A fully convective star of mass and radius is modelled as an polytrope

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 Pressure against gravity: the equation of state, polytropes and the Lane-Emden equation

Problem

A fully convective star of mass M=0.4 M⊙M = 0.4\,M_\odot and radius R=0.4 R⊙R = 0.4\,R_\odot is modelled as an n=3/2n=3/2 polytrope, for which the Lane-Emden constants are ξ1=3.6538\xi_1 = 3.6538, −ξ12θ1′=2.714-\xi_1^2\theta_1' = 2.714 and ρc/ρˉ=5.991\rho_c/\bar\rho = 5.991. Take the mean molecular weight to be μ=0.6\mu = 0.6. ((a)) Compute the central pressure Pc=WnGM2/R4P_c = W_n GM^2/R^4, with Wn=[4π(n+1)(θ1′)2]−1W_n = [4\pi(n+1)(\theta_1')^2]^{-1}. ((b)) Compute the mean density ρˉ\bar\rho and the central density ρc\rho_c. ((c)) Compute the central temperature Tc=μmHPc/(kBρc)T_c = \mu m_H P_c/(k_B\rho_c).

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