The next formalism to be invented was based on conservation of angular momentum. The historical necessity for this alternative, known as the γ-formalism (Nelemans et al. 2000), was to find at least some explanation for formation of the known double-white dwarf (DWD) binaries. There it seemed that the standard energy formalism failed, as it could only explain the observed systems if energy is generated during CEE, i.e. \(\alpha_\rm CE>1\). (More precisely stated, an unknown source of energy appeared to be needed to replace the expected role of the orbital energy source, since the orbital energy actually acts as a further energy sink for these systems.) Apparent violation of energy conservation law is rather stressful for a physicist, so a less obviously troublesome conservation law was called upon to help. Again, as no self-consistent numerical simulations could have been performed at the time, the angular momentum budget had to be parametrized and then its free parameter has been fine-tuned using the observations of several known-to-the-date DWD systems. This did not resolve the apparent energy generation problem, only hid it. Nonetheless, it opened a discussion about the possibility to eject an envelope by some other mechanism other than a standard common envelope event. We will consider this formalism in more detail in Sect. 5. Note that the current explanation for the increase of the binary separation during this first mass-transfer phase is that it is quasi-conservative, such that the mass transfer is driven by nuclear energy input and thermal expansion. So there is no longer any apparent need to resort to unexplained energy generation.
The reaction of the accretor leads to matter filling the binary orbit. For example, if mass transfer proceeds at too great a rate to be accreted by the compact companion, but the system is also unable to quickly expel the mass, then a common-envelope is naturally formed. Potential cases include an envelope temporarily trapped around a neutron-star being fed at super-Eddington rates (Begelman 1979; Houck and Chevalier 1991; King and Begelman 1999), or reincarnation of an accreting white dwarf which tries to form a red giant (Nomoto et al. 1979; Nomoto et al. 2007); or perhaps even in nova systems when the expansion of the nova shell engulfs the companion.
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One potential constraint on CEE that has received little previous attention is the post-CE orbital eccentricity. If we detected post-CE eccentricity then it would be a useful indication in trying to understand the end of the preceding CE phase. However, eccentricity is fragile. For fixed angular momentum, circular orbits have the lowest energy, so energy dissipation can act to circularize orbits following the CE phase. So the effects of tidal circularization (Zahn 1977) largely rule out many binaries from giving us useful information on eccentricities (e.g. the large class of main-sequence + white dwarf binaries). However, binaries in the nuclei of planetary nebulae should still be helpful, since in their case there has been insufficient time since the ejection of the envelope for tides to have had a significant circularizing effect. Another promising exception is a single long-period main-sequence + white dwarf system we mention below.
For the most part observed post-CE systems do not have significant eccentricities. Limits are typically of the order ϵ5σ criterion, there are two cases of significant eccentricity amongst the sdB binaries which are PG1232-136 (ϵ=0.0600.005) and [CW83] 1419-09 (ϵ=0.0390.005) (Edelmann et al. 2005). One other interesting case is G 203-47, an M3.5V star in a 15-day orbit with a white dwarf and having an eccentricity of ϵ=0.0680.004 (Delfosse et al. 1999). With only a few examples, against many non-detections, one should be wary of Kozai-cycle driven eccentricity (Kozai 1962), yet perhaps there is some potential for learning about the CE phase from eccentricities. 2ff7e9595c
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