Ultra-massive DA WD stars are expected to harbor ONe cores resulting from the progenitor evolution through the Super-AGB phase. As evolution proceeds during the WD cooling phase, a crystallization process resulting from Coulomb interactions in very dense plasmas is expected to occur, leading to the formation of a highly crystallized core. Pulsating ultra-massive WDs offer a unique opportunity to infer and test the occurrence of crystallization in WD interiors as well as physical processes related with dense plasmas. We aim to assess the adiabatic pulsation properties of ultra-massive DA WD with ONe cores. We studied the pulsation properties of ultra-massive DA WD stars with ONe cores. We employed a new set of ultra-massive WD evolutionary sequences of models with stellar masses in the range 1.10 $leq M_{star}/M_{sun} leq$ 1.29 computed by taking into account the complete evolution of the progenitor stars and the WD stage. When crystallization set on in our models, we took into account latent heat release and also the expected changes in the core chemical composition that are due to phase separation according to a phase diagram suitable for O and Ne plasmas. We computed nonradial pulsation g-modes of our sequences of models at the ZZ Ceti phase by taking into account a solid core. We explored the impact of crystallization on their pulsation properties, in particular, the structure of the period spectrum and the distribution of the period spacings. We find that it would be possible, in principle, to discern whether a WD has a nucleus made of CO or a nucleus of ONe by studying the spacing between periods. The features found in the period-spacing diagrams could be used as a seismological tool to discern the core composition of ultra-massive ZZ Ceti stars, something that should be complemented with detailed asteroseismic analysis using the individual observed periods.
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