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$\begin{figure} \par\includegraphics[width=5.5cm,clip]{7719f1.ps} \end{figure}$ Figure 1: The completeness of the Hipparcos catalogue in the Sco OB2 region, as a function of V magnitude. The diamonds represent the ratio between the number of stars in the Hipparcos catalogue and the number of stars in the TYCHO-2 catalogue, in each V magnitude bin. The comparison above is made for the Sco OB2 region, and is similar for each of the three subgroups of Sco OB2. The solid line represents the model for the completeness adopted in this paper (Eq. (3)).
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In the text

$\begin{figure} \par\includegraphics[width=16.4cm,clip]{7719f2.ps} \end{figure}$ Figure 2: Properties of the observed binary population in Sco OB2. Only the 521 confirmed members of Sco OB2 are considered. The top-left panel shows the angular separation distribution for visual binaries, at the moment of observation. The other panels show the distribution over radial velocity amplitude K₁, the projected semi-major axis $a_1 \sin i$ , the eccentricity e, the mass function $\mathcal{F}(M)$ , and the period P, derived for the orbits of the spectroscopic binaries (SB1 and SB2), and for HIP 78918, the only astrometric binary in Sco OB2 with an orbital solution. The measurements shown in this figure include those of multiple systems. Above each panel we indicate the number of companions for which the corresponding orbital element is available. Spectroscopic and astrometric binaries without an orbital solution are not included.
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In the text

$\begin{figure} \par\includegraphics[width=6cm,clip]{7719f3.ps} \end{figure}$ Figure 3: The 50% detection limit $\Delta K_{\rm S}$ as a function of angular separation $\rho$ , for the KO5 observations (short-dashed curve), the KO6 observations (dotted curve), the non-coronographic SHT observations (dash-dotted curve), the coronographic SHT observations (long-dashed curve), and the combined SHT observations (solid curve). For the KO5 and KO6 observations the curves represent those for average Strehl ratios of 30% and 24%, respectively.
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In the text

$\begin{figure} \par\includegraphics[width=6cm,clip]{7719f4.ps} \end{figure}$ Figure 4: In the three imaging surveys for binarity discussed in this paper (KO5, KO6, and SHT) the field-of-view is non-circular. Whether a secondary is in the field of view, depends therefore not only on its separation $\rho$ , but also on its position angle $\varphi$ . This figure shows the probability that a secondary is in the field-of-view, as a function of $\rho$ , assuming random orientation of the binary systems, for the KO5 observations (short-dashed curve), the KO6 observations (dotted curve), the non-coronographic SHT observations (dash-dotted curve), the coronographic SHT observations (long-dashed curve), and the combined SHT observations (solid curve). Whether a secondary is detected or not, depends additionally on its brightness and on the brightness difference with the primary (see Fig. 3).
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In the text

$\begin{figure} \par\includegraphics[width=5.1cm,clip]{7719f5a.ps}\hspace*{3mm} ... ...b.ps}\hspace*{3mm} \includegraphics[width=4.8cm,clip]{7719f5c.ps} \end{figure}$ Figure 5: How well can the intermediate-mass binary population in Sco OB2 be described by a mass ratio distribution of the form $f_q(q) \propto q^{\gamma _q}$ ? Left: the goodness-of-fit for the comparison between the observed mass ratio distribution (from the KO5 and SHT datasets) and that predicted by models with different values of $\gamma _q$ . A large value for the goodness-of-fit means that the model predictions are consistent with the observations. The $1\sigma$ , $2\sigma$ and $3\sigma$ confidence limits for rejection of the model are indicated with the horizontal dotted lines. Middle: the goodness-of-fit for the comparison between the observed visual binary fraction (from the KO5 and SHT datasets) and that predicted by the models with an intrinsic binary fraction of 85% (bottom curve), 90%, 95% and 100% (top curve). Right: the predicted (SB1, SB2 and RVV) spectroscopic binary fraction (for the LEV dataset) as a function of $\gamma _q$ . The five curves indicate the results for a model binary fraction of 60% (bottom curve), 70%, 80%, 90% and 100% (top curve). The bottom horizontal dotted line indicates the minimum observed spectroscopic binary fraction, where only the SB1s and SB2s are included, assuming that the radial velocity variations in all RVVs are caused by line-profile variability rather than binarity. The top horizontal line indicates the maximum spectroscopic binary fraction, assuming that all RVVs are indeed binaries. Taken together, the three panels indicate that the mass ratio distribution for intermediate mass stars in Sco OB2 can be described by $f_q(q) \propto q^{\gamma _q}$ with exponent $\gamma _a \approx -0.4$ (vertical dotted line in each panel). An intrinsic binary fraction close to 100% is required in order to produce the large visual and spectroscopic binary fractions that are observed. See Sect. 5 for a further discussion of this figure.
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In the text

$\begin{figure} \par\includegraphics[width=5cm,clip]{7719f6a.ps}\hspace*{4mm} \i... ...b.ps}\hspace*{4mm} \includegraphics[width=4.6cm,clip]{7719f6c.ps} \end{figure}$ Figure 6: How well can the intermediate-mass binary population in Sco OB2 be described by a semi-major axis distribution of the form $f_a(a) \propto a^{\gamma _a}$ ? Left: the goodness-of-fit for the comparison between the observed angular separation distribution (from the KO5 and SHT datasets) and that predicted by the models with different values of $\gamma _a$ . A large value for the goodness-of-fit means that the model predictions are consistent with the observations. The $1\sigma$ , $2\sigma$ and $3\sigma$ confidence limits for rejection of the model are indicated with the horizontal dotted lines. Middle: the goodness-of-fit for the comparison between the observed visual binary fraction (from the KO5 and SHT datasets) and that predicted by the models with an intrinsic binary fraction of 70% (bottom curve), 80%, 90% and 100% (top curve). Right: the predicted (SB1, SB2 and RVV) spectroscopic binary fraction (for the LEV dataset) as a function of $\gamma _a$ . The five curves indicate an intrinsic binary fraction of 60% (bottom curve), 70%, 80%, 90% and 100% (top curve). The bottom horizontal line indicates the lower limit for the observed spectroscopic binary fraction, where only the SB1s and SB2s are included, assuming that the radial velocity variations in all RVVs are caused by line-profile variability rather than binarity. The top horizontal line indicates the upper limit for the spectroscopic binary fraction, assuming that all RVVs are indeed binaries. A combination of the results in these three panels indicates that the semi-major axis distribution for intermediate mass stars in Sco OB2 can be described by $f_a(a) \propto a^{\gamma _a}$ with exponent $\gamma _a \approx -1.0$ (vertical dotted line in each panel), commonly known as Öpik's law. An intrinsic binary fraction close to 100% is required in order to produce the large number of visual and spectroscopic binaries detected in Sco OB2.
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In the text

$\begin{figure} \par\includegraphics[width=5.5cm,clip]{7719f7.ps} \end{figure}$ Figure 7: How well does a model with a log-normal period distribution $f_{P; ~\mu,~\sigma}(P)$ with centroid $\mu$ and width $\sigma$ reproduce the observed angular separation distribution $\tilde{f}_\rho(\rho)$ ? The darkest colors in this figure indicate the best-fitting models. From dark to light, the contours indicate the $\frac{1}{3}\sigma$ , $1\sigma$ , $2\sigma$ , and $3\sigma$ confidence limits for rejection of the model, respectively. The cross at $(\mu =4.8,\sigma =2.3)$ indicates the distribution derived by Duquennoy & Mayor (1991) for solar-type stars in the solar neighbourhood. The comparison with the observed visual and spectroscopic binary fraction is shown in Fig. 8.
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In the text

$\begin{figure} \par\includegraphics[width=16.2cm,clip]{7719f8a.ps}\par\includegraphics[width=16.2cm,clip]{7719f8b.ps} \end{figure}$ Figure 8: How well does a log-normal period distribution with centroid $\mu$ and width $\sigma$ reproduce the observed visual and spectroscopic binary fractions? Top: the goodness-of-fit for the comparison between the observed and predicted visual binary fraction, as a function of $\mu$ and $\sigma$ . Each panel represents a set of models with a different intrinsic binary fraction ranging from $F_{M}=100\%$ ( left) to $50\%$ ( right). The darkest colors indicate the best-fitting models. From dark to light, the three contours indicate the $1\sigma$ , $2\sigma$ , and $3\sigma$ confidence limits for rejection of the model, respectively. The cross at $(\mu =4.8,\sigma =2.3)$ in each panel indicates the distribution derived by Duquennoy & Mayor (1991) for solar-type stars in the solar neighbourhood. Bottom: the predicted (SB1, SB2 and RVV) spectroscopic binary fraction as a function of $\mu$ , $\sigma$ and F_M (note that these panels do not indicate the goodness-of-fit). The spectroscopic binary fraction is indicated with the gray-shade in each panel, where the solid contours indicate the combinations of $\mu$ and $\sigma$ which predict a spectroscopic binary fraction (from black to white) of 80%, 60%, 40% and 20%, respectively. The other curves indicate the extreme constraints imposed by the observations, of $30\pm 6\%$ (dashed contour) and $74\pm 6\%$ (dotted contour), respectively, where the higher value is more likely to represent reality. This figure, combined with the $f_\rho (\rho )$ comparison of Fig. 7, shows that if the binary population can be described with a log-normal period distribution, we require (1) a binary fraction near 100%, and (2) a large value for $\sigma$ (which mimics Öpik's law).
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In the text

$\begin{figure} \par\includegraphics[width=5.7cm,clip]{7719f9a.ps}\hspace*{1cm} \includegraphics[width=9cm,clip]{7719f9b.ps} \end{figure}$ Figure 9: How well do models with a semi-major axis distribution $f_a(a) \propto a^{\gamma _a}$ and a mass ratio distribution $f_q(q) \propto q^{\gamma _q}$ correspond to the observations? This figure is the multi-dimensional equivalent of Figs. 5 and 6, and indicates that with the independent derivation of $\gamma _q$ and $\gamma _a$ we have found a globally best-fitting solution. Left: the consistency between the observed two-dimensional distribution $\tilde{f}_{\rho,~q}(\rho,~q)$ and that of simulated observations, for models with different values of $\gamma _a$ ( horizontal axis) and $\gamma _q$ ( vertical axis), respectively. The darkest colors indicate the best-fitting models. From dark to light, the three contours indicate the $1\sigma$ , $2\sigma$ , and $3\sigma$ confidence limits for rejection of the model, respectively. The cross in each panel indicates the values of $\gamma _a$ and $\gamma _q$ that we adopt to describe the properties of Sco OB2. Right: the consistency between the observed visual binary fraction and that of the simulated observations, for models with different values of $\gamma _a$ and $\gamma _q$ . Each panel corresponds to a set of models with a different binary fraction, which is indicated in the bottom-left corner. The darkest colors indicate the best-fitting models. From dark to light, the three contours indicate the $1\sigma$ , $2\sigma$ , and $3\sigma$ confidence limits for rejection of the model, respectively. Models with a binary fraction smaller than $\approx$ 100% are inconsistent with the observations, as they underpredict the observed number of visual binaries. The cross in each panel indicates the best-fitting values of $\gamma _a \approx -1.0$ and $\gamma _q \approx -0.4$ for Sco OB2.
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In the text

$\begin{figure} \par\includegraphics[width=16cm,clip]{7719f10a.ps}\par\includegraphics[width=16cm,clip]{7719f10b.ps} \end{figure}$ Figure 10: How well do models with a log-normal period distribution $f_{P; ~\mu,~\sigma}(P)$ and a mass ratio distribution $f_q(q) \propto q^{\gamma _q}$ correspond to the observations? Top: a comparison for the observed two-dimensional distribution $\tilde{f}_{\rho,~q}(\rho,~q)$ for models with different values of $\mu$ , $\sigma$ and $\gamma _q$ . The cross in each panel indicates the values observed by Duquennoy & Mayor (1991): $\mu =4.8$ and $\sigma =2.3$ . The four panels correspond to models with a mass ratio distribution exponent $\gamma _q=-0.7$ ( left), -0.6, -0.3 and -0.1 ( right), respectively (cf. Fig. 8). The goodness-of-fit is best for the dark-shaded regions. Contours are plotted at $1\sigma$ , $2\sigma$ and $3\sigma$ confidence regions for rejection of the model. Bottom: the comparison for the observed visual binary fraction for models with different values of $\mu$ , $\sigma$ and $\gamma _q$ . The binary fraction, indicated in the bottom-left corner of each panel, decreases from 100% ( left) to 50% ( right). Contours are plotted at $1\sigma$ , $2\sigma$ and $3\sigma$ confidence regions for rejection of the model.
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In the text

$\begin{figure} \par\includegraphics[width=5.5cm,clip]{7719f11.ps} \end{figure}$ Figure 11: The simulated KO5 data for a realization of the best-fitting model of Sco OB2 (cf. Kouwenhoven et al. 2007, their Fig. 3.8). The plots show the distribution of physical companions (plusses) and background stars (triangles). The horizontal dashed line indicates the criterion used by KO5 (based on the analysis of SHT) to statistically separate companion stars and background stars. The simulated sample consists of 199 targets, next to which we detect 64 companion stars and 28 background stars. Although three companions have $K_{\rm S} > 12$ mag, and two background stars have $K_{\rm S} <12$ mag in this example, the majority of the secondaries is correctly classified if $K_{\rm S}=12$ mag is used to separate companions and background stars.
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In the text