Mind Your Ps and Qs: The Interrelation between Period (P) and Mass-ratio (Q) Distributions of Binary Stars

Multiepoch spectroscopic radial velocity observations are capable of detecting companions to massive MS stars with the shortest orbital periods (Wolff 1978; Garmany et al. 1980; Levato et al. 1987; Abt et al. 1990; Chini et al. 2012; Sana et al. 2012; Kobulnicky et al. 2014). The mass ratio of a double-lined spectroscopic binary (SB2) can be directly measured from the observed ratio of velocity semi-amplitudes q = M₂/M₁ = K₁/K₂. The orbital eccentricity e of an SB2 is derived by fitting the radial velocities as a function of orbital phase.

We initially analyze 44 SB2s with orbital periods P = 2–500 days from four surveys of early-type stars: Levato et al. (1987, 81 B-type primaries; $\langle {M}_{1}\rangle$ ≈ 5 M_⊙; 3 SB2s), Abt et al. (1990, 109 B2–B5 primaries; $\langle {M}_{1}\rangle$ ≈ 7 M_⊙; 9 SB2s), Kobulnicky et al. (2014, 83 B0–B2 primaries; $\langle {M}_{1}\rangle$ ≈ 12 M_⊙; 8 SB2s), and Sana et al. (2012, 71 O-type primaries; $\langle {M}_{1}\rangle$ ≈ 28 M_⊙; 24 SB2s). We note that the Kobulnicky et al. (2014) survey also includes O-type stars, but we consider only their 83 systems with early B primaries in our current analysis.

We list the multiplicity statistics based on the four samples in Table 2. In Figures 3 and 4, we display the eccentricities e and mass ratios q, respectively, of the 44 SB2s as a function of orbital period. Sana et al. (2012) and Kobulnicky et al. (2014) identified additional SB2s with q > 0.55 and P > 500 days, which confirms that spectroscopic observations can reveal moderate mass-ratio binaries at intermediate orbital periods. However, spectroscopic binaries become increasingly less complete and biased toward moderate q at longer P (see Section 3.2 and Figure 4), so we limit our sample selection to SB2s with P < 500 days when discussing these four surveys.

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We also examine the 23 detached SB2s with primary spectral types O and B, luminosity classes III–V, orbital periods P = 8–40 days, and measured mass ratios q = K₁/K₂ and eccentricities e from the Ninth Catalog of Spectroscopic Binaries (SB9; Pourbaix et al. 2004). This sample includes 10 systems with O9–B3 primaries ($\langle {M}_{1}\rangle$ ≈ 14 M_⊙) and 13 systems with B5–B9.5 primaries ($\langle {M}_{1}\rangle$ ≈ 4.5 M_⊙). We report the multiplicity statistics based on these 23 SB2s in Table 3. The spectroscopic binaries contained in the SB9 catalog are compiled from a variety of surveys and samples, and so the cadence and sensitivity of the spectroscopic observations are not as homogeneous. We therefore consider only the SB2s with P < 40 days, which are relatively complete regardless of the instruments utilized. We cannot infer the intrinsic frequency of companions per primary based on the SB9 catalog. We can nonetheless utilize the sample of 23 early-type SB2s to measure the eccentricity and mass-ratio distributions.

For SB2s, the secondary must be comparable in luminosity to the primary in order for both components to be visible in the combined spectrum. Because MS stars follow a steep mass–luminosity relation, SB2s with early-type MS primaries can reveal only moderate mass ratios q > 0.25 (Figures 1 and 4). For single-lined spectroscopic binaries (SB1s) with low-luminosity companions, a lower limit to the mass ratio can be estimated from the observed reflex motion of the primary. A statistical mass-ratio distribution can be recovered for SB1s by assuming that the intrinsic binary population has random orientations (Mazeh et al. 1992). However, SB1s with early-type MS primaries may not necessarily have low-mass A–K type stellar companions. Instead, many SB1s with O-type and B-type primaries may contain 1–3 M_⊙ stellar remnants such as white dwarfs (WDs), neutron stars, or even black holes (Wolff 1978; Garmany et al. 1980). It is imperative to never implicitly assume that early-type SB1s contain two MS components. In the context of spectroscopic binaries with massive primaries, only SB2s provide a definitive uncontaminated census of unevolved companions (Moe & Di Stefano 2015a). At the very least, SB1s can provide a self-consistency check and an upper limit to the frequency of low-mass stellar companions.

The ability to detect spectroscopic binaries depends not only on the resolution of the spectrograph, the signal-to-noise ratio of the spectra, and the cadence of the observations but also on the physical properties of the binary. Early-type stars, including those in binaries with P ≳ 10 days where tidal synchronization is inefficient, are rotationally broadened by $\langle {v}_{\mathrm{rot}}\rangle$ ≈100–200 km s⁻¹ (Abt et al. 2002; Levato & Grosso 2013). The primary's orbital velocity semi-amplitude must therefore be K₁ ≳ 10 km s⁻¹ in order for the orbital reflex motion of the primary to be detectable (Levato et al. 1987; Abt et al. 1990; Sana et al. 2012; Kobulnicky et al. 2014). For SB2s, the atmospheric absorption features of both the primary and secondary components need to be distinguishable. Due to blending of the broad absorption features, early-type SB2s require an even higher threshold of K₁ ≳ 40 km s⁻¹ to be observed. The primary's velocity semi-amplitude K₁ decreases toward wider separations a, smaller mass ratios q, and lower inclinations i. Lower-mass companions at longer orbital periods are more likely to be missed in the spectroscopic binary surveys. Finally, highly eccentric binaries spend only a small fraction of time near periastron while exhibiting appreciable radial velocity variations. Depending on the cadence of the spectroscopic observations, eccentric binaries can be either more or less complete compared to binaries in circular orbits.

To measure the detection efficiencies and correct for incompleteness, we utilize a Monte Carlo technique to generate a large population of binaries with different primary masses M₁, mass ratios q, and orbital configurations P and e. For each binary, we select from a randomly generated set of orientations, i.e., an inclination i distribution such that cos i = [0, 1] is uniform and a distribution of periastron angles ω = [0^o, 360^o] that is also uniform. The velocity semi-amplitude K₁ criterion used in previous studies does not adequately describe the detection efficiencies of eccentric binaries. For each binary, we instead synthesize radial velocity measurements ${v}_{1,r}$ and ${v}_{2,r}$ at 20 random epochs, which is the median cadence of the spectroscopic binary surveys (Levato et al. 1987; Abt et al. 1990; Sana et al. 2012; Kobulnicky et al. 2014). For simplicity, we assume that the systemic velocity of the binary is zero. In an individual spectrum of an early-type star, the radial velocities can be centroided to an accuracy of ≈2–3 km s⁻¹. However, atmospheric variations limit the true sensitivity across multiple epochs to $\delta {v}_{1,r}$ ≈ 3–5 km s⁻¹ (Levato et al. 1987; Abt et al. 1990; Sana et al. 2012; Kobulnicky et al. 2014). In order for a simulated binary to have an orbital solution that can be fitted, and therefore an eccentricity and mass ratio that can be measured, we require a minimum number of radial velocity measurements of the primary ${v}_{1,r}$ to significantly differ from the systemic velocity. We impose that at least 5 of the 20 measurements satisfy $| {v}_{1,r}|$ ≳(3–5)$\delta {v}_{1,r}$ ≈ 15 km s⁻¹ in order to provide a precise and unique orbital solution for the primary.

The rotationally broadened spectral features of both the primary and secondary must also be distinguishable to be cataloged as an SB2. At the very least, the primary's absorption features must not only shift during the orbit but also have velocity profiles that visibly change as a result of the moving absorption lines from the orbiting secondary (De Becker et al. 2006; Sana et al. 2012). For q < 0.8, we require at least 3 of the 20 measurements to satisfy $| {v}_{1,r}-{v}_{2,r}|$ ≳ $\langle {v}_{\mathrm{rot}}\rangle /2$ ≈ 75 km s⁻¹. This velocity threshold is appropriate for both O-type and B-type primaries because the mean rotational velocities $\langle {v}_{\mathrm{rot}}\rangle$ ≈ 150 km s⁻¹ do not significantly vary between these two spectral types (Abt et al. 2002; Levato & Grosso 2013). For twin binaries with q = 1.0, the absorption lines of both the primary and secondary are of comparable depths, and so it is more difficult to distinguish the two components if their absorption features are significantly blended (Sana et al. 2011). For these twin systems, we require at least 3 of the 20 measurements to satisfy $| {v}_{1,r}-{v}_{2,r}|$ > $\langle {v}_{\mathrm{rot}}\rangle$ = 150 km s⁻¹. We linearly interpolate our velocity difference thresholds between 75 km s⁻¹ at q = 0.8 and 150 km s⁻¹ at q = 1.0.

For our simulated population, we keep track of the fraction of binaries that satisfy the above criteria. In this manner, we calculate the detection efficiencies ${ \mathcal D }$(P, M₁, q, e) of SB2s as a function of the physical properties of the binary. In Figure 5, we display the detection efficiencies ${ \mathcal D }$ as a function of orbital period P for various combinations of M₁, q, and e. The sample of SB2s is relatively complete at short orbital periods P < 10 days, while the longer-period systems have detection efficiencies considerably less than unity. As expected, binaries with small mass ratios q ≈ 0.3 are less complete than systems with large mass ratios q ≈ 0.8. Because of Kepler's laws, SB2s with lower-mass mid-B and late B primaries are less complete than SB2s with more massive O-type and early B primaries. Due to the finite cadence of the observations, eccentric binaries with P = 10–200 days have smaller detection efficiencies than their counterparts in circular orbits. At the longest orbital periods P ≳ 200 days, however, only eccentric binaries near periastron produce detectable radial velocity variations, and so ${ \mathcal D }$ becomes larger with increasing eccentricity.

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We also compare in Figure 5 our detection efficiencies ${ \mathcal D }$ to the completeness levels as computed by Kobulnicky et al. (2014, dashed line in their Figure 26). For this calculation, Kobulnicky et al. (2014) assumed a sensitivity threshold of K₁ = 15 km s⁻¹ for both SB1s and SB2s and that the binary population has an underlying flat mass-ratio distribution (γ = 0) and flat eccentricity distribution (η = 0). Although these choices of γ = 0 and η = 0 are consistent with observations of short-period spectroscopic binaries where the observations are relatively complete (see below), the longer-period systems may have different mass-ratio and eccentricity distributions.

To correct for incompleteness, it is better to calculate the detection efficiency ${ \mathcal D }$(P, M₁, q, e) for each individual SB2. The binary's relative contribution to the total sample is then determined by the statistical weight w = 1/${ \mathcal D }$. In other words, for every one system observed with detection efficiency ${ \mathcal D }$(P, M₁, q, e), there are w = 1/${ \mathcal D }$ total systems with similar properties in the intrinsic population after considering selection effects. Given the small sample sizes, this approach is sufficient for recovering the intrinsic binary statistics from the observed SB2 population.

Our calculated detection efficiencies ${ \mathcal D }$ have some level of uncertainty, especially considering the uncertainties in our adopted detection criteria (e.g., five observations with $| {v}_{1,r}|$ > 15 km s⁻¹ and three observations with $| {v}_{1,r}-{v}_{2,r}|$ > 75–150 km s⁻¹, depending on q). We vary our detection criteria within reasonable limits and estimate the uncertainty in ${ \mathcal D }$ to be ≈10%–15%. We propagate this uncertainty into the statistical weights w = 1/${ \mathcal D }$. For example, if ${ \mathcal D }$ ≈ 0.6, then w ≈ 1.7 ± 0.4, while for a smaller detection efficiency ${ \mathcal D }=0.3$, the uncertainty in the weight $w={3.3}_{-0.9}^{+2.3}$ becomes larger and asymmetric.

In Figure 3, we compare the observed SB2s to the detection efficiencies ${ \mathcal D }$(P, M₁, q, e) as a function of P and e while assuming a primary mass M₁ = 10 M_⊙ and mass ratio q = 0.4. Similarly, in Figure 4, we compare the observed SB2s to the detection efficiencies ${ \mathcal D }$ as a function of P and q while assuming an eccentricity e = 0.5${e}_{\max }$ and the same primary mass M₁ = 10 M_⊙. These completeness levels are for illustration purposes only, as we calculate ${ \mathcal D }$(P, M₁, q, e) for each system in the full 4D parameter space. For example, the longest-period SB2 in Figures 3–4 (i.e., the system with the O-type primary, log P ≈ 2.6, q ≈ 0.3, e ≈ 0.4) has a detection efficiency ${ \mathcal D }$ ≈ 0.35 and statistical weight ${\text{}}w\approx {2.9}_{-0.7}^{+1.5}$. Meanwhile, the second-longest-period system (i.e., the SB2 with the mid-B primary, log P ≈ 2.2, q ≈ 0.4, e ≈ 0.6) has an even smaller detection efficiency ${ \mathcal D }$ ≈ 0.24 and therefore larger weight w ≈ ${4.2}_{-1.4}^{+4.2}$. We compute w for each of the 44 early-type SB2s from the four surveys and for the 23 early-type SB2s from the SB9 catalog. By weighting each observed SB2 by w, we can now calculate the intrinsic eccentricity and mass-ratio distributions.

To investigate the intrinsic eccentricity distribution as a function of orbital period, we divide the SB2s from the four surveys into short-period (P = 3–10 days) and long-period (P = 10–500 days) subsamples. We consider only systems with $e\lt {e}_{\max }$, i.e., e < 0.3 and e < 0.6 for the short-period and long-period subsamples, respectively. In Figure 6, we display the cumulative distribution of eccentricities for these two subsamples of SB2s after correcting for incompleteness. The length of each vertical step in the cumulative distribution is proportional to the statistical weight w of the SB2 it represents.

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Our short-period subsample of 20 SB2s with P = 3–10 days and e < 0.3 is relatively complete with statistical weights all below w < 1.4. For this population, we measure the power-law exponent of the eccentricity distribution to be η = −0.3 ± 0.2 (see Table 2 and top panel of Figure 6). The uncertainty in η derives from the quadrature sum of the uncertainties from Poisson sample statistics and the uncertainties in the statistical weights w. Our result of η = −0.3 ± 0.2 is consistent with the measurement of η = −0.4 ± 0.2 by Sana et al. (2012), whose O-type spectroscopic binary sample is dominated by short-period systems.

For our long-period subsample of 13 SB2s with P = 10–500 days and e < 0.6, we measure η = 0.6 ± 0.3 after correcting for selection effects (Table 2 and bottom panel of Figure 6). Although the statistical weights w = 1.2–4.2 of the SB2s in this long-period subsample are relatively large and uncertain, our measurement of η = 0.6 ± 0.3 is still robust. For example, if we were to set all the weights to w = 1, we would still measure a positive exponent η = 0.2. Regardless of how we correct for selection effects, early-type SB2s at intermediate orbital periods are weighted toward larger values of η relative to the short-period systems.

The sample of SB2s from the four spectroscopic surveys is not large enough to measure changes in η as a function of M₁. We therefore investigate the 23 early-type SB2s with P = 8–40 days from the SB9 catalog. To easily compare the eccentricity distributions, we analyze the distributions of e/${e}_{\max }$, where ${e}_{\max }$(P) is determined for each SB2 according to Equation (3). Because we include only SB2s with P < 40 days from the SB9 catalog, the detection efficiencies are ${ \mathcal D }$ > 60% for all our systems, i.e., the individual weights are w < 1.7.

After correcting for incompleteness, we display in Figure 7 the cumulative distributions of e/${e}_{\max }$ for the 10 early (O9–B3) and 13 late (B5–B9.5) SB2s we selected from the SB9 catalog. Early-type SB2s with more massive primaries are clearly weighted toward larger values of η. For the O9–B3 subsample, we measure η = 0.9 ± 0.4. Meanwhile, for the B5–B9.5 subsample, we measure η = −0.3 ± 0.2 (Table 3). The observed differences between these two distributions may suggest that more massive binaries form with systematically larger eccentricities. Alternatively, both early B and late B binaries may initially be born with η ≈ 0.9, but the long-lived late B binaries have had more time to tidally evolve toward smaller eccentricities. Based on the early-type SB2 observations alone, we cannot differentiate which of these two scenarios is the most likely explanation.

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In a magnitude-limited survey, binaries with equally bright twin components are probed across larger distances compared to single stars and binaries with faint companions. This Malmquist bias, typically called the Öpik effect in the context of binary stars, can lead to an artificial peak near unity in the mass ratio distribution (Öpik 1923). Fortunately, the four spectroscopic binary surveys have already accounted for the Öpik effect, either by targeting early-type stars in open clusters/stellar associations with fixed distances or by removing distant twin binaries that do not reside in a volume-limited sample. We can therefore weight each observed SB2 by their respective values of w to correct for incompleteness.

After accounting for selection effects, we show in Figure 8 the corrected cumulative distributions of mass ratios for the 32 short-period (P = 2–20 days) and 10 long-period (P = 20–500 days) SB2s with q > 0.3 from the four spectroscopic surveys. Assuming that the mass-ratio distribution can be described by a power law across 0.3 < q < 1.0, we measure ${\gamma }_{\mathrm{large}q}$ = 0.1 ± 0.3 for the short-period subsample of early-type SB2s. This is consistent with the result of γ = −0.1 ± 0.6 by Sana et al. (2012), whose O-type binary sample is dominated by short-period systems with P = 2–20 days.

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A simple power-law distribution, however, does not fully describe the data. Allowing for an excess fraction of twin components with q > 0.95, we measure ${\gamma }_{\mathrm{large}q}$ = −0.3 ± 0.3 and ${{ \mathcal F }}_{\mathrm{twin}}$ = 0.11 ± 0.04 for our short-period subsample (see Figure 8). For early-type binaries with P = 2–20 days, the relative density ${ \mathcal N }$/${\rm{\Delta }}q$ = 5/(1.00 − 0.95) ≈ 100 of twin companions with q = 0.95–1.00 is larger than the density ${ \mathcal N }$/${\rm{\Delta }}q$ = 5/(0.95 − 0.80) ≈ 30 of companions with q = 0.80–0.95 at the ≈2.6σ significance level (see Figure 4). We emphasize that this twin excess is real, considering that the four spectroscopic binary surveys have already accounted for the Öpik effect. Nonetheless, the excess twin fraction of ${{ \mathcal F }}_{\mathrm{twin}}=0.11$ is quite small; the remaining 89% of companions with P = 2–20 days and q > 0.3 follow a power-law distribution with ${\gamma }_{\mathrm{large}q}$ ≈ −0.3 across the broad interval q = 0.3–1.0.

Based on the four individual SB2 surveys, we do not measure any statistically significant trends in ${\gamma }_{\mathrm{large}q}$ as a function of spectral type. We therefore adopt the average value ${\gamma }_{\mathrm{large}q}$ = −0.3 ± 0.3 for early-type short-period SB2s (Table 2). The short-period SB2s do, however, reveal a change in the excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}$ as a function of primary mass. After combining the 23 O-type and early B SB2s with q > 0.3 and P = 2–20 days, only three have q > 0.95 (i.e., 3/23 ≈ 13%). Using a maximum likelihood method, we measure the excess twin fraction to be ${{ \mathcal F }}_{\mathrm{twin}}$ = 0.08 ± 0.04 for SB2s with P = 2–20 days and O/early B primaries (Table 2). Of the nine short-period SB2s with q > 0.3 and mid-/late B-type primaries, two have q > 0.95 (i.e., 2/9 ≈ 22%). After accounting for incompleteness toward small mass ratios, we measure ${{ \mathcal F }}_{\mathrm{twin}}$ = 0.17 ± 0.09 for these SB2s with lower-mass primaries (Table 2). According to these measurements, there is an indication that the excess twin fraction increases among short-period binaries as the primary mass decreases.

After correcting for selection effects, it is evident from Figure 8 that the long-period subsample of 10 early-type SB2s is weighted toward smaller mass ratios. We measure an excess twin fraction that is consistent with zero, i.e., the 1σ upper limit is ${{ \mathcal F }}_{\mathrm{twin}}\lt 0.05$ (Table 2). Note that the completeness levels for twin systems q ≈ 1.0 at intermediate orbital periods are smaller than the detection efficiencies for q ≈ 0.8 systems (see Figure 4). This is because q ≈ 1.0 binaries have similar absorption features and therefore must have larger radial velocity differences $| {v}_{1,{\rm{r}}}$ − ${v}_{2,{\rm{r}}}|$ > 150 km s⁻¹ to be distinguishable (Section 3.2; Sana et al. 2011). Nonetheless, the completeness levels of twin q = 1.0 binaries across log P = 1.3–2.7 are sufficiently large that an excess twin population would still be detectable. The deficit of q = 0.8–1.0 systems at intermediate orbital periods log P = 1.3–2.7 is therefore intrinsic to the population of early-type binaries.

We also measure the power-law slope ${\gamma }_{\mathrm{large}q}$ = −1.6 ± 0.5 of the mass-ratio distribution to be weighted toward small values for the long-period SB2s (Table 2). The large uncertainty is mainly due to the small sample size, not the uncertainties in the correction factors w. For example, if we were to set the weights w = 1 for all ten long-period SB2s, we would still measure ${\gamma }_{\mathrm{large}q}$ = −0.8. The detection efficiencies of binaries with q = 0.3 are certainly smaller than systems with q = 0.6 (Section 3.2), and so ${\gamma }_{\mathrm{large}q}$  < −0.8 is an upper limit. Hence, early-type SB2s become weighted toward smaller mass ratios q ≈ 0.2–0.4 with increasing orbital period. This trend is already seen in the observed population of early-type SB2s (Figure 4). By correcting for incompleteness, the intrinsic population of binaries with longer orbital periods is even further skewed toward smaller mass ratios. Our result is consistent with the conclusions of Abt et al. (1990), who also find that early B spectroscopic binaries become weighted toward smaller mass ratios with increasing separation. In addition, Kobulnicky et al. (2014) speculate that there may be comparatively more binaries with extreme mass ratios at intermediate orbital periods that are hiding below the spectroscopic detection limits.

We next investigate the mass-ratio distributions of the SB2s with P = 8–40 days we selected from the SB9 catalog. Unlike the four spectroscopic binary surveys, the SB9 catalog is still affected by the Öpik effect. We therefore consider only the SB2s from the SB9 catalog with q < 0.9 in order to remove any bias toward binaries with equally bright components. Although we cannot quantify the excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}$, we can still measure the power-law component ${\gamma }_{\mathrm{large}q}$ of the SB2s in the SB9 catalog.

After correcting for incompleteness, we display in Figure 9 the cumulative distribution of mass ratios across q = 0.35–0.90 for the O9–B3 and B5–B9.5 subsamples from the SB9 catalog. The sample sizes are quite small, and so we cannot discern any statistically significant differences between the two distributions. We measure ${\gamma }_{\mathrm{large}q}$ = −0.6 ± 0.9 for the O9–B3 subsample and ${\gamma }_{\mathrm{large}q}$ = −1.0 ± 0.8 for the B5–B9.5 subsample (Table 3). The measured values of ${\gamma }_{\mathrm{large}q}$ ≈ −1.0–−0.6 for these SB2s at intermediate orbital periods are between the values of ${\gamma }_{\mathrm{large}q}$ ≈ −0.3 and ${\gamma }_{\mathrm{large}q}$ ≈ −1.6 we measured above for the short-period and long-period SB2 subsamples, respectively.

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Now that we have measured the SB2 detection efficiencies, we can calculate the intrinsic frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ (M₁, P) of companions with q > 0.3 per decade of orbital period. In the Sana et al. (2012) sample of ${{ \mathcal N }}_{\mathrm{prim}}=71$ O-type primaries, there are ${{ \mathcal N }}_{\mathrm{SB}2}$ = 17 SB2s with q ≥ 0.3 and P = 2–20 days. The detection efficiencies of these 17 O-type short-period SB2s are nearly 100% (Section 3.2), and so the correction factor due to incompleteness is ${{ \mathcal C }}_{\mathrm{large}q}$ = ${\sum }_{j=1}^{{{ \mathcal N }}_{\mathrm{SB}2}}{w}_{j}/{{ \mathcal N }}_{\mathrm{SB}2}$ = 1.0 ± 0.1. In addition, all 71 O-type primaries in the Sana et al. (2012) sample are in young open clusters, and so the correction factor ${{ \mathcal C }}_{\mathrm{evol}}=1.0$ due to binary evolution is negligible (see Sections 2 and 11). The intrinsic frequency of companions with q > 0.3 per decade of orbital period is ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{SB}2}$ ${{ \mathcal C }}_{\mathrm{large}q}$ ${{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$/Δlog P = ($17\pm \sqrt{17}$) × (1.0 ± 0.1) × 1.0/71/(1.3 − 0.3) = 0.24 ± 0.06 for O-type primaries and short orbital periods (Table 2). In other words, 24% ± 6% of O-type zero-age MS primaries have a companion with P = 2–20 days and q > 0.3.

We perform similar calculations for the samples of short-period companions to B-type MS primaries. For B-type primaries, however, the correction factor ${{ \mathcal C }}_{\mathrm{large}q}$ for incompleteness is slightly larger than unity because the detection efficiencies of SB2s with lower-mass primaries are smaller (see Section 3.2). We find ${{ \mathcal C }}_{\mathrm{large}q}$ = 1.3 ± 0.1, 1.4 ± 0.2, and 1.5 ± 0.2 for the early B, mid-B, and late B subsamples, respectively. The Levato et al. (1987) and Abt et al. (1990) samples are volume limited, and Kobulnicky et al. (2014) targeted systems in the Cyg OB2 association, which has a distribution of ages τ = 1–7 Myr (Wright et al. 2015). We must therefore also account for the correction factor ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.2 ± 0.1 due to binary evolution (Sections 2 and 11). Following the same approach as above, we measure ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.12 ± 0.05, 0.10 ± 0.04, and 0.07 ± 0.04 for the early B, mid-B, and late B subsamples, respectively, at short orbital periods (Table 2). The intrinsic frequency of companions with q > 0.3 and P = 2–20 days is ≈4 times larger for O-type primaries compared to late B primaries (see also Section 9). This trend is consistent with the conclusions of Chini et al. (2012), who find that the observed spectroscopic binary fraction increases by a factor of ≈5–7 between B9 and O5 primaries (see their Figure 3).

In addition to the 17 O-type SB2s with q ≥ 0.3 and P = 2–20 days, Sana et al. (2012) identified one SB2 with q = 0.25–0.30 and three SB1s across the same period interval. Considering the sensitivity of the spectroscopic binary observations and the young nature of the Sana et al. (2012) O-type sample, we expect these ${{ \mathcal N }}_{\mathrm{small}q}$ = 4 binaries to contain low-mass stellar companions with q = 0.1–0.3. Given ${{ \mathcal N }}_{\mathrm{large}q}$ = 17, ${{ \mathcal N }}_{\mathrm{small}q}=4$, ${\gamma }_{\mathrm{large}q}$ = −0.3, ${{ \mathcal F }}_{\mathrm{twin}}$ = 0.08, and our definitions according to Section 2, then ${\gamma }_{\mathrm{small}q}$ = 0.6 ± 0.8 (Table 2). The relative density ${{ \mathcal N }}_{\mathrm{small}q}$/${\rm{\Delta }}q$ = 4/(0.3 − 0.1) = 20 of q = 0.1–0.3 binaries is slightly smaller than the density ${{ \mathcal N }}_{\mathrm{large}q}$/${\rm{\Delta }}q$ = 17/(1.0 − 0.3) = 24 of binaries with q = 0.3–1.0. The mass-ratio probability distribution must flatten and possibly turn over below q ≲ 0.3.

Similarly, in addition to the 15 SB2s with q ≥ 0.3 and P = 2–20 days, the three samples of B-type spectroscopic binaries contain one additional SB2 with q = 0.25–0.30 and 32 SB1s across the same period interval (Levato et al. 1987; Abt et al. 1990; Kobulnicky et al. 2014). At these short orbital periods, the ratio of SB1s to SB2s in the combined B-type sample (32/16 = 2.0) is an order of magnitude larger than the ratio in the O-type sample (3/18 = 0.2). Unlike the O-type SB1s, the 32 B-type SB1s do not all contain low-mass stellar companions with q = 0.1–0.3 due to three effects we discuss below.

First, not all B-type binaries with q = 0.3–1.0 will appear as SB2s, even at these shortest of orbital periods (see blue curve in Figure 5). Although q = 0.3–1.0 binaries may not be distinguishable as SB2s, they will still be detectable as SB1s. Given the correction factors ${{ \mathcal C }}_{\mathrm{large}q}$ = 1.3–1.5 due to incompleteness of short-period B-type SB2s, we expect a total of ${{ \mathcal N }}_{\mathrm{large}q}$ = 22 binaries with q ≥ 0.3 and P = 2–20 days in the combined B-type sample. This implies that 22−15 = 7 of the 32 SB1s actually contain stellar companions with q > 0.3.

Second, the sample of 25 remaining B-type SB1s is contaminated by stellar remnants. In Section 8, we show that ≈30% of close, faint, stellar-mass companions to solar-type stars are actually WDs instead of M dwarfs. We expect similar percentages of contamination by stellar remnants for the B-type stars in our sample (see below and Section 11). We estimate that $\approx 9$ of the B-type SB1s probably contain compact remnant companions. The remaining $\approx 16$ B-type SB1s have stellar companions with q < 0.3.

Finally, the B-type spectroscopic surveys include SB1s with small velocity semi-amplitudes K₁ ≲ 20 km s⁻¹, some as small as K₁ ≈ 6 km s⁻¹ (Levato et al. 1987; Abt et al. 1990; Kobulnicky et al. 2014). If the orientations are sufficiently close to edge-on, these small-amplitude SB1s may be extreme mass-ratio stellar binaries with q < 0.1. Among the three B-type spectroscopic binary surveys, we identify 11 SB1s with P = 2–20 days and sufficiently small velocity semi-amplitudes such that the binary could have q < 0.1. Assuming random orientations, we estimate that $\approx 7$ of these SB1s may indeed have extreme mass ratios q < 0.1.

After accounting for the three effects described above, we estimate that there are ${{ \mathcal N }}_{\mathrm{small}q}$ = 9 ± 4 binaries with q = 0.1–0.3 and P = 2–20 days in the combined B-type sample. Given ${{ \mathcal N }}_{\mathrm{large}q}=22$, then the power-law component across small mass ratios is ${\gamma }_{\mathrm{small}q}$ = −0.5 ± 0.8 for B-type binaries with short orbital periods (Table 2). This is consistent with the value ${\gamma }_{\mathrm{small}q}$ = 0.6 ± 0.8 we measured above for O-type primaries across the same period interval.

In our calculation of ${\gamma }_{\mathrm{small}q}$ = −0.5 ± 0.8 for B-type binaries, we accounted for contamination by $\approx 9$ compact remnant companions in the SB1 sample. Suppose instead that we neglected this effect and assumed that there are ${{ \mathcal N }}_{\mathrm{small}q}=18$ stellar companions with q = 0.1–0.3 and P = 2–20 days in the combined B-type sample. We would then measure ${\gamma }_{\mathrm{small}q}$ = −1.9 ± 0.5, which is discrepant with the O-type sample at the 2.7σ significance level. Either the relative frequency of short-period companions with extreme mass ratios is significantly larger for B-type stars compared to O-type stars, or a sizable fraction of B-type SB1s contain compact remnant companions. We favor the latter scenario for three reasons. First, the power-law component ${\gamma }_{\mathrm{large}q}$ across large mass ratios does not vary between O-type and B-type short-period binaries (Section 3.4). We would naturally expect that ${\gamma }_{\mathrm{small}q}$ does not significantly vary either. Second, other observational techniques demonstrate that ${\gamma }_{\mathrm{small}q}$ is larger than or comparable to ${\gamma }_{\mathrm{large}q}$ (see Sections 4–8). In other words, the mass-ratio probability distribution is always observed to flatten, not steepen, below q < 0.3. Finally, we utilize observations in Section 11 to predict that a non-negligible fraction of early-type stars in a volume-limited survey contain compact remnant companions. We therefore conclude that ${\gamma }_{\mathrm{small}q}$ = −0.5 ± 0.8 for short-period B-type binaries. This self-consistency analysis indicates that ≈30% of an older population of B-type SB1s contain compact remnant companions.

We next utilize the ${{ \mathcal N }}_{\mathrm{SB}2}=6$ SB2s with O-type primaries and log P (days) = 1.3–2.7 to estimate ${f}_{\mathrm{log}P;q\gt 0.3}$ at intermediate orbital periods. We measure the correction factor for incompleteness of q > 0.3 companions to be ${{ \mathcal C }}_{\mathrm{large}q}$ = ${\sum }_{j=1}^{{{ \mathcal N }}_{\mathrm{SB}2}}{w}_{j}$/${{ \mathcal N }}_{\mathrm{SB}2}$ = 2.0 ± 0.5. The intrinsic frequency of companions with q > 0.3 per decade of orbital period is ${f}_{\mathrm{log}P;q\gt 0.3}\,=\,$ ${{ \mathcal N }}_{\mathrm{SB}2}$ ${{ \mathcal C }}_{\mathrm{large}q}$ ${{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$/Δlog P = ($6\pm \sqrt{6}$) × (2.0 ± 0.5) × 1.0/71/(2.7−1.3) = 0.12 ± 0.06 for O-type stars and intermediate orbital periods (Table 2). For O-type primaries, the intrinsic frequency of companions with q > 0.3 decreases with increasing logarithmic orbital period. This is consistent with the results of Sana et al. (2012), who find that the logarithmic period distribution of massive binaries is skewed toward shorter periods.

Finally, we combine the ${{ \mathcal N }}_{\mathrm{SB}2}=4$ SB2s with P =20–200 days from the samples of ${{ \mathcal N }}_{\mathrm{prim}}$ = 109 + 83 = 192 early B and mid-B primaries. We measure a large correction factor ${{ \mathcal C }}_{\mathrm{large}q}$ = ${\sum }_{j=1}^{{{ \mathcal N }}_{\mathrm{SB}2}}{w}_{j}$/${{ \mathcal N }}_{\mathrm{SB}2}$ = 3.0 ± 1.0 because spectroscopic binary surveys of B-type primaries are significantly incomplete at intermediate orbital periods (see Figure 5). The intrinsic frequency of companions with q > 0.3 per decade of orbital period is ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{SB}2}$ ${{ \mathcal C }}_{\mathrm{large}q}$ ${{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$/Δlog P = ($4\pm \sqrt{4}$)(3.0 ± 1.0)(1.2 ± 0.1)/192/(2.3−1.3) = 0.08 ± 0.05 for early/mid-B-type stars at intermediate orbital periods (Table 2). This is consistent with the frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ ≈ 0.10–0.12 at shorter periods log P (days) ≈ 0.8, indicating that the period distribution of companions to early/mid-B-type stars approximately obeys Öpik's law across P = 2–200 days (see Section 9).

We recently became aware of a new direct spectral detection method for identifying binary stars with close to intermediate separations and small mass ratios (Gullikson et al. 2016b). Gullikson et al. (2016a) obtained large signal-to-noise ratio (S/N ∼ 100–1000), high-resolution (R = 80,000) spectra of 341 bright (V < 6 mag) A-type and B-type MS primaries with moderate to large rotation velocities ${v}_{\mathrm{rot}}$ sin i > 80 km s⁻¹. By exploiting the differences in spectral absorption widths between the rapidly rotating primaries and low-mass companions (M₂ ≲ 1.5 M_⊙, ${T}_{\mathrm{eff}}$ ≲ 6500 K) that quickly spin down to ${v}_{\mathrm{rot}}$ sin i ≈ 15 km s⁻¹, Gullikson et al. (2016a) can distinguish the two components in a single-epoch spectrum. Technically, Gullikson et al. (2016a) cross-correlate the observed spectrum with model spectra of cooler companions in order to measure the spectral type of the secondary. Although the orbital periods cannot be measured with a single-epoch spectrum, this technique is sensitive to companions with projected separations $\rho \lesssim 2^{\prime\prime}$ that lie within the slit or optical fiber. Given the median distance d ≈ 100 pc to the 341 systems in their sample, the binaries have separations a ≲ 200 au. In total, Gullikson et al. (2016a) find 64 companions within a ≲ 200 au of the 341 primaries.

Gullikson et al. (2016a) also estimate their detection efficiencies as a function of mass ratio q (see their Figure 6). The completeness estimates peak at ≈90% near q ≈ 0.3. Toward smaller mass ratios q < 0.15, the detection efficiencies plummet because the secondaries become substantially fainter and cannot be distinguished from the more luminous primaries in a combined spectrum. At large mass ratios q > 0.6, the detection efficiencies steadily decrease because the absorption profiles of the secondaries become too similar to those of the primaries.

Utilizing their estimated detection efficiencies, Gullikson et al. (2016a) recover the intrinsic mass-ratio distribution of A-type and B-type binaries with a ≲ 200 au. They find that the mass-ratio distribution peaks at q = 0.3. Fitting our two-component power law to the histogram distribution in Figure 7 of Gullikson et al. (2016a), we estimate ${\gamma }_{\mathrm{small}q}$ ≈ 0.5 and ${\gamma }_{\mathrm{large}q}$ ≈ −1.9.

The Gullikson et al. (2016a) sample covers a broad range of primary masses M₁ ≈ 1.6–10 M_⊙ and orbital separations a ≈ 0.05–200 au. Because the intrinsic binary statistics can significantly vary within such a large parameter space, the inferred mass-ratio distribution may be biased. We therefore cull the Gullikson et al. (2016a) sample according to primary mass and orbital separation as follows. Due to the IMF and short lifetimes of early B stars, the Gullikson et al. (2016a) sample is dominated by A-type and late B stars. We select the ${{ \mathcal N }}_{\mathrm{prim}}$ = 281 primaries with MS spectral types B6–A7, which correspond to primary masses M₁ = 1.6–4.6 M_⊙. From this subset of ${{ \mathcal N }}_{\mathrm{prim}}$ = 281 primaries, Gullikson et al. (2016a) identify 46 companions. We remove the five spectroscopic binaries (HIP 5348, 13165, 16611, 76267, 100221) contained in the SB9 catalog (Pourbaix et al. 2004) with known orbital periods P < 20 days. We also remove the four wide binaries (HIP 12706, 79199, 84606, 115115) in the Washington Double Star (WDS) catalog (Mason et al. 2009) with listed projected separations ρ > 07. The SB9 catalog is relatively complete shortward of P < 20 days (see above), while the WDS catalog contains most bright VBs with q > 0.2 and ρ > 07 (see Section 7). The 37 binaries remaining in our subsample most likely have orbital periods P > 20 days and projected separations ρ < 07. Given the median distance d = 90 pc to the 281 A-type and late B primaries in our subsample, the 37 companions span log P (days) = 1.3–4.9.

Gullikson et al. (2016a) note that their predicted detection efficiencies for large mass ratio q > 0.6 binaries may be overestimated. Utilizing an artificial injection and recovery algorithm, Gullikson et al. (2016a) predict a detection efficiency of ≈80% for hot companions with masses comparable to the primaries. In practice, they detect only 15 of the 25 known binaries (60%) with hot companions in the SB9 and WDS catalogs. Aware of this discrepancy, Gullikson et al. (2016a) multiply their predicted detection efficiencies by a factor of f = 0.8 to roughly match the empirical detection rate of ≈60%. However, the SB9 and WDS catalogs are themselves incomplete, even for binaries with q > 0.6 and especially for binaries with intermediate separations. The ratio 15/25 = 60% is therefore an upper limit to the true detection efficiency for hot companions.

We correct for selection effects as follows. For binaries with 0.15 < q < 0.60, we adopt the completeness levels as displayed in Figure 6 of Gullikson et al. (2016a). Toward larger mass ratios, we interpolate the completeness levels between the Gullikson et al. (2016a) value of ≈83% at q = 0.60 and a smaller value of 60% at q = 0.75. For q > 0.75, the detection efficiencies are too uncertain and may become too small to be included in our statistical analysis. We therefore consider only binaries with q < 0.75, and we remove the one detected system (HIP 14043) in our subsample with q > 0.75. In Figure 10, we display the corrected cumulative distribution of mass ratios for the 36 remaining binaries in our subsample with 0.15 < q < 0.75. The length of each vertical step in Figure 10 is inversely proportional to its corresponding detection efficiency.

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Across q = 0.15–0.75, the binaries in our subsample cannot be described by a single power-law mass-ratio distribution. This is consistent with the conclusions of Gullikson et al. (2016a). Using a maximum likelihood method, we fit our two-component power-law model to the distribution in Figure 10. We measure ${\gamma }_{\mathrm{small}q}$ = 0.7 ± 0.8 and ${\gamma }_{\mathrm{large}q}$ = −1.0 ± 0.5 (Table 4). Although not as narrowly peaked as the mass-ratio distribution presented in Figure 7 of Gullikson et al. (2016a), we confirm their main result that the intrinsic mass-ratio distribution is weighted toward q = 0.3 with a flattening and possible turnover below q < 0.3. This is a robust result considering that we select a subsample from the Gullikson et al. (2016a) data set, apply different correction factors for incompleteness, and examine a narrower interval of mass ratios q = 0.15–0.75.

Of the 36 detected companions with q = 0.15–0.75 in our subsample, 28 have large mass ratios q = 0.30–0.75. By weighting each system according to the inverse of their respective detection efficiency, we estimate a total of 34 ± 9 companions with q = 0.30–0.75. The uncertainty derives from Poisson statistics and the uncertainties in the correction factors. If the power-law component ${\gamma }_{\mathrm{large}q}$ = −1.0 ± 0.5 extends all the way to q = 1.0, then we expect 11 ± 4 companions with q = 0.75–1.00. If instead there is an excess fraction ${{ \mathcal F }}_{\mathrm{twin}}$ ≈ 0.10 of twins, as is indicated by observations of other samples of A-type MS binaries at closer (Section 3.4) and wider (Section 7.1) separations, then there would be 15 ± 5 companions with q = 0.75–1.00. We adopt the average of these two estimates and find a total of ${{ \mathcal N }}_{\mathrm{large}q}$ = 47 ± 12 binaries with large mass ratios q = 0.3–1.0. We account for the correction factor ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.2 ± 0.1 due to binary evolution, whereby 20% ± 10% of A-type/late B primaries in the magnitude-limited sample of Gullikson et al. (2016a) are actually the original secondaries of evolved binaries (see Section 11). We measure a corrected frequency of companions with q > 0.3 per decade of orbital period of ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$/Δlog P = (47 ± 11)(1.2 ±0.1)/281/(4.9 −1.3) = 0.06 ± 0.02.--> = (47 ± 11)(1.2 ±0.1)/281/(4.9−1.3) = 0.06 ± 0.02. We report the binary statistics based on the Gullikson et al. (2016a) survey in Table 4.

For a sample of ${{ \mathcal N }}_{\mathrm{prim}}$ = 58 B0–B5 MS primaries in the Sco-Cen OB association (d ≈ 130 pc), Rizzuto et al. (2013) used LBI to identify 24 companions with angular separations 7–130 mas. They measured the brightness contrasts Δm of the binary components at wavelengths λ = 550–800 nm and then estimated the mass ratios q from Δm according to stellar evolutionary tracks. Rizzuto et al. (2013) report ${{ \mathcal N }}_{\mathrm{comp}}$ = 18 companions with q ≥ 0.3 and projected orbital separations ρ = 1.5–30 au, which correspond to orbital periods 2.3 ≲log P (days) ≲ 4.3. This subsample is relatively complete across the specified mass-ratio and period interval. In Figure 11, we display the cumulative distribution of mass ratios for these 18 systems. The mass-ratio distribution of the 18 binaries is fitted to high accuracy by a single power-law distribution with ${\gamma }_{\mathrm{large}q}$ = −2.4 ± 0.4. The upper limit on the excess twin fraction is ${{ \mathcal F }}_{\mathrm{twin}}$ < 0.03 (Table 6). Companions to early B primaries at intermediate orbital periods are weighted toward extreme mass ratios. For q ≳ 0.3, the binary mass ratios are surprisingly consistent with random pairings drawn from a Salpeter IMF (γ = −2.35).

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Rizzuto et al. (2013) do not directly state that their binaries discovered through LBI strongly favor small mass ratios. They instead compile observations of short-period spectroscopic and long-period visual companions to the 58 early B MS stars in their sample. They then report a mass-ratio distribution with γ ≈ −0.5 that is averaged across all orbital periods. We emphasize that the binary distributions of P and q are not necessarily uncorrelated. LBI offers a unique perspective into the binary properties of massive stars at intermediate periods and should therefore be treated independently.

We wish to evaluate the robustness of our measurement of ${\gamma }_{\mathrm{large}q}$ = −2.4 ± 0.4, and so we estimate the systematic uncertainties in deriving q from Δm. We calculate our own values of q from the measured brightness contrasts Δm reported in Rizzuto et al. (2013) by incorporating different stellar evolutionary tracks, ages, and atmospheric parameters. We also apply this method to the O-type LBI binary sample of Sana et al. (2014), who report only the brightness contrasts Δm (see Section 5.2).

In Figure 12, we compare the brightness contrasts Δm to the mass ratios q calculated by Rizzuto et al. (2013) for the 20 binaries in their sample with ρ = 1.5–30 au. This subsample includes the 18 systems with q ≥ 0.3 incorporated above, as well as two additional systems with q ≈ 0.26–0.29 near the detection limit. Rizzuto et al. (2013) report an uncertainty of 10% in the mass ratios, which we indicate in our Figure 12. Eighteen of the 20 binaries have relatively unevolved MS primaries with spectral types B0–B5 and luminosity classes IV–V. Two systems, -Cen and κ-Sco, have ≈B1III spectral types and are therefore on the upper MS or giant branch (shown in our Figure 12 as the two systems with diamond symbols). Given the same brightness contrast Δm, binaries with older primaries on the upper MS have more massive companions.

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To determine our own values of q from the brightness contrasts Δm, we utilize the solar-metallicity pre-MS and MS stellar evolutionary tracks from Tognelli et al. (2011) and Bertelli et al. (2009), respectively. We consider two primary masses, M₁ = 12 M_⊙ and M₁ = 6 M_⊙, which are representative of B1V and B5V MS stars, respectively. We model the brightness contrasts at two different ages, τ = 5 Myr and τ = 16 Myr, which span the estimated ages of the stellar subgroups within the Sco-Cen OB association (Rizzuto et al. 2013). We incorporate the bolometric corrections and color indices from Pecaut & Mamajek (2013) to calculate the red ${R}_{{\rm{C}}}$-band magnitudes from the stellar luminosities and effective temperatures. In Figure 12, we plot our derived brightness contrasts ${\rm{\Delta }}{R}_{{\rm{C}}}$ as a function of q for the four different combinations of primary mass and age. The four curves are quite similar to each other except for two minor aspects. First, at ages τ = 5 Myr, secondaries with M₂ ≲ 2 M_⊙ are still on the pre-MS. This explains why the blue dashed curve corresponding to our model with M₁ = 6 M_⊙ and τ = 5 Myr is non-monotonic near q = 0.3 (M₂ ≈ 2 M_⊙). Second, our model with an M₁ = 12M_⊙ primary is near the tip of the MS at τ = 16 Myr (green curve). For this older model, a binary with a given mass ratio q is observed to have a slightly larger brightness contrast Δm. In general, our models are consistent with the mass ratios reported by Rizzuto et al. (2013). In fact, the two systems with ≈B1III primaries better match our model with M₁ = 12 M_⊙ and τ = 16 Myr (green curve in Figure 12).

To estimate the systematic uncertainties in ${\gamma }_{\mathrm{large}q}$, we utilize the red line in Figure 12 to calculate our own values of q. Based on this model, we find that the mass ratios q are slightly smaller than those reported by Rizzuto et al. (2013). We find 16 systems (instead of 18) with q > 0.3. By fitting a power-law mass-ratio distribution to these 16 binaries, we measure ${\gamma }_{\mathrm{large}q}$ = −2.2 ± 0.5. This nearly matches the previous result of ${\gamma }_{\mathrm{large}q}$ = −2.4 ± 0.4 from using the mass ratios q directly provided by Rizzuto et al. (2013). The systematic uncertainty is smaller than the measurement uncertainty, and so we adopt the average value ${\gamma }_{\mathrm{large}q}$ = −2.3 ± 0.5 of the two methods (Table 6). Binaries with early B primaries and intermediate orbital periods are indeed weighted toward extreme mass ratios.

We also average the number ${{ \mathcal N }}_{\mathrm{large}q}$ = 17 of binaries with q > 0.3 and 2.3 < log P (days) < 4.3 based on the two methods for measuring q in the Rizzuto et al. (2013) sample. After accounting for the correction factor ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.2 ± 0.1 due to binary evolution, the intrinsic companion frequency is ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}$ ${{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$/Δlog P = ($17\pm \sqrt{17}$)(1.2 ± 0.1)/58/(4.3 − 2.3) = 0.18 ± 0.05 for early B binaries and intermediate orbital periods (Table 6). This is slightly larger than but consistent with our measurements of ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.12–0.16 for binaries with early B primaries and shorter orbital periods based on observations of SB2s (Section 3) and EBs (Section 4).

Sana et al. (2014) recently surveyed $\approx 100$ O-type stars with near-infrared magnitudes H < 7.5 (λ ≈ 1.6 μm) in the southern hemisphere using both LBI and SAM techniques. The combination of these observational methods provides a relatively complete sample of companions with angular separations 2–60 mas and brightness contrasts ΔH < 4.0 mag (see Figure 7 in Sana et al. 2014). Sana et al. (2014) also resolved additional companions at wider separations ≳03 with adaptive optics, which we examine in Section 7.

It is difficult to reliably measure the mass ratios and correct for incompleteness for binaries with supergiant primaries. In our current analysis, we consider only the 56 O-type primaries in the Sana et al. (2014) main sample that were observed by both LBI and SAM methods and have luminosity classes II.5–V (see their Figure 1). Their survey is magnitude limited, so we must correct for the Öpik effect/Malmquist bias toward binaries with equally bright components. We remove the two detected binaries (HD 93222 and HD 123590) with observed total magnitudes H ≈ 7.2–7.5 and brightness contrasts ΔH ≲ 0.3 mag. These two systems would be fainter than the H = 7.5 limit if we were to consider the luminosity from the primaries alone. Our culled sample from the Sana et al. (2014) survey contains ${{ \mathcal N }}_{\mathrm{prim}}$ = 54 O-type MS primaries.

From this subsample of 54 O-type MS primaries, Sana et al. (2014) identified 25 companions with angular separations 2–60 mas and brightness contrasts ΔH < 4.0 mag. Given the typical distances d = 1–2 kpc to the O-type stars with luminosity classes II.5–V in the Sana et al. (2014) sample (see their Figure 3), the angular separations correspond to projected separations ρ ≈ 3–90 au, i.e., 2.5 < log P (days) < 4.7. In Figure 13, we show the measured brightness contrasts ΔH and uncertainties for the 25 binaries as reported in Sana et al. (2014). For the few systems with multiple measurements of ΔH, we adopt a weighted average and uncertainty.

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We employ a method similar to that described in Section 5.1 to measure the binary mass ratios from the observed brightness contrasts ΔH. For our 25 O-type binaries, we utilize the calibrated relations for galactic O-type stars in Martins et al. (2005) to estimate the primary mass, effective temperature, and absolute ${M}_{V}$ magnitude according to the primary's spectral type and luminosity class. We then calculate the near-infrared magnitude H according to the temperature-dependent color indices $(V-H)$ reported in Pecaut & Mamajek (2013). If the secondary is also an O-type star with M₂ ≳ 16 M_⊙, then we use this same technique to estimate its own value of H. For these O-type + O-type binaries, we assume that the secondary is always an MS star with luminosity class V if q < 0.6. For q > 0.6, we smoothly interpolate the secondary's luminosity class between V at q = 0.6 and the luminosity class of the primary at q = 1.0. If the secondary is a B-type star with M₂ ≲ 16 M_⊙, we interpolate the solar-metallicity stellar evolutionary tracks of Bertelli et al. (2009) to determine the secondary's near-infrared magnitude H. For these O-type + B-type binaries, we adopt an age appropriate for the spectral type and luminosity class of the primary.

In Figure 13, we display our modeled brightness contrasts ΔH as a function of mass ratio q for M₁ = 20 M_⊙ (dashed) and M₁ = 40 M_⊙ (solid) primaries. For dwarf stars, these masses correspond to O8.5V (thick dashed blue) and O5V (thick solid magenta), respectively. The magenta model corresponding to the O5V primary is flatter than the dashed blue model of the O8.5V primary. This is because the MS mass–luminosity relation flattens toward larger masses. For example, a 40 M_⊙ MS star is ΔH ≈ 1.0 mag brighter than a 20 M_⊙ MS star (value of magenta curve at q = 0.5). Meanwhile, a 20 M_⊙ MS star is ΔH ≈ 1.7 mag brighter than a 10 M_⊙ MS star (value of dashed blue curve at q = 0.5). The M₁ = 40 M_⊙ primary increases in brightness by ΔH ≈ 0.6 mag as it becomes an O5.5III star (thin green line in Figure 13). This is why the green and magenta solid curves differ by ΔH ≈ 0.6 mag at q ≲ 0.5. Similarly, the M₁ = 20 M_⊙ primary will increase in brightness by ΔH ≈ 1.0 mag as it evolves into an O9.5III giant (thin dashed red).

For the 25 binaries we have selected from Sana et al. (2014), we determine the mass ratios q from our models according to the listed brightness contrasts ΔH and spectral types and luminosity classes of the primaries. We propagate the measurement uncertainties in ΔH, as well as errors of $\approx 0.5$ in both the spectral subtypes and luminosity classes. We display our solutions for the mass ratios of the 25 systems in Figure 13. Twenty-three of the binaries are between our O5V (magenta) and O9.5III (dashed red) models. The two remaining systems have ≈O4V primaries and lie just above the magenta curve.

Five of the 25 O-type binaries resolved with LBI/SAM in our subsample are also long-period SB2s with independent measurements of the mass ratio (Sana et al. 2014). The SB2s with dynamical mass ratio measurements are HD 54662 (q = 0.39; Boyajian et al. 2007), HD 150136 (q = 0.54; Mahy et al. 2012), HD 152246 (q = 0.89; Nasseri et al. 2014), HD 152314 (q = 0.55; Sana et al. 2012), and HD 164794 (q = 0.66; Rauw et al. 2012). We display the five spectroscopic mass ratios as diamond symbols in Figure 13. Three of the five SB2s have dynamical mass ratios consistent with our values. For these three systems, we adopt the SB2 dynamical mass ratios because they are measured to higher precision.

For one of the two remaining systems, HD 150136, we measure q = 0.38 ± 0.05 according to the moderate brightness contrast ΔH = 1.5 mag. Meanwhile, Mahy et al. (2012) report q ≈ 0.54 based on the SB2 dynamics. This apparent discrepancy is because HD 15136 is a triple system. The tertiary component resolved with LBI orbits an inner binary of spectral types O3–3.5V (M₁ ≈ 64 M_⊙) and O5.5–6V (M₂ ≈ 40 M_⊙) in a P = 2.7 day orbit (Mahy et al. 2012). The additional luminosity from the inner companion increases the brightness contrast ΔH between the inner binary and tertiary, which biases our mass ratio measurement toward smaller values. To derive a dynamical mass of M₃ ≈ 35 M_⊙, Mahy et al. (2012) assume that the tertiary component is coplaner with the inner binary. Utilizing the observed spectral type O6.5–7V of the tertiary alone (Mahy et al. 2012), the mass is M₃ ≈ 27 M_⊙ according to the O-type stellar relations provided in Martins et al. (2005). This implies a mass ratio of q = M₃/M₁ ≈ 0.42. We adopt q = 0.48 for the tertiary companion in HD 150136, which is halfway between the dynamical coplaner estimate of q ≈ 0.54 and the measurement of q = 0.42 based on the observed spectral types.

Finally, for HD 54662, we measure q ≈ 0.87 based on the small brightness contrast ΔH = 0.2 mag, while Boyajian et al. (2007) report an SB2 dynamical mass ratio of q = K₁/K₂ = 0.39. The spectroscopic absorption features of the binary components in HD 54662 are significantly blended (Boyajian et al. 2007), and so the velocity semi-amplitudes K₁ and K₂ are rather uncertain. Moreover, Boyajian et al. (2007) fit O6.5V and O9V spectral types to the binary components with an optical flux ratio of F₂/F₁ ≈ 0.5. These spectral types and optical brightness contrast imply a much larger mass ratio of q ≈ 0.7 (Martins et al. 2005). For HD 54662, we adopt this spectroscopic measurement of q = 0.70, which is between the near-infrared brightness contrast measurement of q = 0.89 and the uncertain dynamical measurement of q = 0.39.

For the 20 companions resolved at intermediate orbital periods without spectroscopic mass measurements, we adopt the mass ratios measured from the brightness contrasts ΔH. Given the detection limit ΔH < 4.0 mag, companions to O-type primaries with luminosity classes III–V are relatively complete for q > 0.3 (see Figure 13). For the ${{ \mathcal N }}_{\mathrm{prim}}$ = 54 O-type primaries with luminosity classes II.5–V in Sana et al. (2014), we find ${{ \mathcal N }}_{\mathrm{large}q}$ = 21 companions with q > 0.3 and 2.5 < log P (days) < 4.7. Because we limited our sample to O-type stars with luminosity classes II.5–V, the correction factor ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.1 ± 0.1 due to binary evolution is relatively small. The intrinsic frequency of companions with q > 0.3 per decade of orbital period is ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$/ΔlogP =($21\pm \sqrt{21}$)(1.1 ± 0.1)/54/(4.7 − 2.5) = 0.19 ± 0.05 for O-type stars and intermediate orbital periods (Table 6). This is between the values ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.12–0.24 we measured for O-type SB2s (Section 3).

We display the cumulative distribution of mass ratios for the 21 binaries with q > 0.3 in Figure 14. We fit the mass-ratio probability distribution and measure ${\gamma }_{\mathrm{large}q}$ = −1.4 ± 0.4 and ${{ \mathcal F }}_{\mathrm{twin}}\lt 0.03$ (Table 6). Although not as steep as the slope ${\gamma }_{\mathrm{large}q}$ = −2.3 ± 0.5 measured for LBI companions to early B primaries (Section 5.1), the mass-ratio distribution of LBI/SAM O-type binaries is still weighted toward small mass ratios. In fact, the O-type LBI/SAM measurement of ${\gamma }_{\mathrm{large}q}$ = −1.4 ± 0.4 nearly matches the O-type long-period SB2 measurement of ${\gamma }_{\mathrm{large}q}$ = −1.6 ± 0.5 (Section 3). These two independent measurements of ${\gamma }_{\mathrm{large}q}$ ≈ −1.5 for intermediate-period O-type binaries are smaller than the value ${\gamma }_{\mathrm{large}q}$ = −0.3 ± 0.3 we measured at shorter periods P < 20 days (see Section 3). The LBI/SAM observations of O-type stars confirm that companions at slightly wider separations favor smaller mass ratios.

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We initially examine the ${{ \mathcal N }}_{\mathrm{long}P}$ = 8 long-period companions with 4.1 < log P < 6.5 and q > 0.3 in the Evans et al. (2013) sample. We display in Figure 15 the cumulative distribution of mass ratios q for these eight wide binaries. This subsample is relatively complete across the specified mass-ratio interval. We measure ${\gamma }_{\mathrm{large}q}$ = −2.1 ± 0.5 and ${{ \mathcal F }}_{\mathrm{twin}}$ < 0.07 (Table 7). We find that wide companions to intermediate-mass stars have mass ratios q ≳ 0.3 consistent with random pairings drawn from a Salpeter IMF (γ = −2.35).

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To calculate ${f}_{\mathrm{log}P;q\gt 0.3}$, we account for the fact that many mid-B MS primaries with close companions log P (days) < 2.6 are unlikely to evolve into Cepheids. In the previous sections, we have measured ${f}_{\mathrm{log}P;q\gt 0.3}$ ≈ 0.08–0.14 for mid-B MS primaries and 0.2 < log P < 2.6. We estimate that ${f}_{\mathrm{log}P;q\gt 0.3}$ × Δlog P ≈ (0.11 ± 0.3) × (2.6 − 0.2) ≈ 20%–30% of mid-B MS stars will interact with a binary companion with q > 0.3 and log P < 2.6. Depending on ${\gamma }_{\mathrm{small}q}$, an additional 5%–15% of mid-B stars will interact with a small-mass-ratio (q = 0.1–0.3) companion across log P = 0.2–2.6. By adding these two statistics, we find that 35% ± 10% of mid-B MS primaries have companions with q > 0.1 and log P < 2.6 (see also Section 9). Approximately one-third of these close binaries will probably merge, particularly those with smaller mass ratios and/or shorter separations (Hurley et al. 2002; Belczynski et al. 2008; Section 11). These merger products can continue to evolve into Cepheid giants. Alternatively, the other two-thirds of close binaries will undergo stable MT or survive common envelope (CE) evolution (Hurley et al. 2002; Belczynski et al. 2008), thereby preventing the primary from evolving into a Cepheid. In total, we estimate that 25% ± 15% of mid-B primaries will interact with a close stellar companion in such a manner that it does not evolve into a Cepheid. The remaining ${{ \mathcal F }}_{\mathrm{Cepheid}}$ = 0.75 ± 0.15 of mid-B MS primaries are capable of evolving into Cepheid variables.

Because only a subset ${{ \mathcal F }}_{\mathrm{Cepheid}}$ = 0.75 ± 0.15 of mid-B MS stars evolve into Cepheids, the frequency of wide companions to B-type MS stars is smaller than the frequency of wide companions to Cepheids. We also account for the correction factor ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.2 ± 0.1, whereby 20% ± 10% of observed Cepheids are actually the original secondaries of evolved binaries. We calculate ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{long}P}$ ${{ \mathcal F }}_{\mathrm{Cepheid}}$ ${{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{Cepheid}}$/Δlog P = $(8\pm \sqrt{8}$)(0.75 ± 0.15)(1.2 ± 0.1)/76/(6.5 − 4.1) = 0.04 ± 0.02 for mid-B MS stars and long orbital periods (Table 7).

We next examine the companions to Cepheids at intermediate orbital periods. In a recent follow-up paper, Evans et al. (2015) identified all known spectroscopic binary companions to Cepheids, including those that did not necessarily exhibit a UV excess. Given $\langle {M}_{1}\rangle$ ≈ 6 M_⊙ and the sensitivity K₁ ≈ 2 km s⁻¹ of their radial velocity measurements, the Evans et al. (2015) sample is relatively complete for q > 0.1 and for P ≈ 1–10 yr, i.e., 2.6 < log P (days) < 3.6 (see their Figures 4 and 5). At slightly longer orbital periods P > 10 yr, the Evans et al. (2015) survey most likely missed low-mass companions, due to the limited sensitivity and cadence of the spectroscopic observations. We analyze the 17 companions reported in Table 8 of Evans et al. (2015) that have P = 1–10 yr and measured values or limits on the mass ratios.

Of the 17 intermediate-period companions to Cepheids, ${{ \mathcal N }}_{\mathrm{large}q}=10$ have measured mass ratios q > 0.3 based on the observed UV excess from the hot MS companions. We plot the cumulative distribution of mass ratios for these 10 systems in Figure 16 and measure ${\gamma }_{\mathrm{large}q}$ = −1.7 ± 0.5 and ${{ \mathcal F }}_{\mathrm{twin}}\lt 0.06$ (Table 7). Like the wide binaries, companions to Cepheids at intermediate orbital periods have mass ratios across q = 0.3–1.0 weighted toward small values q ≈ 0.3–0.5.

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Of the seven remaining companions to Cepheids at intermediate orbital periods, only one has a measured mass ratio q = 0.27 near the sensitivity limit. The other six systems do not have detectable UV excesses. Assuming that these six SB1s have MS companions, the lack of UV excess provides an upper limit for the mass ratios q ≲ 0.3 (see Table 8 in Evans et al. 2015). Alternatively, the companions may not exhibit a UV excess because they are faint compact remnants. In either case, the six long-period SB1s must have q ≳ 0.1 given the sensitivity K₁ = 2 km s⁻¹ of the spectroscopic radial velocity observations. In fact, assuming random orientations, we expect one additional system with q = 0.1–0.3 to have escaped detection (see Figure 4 in Evans et al. 2015). Assuming ${{ \mathcal N }}_{\mathrm{small}q}=8$, i.e., all observed and suspected companions with q = 0.1–0.3 are stellar in nature, and given ${{ \mathcal N }}_{\mathrm{large}q}=10$, ${\gamma }_{\mathrm{large}q}$ = −1.7, and ${{ \mathcal F }}_{\mathrm{twin}}=0.0$, then ${\gamma }_{\mathrm{small}q}$ = −0.1 ± 0.7. If instead ≈30% of the six observed SB1s contain compact remnant companions, as found for other populations of SB1s (see Sections 3–4, 8, and 11), then ${{ \mathcal N }}_{\mathrm{small}q}=6$. This smaller number of q = 0.1–0.3 stellar companions implies a positive slope ${\gamma }_{\mathrm{small}q}$ = 0.6 ± 0.8. We adopt the average ${\gamma }_{\mathrm{small}q}$ =0.2 ± 0.9 of these two values (Table 7), where we propagate the uncertainty in the number of compact remnant companions.

Regardless of how we account for selection effects, binary evolution, and contamination by compact remnants, the slope of the mass-ratio distribution across q = 0.1–0.3 must be ${\gamma }_{\mathrm{small}q}$ > −1.4 at the 2σ confidence level. Although some observations of early-type intermediate-period binaries indicate that the power-law component of the mass-ratio distribution across q = 0.3–1.0 is consistent with random pairings drawn from a Salpeter IMF (${\gamma }_{\mathrm{large}q}$ ≈ −2.35), this steep slope cannot continue all the way down to q = 0.1. Instead, the mass-ratio distribution must flatten and possibly turn over below q < 0.3. As a whole, the population of early-type binaries with intermediate orbital periods is inconsistent with random pairings drawn from the IMF.

Finally, we determine the intrinsic companion frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ based on observations of Cepheids with MS companions at intermediate separations. According to Table 6 in Evans et al. (2015), ${{ \mathcal N }}_{\mathrm{Cepheid}}$ = 31 Cepheids brighter than V < 8.0 mag were extensively monitored for spectroscopic radial velocity variations. Of these primaries, ${{ \mathcal N }}_{\mathrm{large}q}=6$ have companions with orbital periods P = 1–10 yr (2.6 < log P (days) < 3.6) and with UV excesses that demonstrate q > 0.3. We again account for the fraction ${{ \mathcal F }}_{\mathrm{Cepheid}}$ = 0.75 ± 0.15 of mid-B MS primaries that evolve into Cepheids and for the correction factor ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.2 ± 0.1 due to evolved secondaries that are observed as Cepheids. We measure ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}{{ \mathcal F }}_{\mathrm{Cepheid}}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{Cepheid}}$/Δlog P = ($6\pm \sqrt{6}$)(0.75 ± 0.15)(1.2 ± 0.1)/31/(3.6 − 2.6) = 0.17 ± 0.08 for mid-B primaries and intermediate orbital periods (Table 7). This is higher than the measurements ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.04–0.14 at shorter (Sections 3–4) and longer (Sections 6.1 and 7) orbital periods, indicating that the period distribution of mid-B binaries peaks at 3 ≲ log P (days) ≲ 5 (see also Section 9).

To fill in the gap between solar-type primaries ($\langle {M}_{1}\rangle$ = 1.0 ± 0.2 M_⊙; see Section 8) and B-type primaries (M₁ = 3–16 M_⊙), we examine the De Rosa et al. (2014) VB survey of A-type primaries ($\langle {M}_{1}\rangle$ = 2.0 ± 0.3 M_⊙). They searched ${{ \mathcal N }}_{\mathrm{prim},\mathrm{AO}}$ = 363 A-type stars within 75 pc for visual companions utilizing adaptive optics in the near-infrared H and K bands. De Rosa et al. (2014) also report the mass ratios q of all their detected VBs according to the observed brightness contrasts. Their adaptive optics survey is complete down to q = 0.3 (ΔK ≲ 4.0 mag) for angular separations ρ ≳ $0\buildrel{\prime\prime}\over{.} 3$, and complete down to q = 0.1 (ΔK ≲ 6.5 mag) beyond ρ ≳ $0\buildrel{\prime\prime}\over{.} 9$ (see their Figures 7–8). At wide separations ρ > 8'', there is a >5% probability that VBs with ΔK ≈ 6 mag are optical doubles based on the observed background stellar densities (De Rosa et al. 2014). At even wider separations ρ > 15'', the adaptive optics survey is incomplete given the limited field of view.

De Rosa et al. (2014) also analyzed ${{ \mathcal N }}_{\mathrm{prim},\mathrm{CPM}}=228$ A-type stars in digitized photographic plates in search of astrometric CPM binaries. They utilized stellar isochrones and infrared colors to confirm that CPM pairs share the same distance and dust reddening. A significant majority of A-type stars in the De Rosa et al. (2014) sample have ages τ > 100 Myr (see their Figure 3). Unlike CPM companions to young B-type and O-type primaries, which may be spurious associations (Abt & Corbally 2000), CPM companions to systematically older A-type stars are most likely gravitationally bound and dynamically stable.

Based on the observational selection effects, we choose three subsamples from the De Rosa et al. (2014) adaptive optics survey and one subsample from their CPM survey. All four of these subsamples are relatively complete and unbiased within the specified mass-ratio and separation intervals (see Table 8). In the following, we measure ${f}_{\mathrm{log}P;q\gt 0.3}$, ${\gamma }_{\mathrm{large}q}$, ${{ \mathcal F }}_{\mathrm{twin}}$, and, if possible, ${\gamma }_{\mathrm{small}q}$ for each of these four subsamples.

We first select the ${{ \mathcal N }}_{\mathrm{large}q}=12$ VBs in the De Rosa et al. (2014) adaptive optics survey with q > 0.3 and small angular separations ρ = 03–09 ($\langle {a}_{\mathrm{proj}}\rangle$ ≈ 25 au; log P ≈ 4.3–4.9). The A-type stars are relatively old (τ ≳ 100 Myr; De Rosa et al. 2014), and so 20% ± 10% are likely to contain compact remnant companions (see Section 11). We account for this correction factor ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.2 ± 0.1 due to stellar evolution within binary systems. The intrinsic frequency of companions with q > 0.3 per decade of orbital period is ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim},\mathrm{AO}}$/Δlog P = ($12\pm \sqrt{12}$)(1.2 ± 0.1)/363/(4.9 − 4.3) = 0.07 ± 0.02 (Table 8).

We display the cumulative distribution of mass ratios q for these 12 VBs in Figure 17 (blue line). By fitting a single power law across q = 0.3–1.0, we measure γ = −0.9 ± 0.4. This is consistent with the value γ = −0.5 reported by De Rosa et al. (2014) for their population of close VBs with ${a}_{\mathrm{proj}}$ = 30–125 au. One of the 12 VBs in our subsample has q = 0.99, so there may be an excess fraction of twins. Allowing for a nonzero excess twin fraction, we measure ${{ \mathcal F }}_{\mathrm{twin}}$ = 0.05 ± 0.05 and ${\gamma }_{\mathrm{large}q}$ = −1.2 ± 0.5 (Table 8).

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Second, we analyze the 54 VBs in the De Rosa et al. (2014) adaptive optics survey with q > 0.1 and separations ρ = 09–80 ($\langle {a}_{\mathrm{proj}}\rangle$ ≈ 150 au; log P ≈ 4.9–6.3). Of these 54 VBs, ${{ \mathcal N }}_{\mathrm{large}q}=24$ have large mass ratios q = 0.3–1.0. The intrinsic companion frequency at these slightly wider separations is ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim},\mathrm{AO}}$/Δlog P = ($24\pm \sqrt{24}$)(1.2 ± 0.1)/363/(6.3 − 4.9) = 0.06 ± 0.01 (Table 8).

In Figure 18, we display the cumulative distribution of mass ratios q = 0.1–1.0 for the 54 VBs. Given such a large sample size, we can clearly see that a single power-law component γ across all mass ratios q = 0.1–1.0 does not adequately describe the data. For instance, there is a small but statistically significant excess twin fraction. Of the 24 VBs with q > 0.3, two have q = 0.99–1.00, implying ${{ \mathcal F }}_{\mathrm{twin}}$ ≈ 2/24 ≈ 0.08. The De Rosa et al. (2014) sample is volume limited within 75 pc, so this excess twin fraction cannot be due to the Öpik effect/Malmquist bias. Even after considering the twin systems with q > 0.95, a single power law does not fit the data across q = 0.10–0.95. For large mass ratios q ≳ 0.3, the data favor a steeper slope of γ ≈ −2 (see Figure 18). Meanwhile, the slope across small mass ratios q ≈ 0.1–0.3 trends toward γ ≈ −1. Fitting our three-parameter model to the data, we measure an excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}$ = 0.07 ± 0.04, a slope ${\gamma }_{\mathrm{large}q}$ = −2.1 ± 0.4 across large mass ratios q = 0.3–1.0, and a slope ${\gamma }_{\mathrm{small}q}$ = −0.8 ± 0.4 across small ratios q = 0.1–0.3 (Table 8; dashed red line in Figure 18).

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Third, we examine the ${{ \mathcal N }}_{\mathrm{large}q}=5$ VBs in the De Rosa et al. (2014) adaptive optics survey with q > 0.3 and wide separations ρ = 80–150 ($\langle {a}_{\mathrm{proj}}\rangle$ ≈ 600 au; log P ≈6.3–6.7). We measure a frequency of companions with q > 0.3 per decade of orbital period of ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim},\mathrm{AO}}$/Δlog P = ($5\pm \sqrt{5}$)(1.2 ± 0.1)/363/(6.7 − 6.3) =0.04 ± 0.02 (Table 8). In Figure 17, we display the cumulative distribution of mass ratios for these five VBs. The five A-type VBs with q > 0.3 and wide separations ρ = 8''–15' (red line in Figure 17) are weighted toward smaller mass ratios compared to the 12 A-type VBs with q > 0.3 and smaller separations ρ = 03–09 (blue line in Figure 17). For the five VBs with wider separations, we measure ${\gamma }_{\mathrm{large}q}$ = −2.5 ± 0.7 and ${{ \mathcal F }}_{\mathrm{twin}}\lt 0.11$ (Table 8). This is consistent with the value γ = −2.3 reported by De Rosa et al. (2014) for their sample of wide VBs with ${a}_{\mathrm{proj}}$ = 125–800 au.

Finally, we choose the 11 VBs in the De Rosa et al. (2014) CPM survey with q > 0.1 and projected separations 2000–8000 au (log P ≈ 7.2–8.2). Of these systems, ${{ \mathcal N }}_{\mathrm{large}q}$ = 5 have large mass ratios q > 0.3. The intrinsic companion frequency is ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim},\mathrm{CPM}}$/Δlog P = ($5\pm \sqrt{5}$)(1.2 ± 0.1)/228/(8.2 − 7.2) = 0.03 ± 0.01 (Table 8). As expected, the frequency of companions per decade of orbital period gradually decreases toward the widest separations.

We display the cumulative distribution of mass ratios for these 11 VBs in Figure 19. We measure an excess twin fraction consistent with zero, i.e., ${{ \mathcal F }}_{\mathrm{twin}}\lt 0.10$ at the 1σ confidence level. We again find that the power-law slope γ is steeper across larger mass ratios and then flattens toward q = 0.1. Statistically, we measure ${\gamma }_{\mathrm{large}q}$ = −2.2 ± 0.6 and ${\gamma }_{\mathrm{small}q}$ = −1.1 ± 0.5 (Table 8).

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For a sample of ${{ \mathcal N }}_{\mathrm{prim}}=115$ B-type stars in the Sco OB2 association (d ≈ 145 pc), Shatsky & Tokovinin (2002) utilized near-infrared adaptive optics to search for visual companions across ρ = 03–64, i.e., a = 45–900 au. Their sample of B0–B9 stars is weighted toward lower primary masses according to the IMF, and so the average primary mass is $\langle {M}_{1}\rangle$ = 5 ± 2 M_⊙. Shatsky & Tokovinin (2002) measured the mass ratios q from the observed infrared colors and brightness contrasts. Adaptive optics are sensitive to q = 0.1 companions for angular separations ρ > 05, while incompleteness and observational biases become important beyond ρ > 4'' (Shatsky & Tokovinin 2002; see their Figure 8). We therefore select the ${{ \mathcal N }}_{\mathrm{comp}}=18$ companions with angular separations ρ = 05–40 and measured mass ratios q > 0.1. These 18 companions represent a relatively complete subsample across separations a = 70–600 au, i.e., 4.9 < log P (days) < 6.3.

We display in Figure 20 the cumulative distribution of mass ratios for the 18 companions. The excess twin fraction is consistent with zero, i.e., the 1σ upper limit is ${{ \mathcal F }}_{\mathrm{twin}}$ < 0.05 (Table 9). If we fit the power-law slope of the mass-ratio distribution across the large range 0.1 < q < 1.0, we measure γ = −0.8 ± 0.3. Our measurement is slightly steeper than the slope γ = −0.5 reported by Shatsky & Tokovinin (2002). This is primarily because Shatsky & Tokovinin (2002) fit the mass-ratio distribution across the entire interval 0.0 < q < 1.0. Like the IMF, the mass-ratio probability distribution p_q cannot be described by a single power law across all mass ratios 0 < q < 1. As we have parameterized in the present study (see Section 2), the mass-ratio distribution is more accurately described by a broken power-law distribution down to some threshold q ≈ 0.1, below which the observations are insensitive and/or incomplete. Allowing for a break in the mass-ratio distribution at q = 0.3, we measure ${\gamma }_{\mathrm{small}q}$ = −0.7 ± 0.5 across q = 0.1–0.3 and ${\gamma }_{\mathrm{large}q}$ = −1.0 ± 0.5 across q = 0.3–1.0 (Table 9).

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Of the 18 companions, ${{ \mathcal N }}_{\mathrm{large}q}=12$ have large mass ratios q > 0.3. The subgroups within the Sco-OB2 association have a range of ages τ = 4–15 Myr (de Zeeuw et al. 1999; Shatsky & Tokovinin 2002), so we account for the correction factor ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.2 ± 0.1 due to binary evolution. The intrinsic frequency of companions with q > 0.3 per decade of orbital period is ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$/Δlog P = ($12\pm \sqrt{12}$)(1.2 ±0.1)/115/(6.3 − 4.9) = 0.09 ± 0.03 for late B MS stars and wide orbital periods (Table 9).

For a sample of ${{ \mathcal N }}_{\mathrm{prim}}=109$ B2–B5 primaries ($\langle {M}_{1}\rangle$ = 7 ± 2 M_⊙), Abt et al. (1990) identified 49 VB companions. These 49 VBs exhibit CPM, have orbital solutions, and/or have sufficiently small angular separations ρ ≲ 5'' to ensure that the systems are physically associated. Their sample is relatively complete down to ΔV < 7.0 mag (q ≳ 0.13) for angular separations >065 (see their Table 5). We therefore select the 10 VBs from Abt et al. (1990) with listed brightness contrasts ΔV ≤ 7.0 mag and angular separations ρ = 065–65, i.e., 5.4 ≲ log P (days) ≲ 6.9. Beyond ρ > 65, the binaries may be spurious associations or dynamically unstable, even if they exhibit CPM (Abt & Corbally 2000; Shatsky & Tokovinin 2002; Duchêne & Kraus 2013).

To estimate the mass ratios q of these 10 VBs from the observed brightness contrasts ΔV, we utilize the empirical relations between spectral type, bolometric corrections, absolute magnitudes, and masses provided in Pecaut & Mamajek (2013, and references therein). We fit the following relation:

applicable only for B2–B5 MS primaries with M₁ ≈ 5–9 M_⊙. The break in the equation above models how the slope of the mass–luminosity relation for K–A type secondaries with M₂ ≈ 0.5–2 M_⊙ is steeper than the slope of the mass–luminosity relation for late/mid-B-type secondaries with M₂ ≈ 3–9 M_⊙ (see also Section 5). Depending on the precise age and spectral type of the primary, the uncertainty in the mass ratio according to Equation (4) is δq ≈ 0.04.

The 10 VBs we selected from Abt et al. (1990) have brightness contrasts ΔV = 1.3–7.0 mag, which correspond to mass ratios q = 0.14–0.63 according to Equation (4). In Figure 21, we display the cumulative distribution of mass ratios for these 10 VBs. The excess twin fraction is consistent with zero, i.e., we measure ${{ \mathcal F }}_{\mathrm{twin}}\lt 0.12$ at the 1σ confidence level (Table 9). Assuming that the sample is complete down to q = 0.13, we measure γ = −2.0 ± 0.4 across the entire interval q = 0.13–1.00. This indicates that wide binaries with mid-B primaries have mass ratios q ≳ 0.13 that are consistent with random pairings drawn from a Salpeter IMF (γ = −2.35). Abt et al. (1990) examined all visual and CPM binaries across 5 ≲ log P (days) ≲ 9 in their sample and also concluded that the mass ratios are consistent with random pairings from a Salpeter IMF. The widest systems, however, may be contaminated by faint spurious associations. Nevertheless, we have shown that γ ≈ −2.0 ± 0.4 applies for relatively close VBs (5.4 < log P < 6.9) that are most probably gravitationally bound and dynamically stable.

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Allowing for a break in the mass-ratio distribution at q = 0.3, we measure ${\gamma }_{\mathrm{small}q}$ ≈ −1.3 and ${\gamma }_{\mathrm{large}q}$ ≈ −2.7. Given the small sample size, the uncertainties in these two statistics are quite large and significantly correlated. It is physically unlikely that either of these slopes is steeper than the value γ = −2.35 implied by random pairings from a Salpeter IMF. If we impose the additional constraint that γ > −2.5, then we measure ${\gamma }_{\mathrm{small}q}$ = −1.7 ± 0.6 and ${\gamma }_{\mathrm{large}q}$ = −2.2 ± 0.6 (Table 9).

Of the 10 VBs, ${{ \mathcal N }}_{\mathrm{large}q}=4$ have mass ratios q > 0.3. As done in Section 3, we account for the correction factor ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.2 ± 0.1 due to binary evolution within the volume-limited sample of Abt et al. (1990). For mid-B primaries and wide separations, the intrinsic frequency of companions with q > 0.3 per decade of orbital period is ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$/Δlog P = ($4\pm \sqrt{4}$)(1.2 ± 0.1)/109/(6.9 − 5.4) = 0.03 ± 0.02 (Table 9).

The binary statistics for wide companions to mid-B primaries (${\gamma }_{\mathrm{large}q}$ = −2.2 ± 0.6, ${{ \mathcal F }}_{\mathrm{twin}}$ < 0.12, ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.03 ± 0.02) nearly matches the statistics we measured in Section 6.1 for wide companions to Cepheids (${\gamma }_{\mathrm{large}q}$ = −2.1 ± 0.5, ${{ \mathcal F }}_{\mathrm{twin}}$ < 0.07, ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.04 ± 0.02). Recall that Cepheids evolve from mid-B primaries with $\langle {M}_{1}\rangle$ = 6 ± 2 M_⊙. The similarity in the statistical parameters validates our ability to measure the properties of mid-B MS binaries with intermediate and wide separations based on observations of Cepheids.

Duchêne et al. (2001) utilized adaptive optics to search for visual companions to massive stars in the young open cluster NGC 6611 (the Eagle Nebula; τ ≈ 2–6 Myr; d ≈ 2.1 kpc). More recently, Peter et al. (2012) performed lucking imaging of intermediate-mass and massive stars within several subgroups of the Cep OB2/3 association (τ ≈ 3–10 Myr; d ≈ 0.8 kpc). A significant fraction of both samples contain B0–B3 primaries. In addition, both surveys are sensitive to low-mass companions across similar ranges of projected separation, and both studies determined the binary mass ratios from the observed brightness contrasts. To more accurately measure the binary statistics, we combine the VB data on the B0–B3 primaries from the Duchêne et al. (2001) and Peter et al. (2012) surveys. In the following, we designate the two samples with subscripts A and B, respectively.

Duchêne et al. (2001) observed ${{ \mathcal N }}_{\mathrm{prim},{\rm{A}}}=30$ B0–B3 stars in NGC 6611, 10 of which were resolved with adaptive optics to have visually resolved companions (see their Tables 1, 3, and 4). Three of these VBs have large near-infrared brightness contrasts ΔK > 3.5 mag (λ ≈ 2.2 μm), indicating very extreme mass ratios q < 0.1. Meanwhile, two other companions have large dust reddenings compared to their primaries, suggesting that they are are background giants. We select the ${{ \mathcal N }}_{\mathrm{comp},{\rm{A}}}=5$ companions with projected separations 05–15, B0–B3 primaries, and estimated mass ratios q > 0.1 (Walker designations 25, 188, 254, 267, and 311). For Walker 311, we adopt the average mass ratio q = 0.63 of the listed range q = 0.47–0.78 in Table 3 of Duchêne et al. (2001). Of the five companions, ${{ \mathcal N }}_{\mathrm{large}q,{\rm{A}}}=2$ have large mass ratios q > 0.3.

From the Peter et al. (2012) survey, we select the ${{ \mathcal N }}_{\mathrm{prim},{\rm{B}}}=56$ primaries with spectral types B0–B3 and luminosity classes III–V (see their Table A.1). Of these primaries, ${{ \mathcal N }}_{\mathrm{comp},{\rm{B}}}=8$ have projected separations 045–15 and brightness contrasts Δz' < 4.3 mag (λ ≈ 900 nm). We adopt the mass ratios presented in Figure 10 of Peter et al. (2012) for our nine selected VBs with B0–B2 primaries. For the two systems with B3 primaries, we utilize the relation between Δz' and q calculated in Peter et al. (2012). Specifically, we estimate q = 0.10 for BD+61 2218 (Δz' = 4.3 mag) and q = 0.12 for HD 239649 (Δz' = 4.2 mag). Of the eight companions, ${{ \mathcal N }}_{\mathrm{large}q,{\rm{B}}}=3$ have large mass ratios q > 0.3.

In Figure 22, we display the cumulative distribution of mass ratios for the ${{ \mathcal N }}_{\mathrm{comp}}$ = ${{ \mathcal N }}_{\mathrm{comp},{\rm{A}}}$ + ${{ \mathcal N }}_{\mathrm{comp},{\rm{B}}}$ = 5 + 8 = 13 binaries we selected from Duchêne et al. (2001) and Peter et al. (2012). We measure an excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}\lt 0.10$ that is consistent with zero (Table 9). By fitting a single power-law slope across the entire interval 0.1 < q < 1.0, we determine γ = −1.6 ± 0.4. This is moderately discrepant with random pairings drawn from a Salpeter IMF (γ = −2.35) at the 1.9σ significance level. We note that Duchêne et al. (2001) conclude that sample data alone, which is smaller than our combined sample, are consistent with γ = −2.35. Meanwhile, Peter et al. (2012) report γ ≈ −1.0, which is inconsistent with random pairings drawn from a Salpeter IMF. However, Peter et al. (2012) fit this exponent down to extreme mass ratios q ≲ 0.05 where the secondary masses M₂ ≲ 0.5 M_⊙ are no longer expected to follow a Salpeter-like slope. In any case, our value of γ = −1.6 ± 0.4 after combining the two samples is between the two independent measurements. Allowing for a break at q = 0.3, we measure ${\gamma }_{\mathrm{small}q}$ = −1.3 ± 0.5 and ${\gamma }_{\mathrm{large}q}$ = −1.9 ± 0.5 (Table 9).

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Given the distance d ≈ 2.1 kpc to NGC 6611, the ${{ \mathcal N }}_{\mathrm{large}q,{\rm{A}}}=2$ companions with ρ = 05–15 and q > 0.3 from the Duchêne et al. (2001) sample have separations a ≈ 1000–3000 au, i.e., 6.4 < log P (days) < 7.1. Meanwhile, the ${{ \mathcal N }}_{\mathrm{large}q,{\rm{B}}}=3$ companions with ρ = 0.45–15 and q > 0.3 from the Peter et al. (2012) sample have separations a ≈ 360–1200 au, i.e., 5.7 < log P < 6.5, given the distance d ≈ 0.8 kpc to the Cep OB2/3 association. The open cluster NGC 6611 is sufficiently young that the correction factor ${{ \mathcal C }}_{\mathrm{evol},{\rm{A}}}$ = 1.0 ± 0.1 due to binary evolution is negligible. For the slightly older Cep OB2/3 association, we adopt ${{ \mathcal C }}_{\mathrm{evol},{\rm{B}}}$ = 1.1 ± 0.1. Combining the statistics, we measure ${f}_{\mathrm{log}P;q\gt 0.3}$ = (${{ \mathcal N }}_{\mathrm{large}q,{\rm{A}}}{{ \mathcal C }}_{\mathrm{evol},{\rm{A}}}$/Δlog ${P}_{{\rm{A}}}$ + ${{ \mathcal N }}_{\mathrm{large}q,{\rm{B}}}{{ \mathcal C }}_{\mathrm{evol},{\rm{B}}}$/Δlog ${P}_{{\rm{B}}}$)/ (${{ \mathcal N }}_{\mathrm{prim},{\rm{A}}}$ + ${{ \mathcal N }}_{\mathrm{prim},{\rm{B}}}$) = 0.08 ± 0.04 for early B primaries and log P (days) = 6.4 ± 0.7 (Table 9).

In addition to LBI and SAM (see Section 5.2), Sana et al. (2014) utilized near-infrared adaptive optics to search for visual companions to the O-type stars in their sample. Aldoretta et al. (2015) recently performed high-contrast imaging in the optical (Δm ≲ 5.0 mag; λ ≈ 583 nm) of O-type stars with the Fine Guidance Sensor aboard the Hubble Space Telescope. Mason et al. (2009) and Caballero-Nieves et al. (2014) have discovered visual companions to other O-type stars, but these two surveys are limited to small optical brightness contrasts ΔV ≲ 2–3 mag and therefore large mass ratios q ≳ 0.4–0.5. In the following, we analyze the VB statistics of O-type primaries based on the Sana et al. (2014) and Aldoretta et al. (2015) samples. These two studies cover slightly different angular separations, incorporate different bandpasses, and report only the observed brightness contrasts. We therefore treat these two surveys independently and model our own values of the mass ratios according to the listed brightness contrasts.

For the Sana et al. (2014) survey, we follow the same procedure as in Section 5.2. We select the ${{ \mathcal N }}_{\mathrm{prim}}=106$ O-type primaries with luminosity classes II.5–V that they observed with near-infrared adaptive optics. This includes the 76 such systems in the Sana et al. (2014) main target list (see their Figure 1) and the 30 additional objects in their supplementary target lists. As shown below and in Figure 23, VBs with brightness contrasts ${\rm{\Delta }}{K}_{{\rm{s}}}$ < 5.8 mag (λ ≈ 2.2 μm) are complete down to q = 0.1. The Sana et al. (2014) adaptive optics survey is sensitive to ${\rm{\Delta }}{K}_{{\rm{s}}}$ ≈ 5.8 mag beyond ρ > 06 (see their Figure 7), while confusion with background/foreground stars becomes non-negligible beyond ρ > 20 (see their Figure 8). From our selected sample of 106 O-type MS primaries, Sana et al. (2014) identified 18 companions with angular separations ρ = 06–20 and brightness contrasts ${\rm{\Delta }}{K}_{{\rm{s}}}$ < 5.8 mag. Based on the observed surrounding stellar densities, Sana et al. (2014) estimate extremely small probabilities ${{ \mathcal P }}_{\mathrm{spur}}$ ≤ 2% that each of these 18 VBs are spurious associations. The total probability that any of the 18 VBs are optical doubles is ∼5%. Our subsample of 18 VBs is therefore relatively unbiased and complete down to q = 0.1.

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Incorporating the same stellar isochrones, relations, and methods adopted in Section 5.2, we calculate the brightness contrasts ${\rm{\Delta }}{K}_{{\rm{s}}}$ as a function of mass ratio q. In Figure 23, we compare q and ${\rm{\Delta }}{K}_{{\rm{s}}}$ for different combinations of primary spectral type and luminosity class. A sample of binaries with O-type primaries and luminosity classes III–V is complete down to q = 0.1 if the observations are sensitive to ${\rm{\Delta }}{K}_{{\rm{s}}}$ ≈ 5.8 mag. The ${\rm{\Delta }}{K}_{{\rm{s}}}$ (λ ≈ 2.2 μm) relations shown in Figure 23 are nearly indistinguishable from the near-infrared ΔH (λ ≈ 1.6 μm) curves presented in Figure 13. We estimate ${\rm{\Delta }}{K}_{{\rm{s}}}$ = 0.96ΔH for a broad range of binary masses and ages.

Sana et al. (2014) measured both brightness contrasts ${\rm{\Delta }}{K}_{{\rm{s}}}$ and ΔH for 14 of our 18 selected VBs. For these systems, we first use the relation ${\rm{\Delta }}{K}_{{\rm{s}}}$ = 0.96ΔH to convert the near-infrared brightness contrasts, and then we calculate a weighted average and uncertainty for ${\rm{\Delta }}{K}_{{\rm{s}}}$. For the four remaining VBs, we adopt the values of and uncertainties in ${\rm{\Delta }}{K}_{{\rm{s}}}$ reported by Sana et al. (2014). We calculate the mass ratios q of the 18 VBs based on the relations presented in Figure 23. As done in Section 5.2, we propagate the measurement uncertainty in ${\rm{\Delta }}{K}_{{\rm{s}}}$, as well as uncertainties of 0.5 in both the primary's spectral subtype and luminosity class.

In Figure 23, we compare the estimated mass ratios q to the observed brightness contrasts ${\rm{\Delta }}{K}_{{\rm{s}}}$ for our 18 VBs. The majority of the VBs have q < 0.35. One system, HD 93161, has a mass ratio q ≈ 0.96 near unity according to the observed brightness contrast ${\rm{\Delta }}{K}_{{\rm{s}}}$ = 0.07 mag. The fainter component HD 93161 B is a O6.5IV–V star, while the brighter component HD 93161 A has a spectral classification of O7.5–8V (Nazé et al. 2005; Sana et al. 2014). The fainter component HD 93161 B should therefore be more massive and luminous. This apparent discrepancy is because component A is itself an SB2 with an orbital period of P ≈ 8.6 days and estimated masses of ${M}_{\mathrm{Aa}}$ = 22.2 ± 0.6 M_⊙ and ${M}_{\mathrm{Ab}}$ = 17.0 ± 0.4 M_⊙ (Nazé et al. 2005). This short-period SB2 with q = ${M}_{\mathrm{Ab}}$/${M}_{\mathrm{Aa}}$ = 0.77 contributes to the statistics of close companions to early-type stars in Section 3. Using the O-type stellar relations provided in Martins et al. (2005), we estimate HD 93161 B to have a mass of M_B = 30 ± 3 M_⊙ according to its observed spectral classification of O6.5IV–V. For the VB HD 93161, we adopt a mass ratio of q ≡${M}_{\mathrm{comp}}$/M₁ = ${M}_{\mathrm{Aa}}$/${M}_{{\rm{B}}}$ = 22.2/30 = 0.74. For the 17 remaining VBs with ${\rm{\Delta }}{K}_{{\rm{s}}}$ < 1.1 mag, the presence of an additional companion to the O-type star will not significantly affect the mass ratio inferred from the brightness contrast. We adopt the mass ratios presented in Figure 23 for the 17 other VBs.

Of our 18 selected VBs, ${{ \mathcal N }}_{\mathrm{small}q}=12$ have q = 0.1–0.3 and ${{ \mathcal N }}_{\mathrm{large}q}=5$ have q = 0.3–1.0. The one remaining VB has a large brightness contrast ${\rm{\Delta }}{K}_{{\rm{s}}}$ = 5.7 mag near our selection limit and a mass ratio q ≈ 0.09 just below our statistical cutoff. In Figure 24, we display the cumulative distribution of mass ratios for the 17 VBs with q > 0.1. We measure an excess twin fraction of ${{ \mathcal F }}_{\mathrm{twin}}$ < 0.10 (Table 9). By fitting a single power-law mass-ratio distribution across q = 0.1–1.0, we determine γ = −1.8 ± 0.3 (see Figure 24). Allowing for a break in the mass-ratio distribution at q = 0.3, the slopes are ${\gamma }_{\mathrm{small}q}$ = −1.5 ± 0.5 and ${\gamma }_{\mathrm{large}q}$ = −2.1 ± 0.5 (Table 9). As found for mid-B (Section 7.3) and early B (Section 7.4) VB samples, the component ${\gamma }_{\mathrm{large}q}$ across large mass ratios q = 0.3–1.0 is consistent with random pairings drawn from the Salpeter IMF, while the slope ${\gamma }_{\mathrm{small}q}$ across small mass ratios q = 0.1–0.3 is slightly flatter.

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As in Section 5.2, we adopt an average distance $\langle d\rangle$ = 1.5 kpc for the O-type binaries in the Sana et al. (2014) survey. The angular separations ρ = 06–20 correspond to separations a = 900–3000 au, i.e., 6.2 < log P (days) < 7.0. We adopt the same correction factor ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.1 ± 0.1 for O-type stars with luminosity classes II.5–V. The intrinsic companion frequency is ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$/Δlog P = ($5\,\pm \sqrt{5}$)(1.1 ± 0.1)/106/(7.0 − 6.2) = 0.07 ± 0.03 (Table 9).

We next analyze the statistics of VBs in the Aldoretta et al. (2015) catalog. They utilized the high resolution of Hubble's Fine Guidance Sensor with the F583W filter (λ ≈ 460–700 nm) to identify 74 close companions to 224 O and early B stars. Their sample is complete down to Δm = 4.0 mag beyond >01, while incompleteness and confusion with background/foreground objects become important at large separations >10. We do not consider their supplementary list of 81 additional targets, as these systems were observed with lower sensitivity Δm < 3.0 mag (Nelan et al. 2004; Caballero-Nieves et al. 2014). From the Aldoretta et al. (2015) main sample, we select the ${{ \mathcal N }}_{\mathrm{prim}}=128$ galactic O-type stars with luminosity classes II.5–V. From these primaries, Aldoretta et al. (2015) identified ${{ \mathcal N }}_{\mathrm{comp}}=21$ VB companions with projected separations 01–10 and optical brightness contrasts Δm < 4.0 mag.

We implement the same technique as above to estimate the mass ratios q. To model the VB brightness contrasts Δm in the broadband F583W filter (λ ≈ 460–700 nm), we average the ΔV and ${\rm{\Delta }}{R}_{{\rm{C}}}$ relations for q. In Figure 25, we display Δm as a function of q for the same primary masses and luminosity classes as in Figures 13 and 23. In the optical, the sensitivity limit of Δm < 4.0 mag is complete down to q > 0.3. For the 21 VBs we selected from Aldoretta et al. (2015), we measure the mass ratios q from Δm and the spectral types and luminosity classes of the primaries. We show our results in Figure 25, where we find that ${{ \mathcal N }}_{\mathrm{large}q}=18$ VBs have mass ratios q > 0.3. The three remaining VBs have large brightness contrasts Δm = 3.6–4.0 mag near the sensitivity limit and mass ratios q = 0.25–0.30 just below our statistical cutoff.

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Given the average distance $\langle d\rangle$ ≈ 2.0 kpc to the ${{ \mathcal N }}_{\mathrm{large}q}=18$ VBs we selected from Aldoretta et al. (2015), the projected angular separations ρ = 01–10 correspond to orbital periods 5.2 < log P (days) < 6.7. The selected VBs have primary luminosity classes II.5–V, and so the correction factor ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.1 ± 0.1 due to binary evolution is relatively small. The intrinsic frequency of companions with q > 0.3 per decade of orbital period is ${f}_{\mathrm{log}P;q\gt 0.3}$ = ${{ \mathcal N }}_{\mathrm{large}q}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$/Δlog P = ($18\pm \sqrt{18}$)(1.1 ± 0.1)/128/(6.7−5.2) = 0.10 ± 0.03 for O-type stars and wide orbital periods (Table 9).

In Figure 26, we display the cumulative mass-ratio distribution for the ${{ \mathcal N }}_{\mathrm{large}q}=18$ VBs from the Aldoretta et al. (2015) main sample with O-type MS primaries, angular separations ρ = 01–10, and mass ratios q > 0.3. We measure a negligible excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}\lt 0.03$ and a power-law component ${\gamma }_{\mathrm{large}q}$ = −1.8 ± 0.4 that is weighted toward small mass ratios q = 0.3–0.5 (Table 9). Our measurement of ${\gamma }_{\mathrm{large}q}$ = −1.8 ± 0.4 based on the Aldoretta et al. (2015) O-type VB sample is consistent with our measurements for VB companions to Cepheids (${\gamma }_{\mathrm{large}q}$ = −2.1 ± 0.5), mid-B stars (${\gamma }_{\mathrm{large}q}$ = −2.2 ±0.6), early B stars (${\gamma }_{\mathrm{large}q}$ = −1.9 ± 0.5), and O-type stars in the Sana et al. (2014) survey (${\gamma }_{\mathrm{large}q}$ = −2.1 ± 0.5). The weighted average and uncertainty of these five values is ${\gamma }_{\mathrm{large}q}$ = −2.0 ± 0.2. For a broad range of primary masses M₁ ≈ 6–40 M_⊙ and orbital periods 5 ≲ log P (days) ≲ 7, the power-law slope of the binary mass-ratio distribution across q = 0.3–1.0 nearly matches the prediction from random pairings drawn from a Salpeter IMF (${\gamma }_{\mathrm{large}q}$ = −2.35).

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As discussed in Section 2, catalogs of VBs with reliable orbital solutions are biased against systems with large eccentricities (Harrington & Miranian 1977; Tokovinin & Kiyaeva 2016). We can nonetheless use VB orbit catalogs to measure a lower limit to the power-law slope η of the eccentricity distribution. We also fit η across eccentricities 0.0 < e < 0.8 that are not as severely affected by this selection effect. Malkov et al. (2012) compiled a set of VBs with known distances, measured primary spectral types, and orbits of high quality from both the Catalog of Orbits and Ephemerides of Visual Double Stars (Docobo et al. 2001) and the Sixth Catalog of Visual Binary Stars (Hartkopf et al. 2001). We consider the early-type VBs in their sample with measured orbital periods P = 10–100 yr. The majority of early-type binaries with shorter orbital periods P < 10 yr remain unresolved and are unsuitable for orbit determinations. VBs with longer periods P > 100 yr generally have incomplete orbits, and so the selection bias against eccentric systems becomes too important.

In the top panel of Figure 27, we show the cumulative distribution of eccentricities 0 < e < 1 for the early-type VBs with P = 10–100 yr from the Malkov et al. (2012) catalog. We compare the 18 VBs with primary spectral types O5–B5 (blue) to the 101 VBs with spectral types B6–A5 (red). The O/early B subsample is weighted toward larger eccentricities, where we measure η = 0.4 ± 0.3. For the late B/early A subsample, we find η = 0.0 ± 0.2. Abt (2005) also reports a uniform eccentricity distribution (η = 0.0) based on a subsample of VBs with orbital solutions, P > 1000 days, and B0–F0 primary spectral types. In all these cases, however, the estimates for η are biased toward smaller values due to the observational selection effects. Our measurements of η > 0.4 and η > 0.0 for the O/early B and late B/early A subsamples, respectively, are lower limits.

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To account for the selection bias against VBs with e > 0.8, we now fit the power-law slope η across 0.0 < e < 0.8. In the bottom panel of Figure 27, we display the cumulative distribution of eccentricities 0.0 < e < 0.8 for our same two subsamples. The power-law slope increases to η = 0.3 ± 0.3 for the late B/early A VBs. For the O/early B subsample, we measure η = 0.8 ± 0.3, which is consistent with a thermal eccentricity distribution (η = 1). The systematic uncertainties in η due to the selection biases are comparable to the measurement uncertainties. We therefore adopt η = 0.8 ± 0.4 and η = 0.3 ± 0.4 for the O/early B and late B/early A VB subsamples, respectively (Table 10).

To extend the baseline toward smaller primary masses, we now investigate the companion properties to solar-type MS primaries (M₁ = 1.00 ± 0.25 M_⊙). The most complete solar-type MS binary sample derives from Raghavan et al. (2010), who updated and extended the sample of Duquennoy & Mayor (1991) (see Section 8.6 for comparison of these two samples of solar-type MS binaries). Raghavan et al. (2010) combined various observational techniques to search for companions around 454 F6–K3 type stars located within 25 pc. The companion properties to solar-type primaries may differ in young star-forming environments (Duchêne et al. 2007; Connelley et al. 2008b; Kraus et al. 2011; Tobin et al. 2016), in dense open clusters (Patience et al. 2002; Köhler et al. 2006; Geller & Mathieu 2012; King et al. 2012), or at extremely low metallicities (Abt 2008; Gao et al. 2014; Hettinger et al. 2015). In Section 10, we compare the corrected solar-type binary statistics of the field MS population to those of younger MS and pre-MS populations in open clusters and stellar associations. For now and for the purposes of binary population synthesis studies, we are mostly interested in the overall companion statistics of typical primaries with average ages. Most solar-type stars are near solar metallicity, in the galactic field, and are several gigayears old. The volume-limited sample of solar-type MS primaries in Raghavan et al. (2010) is therefore most representative of the majority of solar-type MS stars.

We display in Figure 28 the 168 confirmed companions from Raghavan et al. (2010) with measured orbital periods 0.0 < log P (days) < 8.0 and mass ratios 0.1 < q < 1.0. We utilize the same methods as in Raghavan et al. (2010) to estimate the orbital periods P from projected separations and the stellar masses from spectral types. Our Figure 28 is similar to Figures 11 and 17 in Raghavan et al. (2010). However, in the cases of triples and higher-order multiples, we always define the period and mass ratio q ≡ ${M}_{\mathrm{comp}}$/M₁ with respect to the solar-type primary (see Section 2), which differs slightly from the definitions adopted in Raghavan et al. (2010).

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For example, first consider a triple in an (Aa, Ab)–B hierarchical configuration. An ${M}_{\mathrm{Aa}}$ = M₁ = 1.0 M_⊙ primary and ${M}_{\mathrm{Ab}}$ = 0.5 M_⊙ companion are in a short-period orbit of P = 100 days. A long-period tertiary component with ${M}_{{\rm{B}}}$ = 0.4 M_⊙ orbits the inner binary with a period of P = 10⁵ days. This system would contribute two data points in Figure 28: one with q = ${M}_{\mathrm{Ab}}$/${M}_{\mathrm{Aa}}$ = 0.5 and log P = 2.0, and one with q = ${M}_{{\rm{B}}}$/${M}_{\mathrm{Aa}}$ = 0.4 and log P = 5.0. We do not define the mass ratio of the wide system to be q = ${M}_{{\rm{B}}}$/(${M}_{\mathrm{Aa}}$ + ${M}_{\mathrm{Ab}}$), as done in Raghavan et al. (2010).

Next, consider a triple in an A–(Ba, Bb) hierarchical configuration. A solar-type ${M}_{{\rm{A}}}$ = M₁ = 1.0 M_⊙ primary is in a long-period P = 10⁵ day orbit around a close, low-mass binary with ${M}_{\mathrm{Ba}}$ = 0.5 M_⊙, ${M}_{\mathrm{Ba}}$ = 0.4 M_⊙, and P = 100 days. In this situation, only the wide system with q = ${M}_{\mathrm{Ba}}$/${M}_{{\rm{A}}}$ = 0.5 and log P = 5.0 would contribute to our Figure 28. We do not consider the low-mass inner binary with log P = 2.0 and ${M}_{\mathrm{Ba}}$/${M}_{\mathrm{Bb}}$ = 0.8, as done in Raghavan et al. (2010), because neither component Ba nor Bb is a solar-type star. Only if component Ba itself has an F6–K3 spectral type do we include the close (Ba, Bb) pair in our sample. Nearly half of the twins with q > 0.95 in Figures 11 and 17 of Raghavan et al. (2010) are not solar-type twins. They instead contain late K or M dwarf equal-mass binaries in a long-period orbit with a solar-type primary in an A–(Ba, Bb) hierarchical configuration. Our Figure 28 therefore does not contain as many twin components as displayed in Figures 11 and 17 of Raghavan et al. (2010).

The Raghavan et al. (2010) sample is relatively complete except for two regions of the parameter space of P versus q. First, the survey is not sensitive to detecting companions with log P ≈ 5.9–6.7 and q ≈ 0.1–0.2 (green region in our Figure 28). As shown in Figure 11 of Raghavan et al. (2010), companions that occupy this portion in the parameter space are missed by both adaptive optic and CPM techniques. Considering the density of systems in the immediately surrounding regions where the observations are relatively complete, we estimate that $\approx 4$ additional systems occupy this gap in the parameter space (four green systems in Figure 28).

The second region of incompleteness occurs at q ≲ 0.5 and log P ≲ 4.5 (blue region in our Figure 28). The optical brightness contrast between binary components is an even steeper function of mass ratio for F–M type stars. Solar-type SB2s with sufficiently luminous secondaries are observed only if q ≳ 0.40–0.55 (brightness contrasts ΔV < 5.0 mag, depending on the orbital period). Spectroscopic binaries with lower-mass companions will generally appear as SB1s and therefore not have mass ratios that can be readily measured. Of the four spectroscopic binaries with log P < 3.0 and q < 0.4 shown in Figure 28, only one is an SB2 with P ≈ 4 days and a mass ratio q ≈ 0.38 close to the detection limit. The other three systems are SB1s in a hierarchical triple where the tertiary itself has an orbital solution. The total mass of the inner SB1 is measured dynamically, and so the mass of the companion in the SB1 can be estimated. The few observed systems with 3.0 < log P < 4.5 and q < 0.4 are sufficiently nearby and have favorable orientations for the companions to be resolved with adaptive optics. In general, companions below the blue line in our Figure 28 are unresolved, both spatially and spectrally.

We estimate the number of missing systems in the blue region of Figure 28 as follows. Raghavan et al. (2010) identified 26 confirmed and candidate binaries that do not have measurable mass ratios, e.g., SB1s, radial velocity variables, and companions implied through proper-motion acceleration of the primary. A few of the seven radial velocity variables may contain substellar companions with q < 0.1. The SB1s and companions identified through proper-motion acceleration, on the other hand, must have q ≳ 0.1 to produce the measured signal. There is also a small gap at log P = 3.5–4.5 and q = 0.10–0.25 where neither spectroscopic radial velocity surveys nor adaptive optic surveys are complete (see Figure 11 of Raghavan et al. 2010). Considering the density of low-mass companions at slightly longer orbital periods, we estimate ≈3–5 additional systems that escaped detection in this region. Finally, only spectroscopic radial velocity surveys are sensitive to closely orbiting low-mass companions, but only ≈80% of the sample of 454 primaries were searched for such radial velocity variations. We estimate an additional ≈20%, or ≈4–6 SB1s, to be present around primaries that were not surveyed for spectroscopic variability. In total, we estimate $\approx 35$ additional unresolved companions with q > 0.1 and log P ≲ 4.5.

Like early-type binaries, a certain fraction of unresolved solar-type SB1s likely contain compact remnant companions. For solar-type binaries, the majority of unresolved compact remnant companions are WDs instead of neutron stars or black holes (Hurley et al. 2002; Belczynski et al. 2008). Fortunately, we can estimate the frequency of such close Sirius-like systems using three different methods that we discuss below.

8.3.1. UV Excess from WD Companions

8.3.2. Monte Carlo Population Synthesis

8.3.3. Barium Stars

The three independent methods described above result in values ${{ \mathcal F }}_{\mathrm{solar}+\mathrm{WD},\mathrm{log}P\lt 4.5}$ = 2.5% ± 1.5%, 4.4% ± 1.6%, and 3.1% ± 1.7% that are consistent with each other. We adopt a weighted average of ${{ \mathcal F }}_{\mathrm{solar}+\mathrm{WD},\mathrm{log}P\lt 4.5}$ = 3.4% ± 1.0%. In the Raghavan et al. (2010) sample of 454 solar-type MS stars, there should be 454  × (0.034 ± 0.010) = 15 ± 5 WD companions with log P < 4.5. Only one of these suspected systems, HD 13445, was barely resolved to have a WD companion with log P = 4.4. The remaining 14 ± 5 solar-type+WD binaries with log P < 4.5 remain unresolved, but most likely appear as SB1s and/or systems that exhibit proper-motion acceleration.

Raghavan et al. (2010) identified 11 companions with measurable mass ratios 0.1 < q < 0.5 and orbital periods 0.0 < log P < 4.5. In Section 8.2, we estimated $\approx 35$ additional companions across the same period range based on the Raghavan et al. (2010) detection efficiencies and their statistics of SB1s, systems exhibiting proper-motion acceleration, etc. Of the $\approx 35$ additional companions, 14 ± 5 are most likely WDs. We conclude that (14 ± 5)/(11+35) = 30% ± 10% of nearby solar-type primaries that appear as SB1s and/or exhibit proper-motion acceleration actually contain WD companions.

The remaining 35−14 = 21 additional binaries contain M dwarf companions with q ≈ 0.1–0.5 and 0.0 ≲ log P ≲ 4.5. The blue region in Figure 28 is quite large, and so we distribute the estimated 21 additional M dwarf companions based on the nature of the systems. The SB1s have known orbital periods, generally 1.0 < log P (days) < 3.5. There are only two additional SB1s with 0 < log P < 1, and one or both of these systems may contain post-CE WD companions (see the Appendix ). The radial velocity variables and companions implied through proper-motion acceleration have systematically longer orbital periods 2.5 < log P < 4.7 (Raghavan et al. 2010). The ≈3–5 systems that escaped detection lie in the interval 3.5 < log P < 4.7 (Section 8.2). We therefore expect one, four, six, six, and four additional stellar companions in the logarithmic period intervals log P = 0–1, 1–2, 2–3, 3–4, and 4.0–4.7, respectively.

In terms of mass ratios q, we assume that the 21 additional systems are evenly distributed between q = 0.1 and the detection limit at q = 0.40–0.55 indicated by the blue line in Figure 28. Weighting the additional systems toward smaller or larger mass ratios in this interval does not significantly affect our statistical measurements. We display the 21 additional M dwarf companions as the blue diamonds in Figure 28.

In Section 8.3.2, we determined that ${{ \mathcal F }}_{\mathrm{solar}+\mathrm{WD}}$ = 11% ± 4% of solar-type stars in the local solar neighborhood have WD companions. In other words, the correction factor due to binary evolution is ${{ \mathcal C }}_{\mathrm{evol}}$ = 1.11 ± 0.04 for solar-type primaries (see Section 2). Of the 454 solar-type systems in the Raghavan et al. (2010) sample, we estimate that 454 × 0.11 ≈ 50 have WD companions. Some of the brighter WD companions have been identified (Raghavan et al. 2010), but the majority are probably too faint with binary brightness contrasts ΔV ≳ 10 mag that are too large to be detected. The true number of solar-type primaries in the Raghavan et al. (2010) sample, i.e., those that have always been the most massive components of their respective systems, is ${{ \mathcal N }}_{\mathrm{prim}}$ = 454 − 50 = 404. The WD contamination to the total sample (≈11%) is significantly smaller than the WD contamination to SB1s (≈30%). In conclusion, the corrected solar-type sample contains ${{ \mathcal N }}_{\mathrm{prim}}=404$ primaries and ${{ \mathcal N }}_{\mathrm{comp}}=193$ stellar companions with 0.1 < q < 1.0 and 0.0 < log P (days) < 8.0.

In Table 11, we provide the statistics based on the corrected population of solar-type binaries. We initially divide the sample into eight intervals of Δlog P = 1 across 0 < log P (days) < 8. For each logarithmic period interval, we list the number ${{ \mathcal N }}_{\mathrm{small}q}$ of companions with 0.1 < q < 0.3 and the number ${{ \mathcal N }}_{\mathrm{large}q}$ of companions with 0.3 < q < 1.0. The frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ of companions with q > 0.3 per decade of orbital period simply derives from dividing each value of ${{ \mathcal N }}_{\mathrm{large}q}$ by ${{ \mathcal N }}_{\mathrm{prim}}=404$.

To analyze the mass-ratio distribution, we divide the sample into four intervals of Δlog P = 2 across 0 < log P (days) < 8. For each of these four subsamples, we measure ${\gamma }_{\mathrm{small}q}$, ${\gamma }_{\mathrm{large}q}$, and ${{ \mathcal F }}_{\mathrm{twin}}$ (Table 11). In Figure 30, we display the cumulative distribution of mass ratios 0.1 < q < 1.0 for three logarithmic period intervals: 0 < log P < 2, 2 < log P < 6, and 6 <log P < 8.

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The most noticeable trend in the mass-ratio distribution of solar-type binaries is that the excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}$ significantly decreases with orbital period. At short periods 0 < log P < 2, we measure a large excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}$ = 0.29 ± 0.11. This is consistent with the conclusions of Halbwachs et al. (2003), who also find a substantial excess twin population among solar-type spectroscopic binaries. As can be seen in Figures 28 and 30, 5 of the 15 solar-type binaries with 0 < log P < 2 and q > 0.3 have q = 0.95–1.00, implying ${{ \mathcal F }}_{\mathrm{twin}}$ ≈ 5/15 ≈ 0.3. At intermediate orbital periods 2 < log P < 6, the excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}$ = 0.1–0.2 is smaller but definitively nonzero at a statistically significant level (see Figures 28 and 30). Only at the widest orbital separations, i.e., 6 < log P < 8, is the excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}$ < 0.05 negligible.

For solar-type binaries, the power-law components of the mass-ratio distribution exhibit only minor variations with orbital period. In fact, the slopes ${\gamma }_{\mathrm{small}q}$ ≈ 0.3 and ${\gamma }_{\mathrm{large}q}$ ≈ −0.5 are relatively constant and mildly consistent with a uniform distribution γ = 0.0 for 0 < log P < 6. Only at the longest orbital periods 6 < log P < 8 does the power-law slope ${\gamma }_{\mathrm{large}q}$ ≈ −1.1 become weighted toward smaller mass ratios. As can be seen in Figure 28, there is an enhanced concentration of companions with q = 0.1–0.5 across log P = 5.5–8.0. As found for early-type binaries, the excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}$ and power-law component ${\gamma }_{\mathrm{large}q}$ both decrease with increasing orbital period.

Using a Monte Carlo technique, we generate a binary-star population based on random pairings drawn from a Chabrier (2003) IMF (Salpeter power-law slope above 1 M_⊙ and lognormal distribution below 1 M_⊙). From the simulated population, we select the binaries with solar-type primaries M₁ = 0.75–1.25 M_⊙. We display the cumulative distribution of mass ratios for these solar-type binaries in Figure 30. By selecting a narrow interval of primary masses M₁ = 0.75–1.25 M_⊙, the distribution of mass ratios across q = 0.1–1.0 is nearly indistinguishable from the distribution of masses M = 0.1–1.0 M_⊙ according to the IMF (see also Tout 1991; Kouwenhoven et al. 2009). For all orbital periods 0 < log P < 8, the observed population of solar-type MS binaries is inconsistent with random pairings drawn from the IMF. Despite the enhancement of q = 0.1–0.5 companions across log P = 6.0–8.0, the widest solar-type MS binaries have a top-heavy mass-ratio distribution relative to the IMF random pairing prediction (Figure 30). Our results are consistent with Lépine & Bongiorno (2007), who also find that wide solar-type binaries identified via CPM do not match random pairings from the field population.

Knowing that their survey was partially incomplete, especially toward smaller mass ratios q < 0.3, Duquennoy & Mayor (1991) took great lengths to correct for incompleteness in their sample. The more recent analysis by Raghavan et al. (2010) nearly tripled the sample of solar-type primaries. Raghavan et al. (2010) also employed new observational techniques, e.g., adaptive optics and LBI, to discover stellar companions in portions of the $f(P,q)$ parameter space that eluded Duquennoy & Mayor (1991). The raw sample of Raghavan et al. (2010) is certainly larger and more complete than the raw sample of Duquennoy & Mayor (1991). However, believing that their sample was mostly complete, Raghavan et al. (2010) did not conduct as detailed an incompleteness study of their survey. Because of this, there has been tension reported in the literature as to whether the corrected sample of Duquennoy & Mayor (1991) or the raw sample of Raghavan et al. (2010) is a better representation of the true statistics of solar-type binaries (Marks & Kroupa 2011; Geller & Mathieu 2012). Because we conducted our own incompleteness study of Raghavan et al. (2010), we can finally address this concern.

After correcting for incompleteness down to q ≈ 0.1, Duquennoy & Mayor (1991) estimated there to be 91 companions with 0 < log P (days) < 8 in their sample of 164 solar-type primaries (see their Figure 7). We show their corrected frequency of companions per decade of orbital period per primary in Figure 31 (blue histogram). Based on their corrected population, the multiplicity frequency of solar-type primaries is 0.55 ± 0.06. However, Duquennoy & Mayor (1991) included WD companions in their corrected sample. In fact, in their effort to account for incompleteness, Duquennoy & Mayor (1991) artificially added two WDs per decade of orbital period across intermediate separations. We estimate there to be 13 WD companions total in the corrected sample of Duquennoy & Mayor (1991). After removing these 13 systems with WDs, there are 78 companions across periods log P = 0–8 to 151 solar-type primaries. We show in Figure 31 the frequency of companions per decade of orbital period after applying our correction of removing the WDs from the Duquennoy & Mayor (1991) sample (green histogram). By removing the WDs, the multiplicity frequency decreases slightly to 0.52 ± 0.06.

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Raghavan et al. (2010) identified 243 confirmed companions with log P = 0–8 in their sample of 454 solar-type primaries (see their Figure 13). This results in a multiplicity frequency of 0.54 ± 0.04, and we show the period distribution of the raw Raghavan et al. (2010) sample in Figure 31 (red histogram). After accounting for incompleteness and removing extreme mass-ratio companions q < 0.1, companions to M dwarf and late K secondaries in A–(Ba, Bb) hierarchical triples, and systems with WD companions, we find 193 companions with q > 0.1 and log P = 0–8 to 404 solar-type primaries (Table 11; black histogram in our Figure 31). The multiplicity frequency of the Raghavan et al. (2010) sample after accounting for these various selection effects is 0.48 ± 0.04.

The largest difference in the multiplicity frequency stems not from comparing Duquennoy & Mayor (1991) and Raghavan et al. (2010), but by comparing the statistics before and after removing WD companions. Nevertheless, all four different versions of the multiplicity frequency and period distribution displayed in Figure 31 are consistent with each other within the 2σ uncertainty levels. Our analytic fit to the period distribution of companions to solar-type primaries (see Section 9.3) is between the Duquennoy & Mayor (1991) and Raghavan et al. (2010) distributions after applying our corrections and removing WD companions. This fit results in a multiplicity frequency of 0.50 ± 0.04 companions with q > 0.1 and log P = 0–8 per solar-type primary. We conclude that the differences between the corrected Duquennoy & Mayor (1991) sample and raw Raghavan et al. (2010) sample are negligible and, most importantly, smaller than the systematic uncertainties in how corrections for incompleteness and WD companions are applied.

We next measure the eccentricity probability distribution p_e ∝ ${e}^{\eta }$ of solar-type binaries as a function of orbital period P. In Figure 32, we plot e versus P for the 97 solar-type binaries in the Raghavan et al. (2010) sample with spectroscopic and/or visual orbit solutions and 0 < log P (days) < 5. Our Figure 32 is quite similar to Figure 14 in Raghavan et al. (2010). However, we do not include the data points that actually represent the orbits of late K and M dwarf binaries. For example, the two systems near e = 0.12 and log P ≈ 3.9 in Figure 14 of Raghavan et al. (2010) are the orbits of low-mass binaries with tertiary solar-type primaries in A–(Ba, Bb) hierarchical configurations (see Section 8.1). We remove these two systems and four additional low-mass binaries with 4 < log P (days) < 5 in the Raghavan et al. (2010) survey from our sample.

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All of the 44 detected companions to solar-type primaries with 0 < log P (days) < 3 have measured eccentricities according to spectroscopic and/or visual orbit solutions. At 3 < log P < 4 and 4 < log P < 5, however, three and two detected companions, respectively, do not have measured eccentricities. VBs that do not have reliable orbital solutions generally have large eccentricities e ≳ 0.7 (Harrington & Miranian 1977; Tokovinin & Kiyaeva 2016). We assume that the five VBs with intermediate periods 3 < log P < 5 but without orbital solutions are evenly distributed across e ≈ 0.70–0.95 (green systems in Figure 32).

We also display in Figure 32 the maximum eccentricity ${e}_{\max }$ as a function of P according to Equation (3). As expected, all detected systems have $e\lt {e}_{\max }$. In fact, the majority of systems with log P < 1 have been tidally circularized. In this short-period interval, we measure the power-law component η = −0.8 ± 0.2 of the eccentricity distribution to be weighted toward small values (Table 11). Solar-type binaries at longer orbital periods log P > 1 not only contain systems with large eccentricities but also exhibit a deficit of binaries with small eccentricities e ≲ 0.15.

In the top panel of Figure 33, we display the cumulative distributions of e/${e}_{\max }$ for four different logarithmic period intervals. We measure η = −0.4, 0.1, 0.4, and 0.2 for log P = 1–2, 2–3, 3–4, and 4–5, respectively. At these wide separations, the tidal circularization timescales are longer than the ages $\langle \tau \rangle$ ≈ 5 Gyr of solar-type binaries (Zahn 1977; Hut 1981). As expected, we do not find any circularized solar-type MS binaries beyond P > 20 days (see also Meibom & Mathieu 2005). However, solar-type binaries initially born with e > 0.8 and log P > 2 tidally evolve toward smaller eccentricities e < 0.8 on significantly shorter timescales ${\tau }_{\mathrm{tide}}$ ∝ ${(1-{e}^{2})}^{13/2}$ (Zahn 1977; Hut 1981). In the bottom panel of Figure 33, we display the cumulative distribution of eccentricities for only those systems with $e/{e}_{\max }\lt 0.8$ that are not as severely affected by tidal effects. By fitting the power-law component η across 0 < e/${e}_{\max }$ < 0.8, we measure η = −0.4, 0.3, 0.8, and 0.4 for the intervals log P = 1–2, 2–3, 3–4, and 4–5, respectively. We average the two methods for determining η and report η = −0.4 ± 0.3, 0.2 ± 0.3, 0.6 ± 0.4, and 0.3 ± 0.3 for log P = 1–2, 2–3, 3–4, and 4–5, respectively, in Table 11. For solar-type binaries, the power-law component η of the eccentricity distribution increases with orbital period. Nonetheless, the measured values of η ≈ 0.2–0.6 at intermediate periods log P = 2–5 are mildly discrepant with a thermal eccentricity distribution (η = 1).

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Finally, to investigate the eccentricities e of solar-type binaries as a function of age τ, we examine the data set shown in Figure 8 of Meibom & Mathieu (2005). In their Figure 8, Meibom & Mathieu (2005) compare e versus τ for solar-type binaries in eight different environments with various ages. They find that the tidal circularization period increases from ${P}_{\mathrm{circ}}$ ≈ 7 days for pre-MS binaries up to ${P}_{\mathrm{circ}}$ ≈ 15 days for the halo population. In our present study, we are most interested in the eccentricity distributions of binaries with periods log P > 1.2 beyond the circularization period. Observational selection biases become too important in the Meibom & Mathieu (2005) sample toward long orbital periods log P > 2.4 and large eccentricities e > 0.6. We therefore select the 110 solar-type binaries from the Meibom & Mathieu (2005) data set with orbital periods 1.2 < log P < 2.4 and eccentricities e < 0.6. We divide our sample into two subsets: the 49 systems with ages τ < 700 Myr contained in panels (a)–(d) on the left side of Figure 8 in Meibom & Mathieu (2005), and the 61 binaries with ages τ ≈ 3–10 Gyr contained in panels (e)–(h) on the right side of their Figure 8.

In Figure 34, we display the cumulative distribution of eccentricities 0.0 < e < 0.6 of solar-type binaries with intermediate periods for our young and old populations. Even with two large subsamples, the eccentricity distributions of young and old solar-type binaries are surprisingly similar. A K-S test shows that the two populations are consistent with each other at the 92% confidence level. This demonstrates that tidal evolution during the MS is inappreciable for solar-type binaries with intermediate periods 1.2 < log P < 2.4 and modest eccentricities e < 0.6.

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By fitting our power-law distribution p_e ∝ ${e}^{\eta }$ to the data, we measure η = 0.2 ± 0.3 and η = 0.1 ± 0.3 for the young and old populations, respectively (Table 12). These values are consistent with our measurements of η = −0.4 ± 0.3 at slightly shorter periods log P = 1.0–2.0 and of η = 0.2 ± 0.3 at slightly longer periods log P = 2.0–3.0 based on the Raghavan et al. (2010) survey (see Table 11). Upon visual inspection of Figure 34, we find it possible that a single-parameter distribution may not necessarily fully describe the data. For example, both the young and old cumulative eccentricity distributions lie systematically below the η = 0 curve across e = 0.10–0.25, and then they cross above the η = 0 distribution near e ≈ 0.45. A two-parameter eccentricity distribution, e.g., a Gaussian, may better fit the data. However, a K-S test reveals that both the old and young eccentricity distributions are consistent with a power-law distribution at the ≈90% significance level. Moreover, selection biases and tidal evolution at large eccentricities e ≳ $0.8{e}_{\max }$ are not fully understood, and so the initial zero-age MS eccentricity distribution is not well constrained in this portion of the parameter space. With the current samples, we do not find sufficient discrepancy between our adopted single-component power-law distribution and the data to warrant a different probability distribution and/or additional parameters.

After combining all our measurements in Sections 3–8 and Tables 2–12, we display in Figure 35 the excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}$ (top panel) as a function of orbital period P and colored according to primary mass M₁. We group the data points into five spectral subtype intervals: solar-type (M₁ = 0.8–1.2 M_⊙; red), A/late B (M₁ = 2–5 M_⊙; orange), mid-B (M₁ = 5–9 M_⊙; green), early B (M₁ = 9–16 M_⊙; blue), and O-type (M₁ >16 M_⊙; magenta). The excess twin fraction ${{ \mathcal F }}_{\mathrm{twin}}$ exhibits three main aspects. First, for solar-type primaries, the excess twin fraction decreases linearly with respect to log P. Second, for mid-B, early B, and O-type primaries, the excess twin fraction is zero for P ≳ 20 days. Finally, at shorter periods, the excess twin fraction is anticorrelated with respect to log M₁. We find that the following relation fits these three aspects of the data:

where the excess twin fraction at short orbital periods is

and the maximum orbital period of the observed excess twin population is

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We compare the analytic fit (Equation (5); dotted lines) to the data for ${{ \mathcal F }}_{\mathrm{twin}}$(M₁, P) in the top panel of Figure 35. We evaluate the fit according to the average primary mass within each spectral subtype interval, i.e., M₁ = 1, 3.5, 7, 12, and 28 M_⊙ for solar-type, A/late B, mid-B, early B, and O-type primaries, respectively. The fit adequately describes the excess twin fraction as continuous functions of M₁ and P. In fact, the fit nearly coincides with the majority of the observed values. The two nonzero ${{ \mathcal F }}_{\mathrm{twin}}$ measurements for A-type stars (orange data points at log P ≈ 4.6 and 5.6 in the top panel of Figure 35) correspond to $\langle {M}_{1}\rangle$ = 2.0 M_⊙ and therefore lie just above the evaluation of the fit at M₁ = 3.5 M_⊙ (orange dotted line).

There are still uncertainties $\delta {{ \mathcal F }}_{\mathrm{twin}}$ in the excess twin fraction according to the precision of the observational measurements. For all periods and primary masses, we find the 1σ uncertainty in the excess twin fraction to be

In the middle panel of Figure 35, we display the power-law slope ${\gamma }_{\mathrm{large}q}$ of the mass-ratio distribution across large mass ratios q = 0.3–1.0. For all primary masses M₁ and for short orbital periods log P (days) < 1.0, the slope is ${\gamma }_{\mathrm{large}q}$ ≈ −0.5. At longer orbital periods, the power-law component ${\gamma }_{\mathrm{large}q}$ decreases, but the break toward steeper slopes depends significantly on M₁. For solar-type primaries, we find

For the midpoint of A/late B primaries, we measure

Finally, for mid-B, early B, and O-type stars, we fit

For primary stars with masses M₁ = 1.2–3.5 M_⊙, we interpolate between Equations (9) and (10) according to M₁. Similarly, for M₁ = 3.5–6.0 M_⊙, we interpolate between Equations (10) and (11). In this manner, Equations (9)–(11) provide an analytic fit to ${\gamma }_{\mathrm{large}q}$ as continuous functions of M₁ and P.

We compare our fit (dotted lines) to the data for ${\gamma }_{\mathrm{large}q}$ in the middle panel of Figure 35. The analytic functions fit the data reasonably well across all primary masses and orbital periods. There is scatter, but none of the individual observations deviate more than ${\rm{\Delta }}{\gamma }_{\mathrm{large}q}$ = 0.7 from the fit. In fact, the weighted average of observations across a narrow period and primary mass interval match the analytic functions within ${\rm{\Delta }}{\gamma }_{\mathrm{large}q}$ ≈ 0.2. For example, the three measurements of ${\gamma }_{\mathrm{large}q}$ for early-type binaries at intermediate periods log P ≈ 3.4 are −1.7 ± 0.5 (mid-B), −2.3 ± 0.5 (early B), and −1.4 ± 0.4 (O-type). The weighted average and uncertainty is ${\gamma }_{\mathrm{large}q}$ = −1.7 ± 0.3, while evaluation of Equation (11) at log P ≈ 3.4 yields ${\gamma }_{\mathrm{large}q}$ = −1.8. In this parameter space, the analytic function and average of the observations differ by only ${\rm{\Delta }}{\gamma }_{\mathrm{large}q}$ = 0.1. Similarly, the two A/late B measurements at log P ≈ 5.6 are ${\gamma }_{\mathrm{large}q}$ −1.0 ± 0.5 and −2.1 ± 0.4. In this case, the weighted average of ${\gamma }_{\mathrm{large}q}$ = −1.6 ± 0.3 matches the value ${\gamma }_{\mathrm{large}q}$ = −1.6 of Equation (10) evaluated at log P = 5.6. For these two examples, the weighted uncertainties δ${\gamma }_{\mathrm{large}q}$ ≈ 0.3 of the observations are larger than the differences ${\rm{\Delta }}{\gamma }_{\mathrm{large}q}$ < 0.2 between the fits and averages of the observations. Based on these examples and further comparisons between the measurements and fits, we adopt the 1σ uncertainties between the analytic functions and actual values to be

for all primary masses and orbital periods.

In the bottom panel of Figure 35, we display the various measurements of the power-law slope ${\gamma }_{\mathrm{small}q}$ across small mass ratios q = 0.1–0.3. For solar-type primaries, the component ${\gamma }_{\mathrm{small}q}$ ≈ 0.3 is nearly constant across all orbital periods. We adopt

The weighted average of the mid-B, early B, and O-type measurements of ${\gamma }_{\mathrm{small}q}$ near log P ≈ 0.8 is ${\gamma }_{\mathrm{small}q}$ = 0.1 ± 0.5. At log P ≈ 6.5, the weighted average of the mid-B, early B, and O-type observations decreases to ${\gamma }_{\mathrm{small}q}$ = −1.5 ± 0.3. The three A/late B measurements across log P = 5–8 span $-1.1\lt {\gamma }_{\mathrm{small}q}\lt -0.7$. These three values for A/late B binaries are between the solar-type and early-type VB measurements, indicating a natural progression in ${\gamma }_{\mathrm{small}q}$ as a function of primary mass.

Unfortunately, for early B and O-type primaries, there are no direct measurements of ${\gamma }_{\mathrm{small}q}$ across 2 < log P < 4 (see bottom panel in Figure 35). As shown in Figure 1, current observations are insensitive to small mass-ratio binaries at intermediate orbital periods. A simple linear interpolation between the short-period (${\gamma }_{\mathrm{small}q}$ = 0.1 ± 0.5) and long-period (${\gamma }_{\mathrm{small}q}$ = −1.5 ± 0.3) values yields ${\gamma }_{\mathrm{small}q}$ = −0.5 ± 0.5 at log P ≈ 3.0. Based on the observed frequency of SB1 companions to Cepheids, we estimate ${\gamma }_{\mathrm{small}q}$ = 0.2 ± 0.9 for mid-B primaries and log P = 3.1 (Section 6.2). This measurement is quite uncertain, particularly due to the small sample size, but also because some of the SB1 companions may be compact remnants. For early-type binaries and intermediate orbital periods log P ≈ 3, we adopt the average ${\gamma }_{\mathrm{small}q}$ = −0.2 ± 0.6 of the linear interpolation estimate and the Cepheid SB1 estimate.

Based on the above considerations, we determine the following analytic relations. For the midpoint of A/late B primaries, we use

For mid-B, early B, and O-type stars, we fit

As done previously, we interpolate between Equations (13) and (14) for primary stars with masses M₁ = 1.2–3.5 M_⊙. Similarly, for M₁ = 3.5–6.0 M_⊙, we interpolate between Equations (14) and (15) with respect to M₁.

We compare our fit to the data for ${\gamma }_{\mathrm{small}q}$ in the bottom panel of Figure 35. Our fit passes through all the measurements within their respective 1σ uncertainties. The uncertainty in the fit is therefore dominated by the uncertainty in the observations. The 1σ errors in ${\gamma }_{\mathrm{small}q}$ depend primarily on P. They increase from δ${\gamma }_{\mathrm{small}q}$ ≈ 0.4 at log P ≈ 1 to δ${\gamma }_{\mathrm{small}q}$ ≈ 0.6 at log P ≈ 3, and then they decrease to δ${\gamma }_{\mathrm{small}q}$ ≈ 0.3 for log P ≳ 6. For all primary masses M₁, we model the 1σ uncertainties as

This relation accounts for the larger uncertainties in ${\gamma }_{\mathrm{small}q}$ at intermediate orbital periods due to the gaps in the observations.

In Figure 36, we display the power-law slope η of the eccentricity distribution as a function of log P and colored according to spectral type based on all our measurements in Tables 2–12. We divide the data into two spectral type intervals: late-type (M₁ = 0.8–5 M_⊙) and early-type (M₁ > 5 M_⊙). Both late-type (Meibom & Mathieu 2005; Raghavan et al. 2010) and early-type (Sana et al. 2012) populations have similar zero-age MS circularization periods P ≈ 2–6 days (log P ≈ 0.5). In addition, for both late-type and early-type binaries, the eccentricity distribution becomes weighted toward larger values with increasing orbital period P. However, the power-law slope is ${\rm{\Delta }}\eta$ ≈ 0.5 larger for early-type binaries compared to late-type binaries across all orbital periods.

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For close binaries with log P ≲ 1.2, the differences in the eccentricity distribution between the two spectral types can be explained by tidal evolution. Tidal damping via convection is more efficient in cool, late-type stars than is radiative damping in hot, early-type stars (Zahn 1975, 1977; Hut 1981). Moreover, late-type MS binaries live orders of magnitude longer and have had more time to tidally evolve toward smaller eccentricities.

For binaries with intermediate periods 1.2 ≲ log P ≲ 5, however, the differences cannot be explained solely by tidal effects. In Section 8.7, we showed that tidal evolution is negligible for MS solar-type binaries with periods log P = 1.2–2.4 slightly beyond the tidal circularization period. For late-type binaries, even those with orbital periods 3 < log P < 5 where tidal effects are even less significant, the power-law slope η ≈ 0.3–0.6 is discrepant with a thermal eccentricity distribution (see Figure 36). At wider separations $\langle a\rangle$ ≈ 120 au (log P ≈ 5.6), Tokovinin & Kiyaeva (2016) demonstrate that solar-type binaries have an intrinsic eccentricity probability distribution described by p_e = 1.2e+0.4. Fitting our power-law model to their data that account for selection effects (Figure 7 in Tokovinin & Kiyaeva 2016), we measure η = 0.5 ± 0.3. We confirm the conclusion of Tokovinin & Kiyaeva (2016) that the eccentricity distribution of wide solar-type binaries is flatter than a thermal distribution. Meanwhile, for early-type binaries, the zero-age MS eccentricity distribution quickly asymptotes toward a thermal distribution (η = 1) beyond log P ≳ 1 (Figure 36). Early-type binaries with intermediate orbital periods are born onto the zero-age MS with systematically larger eccentricities than their solar-type counterparts.

The differences between the early-type and solar-type intermediate-period eccentricity distributions may stem from the differences in dynamical processing during the earlier pre-MS phase of formation. As noted in Section 1, dynamical interactions and exchanges tend to drive the eccentricities toward a thermal distribution (Ambartsumian 1937; Heggie 1975; Pringle 1989; Kroupa 1995a, 2008; Turner et al. 1995). Considering that massive stars exhibit a larger multiplicity frequency and triple-star fraction (see Section 9.4), massive systems may more efficiently evolve toward η ≈ 1 at early times τ ≲ 1 Myr (see also Goodwin & Kroupa 2005; Pflamm-Altenburg & Kroupa 2006; Oh et al. 2015). Meanwhile, only ≈10% of solar-type MS primaries are observed to be in triple/quadruple systems (Section 9.4). While the triple-star fraction of solar-type stars may be slightly higher during the pre-MS phase, it is still substantially smaller than the triple-star fraction measured for O-type MS stars (see Section 10). The smaller triple-star fraction for solar-type stars may lead to the smaller values of η ≈ 0.5 across intermediate periods.

Another possibility is that both early-type and late-type binaries with intermediate periods are born with η ≈ 1 during the early pre-MS, but then, for solar-type binaries, the eccentricities decrease to η ≈ 0.5 by the zero-age MS. Solar-type binaries have considerably longer pre-MS contraction timescales, and so tidal interactions of much larger, pre-MS components may cause the binaries to evolve toward smaller eccentricities, even at intermediate separations. Solar-type pre-MS stars also have longer disk lifetimes, and so binary-star interactions with the primordial disks could drive the eccentricities downward. Kroupa (1995b) provides analytic models of pre-MS binaries embedded in disks that incorporate both tidal circularization and the preferred accretion onto the secondary component. In addition to orbital circularization, the latter process also drives the binary mass ratio toward unity and may explain the excess fraction of twins observed in solar-type binaries, even those at intermediate separations a ∼ 0.5–100 au. More detailed comparisons with these models, as well as future observations of solar-type pre-MS binaries with intermediate orbital periods, should help in determining whether they are born with or evolve toward η ≈ 0.5.

For zero-age MS binaries, we adopt an analytic function for η that approaches circular orbits for short-period binaries and then asymptotes toward a fixed η at longer periods. For late-type MS binaries, we use

Meanwhile, for early-type MS binaries, we fit

To model a smooth transition between late and early spectral types, we interpolate across M₁ = 3–7 M_⊙ between Equations (17) and (18). We compare our fit to the data in Figure 36.

For the shortest orbital periods log P ≲ 1, the parameter η is not well defined. This is primarily because the maximum eccentricity ${e}_{\max }$ possible without Roche lobe filling (Equation (3)) dramatically decreases toward short periods. It is difficult to reliably measure the power-law slope η of the eccentricity distribution across a narrow interval 0 < e < ${e}_{\max }$. In any case, the combination of ${e}_{\max }$ provided in Equation (3) and our fit to η in Equations (17) and (18) adequately reproduce the statistics of the observed short-period binary populations.

For intermediate-period binaries with 1 ≲ log P ≲ 5, our fit to η describes the observed variations in the eccentricity distribution. The fit passes through the measurement uncertainties of all the data points. For orbital periods P within this interval and for all primary masses M₁, the 1σ uncertainty between the fit and actual value is

For the widest companions with log P (days) ≈ 6–8, we can currently only speculate as to the nature of the eccentricity distribution. For solar-type binaries, the power-law slope η ≈ 0.5 may continue to the widest separations (Tokovinin & Kiyaeva 2016). For early-type primaries, however, wide companions cannot follow a power-law slope η ≈ 0.8 across all eccentricities 0 < e < ${e}_{\max }$ ≈ 1. As shown in Section 9.4, the majority of wide companions to O-type and B-type stars are tertiary components in hierarchical triples. Wide tertiary components have been observed to have systematically smaller eccentricities than their wide-binary counterparts (Tokovinin & Kiyaeva 2016). The maximum eccentricity ${e}_{\max }$ of tertiary components must satisfy not only Equation (3) but also the dynamical stability criterion (Mardling & Aarseth 2001, Equation (8) in Tokovinin 2014). For example, if log ${P}_{\mathrm{inner}}$ = 3 and log ${P}_{\mathrm{outer}}$ = 6, then the eccentricity of the tertiary must be ${e}_{\mathrm{outer}}$ < 0.93 to be dynamically stable. For wide tertiary companions to early-type primaries, the power-law slope η ≈ 0.8 may still continue across 5 < log ${P}_{\mathrm{outer}}$ < 8, but the domain 0 < e < ${e}_{\max }$ of the distribution must be modified so that ${e}_{\max }$ satisfies the stability criterion.

We next investigate the period distribution of MS binaries as a function of primary mass M₁. In the top panel of Figure 37, we display the frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ (M₁, P) of companions with q > 0.3 per decade of orbital period based on all our measurements in Tables 2–12. We group the data into the same five spectral type intervals as done in Section 9.1. For M₁ ≈ 1 M_⊙, ${f}_{\mathrm{log}P;q\gt 0.3}$ follows a lognormal distribution with a peak of ${f}_{\mathrm{log}P;q\gt 0.3}$ ≈ 0.08 near log P ≈ 5. This is consistent with the lognormal period distribution found in previous studies of solar-type MS binaries (Duquennoy & Mayor 1991; Raghavan et al. 2010). For early-type primaries, the companion frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ ≈ 0.1–0.2 is larger at short (log P < 1.0) and intermediate (1.5 < log P < 4.0) orbital periods. Meanwhile, at long orbital periods (5 < log P < 7), our measurements of ${f}_{\mathrm{log}P;q\gt 0.3}$ do not significantly depend on M₁. In the following, we further analyze ${f}_{\mathrm{log}P;q\gt 0.3}$ according to three regimes of orbital period.

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First, at short orbital periods log P < 1, we observe a clear monotonic increase in ${f}_{\mathrm{log}P;q\gt 0.3}$ with respect to M₁ (see top panel of Figure 37). The close binary frequency increases by more than an order of magnitude from ${f}_{\mathrm{log}P;q\gt 0.3}$ ≈ 0.02 for solar-type primaries to ${f}_{\mathrm{log}P;q\gt 0.3}$ ≈ 0.24 for O-type primaries. By fitting the six data points with log P < 1.0 in the top panel of Figure 37, we measure the frequency of short-period binaries with q > 0.3 to be

Second, we examine the companion frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ across intermediate periods 1.5 ≲ log P ≲ 4.0. At these intermediate separations, the observed frequency of companions to mid-B, early B, and O-type primaries is definitively larger than the solar-type binary frequency (see top panel of Figure 37). The weighted average of the two O-type observations in this period interval is ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.16 ± 0.04. The one early B measurement is ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.18 ± 0.05, and the weighted average of the three mid-B values is ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.12 ± 0.03. For solar-type binaries, the companion frequency increases from ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.020 ± 0.007 at log P = 1.5 to ${f}_{\mathrm{log}P;q\gt 0.3}$ =0.056 ± 0.011 at log P = 4.0. At log P ≈ 3, the frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.06 ± 0.02 of companions to A-type/late B primaries is only slightly larger than the frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.035 ± 0.010 of companions to solar-type primaries. We fit the weighted averages of the companion frequencies according to primary mass at log P = 2.7:

Finally, at wide separations, the binary frequency decreases from ${f}_{\mathrm{log}P;q\gt 0.3}$ ≈ 0.08 at log P = 5 to ${f}_{\mathrm{log}P;q\gt 0.3}$ ≈ 0.04 at log P = 8 (see top panel of Figure 37). For long orbital periods, there is a slight non-monotonic trend between primary mass M₁ and the frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ of companions with q > 0.3. By averaging the observations near log P = 6 within each spectral subtype interval, we measure ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.07 ± 0.01, 0.06 ± 0.01, 0.05 ± 0.01, 0.08 ± 0.04, and 0.09 ± 0.02 for solar-type, A/late B, mid-B, early B, and O-type primaries, respectively. By fitting these observations and accounting for the period dependence ${f}_{\mathrm{log}P;q\gt 0.3}$ ∝ exp[−0.3(log P − 5.5)] beyond log P > 5.5, we find the companion frequency at log P = 5.5 to be

According to the top panel of Figure 37, the companion frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ for B-type binaries peaks near log P ≈ 3.5 (a ≈ 10 au). This is consistent with Rizzuto et al. (2013), who find that the period distribution of B-type binaries peaks at projected separations log ${a}_{\mathrm{proj}}$ (au) ≈ 0.9. Interestingly, the magenta data points in the top panel of Figure 37 suggest that the O-type companion frequency is slightly bimodal. There is a dominant peak at short periods log P < 1 (a ≲ 0.3 au). This is consistent with the conclusions of Sana et al. (2012), who find that spectroscopic binary companions to O-type stars are skewed toward shorter periods. However, there is also a secondary peak in the period distribution of O-type binaries near log P ≈ 3.5 (a ≈ 10 au), similar to the peak found in B-type binaries. The companion frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ definitively increases across log P ≈2.0–3.5 for solar-type and B-type stars and may also increase for O-type primaries. We therefore parameterize ${f}_{\mathrm{log}P;q\gt 0.3}$ to increase across these intermediate orbital periods for all spectral types. We set the slope to be α = $\partial {f}_{\mathrm{log}P;q\gt 0.3}$/∂logP = 0.018 across the interval logP = [2.7–ΔlogP, 2.7+ΔlogP], where ΔlogP = 0.7. As future observations better constrain the functional form of the period distribution across intermediate periods, we can adjust α and ΔlogP accordingly. Based on the above considerations, we can now fit ${f}_{\mathrm{log}P;q\gt 0.3}$ as continuous functions of M₁ and P:

where α = 0.018, ΔlogP = 0.7, and the companion frequencies ${f}_{\mathrm{log}P\lt 1;q\gt 0.3}({M}_{1})$, ${f}_{\mathrm{log}P=2.7;q\gt 0.3}({M}_{1})$, and ${f}_{\mathrm{log}P=5.5;q\gt 0.3}({M}_{1})$ are presented in Equations (20), (21), and (22), respectively.

We compare our fit to the data in the top panel of Figure 37. The fit passes through the 1σ uncertainties of all our measurements. The relative measurement uncertainties in ${f}_{\mathrm{log}P;q\gt 0.3}$ depend slightly on primary mass and orbital period. For solar-type primaries, $\delta {f}_{\mathrm{log}P;q\gt 0.3}$/${f}_{\mathrm{log}P;q\gt 0.3}$ decreases from ≈40% at short periods logP < 1 to ≈20% beyond logP > 4. For early-type binaries, $\delta {f}_{\mathrm{log}P;q\gt 0.3}$/${f}_{\mathrm{log}P;q\gt 0.3}$ initially increases from 25% at logP < 1 to 35% at logP = 3 and then decreases to ≈30% beyond logP > 5.5. We model the 1σ uncertainties in ${f}_{\mathrm{log}P;q\gt 0.3}$ for low-mass primaries as

and for early-type primaries as

For primary masses M₁ = 2–6 M_⊙, we interpolate the uncertainties $\delta {f}_{\mathrm{log}P;q\gt 0.3}$ between Equations (24) and (25).

As shown in the top panel of Figure 37, our analytic description for ${f}_{\mathrm{log}P;q\gt 0.3}$ (M₁, P) has many breaks and sharp transitions with respect to logP. The true period distribution may have smooth transitions better modeled by a lognormal or log-polynomial distribution (but see Kobulnicky et al. 2014). Nevertheless, as future observations become available, we can easily update our coefficients in Equations (20)–(22) without having to change the functional form of our entire distribution (Equation (23)). Most importantly, for all M₁ and P, our model is consistent with the true values of ${f}_{\mathrm{log}P;q\gt 0.3}$ (M₁, P) within the 1σ uncertainties provided in Equations (24)–(25).

Given our analytic fits to ${f}_{\mathrm{log}P;q\gt 0.3}$ (M₁, P), ${\gamma }_{\mathrm{small}q}$ (M₁, P), ${\gamma }_{\mathrm{large}q}$ (M₁, P), and ${{ \mathcal F }}_{\mathrm{twin}}$ (M₁, P), we calculate the frequency ${f}_{\mathrm{log}P;q\gt 0.1}$ (M₁, P) of companions with q > 0.1 per decade of orbital period. For example, ${f}_{\mathrm{log}P;q\gt 0.3}$ = 0.14 and a uniform mass-ratio distribution (${\gamma }_{\mathrm{small}q}$ = ${\gamma }_{\mathrm{large}q}$ = ${{ \mathcal F }}_{\mathrm{twin}}=0.0$) gives ${f}_{\mathrm{log}P;q\gt 0.1}$ = 0.18 (see Section 2). We measure the 1σ uncertainties in ${f}_{\mathrm{log}P;q\gt 0.1}$ by propagating in quadrature the 1σ uncertainties in the four parameters used to calculate ${f}_{\mathrm{log}P;q\gt 0.1}$.

In the bottom panel of Figure 37, we display ${f}_{\mathrm{log}P;q\gt 0.1}$ (M₁, P) as a function of logP and colored according to M₁. The frequency ${f}_{\mathrm{log}P;q\gt 0.1}$ of companions with q > 0.1 follows a similar functional form as ${f}_{\mathrm{log}P;q\gt 0.3}$, but ${f}_{\mathrm{log}P;q\gt 0.1}$ is larger owing to the addition of systems with q = 0.1–0.3. For short orbital periods, there is only a slight increase in the companion frequency, i.e., ${f}_{\mathrm{log}P;q\gt 0.1}$ ≈ $1.3{f}_{\mathrm{log}P;q\gt 0.3}$. Meanwhile, for long orbital periods logP = 6–7 and massive primaries M₁ ≳ 6 M_⊙, we measure ${f}_{\mathrm{log}P;q\gt 0.1}$ ≈ $3.1{f}_{\mathrm{log}P;q\gt 0.3}$. This enhanced companion frequency is because the mass-ratio distribution of wider companions to early-type stars is weighted toward smaller mass ratios. A relatively larger fraction of companions with q > 0.1 have q = 0.1–0.3. This effect is so important at long orbital periods that although ${f}_{\mathrm{log}P;q\gt 0.3}$ may be non-monotonic with respect to M₁ (see Equation (22) and right side of Figure 37), ${f}_{\mathrm{log}P;q\gt 0.1}$ is monotonically increasing according to M₁ for all orbital periods.

Not all surveys we have examined in this study are sensitive to binaries with q > 0.1, which is why we have parameterized and measured the frequency ${f}_{\mathrm{log}P;q\gt 0.3}$ of companions with q > 0.3 (see Figure 1 and Section 2). Nonetheless, some samples are complete to q = 0.1, and so we can directly measure the companion frequency ${f}_{\mathrm{log}P;q\gt 0.1}$ down to q = 0.1. For solar-type primaries, the companion frequency is ${f}_{\mathrm{log}P;q\gt 0.1}$ = (${{ \mathcal N }}_{\mathrm{large}q}$ + ${{ \mathcal N }}_{\mathrm{small}q}$)/${{ \mathcal N }}_{\mathrm{prim}}$, where ${{ \mathcal N }}_{\mathrm{prim}}=404$ and both ${{ \mathcal N }}_{\mathrm{large}q}$ and ${{ \mathcal N }}_{\mathrm{small}q}$ are given in Table 11 for each decade of orbital period across 0 < logP < 8. Based on the Sana et al. (2012) SB sample containing ${{ \mathcal N }}_{\mathrm{prim}}$ = 71 O-type primaries, we count ${{ \mathcal N }}_{\mathrm{large}q}=17$ companions with q > 0.3 and ${{ \mathcal N }}_{\mathrm{small}q}=4$ companions with q = 0.1–0.3 across P = 2–20 days (see Section 3.5). This provides ${f}_{\mathrm{log}P;q\gt 0.1}$ = (${{ \mathcal N }}_{\mathrm{large}q}+{{ \mathcal N }}_{\mathrm{small}q}$)/${{ \mathcal N }}_{\mathrm{prim}}$ = (17+4)/71 = 0.30 ± 0.06. Similarly, for the combined SB sample of ${{ \mathcal N }}_{\mathrm{prim}}$ = 81 + 109 + 83 = 273 B-type primaries, we measure ${f}_{\mathrm{log}P;q\gt 0.1}$ = (${{ \mathcal N }}_{\mathrm{large}q}+{{ \mathcal N }}_{\mathrm{small}q}$)${{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$ = (22+9) × 1.2/273 = 0.14 ± 0.03 for logP = 0.8 ± 0.5. Based on observations of EBs with early B primaries (see Section 4), we report in Moe & Di Stefano (2013) a corrected binary frequency of ${f}_{\mathrm{log}P;q\gt 0.1}$ = 0.22 ± 0.05 across P = 2–20 days. For the ${{ \mathcal N }}_{\mathrm{prim}}$ = 31 Cepheids brighter than V < 8.0 mag that were extensively monitored for radial velocity variations, Evans et al. (2015) find ${{ \mathcal N }}_{\mathrm{comp}}=9$ companions with q > 0.1 and P = 1–10 yr (see Section 6.2). These statistics provide ${f}_{\mathrm{log}P;q\gt 0.1}$ = ${{ \mathcal N }}_{\mathrm{comp}}{{ \mathcal F }}_{\mathrm{Cepheid}}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$ = 9 × 0.75 × 1.2/31 = 0.26 ± 0.09. For a sample of ${{ \mathcal N }}_{\mathrm{prim},\mathrm{AO}}$ = 363 A-type primaries, De Rosa et al. (2014) utilized adaptive optics to detect ${{ \mathcal N }}_{\mathrm{comp}}$ = 54 companions with q > 0.1 and projected separations ρ = 09–80 (logP = 4.9–6.3; see Section 7.1). The corrected companion frequency is ${f}_{\mathrm{log}P;q\gt 0.1}$ = ${{ \mathcal N }}_{\mathrm{comp}}{{ \mathcal C }}_{\mathrm{evol}}$/${{ \mathcal N }}_{\mathrm{prim}}$/ΔlogP = 54 × 1.2/363/(6.3−4.9) = 0.13 ± 0.02. We repeat this exercise for the remaining VB surveys examined in Section 7 that are complete to q = 0.1 binaries.

In the bottom panel of Figure 37, we present our measured values of ${f}_{\mathrm{log}P;q\gt 0.1}$ based on the samples that are complete down to q = 0.1. Although we do not fit these data points directly, our analytic function for ${f}_{\mathrm{log}P;q\gt 0.1}$ matches the observed values within their 1σ uncertainties. This demonstrates that we can use our fitted functions to ${f}_{\mathrm{log}P;q\gt 0.3}$, ${\gamma }_{\mathrm{small}q}$, ${\gamma }_{\mathrm{large}q}$, and ${{ \mathcal F }}_{\mathrm{twin}}$ to reproduce the observed values of ${f}_{\mathrm{log}P;q\gt 0.1}$. Most importantly, by measuring ${f}_{\mathrm{log}P;q\gt 0.3}$, ${\gamma }_{\mathrm{large}q}$, and ${{ \mathcal F }}_{\mathrm{twin}}$ for early-type primaries across all orbital periods and by interpolating ${\gamma }_{\mathrm{small}q}$ between the short-period (logP < 1.5) and long-period (logP > 6) regimes (see Section 9.1), we can reliably estimate the total companion frequency ${f}_{\mathrm{log}P;q\gt 0.1}$ across intermediate periods 2 < logP < 5. In our analysis, we have always interpolated, never extrapolated, our binary statistics into the regions of the f(M₁, q, P) phase space we cannot directly observe.

Based on the observations and our fits to the data points, it is quite evident from Figure 37 that the period distribution of companions depends critically on the primary mass. While solar-type MS binaries are weighted toward longer periods logP(days) ≈ 5, companions to massive OB stars peak at intermediate periods logP ≈ 3 and are skewed toward even shorter periods logP ≲ 1 if we focus only on systems with q > 0.3. As done in Oh & Kroupa (2016), it is crucial that future population synthesis studies incorporate a primary-mass-dependent binary period distribution in order to make meaningful predictions.