SIMBAD references

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 10R_☉ and they sometimes reach beyond 1000R_☉ 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 1962R_☉. The progenitor mass is 50M_☉ and 5M_☉ 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 (≥10⁴³erg/s) supernovae in the optical even without energy input from the ⁵⁶Ni nuclear decay when the explosion energy is more than 10⁵²erg. 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 1M_☉. They also have a fast photospheric velocity above 10000km/s 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.