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Astron. Astrophys. 360, 1043-1051 (2000)

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On the energy equation and efficiency parameter of the common envelope evolution

J.D.M. Dewi 1,3,4 and T.M. Tauris 2,1

1 Astronomical Institute "Anton Pannekoek" University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands
2 Nordic Institute for Theoretical Physics (NORDITA), Blegdamsvej 17, 2100 Copenhagen O, Denmark
3 Bosscha Observatory, Lembang 40391, Bandung, Indonesia
4 Department of Astronomy, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia

Received 22 February 2000 / Accepted 23 June 2000


We have investigated the structure of evolved giant stars with masses 3 - 10 [FORMULA] in order to evaluate the binding energy of the envelope to the core prior to mass transfer in close binary systems. This binding energy is expressed by a parameter [FORMULA] which is crucial for determining the outcome of binaries evolving through a common envelope (CE) and spiral-in phase. We discuss the [FORMULA]-parameter and the efficiency of envelope ejection in the CE-phase, and show that [FORMULA] depends strongly on the evolutionary stage (i.e. stellar radius) of the donor star at the onset of the mass transfer. The existence of this relation enables us to introduce a new approach for solving the energy equation. For a given observed binary system we can derive a unique solution for the original mass and age of the donor star, as well as the pre-CE orbital period.

We find that the value of [FORMULA] is typically between 0.2 and 0.8. But in some cases, particularly on the asymptotic giant branch of lower-mass stars, it is possible that [FORMULA]. A high value of [FORMULA] (rather than assuming a high efficiency parameter, [FORMULA]) is sufficient to explain the long final orbital periods observed among those binary millisecond pulsars which are believed to have evolved through a CE-phase.

We also present a tabulation of [FORMULA] as a function of stellar radius and mass, which is useful for a quick estimation of the orbital decay during a common envelope and spiral-in phase.

Key words: stars: binaries: general – stars: evolution – stars: mass-loss – methods: numerical – stars: pulsars: individual: PSR J1454-5846

Send offprint requests to: J.D.M. Dewi (jasinta@astro.uva.nl)

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© European Southern Observatory (ESO) 2000

Online publication: August 23, 2000