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We study the hydrodynamical transition from an hadronic star into a quark ora hybrid star. We discuss the possible mode of burning, using a fullyrelativistic formalism and realistic Equations of State in which hyperons canbe present. We take into account the possibility that quarks form a diquarkcondensate. We also discuss the formation of a mixed phase of hadrons andquarks, and we indicate which region of the star can rapidly convert in variouspossible scenarios. An estimate of the final temperature of the system isprovided. We find that the conversion process always corresponds to adeflagration and never to a detonation. Hydrodynamical instabilities candevelop on the front. We estimate the increase in the conversion's velocity dueto the formation of wrinkles and we find that, although the increase issignificant, it is not sufficient to transform the deflagration into adetonation in essentially all realistic scenarios. Concerning convection, itdoes not always develop. In particular the system does not develop convectionif hyperons are not present in the initial phase and if the newly formed quarkphase is made of ungapped (or weakly gapped) quarks. At the contrary, theprocess of conversion from ungapped quark matter to gapped quarks always allowsthe formation of a convective layer. Finally, we discuss possible astrophysicalimplications of our results

Burning of a Hadronic Star into a Quark or a Hybrid Star