An old open cluster contains a white dwarf member with spectroscopically measured mass

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Stellar structure, energy generation and evolution · Stellar death and compact objects

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

Problem

An old open cluster contains a white dwarf member with spectroscopically measured mass M=1.05 M⊙M = 1.05\,M_\odot, radius R=0.0072 R⊙R = 0.0072\,R_\odot and bolometric luminosity L=1.5×10−3 L⊙L = 1.5\times10^{-3}\,L_\odot; the composition is carbon-oxygen with μe=2\mu_e = 2. Mestel's law puts a cooling white dwarf of 0.6 M⊙0.6\,M_\odot at 10−4 L⊙10^{-4}\,L_\odot at a cooling age of 4.4 Gyr4.4\,\mathrm{Gyr}, and cooling age scales as tcool∝M5/7L−5/7t_{\rm cool} \propto M^{5/7}L^{-5/7} at fixed composition. Isochrone fitting to the cluster's main-sequence turnoff (Chapter 9's method) gives a cluster age of 4.0 Gyr4.0\,\mathrm{Gyr}. Take a star's pre-white-dwarf lifetime as roughly Chapter 9's main-sequence scaling, t≈10 Gyr (M/M⊙)−2.5t \approx 10\,\mathrm{Gyr}\,(M/M_\odot)^{-2.5}. ((a)) Compute the Chandrasekhar mass for μe=2\mu_e = 2 and confirm that this white dwarf's mass is a permitted equilibrium. ((b)) Predict the white dwarf's gravitational redshift czcz. ((c)) Estimate its Mestel cooling age. ((d)) If the white dwarf's progenitor and the rest of the cluster formed together, the sum of the progenitor's pre-white-dwarf lifetime and the white dwarf's cooling age should equal the cluster age. Find the implied pre-white-dwarf lifetime. Then use the chapter's initial-final masses (progenitors of 11 to 8 M⊙8\,M_\odot leave cores of about 0.550.55 to 1.2 M⊙1.2\,M_\odot, a 5 M⊙5\,M_\odot star about 0.9 M⊙0.9\,M_\odot) to estimate this white dwarf's progenitor mass and how long that progenitor lived. Is the white dwarf consistent with a single star that formed with the cluster?

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