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

2. 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

3. 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)

4. 3.2. Blackbody Radiation n rad : (E3.16) P rad : (E3.17) E rad: (E3.18) Fig. 3.1. Planck function

5. 3.3. Ideal Monatomic Gas n: (E3.23, E3.24) P : (E3.27) E : (E3.29) Fig. 3.2. Maxwell-Boltzmann Function

6. 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

7. 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

8. 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

9. 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= ∞

10. 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)

11. 3.7. Mixture of Degenerate and Ideal Gases

12. 3.7.2 Allowing for Chemical Reaction Fig. 3.10. Adiabatic exponent for an ionizing gas

13. 3.7.2 Allowing for Chemical Reaction Fig. 3.11. Temperature Gradient for a ZAMS model Sun Lower gradient, more likely having convection

14. End of Chap. 3 Note: