Ideal-ion white-dwarf cooling timescale

With carbon-ion thermal energy $U=M{k}_B{T}_c/(8m_u)$ and luminosity $L=A(M/M_{\odot })T_c^{7/2}$, the time between two core temperatures is $k_B{M}_{\odot }(T_f^{-5/2}-T_i^{-5/2})/(20m_{u}A)$. Late cooling is slow in this approximation. Extrapolating the formula does not prove a categorical observational absence of ultracool remnants, since atmosphere, crystallization and heat-capacity physics alter the simple model.