A carbon-oxygen white dwarf has

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 0.4 M⊙0.4\,M_\odot carbon-oxygen white dwarf has μe=2\mu_e = 2. Work through the following chain of order-of-magnitude and scaling arguments. ((a)) Starting from hydrostatic balance, Pc∼GM2/R4P_c \sim GM^2/R^4, and the non-relativistic degenerate pressure P∼K1(ρ/μe)5/3P \sim K_1(\rho/\mu_e)^{5/3} with ρ∼M/R3\rho \sim M/R^3, show that balancing the two gives R∝M−1/3μe−5/3R \propto M^{-1/3}\mu_e^{-5/3}. Using the calibrated relation 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}, evaluate RR for this star. ((b)) Repeat the balance with the ultra-relativistic pressure P∼K2(ρ/μe)4/3P \sim K_2(\rho/\mu_e)^{4/3} instead, and show that RR cancels from the balance, leaving a mass M∼(K2/Gμe4/3)3/2M \sim (K_2/G\mu_e^{4/3})^{3/2} that does not depend on RR or ρc\rho_c. Explain in one or two sentences why this makes the corresponding mass, MCh=5.83 μe−2 M⊙M_{\rm Ch} = 5.83\,\mu_e^{-2}\,M_\odot, a maximum rather than an ordinary equilibrium value, and evaluate it for μe=2\mu_e = 2. ((c)) The envelope integration gives L∝MTc7/2L \propto MT_c^{7/2}. Combine this with the ionic thermal energy U∝MTcU \propto MT_c to eliminate TcT_c and show tcool∼U/L∝M5/7L−5/7t_{\rm cool} \sim U/L \propto M^{5/7}L^{-5/7}. Using the calibration that a 0.6 M⊙0.6\,M_\odot carbon-oxygen white dwarf at L=10−4 L⊙L = 10^{-4}\,L_\odot has tcool=4.4 Gyrt_{\rm cool} = 4.4\,\mathrm{Gyr}, find tcoolt_{\rm cool} for this star at L=3×10−4 L⊙L = 3\times10^{-4}\,L_\odot.

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