AN ADAPTIVE OPTICS MULTIPLICITY CENSUS OF YOUNG STARS IN UPPER SCORPIUS

We visually inspected the images to identify all point sources, which include obvious components of multiple systems and several fainter candidate companions. Then, depending on their separation and relative brightness, we computed the position and flux of the detected sources relative to the primary stars using one or more of the three following methods. For the first method, we determined the centroid by fitting a 2D Gaussian+Moffat function and the flux ratio using aperture photometry. For the second, used for well-separated and relatively bright sources, we found the relative position and flux by minimizing the residual noise after subtraction of one component from another. For the third, used for the tightest systems, we determined the parameters by fitting a reference PSF image, taken from a single target star observed shortly before or after the target considered. Whenever possible, we compared the results from different methods and always found them to be consistent with each other. When the sources were clearly detected in the individual images, we made the measurements in the individual images and averaged them together; otherwise, we made the measurements on the averaged image only. We used the dispersion among the individual measurements or the difference between measurements made using two different methods to estimate the measurement errors; these errors are typically very small (a few mas) but can be important for the faintest sources (∼10 mas). The overall error appears to be dominated by systematic geometric distortion residuals, as noted earlier for similar observations (e.g., Lafrenière et al. 2010). A total of 190 companions or candidate companions were identified around the 91 target stars; their separation and contrast are shown in Figure 1, and their measurements are given in Table 3. The (candidate) companions detected span a range of separations from ∼005 up to ∼10'' (i.e., ∼7–1500 AU), although our observations are only mostly complete in the 01–5'' separation range (i.e., ∼15–700 AU); see Section 3.5 for more details on our survey sensitivity and completeness.

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Upon visual inspection, a few of the sources identified appeared to be potentially marginally (un)resolved binaries and were examined more closely. This is the case of the secondary of the HIP 79643 system: it is clearly elongated compared with the other sources in the field of view (including the primary). We thus treat it as a tight binary, and we used a PSF template (that of the primary star) to fit for its separation and contrast. Despite relatively poor AO correction resulting in an elongated PSF, the PSF of the primary star HIP 79374 appears to be significantly more elongated than that of its secondary companion at 134. As there was an interesting wide candidate companion around this star, it was observed again in our follow-up program, and the second-epoch data, of much better quality, clearly confirm the tight binary nature of the primary star, making this a triple system. The PSF of the tertiary component was used as a template to fit the tight binary and determine its parameters. Given the better image quality, the second-epoch measurements are given in the tables. 1RXS J155405.2−29203A is also elongated relative to its secondary companion at 143 and subtraction of the latter from the former leaves significant residuals. This source could thus be itself a tight binary, but the data at hand are insufficient to determine this with confidence. The PSF of RX J155734.4−232112 is significantly elongated relative to the faint point source 46 away, consistent with it being a marginally unresolved binary. This star was resolved into a binary by Kraus et al. (2008) using aperture masking interferometry. We thus treat this star as a binary and use a PSF template (different star observed on same night) to fit for its separation and contrast; our measurements are consistent with those reported by Kraus et al. (2008). The four tightest (marginally resolved) binaries identified in our survey are shown in Figure 2; all the other binaries were clearly resolved.

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We used star counts based on the Two Micron All-Sky Survey (2MASS) Point Source Catalog (PSC) to assess the likelihood that the numerous candidate companions identified in our images are unrelated background stars rather than real companions; a similar approach was used by Correia et al. (2006), and we used the same approach in Lafrenière et al. (2008a) for our survey of Chamaeleon I. For each target in our sample, we first retrieved all sources from the 2MASS PSC that lie within a radius of 15' and built a logarithmic cumulative distribution of the number of sources as a function of K_s magnitude. A straight line can fit these distributions extremely well from K_s ∼ 7 up to the 2MASS completeness limit of K_s ∼ 15, and thus these distributions can reasonably be extrapolated linearly to fainter magnitudes when needed; we have indeed done so for K_s = 15–20.⁸ We next converted the logarithmic cumulative distributions to surface densities as a function of limiting magnitude. Then based on these curves, we calculated, for each of the 190 sources identified, the probability that a background star at least as bright and within the same angular separation would have been found and detected around each of the 91 targets in our survey. Finally, for each candidate companion, by summing these background source detection probabilities over the 91 target stars, we obtained E_det, the expected total number of background sources of the same brightness or brighter and within the same angular separation as the candidate companion that should have been detected by our survey. The value of E_det is indicative of the likelihood of each candidate companion being truly bound or an unrelated background source: candidates with higher values are more likely to be a background source. As a validity check for our approach, for each candidate we have also counted the number of sources with the same characteristics that were actually detected in our survey, N_det; these numbers should be close to E_det if the background source surface densities that we used are correct. As shown in Table 3, N_det and E_det are always of the same order, confirming the legitimacy of our approach.

The sources identified and listed in Table 3 generally fall into two main categories, having either E_det ≪ 1 or E_det ≫ 1 (this is also visible in Figure 1 in the form of a dearth of detected sources with 4 < ΔK < 7). In all likelihood, the former are truly bound companions and the latter are unrelated background stars. For the purpose of this paper, we considered those sources with E_det ⩽ 0.05 as bound, since, according to Poisson statistics, the probability that they are indeed bound is ⩾95%. On the other hand, we considered candidates with E_det ⩾ 3 to be background, again assuming Poisson statistics, as the probability that they are indeed background is ⩾95%. There are 22 sources with an ambiguous status (around 19 stars), having 0.05 < E_det < 3. These include 1RXS J160929.1−210524b and HIP 78530B, both of which we have already followed up and confirmed to be real companions (Lafrenière et al. 2008b, 2010, 2011). Of the 17 other sources, 16 have been observed through our follow-up programs and are discussed in the next subsection (only [PGZ2001] J160545.4−202308 was not re-observed). The properties of all multiple systems—those with probabilities ⩾95% of being bound or those confirmed to be through follow-up—are given in Table 5.

For the ambiguous candidates with follow-up astrometric observations, we compared the measured relative positions of the candidate sources at the different epochs to determine, based on the known proper motion of the primary star, whether they are co-moving with their primary or unrelated background stars. When not available from the revised Hipparcos catalog (van Leeuwen 2007), the proper motions of the stars were taken from the UCAC3 catalog (Zacharias et al. 2010). As mentioned above, our follow-up observations suffered from systematic astrometric errors due to significant distortion residuals combined with largely different field-of-view orientations between the initial and follow-up observations. We assessed the magnitude of this systematic uncertainty using background sources detected in our data at the two epochs and found it to be 15 mas for separations <4'', 25 mas for <8'', and 50 mas for <12''. The systematic uncertainty in position angle is 015. Despite these uncertainties, valid conclusions could be made for the majority of the candidates followed up.

For most candidates (27 from HIP 78104, 53 from HIP 79374, 29 from 1RXS J161949.6−335456, 16 and 124 from 1RXS J161303.8−225745, 33 from 1RXSJ155405.2−292032, 28 and 43 from 1RXS J160355.8−203138, and 53 from [PGZ2001] J160222.4−195653), the relative measurements are in much better agreement with the changes expected for a distant stationary background star; these candidates are thus treated as unrelated stars. For three candidates, however, the measurements are consistent with common proper motion; these are the bright source 89 from HIP 78847, the source 58 from [PGZ2001] J161031.9−191305, and the source 63 from 1RXS J154106.9−265643. Note that the latter two companions were previously detected and considered to be bound companions by Kraus et al. (2008) and Kraus & Hillenbrand (2009).

For the remaining candidates with astrometric follow-up (30 from HIP 79250, 111 from 1RXS J155500.0−234729, 56 from 1RXS J155820.3−183725, and 723 from 1RXS J160251.5−240204), our measurements do not allow us to reach a definitive conclusion because of the significant systematic astrometric errors noted earlier. Of those, the object around 1RXS J155500.0−234729 was confirmed to be a background source by Metchev & Hillenbrand (2009), while the object around 1RXS J160251.5−240204 is indicated to be a known companion in Kraus et al. (2008); the remaining two objects are treated as unrelated for the purpose of this paper. Finally, the candidate 27 from HIP 82319, which was followed up with photometry only, has colors of an early star (J − K = 0.25, H − K = 0.0) and is thus considered to be a background source.

The masses of all targets were estimated from their estimated effective temperature and the predictions of pre-main-sequence evolution models for an age of 5 Myr. This approach is less affected by unresolved binarity, uncertainties in extinction (although small in US), or uncertainties in distance and was thus preferred over a determination based on the source luminosity. To be able to estimate masses in a consistent manner across most of the range of masses observed, we primarily used the models of D'Antona & Mazzitelli (1997),⁹ which cover the mass range 0.017–3 M_☉, sufficient for all but the 14 most massive stars in our sample. For these 14 stars, we used the models from Schaller et al. (1992), which are in excellent agreement with those of D'Antona & Mazzitelli (1997) at 2–3 M_☉, where the transition is made. This combination of models enables determination of the masses of all primary stars observed and all but the lowest mass companion found (1RXS J160929.1−210524b). The estimated masses of all targets are indicated in Table 1.

The mass ratios of the secondary (and tertiary) components of multiple systems were estimated based on their K-band contrast with respect to their primary. More precisely, the observed K-band contrast of the companion was added to the model K-band absolute magnitude of the primary (i.e., derived from the models based on its effective temperature), and the corresponding companion mass was then obtained from the models; this approach was preferred because contrast is not affected by uncertainties in distance and—to first order—extinction. For reference, the model K-band magnitude versus mass (or effective temperature) relation used here is shown in Figure 3. The estimated absolute magnitude of 1RXS J160929.1−210524b falls beyond the faint limit of the evolution models of D'Antona & Mazzitelli (1997), so for this companion we simply used the mass estimated in Lafrenière et al. (2008b).

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For all mass estimates we have assumed a fixed age of 5 Myr, but we note that an uncertainty on the age of ±1 Myr would typically translate into an uncertainty on the masses of ±2% for <0.5 M_☉, ±4% for 0.5–1.5 M_☉ and ±7% for 1.5–3 M_☉, with a smaller effect on the inferred mass ratios.¹⁰ The uncertainties on the primary stars' effective temperature also affect our mass estimates. Assuming a typical uncertainty of ±0.5 subtype for the classification of the primaries leads to typical uncertainties on the temperatures of ±2% for <10,000 K and ±6% for hotter stars. In turn, this leads to uncertainties on the derived masses of ±5%–10% for <10 M_☉ and up to ±20% for higher masses. On top of these uncertainties, one must also consider the uncertainties in the models themselves. These can be quite large. For example, using the models of Baraffe et al. (1998) would lead to masses higher by 25%–50% depending on the effective temperature of the star and the mixing length parameter value used for the models. The effect on the mass ratios is somewhat lower, with differences of 10%–30% between the two models. The actual mass and mass ratio values quoted in this work thus bear large uncertainties and must be used with care, but as shown in Lafrenière et al. (2008a), these uncertainties and the particular choice of models should not significantly affect the main conclusions of this paper. The caveats mentioned above should also be kept in mind when comparing our results with those of other surveys, for which the methodology and models used for the mass estimates could be different.

The detection limits as a function of angular separation from the target stars were obtained by calculating the standard deviation of the residual signal in annuli centered on the target after subtraction of an azimuthally symmetric PSF profile. The detection limits were determined as five times this standard deviation. For multiple systems, the secondary and/or tertiary component were subtracted from the image using an analytic or template PSF model prior to the calculation of the detection limits. Since at the smallest separations the variance of the residual noise may be underestimated by this procedure as there are too few independent spatial elements within the corresponding annuli, we have required that the detection limits within a radius of 2 FWHM be no better than the PSF intensity profile. Mass ratio limits have also been calculated for each target based on their K-band contrast limits using the approach described above, with the only difference here being that, as the contrast limits reached often corresponded to masses below the lowest mass of the models of D'Antona & Mazzitelli 1997 (0.017 M_☉), we extended the models below 0.017 M_☉ using the models with DUSTY atmospheres from Baraffe et al. (2003) and Chabrier et al. (2000). Both sets of models agree very well around the transition point (within 0.017–0.04 M_☉).

The K-band detection limits achieved for each individual target are indicated in Table 2 for six angular separations ranging from 01 to 5'' (∼15–700 AU). They are also summarized, along with contrast and mass ratio limits, in Table 4. The table reports the detection limits achieved simultaneously throughout the 01''−05 range for at least 50% and 90% of the stars in our sample; the latter limits can be viewed as a completeness limit for our sample. In brief, the contrast limits achieved for at least 90% of the sample are 4.1 mag at 025, 7.1 mag at 1'', and 7.5 mag at 5''; the corresponding mass ratio limits are 0.063, 0.017, and 0.015, respectively. Finally, as the limits achieved bear some dependence on the primary star mass (or more specifically on the star brightness), Table 4 also presents the limits corresponding to different primary mass bins. For instance, the contrast limits are not as good for fainter targets and targets observed with LGS due to poorer AO correction, and a brighter apparent magnitude limit was reached for brighter targets in both the contrast- and background-limited regimes, the latter case being due to the use of a narrowband filter.

Table 4. Sample Detection Limits

Mass Bin	Separation
(M☉)	01	025	05	1''	2''	5''
$\Delta K_{50\%}/\Delta K_{90\%}$ (mag)
All	2.78/1.80	5.00/4.05	6.17/5.23	8.22/7.14	10.13/7.43	10.75/7.48
0.2–0.5	2.30/1.32	4.28/3.74	5.46/5.02	7.44/7.14	8.14/7.43	8.19/7.48
0.5–1.2	2.70/2.35	4.34/4.06	5.68/5.32	7.83/7.32	9.86/9.46	10.66/10.22
1.2–3.0	3.17/2.98	5.45/5.10	6.80/6.43	8.81/8.44	10.53/10.34	10.90/10.83
3.0–7.5	3.15/2.95	5.61/5.46	6.95/6.66	9.00/8.75	10.38/10.38	12.23/12.23
7.5–16	3.22/2.94	5.76/5.49	6.91/6.91	9.20/9.10	10.65/10.58	12.72/12.72
$K_{50\%}/K_{90\%}$ (mag)
All	11.3/9.3	13.5/11.5	14.4/13.1	16.2/14.5	17.8/15.6	17.8/15.8
0.2–0.5	12.3/10.6	14.8/13.3	15.9/14.7	17.9/16.6	18.6/17.8	18.6/17.8
0.5–1.2	12.1/11.3	14.1/12.8	15.4/14.2	17.4/16.2	19.4/18.3	20.4/19.2
1.2–3.0	11.1/10.2	13.5/12.5	14.6/13.9	16.5/15.7	18.5/17.6	19.4/18.5
3.0–7.5	9.3/8.7	11.7/11.2	13.1/12.4	15.3/14.5	16.7/16.3	18.3/18.1
7.5–16	8.4/7.7	11.0/10.3	12.2/11.7	14.4/13.9	15.9/15.3	18.1/17.6
$q_{50\%}/q_{90\%}$
All	0.24/0.34	0.034/0.063	0.019/0.038	0.0075/0.017	0.0047/0.015	0.0041/0.015
0.2–0.5	0.14/0.35	0.047/0.12	0.024/0.087	0.012/0.028	0.010/0.020	0.010/0.020
0.5–1.2	0.11/0.22	0.031/0.044	0.019/0.031	0.0070/0.010	0.0042/0.0058	0.0034/0.0049
1.2–3.0	0.18/0.25	0.021/0.033	0.011/0.020	0.0044/0.0073	0.0022/0.0033	0.0019/0.0023
3.0–7.5	0.20/0.21	0.025/0.039	0.0079/0.023	0.0030/0.0072	0.0015/0.0042	0.0009/0.0038
7.5–16	0.19/0.33	0.033/0.081	0.011/0.023	0.0029/0.0034	0.0014/0.0021	0.0005/0.0006

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The raw MF of our sample is 0.37 ± 0.05, while the mean number of companions per star, or companion star fraction (CSF), is 0.43$_{-0.06}^{+0.08}$. The errors quoted are for 67% credibility and assume binomial statistics for the MF and Poisson statistics for the CSF. The raw MF and CSF should be viewed as strict lower limits as they are based on a sample that includes companions across a regime (separations from ∼005 to ∼9'' and contrasts of up to ∼7.5 mag) where our survey is incomplete due to varying levels of sensitivities for our different targets, as well as to a strong dependence of sensitivity with angular separation. Restricting instead to our complete sample, defined as including only the 80 targets for which the K-band magnitude limits achieved at all separations were at least as good as our 90% detection limits (from K < 9.3 at 01 to K < 15.8 at 5''; see Table 4), and considering only companions above those limits and with separations of 01–5'' (∼15–700 AU), the MF and CSF are 0.30 ± 0.05 and 0.30$_{-0.05}^{+0.07}$, respectively. The multiplicity values are significantly lower for the complete sample as several of the companions found fall outside of our completeness separation range.

As our sample was designed to have roughly equal numbers of stars in different logarithmic mass bins, its corresponding MF does not necessarily reflect the stellar population of Upper Sco. We can account for this by applying weights proportional to the initial mass function (IMF) to the systems observed. Doing so using a Kroupa IMF (Kroupa 2001), we obtain an MF and CSF of 0.37 ± 0.05 and 0.42$_{-0.06}^{+0.08}$, respectively, for our raw sample; the effect here is thus negligible. Applying a similar correction to our complete sample, we obtain an MF and CSF of 0.27$_{-0.04}^{+0.05}$ and 0.27$_{-0.05}^{+0.07}$, respectively, close to the uncorrected values. While useful to assess the effect of our mass-based target selection on the multiplicity statistics—which here turns out to be small—we note that making such a correction does introduce additional systematic errors, which stem from uncertainties about the universality and validity of the specific IMF used, as well as from uncertainties on the stellar mass estimates.

As for our survey for Cha I, we compare our results to those of the surveys of ρ Ophiuchus by Ratzka et al. (2005), of Taurus by Kohler & Leinert (1998), of the Orion Nebula cluster by Köhler et al. (2006), and of the IC 348 cluster by Duchêne et al. (1999). The latter two regions are dense star-forming regions, while the former, as well as Cha I and US, are more dispersed. We take the same approach as before and adapt our results to match the sensitivity regimes of the other surveys before making a comparison. For convenience, the statistics from these surveys are reproduced in Table 6, along with our new results. For the reasons given above, we do not attempt to correct our sample for the stellar IMF in making the comparisons. We first compare our new results for US with our previous results for the Cha I region. Considering the detection and separation limits corresponding to the complete Cha I sample of Lafrenière et al. (2008a), 01 < θ < 6'' and ΔK < 1.2 at 01 to <3.1 at 6'', the present survey yields an MF of 0.24$_{-0.04}^{+0.05}$, compared with a value of 0.27$_{-0.04}^{+0.05}$ for Cha I. The MF is thus the same in both regions in this regime of contrast and separation. Taken at face value, the CSF for this restricted sample is marginally lower in US (0.24$_{-0.04}^{+0.06}$) than in Cha I (0.32$_{-0.05}^{+0.06}$), but the difference is not statistically significant. As we find the same results within the statistical uncertainties for these two samples, we also merged them together to improve the statistics. Now considering a more restricted sample (013 < θ < 64, ΔK < 2.5, and K < 10.4), allowing comparison with a larger set of surveys, we obtain an MF of 0.23 ± 0.05 for US, which agrees within the uncertainties with the value of 0.28 ± 0.03 found for the other dispersed regions considered (Cha I+ρ Oph+Taurus). The comparison with the two denser regions, ONC and IC 348 (both requiring an even more restricted sample), does reveal a difference: the MFs in both US and Cha I are higher by factors of 2–3. The statistics from the merged US+Cha I sample indicate that the probabilities that their MF was drawn from the same underlying binomial distribution as for the ONC and IC 348 are only 2% and 6%, respectively.¹¹ The higher occurrence of binaries by factors of two to three in more dispersed regions is thus statistically significant.

Table 6. Multiplicity Fractions of Various Samples

Study	Region	N	B	T	MF	CSF
Raw sample
This work	US	91	29	5	0.37 ± 0.05	0.43$_{-0.06}^{+0.08}$
This work, IMF-weighted	US	91	29.6a	4.4a	0.37 ± 0.05	0.42$_{-0.06}^{+0.08}$
Complete sampleb
This work	US	80	24	0	0.30 ± 0.05	0.30$_{-0.05}^{+0.07}$
This work, IMF-weighted	US	80	18.7a	0	0.27$_{-0.04}^{+0.05}$	0.27$_{-0.05}^{+0.07}$
Restricted sample (01 < θ < 6'' and ΔK < 90% limit of Lafrenière et al. 2008a)
This work	US	88	21	0	0.24$_{-0.04}^{+0.05}$	0.24$_{-0.04}^{+0.06}$
Lafrenière et al. (2008a)	Cha I	110	25	5	$0.27_{-0.04}^{+0.05}$	$0.32_{-0.05}^{+0.06}$
Combined US+Cha I	⋅⋅⋅	198	46	5	0.26 ± 0.03	0.28 ± 0.04
Restricted sample (013 < θ < 64, ΔK < 2.5, and K < 10.4)
This work	US	79	18	0	0.23$_{-0.04}^{+0.05}$	0.23$_{-0.04}^{+0.07}$
Combined Cha I+ρ Oph+Taurusc	⋅⋅⋅	278	72	5	0.28 ± 0.03	0.29 ± 0.03
Restricted sample (0.1 < M/M☉ < 2.0, 60 < θ/AU < 500, and ΔK < 3.5)
This work	US	57	10	0	0.18$_{-0.05}^{+0.06}$	⋅⋅⋅
Lafrenière et al. (2008a)	Cha I	96	13	0	$0.14_{-0.03}^{+0.04}$	⋅⋅⋅
Combined US+Cha I	⋅⋅⋅	153	23	0	0.15 ± 0.03	⋅⋅⋅
Köhler et al. (2006)	ONC	105	5.4	0	$0.05_{-0.02}^{+0.03}$	⋅⋅⋅
Restricted sample (32 < θ/AU < 640 and q > 0.1)
This work	US	83	21	0	0.25$_{-0.05}^{+0.06}$	⋅⋅⋅
Lafrenière et al. (2008a)	Cha I	76	18	0	$0.24_{-0.05}^{+0.06}$	⋅⋅⋅
Combined US+Cha I	⋅⋅⋅	159	39	0	$0.25_{-0.03}^{+0.04}$	⋅⋅⋅
Duchêne et al. (1999)	IC 348	66	8.5	0	$0.13_{-0.04}^{+0.05}$	⋅⋅⋅

Notes. ^aThe effective number is given, after applying the weights as explained in the text. ^bConsidering only targets for which the K-band magnitude limits achieved at all separations were at least as good as our 90% detection limits (see Table 4), and considering only companions above those limits and with separations in 01–5''. ^cSee Table 6 of Lafrenière et al. (2008a).

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Finally, we compare our MF to the value measured for the field. Assuming that the lognormal companion separation distribution observed by Raghavan et al. (2010) for field solar-type stars applies to stars of all masses, our orbital separation range of sensitivity (roughly 15–1000 AU for the combined US+Cha I sample) would contain 44% of all companions. This might be a slight overestimate, however, as the companion separation distribution of M-type stars, which dominate the IMF, has been found to differ from that of solar-type stars (Janson et al. 2012), being shifted toward smaller separations, although their specific distribution is unclear at the moment. Still, using this value in combination with the field IMF-weighted MF of about 0.33 reported by Lada (2006) leads to a field MF of 0.15 within our orbital separation range of sensitivity. The MF in the field is thus a factor of ∼2 below the value for the restricted US+Cha I sample (0.26), as well as a factor of ∼2 below the value for other dispersed star-forming regions per the above remarks.

Besides the overall MF of the sample, we have also divided our sample into four equal logarithmic mass bins and calculated the MF in each bin to investigate how these quantities vary with the primary star mass; the results are shown in Figure 5. Given the uncertainties in each bin, the distribution of MF in US appears to be consistent with being flat over the 0.2–7.5 M_☉ range. Indeed, a straight line fit of the distribution returns a slope of 0.04 ± 0.13. In Figure 5, we also show the MF as a function of mass obtained for our Cha I survey. In the common regime of primary masses, the distributions in both regions agree well within the uncertainties. In the context of the generally accepted trend that multiplicity (integrated over all separations) rises and even becomes dominant at high masses (e.g., Raghavan et al. 2010; Janson et al. 2012; Kouwenhoven et al. 2007b; Delfosse et al. 2004; Siegler et al. 2005; Joergens 2008), the relative flatness of the MF distribution with mass observed in our US sample may be due to our limited orbital separation range, and does not necessarily indicate that the trend observed in the field does not hold for US. Indeed, differences in the companion orbital separation distributions with mass would mean that different fractions of the total population of binaries would be detected in our survey for different masses. For example, Duchêne & Kraus (2013) mention that the companion separation distribution for stars of mass 1.5–5 M_☉ shows two main peaks, one falling well below our minimum separation limit and the other extending beyond our outer limit. On the other hand, the companion separation distributions of 0.2–1.5 M_☉ stars roughly peak within our range of sensitivity. Thus, our completeness to binaries is likely lower for higher mass stars compared to lower mass ones. To illustrate that considering a restricted separation range can indeed affect the observed trend of MF with mass, we consider the results of Raghavan et al. (2010) for (field) G-type stars and of Janson et al. (2012) for (field) M-type stars. For M-type stars, Janson et al. find an MF of 0.27 ± 0.03 for the 3–227 AU separation range. For G-type stars, Raghavan et al. find an MF of 0.46 ± 0.02 over all separations, which, based on their companion separation distribution, translates into an MF of ∼0.21 for the 3–227 AU range. Thus, over this restricted separation range the MFs for G- and M-type stars are comparable, even though they are not when looking at the MF integrated over all separations. The relatively constant MF observed in our sample for a restricted separation range may thus be consistent with the trend observed in the field.

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Surprisingly, we found only one multiple system among the eight primaries more massive than 7.5 M_☉ that we observed (yielding an MF of 2.2%–35%). Various factors could explain this apparent discrepancy. First, as already mentioned above, a large fraction of binaries in this mass range could fall below our inner separation limit, as high-mass stars display a particularly high frequency of short-period binaries (Duchêne & Kraus 2013). Also, this mass limit (7.5 M_☉) coincides with the star brightness at which we switched from using the full detector array to using only a 512 × 512 pixel subarray, and thus the companion search field of view is smaller (<750 AU) for these stars. As the more massive stars harbor companions out to larger orbital separations (≳ 10⁴ AU; Duchêne & Kraus 2013), this could further reduce our completeness to companions for the most massive stars. In addition, as shown in Table 2, because of the different instrumental setup, the sensitivity limits reached for the most massive stars are generally about 2–4 mag worse than for typical targets (see also Table 4), which means that our completeness to low-mass companions is also reduced. In fact, from the mass ratio limits shown in Table 4 we can see that for the most massive stars we did not probe the full stellar companion regime for separations below 1''. Thus, we believe that our results for the highest mass stars should be interpreted with care. For these reasons we have not included this mass bin in Figure 5.

Figure 6 shows the mass ratio distribution for all binary systems in our survey with q > 0.1 (regime in which our completeness is good). The distribution shows a higher incidence of lower mass ratio systems (0.1 < q < 0.4) compared to higher mass ratio ones. The rise is significant, as a Kolmogorov–Smirnov (K-S) test gives a probability of only 8% that the true underlying distribution is flat. For solar-type stars in the field, the observed mass ratio distribution is roughly uniform for q > 0.2 (Raghavan et al. 2010). At first glance, this seems to differ from what we observe here; however, this flat distribution is based on a sample that includes binary companions with separations below our inner separation limit, and these small separation companions were found to preferentially have higher mass ratios. Thus, their absence from our sample could explain the apparent difference. To study further the effect of orbital separation, Figure 6 also shows the mass ratio distributions when our sample is subdivided into binaries with projected orbital separations above or below 07 (∼100 AU). The trend noted above for the overall sample is even more pronounced for systems with wider separations, and a K-S test gives a probability of only 5% that the underlying distribution is uniform. On the other hand, the distribution looks flat for systems with small separations (∼15–100 AU). In this case, the K-S test gives a probability of 74% for a uniform distribution. There is thus a significant difference between the mass ratios of companions at separations of ∼15–100 AU versus those at larger separations, the latter preferentially having lower mass ratios. The exact same trend was also observed in our Cha I study, and in the studies of ρ Oph by Ratzka et al. (2005), of Sco-Cen by Köhler et al. (2000), and of Taurus by Kohler & Leinert (1998). This is also the case for field solar-type binaries (Duchêne & Kraus 2013), where it is observed that the mass ratio distribution of binaries with separations above ∼50 AU peaks at q ∼ 0.3. This is thus a rather common feature of binary systems.

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Figure 7 shows the mass ratio of all binary systems as a function of primary mass. The figure shows an apparent lack of lower mass ratio companions around primaries of lower masses, as no companion with a mass ratio below 0.5 has been detected for primaries with masses below 0.4 M_☉, despite good sensitivity to detect them beyond 025. A similar lack of low mass ratio companions at low primary masses was also observed in our Cha I study. These results seem to differ from what is observed for 0.1–0.5 M_☉ stars in the field, where the observations are consistent with a more uniform companion mass ratio distribution for q > 0.1 (Janson et al. 2012), or only a modest decline at low mass ratios (Duchêne & Kraus 2013). Figure 7 also shows the mass ratios of the systems found in our Cha I study, where it seemed that more massive primaries had gradually lower mass ratio companions. In fact, the results in Cha I suggested a lower envelope on the mass ratio distribution that was roughly consistent with a constant companion mass (roughly coinciding with the stellar/substellar limit), possibly indicating that companion formation may become less efficient below a companion mass near the substellar limit. Given our detections of substellar companions, this is not readily apparent from our sample, although the relative number of substellar versus stellar companions found in our survey points in the same direction. In a regime where our observations are mostly complete to brown dwarf (0.012–0.072 M_☉) companions, i.e., at >05 for 0.2 < M/M_☉ ⩽ 0.5 and at >1'' for 0.5 < M/M_☉ ⩽ 3.0, we have detected 13–14 stellar companions and two to three brown dwarf companions (depending on whether the tight companion to [PGZ2001] J161031.9−191305 is counted as stellar or substellar). The observed ratio of brown dwarf to stellar companions is thus $0.19_{-0.09}^{+0.29}$ (67% credibility). Considering our sample of primary stars and assuming that the companion masses were drawn randomly from a Kroupa IMF (Kroupa 2001), then in the above regime of mass and separation our survey should have found 0.786 times as many brown dwarf companions as stellar ones, which is much higher than the observed ratio (the two values are inconsistent at 90% credibility). Thus, our data indicate that, as compared to formation in isolation, companion formation is less efficient in the brown dwarf mass regime than in the stellar mass regime.

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Figure 7 also suggests that there are fewer similar-mass binaries (q > 2/3) for higher mass primaries, compared to lower mass primaries: only 1 out of 13 binaries with primary masses of 1–3 M_☉ has a mass ratio above 2/3, while it is the case for 7 out of 21 for binaries with primary masses of 0.2–1 M_☉. The same trend was found in our study of Cha I (Lafrenière et al. 2008a), as well as for the A and B stars of Sco-Cen (Kouwenhoven et al. 2005).

As mentioned previously, all of our targets have been observed with Spitzer at 8, 16, 24, and 70 μm by Carpenter et al. (2006, 2009) to search for excess emission arising from circumstellar material. Carpenter et al. (2009) report excess emission due to a circumstellar disk for 18 of our 91 targets: 10 of them are classified as having a debris disk, and the other 8 as having a primordial disk. Among those 18 targets with disks, we find that 5 are multiple systems: HIP 78956 (debris), HIP 79156 (debris), HIP 79643 (debris), 1RXS J160621.5−192851 (primordial), and [PGZ2001] J160643.8−190805 (primordial). The fraction of multiple stars among stars with reported disks is thus 5/18 or 0.27$_{-0.11}^{+0.14}$, which is to be compared with the fraction of multiple stars among stars without disk detection, 29/73 or 0.40 ± 0.06. The incidence of multiplicity may thus be marginally lower among disk-bearing stars, although the result is not statistically significant. Considering only our restricted sample, these fractions would be 4/14 (0.29$_{-0.13}^{+0.17}$) and 20/66 (0.30$_{-0.06}^{+0.07}$), respectively, and thus would be indistinguishable. Alternatively, we can look at the incidence of disks among multiple and single stars. For the raw samples, the numbers are, respectively, 5/34 (0.15$_{-0.06}^{+0.09}$) and 13/57 (0.23$_{-0.06}^{+0.07}$), while for the restricted sample they are, respectively, 4/24 (0.17$_{-0.08}^{+0.11}$) and 10/56 (0.18$_{-0.05}^{+0.06}$). Here again there is no difference. We note that the above comparisons are rather rough in that they do not take into account the various masses of the primaries, which are known to have an effect on the incidence, properties, and evolution of the disks; the statistics in our sample are insufficient to do a more in-depth analysis. We also note that the binary companions found in the systems with disks have projected separations of 40–260 AU, which is typically beyond the dust traced by the Spitzer observations.

The present study found two new low-mass substellar (≲40 M_Jup) companions—1RXS J160929.1−210524b and HIP 78530B (Lafrenière et al. 2008b, 2010, 2011; see also Ireland et al. 2011; Bailey et al. 2013)—and confirmed a previously known one, [PGZ2001] J161031.9−191305B (found by Kraus et al. 2008; see also Lachapelle et al. 2014). The latter is the outer member of a triple system in which the inner companion may itself be substellar as its mass is estimated to be about 0.07 M_☉.

Following the Bayesian approach described in Lafrenière et al. (2007), we estimated the fraction of stars that harbor low-mass substellar companions at intermediate to large orbital separations. For each target, we first computed the completeness of our observations for companions of given masses and semi-major axes. For these Monte Carlo calculations, the companion eccentricities, mean anomalies, and orbital separation projection factors were drawn from a uniform distribution. For stars with a measured parallax, the distance was randomly sampled from a uniform distribution according to the measured value and associated uncertainty, while for the other stars the distance was sampled uniformly between 120 and 170 pc. The age of the stars was fixed at 5 Myr, and the DUSTY evolution models (Chabrier et al. 2000) were used to convert between masses and magnitudes in the K band. The mean completeness of our survey for various masses and semi-major axes is shown on Figure 8. As the figure shows, our observations generally offer good completeness for companions ≳25 M_Jup above ∼50 AU, while for companions of 4–5 M_Jup our completeness is good only for separations larger than ∼200 AU. Our completeness figure is of course affected by the assumed age and model uncertainties, but the effect is relatively small. For example, if the age were 10 Myr instead of 5 Myr, and if we assumed that the models were overluminous by one magnitude for a given mass, then the 50% completeness curve would shift upward by about 3 M_Jup and rightward by about 40 AU.

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Next, to constrain the fraction of stars with low-mass substellar companions, we considered the 5–40 M_Jup mass range, which safely encompasses the masses of the three detected companions mentioned above and safely avoids companions ambiguously near the stellar/substellar boundary. We assumed log-uniform mass and semi-major axis distributions, i.e., dN/dm∝m⁻¹ and dN/da∝a⁻¹. We have also made the calculations for two different semi-major axis ranges, 50–250 AU and 250–1000 AU, which are roughly equivalent in logarithmic space. The results are shown in Table 7. Our data indicate that less than 2.1% of US stars have a companion at 50–250 AU separation while 3.6$_{-1.1}^{+3.3}$ have one at 250–1000 AU. These statistics are insufficient to conclude firmly whether there is a difference between the companion populations at smaller and larger separation.

Table 7. Percentage of Stars Harboring 5–40 M_Jup Companions, at 67% Credibility

Semi-major Axis	Sample
Range (AU)	Up Scoa	Up Sco+b	Cha Ic	GDPSd
Raw
50–250	<2.1	<1.8	<1.2	<1.7
250–1000	3.6$_{-1.1}^{+3.3}$	4.0$_{-1.2}^{+3.0}$	<1.3	<5.7
IMF-weighted
50–250	<1.7	<1.4	<1.1	<2.0
250–1000	1.8$_{-0.5}^{+2.8}$	3.2$_{-1.0}^{+2.8}$	<1.3	<7.7

Notes. ^aThis work. ^bThis work + Kraus et al. (2008) + Ireland et al. (2011). ^cFrom Lafrenière et al. (2008a). ^dFrom Lafrenière et al. (2007).

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To improve the statistics, we merged the survey of US stars performed by Kraus et al. (2008) and Ireland et al. (2011) with ours. This adds 27 new stars to our sample and one substellar companion in a relevant mass and separation range (GSC 06214−00210B, ∼15 M_Jup, ∼320 AU; Ireland et al. 2011). For the augmented sample, the results indicate that less than 1.8% of stars have a low-mass substellar companion at 50–250 AU, while the fraction is 4.0$_{-1.2}^{+3.0}$% for 250–1000 AU. Here the difference is statistically significant: from the posterior distributions we find that, with 84% credibility, the occurrence rate of 5–40 M_Jup companions is higher at large separations than at intermediate ones. This is in line with the trend mentioned earlier that larger separation companions can have lower mass ratios.

For comparison, we repeated the same exercise with two other samples for which we readily had access to the information needed: the AO multiplicity survey of the Chamaeleon I region from Lafrenière et al. (2008a) and the Gemini Deep Planet Survey (GDPS) from Lafrenière et al. (2007). The latter survey targeted nearby young and field stars of spectral types G, K, and M. Although our survey of Cha I was not as deep as the one here, it reached a similar completeness because that region is younger (2 Myr), making substellar objects of corresponding masses significantly brighter. Conversely, GDPS targeted nearby older stars of spectral types G, K, and M, but as it was deeper and reached better contrasts, it achieved similar completeness at 50–250 AU separations. However, it missed sensitivity to companions at 250–1000 AU as the closer target distances often made those separations fall out of the camera field of view. The comparison reveals that, with 88% credibility, the occurrence rate of substellar companions at 250–1000 AU is higher in US (4.0%) than in Cha I (<1.3%). However, the statistics are insufficient to draw any conclusion with respect to the older GDPS sample.

Finally, as done previously for the MF (Section 4.1), we have repeated the above calculations by factoring in the IMF to estimate values that may be more representative of the stellar population as a whole. In practice, this was done by raising each individual probability in the computation of the likelihood function to a power proportional to the probability of drawing the corresponding stellar mass from the IMF. These "powers," or weights, were normalized to have a mean of 1. Again, we used a Kroupa IMF. The results are shown in Table 7. The corrections are all small except for the US sample from this study and the 250–1000 AU separation range. In the latter case, as one of the three detected companions orbits a massive star, with a much lower weight, factoring in the IMF significantly reduced the calculated rate. In any case, all of the above conclusions still hold with the correction.