Convective cores in stellar models

Abstract

The condition for homogeneous radiative stellar models to be marginally stable to convection at the centre is investigated for the family of models where the opacity κ and energy generation ɛ are given by power laws in temperature and density κ = K₀ ρ_α T^−β, ɛ = ɛ₀ ρ T^η. The Naur-Osterbrock (1953) condition 6η > 6 + 10β - 15α is a necessary but not sufficient condition. A better estimate is obtained by taking the effective polytropic index n = dlogP/dlogT - 1 to be a linear function of temperature T throughout the model. This gives the condition

$$6\eta = 10\beta - 15\alpha + \frac{{12 + 4\beta }}{{1 + \alpha }}$$

The predictions of this condition agree well with results for a set of stellar models 0 ≤ α ≤ 1, 0 ≤ β ≤ 5.