AC31 · Heat inside a stellar remnant

White Dwarf Cooling and Crystallization

How does freezing slow the cooling of a white dwarf? A white dwarf shines by losing stored heat. Freezing releases latent heat, and changes in composition can release more energy. That extra supply slows cooling, so brightness gives an age only after the star’s mass, atmosphere and interior physics are specified.

Drag a probe through a partially crystallized core. Watch a shell release latent heat, then compare a published cooling plateau with the same-mass reference.

Core temperature $2.35\times10^{6}\,\mathrm K$

Crystallized core mass $40\%$

Model luminosity $1.87\times10^{-5}\,L_\odot$

Uncalibrated model elapsed time $9.2\,\mathrm{Gyr}$

AC31

Follow the crystal front

Uncalibrated model elapsed time. This is not a stellar age.

Move through this demo

0.00\%

Radial probe

0.630

r/R_{\rm core}

Drag the gold probe through the core. Blue marks solid material; warm colours mark liquid.

Native cases, clocks and model limits

Select a discrete case

The controlled core uses a fixed native density geometry and a new isothermal pure-carbon thermal state. Its clock starts independently. The envelope law is a teaching boundary, and the resulting very long times are not calibrated cooling ages.

The three published surface sequences share one solar mass and the standard 30PR30 composition. Their plateau was smoothed by the authors over about one gigayear. This is not the thin-helium family in Figure 4; no internal phase profile was supplied.

Native density geometries: Bauer (2023), CC BY 4.0 Native cooling tables: Bédard, Blouin and Cheng (2024), CC BY 4.0 Reload with the SVG fallback Recompute the original-data comparison

Model assumptions

The controlled view evolves an isothermal pure-carbon OCP core on four discrete native MESA density geometries, with Debye and degenerate-electron heat capacities, explicit latent heat and a declared Kramers-envelope luminosity. Its independent model clock is uncalibrated and gives very long ages; it is not a stellar age estimate. The displayed phase colours are schematic, and core temperature is not photospheric effective temperature. It omits structural relaxation, a solved atmosphere, residual burning, neutrinos, C/O separation and neon transport. The separate published STELUM view has three 1.00-solar-mass standard-composition cases, with about one-gigayear author smoothing; only supplied surface quantities are shown. It has no radial front or solid-fraction data. No mass/composition interpolation or Galactic population-age inference is provided.

Selected independent clock

A crystal core grows

The section fits the selected core to the view; its actual outer radius is $7.19\times10^{8}\,\mathrm{cm}$

Source total mass $0.608\,M_\odot$

Probe materialLiquid

\rho=9.89\times10^{5}\,\mathrm{g\,cm^{-3}}

\Gamma=151

Core temperature along the model clock

T/(10^6\,\mathrm K)

t/(10^9\,\mathrm{yr})

Scaled core temperature; the gold line marks the selected time in this demo interval.

Stored heat and latent heat leave as light

E/(10^{46}\,\mathrm{erg})

t/(10^9\,\mathrm{yr})

Scaled energy. Green: sensible heat lost. Blue: latent heat released. Gold: photons emitted.

Latent heat released $6.61\times10^{45}\,\mathrm{erg}$

Photon energy emitted $4.46\times10^{46}\,\mathrm{erg}$

Sensible heat capacity

Ionic contribution $2.34\times10^{40}\,\mathrm{erg\,K^{-1}}$

Electron contribution $4.1\times10^{38}\,\mathrm{erg\,K^{-1}}$

These are sensible-energy derivatives; latent release is counted separately. At an exact cell melting temperature, the stored derivative uses the solid-side branch.

Start with the essentials

Focus question: Why does the core temperature pause while the star still radiates?
One-sentence intuition: During freezing, released latent energy can supply the outgoing luminosity without the same drop in sensible energy.

Core mathematical model

The controlled thermal energy

E(T,\{f_i\})=U_{\rm sens}(T)+\sum_i(1-f_i)q_i

Each $f_i$ is the solid fraction of one mass shell; $q_i$ is its declared latent reservoir. The sensible branch contains ionic and electron heat capacities.

Heat leaving the model

\frac{dE}{dt}=-L(T),\qquad L\propto T^{7/2}

The luminosity comes from the stated Kramers-envelope approximation. This is a model core temperature, not the photospheric effective temperature.

A shell freezes at its own temperature

\Gamma_i=\frac{Z^2e^2}{a_i k_BT},\qquad a_i=\left(\frac3{4\pi n_i}\right)^{1/3}

The pure-carbon one-component model freezes at the declared coupling threshold. Denser shells have higher freezing temperatures.

Common difficulties

Support versus heat

Typical misconceptionDegenerate pressure means the electrons supply all the thermal energy.

Better mental modelPressure support and heat capacity answer different questions. The controlled calculation includes ion and electron thermal terms separately.

Radius versus mass

Typical misconceptionA front halfway out contains half the core mass.

Better mental modelMass also depends on the radial density profile. The displayed solid mass is integrated over shells, not inferred from the coloured radius.

One clock for all views

Typical misconceptionThe long controlled-model time can be appended to a published cooling age.

Better mental modelThe models use different physics and origins. The visible uncalibrated elapsed time is not a stellar-age measurement, and the native sequences retain their own clocks.

Run the experiment

01
Cross the phase boundary
Drag the gold probe from the warm liquid into the blue solid.
What to observe: The phase, density and temperature all belong to the selected radial shell and the same saved time.
02
Catch a shell releasing latent heat
Choose Latent plateau, then compare temperature, released latent energy and radiated energy.
What to observe: The shell can continue supplying light while its temperature remains fixed during the phase change.
03
Visit the cold tail
Choose Debye tail, open Sensible heat capacity and compare the ionic and electron terms.
What to observe: The solid ionic heat capacity falls at low temperature. The electron term is computed separately rather than silently discarded.
04
Change the calculation, and read its clock
Open the published plateau, then compare STD, partial and full distillation in the advanced case selector.
What to observe: These are supplied surface sequences with a native origin and about one-gigayear smoothing. They contain no internal front. The controlled core keeps its own uncalibrated clock.

White Dwarf Cooling and Crystallization

A crystal core grows

Core temperature along the model clock

Stored heat and latent heat leave as light

Watch stored heat leave a freezing core

Core mathematical model

The controlled thermal energy

Heat leaving the model

A shell freezes at its own temperature

Common difficulties

Support versus heat

Radius versus mass

One clock for all views

Run the experiment

Cross the phase boundary

Catch a shell releasing latent heat

Visit the cold tail

Change the calculation, and read its clock

Related experiments and articles

The heat and clocks of white dwarfs