Homework 2

339 1 1 y 1 y 2 5 h 2 kt where h is the ionization

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Unformatted text preview: +y = + + = µ µ− µ+ µ0 µ0 (b) [15 pt] Show that, for constant µ0 the density thermal gradient δ is: δ =1+ 1 1+y ∂y ∂ ln T P and by, differentiating the Saha equation (HKT Eqn. 3.39), 1 δ = 1 + y (1 − y ) 2 5 χH + 2 kT where χH is the ionization potential of the H atom. (c) [15 pt] Explain why one can write the internal energy per unit mass as: u= ¯ 3kT χH (1 + y ) + y 2µo mH µo mH and from this derive the specific heat at constant pressure: cP = 5k 1 (1 + y ) 1 + y (1 − y )Φ2 H 2µo mH 5 where ΦH = 5 χH + 2 kT (d) [10 pt] Show that the adiabatic gradient in this case is then: ad = 2 + y (1 − y )ΦH 5 + y (1 − y )Φ2 H (e) [10 pt] Assuming a mass density ρ = 0.01 g/cm3 compute and plot y (using the Saha equation) and ∇ad as a function of log T for 3000 < T < 106 K. Explore this by answering the following questions: (i) What is the minimum ∇ad? At what temperature/ionization fraction does this occur? (ii) What happens if the density increases or decreases by a factor of 10? How does that change the magnitude and temperature for the minimum ∇ad? Qualitatively explain these trends. (iii) What happens if the ionization potential increases or decreases by a factor of 2? How does that change the magnitude and temperature for the minimum ∇ad? Qualitatively explain these trends....
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