Debye cooling of ultramassive white dwarfs
Recent observations of cold and massive white dwarfs are beginning to probe Debye cooling: an accelerated cooling phase driven by the vanishing heat capacity $C_V(T)$ of the crystallized interiors at low temperatures $T$, due to the quantum nature of phonon oscillations. We analyse this regime by calculating, both analytically and numerically, the instantaneous cooling time $τ(L)\equiv -\textrm{d}t/\textrm{d}\ln L$, which is a proxy for the number density of white dwarfs at a given luminosity $L$. We find that cooling accelerates ($τ$ shortens) at $L\lesssim 10^{-4}\,\textrm{L}_{\odot}$ due to a combination of Debye cooling and the simultaneous convective coupling of the degenerate core to the surface, with neither process causing significant acceleration on its own. Specifically, strong magnetic fields can inhibit convective coupling, leading to an almost flat $τ\propto L^{1/7}$ instead. We parametrize the uncertainty in the solid microphysics by generalizing the Helmholtz free energy such that $C_V\propto T^δ$ at low temperatures. The currently observed decreasing luminosity function in the Debye regime is tentatively consistent with $τ(L)$ of non-magnetic white dwarfs with $δ=3$, corresponding to the standard Debye model. Larger samples could probe, in addition to magnetism, deviations $δ<3$, which diverge appreciably before the Galactic disc's age $t\approx 10\textrm{ Gyr}$ for masses $M\gtrsim 1.0\,\textrm{M}_{\odot}$. For $δ>3$, in contrast, the cooling curves converge because the heat capacity becomes dominated by the degenerate electrons $C_V\propto T$.