On the α formalism for the common envelope interaction

Orsola De Marco*, Jean Claude Passy, Maxwell Moe, Falk Herwig, Mordecai Mark Mac Low, Bill Paxton

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

119 Citations (Scopus)


The α formalism is a common way to parametrize the common envelope interaction between a giant star and a more compact companion. The α parameter describes the fraction of orbital energy released by the companion that is available to eject the giant star's envelope. By using new, detailed stellar evolutionary calculations, we derive a user-friendly prescription for the λ parameter and an improved approximation for the envelope binding energy, thus revising the α equation. We then determine α both from simulations and from observations in a self-consistent manner. By using our own stellar structure models as well as population considerations to reconstruct the primary's parameters at the time of the common envelope interaction, we gain a deeper understanding of the uncertainties. We find that systems with very low values of q (the ratio of the companion's mass to the mass of the primary at the time of the common envelope interaction) have higher values of α. A fit to the data suggests that lower-mass companions are left at comparable or larger orbital separations to more massive companions. We conjecture that lower-mass companions take longer than a stellar dynamical time to spiral into the giant's core, and that this is key to allowing the giant to use its own thermal energy to help unbind its envelope. As a result, although systems with light companions might not have enough orbital energy to unbind the common envelope, they might stimulate a stellar reaction that results in the common envelope ejection.

Original languageEnglish
Pages (from-to)2277-2292
Number of pages16
JournalMonthly Notices of the Royal Astronomical Society
Issue number4
Publication statusPublished - Mar 2011

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