ON THE DISTRIBUTION OF ORBITAL ECCENTRICITIES FOR VERY LOW-MASS BINARIES^*

In Table 3, we present a list of all 20 visual binaries that have published orbit determinations and a total system mass <0.2 M_☉. For each binary, we list its distance, semimajor axis (a), and eccentricity (e), as these are the physical parameters that, along with three viewing angle parameters, determine the size and shape of a binary's orbit on the sky. Thus, these are the three key parameters that must be considered when characterizing selection biases. In most cases, we have used orbit parameters as published and distances determined from trigonometric parallax measurements. When parallax measurements were not available, we estimated distances based on the primary components' resolved photometry and spectral types using the updated spectral type–absolute magnitude relations described in Appendix B of Liu et al. (2010).

Table 3. Visual Binaries with Published Orbits

Name	Sp. Type	Distance		Orbit			Orbit Quality		IWA	IWA/a
	Primary	d (pc)	Ref.	a (AU)	e	Ref.	Δtobs/P	σM/M	('')
Gl 569Bab	M8.5	9.65 ± 0.16	16	0.923 ± 0.018	0.316 ± 0.005	8	334%	−3%, +4%	005	0.52
2MASS J0920+3517AB	T0	25.4+0.8 − 0.7	9	1.67+0.06 − 0.05	0.217 ± 0.019	9	144%	3%	006	0.91
LP 349-25AB	M8	13.2 ± 0.3	10	1.94 ± 0.04	0.051 ± 0.003	8	76%	0.6%	010	0.68
DENIS J2252 − 1730AB	L6	15.0 ± 0.3	9	2.2+0.5 − 0.2	0.42+0.11 − 0.08	9	34%–56%	5%	007	0.48
HD 130948BC	L4	18.17 ± 0.11	16	2.26 ± 0.03	0.176 ± 0.006	1	73%	0.8%	006	0.48
2MASS J2132+1341AB	L5	27.5 ± 0.5	9	2.24 ± 0.10	0.34+0.03 − 0.02	9	35%	5%	005	0.61
Indi Bab	T1	3.624 ± 0.004	16	2.43 ± 0.11	0.53 ± 0.03	2	38%	0.9%	008	0.12
LP 415-20AB	M7.5	[27 ± 4]a	1	[2.5 ± 0.4]	0.708+0.016 − 0.014	1	55%	−7%, +9%	012	1.28
2MASS J0746+2000AB	L0	12.21 ± 0.04	5	2.899 ± 0.013	0.487 ± 0.004	1,11	68%	2%	006	0.25
LHS 2397aAB	M8	14.3 ± 0.4	13	3.09 ± 0.10	0.350 ± 0.005	7	83%	3%	012	0.56
2MASS J1534−2952AB	T5	13.59 ± 0.22	15	3.2+0.4 − 0.3	0.10+0.09 − 0.06	1,11	32%–44%	2%	006	0.26
LHS 1070BC	M8.5	7.72 ± 0.15	3	3.56 ± 0.07	0.033+0.003 − 0.002	1,14	82%	0.9%	010	0.22
LHS 1901AB	M6.5	12.9 ± 0.5	12	3.70 ± 0.16	0.194+0.025 − 0.021	8	39%	1.4%	010	0.35
2MASS J2140+1625AB	M8.5	[26 ± 5]a	1	[3.7 ± 0.7]	0.36+0.14 − 0.10	1,11	38%–57%	−27%, +53%	012	0.83
2MASS J0850+1057AB	L7	38+7 − 5	17	4.8+3.9 − 1.4	0.64 ± 0.26	11	10%–49%	100%	006	0.48
2MASS J1728+3948AB	L5	24.1+2.1 − 1.8	17	5.3 ± 0.8	0.28+0.35 − 0.28	11	20%–47%	−27%, +167%	006	0.27
2MASS J2206−2047AB	M8	26.7+2.6 − 2.1	4	5.8+0.8 − 0.7	0.25 ± 0.08	14	20%–28%	2.0%	012	0.56
2MASS J1847+5522AB	M6.5	[25 ± 5]a	1	[5.8 ± 1.5]	0.1+0.5 − 0.1	11	9%–23%	−72%, +194%	008	0.34
2MASS J1750+4424AB	M7.5	[29 ± 8]a	1	[21 ± 13]	0.71 ± 0.18	11	1.3%–9%	60%	012	0.16
2MASS J1426+1557AB	M8.5	[25 ± 5]a	1	[60 ± 40]	0.85+0.10 − 0.41	11	0.2%–20%	−100%, +73%	012	0.05

Notes. This table includes all resolved binaries with a total mass <0.2 M_☉ for which an orbit determination has been published. Orbit quality metrics are given for each binary: (1) fraction of the orbital phase observed (Δt_obs/P), the 1σ range is given where the period uncertainty is >10%; and (2) fractional 1σ error in the total mass from the orbit fit alone (σ_M/M), i.e., independent of the uncertainty in the distance. High-quality orbits have a large fraction of the orbit covered and/or a low mass error. ^aSpectrophotometric distance estimate derived following the method used in the appendix of Liu et al. (2010). References. (1) This work; (2) Cardoso et al. 2009; (3) Costa et al. 2005; (4) Costa et al. 2006; (5) Dahn et al. 2002; (6) Dupuy et al. 2009a; (7) Dupuy et al. 2009c; (8) Dupuy et al. 2010; (9) Dupuy 2010; (11) Konopacky et al. 2010; (10) Gatewood & Coban 2009; (12) Lépine et al. 2009; (13) Monet et al. 1992; (14) Seifahrt et al. 2008; (15) Tinney et al. 2003; (16) van Leeuwen 2007; (17) Vrba et al. 2004.

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For all orbits published by other authors, we have created our own Markov chains using the published astrometry and the same analysis method described in Section 2. However, only in the cases where the astrometric orbit fits yield a reasonable χ² can we trust the MCMC analysis to provide accurate posterior distributions of the orbital parameters. In cases where χ² is unreasonably large, the parameter space is not fully explored and uncertainties can be significantly underestimated by the MCMC method (or in fact by any fitting method). We find such large χ² values (reduced χ² = 2.8–120) for five of the orbits published by Konopacky et al. (2010). For three of these orbits they combined astrometry with radial velocities (2MASS J1847+5522AB, 2MASS J1750+4424AB, 2MASS J1426+1557AB), which may have enabled them to achieve orbit fits with smaller χ² than we found using the astrometry alone. For the other two orbits (2MASS J0850+1057AB and 2MASS J1728+3948AB), it is unclear why we are unable to reproduce their lower χ² values from the same input data (astrometry alone). For all five of these orbits we use the parameters derived by Konopacky et al. (2010). (Note that for reasons unrelated these discrepancies, none of these five binaries ultimately passes the selection criteria that we define below for our unbiased eccentricity sample.)

3.1.1. Selecting a Minimally Biased Sample

With a complete set of the known very low-mass visual binary orbits in hand, we now consider how selection biases may affect the eccentricity distribution of this sample. Visual binaries must pass three independent selection criteria before having their orbits determined: (1) they must be initially resolved by a high-resolution imaging survey, (2) they must be included in subsequent astrometric monitoring programs, and (3) their observed relative motion must yield a unique orbit fit. The first selection cut can result in biases on orbital parameters, as has been shown for direct imaging searches for extrasolar planets (e.g., see Brown 2004). The second cut results from the fact that it is impractical to monitor the orbits of all ≈100 known very low-mass binaries.⁵ Rather, astrometric monitoring programs are inevitably optimized for measuring as many orbits as possible with limited observing resources over a limited amount of time by focusing on the binaries likely to have the shortest orbital periods. Thus, the selection criteria used to define the monitoring sample can result in strong biases on the orbit parameters. Because the orbital period must be estimated from the projected separation of the binary, binaries with wide separations are much less desirable to monitor. This introduces an eccentricity bias because more eccentric binaries at a given period and mass will, on average, have wider projected separations. We will therefore attempt to define a subset of visual binaries that has minimal eccentricity bias due to these selection criteria.

• 1.  
  Discovery bias. We first consider the effect of "discovery bias" which causes the population of binaries resolved by imaging surveys to deviate from the true underlying eccentricity distribution of all binaries. For a given imaging surveys, the extent of this bias depends on how the semimajor axis (a) distribution compares to that survey's resolution limit or inner working angle (IWA). For example, if the a distribution peaks inside the IWA, the discovery of higher eccentricity binaries is favored, as such binaries spend more time at wider separations compared to more circular orbits. To quantify this bias, we have performed a simple Monte Carlo simulation in which a was held fixed, while eccentricity was distributed uniformly from 0 to 1, viewing angles were drawn from appropriate random distributions (p(i)∝sin i; uniform in ω and Ω), and the single-epoch observations were distributed uniformly in orbital phase. Binaries were considered detected if their projected separation at the time of observation was above the resolution limit. The impact on the uniform input eccentricity distribution is illustrated in Figure 1. When the resolution limit falls at or outside the fixed semimajor axis (IWA ⩾a), high-e orbits are strongly preferred as they spend the most time wider than the resolution limit. When the resolution limit falls inside the fixed value of a (IWA <a) both highly eccentric and circular orbits are preferred, with intermediate eccentricities being selected against. This is because in these cases any sufficiently eccentric orbit can be made unresolvable by an unfavorable viewing angle, but this can be compensated by the fact that the most eccentric orbits spend extended periods of time at very wide separations. Thus, high-e orbits are more likely to be detected than intermediate-e orbits, as they spend the most time widely separated.To predict the exact form of the eccentricity discovery bias for our sample would require an assumption for the underlying semimajor axis distribution as well as careful accounting of the resolution limits of the numerous imaging surveys that have contributed to the current sample of very low-mass binaries. However, for our purposes we simply want to ensure that our sample of orbits has an acceptably small amount of bias. Thus, we need only chose a cutoff in the ratio of IWA/a that is small enough to minimize the bias while including as many binaries with orbit determinations as possible. In Table 3, we give our estimate for the IWA of the imaging data used to discover each visual binary as well as the ratio IWA/a. We have chosen a cutoff of IWA/a < 0.75, which Figure 1 shows produces a relatively unbiased selection over a full range of eccentricities from 0 to 1. This cutoff excludes three binaries: LP 415-20AB (IWA/a ≈ 1.28), 2MASS J0920+3517AB (IWA/a ≈ 0.91), and 2MASS J2140+1625AB (IWA/a ≈ 0.83). The highest remaining IWA/a value in our sample is 0.68 for LP 349-25AB. While it is unfortunate to have to exclude any orbits from our sample, we emphasize that this is a very important step in keeping potentially large biases from impacting the following analysis. The case of LP 415-20AB is a prime example of the problem with discovery bias on eccentricity. This binary was discovered at a projected separation of 119 ± 8 mas, right at the resolution limit of the imaging survey of Siegler et al. (2003). Its actual semimajor axis turns out to be 93.5^+3.1 _{− 2.3} mas, so it could have only been discovered by that survey if it had an eccentric orbit, as it indeed does (e = 0.708^+0.016 _{− 0.014}). Thus, while LP 415-20AB does tell us that very low-mass binaries can be quite eccentric, it cannot convey any other information about the eccentricity distribution since the only reason it was discovered is because of its large eccentricity.
• 2.  
  Monitoring sample bias. Next, we consider how to prevent the sample selection of orbital monitoring programs from biasing the resulting eccentricity distribution. This is essentially a matter of choosing a range of semimajor axes for which the monitoring sample is complete at all eccentricities. The lower limit in a is simply defined by the visual binary orbit with the smallest semimajor axis; this is Gl 569Bab with a = 0.923 ± 0.018 AU (Dupuy et al. 2010). There are also no known visual binaries with projected separations (ρ) at discovery smaller than 0.9 AU, so we define this as the lower limit of our sample. The upper limit should be chosen to accurately reflect the completeness of orbit monitoring surveys that tend to focus on the smallest separation binaries, while also including as many of the resulting orbit measurements as possible. For the many binaries without orbit determinations, we must use their minimum possible semimajor axis, which would be the case of a highly eccentric, face-on orbit caught at apoastron (physical separation of a(1 + e)). Since bound orbits have 0 < e < 1, the minimum possible semimajor axis for any binary is therefore half its projected separation (ρ/2). Thus, some binaries that seem to be very wide (i.e., poor monitoring targets) could in fact have much smaller semimajor axes but just be very eccentric. When choosing an upper limit in a, we must therefore ensure that orbit monitoring programs are complete up to projected separations of 2a, otherwise we would introduce a strong selection bias against high eccentricities. We choose an upper limit in a of 3.7 AU in order to include the binaries LHS 1070BC (3.57 ± 0.07 AU; Seifahrt et al. 2008) and LHS 1901AB (3.70 ± 0.16 AU; Dupuy et al. 2010). The next widest binary that would also pass our IWA/a cut is 2MASS 0850+1057AB (4.8^+3.9 _{− 1.4} AU, e = 0.64 ± 0.26). Our chosen 3.7 AU upper limit means that binaries up to projected separations of 7.4 AU should not have been intentionally excluded from orbit monitoring programs. To the best of our knowledge this is the case, as at least a second epoch has been obtained for all binaries with ρ < 7.4 AU, either by our own Keck program or others (e.g., Bouy et al. 2008; Konopacky et al. 2010). A slightly larger upper limit of 4.0 AU would require several more binaries to have undergone orbit monitoring that have not, while adding no additional binaries with orbit determinations to our sample.
• 3.  
  Monte Carlo simulations. With these selection criteria of 0.9 AU < a < 3.7 AU, ρ < 7.4 AU, and IWA/a < 0.75 in hand, we have performed Monte Carlo simulations to assess the level of eccentricity bias that is expected for our sample. For this range of semimajor axes, the very low-mass binary parameter distributions derived by Allen (2007) are appropriate. The semimajor axis distribution is lognormal, with a peak at log (a/AU) = 0.86 and σ_log a = 0.28. To convert a from AU to arcseconds, we randomly drew distances in a Monte Carlo fashion incorporating the selection effects of the seeing-limited surveys that originally discovered the binaries as unresolved sources. We began with objects distributed uniformly in volume. We assigned magnitudes to these objects according to the empirical K_S-band luminosity function of Cruz et al. (2007) which is valid from $9.5\hbox{ mag} < M_{{K_S}} < 13.0$ mag (i.e., ≈M8 to L8, encompassing the spectral type range of the bulk of our sample). We further added flux due to unresolved companions according to the Allen (2007) binary frequency of 20% and mass ratio distribution of q^1.8; we converted mass ratio to flux ratio using the L∝M^2.4 theoretical power-law relation of Burrows et al. (2001). The final distance distribution was then determined by magnitude cuts on this simulated population. We tested both strong ($\mbox{$K_S$}< 12.0$ mag) and weak ($\mbox{$K_S$}< 15.0$ mag) magnitude cuts equivalent to those used by Gizis et al. (2000) and Kirkpatrick et al. (2000) in mining Two Micron All Sky Survey (2MASS) for late-M dwarfs and L dwarfs, respectively. In the latter case, we applied an additional $M_{{K_S}} > 11.0$ mag cut to approximate the J − K > 1.7 mag color cut of Kirkpatrick et al. (2000) that excluded brighter, bluer M dwarfs; we based this cut on the color–absolute magnitude diagrams of Dahn et al. (2002).The strong magnitude cut for the brighter late-M sample resulted in a distance distribution peaking at ≈20 pc, while the weaker cut for the fainter L dwarf sample resulted in similarly shaped distribution peaked at ≈40 pc. We then randomly generated an observed projected separation (ρ) for each binary in a similar manner as described earlier (i.e., random viewing angles and observing times, and a uniform eccentricity distribution). Figure 1 shows the resulting eccentricity distribution after applying the selection criteria of 0.9 AU < a < 3.7 AU, IWA/a < 0.75, ρ < 7.4 AU, and ρ > IWA (i.e., the binary was resolved). For both samples we conservatively assumed the worst-case IWA from the surveys that originally discovered the binaries in our sample. For the late-M dwarfs this is 012 from the Gemini AO surveys (e.g., Close et al. 2002, 2003; Siegler et al. 2003), and for the L dwarfs this is 007 from both Hubble Space Telescope and Keck surveys (e.g., Bouy et al. 2003; Reid et al. 2006; Siegler et al. 2007). By assuming the worst-case IWA we find the largest possible amplitude of the eccentricity bias, since smaller values of the IWA will always yield less bias (i.e., more binaries will always be uncovered for a smaller IWA). The resulting maximal bias we find from our Monte Carlo simulations has an amplitude of less than ±5% for both the late-M and L samples. As expected, the level of eccentricity bias introduced by our selection criteria is negligible, since they were chosen to minimize the eccentricity bias, and as shown in Figure 1 there is no strong systematic trend in the bias. Nearly circular (e < 0.2) and highly eccentric orbits (e > 0.8) are slightly preferred (by ≈4%) as compared to intermediate eccentricities. We neglect this bias in the following analysis.
• 4.  
  Orbit-fitting bias. Finally, we consider one additional selection effect present in our sample: each binary must have undergone astrometric motion suitable for yielding a robust orbit determination. This can potentially introduce an eccentricity bias if orbit fits are more easily determined for certain ranges of eccentricity. For example, it is easy to imagine that an extremely eccentric binary would present a challenge to astrometric monitoring as it spends most of its time at a fixed projected separation before moving rapidly inward in nearly a straight line. Such an orbit would be difficult to observe at just the right times needed for a good fit and would also appear to be less appealing for continued monitoring. Harrington & Miranian (1977) investigated this bias by generating 10³ random orbits and classifying the expected quality of the orbit fit by eye, relying on their extensive experience working with such astrometric data. Such an ad hoc method is necessary because in the marginal cases of visual binary orbit fitting the process of χ² minimization is highly nonlinear and strongly dependent on the initial guess. Harrington & Miranian (1977) found that on average orbits with e < 0.6 all had the same likelihood of yielding a good fit, so they assigned this eccentricity range a normalized completeness factor of 100%. They found that this completeness drops to 77% for eccentricities of 0.6–0.8 and 42% for e > 0.8. In our sample, we have one visual binary with e > 0.8 and none in the 0.6–0.8 range. In the following analysis we consider the impact of applying correction factors of 1.3× and 2.4× due to this orbit-fitting bias.
• 5.  
  Final visual binary sample. In Table 4, we list the orbital periods, eccentricities, and masses of the 11 visual binary orbits from Table 3 that meet our eccentricity bias-minimizing criteria of having 0.9 AU < a < 3.7 AU and a discovery IWA/a < 0.75. (Note that we list orbital period here rather than semimajor axis in order to compare these orbits directly to the unresolved spectroscopic binary orbits.) Although this nominally includes only a little over half of the 20 published visual binary orbit determinations, in fact the majority of the remaining orbits would not contribute much information as they are not as well constrained (see the orbit quality metrics given in Table 3). For example, all orbits in our bias-minimized sample happen to have 1σ eccentricity confidence limits of ≲ 0.1 (median 0.005). However, of the nine orbits not included in the sample, six of them have much more poorly constrained orbits (1σ confidence limits of 0.1–0.5), with a median error of 0.26 that is >50 × larger than the median for the other orbits. In fact, the reason why these six orbits are poorly determined is the same as why they were excluded by our bias minimizing criteria: they are all the widest (i.e., longest period) binaries, and this results in them having orbit monitoring observations that cover much less orbital phase (as little as 0.2% and 1.3% of the orbit is covered in the two most extreme cases).

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Table 4. Very Low-Mass Binary Orbit Sample

Name	Int.-Light	Binary	P	e	Mtot	Mass ratio	Orbit
	Sp. Type	Type	(days)		(M☉)	(q ≡ M2/M1)	Ref.
PPl 15	M6.5	S2	5.8 ± 0.3	0.42 ± 0.05	[0.124 ± 0.004]a	[0.85 ± 0.05]	2
2MASS J0535−0546	M6.5	E	9.77962 ± 0.00004	0.323 ± 0.006	0.088 ± 0.005	0.64 ± 0.04	12
2MASS J0320−0446	M8	S1	246.9 ± 0.5	0.067 ± 0.015	[0.15 ± 0.03a	[0.60–0.87]	3
LSR J1610 − 0040	d/sdM7	A,S1	607 ± 4	0.444 ± 0.017	[0.162 ± 0.015]a	[0.61–0.86]	5
Gl 569Bab	M8.5	Vb	864.5 ± 1.1	0.316 ± 0.005	0.140+0.009 − 0.008	[0.866+0.019 − 0.014]	7
Cha Hα 8	M5.75	S1	1900 ± 130	0.59 ± 0.22	[0.15 ± 0.02]a	[0.3–0.7]	9
LP 349-25AB	M8	V	2834 ± 15	0.051 ± 0.003	0.120+0.008 − 0.007	[0.872+0.014 − 0.018]	7
HD 130948BC	L4	Vb	3760 ± 60	0.176 ± 0.006	0.1095 ± 0.0022	[0.948 ± 0.005]	1
Indi Bab	T2.5	Vb	4090 ± 180	0.53 ± 0.03	0.116 ± 0.001	[0.63 ± 0.02]	4
2MASS J2132+1341AB	L6	V	4000+300 − 200	0.34+0.03 − 0.02	[0.13 ± 0.02]a	[0.76 ± 0.03]	8
DENIS J2252 − 1730AB	L7.5	V	4000+1500 − 700	0.42+0.11 − 0.08	[0.12 ± 0.02]a	[0.70 ± 0.03]	8
2MASS J0746+2000AB	L0.5	V	4640 ± 40	0.487 ± 0.004	0.151 ± 0.003	[0.94 ± 0.02]	1,10
LHS 2397aAB	M8	V	5190 ± 40	0.350 ± 0.005	0.146 ± 0.014	[0.71+0.09 − 0.14]	6
LHS 1901AB	M7	V	5880 ± 180	0.830 ± 0.005	0.194+0.025 − 0.021	[0.958+0.015 − 0.014]	7
LHS 1070BC	M8.5	Vb	6220 ± 16	0.033+0.003 − 0.002	0.156 ± 0.009	[0.84 ± 0.05]	1,11
2MASS J1534−2952AB	T5	V	8400+1500 − 1100	0.10+0.09 − 0.06	0.061 ± 0.003	[0.936 ± 0.012]	1,10

Notes. This table includes visual binaries from Table 3 that pass our eccentricity de-biasing criteria (0.9 AU < a < 3.7 AU and IWA/a < 0.75) as well as all unresolved binaries with a total mass <0.2 M_☉ for which an orbit determination has been published. The type of each binary is given: astrometric (A), spectroscopic single-lined (S1) or double-lined (S2), eclipsing (E), or visual (V). All table values enclosed in brackets "[]" are estimates based on models. Mass ratio estimates for visual binaries are based on Lyon evolutionary models (Chabrier et al. 2000; Baraffe et al. 2003), while the estimates for the SB1s and SB2 are taken from the literature (Basri & Martín 1999; Dahn et al. 2008; Burgasser & Blake 2009; Joergens et al. 2010). ^aNo direct mass measurement is available, so estimates from the literature based on evolutionary models and/or ancillary data are given (Basri & Martín 1999; Siegler et al. 2003; Reid et al. 2006; Siegler et al. 2007; Dahn et al. 2008; Burgasser & Blake 2009; Joergens et al. 2010). ^bBinary is the short-period component in a hierarchical triple system. References. (1) This work; (2) Basri & Martín 1999; (3) Blake et al. 2010; (4) Cardoso et al. 2009; (5) Dahn et al. 2008; (6) Dupuy et al. 2009c; (7) Dupuy et al. 2010; (8) Dupuy 2010; (9) Joergens et al. 2010; (10) Konopacky et al. 2010; (11) Seifahrt et al. 2008; (12) Stassun et al. 2006.

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To date, five unresolved very low-mass binaries have published orbit determinations. Two of these were discovered via radial velocity (RV) monitoring as single-lined spectroscopic binaries (SB1s): Cha Hα 8 (e = 0.59 ± 0.22; Joergens & Müller 2007) and 2MASS J0320−0446 (e = 0.067 ± 0.015; Blake et al. 2008). Shen & Turner (2008) have examined the selection bias introduced by RV surveys on the eccentricity distribution through Monte Carlo simulations. They found that as long as the RV semiamplitude (K₁) of an SB1 is ≳3× larger than the rms of the residuals (σ), its eccentricity does not greatly impact its likelihood of discovery. As shown in their Figure 4, SB1s have a constant detection efficiency up to e = 0.6, beyond which there is a slight downturn. However, even at the highest eccentricities (0.85 < e < 1.0), the detection efficiency is only ≈20% less than for e < 0.6 orbits. The reason for this lack of a strong eccentricity bias in RV data is because eccentricity acts to boost the RV semiamplitude ($K_1 \propto 1/\sqrt{1-e^2}$), counteracting the fact that eccentric orbits are more difficult to adequately sample with times series data. The two SB1s in our sample, Cha Hα 8 and 2MASS J0320−0446, have large semiamplitudes relative to their rms (K₁/σ = 11 and 21, respectively), and thus are expected to harbor little bias (<20%) with respect to eccentricity according to the analysis of Shen & Turner (2008). For this reason and the small size of the sample (two systems), we neglect the eccentricity bias for SB1s.

One double-lined spectroscopic binary (SB2) has a measured orbit: PPl 15 (e = 0.42 ± 0.05; Basri & Martín 1999). It was discovered as an SB2 in three epochs of RV data. The identification of SB2s in spectroscopic data is subject to different eccentricity selection effects than for SB1s because changes in the velocity do not need to be measured to detect the two sets of absorption lines due to the binary. To quantify the eccentricity bias in detecting SB2s, we performed a Monte Carlo simulation similar to the one described above for visual binaries but with the detection criterion that the velocity shift between the binary components must be large enough to detect in high-resolution spectra at one of three random epochs. We fixed the period and semimajor axis to that of PPl 15 (5.8 days, 0.03 AU).⁶ Figure 2 shows the resulting SB2 eccentricity bias predicted by our simulation for a detection threshold of ΔRV = 11 km s⁻¹ (i.e., 3× the spectral resolution of Basri & Martín 1999). We find only a slight (≈3%) preference for the discovery of highly eccentric (e > 0.8) SB2s in such RV data. For comparison, we also show the results if the detection threshold were higher or lower by a factor of two, as would be the case for binaries with RV semiamplitudes different by a factor of two. Higher semiamplitude binaries are less biased, while for lower semiamplitudes the eccentricity bias becomes a larger effect (though still ≲ 15%). Thus, it seems reasonable to neglect eccentricity bias for PPl 15, the one SB2 in our orbit sample.

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LSR J1610−0040 (e = 0.444 ± 0.017) is the only very low-mass astrometric binary known to date, discovered by the USNO parallax program (Dahn et al. 2008). It was later found to be an SB1 in the multi-epoch RV data obtained by Blake et al. (2010). In principle, astrometric time series data is very similar to RV time series data and so would be subject to the same biases as for SB1s. Thus, we assume that the same selection effects apply, namely, that orbits with e > 0.85 have an ≈80% detection efficiency relative to less eccentric orbits, which are all equally accessible by astrometric monitoring.

Finally, just one very low-mass eclipsing binary is known, 2MASS J0535−0546 (e = 0.323 ± 0.006), discovered by Stassun et al. (2006) in a photometric variability survey. In a study of exoplanet transit surveys, Burke (2008) showed that photometric transit/eclipse detections are only slightly biased (<10%) in eccentricity, with the exact correction depending on the underlying e distribution. Thus, we neglect any eccentricity bias for 2MASS J0535−0546.

Table 4 lists the orbital periods, eccentricities, and masses for these five unresolved binaries' orbits. Combined with the visual binary sample, our sample comprises a total of 16 very low-mass binary orbits spanning orbital periods from 5.8 days to 8400 days.

Duquennoy & Mayor (1991) have examined in detail the eccentricity distribution of a well-defined sample of solar-type binaries. In the following analysis, we exclude from consideration three of the binaries in their sample that have an orbit quality grade of 4 or 5 assigned by Worley & Heintz (1983), which indicates an indeterminate orbit. Not surprisingly, these binaries (HD 16895AB, HD 78154AB, and HD 146361AB) are the longest period orbits in the Duquennoy & Mayor (1991) sample, with orbital periods >1000 yr. (We note that Raghavan et al. (2010) have recently updated this volume-limited, solar-type sample, but we cannot use this for comparison since they did not publish the eccentricity values of their sample binaries.)

We find that the overall eccentricity distribution of the Duquennoy & Mayor (1991) solar-type binary sample is very similar to our orbit sample. To assess this quantitatively, we performed Kolmogorov–Smirnov (K-S) tests in a Monte Carlo fashion. We accounted for eccentricity errors in our sample by drawing the data in each trial from the observed distributions (our MCMC chains when possible, Gaussians otherwise). We computed the K-S statistic (D) and the corresponding probability for every trial. The resulting rms scatter in D was relatively small (⩽0.05).⁸ For the final probability that two samples were drawn from the same parent distribution, we used the mean of the probabilities from individual trials. Table 5 shows the resulting probabilities comparing subsamples of our orbits to subsamples of the solar-type binaries from Duquennoy & Mayor (1991). In general, we find good agreement between the two samples' eccentricity distributions. However, a more detailed examination of these samples shows some significant differences.

Table 5. Results of K-S Tests

	Very Low-Mass Binaries
Comparison Sample	All		Visual Only		e < 0.6
	Prob.	D	Prob.	D	Prob.	D
Solar-type binaries ($P > \mbox{$P_{\rm circ}$}$)	0.50	0.23 ± 0.04	0.44	0.28 ± 0.04	0.57	0.23 ± 0.04
Solar-type binaries ($\mbox{$P_{\rm circ}$}< P < 10^3$ days)	0.75	0.22 ± 0.02	0.82	0.22 ± 0.04	0.74	0.23 ± 0.03
Solar-type binaries (P > 103 days)	0.25	0.31 ± 0.05	0.22	0.36 ± 0.04	0.46	0.28 ± 0.03
e2 (Ambartsumian 1937)	9.1 × 10−5	0.55 ± 0.04	2.2 × 10−3	0.52 ± 0.04	5.3 × 10−7	0.71 ± 0.03
Bate (2009a)	0.83	0.20 ± 0.04	0.92	0.19 ± 0.03	0.64	0.26 ± 0.04
Bate (2009a, a ⩽ 3.7 AU)	0.95	0.22 ± 0.04	0.97	0.22 ± 0.03	0.91	0.25 ± 0.05
Stamatellos & Whitworth (2009)	9.9 × 10−3	0.59 ± 0.05	0.034	0.56 ± 0.05	0.19	0.52 ± 0.04
Stamatellos & Whitworth (2009, a ⩽ 3.7 AU)	0.028	0.56 ± 0.05	0.064	0.53 ± 0.05	0.43	0.45 ± 0.05

Notes. Each table entry shows the probability that our sample of very low-mass orbits was drawn from the same parent distribution as the comparison sample along with the D statistic and its rms scatter over 10⁴ Monte Carlo trials.

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Unlike the very low-mass binaries, solar-type binaries follow a strong correlation with orbital period such that short-period (<10³ days) binaries span a different range of eccentricities than do long-period (>10³ days) binaries (Figure 4). Duquennoy & Mayor (1991) suggested that this implies a different formation mechanism and/or dynamical history for these two period regimes of solar-type binaries. The same trend has also been reported for pre-main-sequence, solar-type binaries (Mathieu 1994, and references therein), implying that this period–eccentricity correlation is determined early in a binary's formation history. We confirm the significance of the correlation in the Duquennoy & Mayor (1991) sample by computing Spearman's rank correlation coefficient for solar-type binaries (excluding those that lie below the critical period for tidal circularization; $\mbox{$P_{\rm circ}$}= 11.6$ days). The resulting value of r_S = 0.32 for this sample of 46 solar-type binaries gives a 98.6% confidence that P and e are correlated. As discussed above, our sample shows no such correlation, which is an indication that the solar and very low-mass populations are in fact dynamically distinct from one another.

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We also note that among the large sample of solar-type binaries from Duquennoy & Mayor (1991) there is not a single orbit with e < 0.1 beyond an orbital period of 22 days,⁹ while there are at least three such orbits in our sample of 16 binaries (one more, 2MASS J1534−2952AB, could also have e < 0.1, but the error in its eccentricity makes this uncertain). This feature of solar-type systems has also been pointed out for pre-main-sequence binaries (Mathieu 1994 and references therein). We used a Fisher's exact test to compute the probability that this difference in e < 0.1 orbits between our sample and the Duquennoy & Mayor (1991) sample arises solely by chance (i.e., the P-value). In order to exclude any solar-type binaries that may be affected by tidal circularization but have not reached e = 0 yet, we consider only binaries with P > 100 days (a less conservative, shorter period cutoff could be used but would not change the results). At these orbital periods, 0 of 39 solar-type binaries have e < 0.1, while 3 of 14 very low-mass binaries have eccentricities confidently <0.1. Comparing these two samples, Fisher's exact test finds that this difference is significant with a P-value of 0.0155 (2.2σ). The P-value remains significant (<0.05) for any choice of the eccentricity division point ranging from 0.10 to 0.15. This strong preference for near-circular orbits among very low-mass binaries provides additional evidence that their dynamical history may be different from solar-type binaries.

We next consider the location in the P–e diagram of very low-mass binaries that are members of hierarchical triple systems as compared to their solar-type counterparts. The interpretation given by Duquennoy & Mayor (1991) and Raghavan et al. (2010) is that solar-type binaries that are members of hierarchical triple or quadruple systems tend to lie near the upper envelope of eccentricity at a given orbital period. However, examination of the P–e diagram of solar-type binaries presented in Figure 14 of Raghavan et al. (2010) shows that such members of hierarchical systems actually occupy a wide range of eccentricities at periods >100 days. In fact, members of hierarchical triple and quadruple systems are the only near-circular (e < 0.1) solar-type binaries at these periods, suggesting that their higher order nature may be the cause of their unusually low eccentricities. This is consistent with the fact that the only such long-period, near-circular binary in the pre-main-sequence sample of Mathieu (1994) is now known to have a tertiary companion (GW Ori; Berger et al. 2011). In contrast to solar-type binaries, the higher order multiples in our sample do not populate the upper eccentricity envelope at any period. In fact, the highest eccentricity orbit in our sample is a true binary, and the lowest eccentricity orbit belongs to a member of a triple system. This latter datum may be partially consistent with the fact that near-circular (e < 0.1) solar-type binaries tend to be members of higher order multiples. However, there remains the mystery as to why such near-circular orbits are much more common among the true binaries in our sample than in the solar-type binary sample.

Finally, we note that the shortest period binaries in our sample are not circularized, despite having shorter periods than the empirical limit from field solar-type binaries ($\mbox{$P_{\rm circ}$}= 11.6$ days). This was discussed by Basri & Martín (1999), who used brown dwarf structural models along with the circularization theory of Zahn & Bouchet (1989) to show that PPl 15 (P = 5.8 days) is not expected to ever tidally circularize as its critical period is ∼4.5 days. By extension, the orbit of the eclipsing binary 2MASS J0535−0546 (P = 9.8 days), which has even lower masses and thus a shorter critical period, is also predicted to never circularize, so its nonzero eccentricity is not merely a symptom of its youth (∼1 Myr; Stassun et al. 2006).

Ambartsumian (1937) showed that if binary companions are distributed in phase space solely as a function of energy, whether according to the Boltzmann distribution or any other arbitrary function, the cumulative distribution of eccentricities must be proportional to e² and the mean value of e should be equal to $\frac{2}{3}$. He argued that the observational data for wide stellar binaries at the time were in strong disagreement with this prediction, implying that binaries neither originate from nor relax via dynamical interactions into a state of statistical equilibrium. Duquennoy & Mayor (1991) tested the Ambartsumian (1937) distribution with modern data for solar-type binaries, applying a correction due to eccentricity bias from Harrington & Miranian (1977), and claimed that the e² distribution described the wider binaries (P > 10³ days) reasonably well. However, their data still possess a severe lack of high-eccentricity binaries, especially after excluding the three binaries with >1000 yr orbits, as these happen to include two of the most eccentric orbits in their sample (e ≳ 0.8).

Our sample seems to also contravene the e² distribution, as high-e binaries are notably scarce. To quantify this, we performed K-S tests as described in Section 4.1. Following the discussion in Section 3.1, we corrected for visual binary orbit bias by adding simulated data points uniformly distributed in each of the last two bins (e = 0.6–0.8 and 0.8–1.0). These bins contain 0 and 1 visual binary, respectively, and after accounting for the bias the number in each bin was 0 and 2.4 binaries, averaged over 10⁴ trials. The resulting probability that the null hypothesis is correct (i.e., the observed sample is drawn from the predicted e² distribution) was 9.1 × 10⁻⁵ for the entire sample and 2.2 × 10⁻³ when considering only the visual binaries in our sample. Because of the potential impact of our assumed bias correction on the results, we also performed K-S tests considering only eccentricities below 0.6, for which there is expected to be negligible observational bias. These tests gave probabilities of 5.3 × 10⁻⁷ and 7.1 × 10⁻⁵ for the full sample and visual binaries, respectively. Thus, we conclude that our sample of very low-mass binaries does not follow an Ambartsumian (1937) distribution over orbital periods of 5.8 to 8 × 10³ days.

We have also compared our sample to the predictions of numerical simulations that strive to describe the formation of such low-mass binaries. Bate (2009a) simulated the collapse of a 500 M_☉ molecular cloud, producing sufficiently large numbers of stars and brown dwarfs (>10³) to do statistical comparisons with observations. Bate (2010) reported the parameters of 16 binaries with total masses of <0.2 M_☉ that were generated in this simulation, and we plot their orbital periods and eccentricities in Figure 4.¹⁰ Our observed eccentricity distribution agrees very well with the simulation of Bate (2009a), with a 83% probability of both samples being drawn from the same parent distribution according to a K-S test (92% for visual binaries alone), and this result is not changed if we consider only those simulated binaries within our sample's semimajor axis upper limit of 3.7 AU. However, we note that Bate (2009a) does not produce any high-e (>0.6) binaries over the same orbital period range as we observe, as all of his high-e binaries have very long orbital periods (>10⁶ days). This could be due to the small number statistics of the simulation results, but if not then it is suggestive of a disagreement with the observations that is only made stronger by the fact that the orbit bias correction implies a total of ≈3 binaries with e > 0.6 for our sample. This discrepancy could be due to the fact that at the end of the simulation the Bate (2009a) binaries are still evolving toward shorter orbital periods by dynamical interactions with other protostars and their own accretion disks. Thus, the high-e systems could ultimately reach the orbital periods of our sample. However, their eccentricities would also likely damp down in the process, and thus the discrepancy with observations would remain. Another significant concern is that the very low-mass stars and brown dwarfs in the simulation of Bate (2009a) are formed largely by being ejected from their initial gas reservoirs through dynamical interactions. Bate (2009b) showed that by including a more realistic equation of state and radiative feedback, such ejection events become much less common, which resolves the problem that previous simulations (including Bate 2009a) produced far too many brown dwarfs. It remains to be seen how this improvement in the input physics will impact the orbital elements of binaries produced in these simulations.

In recent simulations by Stamatellos & Whitworth (2009), very low-mass stars, brown dwarfs, and planetary-mass objects are formed via gravitational instability in the outer parts (≳ 100 AU) of massive circumstellar disks (M_disk/M_⋆ = 1.0). The orbital periods and eccentricities of the 12 binaries with $\mbox{$M_{\rm tot}$}< 0.2$ that were generated in these simulations are shown in Figure 4. These binaries are a mix of three field binaries (those that are ejected from the disk) and nine binaries that remain bound to the star in a hierarchical triple. The eccentricities of these two subsets are not statistically different from each other as shown by a K-S test (D = 0.5, 38% chance of being drawn from the same parent distribution), and so we simply compare the full set of simulated binaries to our sample. The simulated binaries have significantly higher eccentricities than we observe, with our K-S test giving a probability of only 9.9 × 10⁻³ that the two samples were drawn from the same parent distribution. This result is not changed significantly if we only consider simulated binaries within our sample's semimajor axis upper limit of 3.7 AU. As discussed by Stamatellos & Whitworth (2009), the binary components formed in these simulations have their own disks, and after the initial 20 kyr their simulations ignore the gas in order to follow the long-term evolution of the system. The tidal interactions associated with these disks that are not accounted for in the simulations at late times could damp the eccentricities of the simulated binaries. Indeed, Bate (2009a) found that including such dissipative interactions was critical in reproducing the stellar eccentricity distribution. Thus, including such additional input physics may solve the disagreement between our observed eccentricities and the simulations of Stamatellos & Whitworth (2009). On the other hand, such large eccentricities may simply be a hallmark of forming in the frenetic dynamical environment of a fragmenting circumstellar disk.

A general problem in the study of visual binary stars is that orbit determinations are often infeasible, so the true semimajor axis (a) of the binary must be estimated from the projected separation on the sky (ρ). In the usual approach, ρ is multiplied by a correction factor that accounts for the fact that the observed separation will typically be smaller than the true semimajor axis. This is always true for circular orbits, but eccentric orbits can be observed at much wider separations than the actual semimajor axis. Abt & Levy (1976) used a correction factor of $a/\rho = \frac{\pi }{2}$, computed analytically by assuming edge-on circular orbits. Fischer & Marcy (1992) performed a Monte Carlo simulation that assumed random inclinations and reported 1.26 as their time-averaged correction factor. Unfortunately, they did not describe the input eccentricity distribution used nor did they report confidence limits for the correction factor. Torres (1999) performed several Monte Carlo simulations assuming random viewing angles and a variety input eccentricity distributions. He showed that the conversion factor not only varies greatly for different eccentricities, but that the probability distributions for these values are quite broad and can display one or more peaks. Thus, it is important to account for these large uncertainties when converting ρ to a.

With empirical eccentricity distributions in hand for solar-type and very low-mass binaries, we have undertaken Monte Carlo simulations to compute appropriate conversion factors for binaries drawn from these populations. We assumed uniformly distributed observation times (0 < t − t₀ < P), uniform arguments of periastron (0° < ω < 360°), and randomly oriented orbits (p(i)∝sin i). From 10⁷ Monte Carlo trials we computed observed projected separations (ρ) and thus arrived at a distribution of conversion factors (a/ρ). The results for various assumptions about the input eccentricity distribution are given in Table 6. In Figure 5, we show the probability distributions of these conversion factors. For comparison, we also show the resulting distribution if e is held constant in the simulations. In this case, the distribution has two peaks: one at 1/(1 + e) corresponding to its apastron passage and one at 1/(1 − e) corresponding to periastron. The empirical distributions of a/ρ have many small localized peaks due to the fact that they are composed of many individual eccentricity measurements that each contribute two peaks. The effect is more pronounced for the smaller sample of the 13 very low-mass binaries as compared to the 46 solar-type binaries.

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Table 6. Conversion Factors from Projected Separation to Semimajor Axis

Assumed e Distribution	Correction Factor (a/ρ)
	Median	68.3% c.l.	95.4% c.l.
No discovery bias
Uniform (0 < e < 1)	1.10	0.75, 2.02	0.57, 5.53
Circular (e = 0)	1.15	1.01, 1.85	1.00, 4.72
e2 (Ambartsumian 1937)	1.02	0.67, 2.07	0.55, 6.00
Solar-type binaries ($P > \mbox{$P_{\rm circ}$}$)	1.15	0.81, 2.01	0.64, 5.11
Solar-type binaries ($\mbox{$P_{\rm circ}$}< P < 10^3$ days)	1.16	0.83, 2.00	0.66, 5.04
Solar-type binaries (P > 103 days)	1.14	0.78, 2.02	0.61, 5.29
Very low-mass (all binaries)	1.16	0.84, 1.97	0.66, 5.08
Very low-mass (visual binaries)	1.16	0.85, 1.97	0.66, 5.11
Moderate discovery bias (IWA = a/2)
Uniform (0 < e < 1)	1.02	0.72, 1.46	0.57, 1.89
Circular (e = 0)	1.11	1.01, 1.46	1.00, 1.88
e2 (Ambartsumian 1937)	0.91	0.65, 1.42	0.55, 1.88
Solar-type binaries ($P > \mbox{$P_{\rm circ}$}$)	1.06	0.79, 1.51	0.63, 1.90
Solar-type binaries ($\mbox{$P_{\rm circ}$}< P < 10^3$ days)	1.07	0.80, 1.51	0.65, 1.90
Solar-type binaries (P > 103 days)	1.04	0.76, 1.50	0.60, 1.90
Very low-mass (all binaries)	1.08	0.82, 1.51	0.66, 1.90
Very low-mass (visual binaries)	1.08	0.82, 1.50	0.64, 1.90
Severe discovery bias (IWA = a)a
Uniform (0 < e < 1)	0.79	0.63, 0.94	0.54, 0.99
e2 (Ambartsumian 1937)	0.74	0.61, 0.91	0.53, 0.99
Solar-type binaries ($P > \mbox{$P_{\rm circ}$}$)	0.83	0.70, 0.95	0.60, 0.99
Solar-type binaries ($\mbox{$P_{\rm circ}$}< P < 10^3$ days)	0.84	0.71, 0.95	0.61, 0.99
Solar-type binaries (P > 103 days)	0.82	0.68, 0.94	0.58, 0.99
Very low-mass (all binaries)	0.84	0.72, 0.96	0.58, 0.99
Very low-mass (visual binaries)	0.85	0.71, 0.96	0.58, 0.99

Note. ^aWhen IWA ⩾a, circular orbits are never detectable.

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We have further considered the impact of discovery bias (see Section 3.1) on these distributions. The resolution limit of an imaging survey will preclude the discovery of binaries when they are at very small separations, which results in truncating the tail of conversion factors at large a/ρ. In Table 6, we give the results when considering two cases of discovery bias. In the moderate case, the IWA is a factor of two smaller than the true semimajor axis (IWA = a/2), which truncates the conversion factor distribution at a/ρ = 2. In the severe case the resolution limit is equal to the semimajor axis, so only eccentric binaries can be discovered, and the conversion factors are never greater than a/ρ = 1. These truncation values are shown as vertical lines in Figure 5.