Smoking Gun of the Dynamical Processing of the Solar-type Field Binary Stars
MMNRAS , 1–17 (2019) Preprint 12 August 2019 Compiled using MNRAS L A TEX style file v3.0
Smoking Gun of the Dynamical Processing of theSolar-type Field Binary Stars
Chao Liu , (cid:63) Key Lab of Optical Astronomy, National Astronomical Observatories, CAS, Beijing 100101, China University of Chinese Academy of Sciences, Beijing 100049, China
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We investigate the binarity properties in field stars using more than 50 000 main-sequence stars with stellar mass from 0.4 to 0.85 M (cid:12) observed by LAMOST and G aia inthe solar neighborhood. By adopting a power-law shape of the mass-ratio distributionwith power index of γ , we conduct a hierarchical Bayesian model to derive the binaryfraction ( f b ) and γ for stellar populations with different metallicities and primarymasses ( m ). We find that f b is tightly anti-correlated with γ , i.e. the populations withsmaller binary fraction contains more binaries with larger mass-ratio and vice versa.The high- γ populations with γ > . have lower stellar mass and higher metallicity,while the low- γ populations with γ < . have larger mass or lower metallicity. The f b of the high- γ group is anti-correlated with [Fe/H] but flat with m . Meanwhile, the f b of the low- γ group displays clear correlation with m but quite flat with [Fe/H]. Thesubstantial differences are likely due to the dynamical processing when the binarieswere in the embedded star clusters in their early days. The dynamical processingtends to destroy binaries with smaller primary mass, smaller mass-ratio, and widerseparation. Consequently, the high- γ group containing smaller m is more effectivelyinfluenced and hence contains less binaries, many of which have larger mass-ratio andshorter period. However, the low- γ group is less affected by the dynamical processingdue to their larger m . These are evident that the dynamical processing does effectivelywork and significantly reshape the present-day binary properties of field stars. Key words: binaries: general – Hertzsprung-Russell and colour-magnitude diagrams– Stars: solar-type – Stars: abundances – Stats:formation – methods:statistical
Binary stars are very common in stellar systems with dif-ferent scales, from star cluster to the whole galaxy. Someliteratures even thought that the most of stars live in bi-nary systems (Heintz 1969; Abt & Levy 1976; Duquennoy &Mayor 1991). The statistical analysis of binary systems playimportant role to constrain star formation in different envi-ronments. The statistical properties that can well describebinary systems include the fraction of binary, the distribu-tion of mass-ratio, the distribution of orbital period, and thedistribution of the orbital eccentricity. These properties areusually compared between populations with different age,metallicity, and primary mass (Duquennoy & Mayor 1991;Henry & McCarthy 1990; Fischer & Marcy 1992; Mason etal. 1998a,b).The association of binary fraction with metallicity hasbeen broadly studied by lots of observational (Carney 1983; (cid:63)
E-mail: [email protected]
Latham et al. 2002; Raghavan et al. 2010; Rastegaev 2010;Moe & Di Stefano 2013; Gao et al. 2014; Hettinger et al.2015; Gao et al. 2017; Tian et al. 2018; Badenes et al.2018; Moe, Kratter & Badenes 2019, etc.) and theoreticalworks (Machida et al. 2009; Kratter et al. 2010; Myers et al.2011; Bate 2014; Tanaka & Omukai 2014, etc.). Recently, lotsof works, especially depending on large volume of datasets,found that binary fraction, at least for close binaries, is anti-correlated with metallicity (e.g. Raghavan et al. 2010; Gaoet al. 2014, 2017; Tian et al. 2018; Badenes et al. 2018; Moeet al. 2019), while a few others claimed that f b is positivelycorrelated (e.g. Carney 1983; Abt & Willmarth 1987; Het-tinger et al. 2015) or not correlated with metallicity (Lathamet al. 2002; Moe & Di Stefano 2013).Among these works, Moe et al. (2019) reanalyzed fivedifferent datasets of close binaries with sample size from sev-eral hundreds to several ten thousands from literatures andconcluded that the binary fraction is strongly anti-correlatedwith metallicity. The datasets used in their studies containclose binaries with separation smaller than AU. As com- © a r X i v : . [ a s t r o - ph . S R ] A ug C. Liu plementary, El-Badry & Rix (2019) discussed about the vari-ation of the binary fraction as a function of metallicity forwide binaries with separation of –
50 000
AU. They foundthe anti-correlation between the binary fraction and metal-licity exists only when the separation of the two companionsis between and AU. It becomes flat for binaries withlarger separations as < a < AU.The binary fraction of main sequence (MS) stars isalso found positively correlated with primary mass in manyworks (see the review by Duchˆene & Kraus 2013, andreferences therein). For very low-mass field binaries ( m < . M (cid:12) ), the binary fraction is only about % (Allen 2007).For low-mass binaries ( . < m < . M (cid:12) ), the fraction in-creases to around % (Delfosse et al. 2004; Dieterich etal. 2012). For solar-type stars ( . < m < . M (cid:12) , whichis also defined as the MS stars with luminosity from . to L (cid:12) by Raghavan et al. (2010)), the fraction further in-creases to around – % (Raghavan et al. 2010). And forthe intermediate-mass ( . < m < M (cid:12) ) and massive stars( m > M (cid:12) ), the fraction is larger than % (Chini et al.2012; Mason et al. 2009; Sana et al. 2013).For the well studied solar-type binaries, the argumentsabout mass-ratio distribution are contradicted. Duquennoy& Mayor (1991) derived a mass-ratio distribution biased tolower mass-ratio for solar-type stars, which makes it verysimilar to the canonical initial mass function (IMF) (Miller& Scalo 1979; Kroupa et al. 1990). However, later stud-ies claimed different shape. Raghavan et al. (2010) foundthat the mass-ratio distribution is rather flat with a peak at q > . , which is not like any of the canonical IMF (Bastian,Covey & Meyer 2010). Goldberg, Mazeh, & Latham (2003),on the other hand, showed a double peaked mass-ratio dis-tribution. Duchˆene & Kraus (2013) compiled many refer-ences and found that if mass-ratio distribution is describedwith a power law, then the power index decreases with pri-mary mass. Again, they demonstrated that the mass-ratiodistribution, which is equivalent with the present-day massfunction of the secondary, is significantly biased from any ofthe canonical IMF.A large fraction of previous studies on the binary frac-tion and mass-ratio distribution were based on small datasetwith number of stars from a dozen to a few hundreds,which suffers from substantially large Poisson noise. In re-cent years, some of the works (Gao et al. 2014, 2017; Het-tinger et al. 2015; Badenes et al. 2018; Moe et al. 2019;El-Badry et al. 2018b; El-Badry & Rix 2019) have made useof large survey data.In light of these studies, we make the measurement ofthe binary fraction and mass-ratio distribution for stars withdifferent metallicity and primary mass using the LAMOSTdata (Zhao et al. 2012) combined with Gaia
DR2 (Gaia Col-laboration et al. 2016, 2018). LAMOST DR5 have collectedmore than 8 million stellar spectra with spectral resolutionof R = and limiting magnitude of r = mag (Deng etal. 2012). It provides stellar effective temperature, surfacegravity, and [Fe/H], with precision of about 110 K, 0.2 dex,and 0.15 dex (Luo et al. 2015; Gao et al. 2015). These pa-rameters enable us to derive the stellar mass for solar-typestars. On the other hand, Gaia
DR2 provides very accuratephotometries in G , BP , and RP bands as well as astrometry(Gaia Collaboration et al. 2018; Luri et al. 2018; Lindegrenet al. 2018). With accurate luminosity derived from parallax, field stars can be mapped in the Hertzprung-Russell (HR)diagram with clear binary sequence located around 0.75 magabove the single main-sequence (see Gaia Collaboration etal. 2018). Hence, one can study the statistical properties ofbinaries directly in HR diagram, similar to the binary stud-ies based on the photometries of star clusters or OB associ-ations (e.g. Kouwenhoven et al. 2007; Milone et al. 2012; Li,de Grijs & Deng 2013; Yang et al. 2018).The paper is organized as follows. Section 2 describehow we select the solar-type stars and how to estimate theirparameters. Section 3 develops the hierarchical Bayesianmodel which can simultaneously derive binary fraction andmass-ratio distribution. Section 4 shows the results for stel-lar populations with various metallicity and primary mass.From the Bayesian model, we also provide probabilities tobe binary of the sample stars with the mass-ratio estimates.Section 5 raises discussions about the caveats of the methodand comparisons with other works. Finally, brief conclusionsare drawn in section 6. We select common stars cross-identified from LAMOST DR5( http://dr5.lamost.org ), Gaia
DR2 (Gaia Collaborationet al. 2018), and 2MASS (Skrutskie et al. 2006). The cross-identification is conducted by comparing the right ascen-sions and declinations in equinox of J2000.0 among thethree catalogs. Because the diameter of the LAMOST fiberis . arcsec and the mean seeing during LAMOST obser-vations is around arcsec (Shi, J-R. private communica-tion), the spatial resolution of LAMOST should be around arcsec, which is used as the radius of circle in the cross-identification. Only the star pairs with nearest positionswithin the circle are selected.First, we derive absolute magnitude in G -band of thestars from Gaia parallax based on the likelihood distribu-tion. which can be written as p ( M (cid:48) G | (cid:36), G ) ∝ exp (cid:32) − ( −( G − M (cid:48) G + )/ − (cid:36) ) σ (cid:36) (cid:33) , (1)where M (cid:48) G is the Gaia G -band absolute magnitude withoutextinction correction, (cid:36) and σ (cid:36) are the parallax and itsuncertainty, respectively. M (cid:48) G equals to M G + A G , where M G and A G are the Gaia G -band absolute magnitude with ex-tinction correction and the interstellar extinction in G -band,respectively. We estimate the maximum likely M (cid:48) G and is un-certainty for each star by arbitrarily drawing 10 000 pointsfrom the distribution defined by Eq. (1).We then select samples using the following criteria:1 M (cid:48) G > mag;2 (cid:36)> so that most of the stars are located within ∼ pc;3 (cid:36) / σ (cid:36) > ;4 phot_bp_mean_flux / phot_bp_mean_flux_error > , phot_rp_mean_flux / phot_rp_mean_flux_error > , and phot_g_mean_flux / phot_g_mean_flux_error > ;5 (cid:112) χ /( ν − ) < . × max ( , exp (− . ( G − . ))) , where χ and ν represent for astrometric_chi2_al and astromet-ric_n_good_obs_al , respectively; MNRAS , 1–17 (2019) olar-type Field Binary Stars K s < . mag and σ K s < . mag;7 T eff derived from LAMOST is in between and < K;8 signal-to-noise ratio at g band of LAMOST spectra islarger than 20.The first criterion ensures that the hot MS stars are re-moved.The second criterion selects the stars located within ∼ pc so that the data comply with the requirement ofthe completeness. The faintest apparent magnitude for astar with mass of . M (cid:12) at distance of pc is between . and . mag depending on different metallicity. This isbright enough for the companion in a binary to contributeluminosity that is able to be detected by LAMOST. There-fore, the volume cut enables the detection of the faintestsecondaries. The criteria 3–6 is set to select stars with ac-curate photometry and astrometry. Among them, criterion5 is adopted from El-Badry & Rix (2018). Criteria 7 and 8help to select high quality spectra from LAMOST so thatthe stellar parameters derived from the spectra are reliable.Meanwhile, the two criteria also comply with the criterion2. After applying these selection criteria, we obtain 101 061MS stars in total. We adopt the present-day mass as the initial mass for theprimary stars. First, because in this work we only focus onthe main-sequence stars with stellar mass of 0.4-1 M (cid:12) , themass loss due to stellar wind is very small and ignorable.Second, the fraction of mass transferring binary system isless than one percent (Rucinski 1994; Chen et al. 2016) forthe solar-mass main-sequence stars, hence the change of thestellar mass due to mass transfer is also ignored.The initial stellar mass of each star is determinedby comparing their T eff , log g , and [Fe/H] with PARSECisochrones (Bressan et al. 2012) by means of likelihood (seethe details in Appendix A).Although the stellar parameters obtained from theLAMOST spectra are dominated by the primary stars ina binary system, the secondary stars may averagely reducethe effective temperature by ∼ K when q > . and in-duce additional 0.1 dex and 0.05 dex dispersions in log g and[Fe/H], respectively, for the case of LAMOST spectra (El-Badry et al. 2018a). Considering these systematics and un-certainties, the total uncertainty of the likelihood approachis about . M (cid:12) . The determination of the interstellar extinction of the starsis described in Appendix B. The uncertainty of the extinc-tion in G band is around . mag.We further select the stars with . < ( G − K s ) < . ,where ( G − K s ) is the dereddened color index, so that thedata can cover stellar mass from . to . M (cid:12) . A few starswithout initial mass estimates due to their unusual photome-tries are excluded. After this cut, the samples are reducedto 76 507 stars. M G ( m a g ) M a ss ( M ) [ F e / H ] ( d e x ) D E C ( d e g r ee ) Figure 1.
The top-left panel displays the distribution of the sam-ple stars in M G vs. distance plane. The top-right panel shows thedistribution of the stars in stellar mass vs. distance plane. Thebottom-left panel shows the distribution of [Fe/H] along distance.The bottom-right panel shows the distribution in right ascensionand declination of the samples. The colors indicate the logarith-mic density of stars in the bins. We adopt the Bayesian distance derived from the parallaxof
Gaia by Bailer-Jones et al. (2018). The top panels ofFigure 1 demonstrate the distributions of the stars in M G and mass vs. distance planes. It shows that M G coveringfrom 4 to 8 mag does not show correlation with distance from50 to 333 pc. Meanwhile, stellar mass in the range from 0.4 to1 M (cid:12) is also independent on distance. The bottom-left panelof the figure shows that [Fe/H] is uncorrelated with distance.These mean that M G , stellar mass, and [Fe/H] of the samplesare unbiased within the volume from 50 to 333 pc.Although LAMOST survey only focuses on northernsky, it basically continuously cover the sky of about 20 000square degrees, from low to high Galactic latitude, as seen inthe bottom-right panel of Figure 1. Therefore, the samplesselected in this work is representative to the stellar popula-tions in the solar neighborhood. Because different [Fe/H] may shift the main-sequence upand down in HR diagram, they can blur out the binarysequence. Therefore, we separate the stars in different[Fe/H] bins. The uncertainty of the LAMOST derived[Fe/H] is around 0.15 dex (Gao et al. 2015). Thus, weset the bin size of [Fe/H] lager than 0.2 dex. On theother hand, we need to keep sufficient number of stars ineach [Fe/H] bin for further separation in primary massin section 4. Therefore, we adopt 8 [Fe/H] bins withranges of − . < [Fe/H] < − . , − . < [Fe/H] < − . , − . < [Fe/H] < − . , − . < [Fe/H] < − . , − . < [Fe/H] < − . , − . < [Fe/H] < + . , − . < [Fe/H] < + . , and + . < [Fe/H] < + . . Notethat the neighboring bins are partly overlapped with eachother such that the number of stars in each bin is as large MNRAS , 1–17 (2019)
C. Liu M G ( m a g ) -1.25<=[Fe/H]<=-0.55 0.00.51.01.52.0 1.50 1.75 2.00 2.25 2.505678 -0.65<=[Fe/H]<=-0.40 0.00.51.01.52.0 1.50 1.75 2.00 2.25 2.505678 -0.50<=[Fe/H]<=-0.25 0.00.51.01.52.0 1.50 1.75 2.00 2.25 2.50( G Ks ) (mag)5678 -0.35<=[Fe/H]<=-0.15 0.00.51.01.52.01.50 1.75 2.00 2.25 2.50( G Ks ) (mag)5678 M G ( m a g ) -0.25<=[Fe/H]<=-0.05 0.00.51.01.52.0 1.50 1.75 2.00 2.25 2.50( G Ks ) (mag)5678 -0.15<=[Fe/H]<=0.05 0.00.51.01.52.0 1.50 1.75 2.00 2.25 2.50( G Ks ) (mag)5678 -0.05<=[Fe/H]<=0.15 0.00.51.01.52.0 1.50 1.75 2.00 2.25 2.50( G Ks ) (mag)5678 0.05<=[Fe/H]<=0.40 0.00.51.01.52.0 Figure 2.
The color coded logarithmic density distributions of the samples in M G vs. ( G − K s ) plane with different [Fe/H] bins, whichranges are indicated in the bottom of the panels. The reddening of the stars have been corrected in color indices. as possible. The overlapping would not harm the finalresults, but only smooth the results to some extent. Thedensity distributions of the sample stars in M G vs. ( G − K s ) plane with different [Fe/H] bins is displayed in Figure 2.The density map is calculated in the grid of ( G − K s ) and M G with size of . × . mag. The colors in the figurecodes the logarithmic density of stars. It is clearly seen twosequences in the color-absolute magnitude diagrams suchthat the unresolved binary sequence is located at upper sideof the single star sequence due to their larger luminositycontributed by both companions.The shift of the absolute magnitude of an unresolvedbinary star from the single sequence reflects the mass-ratioof the binary system, which is denoted as q = m / m , where m and m are the masses of the primary and secondarystars, respectively. When q is small ( (cid:46) . ), the contribu-tion to the luminosity from the secondary star is only around0.1 mag. Therefore, the composed luminosity for such bina-ries is similar to the single stars. When q becomes larger( (cid:38) . ), the contribution from the secondary becomes ob-vious, hence the luminosity of the whole system increasessubstantially. When q is close to 1, the composed luminosityis doubled, which is equivalent to 0.75 mag brighter.To quantify the difference between the single and binarystars in M G , we firstly address the ridge lines of the singleMS stars in a finer [Fe/H] grid with bin size of . dex. Foreach [Fe/H] slice, we further separate the stars into ( G − K s ) bins with size of . mag. At each color index bin, we applya kernel density estimation (KDE) to produce the densitydistribution of M G . Specifically, the density of M G is p ( M G |[ Fe / H ] , ( G − K ) ) = n (cid:213) i = K ( M G − M G , i σ M G , i ) , (2)where K ( t ) = exp (− t / ) is the kernel function; M G , i and G K ) [ F e / H ] M G Figure 3.
The colors show the intrinsic absolute magnitude ofsingle stars in G band as a function of ( G − K s ) and [Fe/H]. Thefive points marked with black frames are interpolated from theneighboring points. σ M G , i are the absolute magnitude and its uncertainty, re-spectively, of the i th star. We then locate the intrinsic M G of single stars at the peak of p ( M G |[ Fe / H ] , ( G − K ) ) . Fig-ure 3 shows the color-coded intrinsic G -band absolute mag-nitude, denoted as M G , single ([ Fe / H ] , ( G − K ) ) , of the sin-gle star in [Fe/H] vs. ( G − K s ) plane. Note that there fivepoints marked as black frames in the figure are failed to de-rive M G , single ([ Fe / H ] , ( G − K ) ) due to the missing data. Wethen provide the linear interpolated values for the five pointsbased on their neighboring points.For a given star with fixed [Fe/H] and ( G − K s ) , wethen calculate the differential absolute magnitude ∆ M G = M G − M G , single ([ Fe / H ] , ( G − K ) ) , in which M G , single is se- MNRAS , 1–17 (2019) olar-type Field Binary Stars lected from Figure 3 with nearest [Fe/H] and ( G − K s ) . Theuncertainty of ∆ M G is contributed from several ways. First,it is mostly from the astrometric and photometric uncer-tainties. Then, the de-reddening process also contributes anadditional uncertainty. Finally, M G , single also brings someuncertainty. Combining all these together, we obtain thetypical uncertainty of ∆ M G as . mag.Figure 4 shows the distributions of the stars in ∆ M G vs. m plane for different [Fe/H] populations. It is seen that alarge fraction of the stars are aligned with ∆ M G = , whichis the location of the single stars. The vertical dispersionof these stars is roughly consistent with the uncertainty of ∆ M G . Another notable groups of stars are located at around ∆ M G ∼ − . mag, which are mostly the binary stars withmass-ratio close to 1. For metal-poor populations ([Fe/H] < − . ), the coverage of m is from around 0.45 to 0.7 M (cid:12) ,while for metal-rich ones ([Fe/H] > − . ), the range of m isfrom 0.6 to 1.0 M (cid:12) .Note that, for single stars, m stands for the stellar massof the single star, while for binaries, it represents for theprimary stellar mass. For convenience, we do not distinguishthem in notation but simply use m for both cases.It is also seen that for stars with super-solar metallic-ity, as shown in the two bottom-right panels of Figure 4,the dispersion of ∆ M G is larger when m > . M (cid:12) . It impliesthat there are more stars located below the single main-sequence in the HR diagram with mass larger than 0.9 M (cid:12) .This larger dispersion, which could be associated with theso called blue sequence discussed by Yang et al. (2018), in-duces larger uncertainties in binary fraction and mass-ratiodistribution estimates. Therefore, we cut off the stars withmass larger than . M (cid:12) to avoid it. A broadly used approach to calculate the fraction of bi-nary stars especially in star clusters is to define an empiricalthreshold in color-absolute magnitude diagram to separatesingle and binary stars. Then, a following Monte Carlo sim-ulations, which adopt some presumed model of binaries, e.g.binary fraction and mass-ratio distribution, are conducted tocompare with the observations. Subsequently, the intrinsicbinary fraction and mass-ratio distribution can be obtainedfrom the simulations which converge their results to the ob-servations (e.g. Li et al. 2013; Sana et al. 2013).In this work, instead to follow the previous method,we propose a hierarchical Bayesian model to simultaneouslysolve out the fraction of binaries and the distribution ofmass-ratio of the binary systems. In Bayesian model, weconcentrate to the posterior probability distribution of thebinary fraction and parameters determining the shape ofthe mass-ratio distribution, rather than derive the best-fitbinary fraction and mass-ratio distribution.Now assume that the distribution of mass-ratio in MSbinary system follows a power law, that is p ( q | γ ) ∝ q γ , (3)where q = m / m represents for the mass ratio of a binaryand γ is the power index. It is noted that the power-lawassumption is not a precise description of the distribution of mass-ratio (Duchˆene & Kraus 2013). However, it is sim-ple and can essentially provide the information about therelative number of binaries with low and high mass-ratio.For each MS star, the observed absolute magnitude isnot only determined by its stellar mass, metallicity, and age,but also affected by two other factors: 1) the luminosity ofthe unresolved companion if it is a binary and 2) the ran-dom error propagated from astrometry and photometry. Toquantify the two effects, we firstly need a prior knowledgeabout whether a star is an unresolved binary. The probabil-ity of a star to be a binary is equivalent with the fraction ofbinary ( f b ). If the star of interest is a binary, we also needto know in a prior about the mass-ratio of the binary sothat we can derive the additional luminosity contributed bythe unresolved companion. The prior knowledge about themass-ratio is provided by the power law of the mass-ratiodistribution with the power index, γ , in Eq. (3). Therefore,there are two unknown parameters in the prior distributions, f b and γ . Then, the posterior distribution of f b and γ giventhe observed differential absolute magnitude ∆ M G i for the i th star can be written as p ( f b , γ | ∆ M i , Z i , m , i ) ∝ p ( f b , γ | Z i , m , i )× ∫ (cid:213) c p ( ∆ M i | c , q , Z i , m , i ) p ( c | f b , Z i , m , i )× p ( q | γ, Z i , m , i ) dq , (4)where Z i and m , i are metallicity and primary mass for the i th star, respectively. ∆ M i stands for the differential absolutemagnitude. c represents for the class of the star, either a sin-gle or a binary. Obviously, p ( c | f b ) follows Bernoulli distribu-tion and thus the term within the integral in the right-handside of Eq. (4) can be rewritten as ∫ (cid:213) c p ( ∆ M i | c , q , Z i , m , i ) p ( c | f b , Z i , m , i ) p ( q | γ, Z i , m , i ) dq = ( − f b ) p s ( ∆ M i ) ∫ p ( q | γ, Z i , m , i ) dq + f b ∫ p b ( ∆ M i | q , Z i , m , i ) p ( q | γ, Z i , m , i ) dq , (5)where p s and p b stand for the likelihood distribution of ∆ M i for single and binary star, respectively. We adopt that bothof them follow normal distribution and then we have p s ( ∆ M i ) = √ πσ i exp (cid:32) − ∆ M i σ i (cid:33) (6)for single star and for binary it becomes p b ( ∆ M i | q , Z i , m , i ) = √ πσ i exp (cid:32) − ( ∆ M i − ∆ M model ( q , Z i , m , i )) σ i (cid:33) , (7)where ∆ M model ( q , m , i , Z i ) is the model differential absolutemagnitude considering that the star has a companion withmass-ratio of q and σ i is the uncertainty of ∆ M i .The posterior distribution of f b and γ for a group ofstars with same (or very similar) [Fe/H] and m can be ob-tained by multiplying all the single star posterior distribu- MNRAS000
The colors show the intrinsic absolute magnitude ofsingle stars in G band as a function of ( G − K s ) and [Fe/H]. Thefive points marked with black frames are interpolated from theneighboring points. σ M G , i are the absolute magnitude and its uncertainty, re-spectively, of the i th star. We then locate the intrinsic M G of single stars at the peak of p ( M G |[ Fe / H ] , ( G − K ) ) . Fig-ure 3 shows the color-coded intrinsic G -band absolute mag-nitude, denoted as M G , single ([ Fe / H ] , ( G − K ) ) , of the sin-gle star in [Fe/H] vs. ( G − K s ) plane. Note that there fivepoints marked as black frames in the figure are failed to de-rive M G , single ([ Fe / H ] , ( G − K ) ) due to the missing data. Wethen provide the linear interpolated values for the five pointsbased on their neighboring points.For a given star with fixed [Fe/H] and ( G − K s ) , wethen calculate the differential absolute magnitude ∆ M G = M G − M G , single ([ Fe / H ] , ( G − K ) ) , in which M G , single is se- MNRAS , 1–17 (2019) olar-type Field Binary Stars lected from Figure 3 with nearest [Fe/H] and ( G − K s ) . Theuncertainty of ∆ M G is contributed from several ways. First,it is mostly from the astrometric and photometric uncer-tainties. Then, the de-reddening process also contributes anadditional uncertainty. Finally, M G , single also brings someuncertainty. Combining all these together, we obtain thetypical uncertainty of ∆ M G as . mag.Figure 4 shows the distributions of the stars in ∆ M G vs. m plane for different [Fe/H] populations. It is seen that alarge fraction of the stars are aligned with ∆ M G = , whichis the location of the single stars. The vertical dispersionof these stars is roughly consistent with the uncertainty of ∆ M G . Another notable groups of stars are located at around ∆ M G ∼ − . mag, which are mostly the binary stars withmass-ratio close to 1. For metal-poor populations ([Fe/H] < − . ), the coverage of m is from around 0.45 to 0.7 M (cid:12) ,while for metal-rich ones ([Fe/H] > − . ), the range of m isfrom 0.6 to 1.0 M (cid:12) .Note that, for single stars, m stands for the stellar massof the single star, while for binaries, it represents for theprimary stellar mass. For convenience, we do not distinguishthem in notation but simply use m for both cases.It is also seen that for stars with super-solar metallic-ity, as shown in the two bottom-right panels of Figure 4,the dispersion of ∆ M G is larger when m > . M (cid:12) . It impliesthat there are more stars located below the single main-sequence in the HR diagram with mass larger than 0.9 M (cid:12) .This larger dispersion, which could be associated with theso called blue sequence discussed by Yang et al. (2018), in-duces larger uncertainties in binary fraction and mass-ratiodistribution estimates. Therefore, we cut off the stars withmass larger than . M (cid:12) to avoid it. A broadly used approach to calculate the fraction of bi-nary stars especially in star clusters is to define an empiricalthreshold in color-absolute magnitude diagram to separatesingle and binary stars. Then, a following Monte Carlo sim-ulations, which adopt some presumed model of binaries, e.g.binary fraction and mass-ratio distribution, are conducted tocompare with the observations. Subsequently, the intrinsicbinary fraction and mass-ratio distribution can be obtainedfrom the simulations which converge their results to the ob-servations (e.g. Li et al. 2013; Sana et al. 2013).In this work, instead to follow the previous method,we propose a hierarchical Bayesian model to simultaneouslysolve out the fraction of binaries and the distribution ofmass-ratio of the binary systems. In Bayesian model, weconcentrate to the posterior probability distribution of thebinary fraction and parameters determining the shape ofthe mass-ratio distribution, rather than derive the best-fitbinary fraction and mass-ratio distribution.Now assume that the distribution of mass-ratio in MSbinary system follows a power law, that is p ( q | γ ) ∝ q γ , (3)where q = m / m represents for the mass ratio of a binaryand γ is the power index. It is noted that the power-lawassumption is not a precise description of the distribution of mass-ratio (Duchˆene & Kraus 2013). However, it is sim-ple and can essentially provide the information about therelative number of binaries with low and high mass-ratio.For each MS star, the observed absolute magnitude isnot only determined by its stellar mass, metallicity, and age,but also affected by two other factors: 1) the luminosity ofthe unresolved companion if it is a binary and 2) the ran-dom error propagated from astrometry and photometry. Toquantify the two effects, we firstly need a prior knowledgeabout whether a star is an unresolved binary. The probabil-ity of a star to be a binary is equivalent with the fraction ofbinary ( f b ). If the star of interest is a binary, we also needto know in a prior about the mass-ratio of the binary sothat we can derive the additional luminosity contributed bythe unresolved companion. The prior knowledge about themass-ratio is provided by the power law of the mass-ratiodistribution with the power index, γ , in Eq. (3). Therefore,there are two unknown parameters in the prior distributions, f b and γ . Then, the posterior distribution of f b and γ giventhe observed differential absolute magnitude ∆ M G i for the i th star can be written as p ( f b , γ | ∆ M i , Z i , m , i ) ∝ p ( f b , γ | Z i , m , i )× ∫ (cid:213) c p ( ∆ M i | c , q , Z i , m , i ) p ( c | f b , Z i , m , i )× p ( q | γ, Z i , m , i ) dq , (4)where Z i and m , i are metallicity and primary mass for the i th star, respectively. ∆ M i stands for the differential absolutemagnitude. c represents for the class of the star, either a sin-gle or a binary. Obviously, p ( c | f b ) follows Bernoulli distribu-tion and thus the term within the integral in the right-handside of Eq. (4) can be rewritten as ∫ (cid:213) c p ( ∆ M i | c , q , Z i , m , i ) p ( c | f b , Z i , m , i ) p ( q | γ, Z i , m , i ) dq = ( − f b ) p s ( ∆ M i ) ∫ p ( q | γ, Z i , m , i ) dq + f b ∫ p b ( ∆ M i | q , Z i , m , i ) p ( q | γ, Z i , m , i ) dq , (5)where p s and p b stand for the likelihood distribution of ∆ M i for single and binary star, respectively. We adopt that bothof them follow normal distribution and then we have p s ( ∆ M i ) = √ πσ i exp (cid:32) − ∆ M i σ i (cid:33) (6)for single star and for binary it becomes p b ( ∆ M i | q , Z i , m , i ) = √ πσ i exp (cid:32) − ( ∆ M i − ∆ M model ( q , Z i , m , i )) σ i (cid:33) , (7)where ∆ M model ( q , m , i , Z i ) is the model differential absolutemagnitude considering that the star has a companion withmass-ratio of q and σ i is the uncertainty of ∆ M i .The posterior distribution of f b and γ for a group ofstars with same (or very similar) [Fe/H] and m can be ob-tained by multiplying all the single star posterior distribu- MNRAS000 , 1–17 (2019)
C. Liu M G ( m a g ) -1.25<=[Fe/H]<=-0.55 0.00.51.01.52.0 0.4 0.6 0.8 1.02.01.51.00.50.00.51.0 -0.65<=[Fe/H]<=-0.40 0.00.51.01.52.0 0.4 0.6 0.8 1.02.01.51.00.50.00.51.0 -0.50<=[Fe/H]<=-0.25 0.00.51.01.52.0 0.4 0.6 0.8 1.0 m ( M )2.01.51.00.50.00.51.0 -0.35<=[Fe/H]<=-0.15 0.00.51.01.52.00.4 0.6 0.8 1.0 m ( M )2.01.51.00.50.00.51.0 M G ( m a g ) -0.25<=[Fe/H]<=-0.05 0.00.51.01.52.0 0.4 0.6 0.8 1.0 m ( M )2.01.51.00.50.00.51.0 -0.15<=[Fe/H]<=0.05 0.00.51.01.52.0 0.4 0.6 0.8 1.0 m ( M )2.01.51.00.50.00.51.0 -0.05<=[Fe/H]<=0.15 0.00.51.01.52.0 0.4 0.6 0.8 1.0 m ( M )2.01.51.00.50.00.51.0 0.05<=[Fe/H]<=0.40 0.00.51.01.52.0 Figure 4.
The color coded density distributions of the stars in ∆ M G vs. m plane for various [Fe/H] bins. The horizontal dot-dashedlines in the panels indicate ∆ M G = , which is the location of single stars. The horizontal dotted lines at ∆ M G = − . mag indicate thelocation of binary stars with mass-ratio of 1. tions, such as p ( f b , γ |{ ∆ M i } , Z , m ) = (cid:214) i p ( f b , γ | ∆ M i , Z , m ) . (8) We separate the stars into different metallicity and stellarmass groups, as shown in Figure 5. The stars are groupedinto [Fe/H] and m bins with various sizes so that we canfind balance between the number of stars in each bin andthe resolution of metallicity and stellar mass. Along m , thecenter positions of the bins are at 0.525, 0.6, 0.65, 0.7, 0.75,and 0.8 M (cid:12) . The width of the bins are 0.15 for the first and0.1 M (cid:12) for the other bins. The bin size of m are a factor of 2larger than the uncertainty of m for individual stars. Along[Fe/H], the centers of the bins are at -0.9, -0.525, -0.375, -0.15, -0.05, 0.1, and 0.225 dex, with bin widths of 0.7, 0.25,0.25, 0.2, 0.2, 0.2, 0.2, and 0.35 dex, respectively. The bin sizeis also larger than the uncertainty of [Fe/H] for individualstars so that the errors of [Fe/H] would not significantlyaffect the results. Note that these bins are partly overlappedwith their adjacent bins such that more stars are fell in thebins and therefore can stabilize the results by smoothing theresults.Table 1 and Figure 5 show the numbers of stars fell intothese bins. According to the Monte Carlo simulation con-ducted in section 5.2, we find that, in principle, the resultsare stable when the number of stars is larger than 2000 starsin a bin. Therefore, in the rest of the analysis we only con-sider the bins containing larger than 2000 stars, which arefilled up with colors in Figure 5 . This cut will lose the starswith [Fe/H] < − . . Then, we apply the hierarchical Bayesian model de-scribed in section 3 to the [Fe/H]– m bins with more than2000 stars. We run Markov chain Monte Carlo simulationsusing EMCEE software package (Foreman-Mackey et al.2013). We adopt the median values and 15% (85%) per-centiles of the random draws in MCMCs as the best fit val-ues and the uncertainties, respectively, for f b and γ .In the hierarchical Bayesian model, we set the lowerlimit of the mass of the secondary at 0.08 M (cid:12) . We do nottake into account the brown dwarfs in this work.The resulting f b and γ for different [Fe/H] and m binsare shown in Figure 6 and Table 1. The left panel of Figure 6 shows that the binary fractionis roughly larger at lower [Fe/H]. Meanwhile, f b is essen-tially larger when m is larger at most [Fe/H] values ex-cept − . dex. A ridge-like valley with f b < . is seen from[Fe/H] ∼ − . dex and m ∼ . M (cid:12) to [Fe/H] ∼ + . dex and m ∼ . M (cid:12) . Within the ridge, f b increases with declining[Fe/H]. In the other regions outside this ridge, the values of f b are larger than . . Most of them are larger than . .The right panel of Figure 6 shows the resulting γ as afunction of [Fe/H] and m . Similar to the distribution of f b ,the location of the bins with γ > . also concentrate in thesame ridge as the low f b bins, i.e. from [Fe/H] ∼ − . dexand m ∼ . M (cid:12) to [Fe/H] ∼ + . dex and m ∼ . M (cid:12) . Fur-thermore, γ roughly increases from . – . at bottom-leftto . – . at top-right in the ridge. On the other hand, thevalues of γ in other bins are around zero. These bins showsopposite trend to those in the ridge such that their valuesincrease when [Fe/H] decreases. MNRAS , 1–17 (2019) olar-type Field Binary Stars m ( M ) l o g N Figure 5.
The figure indicates how the sample stars are separated into sub-groups in m vs. [Fe/H] plane. The range of m covers from0.45 to 0.85 M (cid:12) , while the range of [Fe/H] covers from -1.2 to 0.4 dex. The numbers of stars coded with colors are marked within thecorresponding bins. The filled bins contain stars larger than 2000, while the blank bins contain stars less than 2000. m ( M ) Data f b m ( M ) Data 01234
Figure 6.
The left panel shows the resulting f b , indicated as colors, in the m vs. [Fe/H] plane. The estimated f b value in each bin ismarked on the plot. The right panel shows the resulting γ , indicated as colors, in the m vs. [Fe/H] plane. Similar to the left panel, theestimated γ values are also marked in the corresponding locations. The black frames in both panels indicate the bins with γ > . . Then, we draw the relationship between f b and γ inFigure 7. Surprisingly, we find that they are strongly anti-correlated with each other. This implies that the less thebinary fraction, the more binaries with high mass-ratio andvice versa. To our knowledge, it is for the first time that theanti-correlation is unveiled in the field stars.The right-side panel of Figure 7 displays the distribu-tion of γ using KDE smoothing technique. It shows a gap ataround γ ∼ . , separating the values of γ into two groups:the low- γ group with γ < . , which contains larger binaryfractions, and the high- γ group with γ > . , which con-tains lower binary fractions. The high- γ group is markedwith black frames in Figure 6, which is exactly overlappedwith the ridge regions with low f b and high γ .The gap is a real feature rather than an artifact. The binsizes are quite large than the measurement errors of [Fe/H]and m and each bin contains sufficient number of samples.Therefore, the gap would not be due to the random spikedominated by noise. Moreover, the bins have been smoothed by partly overlapping with the adjacent bins. This makes thegap more robust. However, the nature of the gap is not clear.Since the f b – γ pairs are naturally separated into twogroups, in next sections, we separately discuss the featuresof the two groups. γ group First, although the bins of the low- γ group cover a widerange of [Fe/H] and m , as shown in Figure 6, only with[Fe/H] < − . dex does m continuously cover from 0.5 to0.75 M (cid:12) . In the regime with [Fe/H] > − . dex, most of thebins of the low- γ group have m larger than . M (cid:12) . Two binsat ([Fe/H], m )=( + . dex, . M (cid:12) ) and ( + . dex, . M (cid:12) )are isolated from other low- γ bins.Then, we look at how f b is associated with [Fe/H] and m for the low- γ group in the top panels of Figure 8. Inall four panels of the figure, the resulting f b and γ of low- γ group are drawn in blue. We fit the relationship of f b ( γ ) with [Fe/H] and m respectively using a linear model. MNRAS000
The left panel shows the resulting f b , indicated as colors, in the m vs. [Fe/H] plane. The estimated f b value in each bin ismarked on the plot. The right panel shows the resulting γ , indicated as colors, in the m vs. [Fe/H] plane. Similar to the left panel, theestimated γ values are also marked in the corresponding locations. The black frames in both panels indicate the bins with γ > . . Then, we draw the relationship between f b and γ inFigure 7. Surprisingly, we find that they are strongly anti-correlated with each other. This implies that the less thebinary fraction, the more binaries with high mass-ratio andvice versa. To our knowledge, it is for the first time that theanti-correlation is unveiled in the field stars.The right-side panel of Figure 7 displays the distribu-tion of γ using KDE smoothing technique. It shows a gap ataround γ ∼ . , separating the values of γ into two groups:the low- γ group with γ < . , which contains larger binaryfractions, and the high- γ group with γ > . , which con-tains lower binary fractions. The high- γ group is markedwith black frames in Figure 6, which is exactly overlappedwith the ridge regions with low f b and high γ .The gap is a real feature rather than an artifact. The binsizes are quite large than the measurement errors of [Fe/H]and m and each bin contains sufficient number of samples.Therefore, the gap would not be due to the random spikedominated by noise. Moreover, the bins have been smoothed by partly overlapping with the adjacent bins. This makes thegap more robust. However, the nature of the gap is not clear.Since the f b – γ pairs are naturally separated into twogroups, in next sections, we separately discuss the featuresof the two groups. γ group First, although the bins of the low- γ group cover a widerange of [Fe/H] and m , as shown in Figure 6, only with[Fe/H] < − . dex does m continuously cover from 0.5 to0.75 M (cid:12) . In the regime with [Fe/H] > − . dex, most of thebins of the low- γ group have m larger than . M (cid:12) . Two binsat ([Fe/H], m )=( + . dex, . M (cid:12) ) and ( + . dex, . M (cid:12) )are isolated from other low- γ bins.Then, we look at how f b is associated with [Fe/H] and m for the low- γ group in the top panels of Figure 8. Inall four panels of the figure, the resulting f b and γ of low- γ group are drawn in blue. We fit the relationship of f b ( γ ) with [Fe/H] and m respectively using a linear model. MNRAS000 , 1–17 (2019)
C. Liu
Table 1.
Best-fit f b and γ values at different [Fe/H] and m bins.[Fe/H] . < m < .
60 0 . < m = .
65 0 . < m < . dex N f b γ N f b γ N f b γ (-0.65,-0.40) 2109 . ± .
03 0 . ± . . ± .
03 0 . ± . . ± .
03 0 . ± . (-0.50,-0.25) 2874 . ± .
02 0 . ± . . ± .
02 0 . ± . . ± .
02 0 . ± . (-0.35,-0.15) 7801 . ± .
01 2 . ± . . ± .
01 1 . ± . (-0.25,-0.05) 7984 . ± .
01 2 . ± . . ± .
01 2 . ± . (-0.15,0.05) 5211 . ± .
02 1 . ± . . ± .
02 1 . ± . (-0.05,0.15) 2138 . ± .
07 0 . ± . . ± .
02 1 . ± . (+0.05,0.40) 2212 . ± .
12 0 . ± . [Fe/H] . < m < .
75 0 . < m < .
80 0 . < m < . dex N f b γ N f b γ N f b γ (-0.65,-0.40) 2273 . ± .
03 0 . ± . (-0.50,-0.25) 5085 . ± .
03 0 . ± . . ± .
04 0 . ± . (-0.35,-0.15) 6036 . ± .
03 0 . ± . . ± . − . ± . . ± .
05 0 . ± . (-0.25,-0.05) 8910 . ± .
01 1 . ± . . ± .
04 0 . ± . . ± . − . ± . (-0.15,0.05) 10184 . ± .
01 2 . ± . . ± .
02 1 . ± . . ± . − . ± . (-0.05,0.15) 7501 . ± .
01 3 . ± . . ± .
05 3 . ± . . ± . − . ± . (+0.05,0.40) 4167 . ± .
02 2 . ± . . ± .
02 3 . ± . . ± . − . ± . f b Figure 7.
The figure shows the anti-correlation between the re-sulting f b and γ . The sizes of the circles indicate the number ofsamples. The larger the size the more the samples. The horizon-tal dotted line indicate the location of γ = . . The right- andtop-sides show the KDE smoothed distribution of γ and f b , re-spectively. We find that f b is not related to [Fe/H], because the slope ∂ f b / ∂ [ Fe / H ] = . ± . dex − (see the blue line in the top-left panel) is roughly zero. However, f b is clearly correlatedwith m with substantially positive slope of ∂ f b / ∂ m = . ± . M (cid:12)− (see the blue line in the top-right panel).Finally, as shown in the bottom panels of the figure, γ is tightly anti-correlated with both [Fe/H] and m withnegative slopes of ∂γ / ∂ [ Fe / H ] = − . ± . dex − and ∂γ / ∂ m = − . ± . M (cid:12)− , respectively. γ group As mentioned in Section 4.1, the high- γ group bins con-centrate in the ridge-like region in the m vs. [Fe/H] plane(see the black frames marked in all panels of Figure 6).This means that the high- γ group only contains data with[Fe/H] > − . dex and m smaller than 0.75 M (cid:12) . It is alsonoted that at the most metal-rich regions, the m valuesof the high- γ bins show lower limits, which is 0.6 M (cid:12) at[Fe/H] = + . dex and 0.65 M (cid:12) at [Fe/H] = + . dex.Then, we investigate the correlation of f b with [Fe/H]and m in the top panels of Figure 8. The red stars indicatethe f b values of the high- γ group and the red lines display thebest linear fits of the relationship of f b with [Fe/H] and m .It shows that f b is tightly anti-correlated with [Fe/H] basedon the substantially negative slope of ∂ f b / ∂ [ Fe / H ] = − . ± . dex − . In the mean time, f b is not related with m , sincethe slope ∂ f b / ∂ m = − . ± . M (cid:12)− is not significantlydifferent from zero.Finally, we look at the correlation of γ with [Fe/H] and m for the high- γ group in the bottom panels of Figure 8.We find that γ is weakly correlated with both [Fe/H] and m . Although the slopes ∂γ / ∂ [ Fe / H ] = . ± . dex − and ∂γ / ∂ m = . ± . M (cid:12)− are larger than zero, the signifi-cances are less than - σ . The red stars located in the bot-tom panels also show larger dispersion than the low- γ group(blue rectangles). Therefore, we infer that the correlationsof γ with either [Fe/H] or m is quite marginal.To summarize these complicated relationships, we listthem in Table 2. In this section, we explain the results described in abovesections based on the current theories of binary formationand evolution. We firstly give four assumptions.First, we assume that the mass function of the sec-ondary is universal, i.e. no matter [Fe/H] and m , the sec- MNRAS , 1–17 (2019) olar-type Field Binary Stars f b < 1.2: f b / [Fe/H]=0.08±0.14> 1.2: f b / [Fe/H]=-0.21±0.04< 1.2> 1.2Moe18Gao14 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 m ( M )0.00.20.40.60.8 f b < 1.2: f b / m =1.15±0.25> 1.2: f b / m =-0.18±0.19 < 1.2> 1.2Delfosse04 MRaghaven10 GKRaghaven10 FG m ( M )0123 < 1.2: / m =-3.04±0.77> 1.2: / m =5.05±3.35 DK13 allDK13 close
Figure 8.
The results in various [Fe/H]– m bins are separated into two groups: γ < . (all in blue rectangles) and γ > . (all in redstars). The top-left panel shows the distributions of the two groups in f b vs. [Fe/H] plane. The slopes are displayed in the panel. Theresult of Moe et al. (2019) is indicated as a black dashed line. The results of Gao et al. (2014) are displayed with black rectangles. Thetop-right, bottom-left and bottom-right panels show the distributions for the two groups in f b vs. m , γ vs. [Fe/H], and γ vs. m planes,respectively. The blue and red solid lines indicate the best linear fits for groups with γ < . and > . , respectively, while the dottedlines indicate the 1- σ uncertainties of the fitting. In the top-right panel, the black dashed and solid lines indicate the γ trend for allbinaries and close binaries, respectively, mentioned in Duchˆene & Kraus (2013). Table 2.
Summary of the relationships in γ -low and high groups.Condition γ < . γ > . f b vs. [Fe/H] no relation anti-correlation f b vs. m correlation no relation γ vs. [Fe/H] anti-correlation marginal correlation γ vs. m anti-correlation marginal correlation m larger m smaller m [Fe/H] lower [Fe/H] higher [Fe/H] ondary mass function are all the same. It is not necessarilyto be the canonical IMF given by Chabrier (2005) or Kroupa& Bouvier (2003) or others. Note that although Myers et al.(2011) has discussed theoretically about the invariability ofthe IMF with metallicity, it is not clear whether the IMF ofsecondary is same as that of the primary.Second, according to Moe et al. (2019), the close bina-ries are more likely formed from disk fragmentation, which isassociated with the metallicity of the gas (also c.f. Kratter etal. 2008, 2010; Machida et al. 2009; Tanaka & Omukai 2014).However, for the wide binaries, as found by El-Badry & Rix(2019), they are not related to metallicity, which is consis-tent with the scenario that the wide binaries are formed fromgas fragmentation (Tohline 2002; Offner et al. 2010; Myers et al. 2011; Bate 2012, 2014). Consequently, we assume thatthe binaries are formed both from disk fragmentation, whichcontributes mostly to the close binaries depending on metal-licity, and gas fragmentation, which produces most of thewide binaries not depending on metallicity.Third, we assume that the binary fraction increases withprimary stellar mass based on the simulation by Tanaka &Omukai (2014). This assumption is also supported by manyobservations (Delfosse et al. 2004; Raghavan et al. 2010;Sana et al. 2012, 2013).Finally, it is known that most stars are formed in em-bedded clusters (Lada & Lada 2003). Marks & Kroupa(2011) argued that the dynamical processing in the embed-ded clusters, in which binaries were born and spent theirearly times, prefers to efficiently disrupting binaries withlarger separations, smaller primary mass, and smaller mass-ratios. This is because the binding energy of a binary systemcan be expressed as E b ∝ m qa , (9)where a is the separation of the two companions (Marks,Kroupa & Oh 2011). The dynamical processing occurredin embedded clusters has a very short time scale of a fewmillion years (Belloni et al. 2018). We assume that the dy-namical processing is generally very efficient and can change MNRAS000
Summary of the relationships in γ -low and high groups.Condition γ < . γ > . f b vs. [Fe/H] no relation anti-correlation f b vs. m correlation no relation γ vs. [Fe/H] anti-correlation marginal correlation γ vs. m anti-correlation marginal correlation m larger m smaller m [Fe/H] lower [Fe/H] higher [Fe/H] ondary mass function are all the same. It is not necessarilyto be the canonical IMF given by Chabrier (2005) or Kroupa& Bouvier (2003) or others. Note that although Myers et al.(2011) has discussed theoretically about the invariability ofthe IMF with metallicity, it is not clear whether the IMF ofsecondary is same as that of the primary.Second, according to Moe et al. (2019), the close bina-ries are more likely formed from disk fragmentation, which isassociated with the metallicity of the gas (also c.f. Kratter etal. 2008, 2010; Machida et al. 2009; Tanaka & Omukai 2014).However, for the wide binaries, as found by El-Badry & Rix(2019), they are not related to metallicity, which is consis-tent with the scenario that the wide binaries are formed fromgas fragmentation (Tohline 2002; Offner et al. 2010; Myers et al. 2011; Bate 2012, 2014). Consequently, we assume thatthe binaries are formed both from disk fragmentation, whichcontributes mostly to the close binaries depending on metal-licity, and gas fragmentation, which produces most of thewide binaries not depending on metallicity.Third, we assume that the binary fraction increases withprimary stellar mass based on the simulation by Tanaka &Omukai (2014). This assumption is also supported by manyobservations (Delfosse et al. 2004; Raghavan et al. 2010;Sana et al. 2012, 2013).Finally, it is known that most stars are formed in em-bedded clusters (Lada & Lada 2003). Marks & Kroupa(2011) argued that the dynamical processing in the embed-ded clusters, in which binaries were born and spent theirearly times, prefers to efficiently disrupting binaries withlarger separations, smaller primary mass, and smaller mass-ratios. This is because the binding energy of a binary systemcan be expressed as E b ∝ m qa , (9)where a is the separation of the two companions (Marks,Kroupa & Oh 2011). The dynamical processing occurredin embedded clusters has a very short time scale of a fewmillion years (Belloni et al. 2018). We assume that the dy-namical processing is generally very efficient and can change MNRAS000 , 1–17 (2019) C. Liu the present-day field binary properties to different extentsdepending on the primary mass, separation, mass-ratio etc.Now, we can apply all above assumptions to the twogroups of stars and explain why they show the observed f b and mass-ratio distribution features as listed in Table 2.Firstly, because the high- γ group contains smaller m ,they are easier to be affected by the dynamical evolution.The dynamical processing destroys many binaries with lower q and wider separation since they have smaller binding en-ergy. Consequently, this leads to smaller binary fraction forhigh- γ populations. In the mean time, the loss of lower mass-ratio binaries also results in larger γ since the existing bi-naries are more biased to larger q . Moreover, the loss ofwide binaries tends to increase the fraction of close bina-ries in the present-day samples. According to the second as-sumption that the formation of close binaries is metallicity-dependent, the close binary dominated populations exhibitthe tight anti-correlation between [Fe/H] and f b .Meanwhile, the high- γ group is not only biased to lower m , but also biased to higher metallicity. This means thatthe metal-rich stars may have experienced more effectivedynamical processing. It is not clear how metallicity playsthe role in the formation of the embedded clusters and how itaffect the dynamical processing. This observational evidentmay be helpful in clarifying how metallicity works in thisissue.On the other hand, because the low- γ group containslarger primary masses, it is less affected by the dynam-ical processing. Therefore, the populations in this grouphave relatively larger present-day binary fractions. The flat f b –[Fe/H] relationship is evident that the populations withsmaller γ contain more metallicity-independent wide bina-ries, which smear out the f b –[Fe/H] anti-correlation con-tributed by close binaries. These support that the low- γ group may be less or even not affected by the dynamicalprocessing during the stage of the embedded clusters.The anti-correlations between γ and [Fe/H] and be-tween γ and m in low- γ group may reflect the various star(and also binary) formations influenced by various charac-teristics, e.g. cluster mass, IMF, metallicity, of the embeddedclusters. Future theoretical modeling making use of these ob-servational evidences will be very useful to investigate thenature of star formation.We should emphasize that the explanations of high- andlow- γ groups are based on a number of assumptions compiledfrom various theoretical and observational works. Becausethe theoretical works are usually limited by their ingredi-ents and techniques, they do not exactly reflect the realities.Therefore, these explanations are not exclusive. Although itseems that they are well consistent with the assumption andthe observations, other mechanisms are not ruled out. In each [Fe/H]– m bin, we are able to identify which star islikely a binary, since the resulting f b and γ are the hyper-parameters of the prior in Eq. (4). We can then use Eq. (5)to estimate the probabilities that a star to be either a singleor a binary. Specifically, the probability of the i th star to bea single can be written as P s , i = ( − f b ) p s ( ∆ M i ) ∫ p ( q | γ ) dq . (10) MG (mag)605040302010010 l o g ( P s / P b ) q Figure 9.
This displays the relation between log P s / P b and ∆ M G . The horizontal dashed line indicates P s / P b = . . Thecolors code the mass-ratio if a star is a binary. And the probability to be a binary reads P b , i = f b ∫ p b ( ∆ M | q ) p ( q | γ ) dq . (11)Then, the odd of the star to be a single can be written as P s / P b . When P s / P b is smaller than 1, the star is more likelyto be a binary, while it is larger than 1, it is likely to bea single star. Figure 9 shows that log P s / P b is correlatedwith ∆ M G as expected. Given a star, the more likely to bea binary, the smaller the ∆ M G .When a star is identified as a binary, the mass-ratiocan also be roughly approximated from ∆ M G . We considera posterior distribution for q such that p ( q | ∆ M ) = p ( ∆ M | q ) p ( q | γ ) . (12)We adopt the best-fit q to be at the maximum of the pos-terior distribution and the 15% and 85% percentiles of thedistribution as the uncertainties. Figure 9 also indicates themass-ratios based on Eq. (12).Because only the filled regions in Figure 5 have resulting f b and γ , only the stars located in these regions can be classi-fied as single and binaries. We finally calculate log P s / P b and q for 57 978 stars with [Fe/H] (cid:38) − . dex. Table 3 lists thewhole 57 978 samples with log P s / P b and the best-fit mass-ratio. Note that q estimated from Eq. (12) is only related to M G and γ , no matter whether a star is classified as binaryor single, we provide q value and its uncertainty for it. Wesuggest 5 410 of these samples with log P s / P b < . are verylikely binaries. MNRAS , 1–17 (2019) olar-type Field Binary Stars Table 3.
The MS star catalogue with probability of binary and mass-ratio. The full catalogue is at https://github.com/liuchaonaoc/Binarity/blob/master/table3.txt . LM obsid
Gaia source id ra dec M G M G error A G ∆ M G m m error log P s / P b q q q M (cid:12) M (cid:12) We compare the binary fraction from our results with Moeet al. (2019) (black dashed line in the top-left panel of Fig-ure 8). The slope of the anti-correlation between f b and[Fe/H] is ∂ f b / ∂ [ Fe / H ] ∼ − . dex − from Moe et al. (2019),which is similar to − . ± . dex − from the high- γ group.The binary fraction from Moe et al. (2019) is larger by about0.1 than that of the high- γ group. Compared with the binaryfraction of low- γ group, however, we find that the f b fromMoe et al. (2019) is significantly smaller. Because Moe et al.(2019) studied the close binaries ( a < AU), it is not sur-prising that their f b is more similar to high- γ group, which islikely affected by dynamical evolution and hence more dom-inated by close binaries, than to low- γ group, which containsmore wide binaries.El-Badry & Rix (2019) shows that the anti-correlation issubstantial in binaries with separation smaller than 100 AUbut disappears in wider binaries. This also agrees with ourresults for high- γ and -low groups.We compare the f b – m correlation with previous works.First, Delfosse et al. (2004) and Leinert et al. (1997) foundthat the multiplicity fraction of M dwarf stars ( . – . M (cid:12) )is . . The closest mass range in our work is m ∼ . M (cid:12) ,which has f b of . ± . . This is consistent with Delfosseet al. (2004). Second, Raghavan et al. (2010) selected starsin the mass range slightly larger than us. The multiplicityfraction of F6-G2 stars, which are slightly more massive than ∼ . M (cid:12) , in Raghavan et al. (2010) is . ± . . Only low- γ group contains m ∼ . M (cid:12) and it falls between . and . depending on different metallicity. The fraction of theirG2-K3 samples, which covers . < m < M (cid:12) , is . ± . ,while our results with m = . M (cid:12) at low- γ group is from . to . , which are in the similar ranges. We then compare the resulting γ with previous literatures.Duquennoy & Mayor (1991) argued that the mass-ratio dis-tribution of the G type stars is quite similar to the canonicalIMF (Miller & Scalo 1979; Kroupa, Tout & Gilmore 1990),while Raghavan et al. (2010) suggested a flat distribution ofmass-ratio with a sharp peak at q ∼ . However, we do notfind any literature has discussed about the anti-correlationof the mass-ratio distribution with metallicity as displayedin the bottom-left panel of Figure 8 for the low- γ group. Duchˆene & Kraus (2013) compiled many literatures anddemonstrated that the power index of a power-law fittedmass-ratio distribution substantially changes with the pri-mary mass (see their Figure 2). At m (cid:46) M (cid:12) , the smallerthe primary mass, the larger the power index. This meansthat for low-mass binaries, the secondaries tend to have sim-ilar mass as the primaries, while the more massive binariestend to have more low-mass companions. We fit the γ valuesfor binaries with all range of period and m < M (cid:12) , whichare displayed as the red diamond symbols in the Figure 2 ofDuchˆene & Kraus (2013), using a power law and show it asblack solid line in the bottom-right panel of Figure 8. In therange of m between 0.4 and 0.85 M (cid:12) , the γ s derived fromprevious works prefer to a quite flat relationship with m .Compared to the low- γ group, at around m ∼ . M (cid:12) , ourresult is similar to Duchˆene & Kraus (2013). However, the γ of the low- γ group has stronger anti-correlation with m than Duchˆene & Kraus (2013). Duchˆene & Kraus (2013) alsoprovided γ for populations of binaries with smaller period(the blue squares in their figure 2). We simply connect thetwo points at m ∼ . and . M (cid:12) and show it with a blackdashed line in the bottom-right panel of Figure 8. Their γ is essentially similar to the values of the high- γ group. Thisis also consistent with the hypothesis that high- γ group isdominated by shorter period binaries due to the effectivedynamical processing. In this section, we investigate the effect of the sample sizein the hierarchical Bayesian model.For this purpose, we generate mock samples with var-ious numbers of mock stars and different f b and γ . Forall simulated data, we adopt the primary stellar mass m = . M (cid:12) and [Fe/H] = − . dex. We first select 6 f b values at . , . , ..., . and at 5 γ values at − , , ..., . Given anycombination of f b and γ from the above values, we arbitrar-ily draw three mock datasets with numbers of N = , ,and , respectively. Each mock star has a probability of f b to be a binary. If it is a binary, the mass-ratio q is arbi-trarily drawn from the power law with the given γ . In total,we have × × = groups of mock data with different f b , γ , and samples size.The differential absolute magnitude ∆ M G for the mockdata is calculated from PARSEC stellar model given theadopted m , [Fe/H], and q . We then run the hierarchicalBayesian model described in section 3 via MCMC for eachgroup of the mock data.Figure 10 shows the results of the mock data groupedwith sample size. The first row shows the results for the mock MNRAS , 1–17 (2019) C. Liu f b E s t f b - T r u e f b N=500 1 0 1 2 3True 21012 E s t - T r u e N=500 0.2 0.0 0.2 0.4 0.6 0.8 1.0 f b f b E s t f b - T r u e f b N=2000 1 0 1 2 3True 21012 E s t - T r u e N=2000 0.2 0.0 0.2 0.4 0.6 0.8 1.0 f b f b E s t f b - T r u e f b N=10000 1 0 1 2 3True 21012 E s t - T r u e N=10000 0.2 0.0 0.2 0.4 0.6 0.8 1.0 f b Figure 10.
The rows show the results of simulations to validate the hierarchical Bayesian model with the number of mock stars of 500,2000, and 10000, from top to bottom, respectively. The left column shows the difference of the estimated f b with the true ones. Themiddle columns shows the difference of γ estimates with the true values. The right column shows the difference of the estimated pairs of f b and γ (black dots with error bars) compared with the true pairs (gray hollow circles). data with N = . The top-left panel displays that the esti-mated f b is essentially consistent with the true values (thepresetting binary fraction when generating the mock data, x -axis). However, in the top-middle panel, a few estimated γ are substantially biased, while the others well reproducethe “true” γ . The top-right panel shows the offsets of theestimated (black dots with error bars) parameters from thepresetting parameters (gray circles). It is seen that when thetrue f b is smaller than . , the γ tends to be overestimated.Meanwhile, the binary fractions are slightly overestimatedwith larger error bars to around . when the presettingfraction is larger than . and γ < . The overall system-atic offset of derived f b is − . with uncertainty of . . Thesystematic bias of the estimated γ is . with uncertaintyof . for the sample size of N = .The middle row of Figure 10 shows the results for themock data with N = . Because the sample sizes in-creases, the performance of the estimations of f b and γ aresubstantially improved. The left and middle columns showthat the estimates of f b and γ well reproduce the presettingvalues. The right column shows no significant overestima-tion for γ at f b < . , while the slightly overestimation and larger uncertainty of f b still exists. The γ is also slightly un-derestimated in most of the datasets at f b > . and γ < .At f b < . , the uncertainties of both γ and f b are large. Thesystematic bias of the estimated f b is . with uncertaintyof . . The bias of the deriving γ is − . with dispersionof . . The random errors are significantly smaller than theresults of the N = samples and the systematic bias in γ is reduced by a factor of 2.The bottom row shows the results for samples with N = . The bias of the f b estimates is . with dispersion of . . For γ estimates, they are − . and . . Compared tothe results for N = , the systematics does not improveany more, although the random errors mildly improve.To summarize, the exercises with mock stars are evi-dent that when the sample size is a few hundreds, the binaryfraction estimates is reliable but suffers from large randomerrors. The mass-ratio distribution estimates, however, maybe significantly overestimated when binary fraction is low.When N (cid:38) , the estimations for f b and γ are alreadyrobust. Although increasing sample size can reduce the ran-dom error, a slight systematic bias does not disappear.We therefore conclude that, first, we need to keep the MNRAS , 1–17 (2019) olar-type Field Binary Stars M G (mag)10 N o r m a li z e d c o un t < 1.2> 1.2 Figure 11.
Two distributions of ∆ M G for the data with γ > . (blue line) and γ < . (red). The shadows show − and − σ un-certainties of the distributions. y − axis is displayed in logarithmicscale. The vertical black dot-dashed line indicate the location ofsingle stars and the black dotted line indicate the binaries with q = . sample size to be larger than 2000 so that the random errorsand systematics can be well controlled. This is why we onlyderive f b and γ for the [Fe/H]- m bins with N > . Sec-ond, the readers have to be aware of that the results shownin section 4 may slightly overestimate the binary fractionby around . and underestimate γ by . – . . Finally, al-though the covariance between f b and γ does exist and leadsto larger uncertainties when either f b or γ is small, there isno strong degeneracy between f b and γ , meaning that theycan be determined with reasonable uncertainties and thus,the anti-correlation shown in Figure 7 is real. f b – γ anti-correlation To double-check the anti-correlation between f b and γ , wecompare the mean distributions of ∆ M G for the two groupsof stars in Figure 11. The two distributions are normalizedso that the maximum values are at 1. The low- γ group (blueline) shows more fraction of stars at ∆ M G between − . and − . mag than the high- γ group, while the two groups havesimilar fraction of binaries at ∆ M G ∼ − . mag. This meansthat the low- γ group contains more binaries with moderate q than the high- γ group, although they have similar fractionsof binaries with q ∼ . This leads to a two-fold affect. First,the fraction of binaries for the low- γ group is larger, sinceit contains more binaries with moderate q . Second, γ mustbe smaller for the low- γ group, because that its mass-ratiodistribution must be flatter than the high- γ group since itcontains more binaries with moderate q . This again confirmsthat the anti-correlation between f b and γ is indeed a realfeature. l o g P c u t ( d ) m = 0.6 Mm = 0.8 Mm = 1.0 M f r a c t i o n o f m i ss e d b i n a r y
50 100 150 200 250 300Distance (pc)0.000.020.040.060.08 d i s t r i b u t i o n Figure 12.
The black lines aligned with the left-hand side x -axisshows the logarithmic cut-off period subject to the 2 arcsec angu-lar resolution of Gaia as a function of distance. The green linesaligned with the right-hand side x -axis displays the fraction ofthe missing binaries due to the period cut-off. The solid, dashed,and dotted lines indicate three cases with total masses of . , . ,and . M (cid:12) , respectively. The bottom panel indicates the numberdistribution of the samples along the distance. In this work, we identify unresolved binaries from HR di-agram, which requires the binaries to appear as a singlepoint. When a binary shows recognizable spatial separationbetween the two companions, it would be mistakenly iden-tified as two single stars rather than one binary. Therefore,our method may miss wide binaries in the samples and leadto incompleteness to some extent. In this section we evaluatethe effect of the incompleteness.The incompleteness of period distribution is mainlysubject to the angular resolution of
Gaia ,which is around arcsec (Arenou et al. 2018). The corresponding cut-off pe-riod varies with distance, as shown with black lines in Fig-ure 12. The cut-off in period leads to incompleteness of bi-naries in distribution of period. We then approximate thefraction of missing binaries with large periods beyond thecut-off points by adopting the distribution of period fromRaghavan et al. (2010). The fractions of missing binaries isdisplayed as a function of distance with green lines in Fig-ure 12. It is seen that, in the worst case at 50 pc, about onefourth binaries are missed. However, at 300 pc, only 13%binaries, most of which have wide separations, are cut off.Weighted by the number distribution of the observed starsalong distance (bottom panel in Figure 12), the averagedfraction of the missing binaries is around 15%.The missing wide binaries with period beyond the cut-off points become two single stars to the field. The secon-daries of the missing binaries may have opportunity to beobserved by LAMOST as an independent single star. How-ever, since the secondary is fainter, they have less proba-bility to be targeted in LAMOST survey according to theselection function of LAMOST, which is roughly flat alongapparent magnitude (Carlin et al. 2012; Yuan et al. 2015; MNRAS000
Gaia ,which is around arcsec (Arenou et al. 2018). The corresponding cut-off pe-riod varies with distance, as shown with black lines in Fig-ure 12. The cut-off in period leads to incompleteness of bi-naries in distribution of period. We then approximate thefraction of missing binaries with large periods beyond thecut-off points by adopting the distribution of period fromRaghavan et al. (2010). The fractions of missing binaries isdisplayed as a function of distance with green lines in Fig-ure 12. It is seen that, in the worst case at 50 pc, about onefourth binaries are missed. However, at 300 pc, only 13%binaries, most of which have wide separations, are cut off.Weighted by the number distribution of the observed starsalong distance (bottom panel in Figure 12), the averagedfraction of the missing binaries is around 15%.The missing wide binaries with period beyond the cut-off points become two single stars to the field. The secon-daries of the missing binaries may have opportunity to beobserved by LAMOST as an independent single star. How-ever, since the secondary is fainter, they have less proba-bility to be targeted in LAMOST survey according to theselection function of LAMOST, which is roughly flat alongapparent magnitude (Carlin et al. 2012; Yuan et al. 2015; MNRAS000 , 1–17 (2019) C. Liu
Liu et al. 2017). Moreover, lots of the resolved secondarieshave masses lower than 0.45 M (cid:12) , which have been alreadyexcluded from our samples. We can reasonably assume thatthe completeness of LAMOST survey is around 20%, i.e.,only one fifth of the stars in a field are finally observed byLAMOST. And we can coarsely assume that about half ofthe resolved secondaries observed by LAMOST have masseslarger than 0.45 M (cid:12) and hence are considered as single stars.Combining these two factors, we obtain that only 10% of theresolved secondaries from the 15% missing wide binaries mis-takenly contribute to the single stars. This means that thenumber of the single stars only increase by a few percentsdue to the incompleteness of the wide binary. Therefore, weignore the systematic bias of the number of single stars inour samples.Finally, considering that the resulting binary fraction isslightly overestimated (see section 5.2) by ∼ . , the sys-tematic underestimation of the binary fraction due to theincompleteness of the distribution of period reduces from15% to 13%. Because we identify binaries from their larger luminosity,we cannot distinguish the binary and high-order multiplesystem. Therefore, f b used in the whole paper is actuallyequivalent with the frequency of multiplicity, i.e. how manyobserved stellar objects are multiple stars. This is differentwith the component frequency, which is defined as the aver-aged number of companions in each stellar object (Duchˆene& Kraus 2013).Undiscriminating triple (or higher order multiple) sys-tem would not change the fraction of binaries, but may dis-tort the mass-ratio distribution. The mass-ratio of the bi-nary in this work is defined as the stellar mass of the sec-ondary over the stellar mass of the primary, which is notthe case for high-order multiplicity. In principle, withoutconsidering the tertiary contribution in luminosity, we maymistakenly attribute the total incremental luminosity of thesecondary, tertiary and other high-order companions to amore massive secondary. Therefore, without discriminatinghigh-order multiple system, the mass-ratio distribution maybe systematically bias to larger q .We investigate 27 triple systems with spectral typesor mass studied by Raghavan et al. (2010) and find thatthe mass of tertiary is much smaller than the secondaryin 21 of them. Consequently, they would not significantlychange the total luminosity. For instance, if the primary hasmass of . M (cid:12) , the mass of the secondary is . M (cid:12) , then atertiary with mass of . M (cid:12) only makes ∆ M G brighter by . mag. When the mass of the tertiary increases to . M (cid:12) ,the ∆ M G increases by 0.08 mag, roughly comparable to theuncertainty of ∆ M G . Therefore, when the mass of tertiary issignificantly smaller than that of the secondary, its contri-bution in the mass-ratio distribution can be neglected.We also find 5 of the 27 triple systems, most of whichcontain brown dwarf companions, have the secondary andtertiary with similar masses, which can potentially affectthe mass-ratio distribution. However, since these stars areonly a small fraction in triple systems and triple systems oc-cupy only a small fraction in multiplicities, their effect in themass-ratio distribution is small and hence can be neglected. In this work, we use more than 50 000 solar-type MS starsselected from LAMOST and
Gaia data to derive the binaryfraction and mass-ratio distribution as functions of [Fe/H]and m . We develop a hierarchical Bayesian model to simul-taneously derive both the binary fraction and the mass-ratiodistribution in each small metallicity and primary mass bin.For the first time, we find that the binary fraction istightly anti-correlated with γ , which is the power index ofthe power-law shape mass-ratio distribution. The data canbe separated into two groups with different γ . We find thatthe f b of the low- γ group with γ < . displays clear corre-lation with m but quite flat relationship with [Fe/H]. The γ for this group, on the other hand, shows anti-correlationswith both [Fe/H] and m . For the high- γ group with γ > . , f b shows significant anti-correlation with [Fe/H] but is notcorrelated with m . Although γ of the group is mildly cor-related with [Fe/H] and m , the large uncertainty prefers tothat these correlations are not statistically substantial.Moreover, looking at the distributions of f b and γ inthe m vs. [Fe/H] plane, the two groups are quite different.The high- γ group favorites smaller m and larger [Fe/H],resulting in a concentrated ridge-like regime in the m vs.[Fe/H] plane, while the low- γ group distributes in the regionswith larger m or lower metallicity.These complicated features strongly hint that the fieldbinaries may experience efficient dynamical processing dur-ing the time when they still stayed in the embedded clus-ters in which they were born (Kroupa 1995a,b). As a result,the properties of the present-day binaries have been sub-stantially reshaped, especially for those with lower primarymass and higher metallicity. Using the resulting f b and γ at each [Fe/H] and m bin as a prior, we are able to iden-tify 5 410 binary star candidates with P s / P b < . from the57 978 samples. And we also derived the mass-ratio for all57 978 stars.We find that the number of samples is critical to derivereliable mass-ratio distribution. If the sample size is onlya few hundreds, the derived mass-ratio distribution for thepopulation with small binary fraction would systematicallybias to high mass-ratio. However, when the sample size islarge enough, such as larger than 2000, we can reliably workout both binary fraction and mass-ratio distribution basedon the hierarchical Bayesian model. ACKNOWLEDGEMENTS
CL thanks Changqing Luo, Pavel Kroupa, Zhanwen Han,Lu Li, Cheng-Yuan Li, Xiao-Bin Zhang, Licai Deng, andHaijun Tian for their helpful discussions and suggestions.This work is supported by the National Natural ScienceFoundation of China (NSFC) with grant No. 11835057. Thisproject was developed in part at the 2018 NYC Gaia Sprint,hosted by the Center for Computational Astrophysics ofthe Flatiron Institute in New York City. It was also partlydeveloped at 2018 Gaia-LAMOST sprint workshop sup-ported by the NSFC under grants 11333003 and 11390372.Guoshoujing Telescope (the Large Sky Area Multi-ObjectFiber Spectroscopic Telescope LAMOST) is a National Ma-jor Scientific Project built by the Chinese Academy of Sci-ences. Funding for the project has been provided by the
MNRAS , 1–17 (2019) olar-type Field Binary Stars National Development and Reform Commission. LAMOSTis operated and managed by the National AstronomicalObservatories, Chinese Academy of Sciences. This workhas made use of data from the European Space Agency(ESA) mission
Gaia ( ),processed by the Gaia
Data Processing and Analysis Con-sortium (DPAC, ). Funding for the DPAC has been pro-vided by national institutions, in particular the institutionsparticipating in the
Gaia
Multilateral Agreement.
REFERENCES
Abt, H. A., & Levy, S. G. 1976, ApJS, 30, 273Abt, H. A., & Willmarth, D. W. 1987, ApJ, 318, 786Allen, P. R. 2007, ApJ, 668, 492Arenou, F., Luri, X., Babusiaux, C., et al. 2018, A&A, 616, A17Badenes, C., Mazzola, C., Thompson, T. A., et al. 2018, ApJ,854, 147Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Mantelet, G.,& Andrae, R. 2018, AJ, 156, 58Bastian, N., Covey, K. R., & Meyer, M. R. 2010, ARA&A, 48,339Bate, M. R. 2012, MNRAS, 419, 3115Bate, M. R. 2014, MNRAS, 442, 285Belloni, D., Kroupa, P., Rocha-Pinto, H. J., & Giersz, M. 2018,MNRAS, 474, 3740Bressan, A., Marigo, P., Girardi, L., et al. 2012, MNRAS, 427,127Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345,245Carlin, J. L., L´epine, S., Newberg, H. J., et al. 2012, Research inAstronomy and Astrophysics, 12, 755Carney, B. W. 1983, AJ, 88, 623Chabrier, G. 2005, The Initial Mass Function 50 Years Later, 327,41Chen, X., Deng, L., de Grijs, R., et al. 2016, AJ, 152, 129Chini, R., Hoffmeister, V. H., Nasseri, A., Stahl, O., & Zinnecker,H. 2012, MNRAS, 424, 1925Delfosse, X., Beuzit, J.-L., Marchal, L., et al. 2004, Spectroscop-ically and Spatially Resolving the Components of the CloseBinary Stars, 318, 166Deng, L.-C., Newberg, H. J., Liu, C., et al. 2012, Research inAstronomy and Astrophysics, 12, 735Danielski, C., Babusiaux, C., Ruiz-Dern, L., Sartoretti, P., & Are-nou, F. 2018, A&A, 614, A19Dieterich, S. B., Henry, T. J., Golimowski, D. A., Krist, J. E., &Tanner, A. M. 2012, AJ, 144, 64Ducati, J. R., Bevilacqua, C. M., Rembold, S. B., & Ribeiro, D.2001, ApJ, 558, 309Duchˆene, G., & Kraus, A. 2013, ARA&A, 51, 269Duquennoy, A., & Mayor, M. 1991, A&A, 248, 485El-Badry, K., Rix, H.-W., Ting, Y.-S., et al. 2018a, MNRAS, 473,5043El-Badry, K., Ting, Y.-S., Rix, H.-W., et al. 2018b, MNRAS, 476,528El-Badry, K., & Rix, H.-W. 2018, MNRAS, 480, 4884El-Badry, K., & Rix, H.-W. 2019, MNRAS, 482, L139Foreman-Mackey D., Hogg D. W., Lang D. & Goodman J., 2013,PASP, 125, 306Fischer, D. A., & Marcy, G. W. 1992, ApJ, 396, 178Gaia Collaboration, Babusiaux, C., van Leeuwen, F., et al. 2018,A&A, 616, A10Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016,A&A, 595, A1 Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018,A&A, 616, A1Gao, S., Liu, C., Zhang, X., et al. 2014, ApJ, 788, L37Gao, H., Zhang, H.-W., Xiang, M.-S., et al. 2015, Research inAstronomy and Astrophysics, 15, 2204Gao, S., Zhao, H., Yang, H., & Gao, R. 2017, MNRAS, 469, L68Goldberg, D., Mazeh, T., & Latham, D. W. 2003, ApJ, 591, 397Heintz, W. D. 1969, AJ, 74, 768Henry, T. J., & McCarthy, D. W., Jr. 1990, ApJ, 350, 334Hettinger, T., Badenes, C., Strader, J., Bickerton, S. J., & Beers,T. C. 2015, ApJ, 806, L2Jian, M., Gao, S., Zhao, H., & Jiang, B. 2017, AJ, 153, 5Kouwenhoven, M. B. N., Brown, A. G. A., Portegies Zwart, S. F.,& Kaper, L. 2007, A&A, 474, 77Kratter, K. M., Matzner, C. D., & Krumholz, M. R. 2008, ApJ,681, 375Kratter, K. M., Matzner, C. D., Krumholz, M. R., & Klein, R. I.2010, ApJ, 708, 1585Kroupa, P., Tout, C. A., & Gilmore, G. 1990, MNRAS, 244, 76Kroupa P., 1995, MNRAS, 277, 1491Kroupa P., 1995, MNRAS, 277, 1507Kroupa, P., & Bouvier, J. 2003, MNRAS, 346, 343Lada, C. J., & Lada, E. A. 2003, ARA&A, 41, 57Latham, D. W., Stefanik, R. P., Torres, G., et al. 2002, AJ, 124,1144Leinert, C., Henry, T., Glindemann, A., & McCarthy, D. W., Jr.1997, A&A, 325, 159Li, C., de Grijs, R., & Deng, L. 2013, MNRAS, 436, 1497Lindegren, L., Hern´andez, J., Bombrun, A., et al. 2018, A&A,616, A2Liu, C., Xu, Y., Wan, J.-C., et al. 2017, Research in Astronomyand Astrophysics, 17, 096Luo, A.-L., Zhao, Y.-H., Zhao, G., et al. 2015, Research in As-tronomy and Astrophysics, 15, 1095Luri, X., Brown, A. G. A., Sarro, L. M., et al. 2018, A&A, 616,A9Machida, M. N., Omukai, K., Matsumoto, T., & Inutsuka, S.-I.2009, MNRAS, 399, 1255Marks, M., Kroupa, P., & Oh, S. 2011, MNRAS, 417, 1684Marks, M., & Kroupa, P. 2011, MNRAS, 417, 1702Mason, B. D., Gies, D. R., Hartkopf, W. I., et al. 1998, AJ, 115,821Mason, B. D., Henry, T. J., Hartkopf, W. I., ten Brummelaar, T.,& Soderblom, D. R. 1998, AJ, 116, 2975Mason, B. D., Hartkopf, W. I., Gies, D. R., Henry, T. J., & Helsel,J. W. 2009, AJ, 137, 3358Miller, G. E., & Scalo, J. M. 1979, ApJS, 41, 513Milone, A. P., Piotto, G., Bedin, L. R., et al. 2012, A&A, 540,A16Moe, M., & Di Stefano, R. 2013, ApJ, 778, 95Moe M., Kratter K. M., Badenes C., 2019, ApJ, 875, 61Myers, A. T., Krumholz, M. R., Klein, R. I., & McKee, C. F.2011, ApJ, 735, 49Offner, S. S. R., Kratter, K. M., Matzner, C. D., Krumholz, M. R.,& Klein, R. I. 2010, ApJ, 725, 1485Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010, ApJS,190, 1Rastegaev, D. A. 2010, AJ, 140, 2013Rucinski, S. M. 1994, PASP, 106, 462Sana, H., de Mink, S. E., de Koter, A., et al. 2012, Science, 337,444Sana, H., de Koter, A., de Mink, S. E., et al. 2013, A&A, 550,A107Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131,1163Tanaka, K. E. I., & Omukai, K. 2014, MNRAS, 439, 1884Tian, Z.-J., Liu, X.-W., Yuan, H.-B., et al. 2018, Research inAstronomy and Astrophysics, 18, 052MNRAS000
Abt, H. A., & Levy, S. G. 1976, ApJS, 30, 273Abt, H. A., & Willmarth, D. W. 1987, ApJ, 318, 786Allen, P. R. 2007, ApJ, 668, 492Arenou, F., Luri, X., Babusiaux, C., et al. 2018, A&A, 616, A17Badenes, C., Mazzola, C., Thompson, T. A., et al. 2018, ApJ,854, 147Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Mantelet, G.,& Andrae, R. 2018, AJ, 156, 58Bastian, N., Covey, K. R., & Meyer, M. R. 2010, ARA&A, 48,339Bate, M. R. 2012, MNRAS, 419, 3115Bate, M. R. 2014, MNRAS, 442, 285Belloni, D., Kroupa, P., Rocha-Pinto, H. J., & Giersz, M. 2018,MNRAS, 474, 3740Bressan, A., Marigo, P., Girardi, L., et al. 2012, MNRAS, 427,127Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345,245Carlin, J. L., L´epine, S., Newberg, H. J., et al. 2012, Research inAstronomy and Astrophysics, 12, 755Carney, B. W. 1983, AJ, 88, 623Chabrier, G. 2005, The Initial Mass Function 50 Years Later, 327,41Chen, X., Deng, L., de Grijs, R., et al. 2016, AJ, 152, 129Chini, R., Hoffmeister, V. H., Nasseri, A., Stahl, O., & Zinnecker,H. 2012, MNRAS, 424, 1925Delfosse, X., Beuzit, J.-L., Marchal, L., et al. 2004, Spectroscop-ically and Spatially Resolving the Components of the CloseBinary Stars, 318, 166Deng, L.-C., Newberg, H. J., Liu, C., et al. 2012, Research inAstronomy and Astrophysics, 12, 735Danielski, C., Babusiaux, C., Ruiz-Dern, L., Sartoretti, P., & Are-nou, F. 2018, A&A, 614, A19Dieterich, S. B., Henry, T. J., Golimowski, D. A., Krist, J. E., &Tanner, A. M. 2012, AJ, 144, 64Ducati, J. R., Bevilacqua, C. M., Rembold, S. B., & Ribeiro, D.2001, ApJ, 558, 309Duchˆene, G., & Kraus, A. 2013, ARA&A, 51, 269Duquennoy, A., & Mayor, M. 1991, A&A, 248, 485El-Badry, K., Rix, H.-W., Ting, Y.-S., et al. 2018a, MNRAS, 473,5043El-Badry, K., Ting, Y.-S., Rix, H.-W., et al. 2018b, MNRAS, 476,528El-Badry, K., & Rix, H.-W. 2018, MNRAS, 480, 4884El-Badry, K., & Rix, H.-W. 2019, MNRAS, 482, L139Foreman-Mackey D., Hogg D. W., Lang D. & Goodman J., 2013,PASP, 125, 306Fischer, D. A., & Marcy, G. W. 1992, ApJ, 396, 178Gaia Collaboration, Babusiaux, C., van Leeuwen, F., et al. 2018,A&A, 616, A10Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016,A&A, 595, A1 Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018,A&A, 616, A1Gao, S., Liu, C., Zhang, X., et al. 2014, ApJ, 788, L37Gao, H., Zhang, H.-W., Xiang, M.-S., et al. 2015, Research inAstronomy and Astrophysics, 15, 2204Gao, S., Zhao, H., Yang, H., & Gao, R. 2017, MNRAS, 469, L68Goldberg, D., Mazeh, T., & Latham, D. W. 2003, ApJ, 591, 397Heintz, W. D. 1969, AJ, 74, 768Henry, T. J., & McCarthy, D. W., Jr. 1990, ApJ, 350, 334Hettinger, T., Badenes, C., Strader, J., Bickerton, S. J., & Beers,T. C. 2015, ApJ, 806, L2Jian, M., Gao, S., Zhao, H., & Jiang, B. 2017, AJ, 153, 5Kouwenhoven, M. B. N., Brown, A. G. A., Portegies Zwart, S. F.,& Kaper, L. 2007, A&A, 474, 77Kratter, K. M., Matzner, C. D., & Krumholz, M. R. 2008, ApJ,681, 375Kratter, K. M., Matzner, C. D., Krumholz, M. R., & Klein, R. I.2010, ApJ, 708, 1585Kroupa, P., Tout, C. A., & Gilmore, G. 1990, MNRAS, 244, 76Kroupa P., 1995, MNRAS, 277, 1491Kroupa P., 1995, MNRAS, 277, 1507Kroupa, P., & Bouvier, J. 2003, MNRAS, 346, 343Lada, C. J., & Lada, E. A. 2003, ARA&A, 41, 57Latham, D. W., Stefanik, R. P., Torres, G., et al. 2002, AJ, 124,1144Leinert, C., Henry, T., Glindemann, A., & McCarthy, D. W., Jr.1997, A&A, 325, 159Li, C., de Grijs, R., & Deng, L. 2013, MNRAS, 436, 1497Lindegren, L., Hern´andez, J., Bombrun, A., et al. 2018, A&A,616, A2Liu, C., Xu, Y., Wan, J.-C., et al. 2017, Research in Astronomyand Astrophysics, 17, 096Luo, A.-L., Zhao, Y.-H., Zhao, G., et al. 2015, Research in As-tronomy and Astrophysics, 15, 1095Luri, X., Brown, A. G. A., Sarro, L. M., et al. 2018, A&A, 616,A9Machida, M. N., Omukai, K., Matsumoto, T., & Inutsuka, S.-I.2009, MNRAS, 399, 1255Marks, M., Kroupa, P., & Oh, S. 2011, MNRAS, 417, 1684Marks, M., & Kroupa, P. 2011, MNRAS, 417, 1702Mason, B. D., Gies, D. R., Hartkopf, W. I., et al. 1998, AJ, 115,821Mason, B. D., Henry, T. J., Hartkopf, W. I., ten Brummelaar, T.,& Soderblom, D. R. 1998, AJ, 116, 2975Mason, B. D., Hartkopf, W. I., Gies, D. R., Henry, T. J., & Helsel,J. W. 2009, AJ, 137, 3358Miller, G. E., & Scalo, J. M. 1979, ApJS, 41, 513Milone, A. P., Piotto, G., Bedin, L. R., et al. 2012, A&A, 540,A16Moe, M., & Di Stefano, R. 2013, ApJ, 778, 95Moe M., Kratter K. M., Badenes C., 2019, ApJ, 875, 61Myers, A. T., Krumholz, M. R., Klein, R. I., & McKee, C. F.2011, ApJ, 735, 49Offner, S. S. R., Kratter, K. M., Matzner, C. D., Krumholz, M. R.,& Klein, R. I. 2010, ApJ, 725, 1485Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010, ApJS,190, 1Rastegaev, D. A. 2010, AJ, 140, 2013Rucinski, S. M. 1994, PASP, 106, 462Sana, H., de Mink, S. E., de Koter, A., et al. 2012, Science, 337,444Sana, H., de Koter, A., de Mink, S. E., et al. 2013, A&A, 550,A107Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131,1163Tanaka, K. E. I., & Omukai, K. 2014, MNRAS, 439, 1884Tian, Z.-J., Liu, X.-W., Yuan, H.-B., et al. 2018, Research inAstronomy and Astrophysics, 18, 052MNRAS000 , 1–17 (2019) C. Liu T eff (K) M i n i ( M ) [ F e / H ] Figure A1.
The locations of 10000 arbitrarily selected stars fromthe samples in M ini vs. T eff plane with the color coded [Fe/H].The color lines indicate the PARSEC isochrones with ages of 1,3, 5, 7, 9, and 11 Gyr at each metallicity value, which are codedwith the same color as the stars.Tohline, J. E. 2002, ARA&A, 40, 349Wang, S., & Jiang, B. W. 2014, ApJ, 788, L12Xue, M., Jiang, B. W., Gao, J., et al. 2016, ApJS, 224, 23Yang, Y., Li, C., Deng, L., de Grijs, R., & Milone, A. P. 2018,ApJ, 859, 98Yuan, H.-B., Liu, X.-W., Huo, Z.-Y., et al. 2015, MNRAS, 448,855Zhao, G., Zhao, Y.-H., Chu, Y.-Q., Jing, Y.-P., & Deng, L.-C.2012, Research in Astronomy and Astrophysics, 12, 723 APPENDIX A: INITIAL STELLAR MASSESTIMATION
The stellar mass of stars is determined by comparing the ef-fective temperature ( T eff ), surface gravity ( log g ), and metal-licity ([Fe/H]) of stars with the PARSEC isochrones (Bres-san et al. 2012). For the MS stars in this work, they aremostly not located in the star forming region and henceshould be older than 1 Gyr. Therefore, we remove theisochrones with age younger than 1 Gyr so that the youngisochrones would not affect the mass determination.For each star, the initial stellar mass is determined bylooking for the maximum likelihood in T eff , log g , and [Fe/H]space. The likelihood distribution is written as ln L ∝ − ( T obs − T model ( M ini , a g e ) σ T − ( G obs − G model ( M ini , a g e )) σ G − ( Z obs − Z model ( M ini , a g e )) σ Z , (A1)where T , G , and Z are the effective temperature, surfacegravity, and metallicity, respectively. The uncertainties of T eff (K)0.40.60.81.01.21.41.61.82.0 G B P G R P ( m a g ) Figure B1.
The small dots indicates the locations of the starsin our samples in the G BP − G RP vs. T eff plane. The contoursshows the distribution of these stars. The red stars indicate the5% percentiles, which are adopted as the intrinsic color indices,in the bluest end of G BP − G RP at various T eff . The red solid lineshows the best model of ( G BP − G RP ) as a cubic polynomial of T eff . T eff , log g , and [Fe/H] are σ T = K, σ G = . dex, and σ Z = . dex, according to Gao et al. (2015). Based on El-Badry et al. (2018a), we consider the averaged contributionof the unresolved secondaries in stellar parameter estimationand add systematic bias of 50 K in T eff , uncertainty of 0.1 dexin log g , and uncertainty of 0.05 dex in [Fe/H] to the totaluncertainties and adopt σ T = K, σ G = . dex, and σ Z = . dex.Because that the age for the late type MS stars is verydifficult to be determined, we marginalize the age in Eq. (A1)to derive the likelihood distribution of M ini . We choose thevalue of M ini at the maximum likelihood and determine theuncertainty from the 15% and 85% percentiles. The likeli-hood approach can reach typical precision of about 0.04–0.06 M (cid:12) for the stars with initial stellar mass between 0.4and 1.0 M (cid:12) . For binary system, the stellar parameters aredominated by the primary stars. Therefore, we consider themass estimates for the binaries as the mass of the primary.Figure A1 displays the locations of 10000 randomly se-lected MS stars from the samples in M ini vs. T eff plane withthe color coded [Fe/H]. APPENDIX B: INTERSTELLAR EXTINCTIONCORRECTION
We first derive the color excess of G BP − G RP and then esti-mate the extinction of G band from it with adopted extinc-tion coefficients for G band. Figure B1 shows the distributionof the sample stars in G BP − G RP vs T eff plane. Accordingto Ducati et al. (2001), Wang & Jiang (2014), and Xue etal. (2016), the bluest edge at a given T eff is contributed bythe stars with nearly no reddening in G BP − G RP . Jian et al. MNRAS , 1–17 (2019) olar-type Field Binary Stars (2017) applied this approach and well determined the red-dening of the LAMOST data. We follow the same methodand adopt that the 5% percentiles of G BP − G RP at given T eff bins (red stars displayed in Figure B1) represent for theintrinsic color indices corresponding to T eff . Therefore, wecan estimate the relationship by fitting the intrinsic colorswith a cubic polynomial. We find the best-fit intrinsic colorindex is ( G BP − G RP ) = − . (± . ) × − T + . (± . ) × − T − . (± . ) × − T eff + . (± . ) . (B1)The extinction coefficients for Gaia bands are providedby Danielski et al. (2018) based on
Gaia
DR1 such as k X = c + c ( G BP − G RP ) + c ( G BP − G RP ) + c ( G BP − G RP ) + c A + c A + c ( G BP − G RP ) A , (B2)where k X = A X / A . Given that E ( G BP − G RP ) = ( k BP − k RP ) A , we can solve Eq (B2) for A by using coefficients, c , c , ..., c , corresponding to G BP and G RP , respectively.And then A is brought back to Eq (B2) with coefficientscorresponding to G -band to solve for A G . We adopt the co-efficients, c , c , ..., c for G BP , G RP , and G , respectively,provided in Table 1 of Gaia Collaboration et al. (2018). Asa by-product, we can also obtain A BP and A RP by applyingthe same method. Adopting the extinction law of Cardelli,Clayton & Mathis (1989) with R V = . , we can also esti-mate the color excess of E ( G − K s ) and subsequently give theintrinsic color index ( G − K s ) .It is noted that the estimation of ( G BP − G RP ) doesnot account for the effect induced by metallicity. We tried toderive the intrinsic color index for different metallicity andfound that the difference of the intrinsic color index withmetallicity is only a few hundredth magnitude. Therefore,we ignore the effect of metallicity in the intrinsic color index.The uncertainty of the reddening correction is con-tributed from several channels. First, the cubic polynomialrelation may affected by the uncertainties from the ob-served color index, which is usually as small as a few mill-magnitudes. Then, the uncertainty of T eff from the LAM-OST pipeline, which is around 120 K, is propagated to E ( G BP − G RP ) and contribute to about . mag. Finally,total uncertainty of A G is about . mag. This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS000