An isothermal helium core of mass

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 Stellar evolution: from the Hayashi track to the iron core, and the ages of clusters

Problem

An isothermal helium core of mass McM_c, temperature TcT_c and mean molecular weight μc\mu_c sits under an envelope that exerts a surface pressure PsP_s on it, so that the core's own virial theorem reads 4πRc3Ps=3MckBTcμcmH−35GMc2Rc.4\pi R_c^3 P_s = \frac{3M_ck_BT_c}{\mu_cm_H} - \frac{3}{5}\frac{GM_c^2}{R_c}. ((a)) Find the radius RcR_c at which Ps(Rc)P_s(R_c) is a maximum, and show that this maximum scales as Ps,max⁡∝Tc4μc4Mc2.P_{s,\max} \propto \frac{T_c^4}{\mu_c^4M_c^2}. ((b)) By comparing Ps,max⁡P_{s,\max} with the pressure the envelope exerts (estimated by homology), the fraction of the star's mass the isothermal core can support is qSC≡McM≈0.37(μenvμc)2.q_{\rm SC} \equiv \frac{M_c}{M} \approx 0.37\left(\frac{\mu_{\rm env}}{\mu_c}\right)^2. A star has already processed some of its envelope by rotational or convective mixing so that μenv=0.65\mu_{\rm env} = 0.65 (compared with the unmixed solar value of 0.620.62), while the core has the usual fully ionised helium value μc=1.33\mu_c = 1.33. Evaluate qSCq_{\rm SC} for this star, and state whether the Schonberg-Chandrasekhar limit is reached earlier or later in the star's evolution than for a star with the unmixed envelope value.

Browse the problem library