Abstract
It is likely that at least some planetary nebulae are composed of matter which was ejected from a binary star system during common-envelope (CE) evolution. For these planetary nebulae the ionizing component is the hot and luminous remnant of a giant which had its envelope ejected by a companion in the process of spiralling-in to its current short-period orbit. A large fraction of CE phases which end with ejection of the envelope are thought to be initiated by low-mass red giants, giants with inert, degenerate helium cores. We discuss the possible end-of-CE structures of such stars and their subsequent evolution to investigate for which structures planetary nebulae are formed. We assume that a planetary nebula forms if the remnant reaches an effective temperature greater than 30 kK within 104 yr of ejecting its envelope. We assume that the composition profile is unchanged during the CE phase so that possible remnant structures are parametrized by the end-of-CE core mass, envelope mass and entropy profile. We find that planetary nebulae are expected in post-CE systems with core masses greater than about 0.3 M⊙ if remnants end the CE phase in thermal equilibrium. We show that whether the remnant undergoes a pre-white dwarf plateau phase depends on the prescribed end-of-CE envelope mass. Thus, observing a young post-CE system would constrain the end-of-CE envelope mass and post-CE evolution.