = (S v ,L) and M ij = 1 ∂
Page 65 and 66: 3.4 Nuclear Surface TensionCompared
Page 67 and 68: 1.41.2FRTF II ω 0 , T=0 MeVFRTF II
Page 69 and 70:
For the finite-range term, we use a
Page 71 and 72:
Landau’s quasi-particle formula g
Page 73 and 74:
The symmetry energy in nuclear matt
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Figure 4.2: The left figure shows t
Page 77 and 78:
Figure 4.4: This figure shows the b
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Figure 4.5: In a neutron star, heav
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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
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0.550.5GaussianFRTF IIFRTF I0.450.4
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neutron matter respectively. Those
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Table 5.3: Nuclear properties of si
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which aref Rc = a 0 +a 1 x+a 2 x 2
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parametrized[ht]504540EDF, Z nucFit
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Chapter 6Nuclear Equation of State
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Coulomb, and, symmetry part [24], i
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We can eliminate N s , leading to t
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pressure from pure neutron matter,
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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
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free energy density for the alpha p
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Here s ′ = ∂s(u)/∂u.The minim
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6.5.2 Determination of Coulomb surf
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gives analytic derivatives of therm
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200180160Atomic number, Y p =0.45,
Page 121 and 122:
for Fermi integral. This fitting fu
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A.1 Taylor expansion integrationThe
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thenwhere0u − i+1 = fu− i +J
Page 127 and 128:
We separate the two integrals as in
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Each w i can be obtained[ae −∆/
Page 131 and 132:
By expanding e r 0/a −e −r 0/a
Page 133 and 134:
M 3 2a 0.433 1p 0 (e 2 /a) √ π/3
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of find exact polynomial expression
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Appendix CPhase coexistenceBulk equ
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To summarize, we need to solve 5 eq
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D.1 Non-uniform electron density ap
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Appendix ENuclear Quantities in Non
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Table E.1: Non-relativistic Skyrme
Page 147 and 148:
[20] C.J. Pethick and D.G. Ravenhal
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