How it Works
The simulation integrates the Mestel cooling equation forward in time. The white dwarf interior is modelled as an isothermal ion plasma that slowly loses energy through a thin non-degenerate envelope. The key physics is that Kramers opacity controls the energy leak rate, leading to the luminosity-temperature relation L ∝ T_c^(7/2).
The cooling rate dT/dt is obtained by equating the energy loss L to the rate of change of thermal energy in the ion liquid: E_th ∝ M·k_B·T_c/m_ion. Crystallization latent heat Q_lat is deposited when the Coulomb parameter Γ = 175.
Frequently Asked Questions
What is Mestel cooling theory?
Mestel cooling describes how a white dwarf cools by treating its interior as a nearly-isothermal ion plasma insulated by a thin non-degenerate envelope. The luminosity scales as L ∝ T^(7/2) giving L(t) ∝ t^(-7/5).
Why does L ∝ T^(7/2) in Mestel's model?
In the thin envelope approximation, opacity is dominated by Kramers free-free opacity κ ∝ ρT^(-7/2). This controls how fast energy leaks out, leading to L ∝ T_c^(7/2) where T_c is the core temperature.
What happens when a white dwarf crystallizes?
As the white dwarf cools below a critical temperature (Γ ≈ 175), the ion plasma solidifies into a crystal lattice. This releases latent heat, temporarily slowing the cooling and creating a pile-up of white dwarfs at intermediate luminosities.
What is a white dwarf made of?
Most white dwarfs have a carbon-oxygen core surrounded by a thin helium layer and an even thinner hydrogen atmosphere. The interior is electron-degenerate — the electrons provide pressure independent of temperature.
How long does white dwarf cooling take?
Typical white dwarfs cool to non-detectability (T < 4000 K) in roughly 10–15 billion years. The oldest white dwarfs in the Milky Way set a lower bound on the age of the Galaxy.
What is the Chandrasekhar mass limit?
The Chandrasekhar limit (~1.4 M☉) is the maximum mass a white dwarf can have while supported by electron degeneracy pressure. Above this mass, the star collapses — triggering a Type Ia supernova or neutron star formation.
How is a cooling track plotted on an HR diagram?
A white dwarf cooling track runs from upper left (hot, high luminosity) to lower right (cool, low luminosity). The effective temperature drops from ~100,000 K to below 5,000 K over billions of years.
What is the Coulomb coupling parameter Γ?
Γ = Z²e²/(a·k_B·T) is the ratio of Coulomb potential energy to thermal energy. When Γ < 1 the ions behave as a gas; when Γ ≈ 175 they crystallize into a body-centered cubic lattice.
Do all white dwarfs cool the same way?
No. More massive white dwarfs have higher surface gravity and thinner envelopes, so they cool faster. Hydrogen-rich (DA) and helium-rich (DB) atmosphere white dwarfs also cool at different rates.
Can we use white dwarf cooling as a cosmic clock?
Yes! White dwarf cosmochronology uses the luminosity function to infer the age of stellar populations. The cut-off at low luminosities directly measures the age of the oldest population, giving stellar disk ages of ~8–9 Gyr.