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Is the Common Envelope Ejection Efficiency a Function of the Binary

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Is the Common Envelope Ejection Efficiency a Function of the Binary Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 8 November 2018 (MN LATEX style file v2.2) Is the common envelope ejection efficiency a function of the binary parameters? P. J. Davis,1 U. Kolb,2 C. Knigge3 1Universit´eLibre de Bruxelles, Institut d’Astronomie et d’Astrophysique, Boulevard du Triomphe, B-1050 Brussels, Belgium 2The Open University, Department of Physics and Astronomy, Walton Hall, Milton Keynes MK7 6AA 3University of Southampton, School of Physics and Astronomy, Highfield, Southampton, SO17 1BJ 8 November 2018 ABSTRACT We reconstruct the common envelope (CE) phase for the current sample of observed white dwarf-main sequence post-common envelope binaries (PCEBs). We apply multi- regression analysis in order to investigate whether correlations exist between the CE ejection efficiencies, αCE inferred from the sample, and the binary parameters: white dwarf mass, secondary mass, orbital period at the point the CE commences, or the orbital period immediately after the CE phase. We do this with and without consid- eration for the internal energy of the progenitor primary giants’ envelope. Our fits should pave the first steps towards an observationally motivated recipe for calculating αCE using the binary parameters at the start of the CE phase, which will be useful for population synthesis calculations or models of compact binary evolution. If we do consider the internal energy of the giants’ envelope, we find a statistically significant correlation between αCE and the white dwarf mass. If we do not, a correlation is found between αCE and the orbital period at the point the CE phase commences. Further- more, if the internal energy of the progenitor primary envelope is taken into account, then the CE ejection efficiencies are within the canonical range 0 < αCE ≤ 1, although α > PCEBs with brown dwarf secondaries still require CE ∼ 1. Key words: binaries: close – stars: evolution – methods: statistical – white dwarfs 1 INTRODUCTION giant will expand in response to rapid mass loss. As a result, the giant’s radius expands relative to its Roche lobe radius, The common envelope (CE) phase is a key formation process increasing the mass transfer rate. As a consequence of this arXiv:1106.4741v2 [astro-ph.SR] 21 Sep 2011 of all compact binary systems, such as cataclysmic variables run-away situation, mass transfer commences on a dynam- (CVs) and double white dwarf binaries. Such systems typ- ical timescale. The companion main-sequence star (hence- ically have orbital separations of a few solar radii, and yet forth the secondary) cannot incorporate this material into require orbital separations of between approximately 10 and its structure quickly enough and therefore expands to fill its 1000 R⊙ to accommodate the giant progenitor primary star. own Roche lobe. The envelope eventually engulfs both the Paczynski (1976) first suggested that the CE phase is core of the primary and the main-sequence secondary. responsible for removing large amounts of orbital energy and During the spiral-in phase, if enough orbital energy is angular momentum from the progenitor system, causing a imparted to the CE before the stellar components merge, significant reduction in the orbital separation between the then the CE may be ejected from the system leaving the core two stellar components (for reviews see Iben & Livio 1993; of the primary (now the white dwarf) and the secondary star Webbink 2008). at a greatly reduced separation. We term such systems post- In the pre-CE phase of evolution, the initially more mas- common envelope binaries (PCEBs). In the present study, sive stellar component (which we henceforth denote as the we consider PCEBs which have a white dwarf primary com- primary) evolves off the main sequence first. Depending on ponent, and a main sequence secondary companion. the orbital separation of the binary, the primary will fill Modelling the CE phase presents a major compu- its Roche lobe on either the red giant or asymptotic giant tational challenge, and can only be adequately accom- branch (AGB) and initiate mass transfer. If the giant pri- plished via three-dimensional (3D) hydrodynamical simula- mary possesses a deep convective envelope [i.e. the convec- tions (e.g. Sandquist et al. 2000). Even this approach, how- tive envelope has a mass of more than approximately 50 per ever, still cannot cope with the large dynamic range of time cent of the giant’s mass; Hjellming & Webbink (1987)], the and length scales involved during the CE phase. c 0000 RAS 2 P. J. Davis et al. Clearly, such hydrodynamical simulations are too com- binary PG 1115+166 (Maxted et al. 2002) and the white putationally intensive to be included in full binary or pop- dwarf-main sequence PCEB IK Peg (Davis et al. 2010). This ulation synthesis codes. Instead, such codes resort to pa- indicates that an energy source in addition to gravitational rameterisations of the CE phase. One such parametrization energy is being exploited during the CE ejection process describes the CE phase in terms of a simple energy budget that is not accounted for in the standard energy budget for- argument. A fraction αCE of the orbital energy released as mulation such as thermal energy and recombination energy the binary orbit tightens, ∆Eorb, is available to unbind the of ionized material within the giant’s envelope (Han et al. giant’s envelope from the core. Hence, if the change in the 2002; Dewi & Tauris 2000; Webbink 2008). envelope’s binding energy is ∆Ebind, we have (e.g. de Kool The second issue is that previous studies into the 1992; de Kool & Ritter 1993; Willems & Kolb 2004) formation and evolution of white dwarf binaries (e.g. de Kool & Ritter 1993; Willems & Kolb 2004) have as- ∆Ebind = αCE∆Eorb. (1) sumed that αCE is a global constant, i.e. αCE is the same for The efficiency αCE is a free parameter with values 0 < all progenitor systems, irrespective of their parameters upon αCE ≤ 1 (although values of αCE > 1 are discussed). How- entering the CE phase. This may be a somewhat na¨ıve as- ever, the value of αCE is poorly constrained as a result of sumption. Indeed, Terman & Taam (1996) suggest that αCE our equally poor understanding of the physics underlying may depend upon the internal structure of the Roche lobe the CE phase. filling primary progenitor. An alternative formulation in terms of the angu- In this spirit, Politano & Weiler (2007) calculated the lar momentum budget of the binary was suggested by present-day population of PCEBs and CVs if αCE is a func- Nelemans et al. (2000) from their investigation into the for- tion of the secondary mass. They considered both a power- n mation of double white dwarf binaries, which they assumed law dependence (i.e. αCE ∝ M2 , where n is some power), occurred from two CE phases. They suggested that the rela- and a dependence of αCE on a cut-off mass. If the secondary tive change in the binary’s total angular momentum, ∆J/J, mass was below this cut-off limit, then a merger between and the relative change in the binary’s total mass, ∆M/M, the two stellar components is assumed to be unavoidable during the CE phase are related by (formally αCE = 0 in this case). ∆J ∆M As Davis et al. (2010) found, however, these formalisms = γCE , (2) J M tend to underestimate αCE for PCEBs with low-mass sec- ondaries ( < 0.2 M⊙). For example even though a n = 2 where γCE is the specific angular momentum removed from ∼ dependence could account for IK Peg, it predicts that sys- the binary by the ejected CE in units of the binary’s ini- tems with M2 < 0.4 M⊙ cannot survive the CE phase. This tial specific angular momentum. Nelemans & Tout (2005) ∼ is in conflict with observations that show that PCEBs with found that a value of 1.5 ≤ γCE ≤ 1.75 could account for all secondary masses as low as M2 ≈ 0.05 M⊙ do exist. observed PCEBs. This alternative description was prompted by the fact During the preparation of this paper, we became aware of similar studies being carried out by Zorotovic et al. that for double white dwarf binaries a value of αCE < 0 was needed to describe the first CE phase, which is clearly (2010) and De Marco et al. (2011). Zorotovic et al. (2010) unphysical. This indicates that the orbital energy increases applied the CE reconstruction method described by during the first phase of mass transfer, with a corresponding Nelemans & Tout (2005) to the observed sample of PCEBs. increase in the orbital separation. In light of this, Webbink They investigated whether αCE depended on either the sec- (2008) suggested that the first phase of mass transfer is ondary mass or the orbital period immediately after the CE quasi-conservative, and does not give rise to a CE phase. phase. They concluded that there was no dependence in ei- However, the angular momentum budget approach pre- ther case. However, this conclusion was based on a qualita- dicts PCEB local space densities that are approximately a tive, ‘by-eye’, analysis. factor of ten larger than observed estimates (Davis et al. De Marco et al. (2011), on the other hand, found evi- 2010). It also predicts that the number of systems increases dence that αCE increases with increasing mass of the pro- towards longer orbital periods, with a maximum number genitor primary star, but decreases with increasing mass of of systems at approximately 1000 d (Maxted et al. 2007a; the secondary. However, they only considered PCEBs which Davis et al. 2010). This is contrary to observations, which underwent negligible angular momentum loss since emerg- show that there is a maximum number of PCEBs with or- ing from the CE phase, which limited their sample size to bital periods of about 1 d (Rebassa-Mansergas et al.
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