HST Survey of the Orion Nebula Cluster in the H₂O 1.4 μm Absorption Band. III. The Population of Substellar Binary Companions

As reported in Paper I, saturation in the F130N filter starts at m ≃ 10.9, while the noise floor is at m ≃ 22, setting the magnitude limits of the primaries and companions we are able to analyze.

Our input catalog of targets contained 8210 individual detections (of which 4220 are unique) with m₁₃₀ magnitudes in the range 10.9–22; about 50% of them correspond to repeated detections of the same sources.

Our PSF subtraction technique requires a reference catalog of sources uncontaminated by astrophysical or instrumental noise. We create it from our sample and perform several clean-up steps:

• 1.  
  Visual binary removal. We remove from our catalog 157 unique pairs, for a total of 623 total entries with a neighbor closer than 15 projected distance according to Catalog I. In this way, we avoid contamination from nearby neighbors whose PSF wings may affect the region searched for low-mass companions.
• 2.  
  Bad pixel removal. The data set of full-frame WFC3 images is cleaned from cosmic-ray events in the early stages of standard data processing thanks to the nondestructive sampling of the accumulating signal. Static bad pixels are also flagged by the pipeline. However, we perform an independent check by stacking the images and applying a 10σ threshold to the distribution of median pixel values. We did not find any detection with a flagged pixel closer than ∼08 in any visit.
• 3.  
  ACS catalog matching. The HST/ACS survey of Robberto et al. (2013) provides a high-resolution morphological classification of the sources in the ONC. By cross-matching the ACS catalog with our list of WFC3 detections, we discard all objects flagged as nonstellar: silhouette disk, proplyds, sources with evidence of jets or photoionization, Herbig–Haro objects, or resolved galaxies. We discarded a total of 222 unique objects for a total of 458 entries from the catalog.

Applying these selection criteria, we end up with a catalog of 7129 individual sources, counting multiple observations of the same object separately.

The next step is to create "postage stamps" centered on each source and perform the PSF subtraction inside this area. In setting our 11 × 11 (15 × 15) pixel stamp size, we consider the following factors:

• 1.  
  The area must be large enough to contain the bright wings of the PSF, for sources matching our assumed range of magnitudes.
• 2.  
  The area must have enough pixels to provide a meaningful noise calculation. Detections of close companions are affected by small-number statistics, and a correction to the estimated contrast and signal-to-noise ratio (S/N) has to be applied (Mawet et al. 2014). The following argument shows that the correction is very small for an 11 × 11 stamp. The number of λ/D resolution elements per pixel for WFC3 in the F139M filter is close to 1; that is, WFC3-IR is significantly undersampled. Therefore, an 11 × 11 pixel stamp contains about the same number of resolution elements. The correction factor to the S/N is given by ${\left(\sqrt{1+1/n}\right)}^{-1}$, which for n = 121 is 0.996. Therefore, the sample size does not represent a significant source of uncertainty versus other noise sources, such as photon or read noise.
• 3.  
  The area must be small enough that tiles do not overlap; having rejected from our catalog objects with a nearest companion closer than 15, we find this results in a tile half-size of 07. With a WFC3 pixel scale of 013 pixel⁻¹, the tile half-size translates to a radius of approximately 280 au from a point source in the ONC.

Accurate PSF subtraction depends strongly on the quality of the reference PSF, a task greatly simplified by the stability of the HST, which has enabled the compilation of libraries of PSF models for reference differential imaging (e.g., Choquet et al. 2014). Still, for the most accurate PSF subtraction, one has to deal with the field distortion of WFC3 and the small but nonnegligible time dependence of the HST focus. These effects make the PSF both spatially and time dependent. Our strategy is especially well suited for handling both effects.

The strategy consists of dividing the field of view into 100 equal cells, each cell small enough to neglect local PSF distortion but large enough to build a local PSF library containing enough stars to build an accurate model.

For each cell, PSF subtraction is then performed as follows:

• 1.  
  The postage stamps for all stars in the cell are stacked together into a single data cube.
• 2.  
  Iterating through the data cube, each stamp is assumed as the science image.
• 3.  
  A reference model of the PSF for subtraction is constructed, selecting from the remaining postage stamps those with a photometric error σ_F130N ≤ 0.01.
• 4.  
  The PSF of the target star is then removed using the KLIP algorithm (Soummer et al. 2012).

For each target, we chose the number of modes that simultaneously minimize the standard deviation of the residual image while maximizing the counts of the brightest residual pixel.

To build a preliminary catalog of candidate binaries, we analyze the position of the brightest pixel of the residual images of each target. To be labeled as a candidate detection, at this early stage, we require that

• 1.  
  the pixels with the highest flux in each residual must be within one pixel in both filters and in all available visits when the source is observed with different telescope orientations, and
• 2.  
  the candidate must be detected in at least two different KLIP modes.

The one-pixel distance (rather than zero) is needed to take into account possible misalignments of the center of the stars in the reference library, due to the undersampled PSF and lack of dithering in the survey. This is reflected in an accuracy of our separation estimates of about one-half pixel, that is, 007 or 28 au at the distance of the ONC.

Inspection of the residuals immediately after PSF subtraction reveals a large number of candidate companions, but further down-selection has to be applied to reject sources that presumably do not belong to the ONC. To separate cluster stars from background sources, we use the position of the stars on the CMD. As shown in Paper I, the pair of filters chosen for this survey is sensitive to the depth of the 1.4 μm H₂O absorption band. This temperature-sensitive feature is prominent in the atmosphere of M-type stars and brown dwarfs, down to planetary-mass objects, and can be then used to separate the substellar cluster population of the ONC from background stars and galaxies. Following Paper I, we consider a source to be an ONC member if it lies in the area delimited by the 1 Myr isochrone introduced in Section 2, reddened by A_V = 10 mag. Any companion candidate bluer (redder) than this isochrone is labeled as cluster (background). In Paper I, we found good agreement between this simple approach and a more rigorous Bayesian statistical treatment. At the end of this process, we obtained 2797 multiple-visit cluster sources, with 1392 unique targets for our KLIP PSF subtraction algorithm.

Since the WFC3/IR PSF is highly undersampled, after PSF subtraction we expect most of the flux from a faint candidate companion to be contained within a few pixels. Thus, to derive the total flux, one has to apply a large and rather uncertain aperture correction. To evaluate it, we analyze a sample of isolated bright stars in our catalog, comparing their flux around the brightest pixels with their total flux. This analysis shows that about one-third of the flux is contained within the brightest pixel and ∼60% within the four adjacent brightest pixels. The distribution of relative fluxes for the four brightest pixels is narrower than the distribution for the single pixel. Therefore, we perform our photometry of the companions using a four-pixel aperture, deriving the aperture correction to the total flux through comparison with the Catalog I PSF photometry. Specifically, for each isolated source in Catalog I, we built a square 2 × 2 pixel mask placed so that one pixel always coincides with the brightest pixel of the original image. After probing the four possible mask positions, we record the maximum value of the total counts as c_4p. The magnitude for each primary is then calculated as follows:

$$\begin{eqnarray}&&{m}_{4p}=-2.5{\mathrm{log}}_{10}({c}_{4p})+C\end{eqnarray} \tag{ 1 }$$

where C is a normalization factor between the four-pixel photometry and PSF photometry (${C}_{{m}_{130}}=21.35\pm 0.049$, ${C}_{{m}_{130}-{m}_{139}}=-0.002\pm 0.031$). We then determine C as the mean of the difference between the PSF photometry and the four-pixel photometry of each primary:

$$\begin{eqnarray}&&\langle {m}_{\mathrm{PSF}}-{m}_{4p}\rangle =\,C.\end{eqnarray} \tag{ 2 }$$

Measuring c_4p for each detected companion and using Equation (1) and the value of C from Equation (2), we determine the magnitudes of our candidate companions. Our estimate of the total uncertainty takes into account the uncertainty on the counts of the candidate, on the background counts in the four-pixel aperture, and on the estimated conversion factor between the PSF and the four-pixel system (the standard deviation of the sample we used to evaluate the conversion factor).

Having determined the photometry for each candidate, a new selection is applied while keeping all of the cluster pairs with companion magnitude in the range 10.9 ≥ mag₁₃₀ ≥ 22 (following an approach similar to that mentioned in Section 3.3) and with absolute value of the m₁₃₀–m₁₃₉ color ≤1 to reject noisy outliers. This results in a preliminary selection of 145 cluster candidate binaries.

To assess our ability to separate plausible candidates from instrument-induced false-positive detections, we perform an extensive set of simulations to determine the receiving operating characteristic (ROC) curves (see Appendix A for an explanation of ROC curve construction) for each binary configuration in our preliminary catalog. A configuration is specified by three parameters: (1) brightness of the primary, (2) contrast between primary and companion, and (3) separation and KLIP mode used during the PSF subtraction phase. We use the ROC curves to derive three other quantities we can use to make the following selections on our candidates:

• 1.  
  Area under the curve (AUC) of the ROC. The AUC provides us with a good indication of how well the distribution of the true-positive rate (TPR, i.e., detection of companions injected in our simulations) is separated from the distribution of the false-positive rate (FPR, i.e., detection of noise peaks that may have been erroneously determined to be companions). An AUC curve of 0.5 indicates that there is no possibility of separating the two distributions, whereas an AUC = 1 represents perfect separation. An analysis of the results provided by the simulations led us to select candidates only when the corresponding configuration provides an AUC ≥ 0.7.
• 2.  
  False-positive probability and S/N threshold. As explained in Appendix A, for each given configuration, the ROC curve is built by sliding an S/N threshold across the TPR and FPR distributions. We can therefore invert this process: given the ROC curve for a certain configuration and having determined a limit to the probability for a detection to be a false positive, we find the corresponding S/N that we can use as a threshold for the detection. Because each candidate is found using multiple independent detections (different filters and possibly different locations on the detector for each visit), we multiply the false-positive probabilities of each detection ($\mathrm{FP}^{\prime}$) to obtain an overall false-positive probability for the whole candidate (FP). In particular, if we assume $\mathrm{FP}^{\prime}$ to be the same for each detection, it is

  $$\begin{eqnarray}&&\mathrm{FP}=\mathrm{FP}{{\prime} }^{({N}_{f}\times {N}_{v})},\end{eqnarray} \tag{ 3 }$$

  where N_f is the number of filters and N_v is the number of visits for the candidate. Inverting this relation, we find $\mathrm{FP}^{\prime}$ as a function of FP. Having set $\mathrm{FP}^{\prime}$, we can find the corresponding S/N threshold from the ROC. With 1392 primaries to be searched, assuming an overall false-positive probability FP = 0.2% for each candidate, we expect about three false-positive detections in our final catalog of binaries. We have verified that this probability value represents an optimal trade-off. A further reduction, that is, a more aggressive reduction of false positives, would imply higher detection thresholds, which would lead to rejecting strong, previously known true detections. Vice versa, relaxing the threshold would cause a large increase in the number of false positives beyond the acceptable rate of 50–100 smaller than the expected detection signal (as a point of reference, the expected binary fraction is 10%–20% as per Kraus and Duchene).
• 3.  
  Ratio of true positives over false positives (R). For each candidate detection, we binned the TPR and FPR distributions in bins of 0.5 S/N, and we evaluate the ratio of true positives over false positives in the same bin corresponding to the candidate S/N detection. This parameter gives us an indication of how common the candidate S/N is in the distribution of false positives and true positives. Because each candidate results from multiple detections, we keep only candidates with an R_median ≥ 3.

As a by-product of our simulations, we also obtain the amount of flux lost to oversubtraction (see Pueyo 2016 and references therein), deriving the correction to apply to the photometry of our candidate companions, with the relative errors. Moreover, from the distributions of TPR and FPR, we can also evaluate the contrast curves as a function of the magnitude of the primary, contrast, and separation. Averaging all data, we obtain the contrast curves shown in Figure 2.

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The preceding analysis is not designed to distinguish between true companions and other astrophysical sources of false positives. These include residual contamination from nearby stars and light emitted by circumstellar material. Detector persistence may cause "ghosts" of very bright stars in the subsequent exposures, but they also appear as extended structures that can be easily identified and generally decay within one visit (see Paper I). This is why to conclude this candidate selection we visually inspect all our selected candidates by looking for extended residuals.

To estimate the mass of our substellar companions, we start with an analysis of the primaries and isolated ONC stars. Figure 3 shows the comparison between the masses estimated by da Rio et al. (2012) using the Baraffe et al. (1998) evolutionary models (DR2012_mass) and the masses obtained from our dereddened WFC3 photometry and the 1 Myr isochrone (WFC3_mass, gray points). We use the value of A_V determined by Da Rio et al. when available, otherwise we use the A_V estimate from Paper I, with negative A_V values set to A_V = 0. In the range of WFC3_mass between 0.075 and 1.5M_⊙ (vertical lines in the plot), we observe good correlation with some systematic differences between the two mass estimates. Below this range, the scatter increases, an indication of the difficulty of the DR2012 optical survey in dealing with the reddest and faintest sources of their sample. To reconcile the two data sets, we use an empirical isochrone, fitting the relation between the DR2012_mass and the WFC3_mass in the 0.075–1.5M_⊙ mass range with a spline function as follows:

• 1.  
  We bin the distribution of F130N_mass between 0.075 and 1.5M_⊙. To have bins perpendicular to the DR2012_mass = WFC3_mass relation (blue line in Figure 3), we apply a rotation matrix to the data by an angle of 45°.
• 2.  
  We apply a 3σ cut to the distribution of each bin to exclude outliers.
• 3.  
  We rotate back the data, and we fit a spline matching the 1 Myr isochrone outside the 0.075–1.5M_⊙ WFC3_mass range and the 3-sigma select data otherwise (black dotted line in Figure 3).

In the substellar regime, we decide to rely only on our WFC3 data because of the strong correlation between mass and stellar flux (m₁₃₀), as evidenced by the color–magnitude diagram (Figure 1).

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Finally, to evaluate the mass of our candidate binaries, we assign the same A_V values to both components and then evaluate the mass of the companion using the spline curve.

The completeness of our survey depends on the mass of the primary, the mass ratio of potential candidates, and their separation, that is, the projected SMA. This function, marginalized over the mass of the primaries, can be represented by a set of completeness curves for the mass ratio of the candidate and separation. Completeness as a function of the magnitude of the primary and companion and visual separations can be obtained by direct inspection of the family of ROC curves discussed in Section 3.5. It can then be converted into a completeness as a function of primary mass, mass ratio, and deprojected orbital SMA. This last step is carried out using the following procedure:

• 1.  
  We interpolate over a finer grid both in mass ratio and separation.
• 2.  
  Following Brandt et al. (2014), we integrate over all the possible semimajor axes (s) between 0 and 1.8 using a piecewise function p(s):

  $$\begin{eqnarray}p(s)\simeq \left\{\begin{array}{ll}1.3s & 0\leqslant \ s\ \leqslant \ 1\\ -\tfrac{35}{32}\left(s-\tfrac{9}{5}\right) & 1\lt \ s\ \lt \ 1.8.\end{array}\right.\end{eqnarray} \tag{ 4 }$$

  We then use this completeness map to apply a final selection to our catalog of candidates to reject any detection with completeness smaller than 10% or between 10% and 30% and with only one visit (i.e., the most likely to be one of the few false positives we expect, since our FP analysis was carried out using single visits). At the end of this selection process, we obtain a final catalog of 39 reliable cluster candidate binaries out of 1392 original cluster targets.

Figure 4 shows the final completeness curves as a function of separation in SMA, where the black dots mark the position of our detections on the completeness map. The magenta line shows the 30% completeness cut we apply to our single-visit detection, while the gray area shows the space of parameters in the plot where we always reject candidates because completeness is smaller than 10%.

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The analysis described in Section 3 provides us with a total of 39 candidate cluster binaries with separation in the range 1.26–5.9 pixels (016–077), corresponding to about 66–309 au projected distance from the primary assuming a distance of 403 pc (Kuhn et al. 2019). The primary masses range between 0.015 M_☉ and 1.27 M_☉, while the companions are in the range 0.004 M_☉–0.54 M_☉.

Table 1 shows the physical and photometric properties of the 39 candidates. Column (1) shows the entry number in the catalog; columns (2) and (3) show the R.A. and decl. for Equinox J2000.0; columns (4) to (11) list the m₁₃₀ magnitude and the m₁₃₀−m₁₃₉ color with their relative uncertainties for both primary (P) and companion (C); columns (12) to (15) show the estimated mass from F130N photometry, with its uncertainty, for both primary and companion in units of solar mass. The last three columns list the position angle, the separation between primary and companion, and the distance of the system from the core of the cluster (identified by the position of θ¹Ori-C).

In Appendix B we present a gallery of postage stamps (Figures B1–B2) showing the coadded images before and after KLIP subtraction for each candidate; the companions generally appear as bright single pixels in each residual image, due to the WFC3/IR subsampling. Each postage stamp has dimensions 2'' × 2'' and is rotated so that north is up and east is to the left.

In Figure 5 we show the position of each candidate cluster binary projected against the survey area of the WFC3 survey, while Figure 6 shows the color–magnitude diagram for the entire region with the locus of the KLIP candidate cluster binaries.

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Previously, binary systems that were well resolved in Catalog I were excluded from our analysis, which was designed to discern close companions hidden under the PSF wings of apparently single stars. We now expand our close-companion catalog by adding the wider pairs from Catalog I: 58 systems with projected separation d < 18 (choosing as limit the maximum distance at which we still measure an increase in the number density of stars; see Figure 8 in Section 4.4) and colors compatible with cluster membership for both sources. The brightest star of each pair is generally taken as the primary. Adopting the F130N filter photometry in Catalog I to estimate their masses, we obtain for the primaries values in the range M_P = 0.02–1.08M_☉, while for the companions we find M_C = 0.01–1.04M_☉. Their photometry and resulting physical parameters are listed in Table 2, and a gallery of images is shown in Figure B3 in Appendix B (similar to what we presented for the KLIP pairs).

It should be noted that we did not attempt to find faint substellar companions under the PSF wings of the Paper I binaries, as this goes beyond the current capabilities of our implementation of the KLIP algorithm. Our search strategy, therefore, is generally biased against finding triplets or higher order systems.

Hereafter we will refer to the combination of Tables 1 and 2 as our Master Catalog. The Master Catalog contains 97 pairs of stars with separations between 016 and 173 (corresponding to 66–697 au) and masses in the range ${M}_{P}=0.015\mbox{--}1.27$ M_☉ and M_C = 0.004–1.04 M_☉ for the primary and companion, respectively.

Figure 7 shows the relation between primary and companion masses for all sources in the Master Catalog, with relative error bars and colors identifying the KLIP binaries (black) versus Catalog I binaries (blue). The diagonal lines mark the loci of systems with primary mass equal to 1, 10, and 100 times the mass of the companion, whereas the horizontal and vertical lines indicate the boundaries between stellar, brown dwarfs, and planetary-mass objects. The number of systems in the areas delimited by these lines is given in Table 3. Overall, we observe a primary star-to-brown-dwarf ratio (SBdR) N(0.1–1.27 M_☉)/N(0.014–0.07 M_☉) = 15.16, while the same ratio for isolated stars in the ONC (∼3.8 evaluated from Catalog I or ${3.3}_{-0.7}^{+0.8}$ from Slesnick et al. 2004; Andersen et al. 2008) and in the field (5.2 or 6 from Bihain & Scholz 2016 and Kirkpatrick et al. 2012, respectively) is much smaller. Because the two SBdRs are different from each other (binaries versus singles in ONC/field), this may suggest a preference for companions to form around stellar-mass primaries instead of brown dwarfs in the ONC. This discrepancy may be due to the intrinsic difficulty in detecting companions around fainter primaries, so we evaluated the SBdR from our completed catalog of binaries, obtaining ∼10.6 ± 0.3. Even if we consider the completed distribution of binaries, we still observe a preference for companions to form around primaries in the stellar-mass regime compared to brown dwarf mass.

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Given the increasing stellar density toward the inner regions of the cluster, one may expect to find apparent pairs that are due to chance alignments, that is, cluster members that have small projected separation but are physically unrelated. Assuming a random distribution, one can use estimators like a two-point correlation function to evaluate the probability of observing a pair at a particular separation. Departures from random probability may indicate the presence of really close binaries.

To perform this analysis, we follow Jerabkova et al. (2019), building the so-called elbow plot (Gladwin et al. 1999; Larson 1995), showing the number density of detected targets (Σ) as a function of the separation on-sky (θ). As shown by Gladwin et al. (1999), the presence of an elbow in this distribution graphically indicates the presence of resolved binaries.

Figure 8 shows the elbow plot derived from the cluster selected isolated sources of Catalog I (black dashed–dotted histogram) and the same data where we also add the completed distribution of binaries obtained from the Master Catalog (black solid histogram). To investigate how the excess of binaries varies with the radial distance from the cluster center, the figure also shows the results for four different rings centered around the position of θ¹Ori-C. Overall, the different distributions agree with each other, all showing a clear overabundance of multiple systems starting at ∼10³ au (black vertical line). This result is in agreement with Scally et al. (1999), who suggested, based on a common proper motion study, that there should be no binaries wider than 1000 au. Using GAIA DR2 data in combination with ground-based visible images, Jerabkova et al. (2019) finds for the ONC that the overabundance of multiple systems starts at ∼3000 au. Our data, reaching fainter objects with the diagnostic power to separate cluster members from background sources, lend support to Scally's findings. Moreover, fitting the elbow part of the global Σ(θ) distribution, we find a slope −1.85 ± 0.32 (red line), in excellent agreement with typical values for young clusters (Gladwin et al. 1999), as well as for early studies of the ONC in particular (Bate et al. 1998). These results indicate that the true population of binaries in the ONC has been reliably assessed, and that no overestimate is introduced by our completeness correction.

Reipurth et al. (2007), using HST/ACS Hα images from GO-9825 with 50 mas pixel size (corresponding to about 20 au, 2.5 times smaller than our WFC3-IR data), performed a major survey for visual binaries in the ONC, probing a range of separations similar to ours. More recently, de Furio et al. (2019) used PSF fitting to find close pairs in HST/ACS F555W (V-band) images from GO-10246 to probe separations smaller than 160 au. These surveys, like those performed using ground-based adaptive optics systems, in particular Duchêne et al. (2018), are complementary to our study as they target brighter and bluer (i.e., typically more massive) sources at smaller separations. Comparing the systems in our Master Catalog with those reported in the three aforementioned surveys, we obtain the results listed in Table 4. The columns list the number of targets we identify as cluster members ("Cluster"), those having at least one component classified as background source ("Background"), those appearing unresolved in our data even after KLIP processing ("Unresolved"), and those that do not match any source in our catalog ("Not matched"). If we exclude the binaries that were previously identified in Reipurth et al. (2007) and de Furio et al. (2019) and those identified in Paper I, we are left with 21 new candidate binaries uncovered by the KLIP algorithm. These new candidate detections span a range of primary masses between 0.014 and 0.127 M_☉, companion masses 0.004–0.23 M_☉, separations 016–076, and completeness between 17% and 87% with 49% as median value.

Figure 9 shows a comparison between the separations reported in our Master Catalog versus those given by Reipurth et al. (black) and De Furio et al. (red). Overall, there is excellent agreement between our values and those reported by these surveys, with only one major discrepancy against the Reipurth et al. catalog: their source JW 638 is listed as having a companion at ∼1'' separation, whereas our IR images (as well as the ACS visible images of Robberto et al. 2013) show a closer companion at separation ∼04 (see Figure B1, ID 7). If we exclude this detection, the average scatter of separations between our catalog and the others is ∼005, less than one-half WFC3 pixel.

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The multiplicity fraction (MF) of multiple systems is defined as

$$\begin{eqnarray}&&\mathrm{MF}=\displaystyle \frac{{N}_{\mathrm{mult}}}{{N}_{\mathrm{mult}}+{N}_{\mathrm{single}}}\end{eqnarray} \tag{ 5 }$$

where N_mult and N_single are the number of multiple- and single-star systems in the sample. In Table 5 we report the MF values for (1) the Master Catalog ("all"), (2) the Master Catalog split into two different bins of primary mass ("star" or "BD"), and (3) three different primary mass bins (B0, B1, and B2) having the same number of systems in each bin. Table 5 shows that the fraction of binaries among stellar-mass objects is three times larger than among substellar-mass objects for the separation range we are considering. The deficit of very-low-mass binary systems remains regardless of how the limits are defined, as shown by the bottom half of the table.

A variety of MF values have been previously reported in the literature for the ONC. Petr et al. (1998) looked for binaries in the inner 40'' × 40'' around the Trapezium, finding MF = 5.9% ± 4.0% in the separation range 014–05 (63–225 au). In a similar separation range, we obtain MF = 8.1% ± 0.8%. Köhler et al. (2006) performed a survey of the periphery of the ONC at 5'–15' (0.65–2 pc) from the cluster center, probing separations from 01 to 12 and primary masses from 0.1 to 2M_⊙, finding MF = 5.1% ± 2.7%; for a similar range of mass and separation, we find MF = 13.0 ± 1.1. Reipurth et al. (2007) report MF = 8.8% ± 1.1% in the range of separations 017–169 (67.5–675 au), while we find 10.8% ± 0.9%. In general, we obtain larger MF values than previous ONC studies because the combination of HST/WFC3 and KLIP allows us to unveil a larger number of faint companions at low angular separations. Still, in comparison with other star-forming regions, our multiplicity function is ∼2 times smaller than, for example, Taurus over a similar separation range (Duchêne & Kraus 2013). On the other hand, comparing our result with the binary frequency in the field obtained by Duquennoy & Mayor (1991) for a similar range of separations, we find approximately the same binary frequency between the field and the ONC. This result is also in agreement with de Furio et al. (2019), where the authors find that the low-mass star binary population of ONC is consistent with that of the Galactic Field over mass ratio 0.6–1 and separation 30–160 au.

The left panel of Figure 10 shows the distribution of projected separations in the Master Catalog in bins of 03 before and after completeness correction. The right panel shows histograms of the separations for the three equally populated mass intervals B0, B1, and B2 introduced in Section 5.1. Overall, the separation distribution is peaked toward small values ≲06, or 240 au. At larger distances, the distribution shows a plateau; both results are consistent with what has been already reported by Reipurth et al. (2007).

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Spurzem et al. (2009) have analyzed the disruption of planetary systems in the ONC. Their numerical simulations indicate that moderately close stellar encounters can cause the disruption of planetary systems. They find that the ejected planets have typically low velocity dispersion and in young clusters can be retained by the cluster potential and appear as free floaters. Table 6, based on Spurzem et al. (2009) Equations (36) and (37), shows the typical timescale to get a free floater (τ_ff) for the "close" (01–06) and "wide" (06–15) populations of binaries assuming our typical values for the primary and companion mass and system separation. Considering the total number of systems that may harbor a companion, disruptions can be expected, in particular for the wide binary population in the central region of the cluster, which statistically had enough time to undergo at least one strong gravitational encounter. The observed spectrum of binary separations, in particular the discontinuity between close and wide binaries at 06 (240 au), can thus be attributed to stellar encounters, as anticipated by Reipurth et al. (2007).

In this section, we examine if the close and wide binaries, separated at 240 au, can be isolated as two distinct populations depending on the distance from the cluster core.

To perform this analysis, we study the completeness-corrected cumulative distributions of close and wide binaries, but instead of simply applying a completeness correction to our observations, we estimate the "true" number of underlying objects required to observe an object given the estimated completeness ${ \mathcal C }$. The number of missed detections for each successful detection at completeness ${ \mathcal C }$ is modeled as a negative binomial distribution representing the number of failures f occurring before a number of successes s are observed, assuming a probability p of a single success. We define the specific shape of the negative binomial distribution (for each detection) by using the value $p={ \mathcal C }$ for the individual trial success probability, and s = 1. Using this negative binomial distribution, we extract a random number of "failures," that is, undetected companions, that were not observed due to noise or incompleteness. We then assign to each of these systems a distance from the center similar to that of the actually observed systems. Finally, we iterate over the sample of observed binaries to obtain a single realization of a "complete" binary population and repeat this procedure 1000 times to obtain the completed cumulative distributions shown in Figure 11 for close (green) and wide (blue) binaries. For each iteration, we perform a two-sample Kolmogorov–Smirnov test (K-S test) on the completed populations of close and wide binaries as a function of the distance from the core of the cluster. For ∼48% of the K-S tests we obtain a p-value below 0.01. At this level of confidence, we cannot safely reject the hypothesis that the two samples are drawn from the same distribution. This suggests that the two populations may not be different with respect to their spatial distribution.

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In order to probe the initial mass functions of multiple systems, in Figure 12 we show the histograms of the primary and companion masses. We fit the histograms using broken power laws (i.e., $\sim {m}^{-{{\rm{\Gamma }}}_{i}}$), adopting the peak of each specific sample as the breaking point, and we obtain the results shown in Table 7. Even though the values of Γ₁ are compatible within the errors, both the Γ₂ and the peak of the two populations are not compatible within 1σ. To further characterize the possible differences between the mass distributions of primaries and companions and how they compare to the mass distribution of single stars in the ONC, we show in Figure 13 a set of cumulative mass distributions obtained following the same procedure introduced in Section 5.3. The top left panel shows the comparison between single systems, primaries, and companions. The top right panel shows the comparison between single systems and the full set of masses, both primaries and companions taken individually (we refer to this joint set of mass values as "union"). The bottom panel shows the same comparison where we coadded the mass of the two components of each pair (we refer to this set of mass values as "sum"). In each plot, we also show the cumulative distribution obtained from a Kroupa IMF (Kroupa 2001), a Chabrier IMF for single objects (Chabrier a: Equation (17) in Chabrier 2003), and a Chabrier IMF with unresolved binaries (Chabrier b: Equation (17) in Chabrier 2003). To avoid introducing biases that are due to the saturation limit of our survey, we cut the mass distributions at 1 M_⊙. As explained in Section 5.3, we generate 1000 complete samples for each population. For each combination, we perform a two-sample K-S test. The results, summarized in Table 8, are characterized by the ratio $n=\tfrac{{n}_{i}}{{n}_{\mathrm{tot}}}$, where n_i is the number of times the K-S test provides a p-value ≤ 0.01 (corresponding to a confidence level >99% that the two populations are distinct) and n_tot is the total number of simulations. As the ratio increases, it is safer to reject the hypothesis that the two samples are drawn from the same population. The results suggest that the populations are generally different, in particular (1) the mass distribution of the binaries is different from the mass distribution of single stars, (2) both are different from the Kroupa/Chabrier IMFs, and (3) the primary and companion mass distributions are different from each other (as already noted in Figure 12). The "union" mass distribution is compatible with a Chabrier IMF with unresolved binaries in ∼31% of the tests. The "sum" mass distribution is always incompatible with any Kroupa/Chabrier IMFs.

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We interpret these inconsistencies as a result of a systematic deficiency of companion detections below 100 au. Regardless of our best efforts and of our advanced detection techniques, the technical limit of 1–2 pixels for the closest resolvable pairs is basically insurmountable.

Although in this simple exercise we try to enhance the number of binaries by making use of our completeness tests, it must be remarked that the enhancement is only partial. For every detected binary, we can compute the chance for that binary to be detected at exactly the separation and magnitude contrast at which it is detected, and we can enhance our sample by one minus that chance. However, we cannot account for the truly undetected binaries (i.e., the truly close pairs and those with high flux contrast). A demonstration of this is that our "Single" star sample (blue line in the top left panel of Figure 13) follows the distribution of stellar systems (including unresolved binaries) by Chabrier (2003), an obvious sign that many binaries are actually hiding within our singles.

For the same reason, even the conclusions on dissimilarities of the mass distributions of primaries and companions in detected pairs can only be partial, due to biases affecting which systems are preferentially detected as such.

A more complete exercise, involving modeling the a priori binary mass distribution, SMA, inclinations, eccentricity, and spatial distribution within the cluster, will be the focus of an upcoming paper in this series (L. Pueyo et al. 2020, in preparation).

In this final section, we analyze the mass ratio distribution $q=\tfrac{{M}_{C}}{{M}_{P}}$, grouping binaries in different bins according to the mass of the primary and following the classification adopted to produce Table 5. The results are shown as violin plots (a method for graphically depicting groups of numerical data similar to a box plot with a marker for the median of the data and the addition of a rotated kernel density plot on each side). Overall, we obtain a median value for the mass ratios q ∼ 0.25, indicating a deficiency of similar-mass binaries (which would have q ∼ 1). This result is in agreement with that reported by Duchêne et al. (2018) for smaller separations (10–60 au). To compare our results with other work, we characterize the distribution of mass ratio as a power law f(q) ∝ q^γ. Fitting the completeness-corrected histogram, we determine the median values of q and γ reported in Table 9 for the different mass bins. From a theoretical point of view, we would expect that binaries with separation ≲100 au most likely have formed through fragmentation of the protostellar disk, while wider systems formed via freefall fragmentation during early collapse. Because these two processes occur at different times and through different mechanics, it is reasonable to expect them to produce companions with different mass functions and in turn different distributions of mass ratios. We tested this hypothesis and obtained ${\gamma }_{\lesssim 100\mathrm{au}}=-1.1\pm 0.5$ and ${\gamma }_{\gtrsim 100\mathrm{au}}\,=-0.6\pm 0.2$, finding that the distribution with separation ≳100 au (with a bigger and better constrained sample) is incompatible at 2.5σ with the population of binaries with separation ≲100 au.

Correia et al. (2013) studied eight adaptive optics spatially-resolved binaries in the ONC (along with seven binaries from the literature) in the separation range 85–560 au and primary mass 0.15–0.8, finding γ = 1.03 ± 0.66, γ = 1.11 ± 0.37, and γ = 0.57 ± 0.38 for the B98, PS99, and S00 pre-main-sequence tracks, respectively. The authors find good agreement between their results in the ONC and other star-forming regions (e.g., Taurus-Auriga), while our results seem to disagree with both (see below for our comparison with Taurus-Auriga). We think this discrepancy can be explained by the small number of candidates adopted in their survey and by the large number of close-in, small-mass companions detected in ours. We decided to test this assumption by down-sampling the number of candidates in our catalog, randomly extracting the same number as in Correia et al. in a similar range of masses and separation ≳100 au. We repeated this operation 100 times, finding that in 88%/78%/38% of the cases we agree within 2σ with the results from the PS99/S00/B98 tracks. Note that the candidates we exclude for this test have an average completeness value of 76%, and any candidate with completeness smaller than 30% has been detected through multiple visits. So we conclude that the discrepancy can be attributed to the presence of close-in, small-mass candidate companions we detected through KLIP analysis in our work.

Kraus et al. (2011) conducted a high-resolution imaging survey of the Taurus-Auriga star-forming region by probing the range of separations between 15 and 5000 au for primary and companion masses in the range 0.25–2.5M_☉ and 0.01–1.17 M_☉, respectively, obtaining γ = 0.2 ± 0.2 at separation ≲100 and 0.08 ± 0.2 at separations ≳100, that is, finding an almost flat distribution of q with at most a slight excess of similar-mass binaries. Instead, we find an overabundance of low-q binaries. This result still holds even if we consider a range of overlapping primary and companion masses and separations between the two surveys (0.28–1.27M_☉ and 0.01–1.04 M_☉ and 66–680 au, respectively), obtaining γ_Kraus = 0.3 ± 0.3 and our γ = −0.4 ± 0.2. If instead we limit both data sets at separation ≳100 au and companion masses ≳0.05 M_☉, the gammas obtained from the two surveys are now compatible within ∼1σ, reconciling the difference. Kraus et al. (2011) also remark that their mass-ratio distribution is in stark contrast with Duquennoy & Mayor (1991), who studied field binaries with spectral type between F7 and G9 (∼0.8–1.4 M_⊙) and found a mass-ratio distribution peaked toward low masses (q ∼ 0.3) with few similar-mass companions, a finding very close to our result, q ∼ 0.25. They derived the γ from the Duquennoy & Mayor (1991) data set, obtaining γ_q:0–1.1 = −0.36 ± 0.07 and γ_q:0.2–1.1 = −1.2 ± 0.2. This last value, obtained with a stronger fit (χ_ν = 0.7 with seven degrees of freedom), is in good agreement with the γ we obtain primary masses of 0.5–1.27 ${M}_{\odot }$ (labeled B0 in Table 9). These results, together with the results about the multiplicity fraction presented in Section 5.1, suggest that ONC binaries may represent a template for the typical population of field binaries, upholding the hypothesis that the ONC may be regarded as a most typical star-forming region in the Milky Way.

Figure 15 shows the mass ratio of each pair versus the mass of the primary, that is, the detailed distribution of the data points used to create Figure 14. The shape of each point indicates the mass of the primary (circle = star; hollow diamond = brown dwarf). The limits for substellar and planetary-mass companions are shown as dashed lines. The gray area represents the region of parameter space inaccessible because of our detection limits. Figure 15 shows an overabundance of companions around stellar versus brown dwarf primaries, consistent with the general trend for star-forming regions and young associations (Duchêne & Kraus 2013). When detected, very-low-mass companions tend to have q ≤ 0.4. If present, very-low-mass binary systems with nearly equal mass must have remained unresolved, with a projected SMA smaller than our inner separation limit at the distance of the Orion Nebula. In fact, Winters et al. (2019) find the majority of very-low-mass objects in a local volume with 25 pc radius have q ≳ 0.4, and their separation peaks at ∼20 au. As a comparison, the smallest separation we resolve is ≃50 au with low completeness ${ \mathcal C }\sim 0.1$. On the other hand, our data seem to suggest that very-low-mass binary systems with nearly equal mass and wide separation are exceptionally rare, a possible indication that core fragmentation at the lowest masses favors the formation of asymmetrical systems.

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