Formation of long-period post-common-envelope binaries. II. Explaining the self-lensing binary KOI 3278

Abstract

The vast majority of close binaries containing a compact object, including the progenitors of supernovae\,Ia and at least a substantial fraction of all accreting black holes in the Galaxy, form through common-envelope (CE) evolution. Despite this importance, we struggle to even understand the energy budget of CE evolution. For decades, observed long-period post-CE binaries have been interpreted as evidence of additional energies contributing during CE evolution. We have recently shown that this argument is based on simplified assumptions for all long-period post-CE binaries containing massive white dwarfs (WDs). The only remaining post-CE binary star that has been claimed to require contributions from additional energy sources to understand its formation is KOI 3278. Here, we address in detail the potential evolutionary history of KOI 3278. In particular, we investigate whether extra energy sources, such as recombination energy, are indeed required to explain its existence. We used the 1D stellar evolution code MESA to carry out binary evolution simulations and searched for potential formation pathways for KOI 3278 that are able to explain its observed properties. We find that KOI 3278 can be explained if the WD progenitor filled its Roche lobe during a helium shell flash. In this case, the orbital period of KOI 3278 can be reproduced if the CE binding energy is calculated taking into account gravitational energy and thermodynamic internal energy. While the CE evolution that led to the formation of KOI 3278 must have been efficient -- that is, most of the available orbital energy must have been used to unbind the CE -- recombination energy is not required. We conclude that currently not a single observed post-CE binary requires one to assume that energy sources other than gravitational and thermodynamic energy are contributing to CE evolution. KOI 3278, however, remains an intriguing post-CE binary as, unlike its siblings, understanding its existence requires highly efficient CE ejection.