Why crystals can float inside a white dwarf

Ask where each element goes when a crystal forms

Begin with Bauer’s Figure 3. The plotted oxygen abundance increases in the solid inner region while the liquid immediately outside becomes richer in carbon. The composition profile changes as the front advances; this is not simply a temperature curve with an extra delay attached.

A carbon-enriched liquid layer can be unstable to buoyancy. The implementation mixes the overlying liquid until its composition gradient satisfies the adopted stability condition, while conserving the redistributed species. Read the unequal-zone mass accounting in equations 3–6 before treating the coloured profiles as a result.

Carbon and oxygen both have about two baryons per electron. That makes an ideal electron-pressure argument alone a poor calculator of the separation energy: the smaller composition-dependent plasma terms and the resulting structural adjustment matter. The published calculation evaluates the equation of state and solves stellar structure. Our fixed-density pure-carbon model has neither this composition transport nor its heating.

Sources: Bauer 2023, Carbon-Oxygen Phase Separation in MESA White Dwarf Models, arXiv2303.10110v2, Eq1–8 and §§2–5 · Bauer 2023, fixed MESA inputs and starting models, Zenodo 7846751, CC BY 4.0

Separate the physical phase boundary from a numerical switch

Bauer defines the dimensionless free-energy difference as $\Delta f=f_{i,\rm liquid}-f_{i,\rm solid}$ . A positive difference favours the solid. Equations1–2 introduce the smooth variable below, with $w=10^{-2}$ . The equal-free-energy boundary is $\phi=0.5$ ; the smoothing lets the entropy change contribute latent heat continuously to a stellar-evolution timestep.

The implementation begins composition separation at $\phi_{\rm sep}=0.9$ rather than at one half. This postponement keeps a composition jump from interrupting the smoothed latent-heat release. The paper reports a luminosity change of roughly $0.04\,\mathrm{dex}$ during that passage and an estimated delay overstatement of up to about ten percent. These are numerical choices with consequences, not a new physical freezing criterion.

The corrected smoothed free-energy expression matters: the liquid weight decreases as the solid weight increases. A derivative of this thermodynamics already contains the latent contribution. Adding a second independent latent source to the same complete equation of state would double count it. Our simple model instead uses a continuous sensible-energy branch and an explicit latent reservoir, declared separately.

\phi=\frac{\exp(\Delta f/w)}{1+\exp(\Delta f/w)},\qquad f_{i,\rm smooth}=(1-\phi)f_{i,\rm liquid}+\phi f_{i,\rm solid}

These are Bauer 2023 v2 equations 1–2, including the sign convention of the free-energy difference. This phase variable belongs to the equation of state; the radial model separately integrates the mass that is solid.

Sources: Bauer 2023, Carbon-Oxygen Phase Separation in MESA White Dwarf Models, arXiv2303.10110v2, Eq1–8 and §§2–5

Read the separation heating as an energy calculation

In Bauer’s equation 8, the composition is changed at the start-of-step temperature and pressure. The density adjusts slightly. The specific internal energies before and after this operation determine a heating rate, which is then included in the subsequent stellar-structure solution.

The energy released by rearrangement is additional to latent heat. Calling it gravitational energy is useful as a physical picture, but a guessed gravitational term on an unchanged density profile is not a replacement for this procedure. It would need a consistent structure, composition and energy accounting of its own.

Compare Figure 4 and the later discussion of code differences only after identifying the initial carbon/oxygen profile, hydrogen and helium layers, conductive opacity and sedimentation treatment. Bauer reports different neon-sedimentation delays among MESA, BaSTI and LPCODE calculations. The paper does not establish one universal discrepancy correction. Keeping those model differences visible is more informative than absorbing them into an arbitrary Gaussian error.

\epsilon_k=\frac{e(\rho_{k,\rm start},T_k,\{X_i\}_{k,\rm start})-e(\rho_{k,\rm end},T_k,\{X_i\}_{k,\rm end})}{\delta t}

Here

e

is specific internal energy, so

\epsilon_k

has units of energy per mass per time. The two densities need not be equal. This source enters the subsequent stellar-evolution solution. A global energy test must also include the other energy changes and losses.

Sources: Bauer 2023, Carbon-Oxygen Phase Separation in MESA White Dwarf Models, arXiv2303.10110v2, Eq1–8 and §§2–5 · MIST white dwarf models, arXiv2509.21717v1, cooling clocks, surface boundary and model comparisons

Neon changes the buoyancy question

Now open Bédard, Blouin and Cheng’s main physical picture and Methods. A newly formed crystal can be depleted in neon relative to the liquid. Under the proposed conditions it is buoyant: it rises and melts, while heavier liquid sinks. This repeated separation concentrates neon inward and releases energy. During the distillation stage, the crystal-forming region remains globally liquid rather than accumulating an ordinary solid core.

The authors modify their STELUM evolution calculation to represent this process. The starting oxygen profile and the helium-layer mass matter. Their nominal Figure 4 family uses a thin helium layer and a specified oxygen profile; the three downloadable sequences selected for this course instead have the 30PR30 standard, nonuniform oxygen profile and a thicker helium layer. Both belong to the same study, but their delays cannot be silently exchanged.

Version also matters. A statement that the earlier STELUM grid lacked carbon–oxygen phase separation does not describe every later STELUM calculation. Read the 2024 Methods for the physical ingredients actually used in this paper. The supplied surface tables permit new comparisons of that model’s outputs; they do not make the private evolution implementation independently executable.

Sources: Bédard, Blouin and Cheng 2024, Buoyant crystals halt the cooling of white dwarf stars, arXiv2409.04419v1, author manuscript of the Nature study, main text and Methods · Bédard, Blouin and Cheng 2024, fixed cooling sequences, Zenodo 10201676, CC BY 4.0 · Bauer 2023, Carbon-Oxygen Phase Separation in MESA White Dwarf Models, arXiv2303.10110v2, Eq1–8 and §§2–5

Follow the composition prescription all the way to its endpoint

The Methods fit for distillation onset uses ion number fractions. Its stated low-neon range is $x_{^{22}\mathrm{Ne}}<0.04$ . The model then prescribes a linear evolution of the intermediate composition with the carbon Coulomb coupling parameter toward an endpoint near $\Gamma_{\rm C}=208$ , $x_{\rm C}=0.8$ , $x_{\rm O}=0$ and $x_{^{22}\mathrm{Ne}}=0.2$ . The low-neon onset fit is not a formula to extrapolate up to that endpoint.

Number and mass fractions are different quantities. With only carbon and neon at that endpoint, the neon mass fraction follows from the atomic masses below. It is approximately $0.31$ . In the fixed PDF, the following raised8 is a reference marker, not a third decimal digit. Reading extracted PDF text without checking the printed page can turn a citation into a false numerical disagreement.

The path between onset and the final central composition, and the stopping condition, remain modelling assumptions. The Methods also describe artificially slowing the later crystallization of the inner roughly one quarter by mass for numerical reasons. Keep that treatment beside a claim about the late curve; the table alone cannot tell whether a small feature is resolved interior physics.

Finally, the published outputs have been smoothed over approximately one gigayear. A narrow spike drawn by interpolating densely between rows does not gain physical time resolution. A surface-table comparison therefore retains the native clock and smoothing scale. Drawing an internal front would require additional radial data.

X_{^{22}\mathrm{Ne}}=\frac{22x_{^{22}\mathrm{Ne}}}{12x_{\rm C}+22x_{^{22}\mathrm{Ne}}}=\frac{22(0.2)}{12(0.8)+22(0.2)}\simeq0.314

This conversion checks the meaning of the endpoint fractions. It does not derive the endpoint or validate the prescribed intermediate distillation path.

Make one comparison the supplied data can actually answer

Use the tutorial’s native comparison command on the three fixed one-solar-mass 30PR30 tables. At $\log_{10}(L/L_\odot)=-3$ , the selected partial and full distillation sequences reach the level about $1.764\,\mathrm{Gyr}$ and $5.417\,\mathrm{Gyr}$ later than the standard case. The producer calculates crossings from the actual time/luminosity rows and retains local reversals. These are reproducible differences within this selected smoothed family.

That calculation does not reproduce the paper’s nominal Figure 4 model, recover a radial solid fraction, or measure a star’s missing age. To test the energy source itself, an independent evolution calculation would need the equation of state, phase diagram, transport/mixing rule, structure response and the full heating/loss terms. A three-column surface curve cannot supply those ingredients.

There is a useful next question even before that larger calculation: does an observed accumulation remain after atmospheric classification, binaries and the selection function are treated? The conditional Gaia exercise supplies original-field comparisons and preserves poor fits, but it does not settle that population question. The cooling pause, its physical mechanism and the number of stars caught in it are three connected claims that require different evidence.

python -W error scripts/astronomy/ac31/run_native_comparison.py

Sources: Bédard, Blouin and Cheng 2024, Buoyant crystals halt the cooling of white dwarf stars, arXiv2409.04419v1, author manuscript of the Nature study, main text and Methods · Bédard, Blouin and Cheng 2024, fixed cooling sequences, Zenodo 10201676, CC BY 4.0 · Sandbox Physics, standalone white-dwarf scientific reference, original model inputs and independent tests