A white dwarf has a core of helium ( , ) instead of carbon ( , )

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Astrophysics & Cosmology · Stellar death and compact objects

Stars from the Inside, Ch White dwarfs: the relativistic polytrope, the Chandrasekhar mass and a cooling clock

Problem

A 0.5 M⊙0.5\,M_\odot white dwarf has a core of helium (Z=2Z = 2, A=4A = 4) instead of carbon (Z=6Z = 6, A=12A = 12). Fully ionised helium, like carbon, has one electron per two nucleons, so μe=2\mu_e = 2 for both. Composition enters the chapter's laws in different ways. ((a)) Using R≈1.05×104 km (M/0.6 M⊙)−1/3(μe/2)−5/3R \approx 1.05\times10^4\,\mathrm{km}\,(M/0.6\,M_\odot)^{-1/3}(\mu_e/2)^{-5/3}, find the radius of this star, and say whether a carbon white dwarf of the same mass would be larger, smaller or the same. ((b)) Using Γ=Z2e2/4πϵ0akBT\Gamma = Z^2e^2/4\pi\epsilon_0ak_BT with a=(3/4πni)1/3a = (3/4\pi n_i)^{1/3}, find the ratio ΓHe/ΓC\Gamma_{\rm He}/\Gamma_{\rm C} for helium and carbon at the same mass density and temperature. What does it imply for the temperature at which a helium core crystallises, compared with a carbon core of the same density? ((c)) Mestel's numerical law, tcool≈8.8×106 yr (A/12)−1(M/M⊙)5/7(L/L⊙)−5/7t_{\rm cool} \approx 8.8\times10^{6}\,\mathrm{yr}\,(A/12)^{-1}(M/M_\odot)^{5/7}(L/L_\odot)^{-5/7}, gives 4.4 Gyr4.4\,\mathrm{Gyr} for a 0.6 M⊙0.6\,M_\odot carbon white dwarf at 10−4 L⊙10^{-4}\,L_\odot. Find the cooling age of this helium white dwarf at L=10−2 L⊙L = 10^{-2}\,L_\odot, and explain physically where the factor carrying AA comes from.

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