We show that the luminous supernovae associated with ultra-long gamma-ray bursts can be related to the slow cooling from the explosions of hydrogen-free progenitors that are extended by pulsational pair-instability. We have recently shown that some rapidly-rotating hydrogen-free gamma-ray burst progenitors that experience pulsational pair-instability can keep an extended structure caused by pulsational pair-instability until the core collapse. These types of progenitors have large radii exceeding 10 R⊙ and they sometimes reach beyond 1000 R⊙ at the time of the core collapse. They are, therefore, promising progenitors of ultra-long gamma-ray bursts. Here, we perform light-curve modeling of the explosions of one extended hydrogen-free progenitor with a radius of 1962 R⊙. The progenitor mass is 50 M⊙ and 5 M⊙ exists in the extended envelope. We use the one-dimensional radiation hydrodynamics code STELLA in which the explosions are initiated artificially by setting given explosion energy and 56Ni mass. Thanks to the large progenitor radius, the ejecta experience slow cooling after the shock breakout and they become rapidly evolving (≲10 days), luminous (≳1043 erg s−1) supernovae in the optical even without energy input from the 56Ni nuclear decay when the explosion energy is more than 1052 erg. The 56Ni decay energy input can affect the light curves after the optical light-curve peak and make the light-curve decay slowly when the 56Ni mass is around 1 M⊙. They also have a fast photospheric velocity above 10 000 km s−1 and a hot photospheric temperature above 10 000 K at around the peak luminosity. We find that the rapid rise and luminous peak found in the optical light curve of SN 2011kl, which is associated with the ultra-long gamma-ray burst GRB 111209A, can be explained as the cooling phase of the extended progenitor. The subsequent slow light-curve decline can be related to the 56Ni decay energy input. The ultra-long gamma-ray burst progenitors we proposed recently can explain both the ultra-long gamma-ray burst duration and the accompanying supernova properties. When the gamma-ray burst jet is off-axis or choked, the luminous supernovae could be observed as fast blue optical transients without accompanying gamma-ray bursts.
Model of pulse separation in the gamma light curve of the Vela pulsar
