A New Generation of Cool White Dwarf Atmosphere Models. III. WD J2356$-$209: Accretion of a Planetesimal with an Unusual Composition

WD J2356$-$209 is a cool metal-polluted white dwarf whose visible spectrum is dominated by a strong and broad sodium feature. Although discovered nearly two decades ago, no detailed and realistic analysis of this star had yet been realized. In the absence of atmosphere models taking into account the nonideal high-density effects arising at the photosphere of WD J2356$-$209, the origin of its unique spectrum had remained nebulous. We use the cool white dwarf atmosphere code presented in the first paper of this series to finally reveal the secrets of this peculiar object and details about the planetesimal that polluted its atmosphere. Thanks to the improved input physics of our models, we find a solution that is in excellent agreement with the photometric observations and the visible spectrum. Our solution reveals that the photosphere of WD J2356$-$209 has a number density ratio of $\log\,{\rm Na/Ca}= 1.0 \pm 0.2$, which is the highest ever found in a white dwarf. Since we do not know how long ago the accretion episode stopped (if it has), we cannot precisely determine the composition nor the mass of the accreted planetesimal. Nevertheless, all scenarios considered indicate that its composition is incompatible with that of chondrite-like material and that its mass was at least $10^{21}\,{\rm g}$.