The binarity of the local white dwarf population
aa r X i v : . [ a s t r o - ph . S R ] A p r Astronomy&Astrophysicsmanuscript no. 20pc˙v21 c (cid:13)
ESO 2017April 26, 2017
The binarity of the local white dwarf population
S. Toonen , , , M. Hollands , B.T. G¨ansicke ,T. Boekholt , Anton Pannekoek Institute for Astronomy, University of Amsterdam, 1090 GE Amsterdam, The Netherlandse-mail: [email protected] Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, the Netherlands Department of Astrophysics, IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands Department of Physics, University of Warwick, Coventry CV4 7AL Departamento de Astronom´ıa, Facultad Ciencias F´ısicas y Matem´aticas, Universidad de Concepci´on, Av. Esteban Iturra s / n BarrioUniversitario, Casilla 160, Concepci´on, ChileReceived 31 / / / / ABSTRACT
Context.
As endpoints of stellar evolution, white dwarfs (WDs) are powerful tools to study the evolutionary history of the Galaxy. Inparticular, the multiplicity of WDs contains information regarding the formation and evolution of binary systems.
Aims.
Can we understand the multiplicity of the local WD sample from a theoretical point of view? Population synthesis methodsare often applied to estimate stellar space densities and event rates, but how well are these estimates calibrated? This can be tested bya comparison with the 20 pc sample, which contains ≃
100 stars and is minimally a ff ected by selection biases. Methods.
We model the formation and evolution of single stars and binaries within 20 pc with a population synthesis approach. Weconstruct a model of the current sample of WDs and di ff erentiate between WDs in di ff erent configurations, that is single WDs, andresolved and unresolved binaries containing a WD with either a main-sequence (MS) component or with a second WD. We alsostudy the e ff ect of di ff erent assumptions concerning the star formation history, binary evolution, and the initial distributions of binaryparameters. We compile from the literature the available information on the sample of WDs within 20 pc, with a particular emphasison their multiplicity, and compare this to the synthetic models. Results.
The observed space densities of single and binary WDs are well reproduced by the models. The space densities of the mostcommon WD systems (single WDs and unresolved WD-MS binaries) are consistent within a factor two with the observed value. Wefind a discrepancy only for the space density of resolved double WDs. We exclude that observational selection e ff ects, fast stellarwinds, or dynamical interactions with other objects in the Milky Way explain this discrepancy. We find that either the initial massratio distribution in the Solar neighbourhood is biased towards low mass-ratios, or more than ten resolved DWDs have been missedobservationally in the 20 pc sample. Furthermore, we show that the low binary fraction of WD systems ( ∼ ∼ ≃
50 pc. We provide a detailed estimate of the number of single and binary WDs in theGaia sample.
Key words. binaries: close, stars: evolution, stars: white dwarf
1. Introduction
As most stars end their life as white dwarfs (WDs), they forma significant component of the stellar population and are themost common stellar remnants. As such, WD stars play an im-portant role in the study of the structure and the evolutionaryhistory of stellar ensembles (Fontaine et al. 2001; Althaus et al.2010). They provide us with an e ff ective way to reconstructthe star formation history (SFH) of the Solar neighbourhoodand Galactic disc by analyzing the WD luminosity function(e.g. Tremblay et al. 2014). WDs can also be used to con-strain with good accuracy the age of stellar ensembles, suchas the Solar neighbourhood, stellar clusters, and the Galacticdisc (Torres et al. 2005; Hansen et al. 2007; Bedin et al. 2009).Fundamental for these types of studies are observational sam-ples that are as large and homogeneously-selected as possible.An important, but often complicated aspect in many popu-lation studies, is the level of completeness of the observationalsample and how to compensate for any observational biases. Acomplete sample of WDs is therefore a powerful tool, but as-sembling such a sample can be observationally very demand- ing, as WDs are low-luminosity objects, and the di ff erent WDdiscovery methods, primarily proper motion surveys and ultra-violet excess surveys, have incomplete overlap. Much time ande ff ort has been devoted to create a complete and volume-limitedsample of WDs in the Solar neighbourhood (e.g. Holberg et al.2002; Vennes & Kawka 2003; Kawka et al. 2004; Farihi et al.2005a; Kawka & Vennes 2006; Subasavage et al. 2007, 2008;Holberg et al. 2008b; Sion et al. 2009; Giammichele et al. 2012;Sayres et al. 2012; Limoges et al. 2013, 2015; Sion et al. 2014).The advantage of the Solar neighbourhood is that even thecoolest WDs can be identified with relative ease at these shortdistances from us (e.g. Carrasco et al. 2014). The level of com-pleteness that has been achieved for the WD sample within 20 pcis exceptional, and is estimated to be 80-90% (Holberg et al.2008b; Sion et al. 2009; Giammichele et al. 2012; Holberg et al.2016).Large and homogeneously-selected samples of stellar sys-tems play a vital role in the empirical verification of populationsynthesis studies, such as binary population synthesis (BPS) . See Toonen et al. (2014) for a comparison of four BPS codes. 1. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population
The BPS approach aims to further improve our understandingof stellar and binary evolution from a statistical point of view,and can aid and further motivate observational surveys. It is of-ten used to constrain evolutionary pathways and predict pop-ulation characteristics, such as event rates or the period dis-tribution of stellar populations, including type Ia supernovae(for a review see Wang & Han 2012), post-common envelopebinaries (e.g. Toonen & Nelemans 2013; Camacho et al. 2014;Zorotovic et al. 2014), or AM CVn systems (e.g. Nelemans et al.2001a). Nonetheless, tests on the number densities of a stellarpopulation (e.g. space densities or event rates) predicted by BPSstudies are often not strongly constraining, as the observed num-ber densities are uncertain to (at least) a factor of a few. However,since the 20 pc sample of WDs is volume-limited and nearlycomplete, it allows for a strong test of the number of predictedsystems from the BPS method, which is the aim of this paper.Another important feature of the 20 pc sample is that it con-sists of multiple populations of WD systems. It contains WDsformed by single stellar evolution and from mergers in bina-ries, and WDs in binaries such as double WDs (DWDs) andWD main-sequence binaries (WDMS). The sample provides uswith a rare opportunity to compare multiple stellar populations,formed from very di ff erent evolutionary paths, with the resultsof self-consistent population synthesis models. So far, none ofthe studies of the WD luminosity function have included bina-rity (e.g. Tremblay et al. 2014; Torres & Garc´ıa-Berro 2016), de-spite the expected contribution from binaries (van Oirschot et al.2014).The set-up of this paper is as follows: in Sect. 2, we give anoverview of the observed sample of local WDs. In Sect. 3, wedescribe the BPS simulations. In Sect. 4 the self-consistent sim-ulated WD populations are presented. We compare the numberof systems in the WD population and its subcomponents pre-dicted by the synthetic populations with the observed sample ofSect. 2. For unresolved binaries, we take into account the selec-tion e ff ects against finding a dim star next to a bright star. Wealso predict the number of WD systems within 50 pc in Sect. 5,which will become available with Gaia. In Sect. 6 we discuss thehypothesis of missing WD binaries in the Solar neighbourhood,and in Sect. 7 our results are summarized.
2. Observed sample
Holberg et al. (2002) constructed a local WD sample consist-ing of 109 WD candidates within 20 pc. Holberg et al. (2002)estimated that their sample was approximately 65% complete.Since then the completeness of the local WD sample was esti-mated to have risen to 80–90% (Holberg et al. 2008b; Sion et al.2009; Giammichele et al. 2012). Most recently, the complete-ness level has been estimated to be 86% by Holberg et al. (2016).The local WD sample has been used to derive the local spacedensity [(4 . ± . · − pc − ] and mass density [(3 . ± . · − M ⊙ pc − ] (e.g. Holberg et al. 2002, 2008b; Sion et al. 2009;Holberg et al. 2016). The kinematical properties of the local WDsample have been studied by Sion et al. (2009), who found thatthe vast majority of these stars belong to the thin disk. Finally,Giammichele et al. (2012) performed a systematic model atmo-sphere analysis of all the available data of the local WD popula-tion.The observed sample that we use here is mainly based on thesample of systems from Giammichele et al. (2012) and full de-tails are given in Appendix A and Table A.1. The sample of WDsin binaries is given in Table 1, and WDs in higher-order systemsin Table 2. A good starting point on WD binarity is provided by Farihi et al. (2005b), Holberg et al. (2008b), and Holberg et al.(2013). We note that the latter paper focuses on Sirius-type bi-naries (WDs with companions of spectral K and earlier) in theSolar neighbourhood, but is incomplete with respect to low-masscompanions. Notes on specific WD systems are given below. We report the identification of a new resolved double degen-erate system, comprising WD0648 +
641 and the recently dis-covered WD0649 + / yr and (421, -130) mas / yr, respectively (L´epine & Shara 2005). The trigono-metric distance to WD0648 +
641 has been determined to be33 ± +
639 is about 21 pc (Limoges et al. 2013, 2015).Nevertheless, since the temperatures, spectroscopic masses, andV-band magnitudes of both WDs are very comparable (6220 ± ± . ± . M ⊙ versus 0 . ± . M ⊙ ,and 14.67 versus 15.07 for WD0649 +
639 and WD0648 + The distances given in Table 1 are based on Giammichele et al.(2012) with updates from Limoges et al. (2013), Limoges et al.(2015), and the Discovery and Evaluation of Nearby StellarEmbers (DENSE) project . For a few systems, the derived dis-tances from di ff erent studies are significantly discrepant, suchthat their membership of the 20 pc sample is ambiguous. We dis-cuss these systems here in detail. – WD0019 +
423 has a spectroscopic distance of 12 . ± . V -band magnitude of16.5, e ff ective temperature of 5590 K, and log g of 8.0 fromLimoges et al. (2015) implies an absolute magnitude of 14.5(using the WD models as described in Sect. 3.6) and a dis-tance of 25 pc. This system is therefore removed from the20 pc sample. – WD0454 +
620 is an unresolved WDMS system in whichthe M-dwarf contaminates the WD spectrum. BothLimoges et al. (2013) and Limoges et al. (2015) take specialcare in the fitting procedure of the WD spectral lines, how-ever, the derived distances are distinct. The most recent mea-surement of Limoges et al. (2015) gives a distance of 21 . ± . . ± . +
620 is within20 pc. We adopt the most recent value of Limoges et al.(2015), however, we note that this does not significantly af-fect our conclusions of Sect. 7. – WD1242 −
105 has recently been shown not to be a sin-gle object, but to be part of a double degenerate binary(Debes et al. 2015) with a short period of 2.85 hr. These au-thors find a trigonometric distance of 39 ± −
105 from the 20 pc sample. Previously, thedistance to WD1242 −
105 was estimated to be 23 . ± http: // – Regarding WD1657 + >
50 pc when assuming a log g of 8.0. On theother hand, Kawka & Vennes (2006) derive log g = . ± .
20 and a distance d =
22 pc. Kawka & Vennes (2006) donot provide an uncertainty on the distance. We tentatively as-sume an uncertainty of ± +
321 to be within 20 pc. Even with an uncer-tainty of 2 pc on the distance estimate of Kawka & Vennes(2006) (and subsequently a probability of 20% of being amember of the 20 pc sample,), the space density within 20 pcdoes not change in a significant way. – For WD1912 + ± . ff ectively excludes it from the 20 pc sample. This valueis in agreement with the trigonometric distance found byvan Altena et al. (1995) of 36 . ± . . ± . – WD2011 +
065 has a trigonometric distance of 22 . ± . . ± . ∼
1% chance that WD2011 + +
065 does notsignificantly contribute to the space density within 20 pc. – WD2151 −
015 is part of a binary with a MS compan-ion (Farihi et al. 2005b, 2006; Holberg et al. 2008b). Thebinary has been resolved with an angular separation of1.082 ± ± ± A number of systems are classified as (unresolved) DWD candi-dates in Table 1. These are: – WD0423 +
120 which is overly bright for its parallax(Holberg et al. 2008b) and therefore considered to be a DWDcandidate by these authors. Both the parallax and photomet-ric distances (17.36 pc vs 11.88 pc, respectively), position thesystem within 20 pc from the Sun. – WD0839 −
327 which is classified as a DWD candidate dueto possible radial variations in the DA star (Bragaglia et al.1990). This claim is supported by the marginal di ff erencein the photometric and trigonometric distance (7 pc and8 . ± .
77 pc respectively) found by Kawka et al. (2007).The trigonometric distance as given by DENSE is 8 . ± .
15 pc (see Tbl.1). Holberg et al. (2008a) found a photomet-ric distance of 8 . ± .
11 pc. – WD2048 +
263 which is suspected to be a double-degeneratesystem by Bergeron et al. (2001) based on the low-gravityand mass, as well as the suspected dilution of the Balmer H α profile of the visible DA WD by a possible DC companion. – WD0108 + − − − − +
293 which aresuggested to be double degenerates by Giammichele et al.(2012). This is based on the low mass they derive by meansof the photometric technique. The masses are too low for stars to have evolved as single stars ( . . M ⊙ ). For thesame reason we add WD2322 + − – WD0322 −
019 has been considered a close DWD in the past,however, Farihi et al. (2011b) showed that the source of linebroadening was magnetism and not binarity.A word of caution is necessary for the mass estimates ofWDs in unresolved binaries (and candidates). The mass es-timates in Tables 1 and 2 are taken from Giammichele et al.(2012), who fitted single WD models to all spectra in the 20 pcsample. For example, Giammichele et al. (2012) note that thespectrum of WD0419 −
487 (RR Caeli) is contaminated by thepresence of an M-dwarf companion. As a consequence the WDmass according to Giammichele et al. (2012) is significantlylower (0 . ± . M ⊙ ) than that found by Maxted et al. (2007)(0 . ± . M ⊙ ). Maxted et al. (2007) determined the massand radius of WD0419 −
487 from the combined analysis of theradial velocities and the eclipse light curve.
For eleven WDs, it has been suggested that they are part of abinary or multiple system, however, confirmation or follow-upis lacking. In more detail: – WD0148 +
467 is listed as WD + MS in Holberg et al. (2008b)based on the Hipparcos & Tycho catalogues. We are un-able to find any other objects in these catalogues within twodegrees that have a similar parallax and proper motion toWD0148 + – WD0310 −
688 is suggested to have a second component inthe Washington Double Star catalogue. Stau ff er et al. (2010)suggest the companion does not exist. – Probst (1983) found a possible common proper motion com-panion for WD0341 + – Hoard et al. (2007) report a tentative low mass companionfor WD0357 + – WD0426 +
588 is in a wide binary (Stein2051) with an M-star companion. There is some suggestion that this is a triplesystem (Strand 1977). In their model, the red component isan astrometric binary. – WD0644 +
375 is a single WD now, but Ouyed et al. (2011)speculate it used to be a neutron star-WD binary, where theneutron star transitioned to a quark star during a quark nova,enriching the WD with iron, and stripping some of the WDmass. If this is the case, it should be excluded from the com-parison with the BPS models, as in these models the evolu-tion of neutron stars is not taken into account. – WD0856 +
331 was previously identified as being part ofa common proper motion binary with HD77408 (Wegner1981). However, the magnitudes of the proper motions(L´epine & Shara 2005) and the parallaxes (van Altena et al.1995; van Leeuwen 2007) di ff er significantly. – WD1142 −
645 is listed by Holberg et al. (2008b) as a binary,however, we do not find this to be supported by the associ-ated references or any other literature. – WD1647 +
591 shows possible radial velocity variabilityfor this system (Sa ff er et al. 1998), however, as the par-allax and photometric distance agree to within 1.2 sigma(van Leeuwen 2007; Holberg et al. 2008b), we consider it asingle WD.
3. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population – There is some confusion in the literature as to the multiplic-ity of the system containing WD1917 − ff ers significantlyfrom the others. The star listed as the C component appearsin various literature (Turon et al. 1993; Gould & Chanam´e2004; Lampens et al. 2007) where it is found to have thesame proper motion as the A / B component. However, theB / C components were at the time spatially very close lead-ing to blending, which may have impacted their analyses.Comparison of images between DSS1 and DSS2 surveysshow only the A / B components to have any detectable mo-tion between the two epochs laying to rest any suggestion ofhigher multiplicity. – Sa ff er et al. (1998) found WD2117 +
539 to have possible RVvariability, however Foss et al. (1991) did not find variability.
There are a few WDs found in triples and quadruples (Table 2).The structure of observed multiples tend to be hierarchical, forexample triples consist of an inner binary and a distant compan-ion star (Hut & Bahcall 1983). Despite the distance between thecompanion and the binary, the evolution of these systems canbe di ff erent from that of isolated binary systems (Toonen et al.2016). For example, Thompson (2011) shows that the dynami-cal e ff ect of a third companion on compact DWD binaries canlead to an enhanced rate of mergers and type Ia supernovae. TheBPS models presented in this paper do not include the possibleinteraction of a distant companion. For completeness, we dis-cuss WDs in multiples separately from isolated WDs and bina-ries in the comparison between the synthetic and observed pop-ulations in Sect. 4. Because there are only ∼ – WD0101 +
048 is part of a hierarchical quadruple, consistingof a close DWD binary (Maxted et al. 2000) and an MS-MSbinary (Caballero 2009). The double MS-binary is a visualbinary with a period of ∼
29 yr and an angular separationof ∼ – WD0326 −
273 is a close DWD (Zuckerman et al. 2003;Nelemans et al. 2005) with an M 5 star in a wide or-bit (Sion & Oswalt 1988; Poveda et al. 1994; Garc´es et al.2011). – WD0413 −
077 is part of a resolved WDMS binary, with a K-star companion in a wide orbit (Wegner & McMahan 1988;Tokovinin 2008). – WD0433 +
270 is the outer companion of a spectro-scopic binary of spectral type K2 (Tokovinin et al. 2006;Zhao et al. 2011; Holberg et al. 2013). The K-binary mayalso have a planetary mass companion at 0.025” separation(Lucas & Roche 2002; Holberg et al. 2013). – WD0727 +
482 is in a quadruple system. This system con-sists of a resolved DWD, and an unresolved MS-MS bi-nary of spectral type M (Harrington et al. 1981; Probst 1983;Sion et al. 1991; Andrews et al. 2012; Janson et al. 2014). – WD0743 −
336 is the outer star in a triple system (Tokovinin2012). The inner system, 171 Pup, is an astrometric binaryand is resolved with speckle interferometry. – WD1633 +
572 is in a wide orbit around an eclips-ing MS-MS binary of spectral type M (Silvestri et al.2002; Sion & Oswalt 1988; Poveda et al. 1994;Feiden & Chaboyer 2014). – For WD2054 − – WD2351 −
335 is part of a triple system (Scholz et al. 2004;Farihi et al. 2005b). The inner binary is a visual pair consist-ing of the WD and an M 3.5-star with a separation of 6.6”.The outer star is a M 8.5 star in a wide orbit of about 100”.
WD0939 +
071 is not included in our sample, because itwas mistakenly classified as a WD (Gianninas et al. 2011;Giammichele et al. 2012). The star is also known as GR 431 andPG 0939 +
072 and is reclassified by Gianninas et al. (2011) tobe an MS F-type star. WD0806 −
661 is included as a single starignoring its brown dwarf companion (Luhman et al. 2011).
3. Stellar and binary population synthesis
We employ the population synthesis code SeBa(Portegies Zwart & Verbunt 1996; Nelemans et al. 2001b;Toonen et al. 2012; Toonen & Nelemans 2013) to simulate alarge number of single stars and binaries. We use SeBa to evolvestars from the zero-age main sequence (ZAMS) until and in-cluding the remnant phase. At every timestep, processes such asstellar winds, mass transfer, angular momentum loss, commonenvelope, magnetic braking, and gravitational radiation are con-sidered with appropriate recipes. SeBa is incorporated into theAstrophysics Multipurpose Software Environment, or AMUSE.This is a component library with a homogeneous interfacestructure and can be downloaded for free at amusecode.org (Portegies Zwart et al. 2009).In this paper, we employ 12 BPS models. The BPS mod-els are the 2x2x3 possible permutations of two models for theSFH (BP & cSFR), two models for the initial period distribu-tion (‘Abt’ & ‘Lognormal’), and three models for the common-envelope phase ( γα , αα, & αα ff ect thepredicted space densities most compared to other uncertaintiesregarding the evolution and formation of stars and binaries. Themodels are explained in detail in the following sections and anoverview is given in Table 3. The initial stellar population is generated on a Monte Carlobased approach, according to appropriate distribution functions.The initial mass of single stars and of binary primaries are drawnbetween 0.95–10 M ⊙ from a Kroupa initial mass function (IMF)(Kroupa et al. 1993). Furthermore, Solar metallicities are as-sumed. For binaries, unless specified otherwise, the secondarymass is drawn from a uniform mass ratio distribution between0 and 1 (Duchˆene & Kraus 2013), and the eccentricity from athermal distribution (Heggie 1975) between 0 and 1. For the or-bital period (or equivalently the semi-major axis) distribution,we adopt two models. For model ‘Abt’, the orbits are drawn froma power-law distribution with an exponent of − R ⊙ . For model ‘Lognormal’, periods aredrawn from a lognormal distribution with a mean of 5.03 days,a dispersion of 2.28 (Raghavan et al. 2010), and a maximum pe-riod of 10 d. For Solar-type stars, the latter distribution has
4. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population become the preferred distribution (Duquennoy & Mayor 1991;Raghavan et al. 2010; Duchˆene & Kraus 2013; Tokovinin 2014).
Observational studies have shown that the binary frac-tion depends on the spectral type of the primary star(e.g. Shatsky & Tokovinin 2002; Raghavan et al. 2010;Duchˆene & Kraus 2013). Due to the properties of the IMFand SFH, the average WD progenitor is a ∼ M ⊙ (A-type) starfor the WD systems under consideration in this paper.For G- and F-type stars observed binary fractions are 44 ± ±
2% (Raghavan et al. 2010,more specifically 50 ±
4% for F6–G2 stars and 41 ±
3% for G2–K3 stars). Studies of OB-associations have shown binary frac-tions of over 70% for O- and B-type stars (Shatsky & Tokovinin2002; Kobulnicky & Fryer 2007; Kouwenhoven et al. 2007;Sana et al. 2012). From the most thorough search for com-panions to A-stars (De Rosa et al. 2014), a binary fraction of43 . ± .
3% is estimated.In this paper, we assume an initial binary fraction of 50% un-less specified otherwise. If an initial binary fraction f other than0.5 is shown to be appropriate, the predicted number of systems(see Table 4) can easily be adjusted as follows: the number ofbinaries and merged systems should be multiplied with the cor-rection factor w bin , and the number of single WDs with w sin . Thecorrection factors are given by: w sin = h M sin i + h M bin ih M sin i + h M bin i f / (1 − f ) , (1)and w bin = h M sin i + h M bin ih M sin i (1 − f ) / f + h M bin i , (2)where h M sin i is the average mass of a single star and h M bin i the average (total) mass of a binary system. Assuming the ini-tial distributions as described in Sect. 3.2 and the full range instellar masses of 0.1–100 M ⊙ , h M sin i = . M ⊙ and h M bin i = . M ⊙ for the period distribution of Abt (1983), and h M sin i = . M ⊙ and h M bin i = . M ⊙ for the lognormal period distri-bution.For a lower limit on the binary fraction of 40%, the correc-tion factors are w bin = .
83 and w sin = .
25 for both perioddistributions. For an upper limit of 60%, the correction factorsare w bin = .
15 and w sin = .
77. The uncertainty in the initialbinary fraction therefore induces an error on the BPS results ofabout 15–25%
An important phase in the evolution of many binary systems oc-curs when one or both stars fill their Roche lobes, and matter canflow from the donor star through the first Lagrangian point to thecompanion star. As the evolutionary timescales are shorter formore massive stars, the most massive component of the binarywill reach the giant phase first, and is likely to fill its Roche lobebefore the companion does. If the mass transfer rate from thedonor star increases upon mass loss, a runaway situation ensues,named the common-envelope (CE) phase (Paczynski 1976). TheCE-phase is a short-lived phase in which the envelope of thedonor star engulfs the companion star. If su ffi cient energy andangular momentum is transferred to the envelope, it can be ex-pelled, and a merger of the binary can be avoided. The CE-phase plays an essential role in binary star evolution, in particular,in the formation of short-period systems. The orbital outcomeis one of the aspects of binary evolution that a ff ects the syn-thetic binary populations most (e.g. Toonen & Nelemans 2013).Despite its importance and the enormous e ff orts of the commu-nity, the CE-phase is not understood in detail.The classical model for the CE-phase is the α -formalism,which is based on the energy budget (Tutukov & Yungelson1979). The α -parameter describes the e ffi ciency with which or-bital energy is consumed to unbind the CE according to E gr = α ( E orb , init − E orb , final ) , (3)where E orb is the orbital energy and E gr is the binding energy ofthe envelope. The orbital and binding energy are as defined inWebbink (1984), where E gr is approximated by E gr = GM d M d , env λ R , (4)with M d the donor mass, M d , env the envelope mass of the donorstar, λ the envelope-structure parameter, and R the radius of thedonor star. Due to the uncertainty in the value of both α and λ ,they are often combined into one parameter αλ .An alternative method for CE-evolution, is the γ -formalism(Nelemans et al. 2000), which is based on angular momentumbalance. The γ -parameter describes the e ffi ciency with which or-bital angular momentum is used to expel the CE according to J b , init − J b , final J b , init = γ ∆ M d M d + M a , (5)where J b , init and J b , final are the orbital angular momentum ofthe pre- and post-mass transfer binary respectively, and M a is the mass of the companion. The motivation for the γ -formalism comes from the observed mass-ratio distribution ofDWD systems that could not be explained by the α -formalismnor stable mass transfer for a Hertzsprung gap donor star (seeNelemans et al. 2000). The idea is that angular momentum canbe used for the expulsion of the envelope when there is a largeamount of angular momentum available, such as in binaries withsimilar-mass objects. However, the physical mechanism remainsunclear. Interestingly, Woods et al. (2012, see also Woods et al.2010) suggested an alternative model to create double WDs.This evolutionary path involves stable, non-conservative masstransfer between a red giant and an MS star. The e ff ect on the or-bit is a modest widening with a result alike to the γ -description.Further studies have to take place to see if this path su ffi ces tocreate a significant number of DWDs.In this paper, we adopt three distinct binary evolution modelsthat di ff er in their treatment of the CE-phase. The models arebased on di ff erent combinations of the α - and γ -formalism withdi ff erent values of αλ and γ (see Table 3). In detail: – In model αα , the α -formalism is used to determine the out-come of every CE-phase. The value of the αλ -parameter( αλ =
2) is based on Nelemans et al. (2000), who deducedthis value from reconstructing the second phase of masstransfer for observed DWDs. – For model γα , the γ -prescription is applied unless the binarycontains a compact object or the CE is triggered by a tidal in-stability rather than dynamically unstable Roche lobe over-flow (see Toonen et al. 2012). The value of the αλ -parameteris equal to that in model αα . The value of the γ -parameter( γ = .
75) is based on modelling the first phase of masstransfer of observed DWDs (Nelemans et al. 2000).
5. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population
Fig. 1.
Star formation rate as a function of time for model BP andmodel cSFR. Regarding model BP, the star formation rate at aGalactocentric distance of 8.5 kpc is shown. To convert the localstar formation rate of model cSFR to M ⊙ Gyr − pc − , a Galacticscale height of 300 pc is assumed (Roelofs et al. 2007b,a). – Model αα αα , but with a low valueof αλ ( αλ = . αα ffi ciently than is typically as-sumed in these systems, implying a smaller value for αλ .This finding is based on the concentration of the observedperiod-distribution at short periods ranging from a few hoursto a few days, but a lack of systems at longer periods (e.g.Nebot G´omez-Mor´an et al. 2011). Regarding the assumptions about the Galaxy, two models areadopted that di ff er in their treatment of the SFH. This comprisesthe formation rate of the stars and their assigned positions in theMilky Way.Model BP is taken from Toonen & Nelemans (2013, basedon Nelemans et al. 2004). In this model the star forma-tion rate is a function of time and position in the Galaxy(Boissier & Prantzos 1999). It peaks early in the history of theGalaxy and has decreased substantially since then. We assumethe Galactic scale height of our binary systems to be 300 pc(Roelofs et al. 2007a,b). The Galactic star formation rate as afunction of time (averaged over space) is shown in the leftpanel of Fig. 2 in Nelemans et al. (2004). For this project, onlythe star formation rate in the Solar neighbourhood is relevantwhich is shown in Fig. 1. It peaks around 8 Gyr, and extendsto 13.5 Gyr, which Boissier & Prantzos (1999) assume is theage of the Galactic disk. However, from MS and WD pop-ulations, it has been shown that oldest stars within the diskhave an age of 8-10 Gyr (e.g. Oswalt et al. 1996; Bergeron et al.1997; del Peloso et al. 2005; Salaris 2009; Haywood et al. 2013;Gianninas et al. 2015).Model cSFR is a more simplistic model of the Milky Waywith a constant star formation rate and a homogeneous spa-tial distribution of stellar systems in the Solar neighbourhood.The star formation rate is normalized, such that the total stellar mass in the Galaxy (in the full mass range of 0.1–100 M ⊙ ) is6 · M ⊙ . The spatial distribution is normalized in such waythat a spherical region of radius x centred on the Sun contains afraction of systems in the Galaxy equal to (4 π x ) / (3 V ), where V is the Galactic volume of 5 · pc . We note that from a moreelaborate model distribution of stars in the Galaxy, which is de-pendent on the Galactocentric distance, Nelemans et al. (2001b)found a similar relation between the local space density andthe total number of stars in the Galaxy (their Eq.3), that is, V = . · pc . For model cSFR, we assume star formationhas proceeded for the last 10 Gyr. This time span is appropri-ate for the thin disk, where the majority of objects in the 20 pcsample are located (Sion et al. 2014). The average star formationrate (SFR) in mode cSFR is 6 M ⊙ yr − (see also Fig. 1). The absolute magnitudes (bolometric, as well as ugriz -bands)are taken from the WD cooling curves of pure hydrogen atmo-sphere models (Holberg & Bergeron 2006; Kowalski & Saumon2006; Tremblay et al. 2011, and references therein). For MSstars we adopt the absolute ugriz -magnitudes as given byKraus & Hillenbrand (2007). For both the MS stars and WDs,we linearly interpolate between the brightness models. For thosestars that are not included in the grids of brightness models,the closest gridpoint is taken. V -band magnitudes are calculatedas a transformation from the g - and r -magnitude according toJester et al. (2005) for stars. In this paper we consider six types of stellar systems containingWDs: – Single star: A star that begins and ends its life as a single star. – Merger: A single WD that has formed as a result of a mergerin a binary system. – Resolved WDMS: A binary consisting of a WD and a main-sequence (MS) component in a wide orbit. We assume anorbit can be resolved if the angular separation is larger thanthe critical angular separation s crit :log( s crit ) = . | ∆ V | − . , (6)where ∆ V is the di ff erence in the V -band magnitude of thetwo stellar components of the binary and s crit in arcseconds.The critical angular separation is an empirical limit that takesinto account the brightness contrast between the stars. Itis a fit through the three most compact, resolved binaries(Fig. 2) in our sample of WDMS and DWDs within 20 pc.For our standard model we exclude the multiple systemWD0727 +
482 at 0.656”, as this system is only marginallyresolved (Strand et al. 1976). For our optimistic and pes-simistic scenario of resolving binaries, we translate the criti-cal separation tolog( s crit , opt ) = . | ∆ V | − . , (7)such that a binary similar to WD0727 +
482 would just beresolved in our data, andlog( s crit , pes ) = . | ∆ V | + . , (8)such that a binary with ∆ V =
6. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population angular separation (") 024681012 ∆ V WDMSDWDMultiple
Fig. 2. V -band magnitude di ff erence as a function of angularseparation for the resolved orbits of WDs in Tables 1 and 2.Resolved WDMS are shown with blue circles and DWDs withgreen squares. The resolved orbits in triples and quadruplesare shown with red diamonds. The resolved orbits in multiplesmainly consist of a WD with an MS-companion (see Table2).Overplotted are our empirical estimates of the critical angularseparation s crit . Our standard model of Eq. 6 is shown as theblack solid line, our optimistic model as the grey dashed-dottedline (Eq. 7), and our pessimistic model as the grey dashed line(Eq. 8). – Unresolved WDMS: A binary consisting of a WD and an MSin an orbit with an angular separation less than s crit .This population contains binaries that have undergone aphase of mass transfer (such as post-common-envelope bina-ries) as well as systems in which no mass transfer has takenplace. The observed sample of WDMS is strongly a ff ectedby selection e ff ects. We assume that unresolved WDMS canonly be observed as a WDMS when both components arevisible, that is, when ∆ g ≡ g WD − g MS < , (9)and ∆ z ≡ z WD − z MS > − , (10)where g and z represent the magnitudes in the Sloan g - and z -bands of the WD and MS component. We note that in thispaper the term ’unresolved WDMS’ refers to an unresolvedWDMS in which both components are visible, unless stateddi ff erently. – Resolved DWD: A binary consisting of two WDs in an orbitwith an angular separation larger than s crit . These binariesare all su ffi ciently wide such that mass transfer does not takeplace at any point in their evolution. – Unresolved DWD: A binary consisting of two WDs in anorbit with an angular separation less than s crit . We assume anunresolved DWD can be distinguished from a single WD ifboth stars contribute significantly to the light, that is, when ∆ r ≡ | r WD1 − r WD2 | < , (11)where r represents the magnitudes in the Sloan r -band ofeach of the WD components (WD component 1 and 2). As for unresolved WDMS, the term ’unresolved DWDs’ is usedin this paper for those unresolved DWDs where both com-ponents contribute to the light, unless stated di ff erently.Other types of WD binaries are not taken into account in thisproject, such as binaries that are currently interacting (e.g. cata-clysmic variables or AM CVn systems) or binaries with evolvedstars, neutron stars, or black holes as companions. These sys-tems have not been observed in the Solar neighbourhood, and itis likely that they are much less numerous in general than thebinaries considered in this paper.For the synthetic binaries, the angular separation s on the skyis calculated according to s = a (1 + e / d , (12)where a is the semi-major axis, e is the eccentricity of the orbit,and d the distance from us to the binary given by the Galacticmodel (Sect. 3.5). The time-averaged distance between the twostars for a given orbit is a (1 + e /
4. White dwarfs within 20 pc
Table 4 shows the number of WD systems within 20 pc as pre-dicted by the BPS approach for di ff erent models of the Galaxy,di ff erent initial period distributions, and di ff erent models ofcommon-envelope evolution. The error on the synthetic numberof WD systems in Table 4 represents the statistical error in thesimulations. It is estimated by the square root of the total numberof systems of that stellar type in the simulations. We have sim-ulated multiple realisations of the local WD populations, whichreduces the statistical errors of the BPS models. Besides statisti-cal errors, systematic errors originate due to the uncertainties inbinary formation and evolution. The systematic errors dominateover the statistical errors in our simulations. For this reason, sta-tistical errors are often omitted in BPS studies; instead di ff erentmodels of binary evolution are compared to gain insight into thesystematic errors.In Table 4, we show the e ff ect of di ff erent CE-models, butonly for merger systems, unresolved WDMS, and unresolvedDWDs; as single stars, resolved WDMS and DWDs are nota ff ected by binary evolutionary processes. The most commonsystems are purely single stars, followed by mergers (in a bi-nary leading to a single WD) and resolved WDMS. The pre-dicted population of resolved WDMS is larger than the pop-ulation of resolved DWDs, because not all stars will becomea WD within a Hubble time. On the other hand, the predictedpopulation of unresolved WDMS is smaller than the populationof unresolved DWDs. This is because the observational selec-tion e ff ects on WDMS are much stronger than in DWDs (seeSect. 3.7). In our simulations, 8–19 unresolved WDMS (1 in ∼ and 0.5–2 unresolved DWDs are discarded (1 in 4-5.5)because of the selection e ff ects of Eqs. 9-10. Only very few unre-solved DWDs are discarded, which means that the WD compo-nents of these DWDs tend to have relatively similar brightnesses.We find that this is because the sample is volume-limited insteadof magnitude-limited.For each type of WD system, the observed number of sys-tems within 20 pc is shown in Table 4. This table also gives a There are three candidates for these systems which have been de-tected based on astrometric perturbations of M-dwarfs (Delfosse et al.1999; Winters et al. 2016) within 20 pc. The WD companions have notbeen detected photometrically so far. 7. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population first-order correction for the incompleteness of the 20 pc sam-ple, based on the completeness estimate of Holberg et al. (2016)of 86%. Table 4 also lists the number of WD binaries that arepart of triples and quadruples.The observed number of systems within 20 pc is based onTables 1, 2, and A.1. For each system, we calculate the probabil-ity that the system is within 20 pc with a Monte Carlo approachthat takes into account the uncertainty in the distance as given bycolumn 3 of Tables 1, 2, and A.1. As a consequence, some sys-tems with a mean distance just outside of 20 pc have a non-zeroprobability of being within 20 pc. And equally, some systemsinside, but close to, the 20 pc boundary have a non-zero chanceto fall outside our sample. The number of systems within 20 pcis then estimated by the sum of the probability of each system.The errors on the number of systems within 20 pc are based onthe same Monte Carlo study. These errors do not include anyuncertainty regarding the binarity of the known systems, that is,whether any of the single WDs have an unseen companion ornot. Furthermore, these errors do not take into account the un-certainty due to low number statistics.
Single WDs mostly descend from isolated single stars, but canalso be formed from binaries in which the stellar componentsmerge. Comparing the observations with the combination of thetwo channels (Fig. 3a), our models predict roughly the samenumber of WDs (within a factor of 1.8, i.e. 96.1 and 101–176,respectively). Taking into account an 86% completeness level ofthe observed sample, this factor reduces to 1.6.The fraction of single WDs from mergers is not insignificant(10–30% of all single WDs). This is consistent with estimatesfor the halo (van Oirschot et al. 2014). Additionally, this evolu-tionary channel is interesting in the context of magnetic WDs.A recent hypothesis for strong magnetic fields in single WDsconsiders a magnetic dynamo generation during a CE-mergerin a binary (Tout et al. 2008). The fraction of magnetic WDsamongst all WDs is poorly estimated due to selection e ff ects,but it ranges from 21 ±
8% within 13 pc and 13 ±
4% within20 pc from Kawka et al. (2007), to 8% from Sion et al. (2014).This is consistent with the incidence of mergers in our models,but see Briggs et al. (2015) for a more detailed study.The synthetic number of single WDs is sensitive to the inputassumptions of our models. The di ff erent models for the SFH af-fect the predicted number of single WDs (excluding mergers) bya factor of 1.4. The number of merged systems is most depen-dent on the initial distribution of periods, and to a lesser degreeon the physics of the CE-phase. Regarding the former, in theadopted log-normal distribution, fewer binaries are formed with(relatively short) periods that result in mergers as compared tomodel ‘Abt’. Regarding the latter, when the CE-phase leads toa stronger shrinkage (which increases from model γα , to αα , to αα The selection e ff ects of unresolved WDMS systems a ff ects thepopulation strongly; only in about 1 of 1–8 systems are bothcomponents visible. As a result, our population models predict1.0–2.5 unresolved WDMS systems to be visible within 20 pc.The di ff erent models for the initial period distribution of the bi-naries and SFH hardly a ff ect the number of unresolved WDMS. Our modelling of the selection e ff ects introduces a sys-tematic uncertainty in the synthetic population of WDMS (seeEqs. 9 and 10). Equation 9 distinguishes WDMS from apparentsingle MS; equation 10 distinguishes WDMS from apparent sin-gle WDs. Neither varying the cut between ∆ z > ∆ z > − i -band instead of the z -band significantlya ff ects the number of unresolved WDMS. Varying the cut be-tween ∆ g < ∆ g < ff erentiate between re-solved and unresolved binaries (Eqs. 6-8) do not a ff ect the num-ber of predicted unresolved WDMS significantly. In the op-timistic scenario of Eq. 7, where binaries can be resolved tosmaller angular separations then in the standard scenario ofEq. 6, the number of unresolved WDMS decreases by 7–13%. Inthe pessimistic scenario in which binaries can be resolved onlydown to an angular separation of 2” (Eq. 8), the number of unre-solved WDMS decreases by 14–31%.For compact WDMS that have gone through a CE-phase (i.e.post-common envelope binaries or PCEBs), the preferred CE-model is αα ff ects. This is consistent with the ob-served number of 0 . ± . −
487 orRR Caeli) that is on the edge of 20 pc with d = . ± .
55 pc.Without the distance restriction of the 20 pc sample, theobserved lower limit on the space density of PCEBs is (6 − · − pc − (Schreiber & G¨ansicke 2003). In our models thespace density of visible, unresolved WDMS with P <
100 d(i.e. PCEBs) is (4 . − · − pc − . These space densitiesare calculated in a cylindrical volume with height above theplane of 200 pc and radii of 200 pc and 500 pc centred on theSun. We require both stars to contribute to the light accord-ing to Eqs. 9 and 10, and the WDMS to be brighter than 20thmagnitude in the g -band. Furthermore, the space density is onlycalculated for the BPS models that are based on the SFH ofBoissier & Prantzos (1999) (model BP ), as the homogeneousspatial distribution of stars assumed in model cSFR is not valid atlarge distances from the Galactic plane. In Toonen & Nelemans(2013) the space density of visible PCEBs was simulated us-ing some of the same models as in this paper, that is, basedon the SFH of Boissier & Prantzos (1999) (model BP) andthe initial period distribution from Abt (1983) (model ‘Abt’).Depending on which volume is averaged over, and whethermodel γα , αα or αα . − · − pc − . Both theoretical space densities are ingood agreement with the observed space density of PCEBs. The models presented in this paper predict ≃ − αα In most systems the light of the binary is dominated by that of theMS star, and therefore we ignore those WDMS that appear as singleWDs in the comparison with the observed sample. For model BP the space density of systems goes down when one av-erages over a larger volume (further away from the plane of the Galaxy).8. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population(a) (b)
Fig. 3.
Comparison of the known number of WD systems with that of the synthetic models. On the left, the comparisons for singleWDs and resolved binaries are shown, on the right for unresolved binaries. The lines represent the observations and the markersthe BPS models. The shaded area around the lines represents the statistical error on the observations from the square-root law. Thestatistical error is larger than the error given in Table 4 based on the distance estimate of individual systems.have been classified as DWD candidates (Sect. 2) increasesthe observed number to 5 ±
1, in good agreement with ourmodels. Besides these DWD candidates, there are five systems(WD0141 − − + − + . . M ⊙ ), which might havean undetected companion. Additionally, there are two confirmedDWDs (WD0101 +
048 and WD 0326 − − ff erent models for the SFH or initial period distributionof the binaries hardly a ff ect the number of unresolved DWDs.The major uncertainty is the CE-phase with the three di ff erentmodels varying by about a factor of 3–4. The preferred model ofCE-evolution for DWDs is model γα (Sect. 3.4), which predictsthe highest number of DWDs. Varying the boundary betweenresolved and unresolved DWD a ff ects the number of systemsby less than a factor 2. For the optimistic scenario of resolvingbinaries, the number of unresolved DWDs decreases by 10–30%depending on the CE-model. For the pessimistic scenario, thenumber increases by 16–24% for model γα , 35–46% for model αα , and most strongly for model αα ff ect of the uncertainty in the theoretical selection ef-fects applied to the synthetic population of unresolved DWDs(Eq. 11) is small. Varying the cut ∆ r between 0.5, 1.5, and 2compared to the standard of 1, leads to a decrease of 20–43%,an increase of 8–14%, and an increase of 13–20%, respectively.Overall, the majority of close DWDs satisfy the r -magnitude cri-terion of Eq. 11 in the BPS models. In other words, in most cases both WDs contribute to the light and only a few systems are dis-carded from the synthetic models. Depending on the model, 0.5–2.3 systems (18–30%) are removed from the synthetic modelsto satisfy Eq. 11. Including these systems as an apparent singleWD does not change the number of single WDs significantly.Therefore we refrain from adding these systems to the appar-ently single WDs in the comparison in this paper.When lifting the distance restriction of 20 pc,Maxted & Marsh (1999) find a 95% probability that thefraction of double degenerates among DA WDs lies in the range1.7–19%. Based on the ESO Supernova Type Ia Progenitor sur-veY (SPY) survey, the fraction of unresolved DWDs comparedto all WDs is 7 ±
1% (priv. comm. Tom Marsh). Additionally,the binary fraction of DWDs has been measured from a statis-tical method (Maoz et al. 2012) by measuring the maximumradial velocity shift between observations of the same WD.From the Sloan digital sky survey (SDSS), a binary fractionof 3-20% has been derived for separations less than 0.05AU(Badenes & Maoz 2012), and for the SPY survey a fractionof 10.3% ± ± . − γα . Furthermore, it might indicate that some of the DWDcandidates are indeed DWDs.
9. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population
The predicted number of resolved WDMS and DWDs rangesfrom about 20–40 and 15–30. The uncertainties on the predictedspace densities from the synthetic models are about a factor of ≃
2. This uncertainty comes from the di ff erent models used forthe SFH and initial period distribution. The e ff ect of varyingthe boundary between resolved and unresolved binaries a ff ectsthe number of resolved binaries less strongly than for the unre-solved binaries. In the optimistic scenario for resolving binaries(Eq. 7), the number of resolved WDMS and DWDs increases by3–5% compared to the standard scenario. In the pessimistic sce-nario, the number of resolved binaries decreases by about 10%.Therefore, for resolved binaries the exact value of the criticalangular separation is of little importance. Equally, the cut-o ff at10 d for the lognormal distribution does not a ff ect the numberof resolved binaries significantly (about 1%).The observed number of resolved WDMS is in agreementwith the lower limit of the models, and a factor of 2 below theupper limit (Table 4, Fig. 3a). This is very similar to the case ofsingle WDs. It indicates that our simulations and the adopted starformation histories are adequate in simulating space densities ofthe most common WD populations.In contrast, the observed number of resolved DWDs is signif-icantly lower than the predicted number, by a factor of 7–13. Inother words, the BPS models predict 15–30 (isolated) resolvedDWDs within 20 pc, however, only two such systems are ob-served.Regarding systems with high-order multiplicity, Table 2shows two resolved WDMS in triples (WD0413 −
077 at 5 pcand WD2351 −
335 at 22.9 pc), and three triples with the WDas the outer companion (WD0433 +
270 at 18 pc, WD0743 − +
572 at 14.4 pc). Furthermore, thereis a resolved DWD in a triple (WD0727 + − In this section, we investigate ways to resolve the discrepancy re-garding the number of resolved DWDs between the simulationsand observations, as found in the previous section.
The binaries in our simulations are assumed to evolve inisolation, however, wide binaries can be significantly dis-turbed by dynamical interactions with, for example, other starswhen passing through spiral arms, molecular clouds, or theGalactic tidal field (Retterer & King 1982; Weinberg et al. 1987;Mallada & Fernandez 2001; Jiang & Tremaine 2010). In ex-treme cases, these interactions can lead to the disruption of veryweakly bound binaries. An observational limit to the semi-majoraxis in the Galactic disc is of the order of 0.1 pc (5 · R ⊙ ,e.g. Bahcall et al. 1985; Close et al. 1990; Chanam´e & Gould2004; Kouwenhoven et al. 2007, 2010). Interesting to note inthis context is our new DWD (Sect. 2.1.1), which has a sepa-ration of 0 . ± .
01 pc. Systems with separations out to sev-eral parsec have been identified, although they are extremelyrare (Scholz et al. 2008; Caballero 2009; Mamajek et al. 2010;Shaya & Olling 2011). For models with the initial period distri-bution of Abt (1983), there are no binaries with orbits wider than5 · R ⊙ , and roughly 15% of resolved WDMS and 23% of re- solved DWDs are wider than 1 · R ⊙ . For models with the log-normal distribution there are more wide binaries and the widestbinaries are wider in comparison with the distribution of model‘Abt’. The models with the lognormal distribution of periodspredict that roughly 10% (24%) of resolved WDMS and 15%(31%) of resolved DWDs are wider then 5 · R ⊙ (1 · R ⊙ ).If we assume that a binary will quickly dissolve once its orbitbecome larger than 5 · R ⊙ (1 · R ⊙ ), the number of re-solved binaries is reduced by .
15% ( . ffi cient to resolve the discrepancy between the observedand theoretical number of resolved DWDs. Also, the dissolutionof a binary creates one or two single WDs, such that up to 14(10–30) additional single WDs should be taken into account. Another process that can lead to the disruption of a binary is afast mass-loss event. In our simulations we have made the com-mon assumption that the wind mass loss is slow compared to theorbital period. Within this limit, the change in the orbit becomesadiabatic, and the system remains bound (see Rahoma et al.2009, for a review). If, on the other hand, the mass loss is asudden event, it can lead to the disruption of the system, as dis-cussed in the context of supernova explosions (e.g. Hills 1983).For a wide binary, a fast mass-loss phase can occur during thestrong wind phases in the evolved stages of the star’s evolution,such that the mass-loss interval is short compared to the orbitaltimescale (Hadjidemetriou 1966; Alcock et al. 1986; Veras et al.2011).As a proof of concept, we perform dynamical simulations ofwind mass loss in binaries with four di ff erent mass ratios at arange of orbital separations (Appendix B). We find that the ma-jority of systems will not dissolve due to the stellar winds of theircomponents. Only the orbits of the widest binaries ( a & R ⊙ )will indeed dissolve. The critical separation of order 10 R ⊙ cor-responds to systems in which the orbital period is comparableto the length of the asymptotic giant branch phase. As the criti-cal separation for disruption by stellar winds is similar to that ofdynamical interactions with Galactic objects, one can expect thee ff ect on the population of wide, evolved binaries to be roughlysimilar to that discussed in Sect. 4.5.1. Another possible cause for our overestimation of wide binariescomes from the di ffi culty to identify binaries with large angularseparations as bound objects. For the closest WDs in our sam-ple ( ∼ < <
10. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population
To solve the discrepancy regarding DWDs, instead of a disrup-tion, we consider the possibility that wide (zero-age MS) bina-ries are not formed as regularly as assumed in our models. Weexamine the e ff ect on the WD space densities (of all types) ofour modelling of the SFH, the initial period, and mass-ratio dis-tribution.The local SFH has been studied with a variety of techniques,and these studies have resulted in SFHs that range from con-stant values (e.g. Rocha-Pinto et al. 2000; Reid et al. 2007) topeaked distributions during the last ∼ ff erent model for the SFHthat peaks at recent times can a ff ect the total number of WDs,but does not resolve the discrepancy between theory and obser-vations regarding the ratio of resolved WDMS and DWDs.Regarding the distribution of initial periods, based on obser-vations there are no indications that the distribution is dependenton the mass ratio of the system (e.g. Duchˆene & Kraus 2013).Therefore, a di ff erent model of the initial period distribution islikely to a ff ect the space density of resolved WDMS and DWDsequally, and therefore not solve the discrepancy in the numberof resolved DWDs between observations and models.Regarding the initial mass-ratio distribution, we examinedthe possibility that it is skewed towards unequal masses such thatthe companion star is of low mass and does not evolve far in aHubble time. The observed mass-ratio distributions for di ff erenttypes of stars are approximately uniform down to q ∼ . M & . M ⊙ (see Duchˆene & Kraus 2013, for a review). This isin support of our standard assumption of a uniform mass ratiodistribution, however, we cannot exclude the possibility that theGalactic stellar populations are not representative of the Solarneighbourhood.As an experiment, we constructed an alternative model to‘cSFR’ x ‘Abt’, however, with an uncorrelated initial mass ra-tio distribution; that is, the masses of both stars are randomlydrawn from the IMF. This significantly a ff ects the number andratio of resolved WDMS and resolved DWDs. Where our stan-dard model predicts 22 unresolved WDMS and 15 unresolvedDWDs, the experimental model predicts 60 unresolved WDMSand two unresolved DWDs. This mass-ratio distribution can betested by comparing the synthetic and observed mass distri-bution of MS stars in resolved WDMS (Fig. 4). Our standardmodel (i.e. ‘cSFR’ x ‘Abt’) shows a uniform mass ratio distribu-tion until about 1 M ⊙ , and a decline afterwards, as massive starsevolve into WDs. The observed mass distribution might indicatea slightly steeper distribution favouring low-mass companions, ⊙ )0.00.20.40.60.81.0 C u m u l a t i v e f r a c t i o n ObservedUniform mass ratiosRandom pairing
Fig. 4.
Cumulative distribution of the mass of the MS com-ponents in resolved WDMS within 20 pc. The observedspectral types from Table 1 are converted to masses usingKraus & Hillenbrand (2007). The observed mass distribution isweighted according to the probability for each system to bewithin 20 pc (blue solid line). Both synthetic models shown areunder the assumption of a constant SFR (i.e. model cSFR) anda log-uniform period distribution (i.e. model ‘Abt’), but di ff er intheir treatment of the initial mass ratio distribution. The greendashed line represents the standard assumption in this paper ofa uniform mass ratio distribution, whereas the red dotted linerepresents a random pairing of the primary and secondary mass.however, it is severely hampered by low-number statistics. Withthe current sample, a random-pairing of stellar masses in localbinaries is excluded based on Fig. 4. The observed binary fraction amongst WDs in the 20 pc sam-ple ranges from 18–22% depending on the binarity of the DWDcandidates (Fig. 5a). If we include the triple and quadruple sys-tems, the observed fraction would be 22–26%. This is in goodagreement with the observed binary fraction of 26% found byHolberg (2009). Holberg (2009) focuses on the probability fora WD to be part of a binary or multiple star system, which ishigher (32 ± R ⊙ , the binary fraction becomes 20–25%and 26–30% for model ‘Abt’ and model ‘Lognormal’, respec-tively. Therefore, if wide binaries are e ff ectively destroyed, eventhe models with the lognormal initial period distribution give abinary fraction that is consistent with observations.The current binary fraction for the 20 pc WD populationis dependent on the initial (ZAMS) binary fraction, for which
11. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population
Unresolved DWDs:<4%incl. candidatesResolved DWDs:1.7%Unresolved WDMS:0.4%Resolved WDMS:15%Single WDs + Mergers:78% (a) Observed
Unresolved DWDs:~1-4%Resolved DWDs:~9-14%Unresolved WDMS:~0.5-1%Resolved WDMS:~13-18%Single WDs:~56%Mergers:~7-23% (b) Predicted
Fig. 5.
Current binary fraction for di ff erent WD systems. In thetop panel the observed fractions are shown, on the bottom therange of fractions in the BPS models, based on Table 4. Fromthe BPS models the combination of single WDs and WDs frommergers should be compared with the observed single WDs. Asignificant discrepancy exists between observations and theoryregarding resolved DWDs.we have assumed a value of 50%. Observations have shownthat the initial binary fraction is a function of the primarymass (Sect. 3.3). Lowering the initial binary fraction to 40% de-creases the current binary fraction (see also Eq. 1–2); 18–22%and 25–28% for model ‘Abt’ and model ‘Lognormal’, respec-tively. Similarly increasing the binary fraction to 60%, increasesthe current binary fraction to 28–34% and 39–45% for model‘Abt’ and model ‘Lognormal’, respectively. Unless wide bina-ries are very e ffi ciently destroyed or the observations are verybiased against finding common proper motion binaries, an ini-tial binary fraction of 60% gives a current binary fraction that isnot in agreement with the observations. An initial binary fraction The di ff erence between the initial and current binary fraction hasbeen taken as evidence for missing binaries. See Sect. 6 for a discussionon this. of 40–50% is in agreement with observations of the average WDprogenitor, that is, A-type stars (De Rosa et al. 2014).
5. Outlook to Gaia
Gaia will have a strong impact on our understanding of Galacticstellar populations. The selection e ff ects for the Gaia samplesare clean and homogeneous, and therefore the samples will bevery suitable for statistical investigations such as BPS studies.Regarding WDs, Gaia is expected to increase the known samplesignificantly; from the current ∼ · objects (Kleinman et al.2013; Kepler et al. 2016) to a few 10 WDs (Torres et al. 2005;Robin et al. 2012; Carrasco et al. 2014). In particular, the largesample size provides us with the opportunity to study rare WDs,for example WDs that are pulsating, magnetic, cool, part ofthe halo population, or possible supernova Type Ia progenitors.While the scientific potential of the WD sample has been dis-cussed (e.g. G¨ansicke et al. 2015), little attention has been paidso far to WD binary systems.In the Gaia era, the (relatively) complete sample of WDsis expected to extend from the current 20 pc out to 50 pc(Carrasco et al. 2014; G¨ansicke et al. 2015). Therefore, the ef-fective volume of the complete WD sample increases by morethan an order of magnitude. We predict the number of singleand binary WDs within 50 pc (see Table 5) with a BPS approachsimilar to that used previously in Sect. 4 and described in Sect. 3.Our model of the selection e ff ects for the 50 pc sample are spe-cific to the Gaia sample, and described below. Within 50 pc, Table 5 shows that we expect to detect thousandsof single WDs. This vast number of single WDs in a volume-limited sample will allow for an accurate determination of theluminosity function and the mass function, which will not be af-fected by brightness-related selection e ff ects. These studies havethe potential to teach us about the SFH, initial-final mass relationfor WDs, and the initial mass functions of WDs. Furthermore,our models show that several hundreds of single WDs will bedetected that formed through a merger in a binary system. Withan increasingly more detailed analysis of the complete WD sam-ple, it will become important to understand how to distinguishmerged objects from single stars that evolved completely iso-lated. Due to its high precision astrometry, the Gaia mission is veryproficient in the detection of binaries and systems of higher-order multiplicities. The high precision astrometry leads to im-proved proper motions, and parallax measurements with uncer-tainties of ∼
1% for WDs within 100 pc (Carrasco et al. 2014).This is particularly important for the detection of resolved bi-naries, which can be identified either by their common propermotion (and distance) or astrometrically.The capability of Gaia to resolve a system into two localmaxima depends on the angular separation, magnitude di ff er-ence between the two stars, and the orientation angle of the bi-nary orbit with respect to the scan axis of Gaia (de Bruijne et al.2015). We assume the critical separation for resolving two starsis:log( s crit , gaia ) = . | ∆ G | − . , (13)
12. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population where | ∆ G | is the di ff erence in the G -band magnitude of thetwo stellar components. The functional form of Eq. 13 is verysimilar to the one we derived for the 20 pc sample in Eq. 6.The Gaia G -band magnitudes are calculated using the formal-ism of Jordi et al. (2010). Equation 13 is a fit to the results ofde Bruijne et al. (2015), who calculate the probability for resolv-ing two stars with Gaia as a function of angular separation andmagnitude di ff erence, averaged over all orientation angles (theirFigs. 18 and 19). Gaias resolving power does not vary with themagnitude of the primary for a given | ∆ G | . Equation 13 is a fitto the contour of 50% probability, with the idea that the systemwill be resolved in at least one of the transits observed during themission.Overall, the critical angular separation is about 0.3 arcsec.This is a vast improvement compared to that of the current sam-ple (Sect. 3.7), and as a result one would expect the ratio of re-solved binaries to unresolved binaries to increase compared tothat of the 20 pc sample. However, as the typical distances forthe Gaia sample are larger than for the 20 pc sample, the relativenumber of resolved binaries remains approximately the same.The total number of resolved binaries is very similar to whatis expected from solely the increase in e ff ective volume, whichgives an increase of about a factor of 15.In absolute numbers, Table 5 shows that the BPS models pre-dict that hundreds of resolved binaries can be observed within50 pc. The Gaia sample is, therefore, expected to overcome thesmall number statistics by which the 20 pc sample is hampered.Consequently, the Gaia sample will shed more light on the cur-rent discrepancy between the observations and models regardingthe space density of resolved DWDs. Additionally, the sampleof resolved WDMS will expose the initial mass-ratio and pe-riod distribution of wide binaries, and show if these can resolvethe just mentioned discrepancy. Lastly, the widest binaries withseparation above 10 R ⊙ will give insights into the formation ofwide binaries. Unresolved binaries can be recognized within the Gaia databased on their odd colours, odd absolute magnitudes, or due totheir poor fit to an astrometric model of a single star. Regardingthe colours, we model the selection e ff ects in a way similar toEqs. 9-11, but based on Gaia colours. This has the advantagethat it guarantees that the relevant photometry and astrometry isavailable for all stars in a homogeneous way. We assume thatunresolved WDMS can be recognized as a binary when: ∆ G BP ≡ G BP , WD − G BP , MS < , (14)and ∆ G RP ≡ G RP , WD − G RP , MS > − , (15)where G BP and G RP represent the magnitudes in the Gaia BPand RP bands for the WD and MS component. For DWDs, werequire: ∆ G ≡ | G WD1 − G WD2 | < . (16)Alternatively binaries could be detected by their odd abso-lute magnitude. If the photometric or spectroscopic distance issignificantly di ff erent from the trigonometric distance, the sys-tem can be flagged as a binary candidate. Due to the high preci-sion astrometry of Gaia, the error on the trigonometric distance is negligible. Assuming a 10% accuracy for the WD spectro-scopic distances, there would be a discrepancy with the trigono-metric distance if: G WD , bright − G total > − . , (17)where G WD , bright and G total are the G -band magnitude of thebrightest WD component and that of the binary as a whole, re-spectively. This is equivalent to ∆ G < .
57 (see also Eq. 16).As the mass-radius relationship is less strict for MSs than forWDs, we assume the accuracy for the distance determination toMSs is lower, that is, 20%. An MS can be discovered to host acompanion, if: G MS − G total > − . , (18)where G MS is the G -band magnitude of the MS.Similar to single WDs and resolved WD binaries, the largestvolume-limited sample of unresolved WD binaries is about 15times as large as the current sample (Table 5). The BPS mod-els predict that about 10–30 unresolved WDMS and 20–130unresolved DWDs can be observed with Gaia. For the visibleWDMS, 94% of the systems are selected based on their oddcolours; that is, these systems fulfil Eqs. 14 and 15. Similarlyfor the visible DWDs, the majority of binaries have odd colours;90% for models γα and αα , and 84% for model αα
2. Assumingthat accurate periods can be determined by the radial veloc-ity method up to ten days, the number of close DWDs withknown periods are reduced to less than ten for model αα
2, a fewtens for model αα , and several tens for model γα (last columnTable 5). These DWDs will be extremely useful to constrain theCE-phase, for example by modelling the specific evolution ofeach system as in Nelemans et al. (2000). Furthermore, as thenumber of unresolved DWDs (with and without known periods)in the complete 50 pc sample is strongly dependent on the mod-elling of the CE-phase, the number of systems provides an extraconstraint for the CE-phase.Lastly, unresolved astrometric binaries can be recognizedfrom their poor fit to a standard single star astrometric model.For many it should be possible to determine a photocentre or-bit with semi-major axis a photo (Gontcharov & Kiyaeva 2002;Sahlmann et al. 2015): a photo = (cid:18) M faint M bright + M faint − L faint L bright + L faint (cid:19) a , (19)where L bright and L faint are the luminosities of the bright andfaint stellar component. A common detection criterion for as-trometric binaries is a photo /σ > σ is the astrometric precision ofGaia (de Bruijne et al. 2014). The precision is a function of the G -band magnitude and the V − I -colour of the system, where forthe latter we use the transformations of Jordi et al. (2006). Forfaint sources, such as the WDs in our sample, the precision is afew hundred µ as. From the astrometric motion of the binary pho-tocentre, it will be possible to derive the orbital period, however,it will be di ffi cult to work out the nature of the unseen companionof the unresolved binary. For WD primaries with an astrometricperturbation, there is a good possibility that the companion is aWD as well, and therefore we focus on DWDs. The BPS mod-els predict 20–45 unresolved astrometric DWD binaries within50 pc. The majority of these have orbital separations just below s crit , gaia . Only three–six DWDs are compact enough to have expe-rienced one or more phases of mass transfer during their forma-tion. If an unresolved astrometric DWD is observed for whichboth masses can be measured spectroscopically, it would be a
13. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population very interesting system to constrain CE-evolution, in particularbecause the astrometric method to determine periods is sensitiveto longer periods than is feasible with the spectroscopic method.
6. Discussion on missing binaries
Ferrario (2012) noted a tension between the high binary frac-tion of Solar-type MS stars (here initial binary fraction, ∼ ∼ ∼
25% of as yet undiscovered WDs hiding inunresolved binaries. However, we find that when taking into ac-count the full binary evolution and including selection e ff ects,this tension is largely removed. The dominant reason in mostBPS models is that the binaries may merge during their evolu-tion. A secondary reason is that a WD may hide in the glareof the primary star. In our models, for every (detectable) unre-solved WDMS, there are eight WDMS systems that would notbe recognized as a WDMS due to the luminosity contrast.Another claim of missing binaries with WD components hascome from Katz et al. (2014), based on the luminosity functionof the resolved WDMS in the 20 pc sample. With a similar rea-soning as Ferrario (2012), Katz et al. (2014) argue there is adeficit of up to 100 WDs in binary systems within 20 pc. Theyconclude that it is likely that the number of WDMS is roughlyequal to or higher than that of single WDs. This conclusion isnot supported by our results; we find approximately five timesas many single WDs (both from single stellar evolution as frombinary mergers) as WDMS, which is consistent with the obser-vations (Table. 4).Beyond 20 pc, it has been claimed by Holberg (2009) that asignificant number of Sirius-like systems (resolved WDMS withcompanions of spectral type K or earlier) are missing. This isbased on a comparison of space densities at di ff erent distancesfrom the Sun. A comparison with BPS models is outside thescope of this paper. Our simulations show a discrepancy with the observations forthe number of resolved DWDs. The BPS models predict a factorof 7–13 more systems than what is observed. This large factoris remarkable as resolved binaries are too wide for mass transferto take place. The stars have practically evolved as if they wereisolated stars. Therefore, there are only a few physical processesthat a ff ect the number density of resolved binaries.The (apparent) disruption of wide binaries is not likely tosolve the discrepancy. We considered disruptions due to dy-namical interactions with other stars, molecular clouds, or theGalactic tidal field, and due to stellar winds that are short-livedcompared to the binary period. In addition, we studied the ap-parent disruption of wide binaries from selection biases againstfinding common proper motion pairs.It is possible that the progenitors of wide DWDs are not ascommonly formed as previously assumed. We considered threeoptions: – The star formation rate and initial stellar space density arelikely not the cause for the discrepancy, as the space densityof single WDs and resolved WDMS are modelled correctlywithin a factor of 2. – The binary fraction decreases as the primary mass increases.In this case, fewer binaries with massive stars are born thatcan form WDs in a Hubble time. This does not seem likely as the binary fraction is observed to increase with primary mass(e.g. Duchˆene & Kraus 2013), however, we cannot discardthe possibility that locally it could be di ff erent. – In this study we have assumed a uniform mass-ratio dis-tribution for the ZAMS-binaries, which is the current con-sensus among surveys of di ff erent types of field stars (e.g.Duchˆene & Kraus 2013). However, there are observational(Raghavan et al. 2010; De Rosa et al. 2014) indications thatthe mass-ratio distribution of close and wide binaries are dis-tinct and that for wide binaries ( >
125 AU) the distributiontends towards unequal masses. In this scenario, the compan-ion stars are biased to low masses and would not evolve farin a Hubble time. This would decrease the number of ex-pected DWDs, but increase the number of WDMS. Eventhough the 20 pc sample is severely hampered by small num-ber statistics, the mass distribution of the MS-component ofresolved WDMS might indicate a mass-ratio distribution thatis slightly steeper than uniform, that is, one which favourslow mass companions. Our BPS models predict that thesmall number statistics can be overcome with the 50 pc sam-ple based on Gaia (Table 5).The last option we consider is that at least ten resolved DWDsystems have been missed observationally. The chance that thisis due to Poisson fluctuations is less than 0.005%.
7. Conclusion
The sample of white dwarfs within 20 pc of the Sun is extraordi-nary due to its high level of completeness of 80–90%. It is alsorelatively unbiased with respect to WD luminosity and cooling.From a literature study, we compiled the most up-to-date sam-ple and divided it into di ff erent binary types. We compared thesample with the results of a binary population synthesis studyin which the evolution of binaries is modelled starting from thezero-age main-sequence. Where many BPS studies focus on asingle binary population, the 20 pc sample allows for a consis-tent and simultaneous study of the six most common WD sys-tems. Moreover, the 20 pc sample allows for a strong test on thesynthetic space density estimates of the local WD populations,and in turn the synthetic event rates and space density estimatesof other stellar populations as well.We have constructed (2x2x3 = ) 12 BPS models that di ff er intheir treatment of the SFH, initial period distribution of the bi-naries, and the CE-phase for interacting binaries. The statisticalerror on the BPS results is small, for example the uncertainty onthe space densities is <
10% . The main source of uncertaintyin BPS simulations comes from the uncertainty in the input as-sumptions (and not from numerical e ff ects, see also Toonen et al.2014): – The di ff erent models of the SFH a ff ect the WD space densi-ties by ∼ – The di ff erent models of the initial binary period distributiona ff ect most strongly the space densities of single WDs thatare formed through mergers of binary systems. It a ff ects theirspace density by a factor of ∼ – The space densities of unresolved binaries are most stronglya ff ected by the uncertainty in the common-envelope phase,by about a factor of 2 and 4, for WDMS and DWDsrespectively.Our main results can be summarized as follows: – Overall, we find that the number of systems predicted by theBPS models for the di ff erent types of WD systems are in
14. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population good agreement with the observations. We show that the BPSestimates of the number of WDs within 20 pc are well cali-brated, which gives confidence in the synthetic space densi-ties and event rates for other populations. – With an initial binary fraction of 50%, the number of ob-served and predicted single WDs and resolved WDMSagrees within a factor of 2. This may indicate that the lo-cal star formation rate is somewhat overestimated, in par-ticular model BP where the model of the Galaxy is basedon Boissier & Prantzos (1999). In this model of the Galactichistory, star formation has proceeded for 13.5 Gyr in the disc,however from MS and WD populations in the Galactic disca maximum age of 8–10 Gyr seems more appropriate. – We find that the number of single WDs that are formed frommergers in binaries is significant, about 10–30%. Therefore,it is important to take mergers into account in studies thatderive the SFR and initial mass function from observed WDsamples. – Regarding the space densities of unresolved binaries, we findthat the BPS models are consistent with the observations,however, the errors on both measurements are large. Themain source of uncertainty on the synthetic numbers comesfrom the uncertainty in the common-envelope phase and themodelling of the selection e ff ects. The observations are ham-pered by low number statistics and the fact that the binarity isnot confirmed for all DWD candidates. Larger number statis-tics, such as expected for Gaia, would allow for stronger con-straints on the BPS models. – We find a discrepancy between the observed and syntheticnumber of resolved DWDs. Our models overpredict thenumber of resolved DWDs by a factor of 7–13. We havestudied several possible mechanisms for the (apparent) dis-ruption of wide binaries, but show that these are not likelyto solve the discrepancy (Sect. 4.5). Either more than ten re-solved DWDs have been missed observationally in the Solarneighbourhood, or the initial mass-ratio distribution is biasedtowards low-mass ratios, of which there are some indicationsin the 20 pc sample (see also Sect. 6.1 for a full discussion). – We predict the number of single and WD binary systemswithin 50 pc of the Sun. This is the largest volume-limitedsample that can be fully observed by Gaia. We predict it willcontain thousands of single WDs, hundreds of single WDsthat are formed due to a merger in a binary, hundreds of widebinaries, and several dozen unresolved binaries. The largedata set of single WDs allows for detailed studies of e.g.the space density, mass function, and luminosity function.The large population of wide binaries in the 50 pc samplecan provide stringent tests of WD evolutionary models, forexample the age of the stellar components, the initial-finalmass relation of WDs, or the mass-radius relation of WDs,and in particular the discrepancy between the observed andsynthetic number of resolved DWDs. The population of re-solved and unresolved binaries can provide additional infor-mation, for example on the period- and mass-ratio distribu-tions of the WD binaries. As such the 50 pc sample has thepotential of breaking the degeneracy between the syntheticmodels.
Acknowledgements.
We thank Elme Breedt, Roberto Raddi, and Silvia Catalanfor the discussions on WDs, Pierre Maxted & Tom Marsh for the discussions onwide binaries, and Carmen Martinez Barbosa, Gr´ainne Costigan, and AnthonyBrown for the discussions on Gaia. This work was supported by the NetherlandsResearch Council NWO (grant VIDI [ log semi-major axis (RSun)0.80.91.01.11.21.31.41.5 f r a c t i o n a l e n e r g y c h a n g e Fig. B.1.
Fractional energy change of the orbit due to winds asa function of initial orbital separation for initially circular or-bits. The di ff erent lines represent four di ff erent systems. Low-mass ratios are shown in black, high-mass ratios in grey. Theblack, grey, black-dashed, and grey-dashed lines represent sys-tems with initial masses of (2.5 & 1 M ⊙ ), (2.5 & 2 M ⊙ ), (5 &1 M ⊙ ), and (5 & 4 M ⊙ ), respectively. If the fractional energychange is larger than unity, the system dissolves. Seventh Framework Programme (FP / / ERC Grant Agreement n.320964 (WDTracer).
Appendix A: Sample of observed single WDs
Table A.1 shows the sample of observed single WDs. Thisis mainly based on Giammichele et al. (2012), with additionsof Limoges et al. (2013), Sion et al. (2014), and Limoges et al.(2015).
Appendix B: The effect of stellar wind on widebinaries
We simulate the dynamical and stellar evolution of wide bi-nary stars using the Astrophysical Multi-purpose SimulationEnvironment (AMUSE). For the dynamical evolution we usea direct, fourth-order Hermite integrator (Makino & Aarseth1992), and for the stellar evolution we use the same code as usedfor the BPS simulations in this paper (SeBa). Every integrationtime step, we evolve the dynamics and stellar evolution inde-pendently, after which we synchronize the data with the new up-dated masses, positions, and velocities. The time step criterion isbased on changes in the masses of the stars, such that more stepsare taken during events of rapid mass loss. The dynamical codehas its own internal time step criterion to resolve close encoun-ters, but will always finish on the prescribed integration time. Weevolve the binary stars until the primary component has becomea WD, and then we measure the final orbital energy of the sys-tem. If the fractional energy change − ( E orb , final − E orb , init ) / E orb , init exceeds unity, then the system dissolves. The four binary sys-tems in Fig. B.1, chosen to represent a wide range in WD binaryprogenitors, all dissolve if the initial separation is wide enough.The critical separation is of the order of 10 R ⊙ . For eccentricsystems the outcome can be di ff erent and it is likely dependenton the orbital phase.
15. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population
References
Abt, H. A. 1983, ARA&A, 21, 343Alcock, C., Fristrom, C. C., & Siegelman, R. 1986, ApJ, 302, 462Alexander, J. B. & Lourens, J. V. B. 1969, Monthly Notes of the AstronomicalSociety of South Africa, 28, 95Althaus, L. G., C´orsico, A. H., Isern, J., & Garc´ıa-Berro, E. 2010, A&A Rev.,18, 471Andrews, J. J., Ag¨ueros, M. A., Belczynski, K., et al. 2012, ApJ, 757, 170Badenes, C. & Maoz, D. 2012, ApJ, 749, L11Bahcall, J. N., Hut, P., & Tremaine, S. 1985, ApJ, 290, 15Balega, I. I., Balega, Y. Y., Hofmann, K.-H., et al. 2006, A&A, 448, 703Bedin, L. R., Salaris, M., Piotto, G., et al. 2009, ApJ, 697, 965Bergeron, P., Leggett, S. K., & Ruiz, M. T. 2001, ApJS, 133, 413Bergeron, P., Ruiz, M. T., & Leggett, S. K. 1997, ApJS, 108, 339Bessell, M. S. & Wickramasinghe, D. T. 1979, ApJ, 227, 232Boissier, S. & Prantzos, N. 1999, MNRAS, 307, 857Bragaglia, A., Greggio, L., Renzini, A., & D’Odorico, S. 1990, ApJ, 365, L13Briggs, G. P., Ferrario, L., Tout, C. A., Wickramasinghe, D. T., & Hurley, J. R.2015, MNRAS, 447, 1713Bruch, A. & Diaz, M. P. 1998, AJ, 116, 908Buscombe, W. & Foster, B. E. 1998, MK spectral classifications. Thirteenth gen-eral catalogue, epoch 2000, including UBV photometry.Caballero, J. A. 2009, A&A, 507, 251Camacho, J., Torres, S., Garc´ıa-Berro, E., et al. 2014, ArXiv:1404.5464Carrasco, J. M., Catal´an, S., Jordi, C., et al. 2014, A&A, 565, A11Casertano, S., Lattanzi, M. G., Sozzetti, A., et al. 2008, A&A, 482, 699Chanam´e, J. & Gould, A. 2004, ApJ, 601, 289Cignoni, M., Degl’Innocenti, S., Prada Moroni, P. G., & Shore, S. N. 2006,A&A, 459, 783Close, L. M., Richer, H. B., & Crabtree, D. R. 1990, AJ, 100, 1968Dahn, C. C. & Harrington, R. S. 1976, ApJ, 204, L91Dahn, C. C., Harrington, R. S., Riepe, B. Y., et al. 1982, AJ, 87, 419Davison, C. L., White, R. J., Henry, T. J., et al. 2015, AJ, 149, 106de Bruijne, J. H. J., Allen, M., Azaz, S., et al. 2015, A&A, 576, A74de Bruijne, J. H. J., Rygl, K. L. J., & Antoja, T. 2014, in EAS Publications Series,Vol. 67, EAS Publications Series, 23–29De Rosa, R. J., Patience, J., Wilson, P. A., et al. 2014, MNRAS, 437, 1216Debes, J. H., Kilic, M., Tremblay, P.-E., et al. 2015, ArXiv e-printsdel Peloso, E. F., da Silva, L., & Arany-Prado, L. I. 2005, A&A, 434, 301Delfosse, X., Forveille, T., Beuzit, J.-L., et al. 1999, A&A, 344, 897Dieterich, S. B., Henry, T. J., Golimowski, D. A., Krist, J. E., & Tanner, A. M.2012, AJ, 144, 64Duchˆene, G. & Kraus, A. 2013, ARA&A, 51, 269Duquennoy, A. & Mayor, M. 1991, A&A, 248, 485Eggen, O. J. 1956, AJ, 61, 405Farihi, J., Becklin, E. E., & Zuckerman, B. 2005a, ApJS, 161, 394Farihi, J., Becklin, E. E., & Zuckerman, B. 2005b, ApJS, 161, 394Farihi, J., Burleigh, M. R., Holberg, J. B., Casewell, S. L., & Barstow, M. A.2011a, MNRAS, 417, 1735Farihi, J., Dufour, P., Napiwotzki, R., & Koester, D. 2011b, MNRAS, 413, 2559Farihi, J., Hoard, D. W., & Wachter, S. 2006, ApJ, 646, 480Feiden, G. A. & Chaboyer, B. 2014, A&A, 571, A70Ferrario, L. 2012, MNRAS, 426, 2500Fontaine, G., Brassard, P., & Bergeron, P. 2001, PASP, 113, 409Foss, D., Wade, R. A., & Green, R. F. 1991, ApJ, 374, 281Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2016, A&A, 595, A2G¨ansicke, B., Tremblay, P.-E., Barstow, M., et al. 2015, ArXiv e-printsGarc´es, A., Catal´an, S., & Ribas, I. 2011, A&A, 531, A7Gatewood, G. & Coban, L. 2009, AJ, 137, 402Gatewood, G. D. & Gatewood, C. V. 1978, ApJ, 225, 191Giammichele, N., Bergeron, P., & Dufour, P. 2012, ApJS, 199, 29Gianninas, A., Bergeron, P., & Ruiz, M. T. 2011, ApJ, 743, 138Gianninas, A., Curd, B., Thorstensen, J. R., et al. 2015, MNRAS, 449, 3966Giclas, H. L., Slaughter, C. D., & Burnham, R. 1959, Lowell ObservatoryBulletin, 4, 136Gliese, W. & Jahreiß, H. 1991, Preliminary Version of the Third Catalogue ofNearby Stars, Tech. rep.Gontcharov, G. A. & Kiyaeva, O. V. 2002, A&A, 391, 647Gould, A. & Chanam´e, J. 2004, ApJS, 150, 455Greenstein, J. L. 1970, ApJ, 162, L55Greenstein, J. L. 1986a, AJ, 92, 867Greenstein, J. L. 1986b, AJ, 92, 859Hadjidemetriou, J. 1966, in IAU Symposium, Vol. 25, The Theory of Orbits inthe Solar System and in Stellar Systems, ed. G. I. Kontopoulos, 129Hansen, B. M. S., Anderson, J., Brewer, J., et al. 2007, ApJ, 671, 380Harrington, R. S., Christy, J. W., & Strand, K. A. 1981, AJ, 86, 909Hartkopf, W. I., Tokovinin, A., & Mason, B. D. 2012, AJ, 143, 42 Hartmann, L., Dussault, M., Noah, P. V., Klimke, A., & Bopp, B. W. 1981, ApJ,249, 662Hawley, S. L., Gizis, J. E., & Reid, I. N. 1996, AJ, 112, 2799Haywood, M., Di Matteo, P., Lehnert, M. D., Katz, D., & G´omez, A. 2013, A&A,560, A109Heggie, D. C. 1975, MNRAS, 173, 729Heintz, W. D. 1990, AJ, 99, 420Henry, T. J., Walkowicz, L. M., Barto, T. C., & Golimowski, D. A. 2002, AJ,123, 2002Hills, J. G. 1983, ApJ, 267, 322Hoard, D. W., Wachter, S., Sturch, L. K., et al. 2007, AJ, 134, 26Holberg, J. B. 2009, Journal of Physics Conference Series, 172, 012022Holberg, J. B. & Bergeron, P. 2006, AJ, 132, 1221Holberg, J. B., Bergeron, P., & Gianninas, A. 2008a, AJ, 135, 1239Holberg, J. B., Oswalt, T. D., & Barstow, M. A. 2012, AJ, 143, 68Holberg, J. B., Oswalt, T. D., & Sion, E. M. 2002, ApJ, 571, 512Holberg, J. B., Oswalt, T. D., Sion, E. M., Barstow, M. A., & Burleigh, M. R.2013, MNRAS, 435, 2077Holberg, J. B., Oswalt, T. D., Sion, E. M., & McCook, G. P. 2016, MNRAS, 462,2295Holberg, J. B., Sion, E. M., Oswalt, T., et al. 2008b, AJ, 135, 1225Hut, P. & Bahcall, J. N. 1983, ApJ, 268, 319Janson, M., Bergfors, C., Brandner, W., et al. 2014, ApJ, 789, 102Jester, S., Schneider, D. P., Richards, G. T., et al. 2005, AJ, 130, 873Jiang, Y.-F. & Tremaine, S. 2010, MNRAS, 401, 977Jordi, C., Gebran, M., Carrasco, J. M., et al. 2010, A&A, 523, A48Jordi, K., Grebel, E. K., & Ammon, K. 2006, A&A, 460, 339Katz, B., Dong, S., & Kushnir, D. 2014, ArXiv e-printsKawka, A. & Vennes, S. 2006, ApJ, 643, 402Kawka, A., Vennes, S., Schmidt, G. D., Wickramasinghe, D. T., & Koch, R.2007, ApJ, 654, 499Kawka, A., Vennes, S., & Thorstensen, J. R. 2004, AJ, 127, 1702Kepler, S. O., Pelisoli, I., Koester, D., et al. 2016, MNRAS, 455, 3413Kleinman, S. J., Kepler, S. O., Koester, D., et al. 2013, ApJS, 204, 5Kobulnicky, H. A. & Fryer, C. L. 2007, ApJ, 670, 747Kouwenhoven, M. B. N., Brown, A. G. A., Portegies Zwart, S. F., & Kaper, L.2007, A&A, 474, 77Kouwenhoven, M. B. N., Goodwin, S. P., Parker, R. J., et al. 2010, MNRAS,404, 1835Kowalski, P. M. & Saumon, D. 2006, ApJ, 651, L137Kraus, A. L. & Hillenbrand, L. A. 2007, AJ, 134, 2340Kroupa, P., Tout, C. A., & Gilmore, G. 1993, MNRAS, 262, 545Lampens, P., Strigachev, A., & Duval, D. 2007, A&A, 464, 641L´epine, S. & Shara, M. M. 2005, AJ, 129, 1483Liebert, J. 1976, ApJ, 210, 715Liebert, J., Fontaine, G., Young, P. A., Williams, K. A., & Arnett, D. 2013, ApJ,769, 7Limoges, M.-M., Bergeron, P., & L´epine, S. 2015, ArXiv e-printsLimoges, M.-M., L´epine, S., & Bergeron, P. 2013, AJ, 145, 136Lucas, P. W. & Roche, P. F. 2002, MNRAS, 336, 637Luhman, K. L., Burgasser, A. J., & Bochanski, J. J. 2011, ApJ, 730, L9Luyten, W. J. 1949, ApJ, 109, 528Luyten, W. J. & Hughes, H. S. 1980, Proper Motion Survey, University ofMinnesota, 55Makino, J. & Aarseth, S. J. 1992, PASJ, 44, 141Mallada, E. & Fernandez, J. A. 2001, in Revista Mexicana de Astronomia yAstrofisica, vol. 27, Vol. 11, Revista Mexicana de Astronomia y AstrofisicaConference Series, 27Mamajek, E. E., Kenworthy, M. A., Hinz, P. M., & Meyer, M. R. 2010, AJ, 139,919Maoz, D., Badenes, C., & Bickerton, S. J. 2012, ApJ, 751, 143Maoz, D. & Hallakoun, N. 2016, ArXiv e-printsMaxted, P. F. L. & Marsh, T. R. 1999, MNRAS, 307, 122Maxted, P. F. L., Marsh, T. R., & Moran, C. K. J. 2000, MNRAS, 319, 305Maxted, P. F. L., O’Donoghue, D., Morales-Rueda, L., Napiwotzki, R., &Smalley, B. 2007, MNRAS, 376, 919Morales, J. C., Ribas, I., Jordi, C., et al. 2009, ApJ, 691, 1400Mugrauer, M. & Neuh¨auser, R. 2005, MNRAS, 361, L15Nebot G´omez-Mor´an, A., G¨ansicke, B. T., Schreiber, M. R., et al. 2011, A&A,536, A43Nelemans, G., Napiwotzki, R., Karl, C., et al. 2005, A&A, 440, 1087Nelemans, G., Portegies Zwart, S. F., Verbunt, F., & Yungelson, L. R. 2001a,A&A, 368, 939Nelemans, G., Verbunt, F., Yungelson, L. R., & Portegies Zwart, S. F. 2000,A&A, 360, 1011Nelemans, G., Yungelson, L. R., & Portegies Zwart, S. F. 2004, MNRAS, 349,181Nelemans, G., Yungelson, L. R., Portegies Zwart, S. F., & Verbunt, F. 2001b,
16. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population
A&A, 365, 491Newton, E. R., Charbonneau, D., Irwin, J., et al. 2014, AJ, 147, 20Oswalt, T. D., Hintzen, P. M., & Luyten, W. J. 1988, ApJS, 66, 391Oswalt, T. D., Smith, J. A., Wood, M. A., & Hintzen, P. 1996, Nature, 382, 692Ouyed, R., Sta ff , J., & Jaikumar, P. 2011, ApJ, 743, 116Paczynski, B. 1976, in IAU Symposium, Vol. 73, Structure and Evolution ofClose Binary Systems, ed. P. Eggleton, S. Mitton, & J. Whelan, 75– + Portegies Zwart, S., McMillan, S., Harfst, S., et al. 2009, New A, 14, 369Portegies Zwart, S. F. & Verbunt, F. 1996, A&A, 309, 179Poveda, A., Herrera, M. A., Allen, C., Cordero, G., & Lavalley, C. 1994, Rev.Mexicana Astron. Astrofis., 28, 43Probst, R. G. 1983, ApJS, 53, 335Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010, ApJS, 190, 1Rahoma, W. A., Abd El-Salam, F. A., & Ahmed, M. K. 2009, Journal ofAstrophysics and Astronomy, 30, 187Reid, I. N., Turner, E. L., Turnbull, M. C., Mountain, M., & Valenti, J. A. 2007,ApJ, 665, 767Retterer, J. M. & King, I. R. 1982, ApJ, 254, 214Robin, A. C., Luri, X., Reyl´e, C., et al. 2012, A&A, 543, A100Rocha-Pinto, H. J., Scalo, J., Maciel, W. J., & Flynn, C. 2000, A&A, 358, 869Roelofs, G. H. A., Groot, P. J., Benedict, G. F., et al. 2007a, ApJ, 666, 1174Roelofs, G. H. A., Nelemans, G., & Groot, P. J. 2007b, MNRAS, 382, 685Sa ff er, R. A., Livio, M., & Yungelson, L. R. 1998, ApJ, 502, 394Sahlmann, J., Triaud, A. H. M. J., & Martin, D. V. 2015, MNRAS, 447, 287Salaris, M. 2009, in IAU Symposium, Vol. 258, IAU Symposium, ed. E. E.Mamajek, D. R. Soderblom, & R. F. G. Wyse, 287–298Salim, S. & Gould, A. 2003, ApJ, 582, 1011Sana, H., de Mink, S. E., de Koter, A., et al. 2012, Science, 337, 444Sayres, C., Subasavage, J. P., Bergeron, P., et al. 2012, AJ, 143, 103Scholz, R.-D., Kharchenko, N. V., Lodieu, N., & McCaughrean, M. J. 2008,A&A, 487, 595Scholz, R.-D., Lodieu, N., Ibata, R., et al. 2004, MNRAS, 347, 685Scholz, R.-D., Szokoly, G. P., Andersen, M., Ibata, R., & Irwin, M. J. 2002, ApJ,565, 539Schreiber, M. R. & G¨ansicke, B. T. 2003, A&A, 406, 305Shatsky, N. & Tokovinin, A. 2002, A&A, 382, 92Shaya, E. J. & Olling, R. P. 2011, ApJS, 192, 2Silvestri, N. M., Oswalt, T. D., & Hawley, S. L. 2002, AJ, 124, 1118Sion, E. M., Holberg, J. B., Oswalt, T. D., McCook, G. P., & Wasatonic, R. 2009,AJ, 138, 1681Sion, E. M., Holberg, J. B., Oswalt, T. D., et al. 2014, AJ, 147, 129Sion, E. M. & Oswalt, T. D. 1988, ApJ, 326, 249Sion, E. M., Oswalt, T. D., Liebert, J., & Hintzen, P. 1991, AJ, 101, 1476Sozzetti, A., Giacobbe, P., Lattanzi, M. G., et al. 2014, MNRAS, 437, 497Stau ff er, J., Tanner, A. M., Bryden, G., et al. 2010, PASP, 122, 885Strand, K. A. 1977, AJ, 82, 745Strand, K. A., Dahn, C. C., & Liebert, J. W. 1976, in BAAS, Vol. 8, Bulletin ofthe American Astronomical Society, 506Subasavage, J. P., Henry, T. J., Bergeron, P., Dufour, P., & Hambly, N. C. 2008,AJ, 136, 899Subasavage, J. P., Henry, T. J., Bergeron, P., et al. 2007, AJ, 134, 252Subasavage, J. P., Jao, W.-C., Henry, T. J., et al. 2009, AJ, 137, 4547Tamazian, V. S. & Malkov, O. Y. 2014, Acta Astron., 64, 359Thompson, T. A. 2011, ApJ, 741, 82Tokovinin, A. 2008, MNRAS, 389, 925Tokovinin, A. 2012, AJ, 144, 56Tokovinin, A. 2014, AJ, 147, 87Tokovinin, A., Hartung, M., Hayward, T. L., & Makarov, V. V. 2012, AJ, 144, 7Tokovinin, A., Thomas, S., Sterzik, M., & Udry, S. 2006, A&A, 450, 681Toonen, S., Claeys, J. S. W., Mennekens, N., & Ruiter, A. J. 2014, A&A, 562,A14Toonen, S., Hamers, A., & Portegies Zwart, S. 2016, ComputationalAstrophysics and Cosmology, 3, 6Toonen, S. & Nelemans, G. 2013, A&A, 557, A87Toonen, S., Nelemans, G., & Portegies Zwart, S. 2012, A&A, 546, A70Torres, S. & Garc´ıa-Berro, E. 2016, A&A, 588, A35Torres, S., Garc´ıa-Berro, E., Isern, J., & Figueras, F. 2005, MNRAS, 360, 1381Tout, C. A., Wickramasinghe, D. T., Liebert, J., Ferrario, L., & Pringle, J. E.2008, MNRAS, 387, 897Tremblay, P.-E., Bergeron, P., & Gianninas, A. 2011, ApJ, 730, 128Tremblay, P.-E., Gentile-Fusillo, N., Raddi, R., et al. 2017, MNRAS, 465, 2849Tremblay, P.-E., Kalirai, J. S., Soderblom, D. R., Cignoni, M., & Cummings, J.2014, ApJ, 791, 92Turon, C., Creze, M., Egret, D., et al. 1993, Bulletin d’Information du Centre deDonnees Stellaires, 43, 5Tutukov, A. & Yungelson, L. 1979, in IAU Symposium, Vol. 83, Mass Loss andEvolution of O-Type Stars, ed. P. S. Conti & C. W. H. De Loore, 401–406van Altena, W. F., Lee, J. T., & Ho ffl eit, E. D. 1995, The general catalogue of trigonometric [stellar] parallaxesvan Biesbroeck, G. 1961, AJ, 66, 528van Leeuwen, F. 2007, A&A, 474, 653van Oirschot, P., Nelemans, G., Toonen, S., et al. 2014, A&A, 569, A42Vennes, S. & Kawka, A. 2003, ApJ, 586, L95Veras, D., Wyatt, M. C., Mustill, A. J., Bonsor, A., & Eldridge, J. J. 2011,MNRAS, 417, 2104Vergely, J.-L., K¨oppen, J., Egret, D., & Bienaym´e, O. 2002, A&A, 390, 917Vornanen, T., Berdyugina, S. V., Berdyugin, A. V., & Piirola, V. 2010, ApJ, 720,L52Wang, B. & Han, Z. 2012, New A Rev., 56, 122Webbink, R. F. 1984, ApJ, 277, 355Wegner, G. 1973, MNRAS, 165, 271Wegner, G. 1981, AJ, 86, 264Wegner, G. & McMahan, R. K. 1988, AJ, 96, 1933Weinberg, M. D., Shapiro, S. L., & Wasserman, I. 1987, ApJ, 312, 367Winters, J. G., Sevrinsky, R. A., Jao, W.-C., et al. 2016, ArXiv e-printsWoods, T. E., Ivanova, N., van der Sluys, M., & Chaichenets, S. 2010, inAmerican Institute of Physics Conference Series, Vol. 1314, AmericanInstitute of Physics Conference Series, ed. V. Kologera & M. van der Sluys,24–25Woods, T. E., Ivanova, N., van der Sluys, M. V., & Chaichenets, S. 2012, ApJ,744, 12Zhao, J. K., Oswalt, T. D., Rudkin, M., Zhao, G., & Chen, Y. Q. 2011, AJ, 141,107Zorotovic, M., Schreiber, M. R., G¨ansicke, B. T., & Nebot G´omez-Mor´an, A.2010, A&A, 520, A86Zorotovic, M., Schreiber, M. R., Garc´ıa-Berro, E., et al. 2014, A&A, 568, A68Zuckerman, B., Becklin, E. E., Macintosh, B. A., & Bida, T. 1997, AJ, 113, 764Zuckerman, B., Koester, D., Reid, I. N., & H¨unsch, M. 2003, ApJ, 596, 477
17. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population
Table 1.
Known WDs in binary systems in the Solar neighbourhood. The distances, spectral types, masses, and luminosities aretaken from Giammichele et al. (2012). References for the binarity of the system are given in the last column. For the unresolvedsystems, the period P is given in days instead of angular separation, if available. WD name Distance [pc] Spectral Mass [ M ⊙ ] log L / L ⊙ Companion Spectral Angular Referencestype name type separation [”]
Resolved WDMS +
641 17.35 (0.15) DA5.6 0.66 (0.03) -3.08 GJ 3117 A M2 12.1 1, 2, 3, 4, 50208 −
510 10.782 (0.004) DA6.9 0.59 (0.01) - GJ86A K0 1.9 3, 6, 70415 −
594 18.46 (0.05) DA3.3 0.60 (0.02) - eps. Reticulum A K2 12.8 7, 9,100426 +
588 5.51 (0.02) DC7.1 0.69 (0.02) -3.52 GJ 169.1 A M4.0 9.2 4, 5, 8, 11, 12, 13, 140628 −
020 20.49 (0.46) DA7.2 0.62 (0.01) - LDS 5677B M 4.5 15, 16, 170642 −
166 2.631 (0.009) DA2.0 0.98 (0.03) -1.53 Sirius A A0 7.5 8, 10, 180736 +
053 3.50 (0.01) DQZ6.5 0.63 (0.00) -3.31 Procyon A F5 53 10, 190738 −
172 9.096 (0.046) DZA6.6 0.62 (0.02) -3.35 GJ 238 B M6.5 21.4 3, 8, 20, 210751 −
252 17.78 (0.13) DA9.9 0.59 (0.02) -4.02 LTT2976 M0 400 3, 4, 5, 22, 231009 −
184 18.3 (0.3) DZ8.3 0.59 (0.02) -3.74 LHS 2031 A K7 400 3, 10, 24, 25, 261043 −
188 19.01 (0.18) DQpec8.7 0.53 (0.11) -3.77 GJ 401 A M3 8 3, 4, 5, 151105 −
048 17.33 (3.75) DA3.5 0.54 (0.01) - LP 672-2 M3 279 2, 17, 27, 281132 −
325 9.560 (0.034) DC10 - - HD 100623 K0 16 8, 10, 17, 291327 −
083 16.2 (0.7) DA3.5 0.61 (0.03) -2.16 LHS 353 M4.5 503 3, 28, 30, 311345 +
238 12.1 (0.3) DC11.0 0.45 (0.02) -4.08 LHS 362 M5 199 3, 31, 321544 −
377 15.25 (0.12) DA4.8 0.55 (0.03) -2.67 GJ 599 A G6 15.2 3, 4, 5, 8, 10, 331620 −
391 12.792 (0.062) DA2.1 0.61 (0.02) -1.12 HD 147513 G5 345 8, 10, 27, 31, 341917 −
077 10.1 (0.3) DBQA4.8 0.62 (0.02) -2.81 LDS 678B M6 27.3 3, 20, 312011 +
065 22.4 (1.0) DC7.6 0.7 (0.04) -3.68 LHS 3533 M3.5 101 13, 26, 352151 −
015 24.5 (1.0) DA5.5 0.58 (0.03) -2.96 LTT 8747B M8 1.082 3, 31, 362154 −
512 15.12 (0.12) DQP8.3 0.60 (0.04) -3.44 GJ841 A M2 28.5 3, 4, 5, 30, 37, 382307 +
548 16.2 (0.7) DA8.8 0.58 (-) - G233-42 M5 6 13, 17, 39, 402307 −
691 20.94 (0.38) DA5 0.57 (-) - GJ 1280 K3 13.1 172341 +
322 17.61 (0.55) DA4.0 0.56 (0.03) -2.3 G130-6 M3 175 20, 41, 42
Unresolved WDMS −
487 20.13 (0.55) DA7.8 0.22 (0.05) -3.14 - M4 P = . +
620 21.6 (1.2) DA4.6 1.14 (0.07) - - - - 13, 39
Resolved DWD +
641 33.3 (5.9) DA8.3 0.98 (0.09) -4.09 WD0649 +
639 DA8.1 490 13, 28, 46, this work0747 + + +
734 21.2 (0.8) DA3.2 0.60 (0.03) -1.97 - DC10 1.4 13, 31, 482226 −
754 13.5 (0.9) DC12.0 0.58 (0.00) -4.32 WD2226 −
755 DC12.0 93 3, 49
Unresolved DWD −
052 12.3 (0.4) DA6.9 0.24 (0.01) -3.00 - DA6.9 P = .
56 27, 500532 +
414 22.4 (1.0) DA6.5 0.52 (0.03) -3.20 - - - 3, 49
Unresolved DWD candidate +
277 28.0 (1.5) DA7.8 0.59 (0.00) -3.60 - - - 30121 −
429 18.3 (0.3) DAH8.0 0.41 (0.01) -3.46 - - - 30423 +
120 17.4 (0.8) DC8.2 0.65 (0.04) -3.75 - - - 3,250503 −
174 21.9 (1.9) DAH9.5 0.38 (0.07) -3.75 - - - 30839 −
327 8.80 (0.15) DA5.6 0.44 (0.07) -2.84 - - - 3, 82048 +
263 20.1 (1.4) DA9.9 0.24 (0.04) -3.65 - - - 32248 +
293 20.9 (1.9) DA9.0 0.35 (0.07) -3.62 - - - 32322 +
137 22.3 (1.0) DA9.7 0.35 (0.03) -3.75 - - - 3
Notes. Greenstein (1970); Wegner (1981); Giammichele et al. (2012); Tremblay et al. (2017); Gaia Collaboration et al. (2016); Mugrauer & Neuh¨auser (2005); van Leeuwen (2007); http: // Farihi et al. (2011a); Liebert(1976); Heintz (1990); Limoges et al. (2015); Dieterich et al. (2012); Oswalt et al. (1988); Subasavage et al. (2009); Holberg et al.(2016); Gatewood & Gatewood (1978); Liebert et al. (2013); Luyten (1949); Davison et al. (2015); Subasavage et al. (2008); Luyten & Hughes (1980); Henry et al. (2002); Holberg et al. (2008b); Hawley et al. (1996); Sion et al. (2014); van Altena et al. (1995); Poveda et al. (1994); Eggen (1956); Farihi et al. (2005b); Dahn & Harrington (1976); Wegner (1973); Alexander & Lourens (1969); Giclas et al. (1959); Farihi et al. (2006); Vornanen et al. (2010); Tamazian & Malkov (2014); Limoges et al. (2013); Newton et al.(2014); Sion & Oswalt (1988); Garc´es et al. (2011); Bessell & Wickramasinghe (1979); Bruch & Diaz (1998); Maxted et al. (2007); L´epine & Shara (2005); Greenstein (1970); Zuckerman et al. (1997); Scholz et al. (2002); Sa ff er et al. (1998) Zuckerman et al. (2003).18. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population
Table 2.
Known WDs in the Solar neighbourhood that are part of triples and quadruples. The distances, spectral types, masses,and luminosities are taken from Giammichele et al. (2012). For the unresolved systems, the period P is given instead of angularseparation. Distance [pc] Spectral Mass [ M ⊙ ] log L / L ⊙ Companion Spectral Angular Referencestype name type separation [”]0101 +
048 21.3 (1.7) DA6.3 0.36 (0.05) -2.96 - DC see text 1, 2, 3, 4HD 6101 K3 + K8 12760326 −
273 17.4 (4.3) DA5.9 0.45 (0.18) -2.97 - DC8 P = . d
4, 5, 6GB 1060B M3.5 70413 −
077 4.984 (0.006) DA3.1 0.59 (0.03) -1.85 40 Eri A K0.5 83.4 7, 8, 9, 1040 Eri C M4.5 11.90433 +
270 17.48 (0.13) DA9 0.62 (0.02) -3.87 V833 Tau K2 + + ∗ −
336 15.2 (0.1) DC10.6 0.55 (0.01) -4.23 171 Pup A F9 870 8, 18, 19, 201633 +
572 14.4 (0.5) DQpec8.1 0.57 (0.04) -3.75 CM draconis M4.5 ∗
26 20, 212054 −
050 16.09 (0.14) DC11.6 0.37 (0.06) -4.11 Ross 193 M3.0 15.1 4, 10, 11, 12, 22, 232351 −
335 22.90 (0.75) , DA5.7 0.58 (0.03) -3.03 LDS826B M3.5 6.6 4, 6, 24, 25, 26LDS826C M8.5 103
Notes. Sa ff er et al. (1998) Maxted et al. (2000); Caballero (2009); Giammichele et al. (2012); Nelemans et al. (2005); Luyten (1949); Holberg et al. (2012); Holberg et al. (2013); Sion et al. (2014); Discovery and Evaluation of Nearby Stellar Embers (DENSE) project,http: // Tremblay et al. (2017); Gaia Collaboration et al. (2016); Hartmann et al. (1981); Tokovinin et al. (2006); Strand et al. (1976); Harrington et al. (1981); Buscombe & Foster (1998); Hartkopf et al. (2012); Tokovinin et al. (2012); Limoges et al.(2015); Morales et al. (2009); van Biesbroeck (1961); Tamazian & Malkov (2014); Scholz et al. (2004); Farihi et al. (2005b); Subasavage et al. (2009); ∗ Spectral type corresponds to an unresolved binary;
Table 3.
Overview of di ff erent BPS models. There are two models for the SFH, two for the period distribution, and three for theCE-phase, giving 12 models in total. Model Description ReferenceStar formation history BP Star formation rate and space density depends on time and location 1in the Galaxy. SFR peaks at early times, declines afterwardscSFR Constant space density and SFR for 10 Gyr -Initial period distribution Abt Log-uniform 2Lognormal Lognormal distribution with a mean of 5.03 d 3Common-envelope phase γα γ = . αλ =
2; Preferred for unresolved DWDs 4,5,6 αα αλ = αα αλ = .
25; Preferred for unresolved WDMS 7,8,9
Notes. Boissier & Prantzos (1999); Abt (1983); Raghavan et al. (2010); Nelemans et al. (2000); Nelemans et al. (2001b); Toonen et al.(2012); Zorotovic et al. (2010); Toonen & Nelemans (2013); Camacho et al. (2014). 19. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf population
Table 4.
Number of systems with WDs components within 20 pc, see also Fig. 3. The observed sample is based onGiammichele et al. (2012), but see Sect. 2 for adaptations. For unresolved DWDs, we list two numbers. The first number repre-sents confirmed DWD systems, whereas the number in brackets represents the number of confirmed plus candidate DWDs. Thethird line lists the number of WD systems in triples and quadruples, which are not included in the first line. The evolution of thesesystems has not been simulated in the BPS models. The di ff erent BPS models are described in Sect. 3 and an overview is given inTable 3. The selection e ff ects described in Sect. 3.7 have been applied to the BPS models. Single WDs are formed by single stellarevolution and mergers in binaries. As such, for a given BPS model, the sum of the ‘Single stars’ column and the ‘Mergers’ columnshould be compared with the observed number of single WDs. The statistical errors on the BPS simulations are given in brackets. Observations
Single WDs WDMS DWDResolved Unresolved Resolved Unresolved
Observed . ± . . ± . . ± . . ± . . ± . . ± . .
58 2 . . . . ± .
01 0 1 . ± . . ± . . ± . BPS models
Single WDs Mergers WDMS DWDSFH Period distr. CE Resolved Unresolved Resolved UnresolvedBP Abt γα
126 (3.5) 36 (1.9) 30 (0.8) 2.4 (0.21) 20 (0.63) 8.2 (0.40) αα
43 (2.1) 2.3 (0.21) 4.0 (0.28) αα γα
126 (3.5) 15 (1.2) 40 (0.9) 2.5 (0.22) 28 (0.75) 8.0 (0.40) αα
19 (1.4) 2.4 (0.22) 4.0 (0.28) αα γα
89 (0.5) 26 (0.1) 22 (0.23) 1.8 (0.07) 15 (0.06) 6.1 (0.04) αα
30 (0.1) 1.9 (0.07) 3.1 (0.03) αα γα
89 (0.5) 12 (0.05) 29 (0.27) 1.9 (0.07) 21 (0.07) 5.8 (0.04) αα
14 (0.06) 2.0 (0.07) 3.0 (0.03) αα Table 5.
Number of systems with WDs components within 50 pc for di ff erent BPS models. The Gaia WD sample is expected to beroughly complete out to approximately 50 pc (Sect. 5). Here a limiting angular separation of 0.3” is assumed to di ff erentiate betweenresolved and unresolved binaries. The table layout is the same as Table 4 with one extra column. The column on the far right showsthe number of unresolved DWDs with periods less than ten days. The statistical error is omitted, as it is smaller than the systematicerror, that is, variation between the di ff erent BPS models. BPS models
Single stars Mergers WDMS DWDSFH Period distr. CE Resolved Unresolved Resolved Unresolved P < γα αα
640 31 65 38 αα γα αα
297 32 65 43 αα γα αα
467 23 50 31 αα γα αα
219 24 48 26 αα Table A.1.
Known single WDs in the Solar neighbourhood. This sample is mostly based on Giammichele et al. (2012) with additionsand modifications from papers indicated in the last column.
Distance [pc] Spectral type Mass [ M ⊙ ] log L / L ⊙ References0000 −
345 13.2 (1.6) DAH 0.88 (0.10) -3.820004 +
122 21.0 (3.4) 0.57 (0.15) -4.02 10005 +
395 20.21 (1.25) 0.58 (-) - 20008 +
424 21.4 (1.1) DA 0.64 (0.04) -3.450009 +
501 11.0 (0.5) DAP 0.73 (0.04) -3.720011 −
134 19.5 (1.5) DAH 0.72 (0.07) -3.850011 −
721 17.6 (0.7) DA 0.59 (0.00) -3.630019 +
423 Sect. 2.1.2 0.58 (0.15) -3.85 10025 +
054 21.12 (1.71) 0.58 (-) - 20038 −
226 9.05 (0.10) DQpec 0.53 (0.01) -3.940046 +
051 4.297 (0.033) DZ 0.68 (0.02) -3.77 3, 4, 50053 −
117 20.7 (1.3) DA 0.67 (0.05) -3.490115 +
159 15.4 (0.7) DQ 0.69 (0.04) -3.10123 −
262 21.7 (0.8) DC 0.58 (0.00) -3.40136 +
152 21.2 (0.8) 0.72 (0.03) -3.34 10141 −
675 9.73 (0.080) DA 0.48 (0.06) -3.55 60148 +
467 15.5 (0.8) DA 0.63 (0.03) -2.26 2, 7, 80208 +
396 16.7 (1.0) DAZ 0.59 (0.05) -3.390213 +
396 20.9 (0.9) DA 0.8 (0.03) -3.140213 +
427 19.9 (1.6) DA 0.64 (0.08) -3.930230 −
144 15.6 (1.0) DA 0.66 (0.06) -3.960233 −
242 16.7 (0.7) DC 0.58 (0.00) -3.940236 +
259 21.8 (0.8) DA 0.59 (0.00) -3.830243 −
026 21.2 (2.3) DAZ 0.7 (0.10) -3.620245 +
541 10.3 (0.3) DAZ 0.73 (0.03) -4.130252 +
497 17.99 (2.9) 1.2 (0.11) - 20255 −
705 27.8 (1.1) DA 0.57 (0.03) -2.670310 −
688 10.15 (0.15) DA 0.67 (0.03) -1.97 2, 3, 4, 50322 −
019 16.8 (0.9) DAZ 0.63 (0.05) -4.020340 +
198 19.5 (0.83) 0.94 (0.05) - 20341 +
182 19.0 (1.1) DQ 0.57 (0.06) -3.570344 +
014 20.6 (1.2) DC 0.58 (0.00) -3.990357 +
081 17.8 (1.2) DA 0.61 (0.06) -3.910414 +
420 23.8 (3.6) 0.58 (-) - 20423 +
044 20.9 (1.7) 0.67 (0.08) -4.22 9, 100435 −
088 9.51 (0.24) DQ 0.53 (0.02) -3.590457 −
004 28.7 (1.4) DA 1.07 (0.03) -3.090511 +
079 20.3 (0.6) 0.8 (0.08) -3.75 9, 10, 110541 +
620 20.4 (3.2 ) 0.58 (-) - 20548 −
001 11.1 (0.3) DQP 0.69 (0.03) -3.80552 −
041 6.412 (0.032) DZ 0.82 (0.01) -4.21 3, 5, 60553 +
053 8.0 (0.23) DAH 0.72 (0.03) -3.910618 +
067 22.6 (2.1) 0.93 (0.17) -4.05 10620 −
402 25.3 (4.0) - - 9, 120644 +
025 18.4 (1.9) DA 1.01 (0.07) -3.790644 +
375 15.276 (0.423) DA 0.69 (0.03) -1.48 3, 4, 50655 −
390 17.1 (0.7) DA 0.59 (0.00) -3.640657 +
320 18.7 (0.3) DA 0.6 (0.02) -4.10659 −
063 12.3 (1.3) DA 0.82 (0.07) -3.770708 −
670 17.3 (0.6) DC 0.57 (0.00) -4.020728 +
642 18.4 (0.5) DAP 0.58 (0.00) -4.00749 +
426 24.6 (0.8) DC 0.58 (0.00) -4.20752 −
676 7.898 (0.082) DA 0.73 (0.06) -3.94 3, 5, 60802 +
387 20.8 (1.8) 0.73 (0.02) -4.13 10805 +
356 24.5 (0.8) 0.83 (0.03) -3.25 9, 130806 −
661 19.2 (0.6) DQ 0.58 (0.03) -2.800810 +
489 18.3 (0.6) DC 0.57 (0.00) -3.550816 −
310 22.1 (1.6) DZ 0.57 (0.00) -3.610821 −
669 10.7 (0.1) DA 0.66 (0.01) -4.080827 +
328 22.3 (1.9) DA 0.84 (0.07) -3.640840 −
136 13.9 (0.8) DZ 0.57 (0.0) -4.10843 +
358 27.0 (1.5) DZA 0.58 (0.0) -3.020856 +
331 20.5 (1.4) DQ 1.05 (0.05) -3.320912 +
536 10.3 (0.2) DCP 0.75 (0.02) -3.570939 +
071 18.9 - -0946 +
534 23.0 (1.9) DQ 0.74 (0.08) -3.350955 +
247 24.4 (2.7) DA 0.76 (0.10) -3.24 21. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf populationDistance [pc] Spectral type Mass [ M ⊙ ] log L / L ⊙ References1008 +
290 14.8 (0.1) DQpecP 0.68 (0.01) -4.311019 +
637 16.4 (1.0) DA 0.57 (0.05) -3.51033 +
714 19.6 (0.8) DC 0.58 (0.00) -4.151036 −
204 14.3 (0.1) DQpecP 0.6 (0.01) -4.191055 −
072 12.2 (0.5) DC 0.85 (0.04) -3.61116 −
470 17.5 (0.7) DC 0.57 (0.00) -3.81121 +
216 13.4 (0.5) DA 0.71 (0.03) -3.461124 +
595 27.6 (1.3) DA 0.98 (0.03) -3.091134 +
300 15.3 (0.7) DA 0.97 (0.03) -1.781142 −
645 4.634 (0.008) DQ 0.61 (0.01) -3.27 3, 4, 5, 6, 7, 81143 +
633 21.3 (3.4) 0.58 (0.15) -3.95 1, 141145-451 22.94 (2.08) 0.58 (0.12) - 21148 +
687 18.0 (0.6) 0.69 (0.04) -3.64 9, 151202 −
232 10.83 (0.11) DAZ 0.59 (0.03) -3.05 61208 +
576 20.4 (1.9) DAZ 0.56 (0.09) -3.741223 −
659 10.26 (0.31) DA 0.45 (0.02) -3.16 61236 −
495 16.4 (2.6) DAV 1.0 (0.11) -2.971257 +
037 16.6 (1.0) DA 0.7 (0.06) -3.951309 +
853 16.5 (0.3) DAP 0.71 (0.02) -4.011310 +
583 24.9 (1.0) DA 0.66 (0.03) -2.771310 −
472 15.0 (0.5) DC 0.63 (0.04) -4.421315 −
781 19.2 (0.3) DC 0.69 (0.02) -3.941334 +
039 8.24 (0.23) DA 0.54 (0.03) -4.021339 −
340 21.0 (1.2) DA 0.58 (0.00) -3.961344 +
106 20.0 (1.5) DAZ 0.65 (0.07) -3.491344 +
572 25.8 (0.8) 0.53 (0.03) -4.02 9, 111350 −
090 25.3 (1.0) DAP 0.68 (0.03) -2.981425 −
811 26.9 (1.0) DAV 0.61 (0.03) -2.461443 +
256 17.5 (2) 0.58 (-) - 21444 −
174 14.5 (0.8) DC 0.82 (0.05) -4.271524 +
297 22.4 (2.6) 0.58 (-) - 21532 +
129 19.17 (0.38) 0.57 (0.15) -3.99 11538 +
333 29.1 (1.1) DA 0.63 (0.03) -3.061540 +
236 19.6 (0.8) 1.11 (0.1) -4.2 1, 161609 +
135 18.4 (1.6) DA 1.07 (0.06) -3.51626 +
368 15.9 (0.5) DZA 0.58 (0.03) -3.131630 +
089 13.8 (0.4) 0.59 (0.15) -3.81 1, 9, 171632 +
177 18.7 (0.7) DA 0.46 (0.02) -2.641633 +
433 15.1 (0.7) DAZ 0.68 (0.04) -3.631639 +
537 21.2 (1.6) 0.62 (0.11) -3.4 9, 181647 +
591 10.98 (0.07) DAV 0.76 (0.03) -2.55 3, 4, 5, 7, 81653 +
385 30.7 (1.2) DAZ 0.59 (0.00) -3.771655 +
215 23.3 (1.7) DA 0.52 (0.06) -2.91657 +
321 51.7 (2.5) DA 0.59 (0.00) -3.621705 +
030 17.5 (1.7) DZ 0.68 (0.09) -3.671729 +
371 50.3 (2.2) DAZB 0.64 (0.03) -2.81748 +
708 6.07 (0.09) DXP 0.79 (0.01 -4.071756 +
143 20.5 (1.2) DA 0.58 (0.00) -3.991756 +
827 15.7 (0.7) DA 0.58 (0.04) -3.391814 +
134 14.2 (0.2) DA 0.68 (0.02) -4.051820 +
609 12.8 (0.7) DA 0.56 (0.05) -4.061829 +
547 15.0 (1.3) DXP 0.9 (0.07) -3.941900 +
705 13.0 (0.4) DAP 0.93 (0.02) -2.881912 +
143 35.0 (6.6) 1.03 (0.09) -3.89 1, 91917 +
386 10.51 (0.06) DC 0.75 (0.04) -3.77 7, 81919 +
145 19.8 (0.8) DA 0.74 (0.03) -2.211935 +
276 18.0 (0.9) DAV 0.6 (0.03) -2.41 151953 −
011 11.4 (0.4) DAH 0.73 (0.03) -3.382002 −
110 17.3 (0.2) DC 0.72 (0.01) -4.292007 −
303 15.4 (0.6) DA 0.6 (0.02) -1.972008 −
600 16.6 (0.2) DC 0.44 (0.01) -3.972032 +
248 14.6 (0.4) DA 0.64 (0.03) -1.56 3, 4, 52039 −
202 21.1 (0.8) DA 0.61 (0.03) -1.582039 −
682 19.6 (0.9) DA 0.98 (0.03) -2.272040 −
392 22.6 (0.5) DA 0.61 (0.03) -2.62 62047 +
372 17.3 (0.3) DA 0.81 (0.03) -2.342048 −
250 28.2 (1.1) DA 0.59 (0.00) -3.312058 +
550 22.6 (2.5) 0.58 (-) - 222. Toonen, M. Hollands, B.T. G¨ansicke,T. Boekholt: The binarity of the local white dwarf populationDistance [pc] Spectral type Mass [ M ⊙ ] log L / L ⊙ References2105 −
820 17.1 (2.6) DAZH 0.74 (0.13) -2.932115 −
560 26.5 (1.0) DAZ 0.58 (0.03) -2.832117 +
539 17.3 (0.2) DA 0.56 (0.03) -2.1 7, 82133 −
135 20.4 (3.5) - - 3, 62138 −
332 15.6 (0.3) DZ 0.7 (0.02) -3.482140 +
207 12.5 (0.5) DQ 0.48 (0.04) -3.092159 −
754 21.0 (1.1) DA 0.92 (0.04) -3.352210 +
565 22.3 (1.4) 0.68 (0.03) -1.97 7, 18, 192211 −
392 18.7 (0.9) DA 0.8 (0.04) -3.882215 +
368 23.5 (1.8) DC 0.58 (0.00) -4.052246 +
223 19.1 (1.5) DA 0.96 (0.06) -3.132251 −
070 8.520 (0.069) DZ 0.58 (0.03) -4.45 3, 5, 62326 +
049 13.6 (0.8) DAZ 0.63 (0.03) -2.52336 −
079 15.9 (0.4) DAV 0.76 (0.02) -2.822345 +
027 22.7 (3.6) 0.58 (-) - 22347 +
292 21.5 (1.9) DA 0.49 (0.08) -3.692359 −
434 8.169 (0.074) DA 0.78 (0.03) -3.26 3, 5, 6
Notes. van Altena et al. (1995); van Leeuwen (2007); Discovery and Evaluation of Nearby Stellar Embers (DENSE) project,http: // Subasavage et al. (2009); Sion et al. (2014); Gatewood & Coban (2009); Tremblay et al. (2017); Gaia Collaboration et al. (2016); Gianninas et al. (2011); Subasavage et al. (2008); Tremblay et al. (2011); Holberg et al. (2013); Limoges et al. (2013); Limoges et al. (2015); van Altena et al. (1995); Sayres et al. (2012); Salim & Gould (2003); Gliese & Jahreiß(1991);19