(a) A fluid element is displaced upward by inside a star and equilibrates in pressure…

PHYSICS · Subjective · Fermi University Physics · Astrophysics & Cosmology · Stellar structure, energy generation and evolution · Stellar structure, energy generation and evolution

Stars from the Inside, Ch Energy transport and the complete stellar model

Problem

(a) A fluid element is displaced upward by Δr>0\Delta r > 0 inside a star and equilibrates in pressure with its surroundings faster than it exchanges heat, so it evolves adiabatically. Its fractional density excess over the surroundings after the displacement is Dρρ=[δ(∇ad−∇)+φ∇μ]ΔrHP,\frac{D\rho}{\rho} = \Big[\delta(\nabla_{\rm ad} - \nabla) + \varphi\nabla_\mu\Big]\frac{\Delta r}{H_P}, where δ≡−(∂ln⁡ρ/∂ln⁡T)P>0\delta \equiv -(\partial\ln\rho/\partial\ln T)_P > 0, φ≡(∂ln⁡ρ/∂ln⁡μ)P,T>0\varphi \equiv (\partial\ln\rho/\partial\ln\mu)_{P,T} > 0, ∇\nabla is the local (surrounding) logarithmic temperature gradient, and ∇μ≡dln⁡μ/dln⁡P\nabla_\mu \equiv \mathrm{d}\ln\mu/\mathrm{d}\ln P. A displaced element that ends up denser than its surroundings (Dρ>0D\rho > 0) sinks back and the layer is stable. Derive the Ledoux criterion for convective stability, and show that it reduces to the Schwarzschild criterion when the composition is uniform. (b) Consider a family of stars supported by an ideal gas of mean molecular weight μ\mu, in hydrostatic and thermal equilibrium, radiative throughout so ∇=∇rad\nabla = \nabla_{\rm rad} everywhere, whose opacity follows a general power law κˉ=κ0ρaTb\bar\kappa = \kappa_0\rho^aT^b with constant exponents aa and bb. Using the homology assumption that all stars of this family have structurally similar ρ(r/R)\rho(r/R), T(r/R)T(r/R) profiles, so that central quantities scale as ρc∝M/R3\rho_c \propto M/R^3, Pc∝GM2/R4P_c \propto GM^2/R^4 and Tc∝μPc/ρcT_c \propto \mu P_c/\rho_c, find how LL scales with μ\mu, MM, RR and κ0\kappa_0. Check that your result reproduces both homology relations of the text, and find the condition on aa and bb under which RR drops out.

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