SLoWPoKES-II: 100,000 WIDE BINARIES IDENTIFIED IN SDSS WITHOUT PROPER MOTIONS

The tightest binary we were able to identify was ≳1'' (see Figure 7 below), suggesting a fundamental limitation in our technique at that value. However, with a plate scale of 0396 pixel⁻¹, SDSS should be able to distinguish tighter binaries. Both unresolved and partially resolved binaries, with separations smaller than the resolution limit or the plate scale, have been identified from both the 2MASS (Kraus & Hillenbrand 2007b) and the Palomar Transient Factory (Terziev et al. 2013) photometric catalogs, albeit aided by multiepoch data in the latter case. Therefore, we conducted an examination of (1) what was restricting us from identifying binaries at <1'' and (2) what fraction of resolved binaries we were missing in our resultant sample because of this limitation.

First, we briefly describe the SDSS point source classification scheme. The algorithm is fully documented on the SDSS web pages⁷ and is beyond the scope of this paper, thus we only outline the scheme that classifies identified sources into the STAR table. The SDSS pipeline uses Resolve to identify unique sources and DEBLEND to deblend them when multiple sources are present. All of the identified sources, for which photometric parameters have been measured, are cataloged in the PhotoObjAll table. This table has several subtables, known as views in SDSS: PhotoObj, which includes all primary and secondary objects; PhotoPrimary, which includes all of the primary detections or ones classified as the best version of the objects; PhotoSecondary, which includes the duplicate detection(s); and PhotoFamily, which includes the original undeblended source as well as the sources for which the deblending failed. Based on the morphology, the PhotoPrimary is further divided into Star (point sources), Galaxy (extended sources), Sky (sky samples), and Unknown (unclassified sources). A PhotoPrimary object is classified as a point source and included in Star when a PSF fit provides a good approximation to its light profile; otherwise, it is classified as an extended source.⁸

As the table with all of the unique point-source objects, Star is the repository used by all stellar studies that use the SDSS survey (e.g., Covey et al. 2007; Jurić et al. 2008; Bochanski et al. 2010), including this study. Therefore, it is troublesome that we detected no resolved companions within 1'' in our search. Given the scope of Star, it is of utmost importance to understand its completeness level. In particular, it is necessary to quantify the sources that were detected in SDSS but are missing from Star, i.e., the sources that were misclassified. This would allow any individual study to apply the relevant corrections for the sources that were truly missing from the SDSS survey (e.g., faintness limit, unresolved binarity).

To test the completeness of Star, a sample of bona fide stellar sources needs to be used. Fortunately, there exist large spectroscopic samples of confirmed stellar sources in the SDSS catalog. We chose the DR7 catalog of 70,841 M dwarfs from the SDSS (West et al. 2011). Each spectrum in this catalog was inspected by eye and verified to be a bona fide M dwarf with a relatively high signal-to-noise ratio. Therefore, looking at how these bona fide M dwarfs are cataloged in Star using only photometric information and how they are flagged should help us understand why we were unable to find binaries tighter than 1''. Because the selection algorithm for M dwarfs in the SDSS spectroscopic survey was not based on Star (West et al. 2011), this is an ideal sample to investigate how many of the 70,841 bona fide M dwarfs are not included in Star.

3.1.1. What is the Completeness of the DR7 M Dwarf Sample?

Main-sequence dwarfs: We determined distances to main-sequence dwarfs using photometric parallax relations that are calibrated from directly measured trigonometric parallaxes. Even though there is no spectroscopic confirmation, MS dwarfs dominate the stellar sources in the SDSS catalog (Covey et al. 2007). Three separate photometric parallax relations were used, such that the chosen magnitude and colors trace the stellar temperature monotonically in that regime: M_g versus $g-i$ for F0–K4 (Covey et al. 2007), M_r versus $r-z$ for K5–M9 (Bochanski et al. 2010), and M_i versus $i-z$ for L0–L9 (Schmidt et al. 2010) dwarfs. The photometric parallax relations are given in Table 1.

Table 1.  Photometric Parallax Relations Used in This Paper

Type	Locus	Photometric Parallax Relation	References
F0–K4	−0.01 ≤ (r−z) < 0.50	Mg = 2.845 + 1.656 (g−i) + 3.863 (g−i)2 − 1.795 (g−i)3	Covey et al. (2007)
K5–M9	0.50 ≤ (r−z) < 4.53	Mr = 5.190 + 2.474 (r−z) + 0.4340 (r−z)2 − 0.08600 (r−z)3	Bochanski et al. (2010)
L0–L9	1.70 ≤ (i−z) ≤ 3.20	Mi = −23.27 + 38.40 (i−z) − 11.11 (i−z)2 + 1.064 (i−z)3	Schmidt et al. (2010)
M subdwarf	(g−r) > 1.50; 0.8 < r−z < 2.5	Mr = 7.9547 + 1.8102 (r−z) − 0.17347 (r−z)2 + 7.7038 ${\delta }_{g-r}$ − 1.4170 (r−z) ${\delta }_{g-r}$	Bochanski et al. (2013)
White dwarf	Girven et al. (2011)	Iterative fits to DA cooling models	Harris et al. (2006)

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Magnitudes corrected for extinction, with values from Schlegel et al. (1998) as tabulated in the SDSS database, were used in all cases. We adopted an error of 0.3 mag in the calculated absolute magnitudes, which implies a 1σ error of 14% in distance. This scatter is caused by a combination of metallicity, magnetic activity, and unresolved binarity (West et al. 2005; Sesar 2008; Bochanski et al. 2011). The presence of magnetic activity and higher metallicity increases the intrinsic brightness of a star by as much as a magnitude (Bochanski et al. 2011). An unresolved binary companion increases the apparent magnitude by as much as 0.75 mag. Based on high-resolution imaging studies, the components of wide binaries are highly likely to harbor close companions (Law et al. 2010). Therefore, we are selecting against triple systems that contain an unresolved binary. However, none of these parameters can be measured or even estimated from the photometry alone. Ivezić et al. (2008) used a photometric metallicity in their photometric parallax relation, but that is applicable only to FGK dwarfs. No such relation is known for M dwarfs, except when they are significantly metal-poor subdwarfs (see below). Therefore, we chose to ignore the metallicity dependence.

M subdwarfs: M subdwarfs are low-metallicity, main-sequence dwarfs typically associated with the thick disk or the halo. In the M spectral type, the subdwarfs can be differentiated from the dwarfs by the depleted TiO feature in the optical spectrum (Gizis 1997). In Paper I, we used the reduced proper motions to select subdwarf candidates and identify 70 CPM binaries as subdwarf systems. Because a photometric parallax relation had not been calibrated for M subdwarfs, we used the one for M dwarfs. As a result, the distances were overestimated, and we were able to identify only a subset of the subdwarf pairs. Bochanski et al. (2013) have since developed a statistical parallax relation using the SDSS DR7 subdwarf catalog (Savcheva et al. 2014) based on their redder $g-r$ colors. Likely caused by increased hydride absorption, the redder $g-r$ color causes the M subdwarf locus to clearly separate from the M dwarf locus in the ($r-z$, $g-r$) space (West et al. 2004, 2011; Lépine and Scholz 2008).

Distinguishing the subdwarfs from the dwarfs, however, is not trivial. As shown in Figure 2, while the loci clearly separate in the ($r-z$, $g-r$) space, the tail of the M dwarf distribution scatters into the subdwarf locus. Given their vastly larger number in the Solar neighborhood, the dwarfs overwhelm the subdwarf population. However, we found that the scatter is in large part due to the uncertainty in the magnitude measurements. In Figure 2, the spectroscopically confirmed M dwarfs (West et al. 2011) are plotted as black dots, and the spectroscopically confirmed subdwarfs are plotted as purple (subdwarfs), green (extreme subdwarfs), and red (ultrasubdwarfs). In the bottom panel, where all the stars plotted, the dwarfs scatter into the subdwarf locus (blue box; Bochanski et al. 2013) significantly. However, when only the dwarfs with uncertainties $\leqslant 0.05$ (middle) and $\leqslant 0.02$ mag (top) in the griz bands are plotted, the scatter decreases. While the photometric accuracy for SDSS is ∼1% in the griz bands, imposing a 1% or 2% cut seems to exclude a large number of sources. This also biases the sample against fainter and more distant stars like the M subdwarfs. However, when available, exquisite photometry can be used to select subdwarfs without the need for spectra. This could possibly be extremely beneficial to future photometric surveys like the Large Synoptic Survey Telescope.

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We chose to accept photometry with uncertainties $\leqslant 0.05$ mag to identify subdwarf candidates. We recognized that this will include some M dwarf interlopers and increase the rate of false positives. However, given the low number of subdwarf binaries known, this is an acceptable risk. We did not include the identified subdwarf systems in the statistical analysis for this paper. We did not include the subdwarf binaries in the analysis for this paper because they are less likely to be bona fide systems compared to the rest of our sample.

We calculated photometric distances to subdwarf candidates that fit the following criteria:

$$\begin{eqnarray}g-r & \gt & 1.5\\ 0.8\lt r-z & \lt & 2.5\\ {\mathrm{psfMagErr}}_{{\text{}}{griz}} & \lt & =0.05\end{eqnarray} \tag{ 1 }$$

from the Bochanski et al. (2013) relations. The uncertainty in the absolute magnitude is ∼0.41 mag, which translates to a 20% uncertainty in the distance. There were over six million subdwarf candidates, as defined by Equation (1), in the SDSS photometric catalog.

White dwarfs: In Paper I we used proper motions to calculate the reduced proper motions and identify potential WD candidates and search for WD companions to the low-mass star sample. Based on ugriz photometry alone, there is no conclusive way to identify potential WD companions around our low-mass dwarf sample. However, in the color–color diagrams, WDs segregate from the main-sequence stars and the quasars. In particular, Girven et al. (2011) have defined a ($g-r$, $u-g$) locus for hydrogen-atmosphere WDs (DAs) based on a sample of spectroscopically identified DAs (Eisenstein et al. 2006) in SDSS DR7. The efficiency of the photometric selection for the spectroscopic sample was only ∼62.3% after non-DA WDs, including WD+MS pairs (∼8.9%), early-type MS stars and subdwarfs (∼11.3%), and quasars (∼17.2%), were removed. However, because quasars were specifically targeted for the SDSS spectroscopic sample, they are disproportionately represented. In the larger photometric sample, the contamination due to quasars is likely to be minor (Girven et al. 2011). Therefore, we use the Girven et al. (2011) ($g-r$, $u-g$) locus to identify potential WDs but note that >20% of the WD candidates will be interlopers and will require spectroscopic confirmation.

We calculated the photometric distances to candidate DAs by fitting the ugriz photometry to WD cooling models (Bergeron et al. 1995), as specified in Harris et al. (2006). This algorithm fits the photometry to the model in an iterative manner to derive a bolometric luminosity and a distance. However, the composition and mass/gravity is degenerate when only the photometry is available and cannot be determined. So we used the models with pure hydrogen atmospheres and a gravity of $\mathrm{log}\;g=8.0$. Because our adopted ($g-r$, $u-g$) locus selected only DAs, the composition should introduce significant uncertainties in the distances. However, incorrect distances will be derived for WDs with unusually high mass/gravity (∼15% of all WDs), unusually low mass/gravity (∼10% of all WDs), or helium-dominated atmospheres (Harris et al. 2006).

A comparison of our photometric distances and the spectroscopic distances for the DA WDs in the SDSS DR7 catalog (Kleinman et al. 2013) showed a 14–20% scatter (J. Andrews 2015, private communication). To be conservative in our matching process, we adopted 14% as the error in our photometric distances. There were 56,505 WD candidates, as selected using the Girven et al. (2011) ($g-r$, $u-g$) locus, in the SDSS photometric catalog.

We searched for stellar sources that had been classified as Primary detections within angular separations of 1–20'' in our sample of 24,036,982 low-mass stars. The inner search radius of 1'' was determined by the SDSS database, as discussed in Section 3.1. We chose the outer search radius to be 20'', after which the number of chance alignments for pairs without proper motions were significant in Paper I (see Figure 1). The search was conducted using the Neighbors table in the SDSS CasJobs Query interface. Photometric quality cuts, as described above, were performed. We then matched the photometric distances (d) to within 1σ. However, because the error in the distances is a percentage error, we also required the difference in the distance to be less than 100 pc to be classified as a candidate binary pair. Thus, our binary candidate pairs matched the following criteria:

$$\begin{eqnarray}\theta & = & 1-20^{\prime\prime} \\ {\rm{\Delta }}\ d & \leqslant & \mathrm{min}(1{\sigma }_{{\rm{\Delta }}\;d},100\;\mathrm{pc})\\ d & \leqslant & 2500\;\mathrm{pc}\\ | b| & \geqslant & 20^\circ.\end{eqnarray} \tag{ 2 }$$

All stars were selected to have good photometry and psfMagErr $\leqslant 0.10$ mag for the bands that are used in their selection and analysis. We identified 514,424 dM+MS, 1212 sdM+sdM, and 642 WD+dM candidate pairs.

Despite the rigorous nature of our candidate selection, chance alignments will be present in a sample of wide visual binaries. Such chance alignments arise from the measurement uncertainties in the parameters used in the selection criteria, as well as from the inherent spreads in these parameters in the Galaxy (Paper I). The number of chance alignments grows as a function of the angular separation ($\propto \;{\theta }^{2}$) and distance. Because our search does not include any kinematic matching, the probability of chance alignment is particularly high and, therefore, needs to be rigorously assessed for each and every candidate binary pair. Such a quantitative assessment sifts out false positives from the sample. In Paper I we built a Monte Carlo–based Galactic model that recreated the stellar populations along the LOS of a candidate binary and calculated the probability of chance alignments. Based on empirically measured parameters for the Milky Way, the model accounted for the variations in the stellar number density and space velocities, which become important beyond the Solar neighborhood. We employed the same model to assess the probability of chance alignment for the binary candidates identified here. The model was described in detail in Paper I; here we only provide a brief synopsis.

Instead of the computationally implausible task of simulating the entire Galaxy with $\gtrsim {10}^{11}$ stars, we recreated a $30^{\prime\prime} \times 30^{\prime\prime}$ cone centered at the (α, δ) of each primary out to a distance of 2500 pc. First, we calculated the total number of stars in the conical volume by integrating stellar number density profiles that assume a bimodal disk (Bochanski et al. 2010) and an ellipsoidal halo (Jurić et al. 2008). The model parameters are given in Table 2. While a two-component model is an oversimplification of the complicated scale height distribution (Bovy et al. 2012), it is easy to model and suffices for our purpose of recreating a random Galaxy. Each LOS was then repopulated with stars using the rejection method (Press et al. 1992). The rejection method ensured that the stars were randomly redistributed while following the overlying stellar number density distribution function. Each star that is generated has a α, δ, and distance.¹⁰ The total number of stars in an LOS ranged between 2,000 and 10,000, enough to both produce small-scale density variations as seen in the Milky Way and perform calculations with relative ease. However, because the number density profiles are smoothed functions that were made to fit the Galactic field, our model does not produce larger density variations like moving groups, open clusters, or streams.

Table 2.  Galactic Structure Parameters

Component	Parameter Name	Parameter Description	Adopted Value
	$\rho \ ({R}_{\odot },0)$	Stellar density	0.0064
	${f}_{\mathrm{thin}}$	Fractiona	1-${f}_{\mathrm{thick}}$-${f}_{\mathrm{halo}}$
Thin disk	${H}_{\mathrm{thin}}$	Scale height	260 pc
	${L}_{\mathrm{thin}}$	Scale length	2500 pc
	${f}_{\mathrm{thick}}$	Fractiona	9%
Thick disk	${H}_{\mathrm{thick}}$	Scale height	900 pc
	${L}_{\mathrm{thick}}$	Scale length	3500 pc
	${f}_{\mathrm{halo}}$	Fractiona	0.25%
Halo	${r}_{\mathrm{halo}}$	Density gradient	2.77
	$q(=c/a)$ b	Flattening parameter	0.64

Note. The parameters were measured using M dwarfs for the disk (Bochanski et al. 2010) and main-sequence turn-off stars for the halo (Jurić et al. 2008) in the SDSS footprint.

^aEvaluated in the solar neighborhood. ^bAssuming a biaxial ellipsoid with axes a and c.

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With the simulated galaxy, we then counted how many stars were found in the ellipsoid that is centered at the α and δ of the binary candidate and defined by its angular separation and its distance errors. This was the same criteria that we used in the search for candidate systems in the SDSS data. Indeed, to conduct the exact same search, we chose to not convolve the luminosity or mass function into the model and rather searched for all stars. Because all stars in our simulation are single stars, any star that satisfied the search criteria is a chance alignment. Therefore, the average number of chance alignments is an assessment of the probability that our candidate binary is a false positive. For each candidate pair, we ran 1,000 Monte Carlo realizations. If the probability of chance alignment is P${}_{{\rm{f}}}\leqslant 0.10$, we ran a further 4,000 realizations for better resolution.

Figure 4 shows the number density of the 514,424 binary candidates, identified in Section 3.3, as a function of the probability of chance alignment, P_f, and angular separation. As would be expected, the number of candidate pairs grows sharply with angular separation. There is also a large spread in P_f at a given separation, reflecting the varying stellar densities along different LOS. Especially at the large angular separations, the P_f is quite high for a large number of candidates, indicating that they are chance alignments. Therefore, we applied the P_f $\leqslant$ 5% cut, shown as the red line, for the candidates to be included in the SLoWPoKES-II catalog. This is a fairly conservative cut but serves to minimize the number of false positives. It is also the same threshold that we used in Paper I, although the SLoWPoKES systems had proper motions and, hence, should have a much lower rate of false positives. The middle panel shows the histogram of the fraction of candidate binaries that passed this threshold as a function of angular separation. All of the candidate pairs are within 3'', but only ∼2% of the pairs at 20'' were vetted to be real binaries by the Galactic model. The bottom panel shows the angular separation histogram for the 105,537 binaries in the SLoWPoKES-II catalog. This distribution is strongly skewed toward smaller separations, with 51% of the pairs at 1''–7'' and 72% at 1''–10''.

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Figure 5 shows the distribution of the fraction of binary candidates that pass various P_f thresholds: $\leqslant$ 1%, 5%, 10%, and 50%, with the total number of candidates also shown. Obviously, as we discussed in Paper I, the choice of a threshold is subjective and depends on the need. For example, if a follow-up study needed to minimize the number of false positives, a sample with P_f $\leqslant$ 1% should be chosen. However, if the study needed bright, rare binaries (e.g., WD+dM binaries), it might need to tolerate the high number of false positives and select candidates with P_f $\leqslant$ 10%. With such a large number of binary candidates, there is a flexibility in the number and types of binary systems that are available for follow-up studies.

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Figure 6 shows the number density of all candidate pairs as a function of P_f and photometric distance. In general, P_f is flat as a function of the distance, indicating that the distance of a given pair has no significant effect on how likely it is to be a chance alignment. More significantly, the number of candidate pairs peaks around ∼800 pc and declines smoothly afterward. This is because of an additional selection criteria, ${\rm{\Delta }}\;d\;\leqslant$ 100 pc (Equation (2)), which becomes effective at large distances and rejects pairs with ${\rm{\Delta }}\;d$ within 1${\sigma }_{\;{\rm{\Delta }}d}$ but larger than 100 pc. So even as the search volume grows larger, the number of candidates actually decreases. However, the resulting candidates are more likely to be binaries, as shown in the middle panel: the fraction of candidate binaries that pass the P_f $\leqslant$ 5% threshold reaches a minimum at ∼700 pc and increases sharply afterward. In fact, the acceptance rate for binaries at ∼2200–2500 pc is higher than for binaries at any other distance. At these distances, ${\rm{\Delta }}\;d\;\leqslant$ 100 pc implies that the distances match within 6%–7% of each other, explaining the low rate of false positives. Therefore, as shown in the lower panel of Figure 6, there are a relatively large number of identified binaries at large distances.

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Figure 7 shows the distribution of projected physical separations of the SLoWPoKES and SLoWPoKES-II binaries.¹¹ The SLoWPoKES distribution exhibited a bimodal distribution, which we interpreted as either the presence of two formation modes or the preferential destruction of the widest pairs in the timescale of 1–2 Gyr (Paper I). These binaries were 7–180'' systems, within an average distance of 1000 pc. In SLoWPoKES-II, we are not sensitive to binaries as wide because the lack of proper motions limits us to angular separations of 1–20''; however, the binaries are up to 2500 pc away. Clearly, the selection biases and the resultant distributions of the identified binaries involved are markedly different. For example, the steep falloff of the SLoWPoKES-II distribution at $a\gt {10}^{4.1}$ AU is due to our insensitivity to binaries at those separations.

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In Paper I we found a dip in the distribution of binary separations at $a\sim {10}^{4}$ AU (bottom panel of Figure 7), but at first glance this is not apparent in the new SLoWPoKES-II distribution (top panel). However, there is an inflection point at $a\sim {10}^{3.4}$ AU (2500 AU), where the slope of the distribution becomes noticeably shallower. In Figure 7, two power-law fits are shown on either side of the inflection at $a\sim {10}^{3.4}$ AU. This inflection is at a separation scale similar to that seen in previous samples of binary stars (Allen et al. 2000; Lépine & Bongiorno 2007), which were interpreted as wider binaries being disrupted beyond a critical separation and being less common, as was originally predicted by Öpik (1924). However, it is much smaller than the bimodality seen in Paper I at ∼20,000 AU. Whether present at formation or sculpted later in life, the multiple modes in binary populations are intriguing and suggest that interstellar interactions are more common than previously thought.

Figure 8 shows the $g-i$ and $r-z$ color distributions for the primary (left) and secondary (right) components of the SLoWPoKES-II binaries. The primary is defined as the component with the bluer $r-z$ color. The histograms for the color distributions are shown along the top and sides of each panel. The inferred spectral types, based on the median color–spectral type relations (Covey et al. 2007; West et al. 2011), are shown along the top axes. In general, the SLoWPoKES-II binaries reflect the SDSS low-mass dwarf population in both $g-i$ and $r-z$ colors. The distributions peak between the M2 and M4 spectral types, same as the field mass function (Bochanski et al. 2010). The primary color distributions also exhibit a pileup at the bright end of the distribution, which is most likely a selection bias. SLoWPoKES-II binaries span a large range of colors (and masses). There is a significant number of primary components from early-mid G to the mid-M dwarfs, while the secondaries extend from K5 to the M9 spectral types. The blue end of the secondary component distributions are defined by the color cuts imposed on our initial target sample ($r-i\geqslant \;$ 0.3, $i-z\;\geqslant$ 0.2). While SLoWPoKES contained only a handful of binaries after ∼M6–M7, there is now a significant number of binaries at the low-mass end of the stellar main sequence at large distances.

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Figure 9 shows the mass ratio distribution of SLoWPoKES-II binaries as a histogram (top) and as a cumulative distribution function (bottom). The black line and dots represent the entire catalog, and the red, purple, blue, and cyan show the distributions for different primary masses. The masses were calculated by interpolating from their $g-i$, $r-z$, or $i-z$ colors based on Kraus & Hillenbrand (2007a). We set the lowest mass at 0.075 ${M}_{\odot }$, the hydrogen burning minimum mass (Burrows et al. 1997), because the $i-z$–mass relation is even less constrained at lower masses. All of the distributions are skewed toward similar masses, with a notable deficit at the lowest mass ratios, below $q\;\sim$ 0.4. This is mostly due to an additional selection criterion that at least one component be K5 or later (≤0.70 ${M}_{\odot }$), which means that the lowest mass ratio possible for a K5 primary is ∼0.11. Moreover, the steep mass–luminosity ratio among the M spectral types means that SDSS photometry would only detect an M6 or earlier companion for the K5 primary, restricting the lowest possible mass ratio to ∼0.17. Similarly, for an M4 primary (∼0.20 ${M}_{\odot }$), the observed mass ratio is always greater than 0.39. After $q\;\sim$ 0.4, the various cumulative distributions show a relatively smooth progression that is similar for the different masses. There is a preference for a larger q at the lowest masses and a smaller q at the highest masses, but those are most likely due to selection biases, as discussed above.

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In addition to being the largest catalog of wide binaries, SLoWPoKES-II contains a diversity of systems—in mass, mass ratios, metallicity, separations, evolutionary states, and Galactic positions—that should facilitate follow-up studies to characterize the properties of low-mass stars. In addition, we have identified three specific subsets of candidate binaries that are rare: (1) wide VLM systems, (2) WD+dM systems, and (3) sdM+sdM binaries. They could prove uniquely useful in characterizing those stellar types.

The diversity of systems should allow for in-depth probes of different aspects of low-mass stellar physics. Specifically, the flexibility offered by the large sample size and all-sky nature of SLoWPoKES-II could be exploited by large spectroscopic surveys where fibers are often available in certain parts of the sky. For example, we were awarded 1,000 fibers in the Baryon Oscillation Spectroscopic Survey of SDSS-III (BOSS; Dawson et al. 2013) to acquire spectra from 500 SLoWPoKES-II binaries. This was in addition to the 1,000 fibers awarded earlier for the SLoWPoKES binaries. We are using the combined sample of ∼1,000 binaries to ascertain the age–activity and metallicity relationships in low-mass M dwarfs (A. Massey et al. 2015, in preparation.).

In Paper I we noted the presence of a bimodality in the physical separation distribution of the binaries (Figure 7). When compared to the dynamical dissolution timescales (Weinberg et al. 1987), the bimodality in the SLoWPoKES distribution led us to suggest that it comprised two different populations of binaries with a break at ∼20,000 AU: (1) a "wide" population that is dynamically stable over ∼10 Gyr and (2) an "ultrawide" population of young, loosely bound systems that will dissipate in 1–2 Gyr. In a follow-up, high-resolution imaging study with the Keck II and Palomar Laser Guide Systems with Adaptive Optics, we found that the frequency of a close companion was higher in wide binaries than in single stars (Law et al. 2010). Moreover, the frequency increased significantly with wide binary separation, suggesting that different formation modes were at work.

In SLoWPoKES-II we have found further evidence of multiple populations in the distribution of physical separations (Figure 7). While more subtle than in Paper I, a break is clearly evident at ∼2500 AU. This is much smaller than the critical separation in Paper I but approximately where we saw the first peak in the SLoWPoKES sample. It is also at the same separation scale as seen in previous studies (Öpik 1924; Allen et al. 2000; Lépine & Bongiorno 2007). That an inflection in the separation distribution stands out among the myriad of selection biases and incompletenesses is quite significant. Further investigation into these multiple populations and their origins is clearly warranted. Whether wide binary distributions are segregated at birth (e.g., Reipurth & Mikkola 2012) or sculpted as they traverse around the Milky Way (e.g., Weinberg et al. 1987; Jiang & Tremaine 2010) has important implications for our interpretation of observed binary populations and, consequently, on our understanding of binary star formation.

There are no signatures of multiple populations in the color (a proxy for mass; Figure 8) or mass ratio (Figure 9) distributions. Even if such signatures were present, identifying them from among the selection biases without further data would be extremely difficult. Recently, there has been an extensive discussion on the formation and stability of the extremely wide systems. This has partly been motivated by the large samples of wide binaries that have become available over the past decade (e.g., Chanamé & Gould 2004; Lépine & Bongiorno 2007; Sesar et al. 2008; Dhital et al. 2010). In addition, such wide binaries have been identified at very young ages in Orion (Connelley et al. 2009) and in Taurus and Upper Sco (Kraus & Hillenbrand 2009b). Numerical simulations have explored mechanisms that would form wide binaries primordially or via dynamical interactions, with three different pathways suggested. First, observations of protostars at the end of long, extended filaments of molecular gas (Tobin et al. 2010) argue that wide binaries can form with primordial separations of ∼0.1 pc. Second, primordial triple or quadruple systems become extremely hierarchical by scattering or even ejecting one of the components (Reipurth & Mikkola 2012). Third, simulations have shown wide binaries forming in the halos of clusters via dynamical interactions or as they escape the cluster (Bate & Bonnell 2005; Kouwenhoven et al. 2010; Moeckel & Bate 2010; Moeckel & Clarke 2011). These pathways are unique and are expected to imprint distinct signatures on the resultant population. For example, the Reipurth & Mikkola (2012) pathway would produce a high frequency of triples and quadruples, while dynamical interactions would produce a very high proportion of low-mass binaries.

There is no clear evidence from the SLoWPoKES samples that any of these pathways are dominant. Rather, there is reason to believe that all of the pathways could be operational. Without a reliable age indicator, there is no way to observationally determine if the components of a wide binary formed together. In fact, it might be impossible to do so, even with an age indicator, because the age difference of a few million years can be indistinguishable once the stars are in the Galactic field. We suggest that there is no principal pathway for wide binary formation. Determining the relative efficiencies of the different formation modes would be more valuable.

SLoWPoKES did not contain wide VLM/BD binaries. They were too red and too faint to be detected in the USNO-B survey and, therefore, did not have measured proper motions. Finding more VLM/BD systems was one of our primary motivations for identifying binaries without proper motions. Only 11 wide (≳100 AU) VLM/BD systems are currently known, including a VLM triple at 820 AU (Burgasser et al. 2012) and binaries at 5100 AU (Artigau et al. 2007) and 6700 AU (Radigan et al. 2009). With the formation processes and techniques to measure the properties of VLM/BDs not completely understood, the value of a larger sample could not be overstated.

Until recently, the processes by which VLM/BDs form were thought to be different from those of higher-mass stars. Reipurth & Clarke (2001) suggested that VLM/BDs are protostars that are ejected from their natal cores before they can accrete and grow to stellar masses. Such ejections were seen in numerical simulations (e.g., Bate et al. 2002), and all observed VLM/BD binaries had enough binding energy to have survived such ejections (Burgasser et al. 2003; Close et al. 2003). However, with the discovery of each subsequent VLM/BD binary wider than ∼100 AU, the viability of the ejection hypothesis has been called into question (e.g., Dhital et al. 2011; Burgasser et al. 2012).

Figure 10 shows the projected physical separation distribution for a sample of 141 binaries with $i-z\geqslant 1.14$, the median color of an M8 dwarf (West et al. 2011). While spectroscopic follow-up is ongoing to confirm their spectral types (a proxy for mass), it seems clear that a significant number of systems with separations of 1000–5000 AU exist in the Galactic field. We note that photometric colors can be particularly deceiving in the VLM/BD regime, so the need for spectra cannot be overemphasized. However, such a large number of wide VLM binary candidates can give us a lot of information about their formation.¹² These systems most definitely did not form primordially via ejection because their binding energy is too small to have survived a dynamical kick. With the 11 previous systems and 141 candidates presented here, we cannot explain away the wide VLM binary population as exceptions.

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The alternative pathway to VLM/BD formation is that they form in a manner similar to stars via gravoturbulent fragmentation (Hennebelle & Chabrier 2011; Jumper & Fisher 2013). The detection of the first pre-BD core, Oph B-11 (André et al. 2012), has helped further that idea significantly. Our sample does not allow for rigorous testing of this hypothesis; we will wait for follow-up observations that will better characterize the binaries. However, neither the separation distribution nor the mass ratio distribution shows any significant change as a function of mass. The systems with less-massive primaries are more likely to be equal in mass (Figure 9). This characteristic is a bias of our sample that is mostly due to the insensitivity to lower-mass, fainter companions and to setting the minimum mass at 0.075 ${M}_{\odot }$. The trend of more equal-mass binaries is also a gradual change at all masses, with no significant break at the VLM/BD regime.

There is no suggestion of multiple populations of wide VLM binaries, as was observed for wide stellar binaries in Section 5.2. However, we cannot discount the possibility that these wide VLM binaries were bound postformation (Kouwenhoven et al. 2010; Moeckel & Bate 2010; Moeckel & Clarke 2011). Numerical simulations show that a molecular core typically fragments into three to five cores, whence the smaller cores are ejected before they can grow into stellar masses (Bate 2012). Open clusters could have hundreds of these VLM protostars floating around, increasing the likelihood of multiple ones interacting at the same time and forming wide, gravitationally bound systems. The efficiency of these capture-like processes could be much higher in the VLM/BD regime.