On rare core collapse supernovae inside planetary nebulae

We conduct simulations using MESA of the reverse formation of a white dwarf (WD)-neutron star (NS) binary system in which the WD forms before the NS. We conclude that a core collapse supernova (CCSN) explosion might occur inside a planetary nebula (PN) only if a third star forms the PN. In this WD-NS reverse binary evolution, the primary star evolves and transfers mass to the secondary star, forms a PN, and leaves a WD remnant. If the mass-transfer brings the secondary star to have a mass of >8Mo before it develops a helium core, and if the secondary does not suffer an enhanced mass-loss before it develops a massive helium core, e.g., by mass-transfer, it explodes as a CCSN and leaves a NS remnant. The time period from the formation of the PN by the primary to the explosion of the secondary is >10^6 year. By that time the PN has long dispersed into the interstellar medium. In a binary system with nearly equal mass components, the first mass-transfer episode takes place after the secondary star has developed a helium core and it ends its life forming a PN and a WD. The formation of a CCSN inside a PN (CCSNIP) requires the presence of a third star. The third star should be less massive than the secondary star but by no more than few times 0.01Mo. We estimate that the rate of CCSNIP is ~10^{-4} times the rate of all CCSNe.