On the α formalism for the common envelope interaction

Abstract

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