 Solution

The given cooling law yields $T_{c,i}=[0.056L_{\odot }/0.09{]}^{2/7}\simeq 3.4\times 10^7$ K. Combine it with the ion thermal energy to obtainFor $T_{c,f}=0.1T_{c,i}$ this is about $1.32\times 10^{10}$ years; the exact value is retained in the independent numerical check. At approximately fixed radius, $T_{\mathrm{eff}}\propto L^{1/4}\propto T_c^{7/8}$, giving $\boxed{T_{\mathrm{eff},f}\simeq 3.3\times 10^3\mathrm{K}}$. The required cooling time is comparable to the cosmic age and leaves little time for progenitor evolution. Such objects would also be very faint. This explains scarcity in the toy model, rather than a universal prohibition on observing ultracool white dwarfs. The source's categorical absence claim is too strong; detailed cooling histories can differ. The ideal-ion heat capacity and luminosity law are being extrapolated into regimes where crystallization, atmosphere physics and other corrections matter, so this is not a precision cooling age.

A concrete observational counterexample to a categorical absence in this temperature regime is the observed ultracool population including LHS 1402, LHS 3250 and SDSS J1337+00, with published atmosphere fits spanning about 3240–3480 K. ^[ref]