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CSI661/ASTR530 Spring, 2009 Chap. 3 Equations of State Feb. 25, 2009 Jie Zhang Copyright ©

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Outline Distribution Functions Blackbody Radiation Ideal Monatomic Gas Saha Equations Fermi-Dirac Equations of State Complete Degenerate Gas Application to White Dwarf Effect of Temperature Thermal Dynamic Derivatives -- Adiabatic Exponents Mixtures of Ideal Gases and Radiations Mixtures of Degenerate and Ideal Gases Allowing for Chemical Reactions

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3.1. Distribution Function Distribution Function n(p): (E3.9) State parameter n: (E3.10) State parameter P: (E3.13) State parameter E: (E3.14)

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3.2. Blackbody Radiation n rad : (E3.16) P rad : (E3.17) E rad: (E3.18) Fig. 3.1. Planck function

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3.3. Ideal Monatomic Gas n: (E3.23, E3.24) P : (E3.27) E : (E3.29) Fig. 3.2. Maxwell-Boltzmann Function

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3.4. Saha Equation Saha equation: (E3.35), (E3.39) Ionization zone: ionization sensitive to temperature because of exponential dependence Fig. 3.3. Half-ionization curve

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3.5. Fermi-Dirac Equation Fermi energy: Eq (3.48) Dimensionless Fermi momentum n: (E3.49) P: (E3.53) E: (E3.56) Non-relativistic limit: x<<1 Extreme relativistic limit: x>>1 γ-law equation of state Fig. 3.5. Fermi-Dirac Function

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3.5. Fermi-Dirac Equation White dwarf Mass-radius relation: (E.3.62), (E3.63) mass increases, radius decreases Chandrasekhar limit In the case of extremely relativistic limit Demarcation of degeneracy and non-degeneracy: (E3.70) Fig. 3.7. Degenerate Dependence

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3.7. Adiabatic Exponents Specific heat Cv: (E3.85) Specific heat Cp: (E3.86) χ T : (E3.88) χ ρ : (E3.89) γ: ratio of specific heats: (E3.92) First adiabatic exponent Γ 1 : (E3.93) Second adiabatic exponent Γ 2 : (E3.94) Third adiabatic exponent Γ 3 : (E3.95) Ideal gas: Γ 1 =Γ 2 =Γ 3 = γ=5/3 Radiation “gas”: Γ 1 =Γ 2 =Γ 3 = 4/3, and γ= ∞

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3.7. Mixture of Ideal Gases and Radiation P=Pg+Prad: (E3.104) E=Eg+Erad: (E3.105) Gas β: (E3.106) Cv: (E3.108) Adiabatic exponents: (E3.109), (E3.110), (E3.111), and (E3.112)

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3.7. Mixture of Degenerate and Ideal Gases

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3.7.2 Allowing for Chemical Reaction Fig. 3.10. Adiabatic exponent for an ionizing gas

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3.7.2 Allowing for Chemical Reaction Fig. 3.11. Temperature Gradient for a ZAMS model Sun Lower gradient, more likely having convection

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End of Chap. 3 Note:

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