The Close Binary Fraction of Solar-type Stars is Strongly Anti-correlated with Metallicity

The Close Binary Fraction of Solar-type Stars is Strongly Anti-correlated with Metallicity

Abstract

There is now strong evidence that the close binary fraction (  10 days;   10 AU) of solar-type stars (  0.6 - 1.5 M) decreases significantly with metallicity. Although early surveys showed that the observed spectroscopic binary (SB) fractions in the galactic disk and halo are similar (e.g., Carney-Latham sample), these studies did not correct for incompleteness. In this study, we examine five different surveys and thoroughly account for their underlying selection biases to measure the intrinsic occurrence rate of close solar-type binaries. We re-analyze: (1) a volume-limited sample of solar-type stars (Raghavan et al. 2010), (2) the SB survey of high-proper-motion stars (Latham et al. 2002), (3) various SB samples of metal-poor giants (Carney et al. 2003; Hansen et al. 2015, 2016), (4) the APOGEE survey of radial velocity (RV) variables (Badenes et al. 2018), and (5) eclipsing binaries (EBs) discovered by Kepler (Kirk et al. 2016). The observed APOGEE RV variability fraction and Kepler EB fraction both decrease by a factor of  4 across 1.0  [Fe/H]  0.5 at the 22 and 9 confidence levels, respectively. After correcting for incompleteness, all five samples / methods exhibit a quantitatively consistent anti-correlation between the intrinsic close binary fraction (  10 AU) and metallicity: = 53%  12%, 40%  6%, 24%  4%, and 10%  3% at [Fe/H] = 3.0, 1.0, 0.2 (mean field metallicity), and +0.5, respectively. We present simple fragmentation models that explain why the close binary fraction of solar-type stars strongly decreases with metallicity while the wide binary fraction, close binary fraction of OB stars, and initial mass function are all relatively constant across 1.5  [Fe/H]  0.5. The majority of solar-type stars with [Fe/H]  1.0 will interact with a stellar companion, which has profound implications for binary evolution in old and metal-poor environments such as the galactic halo, bulge, thick disk, globular clusters, dwarf galaxies, and high-redshift universe.

Subject headings:
binaries: close, spectroscopic, eclipsing; stars: formation, abundances, solar-type
4

1. Introduction

Variations in the close binary fraction (  10 AU) with respect to metallicity have been continuously debated over the years (Carney, 1983; Latham et al., 2002; Carney et al., 2005; Machida et al., 2009; Raghavan et al., 2010; Rastegaev, 2010; Moe & Di Stefano, 2013; Bate, 2014; Badenes et al., 2018, additional references below). Some observations indicate no dependence on metallicity (Latham et al., 2002; Carney et al., 2005; Moe & Di Stefano, 2013), others find the close binary fraction and metallicity are positively correlated (Carney, 1983; Abt & Willmarth, 1987; Hettinger et al., 2015), while yet others have found that the close binary fraction decreases with metallicity (Grether & Lineweaver, 2007; Raghavan et al., 2010; Gao et al., 2014; Badenes et al., 2018). Studying how the close binary fraction varies with primary mass, metallicity, age, and environment provides significant insight into the processes of protobinary fragmenation, accretion, and orbital migration (Kratter et al., 2008, 2010a; Duchêne & Kraus, 2013; Moe & Di Stefano, 2017; Moe & Kratter, 2018). The close binary fraction is also a crucial input parameter in population synthesis studies of blue stragglers, chemically peculiar stars, cataclysmic variables, Type Ia and Ib/c supernovae, X-ray binaries, mergers of compact objects, short gamma-ray bursts, and sources of gravitational waves (Hurley et al., 2002; Eggleton, 2006; Belczynski et al., 2008; Sana et al., 2012; De Marco & Izzard, 2017) A substantial change in the close binary fraction with respect to metallicity would have dramatic consequences for the predicted rates and properties of various channels of binary evolution. The apparent discrepancies in the inferred close binary fraction as a function of metallicity must be reconciled in order to more fully understand binary star formation and to make reliable predictions for binary evolution.

The primary goal of this study is to reconcile the conflicting results reported in the literature in order to accurately measure the bias-corrected close binary fraction of solar-type stars as a function of metallicity. In §2, we overview the methods, results, and potential caveats associated with previous results. In §3, we correct for incompleteness within the Carney-Latham sample and other spectroscopic binary surveys to determine if a large change in the close binary fraction with respect to metallicity is apparent in these earlier datasets. In §4, we analyze the Badenes et al. (2018) sample of APOGEE stars to measure precisely how the radial velocity variability fraction and bias-corrected close binary fraction change as a function of metallicity. We next measure the eclipsing binary fraction of solar-type dwarfs in the Kepler sample, providing a new and independent method for determining how the close binary fraction varies with metallicity (§5). We combine and summarize the observational constraints in §6, where we show all five samples / methods investigated in this study exhibit a remarkably consistent anti-correlation between metallicity and close binary fraction. We also discuss the overall binary fraction and period distribution as a function of mass and metallicity, and highlight the resulting implications for binary evolution. In §7, we investigate fragmentation models to explain why the close binary fraction of solar-type stars strongly decreases with metallicity while the wide binary fraction, close binary fraction of massive stars, and initial mass function are relatively constant. We conclude in §8.

2. Overview of Previous Observations

Carney-Latham Sample. For solar-type (FGK) dwarfs, early observations indicated the spectroscopic binary (SB) fraction of metal-poor halo stars was slightly lower than that of metal-rich stars in the galactic disk (Carney, 1983; Abt & Willmarth, 1987). Subsequent surveys instead found the SB fraction was relatively independent of metallicity (Stryker et al., 1985; Ryan, 1992; Latham et al., 2002; Carney et al., 2005). In particular, Latham et al. (2002) and Carney et al. (2005) investigated a large sample of 1,464 FGK stars with high proper motion in the disk and halo. They identified SBs as stars that exhibited larger radial velocity (RV) variations compared to their RV measurement uncertainties. They obtained a median of = 12 RV measurements per star, and so they were able to fit robust orbital parameters for the majority of their SBs. Latham et al. (2002) measured the halo and disk SB fractions to be 14.5%  1.8% and 15.6%  1.5%, respectively, which are consistent with each other within the uncertainties. They also showed the observed SB period distributions in the disk and halo are similar (see their Fig. 8). Carney et al. (2005) refined the sample by excluding stars with too few RV measurements or large uncertainties in the RVs or metallicities, leaving 994 systems. Carney et al. (2005) measured a slightly larger SB fraction of 24%  2% for their refined sample, but still found the SB fraction was nearly constant across 2.5  [m/H]  0.0 (see their Fig. 2).

However, Latham et al. (2002) and Carney et al. (2005) did not correct for incompleteness. Although the observed SB fraction appears to be independent of metallicity, the true bias-corrected close binary fraction could be substantially different. In fact, to explain the small deficit in the halo SB fraction (14.5%) compared to the disk SB fraction (15.6%), Latham et al. (2002) hinted at the likelihood that their halo measurement was more incomplete. They stated, “This might be the result of an observational bias, because halo binaries have lower metallicity and therefore weaker lines, with a corresponding poorer velocity precision and higher threshold for the detection of binaries.” This effect likely explains why the earlier observations by Carney (1983) and Abt & Willmarth (1987) found a smaller SB fraction for metal-poor stars. In §3.1, we demonstrate that this selection bias reverses the inferred trends in the Carney-Latham SB samples, and therefore the intrinsic close binary fraction of metal-poor halo stars is actually larger than that of metal-rich disk stars.

Volume-limited Samples. Grether & Lineweaver (2007) and Raghavan et al. (2010) provided the earliest statistically significant evidence that the binary fraction of solar-type stars is anti-correlated with metallicity. Raghavan et al. (2010) utilized spectroscopic RV observations, long-baseline and speckle interferometry, adaptive optics, and common proper motion to investigate the multiplicity statistics of 454 FGK dwarfs within 25 pc. In their sample, 411 stars have reliable metallicity measurements across 0.9  [Fe/H]  0.4. As shown in their Fig. 19, Raghavan et al. (2010) found the overall binary fraction decreases from 66%  7% across 0.9  [Fe/H]  0.4 ( = 44 systems) to 39%  3% across 0.3  [Fe/H]  0.4 ( = 343; uncertainties derive from binomial statistics). The overall binary fraction decreases with metallicity by a factor of 1.7  0.2, statistically significant at the 3.8 level. Although the Raghavan et al. (2010) survey is slightly incomplete (Chini et al., 2014; Moe & Di Stefano, 2017), it is difficult to explain how selection biases alone could cause the observed anti-correlation between binary fraction and metallicity.

Close versus Wide Solar-type Binaries. The anti-correlation between metallicity and binary fraction appears to be limited to shorter orbital separations. Of the 44 systems in the Raghavan et al. (2010) sample with 0.9  [Fe/H]  0.4, 22 (50%  8%) have companions with log  (days)  6 (  200 AU) and 7 (16%  5%) are wide binaries with log  (days)  6 (  200 AU). Meanwhile, of the 343 systems with 0.3  [Fe/H]  0.4, 87 (25%  2%) and 47 (14%  2%) have companions below and above   200 AU, respectively. Hence, the very wide binary fraction (  200 AU) remains constant within the uncertainties. Common proper motion and CCD imaging surveys also demonstrate the wide binary fraction of solar-type stars is independent of metallicity (Chanamé & Gould, 2004; Zapatero Osorio & Martín, 2004). Meanwhile, the binary fraction below   200 AU in the Raghavan et al. (2010) sample decreases by a factor of 2.0  0.3 between [Fe/H]  0.6 and 0.0, statistically significant at the 3.2 level.

Rastegaev (2010) combined spectroscopy, speckle interferometry, and visual observations to measure the full multiplicity properties of metal-poor FGK stars ([m/H]  1). After correcting for incompleteness, they measured an overall binary fraction of  40%, which is consistent with the binary fraction of 46%  2% measured by Raghavan et al. (2010) for solar-type stars within 25 pc. Compared to metal-rich systems, however, Rastegaev (2010) showed metal-poor binaries are significantly skewed toward close to intermediate separations, exhibiting a factor of  2 - 3 excess across log  (days) = 1 - 4 ( 0.1 - 10 AU; see their Fig. 10). Their combined spectroscopic and speckle interferometric survey is relatively complete across this parameter space, and so the factor of  2 - 3 excess observed across   0.1 - 10 AU for metal-poor FGK binaries is likely a real effect.

Wide Companions to KM Subdwarfs. Speckle, HST, and adaptive optics imaging of metal-poor KM subdwarfs all indicate a lower wide binary fraction compared to their solar-metallicity counterparts (Riaz et al., 2008; Jao et al., 2009; Lodieu et al., 2009; Ziegler et al., 2015). However, these surveys specifically targeted metal-poor stars based on their photometric colors and absolute magnitudes, i.e., KM subdwarfs in the HR diagram that lie well below the main-sequence relation of solar-metallicity dwarfs. A metal-poor subdwarf with an equally bright companion would appear photometrically as a normal metal-rich dwarf, and so would not have been included in their samples. Late-K and M-type binaries are weighted toward equal-mass companions (Janson et al., 2012; Dieterich et al., 2012; Duchêne & Kraus, 2013). A bias against equally bright companions would dramatically reduce the inferred binary fraction of metal-poor KM subdwarfs. In their adaptive optics survey of metal-poor KM subdwarfs, Ziegler et al. (2015) specifically noted a substantial shortage of low-contrast companions with  2 mag compared to metal-rich KM dwarfs (see their Fig. 10). A deficit of binaries with nearly equal brightnesses is naturally explained by their subdwarf photometric selection criteria. These surveys are heavily influenced by this selection bias and we conclude there is little or no change in the wide binary fraction of KM stars as a function of metallicity.

Recent Wide-field Surveys. Over the past few years, there have been several wide-field spectroscopic surveys that measured the chemical abundances and radial velocities of hundreds of thousands of stars. Some of these spectroscopic surveys obtained multiple epochs of individual stars, allowing for a statistical measurement of the RV variability fraction as a function of metallicity. Utilizing multi-epoch SDSS spectra of F-type dwarfs (resolution R  2,000), Hettinger et al. (2015) measured the RV variability fraction increases by  30% between [Fe/H] = 1.7 and 0.5 (see their Fig. 5). Based on SEGUE and LAMOST spectra of FGK dwarfs (R  2,000), Gao et al. (2014), Gao et al. (2017), and Tian et al. (2018) instead found the RV variability fraction decreases by a factor of  2 between their metal-poor ([Fe/H] 1.1) and metal-rich ([Fe/H] 0.6) samples. They also determined the RV variability fraction increases by a factor of  2 between K-type and F-type dwarfs, consistent with other studies that show the close binary fraction strongly increases above   1M (Abt et al., 1990; Raghavan et al., 2010; Sana et al., 2012; Duchêne & Kraus, 2013; Moe & Di Stefano, 2017; Murphy et al., 2018). Utilizing SEGUE spectra of extremely metal-poor stars with [Fe/H]  3.0, Aoki et al. (2015) estimated the binary fraction below   1,000 days is  20%, nearly double that of their metal-rich counterparts.

Most recently, Badenes et al. (2018) analyzed multi-epoch APOGEE spectra of  90,000 FGK stars, which had superior spectral resolution R  22,500 and higher signal-to-noise ratios S/N  40. They searched for RV variables that exhibited large enough amplitudes RV  10 km s between epochs to be nearly 100% certain they were real binary stars. Badenes et al. (2018) demonstrated the RV variability fraction decreases by a factor of  2 - 3 between their low-metallicity tercile ([Fe/H] 0.3) and high-metallicity tercile ([Fe/H]  0.0). They observed this factor of  2 - 3 metallicity effect for stars of varying surface gravities 0.0  log  (cm s 5.0 (see their Fig. 13). This suggests the anti-correlation between binary fraction and metallicity occurs for both close companions orbiting small main-sequence stars and for slightly wider companions orbiting large giants. We investigate a subset of the APOGEE data in §4 to quantify more precisely how the RV variability fraction and close binary fraction change as a continuous function of metallicity.

Other observational methods corroborate that the binary fraction of FGK stars decreases with metallicity, but to a lesser extent than the factor of  2 - 3 effect determined by Badenes et al. (2018). For example, Yuan et al. (2015) analyzed the properties of binaries discovered through the stellar locus outlier method. These are unresolved binaries in which the companions are bright enough to sufficiently shift the combined photometric colors to be inconsistent with single stars. They found the unresolved binary fraction decreases by a factor of  1.4 between [Fe/H]  1.7 and 0.3. Similarly, El-Badry et al. (2018) identified double-lined spectroscopic binaries (SB2s) with luminous secondaries in the APOGEE dataset. For SB2s that exhibited significant orbital motion between epochs, i.e., RV  10 km s as adopted in Badenes et al. (2018), El-Badry et al. (2018) confirmed the close binary fraction decreases by a factor of  1.6 between their low-metallicity tercile ([Fe/H] 0.2) and high-metallicity tercile ([Fe/H] 0.0). However, for their larger population of wider SB2s that did not show RV variability, El-Badry et al. (2018) found the binary fraction was consistent with being constant with respect to metallicity. Taken as a whole, these recent observations suggest the close binary fraction of solar-type stars is strongly anti-correlated with metallicity while the wide binary fraction is independent of metallicity. Photometric binaries (Yuan et al., 2015) and SB2s (El-Badry et al., 2018), which include both close and wide binaries, exhibit a weaker trend with metallicity compared to close binaries exclusively.

Close Massive Binaries. Meanwhile, the close binary fraction of massive stars does not vary significantly with metallicity (Moe & Di Stefano, 2013; Dunstall et al., 2015; Almeida et al., 2017). Moe & Di Stefano (2013) measured the eclipsing binary (EB) fraction of early-B stars (  6 - 16 M) based on OGLE observations of the Small ([Fe/H]  0.7) and Large ([Fe/H]  0.4) Magellanic Clouds (SMC/LMC) and Hipparcos observations of nearby systems in the Milky Way (MW; [Fe/H]  0.0). They found the fraction of early-B stars that have eclipsing companions across orbital periods  = 2 - 20 days and eclipse depths m = 0.25 - 0.65 mag is 0.70%  0.06%, 0.69%  0.03%, and 1.00%  0.25% for the SMC, LMC, and MW, respectively (see their Table 1). Although EB observations are less complete due to geometrical selection effects, they are not affected by the spectroscopic selection bias discussed above and are therefore more robust in detecting variations in the close binary fraction with respect to metallicity. Nevertheless, after correcting for incompleteness in their spectroscopic RV observations, the close binary fraction of O stars (Almeida et al., 2017) and early-B stars (Dunstall et al., 2015) in the LMC is consistent with their solar-metallicity counterparts in the MW. For massive stars (  6 M), the close binary fraction is relatively independent of metallicity, at least within the / 30% measurement uncertainties and across the range of metallicities 0.7  [Fe/H]  0.1 probed by the observations.

Initial Mass Function. Similarly, the initially mass function (IMF) is fairly universal across two orders of magnitude in metallicity 1.5  [Fe/H]  0.5 (Bastian et al., 2010; Kroupa et al., 2013, references therein). Young metal-poor associations and clusters in the LMC ([Fe/H]  0.4; Da Rio et al. 2009), in the SMC ([Fe/H]  0.7; Sirianni et al. 2002; Schmalzl et al. 2008), and in the outer regions of the MW ([Fe/H]  0.8; Yasui et al. 2016b, a) all have IMFs consistent with the canonical IMF. The low-mass end of the IMF (  0.1 - 0.9M) is invariant across galactic open clusters and globular clusters that span a wide range of metallicities 2.3  [Fe/H]  0.3 (von Hippel et al., 1996; De Marchi et al., 2010; Bastian et al., 2010). Although some observations indicate the IMF becomes top-heavy toward lower metallicities (Marks et al., 2012; Geha et al., 2013; Kroupa et al., 2013), this trend is not statistically significant until the metallicity falls below at least [Fe/H]  1.5.

3. Spectroscopic Versus Intrinsic Close Binary Fraction

3.1. Carney-Latham Sample

Description of Observations

Of the 1,464 stars with high proper motion in the Carney-Latham sample, Latham et al. (2002) cataloged detailed information for 1,359 single-lined stars. They listed the stellar properties, e.g., metallicity [m/H], effective temperature , and rotational velocity , of the template spectrum that most closely matched the observed spectra. The full temperature range is   3,800 -7,700 K, but 1,301 of the systems (96%) have   4,500 - 6,300 K, corresponding approximately to F7 - K4 spectral types. The template spectra are in large metallicity increments of [m/H] = 0.5, but 1,349 of their 1,359 single-lined stars span a large range of 3.0  [m/H]  0.5 to provide sufficient leverage for investigating metallicity effects. Latham et al. (2002) derived robust orbital solutions for 156 SB1s (all with   7,000 days) and presented preliminary orbits for an additional 15 SB1s (mostly with  = 5,000 - 10,000 days). They also cataloged 17 large-amplitude RV variables that likely have wide stellar companions but lack the necessary phase coverage to measure orbital elements (see their Fig. 6). In a separate study, Goldberg et al. (2002) measured stellar parameters and orbital solutions for 34 SB2s from the Carney-Latham sample, all of which have   5,000 days and 2.5  [m/H]  0.0. Neither Latham et al. (2002) nor Goldberg et al. (2002) fitted the surface gravities log  directly, but instead adopted log  = 4.5 for cooler stars ( 6,000K) and log  = 4.0 for hotter stars ( 6,000K). About 10% of the high-proper-motion stars in the Carney-Latham sample are likely subgiants or giants (Laird et al., 1988; Carney et al., 1994), and the fraction is probably larger for systematically older halo stars.

Latham et al. (2002) listed the Julian dates, RVs, and RV uncertainties for each of the observations of each single-lined star. We compile their data and compute the mean RV uncertainty for each system. In Fig. 1, we show the average of and 1 spread in as a function of metallicity [m/H]. As indicated in Latham et al. (2002), the metal-poor stars in their sample have systematically larger RV uncertainties due to their weaker absorption lines. The mean uncertainties double from  = 0.5 km s for solar-metallicity to  = 1.0 km s for metal-poor stars with [m/H]  2.0.

Latham et al. (2002) observed their single-lined stars with varying cadence (see their Fig. 3). For their full sample, the median number of RV measurements is = 12, and the 10 - 90 percentile range spans = 8 - 39. Similarly, the median timespan is  = 9 yr between first and final visits, and the 10 - 90 percentile interval is  = 8 - 14 yr. There is no trend in the number or timespan of RV measurements as a function of metallicity. The median number of RV observations is  = 13 for the 544 metal-poor single-lined stars with 3.0  [m/H]  0.8 and  = 11 for the 805 metal-rich stars with 0.8  [m/H]  0.5. The median timespan, which is most important parameter for estimating completeness rates (see below and §4), is  = 9 yr for both the metal-poor and metal-rich subsamples.

Corrections for Incompleteness

We next perform Monte Carlo simulations to determine the probability of detecting SBs as a function of . In our simulations, we fix the mass of the primary to be  = 1.0 M  and draw period, mass-ratio, and eccentricity distributions consistent with solar-type binaries in the field (Duquennoy & Mayor, 1991; Raghavan et al., 2010; Tokovinin, 2014; Moe & Di Stefano, 2017). Specifically, we adopt a log-normal period distribution with a peak at log  (days) = 4.9 and dispersion of = 2.3, but only select binaries from the short-period tail across the interval 0.0  log  (days)  4.0 (  10 AU) we are investigating. We assume a uniform mass-ratio distribution across  = / = 0.1 - 1.0 and that very close binaries with    = 10 days are tidally circularized. Toward longer periods   , we adopt a uniform eccentricity distribution across the interval 0.0    (), where the upper envelope of the eccentricity versus period distribution derives from conservation of orbital angular momentum during tidal evolution (Badenes et al., 2018):

(1)

We assume random orientations, which have an inclination probability distribution of = sin  and a uniform distribution for arguments of periastron. Reasonable variations in the period, mass-ratio, or eccentricity distributions yield only minor changes in the simulated detection efficiencies.

Figure 1.— The mean RV uncertainty as a function of metallicity [m/H] in the Latham et al. (2002) sample (black). We simulate the completeness rates for a population of binaries with   10 days (dotted red). As the RV uncertainties decrease from = 1.0 km s to 0.5 km s between metal-poor and metal-rich stars, the completeness fractions increase from  40% to  70%.

For each binary, we generate RVs at  = 12 epochs randomly distributed across a timespan of  = 9 yr, matching the median cadence and median baseline of the Latham et al. (2002) observations. For each RV measurement, we add Gaussian random noise according to . A large-amplitude RV variable will exhibit a larger variance of RVs compared to the variance implied by its measurement uncertainties. We therefore use an F-variance test to measure the probability that each generated system has a constant RV. In the Latham et al. (2002) catalog, the majority of constant RV stars have   510 while nearly all systems with   510 are cataloged as SBs, the majority of which have measured orbital parameters. We adopt the criterion that   510 for a simulated binary to be considered an RV variable, corresponding to a 5.0 level of significance.

We show the results of our Monte Carlo simulations in Fig. 1. Given a small RV uncertainty  = 0.2 km s,  90% of the binaries with   10 days would appear as spectroscopic RV variables with   510. The remaining  10% of the binaries are generally in wide orbits (  5,000 - 10,000 days) with low-mass companions ( 0.1 - 0.3). Meanwhile, given a mean uncertainty of  = 1.3 km s and 12 random epochs across 9 years, only  30% of binaries with   10 days would appear as RV variables. Across the interval of interest, the completeness rate increases from  40% for metal-poor halo stars ([m/H]  2.0;   1.0 km s) to  70% for metal-rich disk stars ([m/H]  0.0;   0.5 km s). The Latham et al. (2002) spectroscopic survey is  1.8 times more complete in detecting close binary companions to metal-rich disk stars compared to metal-poor halo stars.

Binary Mass Functions

The observed distribution of binary mass functions = ( sin )/( + ) =  (1 )/(2G) also demonstrates that metal-poor SBs are less complete. In Fig. 2, we show the measured binary mass functions versus orbital periods for the 169 SB1s with   10 days in the Latham et al. (2002) sample. We also display with slightly larger symbols the 34 SB2s from Goldberg et al. (2002), which concentrate toward larger binary mass functions = 0.007 - 0.2 M as expected. We divide the sample into a metal-poor subset with 3.0  [m/H]  0.8 (red crosses;  = 91 SBs with measured orbital elements;  = 562 stars) and a metal-rich subset with 0.8  [m/H]  0.5 (blue squares;  = 114,  = 821). Both subsamples are measurably incomplete toward wide separations and small ratios. However, the metal-rich SB1s, which have systematically smaller RV uncertainties, extend toward smaller binary mass functions and longer orbital periods. A KS test demonstrates that the observed 71 SBs with   100 days in our metal-rich subset are weighted toward smaller velocity semi-amplitudes compared to the 57 metal-poor SBs with   100 days at the 2.7 confidence level ( = 0.004). For reference, we also show as a function of for a fixed eccentricity of  = 0.5 and a velocity semi-amplitude of  = 6, corresponding to  = 3 km s for metal-rich stars (dashed blue line in Fig. 2) and  = 6 km s for metal-poor stars (dashed red). The Latham et al. (2002) SB1 sample is measurably incomplete below these relations.

The samples of SB1s and SB2s with measured orbital solutions are relatively complete across  = 20 - 2,000 days and above binary mass functions corresponding to  = 6 km s and  = 0.5. We display this relatively complete parameter space by solid black lines in Fig. 2. Enclosed within this area, the SB fraction is 49/554 = 8.7%  1.2% for our metal-poor subsample (3.0  [m/H]  0.8). Meanwhile, the SB fraction within the same region of and is only 38/821 = 4.6%  0.7% for our metal-rich subsample (0.8  [m/H]  0.5). By focusing on this relatively complete parameter space, we demonstrate that the close binary fraction decreases by a factor of 1.9  0.4 at the 3.0 significance level between our metal-poor and metal-rich subsamples.

The sample of SBs with measured orbital solutions is incomplete beyond   2,000 days (right of black dashed line in Fig. 2). The handful of systems in this part of the parameter space required substantially more RV measurements and longer timespans to fit the orbits. For example, the median number and timespan of RV measurements for the 15 long-period SB1s with preliminary orbits are  = 57 and  = 18 yr, respectively, which are considerably larger than the median values of = 12 and  = 9 yr for the Latham et al. (2002) sample as a whole. In addition, the 17 SB1s without orbital solutions in the Latham et al. (2002) catalog likely have   2,000 days, but simply lack the number of observations and/or timespan to fit the RVs (see their Fig. 6).

Figure 2.— The measured binary mass functions and orbital periods for the 169 SB1s with   10 days from Latham et al. (2002, smaller symbols) and 34 SB2s from Goldberg et al. (2002, larger symbols) divided into metal-poor (3.0  [m/H]  0.8; red crosses) and metal-rich (0.8  [m/H]  0.5; blue squares) subsets. The samples are biased against very close binaries with   20 days (left of dotted black line) due to contamination by subgiants and giants while wide binaries beyond   2,000 days (right of dashed black line) are incomplete given the median number  = 12 and timespan  = 9 yr of the RV observations. SBs with small velocity semi-amplitudes   6 are also incomplete, corresponding to   3 km s for metal-rich systems (dashed blue) and   6 km s for metal-poor systems (dashed red). Within the relatively complete and unbiased parameter space (solid black lines), the SB fraction decreases by a factor of 1.9  0.4 from 8.7%  1.2% for the metal-poor subsample to 4.6%  0.7% for the metal-rich subsample at the 3.0 significance level.

The Carney-Latham SB sample is also slightly biased against very close binaries with   20 days due to contamination by subgiants and giants. As stars in very close binaries expand beyond the main-sequence (MS), they undergo Roche-lobe overflow, thereby preventing evolution toward the giant stage. Badenes et al. (2018) thoroughly discussed this effect of giant evolution truncating the short-period tail of the binary period distribution as a function of giant surface gravity, an indicator of radius. In volume-limited samples of solar-type dwarfs, the very close binary fraction below   20 days is 4%  1% (Duquennoy & Mayor, 1991; Raghavan et al., 2010; Tokovinin, 2014; Moe & Di Stefano, 2017). In our metal-rich subsample with 0.8  [m/H]  0.5, however, the observed very close binary fraction is only 13/821 = 1.6%  0.4% (see systems left of dotted black line in Fig. 2). The very close binary fraction in our metal-poor subsample with 3.0  [m/H]  0.8 is lower still at 6/562 = 1.1%  0.4%, likely due to a larger contamination by giants for systematically older halo stars. We estimate that the close binary fraction should increase by 1% and 2% for our metal-rich and metal-poor subsamples, respectively, in order to correct for this selection bias.

Intrinsic Close Binary Fraction

In Fig. 3, we show the observed SB fraction as a function of metallicity for the combined Latham et al. (2002) and Goldberg et al. (2002) samples (dotted black data points). The observations are consistent with a constant  15% - 20% SB fraction across the full metallicity range 3.0  [m/H]  0.5 as reported in Latham et al. (2002) and Carney et al. (2005). We correct the observed distribution according to our simulated completeness rates displayed in Fig. 1. For example, the observed SB fraction for [m/H] = 0.0 is 14%  2%. For this metallicity, we estimate  70% of binaries with   10 days are detectable as SBs (Fig. 1), implying a corrected close binary fraction of (0.14  0.02)/0.70 = 20%  3%. We add the 1% of very close metal-rich binaries (  20 days) that were excluded due to contamination by subgiants and giants, resulting in our final value of  = 21%  3% for [m/H] = 0.0. We repeat this procedure for each of the metallicity intervals, but add 2% to the close binary fraction of metal-poor stars ([m/H]  1) to account for the increased contamination by evolved giants in the older metal-poor populations.

Figure 3.— As a function of metallicity, the observed SB fraction (dotted) versus the intrinsic close binary fraction (  10 days;   10 AU) after correcting for incompleteness and the removal of very close binaries due to giant evolution (solid). We compare the samples of extremely metal-poor giants (red; Hansen et al., 2015, 2016a), metal-poor giants (blue; Carney et al., 2003), and solar-type stars (mostly dwarfs) with high proper motion in the Carney-Latham survey (black; Latham et al., 2002; Goldberg et al., 2002). Although the observed SB fraction of  15% - 20% is relatively independent of metallicity, the true bias-corrected close binary fraction decreases from 35% - 55% across 3.5  [Fe/H]  1.0 to   20% at [Fe/H] = 0.0.

We display in Fig. 3 our bias-corrected close binary fraction as a function of metallicity based on the Carney-Latham sample (solid black). The corrected close binary fraction decreases by a factor of 3.2 from = 54%  12% at [m/H]  2.7 to = 17%  6% at [m/H]  0.5. Attempting to fit a constant close binary fraction to the seven black data points in Fig. 3 results in a reduced / = 3.5 with  = 6 degrees of freedom. The probability to exceed this value is  = 0.0016, i.e., the bias-corrected close binary fraction decreases with metallicity at the 3.0 significance level. This is identical to the level of significance determined by comparing the metal-poor and metal-rich SB fractions across the parameter space in Fig. 2 that was relatively complete.

Focusing on a narrower metallicity interval, the close binary fraction decreases by a factor of 2.2 between [m/H] = 1.0 and +0.5 in Fig. 3. A factor of  2 - 4 decrease in the close binary fraction across this metallicity interval, as indicated in Badenes et al. (2018) and measured by us in §4, is fully consistent with the Carney-Latham observations. We conclude that once corrections for incompleteness and selection biases are considered, the Carney-Latham sample is not only consistent with a large anti-correlation between metallicity and the close binary fraction, but actually supports such a trend at the 3.0 significance level.

3.2. Metal-poor Giants

The SB fractions of metal-poor giants (Carney et al., 2003) and extremely metal-poor giants enriched with r-process elements or carbon (Hansen et al., 2015, 2016a) are  15% - 20%. These values are consistent with the observed SB fractions of metal-poor dwarfs in the halo (Latham et al., 2002; Carney et al., 2005). We re-emphasize that the observed SB fractions are lower limits to the true close binary fractions, especially for metal-poor stars that have weaker absorption lines. In the following, we account for incompleteness within these additional samples of metal-poor stars in order to compute their intrinsic close binary fractions.

Carney et al. (2003) Sample

Carney et al. (2003) obtained a median of  = 13 RV measurements of 91 metal-poor field giants with an average precision of  = 0.65 km s and a median timespan of  = 13.8 yr. This is similar in frequency but with improved sensitivity and duration compared to the Latham et al. (2002) survey of metal-poor dwarfs in the halo. The metallicities of the giants span 4.0  [Fe/H]  0.9, resulting in a mean and 1 spread of [Fe/H] = 2.0  0.5. These metal-poor giants are some of the oldest stars in the galaxy, and therefore have masses 0.8 - 1.1 M corresponding to MS-turnoff ages of   7 - 13 Gyr. Carney et al. (2003) identified 16 SB1s in their sample and measured robust orbital periods spanning   40 - 5,200 days for 14 of them. As shown in Fig. 3, the observed SB fraction is 16/91 = 18%  4%.

The most luminous giants in the Carney et al. (2003) sample exhibit significant RV jitter due to radial pulsations, convective instabilities in the tenuous upper layers, or intermittent starspots modulated by rotation. They found  40% of giants with absolute magnitudes   1.4 display detectable RV jitter   1 km s. Hekker et al. (2008) later showed that non-periodic RV jitter occurs in smaller, less luminous giants, but simply the magnitude increases from  = 0.03 km s at log   3.0 to  = 0.3 km s at log   1.5. Stochastic variations in the RVs due to intrinsic fluctuations in the atmospheres inhibit the detection of SBs with small velocity semi-amplitudes. We therefore remove the nine giants in the Carney et al. (2003) sample that exhibit significant RJ jitter (dark systems in their Fig. 8). One of these objects, HD 218732, is also an SB in which the velocity semi-amplitude  = 2.9 km s induced by the companion is larger than the RV jitter   1 km s. The observed SB fraction for our refined subsample remains unchanged at 15/82 = 18%  4%.

The metal-poor giants in the Carney et al. (2003) sample also span a broad range of radii  = 4.3 - 112 R, providing a mean of = 23 R. Adopting typical parameters   1.0 M and  = 0.5, then very close binaries with   35 days would have already filled their Roche lobes by the time the primaries evolved to = 23 R (Eggleton, 1983). The Carney et al. (2003) sample is therefore significantly biased against very close binaries with   35 days. Their closet binary, i.e., BD +133683 with   40 days, happens to contain the smallest giant ( = 4.3R) in their sample. We correct for incompleteness and this selection bias using two different methods described below.

First, we perform a Monte Caro simulation as done in §3.1.2 to measure the completeness rate, but adopt  = 13,  = 13.8 yr, and  = 0.65 km s to match the median cadence and sensitivity of the Carney et al. (2003) observations. We increase the circularization period to  = 100 days in Eqn. 1 to account for the larger tidal radius of the giants. We also generate close binaries across the interval  = 35 - 10 days because very close binaries with   35 days have effectively been removed from the Carney et al. (2003) sample of giants. Of all the metal-poor giants with companions across  = 35 - 10 days, we calculate 55% would have been detected as SBs by Carney et al. (2003) at the 5 significance level. This is slightly lower than the completeness rate of 62% for  = 0.65 km s inferred from Fig. 1. Despite the increased timespan of the Carney et al. (2003) observations, the removal of very close binaries with   35 days, which are easier to detect, causes the overall completeness rate to decrease. The details of tidal circularization during the giant phase have a negligible effect on our corrections for incompleteness; we repeat our Monte Carlo simulation with  = 20 and 500 days, and calculate completeness rates of 54% and 56%, respectively. The corrected binary fraction of metal-poor giants in the Carney et al. (2003) sample is (0.18  0.04)/0.55 = 33%  7% across  = 35 - 10 days. According to our adopted log-normal period distribution for solar-type binaries, 17% of close binaries with log  (days) = 0 - 4 have very short periods  = 1 - 35 days. The close binary fraction (log  = 0 - 4;  10 AU) of metal-poor dwarfs is therefore = (0.33  0.07)/0.83 = 40%  8% after accounting for the bias against very close binaries in giant systems.

Second, we examine in Fig. 4 the binary mass functions and periods of the 13 SBs with measured orbital elements and no significant RV jitter in Carney et al. (2003), similar to our analysis of the the Carney-Latham SBs (see Fig. 2). We also show in Fig. 4 a random subset of 1,000 binaries spanning  = 35 - 10 days from our Monte Carlo simulation with  = 100 days, indicating those that were detectable above the 5 level with darker, thicker symbols. The observed density of SBs in the versus parameter space follow our simulated detections quite well. Our analysis confirms that the Carney et al. (2003) SB survey is incomplete toward long periods and small binary mass functions. In our Monte Carlo model, 37% of binaries have  = 35 - 3,000 days and binary mass functions greater than that corresponding to  = 7 km s and  = 0.5. We indicate this parameter space, which is  95% complete, in Fig. 4. We find eight of the SBs from the Carney et al. (2003) sample are located within this relatively complete region, indicating a corrected binary fraction of 8/82/0.37/0.95 = 28%  10%. After accounting for the bias against very close binaries with   35 days, the close binary fraction of metal-poor dwarfs is =(0.28  0.10)/0.83 = 34%  12%.

Figure 4.— Similar to Fig. 2, but for the 13 SBs with orbital solutions and no significant RV jitter in the Carney et al. (2003) sample of metal-poor giants (blue squares) and 6 SBs with orbital solutions in the Hansen et al. (2015, 2016a) samples of extremely metal-poor giants chemically enriched with r-process elements or carbon (red crosses). We also display a random subset of 1,000 binaries from our Monte Carlo simulations (green pluses) that match the cadence and sensitivity of the Carney et al. (2003) observations. The simulated binaries that exhibit RV variability above a 5 significance level are indicated with darker, larger symbols. The observations are  95% complete across  = 35 - 3,000 days and above binary mass functions corresponding to  = 7 km s and  = 0.5 (black lines).

The bias-corrected close binary fraction determined from our forward-modeling method ( = 40%  8%) is consistent with our inversion technique ( = 34%  12%). We adopt an average of  = 37%  10%, and present the result in Fig. 3. The bias-corrected close binary fraction measured for the Carney et al. (2003) sample of metal-poor giants matches the close binary fraction determined for metal-poor halo stars with high proper motion in the Carney-Latham sample.

Hansen et al. (2015, 2016a) Samples

We next combine the samples of extremely metal-poor giants enriched with r-process elements (Hansen et al., 2015) and with carbon (Hansen et al., 2016a). We do not include extremely metal-poor giants enriched with s-process elements, e.g., barium, which exhibit a very large SB fraction of 80% and are clearly the result of post-MS binary mass transfer (Jorissen et al., 1998; Lucatello et al., 2005; Hansen et al., 2016b). Hansen et al. (2015, 2016a) concluded the abundances of extremely metal-poor giants enriched with r-process elements and carbon are primordial, i.e., the enhanced elements were imprinted on their natal molecular clouds. Our combined sample contains 41 extremely metal-poor giants that span 5.8  [Fe/H]  1.6, providing a mean and 1 spread of [Fe/H] = 3.0  0.7. Within this sample, Hansen et al. (2015, 2016a) found seven SBs, six of which have orbital solutions. We display the observed SB fraction of 7/41 = 17  6% in Fig. 3.

Hansen et al. (2015, 2016a) observed their 41 targets with varying cadence. In particular, 11 of their extremely metal-poor giants were observed only = 2 - 7 times. For comparison, both Latham et al. (2002) and Carney et al. (2003) obtained at least 7 measurements for each of their targets,  90% of which were observed 9 times. A small number = 2 - 7 of RV measurements reduces the probability of detecting RV variability, and makes it nearly impossible to fit robust orbital solutions. We therefore remove the 11 objects with = 2 - 7, none of which were identified as SBs, leaving 30 extremely metal-poor giants in our culled sample.

The mean RV precision of the extremely metal-poor giants in the Hansen et al. (2015, 2016a) samples ranged significantly from = 0.012 km s to 2.5 km s. With such a large variance in , a Monte Carlo simulation with a single value of is no longer valid. We instead rely on the measured binary mass functions and periods of the 6 SBs with orbital solutions, which are displayed in Fig. 4. One of the SBs, HE 1523−0901, has an extremely small binary mass function of = 1.310 M (Hansen et al., 2015). This object was observed with superior precision = 0.016 km s and more times ( = 34) than any other targets in the Hansen et al. (2015, 2016a) samples. If the other targets were SBs with such small binary mass functions, they would not be detected.

Meanwhile, the other five SBs with orbital solutions in the Hansen et al. (2015, 2016a) survey extend across the upper middle region in Fig. 4, spanning 0.04 - 0.14 M and 37 - 2,500 days. The fact that 5 of the 30 extremely metal-poor giants with   8 are SBs with such large binary mass functions strongly suggests the intrinsic close binary fraction is particularly large. These five SBs occupy the same parameter space that is  95% complete according to our Monte Carlo model that simulates the cadence and sensitivity of the Carney et al. (2003) observations. Although the Hansen et al. (2015, 2016a) surveys had variable precision, we also expect this parameter space to be  95% complete. We therefore use the same inversion technique to correct for incompleteness, resulting in an intrinsic binary fraction of 5/30/.37/.95 = 47%  19% across  = 35 - 10 days. After accounting for the bias against very close binaries with   35 days, the primordial close binary fraction of extremely metal-poor dwarfs is = (0.47  0.19)/0.83 = 57%  22%. We display our result in Fig. 3, which is consistent with our measurement of = 54%  12% for extremely metal-poor stars with [m/H] = 2.7  0.7 selected from the Carney-Latham sample. The close binary fraction of metal-poor dwarfs, metal-poor giants, and extremely metal-poor giants are all   35% - 55%, substantially larger than the close binary fraction   20% of solar-metallicity FGK dwarfs in the disk.

4. APOGEE RV Variables

4.1. Sample Selection and Description

The SDSS-IV/APOGEE near-infrared spectroscopic survey (data release 13) measured the effective temperatures, surface gravities, metallicities, and RVs of  164,000 stars in various environments including the galactic disk, bulge, and halo (Zasowski et al., 2013; Holtzman et al., 2015; Nidever et al., 2015; Albareti et al., 2017). After calibrating their observations to both synthetic spectra and empirical relations, APOGEE measured the stellar parameters to high precision, e.g.,   90 K, log   0.11 dex, and [Fe/H]  0.15 dex (Holtzman et al., 2015). In their study, Badenes et al. (2018) removed targets in open clusters and stars with effective temperatures or surface gravities that were inadequately measured, leaving 122,141 objects. They then examined the spectra and RV measurements for each star, keeping only the individual visits with spectral S/N  40. A total of 91,246 stars with   2 high-quality RV measurements (78% which have   3 epochs) were included in the Badenes et al. (2018) analysis. We further remove the 2,893 stars (mostly giants) with [Fe/H]  0.9 and 7 systems with [Fe/H]  0.5, leaving 88,346 stars across the interval 0.9  [Fe/H]  0.5 in our final sample. The metallicity distribution is adequately modeled by a Gaussian with mean of [Fe/H] = 0.16 and dispersion of  = 0.26 dex (see Fig. 5).

We divide our sample according to the measured surface gravities and effective temperatures. Of the 88,346 stars in our full sample, 20,649 are MS dwarfs or Hertzsprung gap (HG) subgiants with 3.2  log   5.0 while the remaining 67,697 are giants with 0.1  log   3.2. The giants mostly have primary masses   1.1 - 2.0 M with an average of   1.5 M (see Fig. 2 and Fig. 4 in Badenes et al. 2018). Our giant subsample includes both normal and red clump giants. APOGEE red clump giants were targeted differently (Zasowski et al., 2013), and as a result are slightly biased against close binaries (Badenes et al., 2018). Fortunately, only  20% of the APOGEE giants occupy the red clump (Badenes et al., 2018), and so the bias in the RV variability fraction can at most be 20% for our overall giant subsample. For our MS/HG stars, a majority (13,864 objects; 67%) have effective temperatures  = 4,000 - 5,000 K, corresponding roughly to K IV/V stars with primary masses   0.6 - 1.1 M. Another 5,375 MS/HG stars (26%) have  = 5,000 - 6,000 K, corresponding approximately to G IV/V stars with   0.9 - 1.4 M. The remaining 1,410 MS/HG stars (7%) are either cool early-M dwarfs ( = 3,500 - 4,000 K) or hot late-F stars ( = 6,000 - 6,500 K). In the following, we separately analyze our three main subsamples: giants ( = 67,697), K IV/V stars ( = 13,864), and G IV/V stars ( = 5,375). As shown in Fig. 5, giants dominate the total sample and peak at [Fe/H]  0.2. Meanwhile, K IV/V and G IV/V stars are systematically younger and peak at slightly larger metallicities [Fe/H]  0.0.

Figure 5.— Metallicity distribution of APOGEE stars in our selected total sample (black) and giant (green), K IV/V (red), and G IV/V (blue) subsamples.

The resolution R  22,500 (13 km s) of the APOGEE spectra is similar to the Latham et al. (2002) and Carney et al. (2005) observations (R  35,000; 9 km s). However, our selected subsample of high-quality APOGEE spectra has an average S/N  110, which is a factor of six times larger than the mean S/N  15 - 20 of the Carney-Latham observations. The average RV measurement uncertainties are  = 0.02 km s, 0.04 km s, and 0.05 km s for our giant, K IV/V, and G IV/V subsamples, respectively. For our K IV/V subsample, the 1 - 99 percentile range in the RV measurement uncertainties is  = 0.006 - 0.152 km s. The APOGEE RVs are substantially more precise than the mean RV uncertainties  = 0.5 - 1.0 km s in the Latham et al. (2002) sample (see Fig. 1).

The number and timespan of the APOGEE RV observations are comparatively smaller, but fortunately they do not vary significantly with metallicity. For metal-poor (0.9  [Fe/H]  0.7) and metal-rich (0.3  [Fe/H]  0.5) K IV/V stars, the mean numbers of RV measurements are = 2.93 and 3.04, respectively, and the median timespans are = 33 days and 37 days, respectively. We find similar results for the giant and G IV/V subsamples. The APOGEE sample is incomplete toward SBs with longer periods due to the limited timespan, but the superior RV precision helps significantly to offset this effect. The timespans of the APOGEE observations vary substantially from system to system. For K IV/V stars, the 15 - 85 percentile range in the timespan is  = 23 - 305 days. When correcting for incompleteness (see below), we assume the cadence is independent of metallicity but account for the small number of observations and wide distribution in the timespans.

The RV uncertainties in our APOGEE sample decrease with metallicity, similar to the trend in the Carney-Latham sample. In particular, the mean RV measurement uncertainty for K IV/V stars decreases by a factor of  2.9 from  = 0.08 km s across 0.9  [Fe/H]  0.7 to  = 0.03 km s across 0.3  [Fe/H]  0.5. It is therefore crucial that we do not follow Latham et al. (2002) and Carney et al. (2005) by defining the SB fraction according to those systems that exhibit RV variability above some statistical significance.

Another reason to avoid this definition is because a substantial fraction of our giants are RV variables due to RV jitter. The mean surface gravity of giants in our sample is log  = 2.4, which exhibit an average RV jitter of = 0.07 km s according to Fig. 3 in Hekker et al. (2008). In addition, we find the APOGEE pipeline underestimates the true RV uncertainties for systems with very small measurement uncertainties   0.1 km s. Many RV variables with very small amplitudes are actually spurious. To account for both RV jitter and systematic effects in the APOGEE pipeline, we add a systematic uncertainty of in quadrature with each of the measurement uncertainties . As shown in Fig. 6, the fraction of systems that exhibit RV variability above the 5 significance level decreases as the assumed value for increases. The curves in Fig. 6 rapidly decline and then begin to flatten beyond   0.08 km s. We therefore adopt a systematic uncertainty of  = 0.08 km s for all three subsamples. Systems that exhibit statistically significant RV variability well above the total RV uncertainty = ( + ) are real SBs.

Figure 6.— The fraction of APOGEE stars that exhibit RV variability above the 5 significance level as a function of an assumed value of systematic uncertainty for our giant (green), K IV/V (red), and G IV/V (blue) subsamples. Assuming no systematic uncertainty, a significant fraction of APOGEE stars are spurious RV variables due to either RV jitter and/or the APOGEE pipeline underestimating the true RV uncertainties. The curves begin to significantly flatten beyond   0.08 km s, and so we add a systematic uncertainty of   0.08 km s (dotted) in quadrature with all the measurement uncertainties.

4.2. RV Variability Fractions

As advocated in Badenes et al. (2018), we instead measure the RV variability fraction according to the fraction of stars that exhibit a maximum difference in radial velocities RV between any two epochs above a certain threshold RV. Based on this definition, the close binary fraction is directly proportional to the observed RV variability fraction, i.e., corrections for incompleteness are independent of metallicity. In Fig. 7, we show the RV variability fraction as a function of RV for our giant, K IV/V, and G IV/V subsamples. For the K IV/V and G IV/V subsamples, the RV variability fraction increases from  4.4% for RV  10 km s to  12% - 13% for RV  1 km s. The similarity in their RV variability distributions, both in terms of functional form and normalization, suggests K IV/V stars and G IV/V stars have the same close binary fraction and period distribution. The relative change in the close binary fraction between these two subsamples can at most be / 20% (2 confidence level). This is consistent with previous studies that show the close binary fraction changes only slightly between early-M dwarfs and G-dwarfs (Fischer & Marcy, 1992; Raghavan et al., 2010; Clark et al., 2012; Duchêne & Kraus, 2013; Murphy et al., 2018). The RV variability fraction for our giant subsample is considerably lower, increasing from  1.3% for RV  10 km s to  6.9% for RV  1 km s. As discussed in §3 and Badenes et al. (2018), giant evolution truncates the short-period tail of the binary period distribution, thereby removing SBs with large RV amplitudes.

We display the false positive rate in Fig. 7, i.e., the fraction of systems that have both RV  RV and a difference in RVs that are discrepant with each other by less than 5. We also display the difference between the RV variability fraction and false positive rate, which provides the real SB fraction. Badenes et al. (2018) chose a very conservative threshold of RV  10 km s in order to be certain all of their RV variables were real binaries (see their Fig. 9). Indeed, we find 100% of RV variables with RV  10 km s are real, i.e., the false positive rate is zero for all three subsamples (see Fig. 7). The false positive rate remains zero down to RV  2 km s and then steadily increases below RV  1 km s. Systems with RV  0.4 km s are consistent with constant RV or exhibit RV jitter.

Figure 7.— The fraction of APOGEE stars that exhibit RV variability above RV  RV (solid), the fraction of systems that have RV  RV but are consistent with constant RV within the 5 tolerance level (dashed), and the difference between these two distributions (dotted) for our giant (green), K IV/V (red) and G IV/V (blue) subsamples. Badenes et al. (2018) adopted a conservative threshold of RV = 10 km s (right dash-dotted line) to be 100% certain all RV variables were real SBs. We adopt a threshold of RV = 1 km s (left dash-dotted line) in order to retain a significant majority of the real SBs while simultaneously keeping the false positive rate below  1% for all three subsamples and across all metallicities.

We adopt a threshold of RV = 1 km s (Fig. 7), but we also keep track of large-amplitude RV variables with RV  3 km s and RV  10 km s to perform consistency checks (see below). A significant majority ( 70% - 80%) of the real SBs have RV  1 km s. The false positive rate is also negligible above RV  1 km s, e.g., 0.0%, 0.1% and 0.3% for our giant, K IV/V, and G IV/V subsamples, respectively. Our threshold of RV = 1 km s is well above the systematic uncertainty   0.08 km s. The few false positives with RV  1.0 - 1.5 km s simply have larger measurement uncertainties   0.2 km s compared to average. The false positive rate increases slightly toward lower metallicities for our adopted threshold. Nonetheless, the false positive rate is extremely small across all metallicities, especially compared to the RV variability fraction. For instance, the false positive rate for metal-poor K IV/V stars with 0.9  [Fe/H]  0.5 is 0.8% above RV  1 km s. For this same metal-poor subset, the ratio of the false positive rate to RV variability fraction is only 4.3%. In other words,  96% of metal-poor K IV/V RV variables with RV  1 km s are real SBs. A systematic uncertainty of /  4% in the inferred close binary fraction due to spurious RV variables is much smaller than the measurement uncertainties and other sources of systematic error (see below).

4.3. Variations with Metallicity

As displayed in Fig. 8, the fraction of APOGEE stars that exhibit RV variability above RV  1 km s decreases dramatically with metallicity for all three subsamples. For K IV/V stars, the RV variability fraction decreases by a factor of 3.8 from 25%  5% across 0.9  [Fe/H]  0.7 to 6.6%  1.3% across 0.3  [Fe/H]  0.5. Attempting to fit a uniform RV variability fraction for K IV/V stars across the seven metallicity bins in Fig. 8 results in a reduced / = 19.7 with  = 6 degrees of freedom. The probability to exceed this value is = 410, i.e., the RV variability fraction of K IV/V stars decreases with metallicity at the 9.9 significance level. The G IV/V histogram in Fig. 8 is consistent with the K IV/V histogram, but has larger uncertainties due to the smaller sample size. The RV variability fraction of giants is measurably smaller due to the effective removal of very close binaries, but nonetheless exhibits the same metallicity trend. The giant RV variability fraction decreases by a factor of 4.4 from 14.5%  0.9% at [Fe/H] 0.8 to 3.3%  0.5% at [Fe/H]  0.4. A model of a uniform RV variability fraction for giants results in an even larger reduced / = 62.1 that can be rejected at the 18.6 confidence level ( = 210). By combining the results from our three independent subsamples, the RV variability fraction decreases by a factor of 4.0  0.5 across 0.9  [Fe/H]  0.5 at the 21.9 significance level.

Figure 8.— The fraction of APOGEE stars that exhibit RV variability above RV  1 km s (solid) for our giant (green), G IV/V (blue), and K IV/V (red) subsamples. We also display the fraction of K IV/V stars with RV  3 km s (dashed red) and RV  10 km s (dotted red). The RV variability fraction decreases with metallicity at a similar rate for all three subsamples and RV thresholds. Combining the giant, G IV/V, and K IV/V subsamples, the RV variability fraction decreases by a factor of 4.0  0.5 across 0.9  [Fe/H]  0.5 at the 22 confidence level.

The relative decrease in the RV variability fraction as a function of metallicity is consistent among our K IV/V, G IV/V, and giant subsamples. This indicates the slope of the anti-correlation between the close binary fraction and metallicity is similar across primary masses   0.6 - 1.5 M. The consistency also suggests the binary fraction decreases with metallicity at a similar rate for both very close companions orbiting small MS/HG stars and for slightly wider companions orbiting larger giants.

We also display in Fig. 8 the fraction of K IV/V stars with RV  3 km s and RV  10 km s, which both exhibit the same metallicity trend as K IV/V binaries with smaller RV amplitudes. Utilizing the K IV/V histogram with RV  1 km s as a template, we multiply this distribution by a reduction factor to fit the other K IV/V histograms. We measure = (RV  3 km s)/(RV  1 km s) = 0.65  0.03 with goodness-of-fit parameter / = 0.43 ( = 0.86). Similarly, we fit = (RV  10 km s)/(RV  1 km s) = 0.38  0.02 with / = 1.9 ( = 0.08). If spurious RV variables with RV = 1 - 3 km s had significantly contaminated metal-poor systems, we would have expected the RV  1 km s distribution to be steeper than the RV  3 km s distribution. Instead, all three K IV/V histograms in Fig. 8 have the same slope, which further demonstrates false positives negligibly affect the distribution with RV  1 km s. The consistency also suggests the frequency of very close binaries, which dominate the large-amplitude RV tail with RV  10 km s, decreases with metallicity at a similar rate as slightly wider binaries.

As discussed in Badenes et al. (2018), systematic uncertainties can potentially bias the measured relation between the RV variability fraction and metallicity, but to a substantially smaller degree than the observed anti-correlation. For example, metal-poor stars are systematically older and therefore have a larger fraction of close white dwarf (WD) companions. In the field, 20% of close companions to solar-type stars are WDs (Moe & Di Stefano, 2017; Murphy et al., 2018). The close binary fraction therefore increases by / 5% - 10% between metal-rich field stars and slightly older metal-poor field stars due to the larger frequency of close WDs. Similarly, older metal-poor binaries have had more time for tidal friction and magnetic braking to harden the orbit, thereby boosting the RV variability fraction. However, only 2% of solar-type stars in volume-limited samples have   10 days (Duquennoy & Mayor, 1991; Raghavan et al., 2010; Tokovinin, 2014; Moe & Di Stefano, 2017), and so tidal friction and magnetic braking alone cannot explain the observed RV variability fraction of 25%  5% for metal-poor K IV/V stars. Finally, we selected our giant, K IV/V, and G IV/V subsamples according to fixed intervals of surface gravity and temperature, not mass. By interpolating the Dartmouth stellar evolutionary tracks (Dotter et al., 2008), we find a  = 0.9 M star with [Fe/H] = 0.4 and age  = 5 Gyr has log   4.53 and   5,100 K. Meanwhile, a metal-poor star with [Fe/H] = 0.8 of the same mass and age is substantially smaller (log  = 4.43) and hotter ( = 6,300 K) due to the decreased opacities. To extend down to   5,100 K, a metal-poor dwarf with [Fe/H] = 0.8 must be   0.67 M. Given the same cuts in log  and , the metal-poor stars in our APOGEE subsamples are   0.2 M less massive than their metal-rich counterparts. The close binary fraction increases slightly with primary mass across   0.5 - 1.5 M. Our selection criteria therefore leads to a  10% bias in the metallicity versus binary relation in the positive direction. This effect is opposite the observed anti-correlation, i.e., consideration of this particular selection bias strengthens our overall conclusion. In any case, the systematic uncertainty / 10% in the inferred close binary fraction is insignificant compared to the observed factor of 4.0  0.5 decrease across 0.9  [Fe/H]  0.5. We confirm the conclusion of Badenes et al. (2018) that the RV variability fraction and thus the intrinsic close binary fraction strongly decreases with metallicity.

4.4. Cumulative Metallicity Distributions

In Fig. 9, we display the cumulative distribution of metallicities for our giant, K IV/V, and G IV subsamples. For each subsample, we show the metallicity distributions for large-amplitude RV variables with RV  10 km s, small-amplitude RV variable with RV  1 km s, and constant RV stars with RV  0.4 km s. The distributions of small-amplitude and large-amplitude RV variables are consistent with each other. For K IV/V stars, a KS test shows the probability the RV  1 km s and RV  10 km s histograms are drawn from the same parent distribution is  = 0.20. For giants and G IV/V stars, the RV variability distributions are even closer, resulting in  = 0.71 and  = 0.99, respectively. This further demonstrates that false positives negligibly affect RV variables with RV = 1.0 - 2.0 km s and that very close binaries that produce large-amplitude RV variations follow the same metallicity trend as slightly wider binaries.

Figure 9.— Cumulative metallicity distributions of giants (top), K IV/V stars (middle) and G IV/V stars (bottom) for the total populations (magenta), RV variables with RV  10 km s (red), RV variables with RV  1 km s (blue), and constant RV stars with RV  0.4 km s (green). We indicate in cyan the probability the blue and green distributions are drawn from the same parent distribution according to a KS test, the corresponding level of significance, and the difference in their median metallicities. After correcting for incompleteness, the median metallicities of close binaries are [Fe/H] = 0.13  0.03 smaller than single stars.

Meanwhile, RV variables are noticeably shifted toward smaller metallicities compared to both the total population and especially the constant RV stars. KS tests demonstrate the RV  1 km s and RV  0.4 km s distributions are discrepant with each other at the 17.9 ( = 610), 11.8 ( = 1.310), and 6.3 ( = 1.410) confidence levels for the giant, K IV/V, and G IV/V subsamples, respectively. These levels of statistical significance are similar to those found above, but are based on the discrete metallicity distributions instead of the binned RV variability fractions. Both the and KS tests confirm the close binary fraction decreases with metallicity at the  20 confidence level.

Close binaries have systematically smaller metallicities compared to single stars and wide binaries. We measure the differences between the median metallicities of the RV  1 km s and total populations to be [Fe/H] = 0.068, 0.067, and 0.051 for the giant, K IV/V and G IV/V subsamples, respectively. The metallicity differences between the RV  1 km s and RV  0.4 km s distributions are slightly larger at [Fe/H] = 0.087, 0.089, and 0.073. Constant RV stars mainly consist of single stars and wide binaries, but also include close binaries that have small velocity amplitudes or were observed with unfavorable cadence to detect RV variations. As we calculate in §4.5, the fraction of close binaries (  10;   10 AU) that are detectable as APOGEE RV variables with RV  1 km s is  60%. The median metallicities of close binaries are therefore [Fe/H] = 0.11  0.02 smaller than single stars and wide binaries with   10 AU. Very wide binaries with   200 AU do not depend significantly on metallicity, while solar-type binaries with intermediate separations   10 - 200 AU likely exhibit a weak metallicity anti-correlation (see §2 and §6). We estimate the median metallicities of close binaries are [Fe/H] = 0.13  0.03 smaller than single stars and very wide binaries with   200 AU. This difference may seem relatively small compared to the broad metallicity distribution of solar-type stars. However, the mean metallicities of large stellar populations, such as the APOGEE sample, are measured to extremely high precision [Fe/H]  0.02 dex. A metallicity difference of [Fe/H] = 0.13  0.03 between close binaries and single stars therefore represents a relatively substantial offset.

4.5. Corrections for Incompleteness

We next correct for incompleteness to recover the intrinsic close binary fraction from the observed RV variability fraction. Accounting for the distribution of giant surface gravities, how close binaries evolve during giant expansion, the larger RV jitter associated with very luminous giants, and the differences in target selection of red clump versus normal giants is beyond the scope of this paper (see Badenes et al. 2018). A more detailed analysis of RV variability in APOGEE giants utilizing the more recent data release 14 is the subject of a future study (Mazzola et al., in prep.). In the present study, we combine our K IV/V and G IV/V subsamples, and we account only for incompleteness to measure the close binary fraction.

We modify our Monte Carlo model in §3.1.2 to compute the completeness fraction of close binaries with  = 1 - 10 days that are detectable as APOGEE RV variables. We adopt a primary mass of  = 0.9 M appropriate for the combined GK IV/V subsample. We calculate the probability to detect RV variations as a continuous function of timespan . We generate RVs at = 2, 3 (average) and 4 epochs. For  = 2, the two epochs span , while for  = 3 and 4 the additional epochs are randomly distributed across . We do not add noise to the simulated RVs because the RV uncertainties are below our adopted RV thresholds. We simply calculate the fraction of close binaries that have RV  RV for RV = 1, 3, and 10 km s.

We display in Fig. 10 the simulated completeness fractions as a function of for the different values of and RV. The fraction of close binaries that are detectable as RV variables increases nearly linearly with respect to log . Given  = 3, the fraction of close binaries that have RV  1 km s increases from  = 37% for  = 10 days to  = 88% for  = 1,000 days. The number of RV observations only slightly affects the detection rates. In particular, a fourth RV measurement negligibly increases the completeness fraction unless it also extends the timespan between first and final visits. The completeness curves for RV  3 km s and RV  10 km s are substantially smaller, and the latter is also flatter with respect to . Even with an infinite number and timespan of RV observations, only   45% of close binaries with  = 1 - 10 days produce large-amplitude RV variations above RV  10 km s.

Figure 10.— The simulated fraction of close binaries below   10 days (  10 AU) that exhibit RV variability above RV  1 km s (solid), 3 km s (dashed), and 10 km s (dotted) given = 2 (blue), 3 (red), and 4 (green) RV measurements as a function of timespan between first and final visits. APOGEE observed the 19,239 GK IV/V stars in our sample with varying cadence, and we indicate the 15-percentile, median, and 85-percentile in timespans with vertical dash-dotted lines.

For our combined GK IV/V subsample, the 15-percentile, median, and 85-percentile in timespans are = 19, 42, and 303 days, respectively, which we indicate in Fig. 10. Given the wide spread in timespans, we do not adopt the median but instead weight our Monte Carlo models according to the actual cadence of the APOGEE observations. We calculate weighted completeness fractions of = 0.57, 0.40, and 0.24 for RV  1, 3, and 10 km s, respectively.

Our Monte Carlo model, which incorporates the short-period tail of a log-normal period distribution (see §3.1.2), accurately reproduces the observed distribution of RV amplitudes. For example, the modeled ratio = (RV  3 km s)/(RV  1 km s) = 0.40/0.57 = 0.70 between the completeness fractions is consistent with the observed ratio = 0.65  0.03 between the corresponding number of RV variables (see §4.3 and Fig. 8). Similarly, the simulated ratio = 0.24/0.57 = 0.42 is slightly larger than but still consistent with the observed ratio = 0.38  0.02 between the number of large-amplitude and small-amplitude RV variables. If we instead adopt a uniform distribution in log , i.e., Opik’s law, then we simulate larger completeness fractions of = 0.75, 0.63, and 0.47 for RV  1, 3, and 10 km s, respectively, because more of the close binaries are weighted toward shorter periods. However, Opik’s law predicts ratios = 0.63/0.75 = 0.84 and = 0.47/0.75 = 0.63 that are clearly discrepant with the observed ratios 0.65  0.03 and 0.38  0.02, respectively. Both metal-poor and metal-rich solar-type binaries therefore follow the same short-period tail of a log-normal period distribution. Metal-poor solar-type stars simply have a larger close binary fraction.

Similar to Fig. 8, we display in Fig. 11 the fraction of GK IV/V stars with RV  1 km s and RV  3 km s as a function of metallicity. Of the 19,239 GK IV/V stars in our combined sample, 5,394 (28%) were observed by APOGEE during a timespan of at least   100 days. As shown in Fig. 11, this subset exhibits a noticeably higher fraction of RV variables with RV  1 km s compared to the total GK IV/V sample. By fitting across all metallicities, we find the RV variability fraction of GK IV/V stars observed with longer timespans is = 1.37  0.05 times larger than the total GK IV/V population (/ = 0.49,  = 0.82). With increased timespans, the APOGEE observations become more complete toward detecting SBs with longer periods (see Fig. 10). We weight our Monte Carlo model according to the cadence of RV observations for the 5,394 GK IV/V stars with   100 days. The resulting completeness fraction of = 0.76 is = 0.76/0.57 = 1.33 times larger than the completeness fraction for the total GK IV/V population. The simulated ratio nearly matches the observed ratio, providing another confirmation our Monte Carlo model accurately describes close solar-type binaries.

Figure 11.— As a function of metallicity, the observed fraction of all GK IV/V APOGEE stars in our sample that exhibit RV variability above RV 1 km s (dotted red) and 3 km s (dotted green), and the observed fraction of GK IV/V APOGEE stars monitored during a timespan of at least   100 days that exhibit RV variability above RV 1 km s (dotted blue). We divide these three histograms by their respective completeness fractions of = 0.40, 0.57, and 0.76, resulting in the bias-corrected close binary fractions (thin colored lines). We adopt a weighted average and a systematic uncertainty of / = 10%, providing an intrinsic close binary fraction that decreases from  = 41%  7% at [Fe/H] = 0.8 to  = 11%  2% at [Fe/H] = +0.4 (thick solid black).

In Fig. 11, we divide the observed RV variability fractions by their corresponding completeness fractions. The three resulting completeness-corrected close binary fractions are all consistent with each other. We adopt a weighted average of the three histograms and the measurement uncertainties from the distribution based on all GK IV/V RV variables with RV 1 km s. For each metallicity bin, we add a systematic uncertainty of / = 10% in quadrature with the measurement uncertainties to account for the small selection biases discussed in §4.3. We present our final completeness-corrected close binary fraction of GK IV/V stars as the thick black histogram in Fig. 11. The intrinsic close binary fraction (  10 days;   10 AU) decreases from = 41%  7% at [Fe/H] = 0.8 to = 11%  2% at [Fe/H] = +0.4. The metallicity-dependent close binary fraction inferred from the APOGEE RV variables and the Carney-Latham SB samples (see Fig. 3) are consistent with each other. Our APOGEE RV sample of 19,239 GK IV/V stars is a factor of 14 times larger than the Latham et al. (2002) sample. Moreover, APOGEE measured the RVs and metallicities of their targets to substantially higher precision. The anti-correlation between the close binary fraction and metallicity is therefore even more pronounced and measured to much higher statistical significance with the APOGEE dataset.

5. Kepler Eclipsing Binaries

5.1. Sample Selection and Description

The primary Kepler mission monitored nearly 200,000 solar-type stars for four years with exquisite photometric precision. Designed to discover transiting exoplanets, Kepler also identified and characterized 2,878 EBs and non-eclipsing binary ellipsoidal variables (Prša et al., 2011; Kirk et al., 2016). About a third of the systems in the Kepler EB catalog have very short periods   1 day, the majority of which are evolved contact or ellipsoidal binaries. Most of the Kepler EBs with longer periods are in pre-mass-transfer detached configurations. A few EBs have especially long periods  = 1,000 - 1,100 days, but geometrical selection effects and the four-year lifetime of the main Kepler mission severely limited the discovery of such wide binaries. We initially select the 1,924 EBs with  = 1 - 1,000 days in the third revision of the Kepler EB catalog (Kirk et al., 2016).

Sample with Photometric Metallicities

Brown et al. (2011) utilized photometry, stellar isochrones, and a Bayesian model of the galactic stellar population to estimate , log , and [Fe/H] for all stars in the Kepler input catalog. Specifically, they measured the spectral energy distribution (SED) of each Kepler star based on broadband optical photometry (griz), 2MASS near-infrared photometry (JHK), and an intermediate-band filter (D51) centered on the Fraunhofer b absorption lines near 515 nm that are associated with Mg and Fe. Brown et al. (2011) then fitted the measured SEDs to synthetic colors from ATLAS9 model atmospheres (Castelli & Kurucz, 2004) assuming the dust extinction varied as a simple function of distance and galactic latitude. They also incorporated Bayesian priors in , log , and [Fe/H] according to the observed distributions in the solar neighborhood. Huber et al. (2014) revised and significantly improved the measured parameters of 196,468 Kepler stars. They updated the photometry with recent observations, calibrated according to empirical relations, incorporated more accurate stellar isochrones from the Dartmouth evolutionary tracks (Dotter et al., 2008), and treated dust extinction in a more realistic manner. Huber et al. (2014) adopted Bayesian priors in log  and [Fe/H] similar to those in Brown et al. (2011), but developed a slightly more sophisticated method for sampling the distributions.

Brown et al. (2011) and Huber et al. (2014) stressed the measured surface gravities and metallicities in their catalogs are highly uncertain and should not be used on a star-by-star basis. Nevertheless, they argued the distributions of surface gravities and metallicities are statistically accurate and can therefore be utilized to study broad trends across these parameters. Brown et al. (2011) and Huber et al. (2014) also identified regions in the HR diagram where the photometric solutions for log  and [Fe/H] are highly degenerate and most uncertain, notably for subgiants and cool late-K and M-type dwarfs. We therefore select the = 142,951 solar-type dwarfs in the Huber et al. (2014) catalog with photometric parameters = 4,800 - 6,800 K, log  = 4.0 - 5.0, and 1.7  [Fe/H]  0.5, corresponding approximately to F3V - K3V stars.

Berger et al. (2018) recently utilized Gaia parallactic distances to measure the stellar radii of Kepler stars, and found  65%, 23%, and 12% are MS stars, subgiants, and giants, respectively. They concluded contamination by subgiants in the Kepler sample is smaller than previously thought. Moreover, a non-negligible fraction of the Berger et al. (2018) subgiants, which were identified because they lie slightly above the MS relation in the HR diagram, are actually twin binaries with MS components of comparable luminosity. Thus a significant majority of the solar-type dwarfs in our photometric sample are truly MS stars.

The metallicity distribution of our photometric sample of Kepler solar-type dwarfs follows a Gaussian with mean of [Fe/H] = 0.17 and dispersion of  = 0.26 dex. Huber et al. (2014) estimated the uncertainties in the photometric metallicities of Kepler stars is [Fe/H]  0.3 dex. We can therefore examine metallicity trends across the much broader interval 1.7  [Fe/H]  0.5. Of the 1,924 Kepler EBs with  = 1 - 1,000 days, = 1,292 systems satisfy our selection criteria of = 4,800 - 6,800 K, log  = 4.0 - 5.0, and 1.7  [Fe/H]  0.5 according to the Huber et al. (2014) photometric catalog. The observed EB fraction in our photometric sample of Kepler solar-type dwarfs is = 1,291/142,951 = 0.90%  0.03%.

The presence of a binary companion can potentially bias the metallicities inferred from fitting single-star isochrones to the measured photometry. The photometric metallicities of EBs in particular may be substantially inaccurate if the observations in the different filters correspond to different orbital phases, e.g., during versus outside of eclipse. In addition, the majority of very close binaries with   7 days have tertiary companions (Tokovinin et al., 2006), and so most EBs also have third light contamination.

We assess the significance of these potential biases by fitting isochrones to simulated photometry of solar-type binaries. We download a dense grid of Dartmouth stellar evolutionary tracks (Dotter et al., 2008) spanning masses  = 0.15 - 1.7 M, ages = 1 - 13 Gyr, and metallicities 2.4  [Fe/H]  0.5. We simulate binaries with metallicities [Fe/H] = 1.3, 0.8, 0.3, and +0.2 at representative ages of = 11 Gyr, 8 Gyr, 5 Gyr, and 2 Gyr, respectively. We select G8V primaries with  = 5,500 K, corresponding to primary masses = 0.65, 0.71, 0.82, and 0.98 M for the four combinations of metallicities and ages. We also consider hotter F8V primaries with = 6,200 K, corresponding to slightly higher masses of = 0.75, 0.84, 0.99, and 1.22 M. For different combinations of mass ratios = /, we add the fluxes of both binary components for all eight filters (D51grizJHK) utilized in Brown et al. (2011) and Huber et al. (2014). We add a dust extinction of A = 0.2 mag and adopt a dust reddening law from Schlafly & Finkbeiner (2011) such that A/A = 1.45, 1.31, 0.74, 0.55, 0.31, 0.20, 0.13 for bands = g, D51, i, z, J, H, and K, respectively. We do not fit the distances to our simulated binaries, and so we consider only the seven unique color combinations. Brown et al. (2011) measured the bright Kepler stars to a precision of  0.02 mag in the D51griz filters, and so we adopt uncertainties of 0.03 mag in all the colors. We measure the photometric masses , ages , metallicities [Fe/H], and dust extinctions by minimizing the statistic between the seven colors of our simulated binaries and the isochrones of single stars. We assume uniform priors in our four photometric parameters. In this manner, our fits are not dominated by short-lived phases of stellar evolution that provide only marginally smaller values.

Figure 12.— The photometric metallicities [Fe/H] determined by fitting single-star isochrones to simulated broad-band photometry of binaries as a function of mass ratio . We consider binaries with cooler primaries ( = 5,500 K; top) and hotter primaries ( = 6,500 K; bottom) for four different metallicities [Fe/H] = 1.3 (magenta), 0.8 (green), 0.3 (blue), and +0.2 (red), where we list the corresponding ages and primary masses . For some combinations (e.g.,   0.6), the fitted photometric metallicities underestimate the true metallicities (dotted) by as much as  0.5 dex. In general, however, the measurement uncertainties simply increase from  0.25 dex near [Fe/H] = 0.2 to  0.45 dex near [Fe/H] = 1.3 with negligible bias between the true and photometric metallicities.

We measure the mean and 1 uncertainties in the photometric metallicities [Fe/H] by marginalizing across the other parameters. We display the measured values of [Fe/H] in Fig. 12 for the various combinations of [Fe/H], , and . The measurement uncertainties increase from [Fe/H] = 0.25 dex near [Fe/H] = +0.2 to [Fe/H] = 0.45 dex near [Fe/H] = 1.3, consistent with the average uncertainty of [Fe/H] = 0.3 dex reported in Huber et al. (2014). Compared to their primaries, low-mass companions with   0.4 contribute negligible flux across the optical and near-infrared bands. For such extreme mass-ratio binaries, the photometric metallicities [Fe/H] determined by fitting single-star isochrones are close to the true metallicities [Fe/H]. Similarly, companions with   0.8 have SEDs similar to their primaries, and so the photometric metallicities of twin binaries are consistent with their actual values. For   0.4 - 0.8, however, there are certain combinations of [Fe/H] and for which the photometric metallicities underestimate the true metallicities. In particular, Fig. 12 shows that binaries with  = 5,500 K, [Fe/H]  0.0, and 0.6 - 0.8 and binaries with  = 6,200 K, [Fe/H]  1.3, and   0.5 - 0.7 are biased by [Fe/H]  0.5 dex toward smaller metallicities. Fortunately, only