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(b) [15 pt] Show that, for constant µ0 the density thermal gradient δ is: δ =1+ 1
∂ ln T P and by, differentiating the Saha equation (HKT Eqn. 3.39), 1
δ = 1 + y (1 − y )
2 5 χH
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=
(1 + y ) + y
µo mH and from this derive the specific heat at constant pressure: cP = 5k
(1 + y ) 1 + y (1 − y )Φ2
5 where ΦH = 5 χH
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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