Unformatted text preview: cP show that:
ad 1 + (4 + β )(1 − β )/β 2
5/2 + 4(4 + β )(1 − β )/β 2 Plot ∇ad as a function of β and determine the limits for β = 0 and 1.
(e) [10 pt] Based on part (d), it appears that a star supported by radiation
pressure might nevertheless transport heat through convection (because of
the reduced adiabatic gradient). Using our previously derived expression for
∇rad and assuming the opacity is dominated by Thompson (free-free)
scattering, κ ≈ 0.35 cm2/g, in what mass/luminosity regime would this be the
case? Explain the relevance of this constraint to the maximum mass a star
can have (the Eddington limit).
(3) Adiabatic gradient for a partially ionized gas. We’ll now repeat the
above analysis for a partially ionized ideal gas, to show how ionization can
influence thermal transport. Consider a pure H gas, with pressure P= ρkT
µmH and ionization fraction n−
1 − n0
where n+, n-, n0 and n are the number densities of free protons, free
electrons, neutral H atoms and total particles, respectively; and n+ = n-.
(a) [10 pt] First compute the mean molecular weight, and show that: 1
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