LAUSR

We study compact stars with hybrid equations of state consisting of a nuclear outer region and two nested quark phases, each separated from the lower density phase by a strong first-order phase transition. The stability of these models is determined by calculating their radial oscillation modes with different conversion rates between adjacent phases and hence junction conditions for the modes at the phase separation interface between them. In the case when the timescale of transition is faster than the period of oscillations, we recover the traditional stability criterion implying that $\partial M/\partial\rho_c>0$ on the stable branch(es), where $M$ is the mass and $\rho_c$ is the central density. In the opposite limit of slow conversion, we find stable stellar multiplets beyond triplets consisting of stars that are stable by the usual criterion plus slow-conversion (denoted by $s$) hybrid stars with $\partial M/\partial\rho_c<0$ that are stabilized due to an alternative junction condition on the fluid displacement field at the interface reflecting the slow rate of conversion. We also study the properties of the reaction mode, the radial mode that only exists for stars with rapid (abbreviated by $r$) phase transitions, in stars with either two rapid phase transitions or alternating rapid and slow phase transitions for the first time. The implications of alternative junction conditions are also examined, with these conditions generally being found to provide stability properties similar to those for a slow conversion rate.