White dwarf cooling - 2D HR diagram

Mestel cooling

The degenerate core is an almost isothermal reservoir of ions; the thin non-degenerate envelope controls the leak. With Kramers opacity the luminosity is L = C M Tc7/2, calibrated here so a 0.6 M_sun dwarf with T_c = 10⁷ K has L = 0.01 L_sun. The heat content is the ion Dulong-Petit value Cv = 3 N kB with N = M/(A m_u) ions, so dTc/dt = -L / Cv = -a Tc7/2. This integrates exactly: Tc(t) = (Tc0-5/2 + 5at/2)-2/5, hence at late times L ∝ t-7/5 and the age to reach a luminosity scales as t ∝ M5/7 L-5/7 (L ∝ M and C_v ∝ M make the coefficient a independent of mass).

Effective temperature and radius

T_eff follows from L = 4πR2σTeff4 with the degenerate mass-radius relation R ∝ M-1/3, anchored at R = 0.0127 R_sun for 0.6 M_sun. Heavier dwarfs are smaller and denser.

Crystallization

The Coulomb coupling is Γ = Z2e2 / (4πε0 a kB T) with ion sphere radius a = (3/(4πn))^1/3. The ion plasma freezes at Γ ≈ 175. Denser (heavier) dwarfs and higher Z cores reach it at higher T, so oxygen-neon cores freeze early and helium cores essentially never. The optional latent heat adds a pause of Δt = 0.77 N kB Tx / L(Tx) during which T_c stays at T_x.

Checks

The closed-form core temperature is compared with an independent adaptive RK4 integration of the ODE; the late-time slope of log L against log t is measured from that RK4 run and compared with -7/5; the mass scaling of the cooling age is compared with 2^5/7.

Honest simplifications: a single isothermal core with constant C_v; no neutrino losses (they matter above about 10⁵ K), no gravitational settling, no Debye cooling at the end, no envelope-opacity differences between hydrogen and helium atmospheres, no mass-dependent envelope mass. The latent-heat pause is a simple estimate and is longer than detailed models give (real delays are around 1 Gyr for a 0.6 M_sun carbon-oxygen dwarf). Use the tracks for scaling and intuition, not for age dating.

• Press the Light 0.4 Msun and Massive 1.0 Msun presets: the heavy dwarf is dimmer in radius but hotter for longer and freezes sooner.
• Switch the core to ONe, then He, and watch the blue crystallization line move to higher and lower luminosity.
• Move the reference luminosity and see the age in the right graph follow the dotted M^5/7 law.
• Tick the latent heat box: the star hovers at the freezing point before continuing down the track.