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Open Access
Issue
A&A
Volume 687, July 2024
Article Number A12
Number of page(s) 6
Section Stellar structure and evolution
DOI https://doi.org/10.1051/0004-6361/202449320
Published online 26 June 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1. Introduction

A significant fraction of all compact binaries, which includes low-mass X-ray binaries and double degenerate objects, are thought to form through common-envelope (CE) evolution (e.g. Paczynski 1976; Ivanova et al. 2013; Belloni & Schreiber 2023). Despite some recent progress, we still struggle to reliably predict the outcome of this fundamental evolutionary phase from hydrodynamic simulations. Therefore, typically a simple energy equation, relating the binding energy of the envelope to the change in orbital energy scaled with an efficiency, αCE, is used in binary population models. Observed post-CE binaries have frequently been used to derive constraints on the importance of different terms in this energy balance and on the CE efficiency.

For the vast majority of observed post-CE systems, with typical orbital periods of hours to a few days, it has been found that CE evolution with an efficiency of αCE ≃ 0.3, and only taking into account thermal and gravitational energies, convincingly explains the observations (Zorotovic et al. 2010; Zorotovic & Schreiber 2022; Hernandez et al. 2022). However, two types of post-CE binaries have challenged this picture; namely, the Kepler Object of Interest 3278 (KOI 3278; Zorotovic et al. 2014b) and long-period systems with oxygen-neon white dwarfs (WDs; Davis et al. 2010; Zorotovic et al. 2010; Yamaguchi et al. 2024). These two types of systems have been claimed to require contributions from additional energy sources, such as recombination energy, during CE evolution to explain their current characteristics, in particular their long orbital periods.

We have recently shown that there is no need to invoke extra energy sources to explain the long orbital period of post-CE binaries with oxygen-neon WDs if CE evolution was triggered by dynamically unstable mass transfer from a highly evolved thermally pulsing asymptotic giant branch (TP-AGB) star (Belloni et al. 2024). In this case, at the onset of mass transfer, the mass of the envelope of the donor is comparable to the mass of its core, which means that the envelope is sufficiently loosely bound and can be ejected due to the input of relatively little orbital energy, resulting in a long-period post-CE binary with a massive WD. The single remaining system that has been claimed to provide evidence for contributions from additional energy sources is thus KOI 3278.

KOI 3278 consists of a G-type main-sequence (MS) star eclipsed by a low-mass carbon-oxygen WD companion. It was classified as a candidate system for showing a planetary transit signal due to the repeated occultation of the WD as it passes behind the G-type MS star (Burke et al. 2014; Tenenbaum et al. 2014). Data from the Quasiperiodic Automated Transit Search algorithm (Carter & Agol 2013) revealed 16 occultations as well as 16 pulses of brightening occurring almost half an orbital period later with the same period and duration as the occultations. Kruse & Agol (2014) interpreted these brightenings as the gravitational microlensing effect, which produces a magnification of the G-type star as a WD passes in front of it in a nearly circular orbit. The most recent robust measurements of the parameters of KOI 3278 were made by Yahalomi et al. (2019), who used a joint Einsteinian microlensing and Newtonian radial velocity model.

In this work, which is the second of a series of papers dedicated to long-period post-CE binaries, we investigate whether a formation pathway that includes detailed calculations of the TP-AGB phase can explain not only the existence of the long period post-CE binaries containing oxygen-neon WDs (Belloni et al. 2024), but also the properties of KOI 3278, which contains a WD of a much lower mass, without additional energy sources. We find that for post-CE binaries with low-mass carbon-oxygen WDs (≲0.55 M⊙) and sufficiently long orbital periods, like KOI 3278, the onset of CE evolution has to occur during a helium shell flash, when the WD progenitor is at the beginning of the TP-AGB phase, to explain their current orbital periods without considering energy sources in addition to orbital and thermodynamic internal energy.

2. Model assumptions

We used the MESA code (Paxton et al. 2011, 2013, 2015, 2018, 2019; Jermyn et al. 2023), version r15140, to simulate binary evolution1. We adopted the grey Eddington T(tau) relation to calculate the outer boundary conditions of the atmosphere, using a uniform opacity that is iterated to be consistent with the final surface temperature and pressure at the base of the atmosphere. We assumed an auto-extended scheme for the nuclear network, which automatically extends the net as needed. We included mass loss through stellar winds, adopting the Reimers (1975) prescription during pre-AGB evolution with a wind efficiency parameter equal to 0.5, and the prescription proposed by Vassiliadis & Wood (1993) for wind mass-loss during AGB evolution. Although we set the rotation of the zero-age MS stars to zero, we allowed it to vary during binary evolution according to the MESA standard prescription. For the stability criterion of convection, we adopted that proposed by Ledoux (1947). We treated convective regions using the scheme by Henyey et al. (1965) for the mixing-length theory. We used the predictive mixing scheme with the parameters suggested by Ostrowski et al. (2021). We also included rotationally and non-rotationally-induced mixing, namely angular momentum mixing, Solberg-Hoiland, secular shear instability, Eddington-Sweet circulation, Goldreich-Schubert-Fricke, and Spruit-Tayler dynamo (Heger et al. 2000, 2005). We allowed changes in the angular velocities of the stars due to tidal interactions. The Roche lobe radius of each star was computed using the fit of Eggleton (1983). The mass transfer rates due to Roche lobe overflow were determined following the prescription of Ritter (1988). We adopted the Bondi-Hoyle-Lyttleton prescription (Hoyle & Lyttleton 1939; Bondi & Hoyle 1944) for wind accretion. All other MESA parameters not mentioned here were fixed and set to their default values in version r15140.

For CE evolution, we adopted the so-called energy formalism and the relation put forward by Iben & Livio (1993), in which the outcome of CE evolution is approximated by the balance between the change in the orbital energy and the envelope binding energy, given by

E bind = α CE Δ E orb = − α CE ( G M d , c M a 2 a f − G M d , c M a 2 a i ) , $$ \begin{aligned} E_{\rm bind} \ = \ \alpha _{\rm CE} \ \Delta E_{\rm orb} = - \ \alpha _{\rm CE} \ \left( \, \frac{G\,M_{\rm d,c}\,M_{\rm a}}{2\,a_f} \ - \frac{G\,M_{\rm d,c}\,M_{\rm a}}{2\,a_i} \, \right), \end{aligned} $$(1)

where Ebind is the envelope binding energy, Eorb is the orbital energy, G is the gravitational constant, Md, c is the mass of the hydrogen-free core of the donor, ai is the semimajor axis at the onset of the CE evolution, af is the semimajor axis after CE ejection, and αCE is a parameter corresponding to the fraction of the difference in orbital energy (before and after CE evolution) that unbinds the envelope.

We calculated the binding energy by integrating over the star envelope from the helium core boundary (i.e., the radius at which the helium mass fraction is 0.1) to the surface of the star; that is,

E bind = − ∫ M d , c M d G m r ( m ) d m + ∫ M d , c M d ε int ( m ) d m , $$ \begin{aligned} E_{\rm bind} = - \int _{M_{\rm d,c}}^{M_{\rm d}} \frac{G \; m}{r(m)} \; \mathrm{d}m + \int _{M_{\rm d,c}}^{M_{\rm d}} \varepsilon _{\rm int}(m) \; \mathrm{d}m, \end{aligned} $$(2)

where m is the mass coordinate, r is the radius at a given mass coordinate, and εint is the thermodynamic internal energy per unit mass, which is related to the equation of state and contains terms such as the thermal energy of the gas and the radiation energy, but not the recombination energies2. We assumed that the onset of the CE evolution takes place when the mass transfer rate exceeds 10−2 M⊙ yr−1.

3. Numerical procedure

Our search toward an evolutionary model of KOI 3278 can be divided into three steps: (i) we varied the metallicity and mixing length of single-star models to find a fit for the secondary star; (ii) we ran a grid of binary models to identify a region in the parameter space that could offer a solution; and (iii) we used a finer grid to find an evolutionary track that produces a binary resembling KOI 3278.

For the first step, we used the properties of the MS star. We ran a grid of single-star models, varying the metallicity and the mixing length, as these two parameters are the most important in determining the stellar effective temperature, radius, and log g. For the mixing length, in units of the local pressure scale height, Hp, we adopted values from 1.7 to 2.3, in steps of 0.1. We adopted metallicities ranging from 0.020 to 0.030, in steps of 0.001. We assumed an initial mass of 0.912 M⊙ for the MS star, which is close to the observed value because the secondary is expected to accrete only a negligible amount of mass due to either wind accretion or atmospheric Roche lobe overflow before the WD progenitor fills its Roche lobe. We found that for a metallicity comparable to the observed one (i.e., Z ≈ 0.024), the mixing length has to be ≈2.0 Hp.

Having determined the metallicity and the mixing length, we searched for a binary model able to explain the present-day properties of KOI 3278 by running first a broad grid of models and subsequently finer grids to properly cover the parts of the parameter space that might offer solutions for KOI 3278. In all of the simulations, we initially assumed a binary consisting of two zero-age MS stars in a circular orbit. We fixed the initial mass of the WD progenitor to 1.5 M⊙, which is a value high enough to avoid dynamically stable mass transfer when it fills its Roche lobe and low enough for the core mass to not exceed the observed WD mass. We varied the mass of the zero-age MS companion around 0.911 M⊙ and the zero-age orbital period around 1000 d. The metallicity and the mixing length were set to the values we found in the previous step; that is, 0.024 and 2.00, respectively.

4. Potential formation pathway for KOI 3278

In what follows, we discuss a formation pathway for KOI 3278 that does not rely on the inclusion of recombination energy (or other additional energy sources) during CE evolution. The mass estimated by Yahalomi et al. (2019) for the WD in KOI 3278 is ≈0.53 M⊙, which is low for a carbon-oxygen WD. This means that the onset of the CE evolution cannot take place after several thermal pulses as the core of the WD progenitor would grow so that the resulting post-CE WD would be more massive than observed. In addition, for the initial WD progenitor that we assumed, the onset of CE evolution has to occur no earlier than the beginning of the TP-AGB phase. If the mass transfer starts when the WD progenitor is in the early-AGB phase, it does not become dynamically unstable (e.g. Ge et al. 2020; Temmink et al. 2023) as the donor star (i.e., the WD progenitor) is not much more massive than its companion (≈0.911 M⊙), resulting in a WD binary with a much longer orbital period than that observed. To have dynamically unstable mass transfer with such a low mass ratio, as is required to explain the observed orbital period, the onset of mass transfer has to occur during a helium shell flash, during the first thermal pulses. This is needed because only during the thermal pulse are the changes in the radius of the TP-AGB star large and do they occur on a very short timescale, which causes mass transfer to be dynamically unstable.

We provide in Table 1 an example of a zero-age MS binary that evolves into a binary with properties comparable to those of KOI 3278. We emphasize that this corresponds to an example because other solutions most likely exist when different values of stellar and binary evolution parameters as well as different initial binary properties are adopted. The initial masses of the WD progenitor and its companion are 1.5 and 0.9 M⊙, respectively. The orbit is circular with a period of 920 d. When the WD progenitor becomes an evolved first giant branch star, wind accretion is no longer negligible and the companion mass slightly increases. As soon as the WD progenitor becomes a TP-AGB star, it fills its Roche during the first helium shell flash, when its mass is ≈1.18 M⊙ and its hydrogen-free core has a mass of ≈0.53 M⊙. At this moment, the orbital period is ≈978 d and the companion mass increases to ≈0.91 M⊙ due to wind accretion. This pre-CE evolution is shown in Fig. 1, in which we include the evolution of the orbital period and the MS companion mass with the core mass of the WD progenitor.

thumbnail Fig. 1.

Pre-CE evolution of the orbital period with the core mass of the WD progenitor, color-coded with the companion mass. The gray strip in the vertical direction represents the observed WD mass in KOI 3278, while the red solar symbol shows the onset of the CE evolution. When the WD progenitor becomes an evolved first giant branch star, the companion starts to accrete a non-negligible amount of mass. The WD progenitor fills its Roche lobe during the first helium shell flash, as soon as it becomes a TP-AGB star.

Table 1.

Evolution of a zero-age MS binary toward KOI 3278.

During CE evolution, different energy sources can contribute to the integrated envelope binding energy. At the onset of the CE evolution, the gravitational energy is ≈ − 4.9 × 1046 erg. The thermodynamic internal energy is ≈4.125 × 1046 erg, with the contribution from thermal energy being ≈2.426 × 1046 erg. The hydrogen and helium recombination energies are ≈1.577 × 1046 and ≈3.928 × 1045 erg, respectively. We illustrate in Fig. 2 the profiles at the onset of the CE evolution of these energies. The CE binding energy we adopted here, which is defined in Eq. (2), is the gravitational energy plus the thermodynamic internal energy; that is, ≈ − 7.753 × 1045 erg.

thumbnail Fig. 2.

Profiles of different specific energies (i.e., gravitational, thermal, thermodynamic internal, and hydrogen and helium recombination) that contribute to the CE binding energy. The vertical line indicates the core mass.

The observed orbital period of ≈88 d can be reproduced if a fraction of ≈0.97 of the change in the orbital energy is used to unbind the CE. Afterward, when the post-CE binary is ≈2.2 Gyr old and the MS companion is ≈5.2 Gyr old, the properties of the MS star resemble the observed ones. We show in Fig. 3 the pre- and post-CE evolution of the radius of the MS companion as a function of its effective temperature. For the post-CE evolution, the different colors indicate its log g. During pre-CE evolution, both the effective temperature and the radius of the MS companion initially increase. This trend is reversed due to the accretion of a fraction of the stellar winds from the WD progenitor for a short period of time. After ≈2.2 Gyr, the properties of the MS companion are quite similar to those of the G-type star in KOI 3278. The properties we predict in our modeling with MESA are compared with the observed ones in Table 2.

thumbnail Fig. 3.

Evolution of the MS companion radius with its effective temperature. The black line corresponds to the pre-CE evolution, while the dashed line represents the post-CE evolution. The colors indicate its log g during post-CE evolution. The red solar symbol indicates the onset of the CE evolution and the gray star the observed properties. During pre-CE evolution, the effective temperature and radius increase, but this is reversed due to accretion of a portion of the stellar winds of the WD progenitor. After ≈2.2 Gyr of post-CE evolution, the MS companion has properties that are similar to those of the G-type star in KOI 3278.

Table 2.

Predicted and observed (Yahalomi et al. 2019) parameters of KOI 3278.

We searched for a reasonable formation pathway for KOI 3278 based on a set of assumptions about the parameters of stellar and binary evolution. By varying the stellar evolution parameters such as the mixing length and the amount of core overshooting, or even by including others not considered here such as semiconvective and thermohaline mixing, gravitational settling, and chemical diffusion, we most likely would be able to reproduce KOI 3278, although different parameters of the zero-age MS binary would be required. For instance, some of these parameters affect the core mass of the WD progenitor at the onset of the TP-AGB phase, which would then require a zero-age star with a different mass. Additionally, the TP-AGB phase is a notoriously difficult phase of stellar evolution to model. It is inherently three-dimensional and out of equilibrium, and many previous works have demonstrated that the physical properties of TP-AGB star models are quite sensitive to the input physics and numerics. Despite that, we believe the uncertainties in stellar and binary evolution should not affect the validity of our scenario. A more thorough investigation of mass transfer with TP-AGB donors and the impact of assumptions about the stellar and binary evolution parameters will be presented in another paper.

5. Discussion

The formation of close binary stars through CE evolution has been studied with hydrodynamic simulations and has been incorporated in binary population codes such as BSE, SeBa, StarTrack, and binary_c (e.g. Toonen et al. 2014). In order to explain the existence of long-period post-CE binaries and close double helium core WDs, both of which require a first mass transfer phase with little or no spiral-in, two approaches have been discussed in the literature.

The first one, largely designed to explain the existence of close double helium core WDs, assumes angular momentum conservation instead of energy conservation (Nelemans et al. 2000), but the physical interpretation of this approach remains unclear (Webbink 2008; Woods et al. 2012; Ivanova et al. 2013). The second idea assumes that recombination energy stored in the envelope, which was used to explain planetary nebula around single stars decades ago (Lucy 1967; Roxburgh 1967; Paczyński & Ziółkowski 1968), contributes to the CE ejection process (Han et al. 1994, 1995). The question of whether this assumption is reasonable or not has been intensively discussed (e.g. Soker & Harpaz 2003; Webbink 2008; Ivanova et al. 2013, 2015; Ivanova 2018; Zorotovic et al. 2014a; Nandez et al. 2015; Soker et al. 2018; Grichener et al. 2018; Reichardt et al. 2020; Kramer et al. 2020; López-Cámara et al. 2022; González-Bolívar et al. 2022; Belloni & Schreiber 2023; Röpke & De Marco 2023, and references therein) and observed post-CE binary stars with measured stellar and orbital parameters that seem to require contributions from energies in addition to gravitational energy play a fundamental role in this process (Davis et al. 2010; Zorotovic et al. 2014b; Miszalski et al. 2019; Yamaguchi et al. 2024).

The first attempt to investigate the formation of KOI 3278 was performed by Zorotovic et al. (2014b). These authors reconstructed its evolutionary history and predicted its future using the BSE code, based on the stellar and binary parameters measured by Kruse & Agol (2014). They found that a small amount of recombination energy, or any other source of extra energy, is required to explain the formation of KOI 3278 through CE evolution; that is, KOI 3278 seemed to provide observational evidence of additional energy playing a role.

Over the past few decades, mass transfer from AGB stars leading to CE evolution has been frequently modeled (e.g. Izzard & Jermyn 2018). However, the intricate details of TP-AGB evolution have often been ignored, possibly due to the high computational costs involved (e.g. Pols et al. 1998; Dewi & Tauris 2000; Bonačić Marinović et al. 2008; Davis et al. 2010; Xu & Li 2010; Loveridge et al. 2011; Temmink et al. 2023). Nonetheless, there have been some hydrodynamic simulations with TP-AGB donors (e.g. Reichardt et al. 2020; González-Bolívar et al. 2022), and TP-AGB evolution is included to some extent in the binary population synthesis code binary_c (Izzard et al. 2004).

Belloni et al. (2024) have recently suggested that highly evolved TP-AGB stars filling their Roche lobe could be the progenitors of long-period post-CE binaries consisting of massive (oxygen-neon) WDs with AFGK-type MS companions. This scenario does not work for systems like KOI 3278 because mass transfer is likely to become dynamically stable if the WD progenitor is allowed to lose a significant amount of mass. Therefore, long-period post-CE binaries with low-mass carbon-oxygen WDs (≲0.55 M⊙) can only originate from unevolved TP-AGB stars. We here have shown that a solution without contributions from recombination energy also exists for KOI 3278 if the extension and the binding energy of the envelope are consistently calculated during helium shell flashes. Codes such as BSE and binary_c are unable to provide the structure of the donor during helium shell flashes, and therefore extra energy during CE evolution is required to reproduce the properties of KOI 3278 if BSE is used (and possibly this remains true for binary_c).

The results presented in this paper and in Belloni et al. (2024) show that the currently available sample of post-CE binaries with accurately measured parameters consisting of a WD and a MS companion star can be explained without contributions from recombination energy. An interesting follow-up project would be to compare the predictions of population models with the large recently established samples of WD plus MS star binaries (e.g. Nebot Gómez-Morán et al. 2011; Rebassa-Mansergas et al. 2012; Jorissen et al. 2016, 2019; Oomen et al. 2018; Escorza et al. 2019; Roulston et al. 2021; Shahaf et al. 2023, 2024; Hallakoun et al. 2023; Brown et al. 2023; Yamaguchi et al. 2024).

6. Conclusions

We recently showed that long-period post-CE binaries containing massive oxygen-neon WDs can form through CE evolution without assuming extremely efficient CE evolution and without assuming that additional energy sources contribute to expelling the envelope. This exercise left KOI 3278 as the one and only observed post-CE binary that has been claimed to provide evidence for contributions from additional energy sources such as recombination energy. We here have presented binary evolution simulations with the MESA code showing that the existence of KOI 3278 can be explained if the WD progenitor fills its Roche lobe during a helium shell flash occurring with the first thermal pulses on the AGB and that in this case the available gravitational and thermodynamic energy are sufficient to expel the envelope.

Our results have two fundamentally important implications for understanding CE evolution. First, incorporating the detailed evolution of the early and late AGB in reconstructing CE evolution is fundamental to avoid drawing wrong conclusions. Second, given our result on KOI 3278, not a single post-CE binary consisting of a WD with a stellar companion provides evidence for additional energy sources such as recombination energy playing a role during CE evolution. Nevertheless, KOI 3278 remains an intriguing system. In contrast to the vast majority of observed post-CE binaries, understanding the existence of KOI 3278 requires one to assume that nearly all the available orbital energy is used to unbind the CE.

1

The files run_star_extras.f90 and run_binary_extras.f90 as well as the inlists needed for the simulations are available at https://zenodo.org/records/10841636

2

We computed the thermodynamic internal energy following closely Hirai & Mandel (2022), who made their MESA files available in https://zenodo.org/records/7066430

Acknowledgments

We would like to thank an anonymous referee for the comments and suggestions that helped to improve this manuscript. We thank the Kavli Institute for Theoretical Physics (KITP) for hosting the program “WDs as Probes of the Evolution of Planets, Stars, the Milky Way and the Expanding Universe”. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. This research was partially supported by the Munich Institute for Astro-, Particle and BioPhysics (MIAPbP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC–2094 – 390783311. DB acknowledges financial support from FONDECYT grant number 3220167. MRS and MZ were supported by FONDECYT grant number 1221059. MRS was supported by ANID, – Millennium Science Initiative Program – NCN19_171.

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All Tables

Table 1.

Evolution of a zero-age MS binary toward KOI 3278.

In the text
Table 2.

Predicted and observed (Yahalomi et al. 2019) parameters of KOI 3278.

In the text

All Figures

thumbnail Fig. 1.

Pre-CE evolution of the orbital period with the core mass of the WD progenitor, color-coded with the companion mass. The gray strip in the vertical direction represents the observed WD mass in KOI 3278, while the red solar symbol shows the onset of the CE evolution. When the WD progenitor becomes an evolved first giant branch star, the companion starts to accrete a non-negligible amount of mass. The WD progenitor fills its Roche lobe during the first helium shell flash, as soon as it becomes a TP-AGB star.
In the text
thumbnail Fig. 2.

Profiles of different specific energies (i.e., gravitational, thermal, thermodynamic internal, and hydrogen and helium recombination) that contribute to the CE binding energy. The vertical line indicates the core mass.
In the text
thumbnail Fig. 3.

Evolution of the MS companion radius with its effective temperature. The black line corresponds to the pre-CE evolution, while the dashed line represents the post-CE evolution. The colors indicate its log g during post-CE evolution. The red solar symbol indicates the onset of the CE evolution and the gray star the observed properties. During pre-CE evolution, the effective temperature and radius increase, but this is reversed due to accretion of a portion of the stellar winds of the WD progenitor. After ≈2.2 Gyr of post-CE evolution, the MS companion has properties that are similar to those of the G-type star in KOI 3278.
In the text

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