In a stellar layer the Rosseland mean opacity follows with and constant across the layer

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Stellar structure, energy generation and evolution · Stellar structure, energy generation and evolution

Stars from the Inside, Ch Pulsating stars: the kappa mechanism, the Leavitt law and asteroseismology

Problem

In a stellar layer the Rosseland mean opacity follows κ∝ρnT−s\kappa\propto\rho^n T^{-s} with nn and ss constant across the layer. A fluid element there is compressed adiabatically. ((a)) Using the definition Γ3−1≡(∂ln⁡T/∂ln⁡ρ)s\Gamma_3-1\equiv(\partial\ln T/\partial\ln\rho)_s, show that the fractional opacity change obeys δln⁡κ=[n−s(Γ3−1)] δln⁡ρ\delta\ln\kappa=[n-s(\Gamma_3-1)]\,\delta\ln\rho. ((b)) A partial-ionisation layer has locally κ∝ρ0.5T−2.0\kappa\propto\rho^{0.5}T^{-2.0} and Γ3−1=0.15\Gamma_3-1=0.15 there. Evaluate the bracket and state whether κ\kappa rises or falls when the layer is compressed. ((c)) The kappa mechanism requires opacity to rise on compression, i.e. the bracket in part (a) to be positive, equivalently Γ3−1<n/s\Gamma_3-1<n/s. Give the threshold n/sn/s for this layer, compare it with the ordinary-gas value Γ3−1≈2/3\Gamma_3-1\approx2/3 quoted for a fully ionised deep interior, and explain in one sentence, using the Saha equation, why partial ionisation lowers Γ3−1\Gamma_3-1 enough to reverse the sign of the bracket.

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