A pre-main-sequence star of mass contracts quasi-statically down its Hayashi track

PHYSICS · Subjective · 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

A pre-main-sequence star of mass MM contracts quasi-statically down its Hayashi track, generating no nuclear energy, so that L=−12 dΩ/dtL = -\tfrac12\,\mathrm{d}\Omega/\mathrm{d}t. Being fully convective, it is an n=3/2n = 3/2 polytrope, with Ω=−67 GM2/R\Omega = -\tfrac67\,GM^2/R from Chapter 5. (a) On the Hayashi track TeffT_{\rm eff} stays nearly fixed, so L=4πR2σTeff4L = 4\pi R^2\sigma T_{\rm eff}^4 falls as R2R^2. Show that the time taken to contract from a much larger radius down to RR is t≈GM27RL,t \approx \frac{GM^2}{7RL}, where LL is the luminosity at radius RR, and evaluate it for M=1 M⊙M = 1\,M_\odot, R=2 R⊙R = 2\,R_\odot and Teff=4000 KT_{\rm eff} = 4000\ \mathrm{K}. (b) Using the order-of-magnitude estimate Ω∼−GM2/R\Omega \sim -GM^2/R together with hydrostatic equilibrium (Pc∼GM2/R4P_c \sim GM^2/R^4) and the ideal gas law, show that the central temperature obeys Tc∝μM/RT_c \propto \mu M/R (drop all numerical factors of order unity). (c) For M=1 M⊙M = 1\,M_\odot and μ=0.6\mu = 0.6, the star has radius R1=4 R⊙R_1 = 4\,R_\odot at one epoch and R2=2 R⊙R_2 = 2\,R_\odot at a later one. Using the proportionality of part (b) with the constant of proportionality taken as GmH/kBGm_H/k_B (that is, Tc≈GMμmH/(kBR)T_c \approx GM\mu m_H/(k_BR)), evaluate TcT_c at both radii and give the ratio Tc(R2)/Tc(R1)T_c(R_2)/T_c(R_1).

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