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We calculate the evolution of massive stars, which undergo pulsational pair-instability (PPI) when the O-rich core is formed. The evolution from the main-sequence through the onset of PPI is calculated for stars with the initial masses of $80 - 140$ $M_{odot}$ and metallicities of $Z = 10^{-3} - 1.0$ $Z_odot$. Because of mass loss, $Z leq 0.5$ $Z_odot$ is necessary for stars to form He cores massive enough (i.e., mass $>40 ~M_odot$) to undergo PPI. The hydrodynamical phase of evolution from PPI through the beginning of Fe core collapse is calculated for the He cores with masses of $40 - 62 ~M_odot$ and $Z = 0$. During PPI, electron-positron pair production causes a rapid contraction of the O-rich core which triggers explosive O-burning and a pulsation of the core. We study the mass dependence of the pulsation dynamics, thermodynamics, and nucleosynthesis. The pulsations are stronger for more massive He cores and result in such a large amount of mass ejection such as $3 - 13$ $M_odot$ for $40 - 62 ~M_odot$ He cores. These He cores eventually undergo Fe-core collapse. The $64 ~M_odot$ He core undergoes complete disruption and becomes a pair-instability supernova. The H-free circumstellar matter ejected around these He cores is massive enough for to explain the observed light curve of Type I (H-free) superluminous supernovae with circumstellar interaction. We also note that the mass ejection sets the maximum mass of black holes (BHs) to be $sim 50$ $M_{odot}$, which is consistent with the masses of BHs recently detected by VIRGO and aLIGO.
Pair-instability and pulsational pair-instability supernovae (PPISN) have not been unambiguously observed so far. They are, however, promising candidates for the progenitors of the heaviest binary black hole (BBH) mergers detected. If these BBHs are
Massive stars that end their lives with helium cores in the range of 35 to 65 Msun are known to produce repeated thermonuclear outbursts due to a recurring pair-instability. In some of these events, solar masses of material are ejected in repeated ou
Massive stars of helium cores of 35-65 Msun eventually encounter the electron/positron creation instability, and it triggers explosive carbon or oxygen burning that produces several thermonuclear eruptions. The resulting catastrophe collisions of eru
Massive stars having a CO core of $sim$40-60 M$_odot$ experience pulsational pair-instability (PPI) after carbon-burning. This instability induces strong pulsations of the whole star and a part of outer envelope is ejected. We investigate the evoluti
In certain mass ranges, massive stars can undergo a violent pulsation triggered by the electron/positron pair instability that ejects matter, but does not totally disrupt the star. After one or more of these pulsations, such stars are expected to und