Two shells of a model star lie at (where , ) and (where , )

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 Energy transport and the complete stellar model

Problem

Two shells of a model star lie at r1=0.60 R⊙r_1 = 0.60\,R_\odot (where T1=3.4×106 KT_1 = 3.4\times10^{6}\,\mathrm{K}, P1=4.5×1013 PaP_1 = 4.5\times10^{13}\,\mathrm{Pa}) and r2=0.65 R⊙r_2 = 0.65\,R_\odot (where T2=3.0×106 KT_2 = 3.0\times10^{6}\,\mathrm{K}, P2=3.0×1013 PaP_2 = 3.0\times10^{13}\,\mathrm{Pa}). Someone argues: "The temperature drops steeply between these shells. Taking Δln⁡T/Δln⁡r\Delta \ln T/\Delta \ln r between them gives −1.56-1.56; in size that is nearly four times the critical value ∇ad=0.4\nabla_{\rm ad} = 0.4, so this layer must be convectively unstable." ((a)) Find the error in this reasoning. ((b)) Compute the correct local gradient ∇=dln⁡T/dln⁡P\nabla = \mathrm{d}\ln T/\mathrm{d}\ln P between the two shells (using the finite differences given) and decide whether the layer is in fact stable or unstable against convection.

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