WISE Y DWARFS AS PROBES OF THE BROWN DWARF–EXOPLANET CONNECTION

The WISE mission had three distinct phases: the four-band cryogenic period, the three-band cryogenic period, and two-band warm mission. Depending on the position on the sky, especially ecliptic longitude, sources were observed in one or more of these phases. We determined positions and magnitudes for each period separately with a median date of observation spanning one to two days. Positions and associated uncertainties were obtained by averaging the source positions in the multi-band extractions from each individual orbit. The uncertainty in the WISE astrometric frame is approximately 80 mas, based on the input 2MASS catalog used for WISE position reconstruction (Cutri et al. 2011). Typically, however, the positional uncertainties in the WISE detections are much larger than this, ∼250–500 mas, due to its large beamsize and detection at only one, or at most two, wavelengths or to the effects of confusion with other nearby objects.

In Table 3 we report averages of the 4.5 μm magnitudes (W2) for the various epochs and include 3.5 μm (W1) when available. Upper limits in the two longer wavelength bands, 12 (W3) and 22 μm (W4), are high and do not significantly constrain the SEDs. We converted the magnitudes to flux densities using the zero points from Wright et al. (2010), but because of the unknown and extremely non-blackbody-like nature of brown dwarf SEDs, we have not color-corrected these flux densities. Table 4 indicates that there is no evidence for variability in the [4.6] mag at the 2%–3% level for any of these objects. While not varying in the Spitzer bands, WISE 2220−3628 shows evidence for variability in the comparison of the ground-based J and HST/F125W photometry, with a nearly 1 mag difference between the two bands. Further monitoring of this object may be warranted.

Nine objects were imaged with HST's WFC3/IR in the F105W, F125W, or F140W filters as precursor observations in support of subsequent grism measurements. The final images are quite heterogeneous, consisting of one to four dithered exposures, with exposure times ranging from 312 to 2412 s. In some cases multiple exposures were taken with small offsets to reduce the effects of cosmic rays and the undersampling of the individual frames. The Space Telescope Science Institute's (STScI) "AstroDrizzle" mosaic pipeline was used to process these data to produce final mosaicked images. The pipeline corrects for the geometric distortion of the WFC3-IR camera to a level estimated to be ∼5 mas (Kozhurina-Platais et al. 2009), which is of the same order or less than the extraction uncertainties of the faint target. Sources were extracted using the Gaussian-fitting IDL FIND routine to determine centroid positions and the APER routine¹⁰ with a 3 pixel radius for photometric measurements. The FWHM for the undersampled data is ∼2 pixels or 026 consistent with STScI analyses (Kozhurina-Platais et al. 2009). Analysis of fields with multiple HST observations (e.g., WISE 1541−2250) shows that after registration onto a common reference frame, the repeatability of individual source positions is ∼5 mas for bright objects located within 90'' of the brown dwarf. The photometry was calibrated using the appropriate zero points for Vega magnitudes¹¹ from the WFC3 Handbook (Rajan et al. 2010).

Observations with the Spitzer Space Telescope were made using a variety of General Observer (GO) programs (PI: D. Kirkpatrick) and some Director's Discretionary Time (A. Mainzer and T. Dupuy). In all cases the observations were obtained during the Warm Mission phase using the IRAC camera (Fazio et al. 2004) in its full array mode to make observations at 3.6 (Channel 1) and/or 4.5 μm (Channel 2). We analyzed post-BCD mosaics from the Spitzer Science Center (SSC) to make photometric and astrometric measurements, extracting sources using a 4 pixel radius aperture, a 4–12 pixel annulus for sky subtraction, and normalizing the resultant counts using SSC-recommended aperture corrections.¹²

For each target we put all epochs of Channel 1 and Channel 2 onto a common reference frame by averaging the positions of all bright sources within ∼60''–90'' of the target, typically 25–50 objects per frame, and calculating small offsets from one epoch to the next to register all frames to the average value. The largest offsets were of the order 200 mas and typically much smaller, around 50 mas. We kept the size of the overlap region smaller than the overall size of the IRAC field of view to minimize the effects of optical distortion. The dispersion around the average bright source position is typically 60 mas in both right ascension and declination, or one-twentieth of the native 12 pixel (Figure 1). These values are less than 100 mas distortions quoted by the SSC¹³ in part because we have confined our observations to the small regions at the center of the IRAC arrays. Figure 1 shows the positional uncertainty in multiple observations (N_obs = 2–13) for 800 reference sources from all of our target fields as a function of IRAC [4.6] mag. These single axis uncertainties have been normalized to a single epoch according to $N_{{\rm obs}}^{1/2}$ and are thus representative of the uncertainties for our single epoch brown dwarf measurements. The final positions for reference sources are improved relative to these values by $N_{{\rm obs}}^{1/2}$. The solid line shows a simple model to the positional uncertainty, with a constant value of 58 ± 8 mas for sources brighter than [4.6] = 17.6 ± 0.2 mag and a value that increases monotonically as S/N⁻¹ to fainter levels (Monet et al. 2010). Our bright brown dwarf targets are always in the flat part of the uncertainty distribution.

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Targets were observed in the H band using NIRC2 on the Keck-II telescope with the laser guide star AO system (Wizinowich et al. 2006; van Dam et al. 2006) and tip-tilt stars located 10''–50'' away. The wide-field camera (40 mas pixel⁻¹ scale; 40'' field of view) was used to maximize the number of reference stars for astrometry. At each epoch, dithered sequences of images with offsets of 15–3'' in right ascension or declination and total integration times of 1080 s were obtained at an airmass of 1.0 to 2.0. The majority of sources were observed at airmasses of <1.5. The individual images were sky-subtracted with a sky frame created by the median of the science frames and flat-fielded with a dome flat using standard and custom IDL routines. Individual images were "de-warped" to account for optical distortion in the NIRC2 camera (Beichman et al. 2013). The reduced images were shifted to align stars onto a common, larger grid and the median average of overlapping pixels was computed to make the final mosaic. The source positions obtained from the Keck images were corrected for the effects of differential refraction relative to the center of the field using meteorological conditions available at the Canada–France–Hawaii Telescope weather archive¹⁴ to determine the index of refraction corrected for wavelength, local temperature, atmospheric pressure and relative humidity (Lang 1983), and standard formulae (Stone 1996). As discussed in Beichman et al. (2013), for the small field of view of the NIRC2 images and the relatively low airmasses under consideration here, the first-order differential corrections are small, <10 mas across the ±20'' field, and proportionately less at smaller separations.

The effects of optical distortion in the wide-field NIRC2 camera were corrected using a distortion map derived by comparing Keck data of the globular cluster M15 (Alibert et al. 2005). Details of this distortion mapping are described in Beichman et al. (2013), but the correction amounts to <1 pixel (40 mas) across most of the array and up to 2 pixels at the edges of the array. After our correction procedure the residual distortion errors are less than 10 mas over the entire field.

Table 2 lists the positions for the WISE targets for each available epoch. The right ascension and declination data were fitted to a model incorporating proper motion and parallax (Smart 1977; Green 1985):

$$\begin{eqnarray} \alpha ^\prime &\equiv & \alpha _0+\mu _\alpha (t-T_0)/\cos (\delta ^\prime)\nonumber \\ \delta ^\prime &\equiv & \delta _0+\mu _\delta (t-T_0) \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \alpha (t)&=& \alpha ^\prime +\pi (X(t) \sin \alpha ^\prime - Y(t) \cos \alpha ^\prime)/\cos \delta ^\prime \nonumber \\ \delta (t)&=& \delta ^\prime +\pi (X(t) \cos \alpha ^\prime \sin \delta ^\prime\nonumber\\ &&+ Y(t) \sin \alpha ^\prime \sin \delta ^\prime -Z(t) \cos \delta ^\prime), \end{eqnarray} \tag{ 2 }$$

where (α₀, δ₀) are the source position for equinox and epoch T₀ = J2000.0, μ_{α, δ} are proper motion in the two coordinates in arcsec yr⁻¹, and π is the annual parallax in arcsec. The coefficients X(t), Y(t), and Z(t) are the rectangular coordinates of the observatory as seen from the Sun in AU. Values of X, Y, Z for the terrestrial or Earth-orbiting observatories are taken from the IDL ASTRO routine XYZ, whereas X, Y, Z values for the earth-trailing Spitzer observatory are obtained from the image headers provided by the SSC. Equations (1) and (2) are solved simultaneously using the Mathematica routine NonLinearModelFit¹⁵ incorporating appropriate uncertainties for each data point.

The solutions are given in Table 6 with precisions for the derived parallax values ranging from 5% (WISE 1541−2250) up to indeterminate values with uncertainties of 50% (WISE 0836−1859). Figures 17–24 show the fit to the total motion of the sources (proper motion plus parallax) as well as the fit to the motion with both proper motion and the effect of observatory location (terrestrial or Earth-trailing) removed. Our determinations are robust for 12 objects (uncertainties < 15%), with an average distance of 8.7 pc and a maximum distance for a well-determined distance of 15 pc. Three objects (i.e., WISE 0836−1859, WISE 1542+2230, WISE 2220−3628) have low precision parallaxes due to either a small number of measurements, in particular with Keck or HST, and/or a sparse set of reference stars. For the first two objects it is likely that the true distance for these T dwarfs is greater than 15 pc and thus more challenging to determine. For the 13 objects showing uncertainties less than 20% (and 12 objects with uncertainties <15%) we are relatively immune to the Lutz–Kelker bias in our determination of absolute magnitudes or other derived quantities. The bias occurs when more objects at larger distances are scattered into a sample than when objects of smaller distances are scattered out of the sample (Lutz & Kelker 1973).

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Table 6. Parallax and Proper Motion Solutions

WISE Designation	R.A.a	Decl.a	μα	μδ	π	Dist	Vtan	χ2 c	χ2 d
	(J2000.0)	(J2000.0)	('' yr−1)b	('' yr−1)	('')	(pc)	km s−1
J014656.66+423410.0	1h46m570940 ± 00112	42°34'10214 ± 0215	−0.441 ± 0.013	−0.026 ± 0.016	0.094 ± 0.014	10.6 ± 1.5	22 ± 3	23.0(27)	61.6(28)
J031358.93+780748.9	3h13m258000 ± 00097	78°7'43524 ± 0705	0.080 ± 0.012	0.072 ± 0.057	0.153 ± 0.015	6.5 ± 0.6	3 ± 1	21.4(17)	104.7(18)
J033515.01+431045.1	3h35m142520 ± 00096	43°10'53405 ± 0196	0.826 ± 0.011	−0.803 ± 0.015	0.070 ± 0.009	14.3 ± 1.7	78 ± 10	21.5(27)	71.4(28)
J041022.71+150248.4	4h10m220630 ± 00110	15°3'11053 ± 0158	0.966 ± 0.013	−2.218 ± 0.013	0.160 ± 0.009	6.2 ± 0.4	72 ± 4	23.7(33)	232.7(34)
J071322.55-291751.9	7h13m222510 ± 00166	−29°17'47558 ± 0277	0.388 ± 0.020	−0.419 ± 0.022	0.106 ± 0.013	9.4 ± 1.2	26 ± 3	17.4(19)	75.4(20)
J083641.10-185947.0	8h36m412030 ± 00061	−18°59'45080 ± 0086	−0.038 ± 0.007	−0.144 ± 0.006	0.020 ± 0.008	48.9 ± 20.0	35 ± 14	3.9(13)	5.6(14)
J131106.20+012254.3	13h11m60538 ± 00135	01°23'2840 ± 02	0.280 ± 0.016	−0.838 ± 0.016	0.062 ± 0.012	16.1 ± 3.0	68 ± 13	13.0(17)	34.8(18)
J154151.65-225024.9	15h41m522500 ± 00100	−22°50'24540 ± 0162	−0.857 ± 0.012	−0.087 ± 0.013	0.176 ± 0.009	5.7 ± 0.3	23 ± 1	16.8(19)	354.0(20)
J154214.00+223005.2	15h42m147040 ± 00200	22°30'9098 ± 033	−0.960 ± 0.024	−0.374 ± 0.026	0.096 ± 0.041	10.4 ± 4.5	51 ± 22	23.6(13)	32.9(14)
J173835.53+273259.0	17h38m352890 ± 00076	27°33'2091 ± 0128	0.317 ± 0.009	−0.321 ± 0.011	0.128 ± 0.010	7.8 ± 0.6	17 ± 1	19.4(27)	122.9(28)
J180435.37+311706.4	18h4m355700 ± 00082	31°17'6105 ± 0143	−0.269 ± 0.010	0.035 ± 0.011	0.080 ± 0.010	12.6 ± 1.6	16 ± 2	23.8(27)	69.7(28)
J182831.08+265037.7	18h28m302950 ± 00059	26°50'36030 ± 0075	1.024 ± 0.007	0.174 ± 0.006	0.106 ± 0.007	9.4 ± 0.6	46 ± 3	34.5(39)	234.6(40)
J205628.91+145953.2	20h56m283190 ± 00072	14°59'47804 ± 0097	0.812 ± 0.009	0.534 ± 0.008	0.140 ± 0.009	7.1 ± 0.5	33 ± 2	27.1(35)	201.7(36)
J220905.73+271143.9	22h9m47813 ± 00111	27°11'58336 ± 0185	1.217 ± 0.013	−1.372 ± 0.015	0.147 ± 0.011	6.8 ± 0.5	59 ± 4	15.1(19)	148.1(20)
J222055.31-362817.4	22h20m550650 ± 00122	−36°28'16312 ± 0227	0.283 ± 0.013	−0.097 ± 0.017	0.136 ± 0.017	7.4 ± 0.9	10 ± 1	16.8(17)	69.0(18)

Notes. ^aEquinox and Epoch J2000. ^bProper motion in right ascension is given in units of '' yr⁻¹ and includes the correction for cos (δ). ^cχ² value with degrees of freedom in parentheses. Fit includes parallax. ^dχ² value with degrees of freedom in parentheses. Fit does not include parallax.

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These parallaxes are determined relative to small groups of objects (typically stars) and not tied directly to an absolute reference frame. Thus our parallaxes are relative measurements and may have biases at the <5 mas level (Dupuy & Liu 2012), which represents a limiting floor to the accuracy of our quoted distances. Mitigating against this problem are the large parallactic values (∼100 mas) for sources located within 10–15 pc, as well as the fact that each field typically contains one or more extragalactic sources that help to anchor the coordinate system in an absolute sense (Mahmud & Anderson 2008).

The distance estimates determined herein are, on average, close to those presented in Kirkpatrick et al. (2012). A source-by-source comparison (Table 7) gives the ratio of the Kirkpatrick et al. (2012) values to the ones determined here. For the late T and Y dwarfs, Kirkpatrick et al. (2012) list only a few trig parallaxes (Marsh et al. 2013), with the majority coming from photometric distances determined by comparing source brightness in the H and WISE W2 bands with color–magnitude diagrams for those few T and Y dwarf objects with measured parallaxes. For the 12 sources with Keck distance errors <20%, the Kirkpatrick/Keck distance ratio is 0.9, with a dispersion of 0.2 and mean uncertainty of 0.06. This close agreement indicates that the photometric parallaxes are, in general, adequate to predict a distance within 25%. More importantly, the agreement in the average distances implies that the conclusions about the luminosity and mass functions for these ultra-low mass late T and Y dwarfs presented in Kirkpatrick et al. (2012) remain valid and now rest on more solid footing with these more precise distances.

Finally, we note that Dupuy & Kraus (2013) have recently published parallaxes for six objects in common with our sample. Table 8 demonstrates good agreement (1σ–2σ) between their parallax and proper motions in all but one case, WISE 1541−2250, which differs by 3σ. Examination of the Spitzer data for this object shows significant contamination with a nearby star as the WISE object approaches the star. We simultaneously fitted Gaussian profiles of the same width to the two sources for sightings when the sources were far enough apart to distinguish cleanly. For observations after MJD = 56066, we were unable to make an accurate determination and did not use Spitzer data in our fitting. However, the WISE object and the star are cleanly delineated in the early Spitzer observations and most importantly in our high-resolution Keck and HST data, leading us to trust our solution, which puts the object at 5.7 ± 0.3 pc instead of Dupuy & Kraus's more distant 13.5 ± 5.6 pc. A few more observations, especially after the object clears the offending star, will put the distance to this object on a firm footing.

We have used published models that predict the SEDs of our sources to investigate their physical properties. We acknowledge at the outset that this discussion is fraught with danger, given the known difficulties of modeling brown dwarfs with effective temperatures T_eff ≪ 1000 K and sometimes <400 K. Developing models at these low temperatures is very much an ongoing task requiring new gas and dust opacities, as well as incorporating clouds of water and metallic precipitates, and possibly non-equilibrium chemistry (Baraffe et al. 2003; Morley et al. 2012; Marley et al. 2007). In addition to the intrinsic model uncertainties, the models are degenerate between mass and age because the temperature and luminosity of a brown dwarf decrease slowly with time. Thus a source with a particular SED (i.e., with some T_eff) could be either a young, low-mass object or an older, more massive one. With these caveats in mind we examined two different sets of models, the dust-free BT-Settl models (Allard et al. 2003, 2012) with opacities updated relative to the older COND models (Baraffe et al. 2003) and a series of models (hereafter denoted "Morley" models) incorporating sulfide and chloride clouds, as well as a cloud-free case (Morley et al. 2012; Leggett et al. 2012; Saumon & Marley 2008). The Morley models include a cloud-free case, or are characterized by the amount of sedimentation of the precipitated material, according to a parameter f_sed, ranging from 2 < f_sed < 5. A higher value of f_sed corresponds to optically thinner clouds, whereas a lower f_sed corresponds to optically thicker clouds. Neither of these models include non-equilibrium chemistry or the influence of water clouds, although the effects of water condensation are included in the model.

The models tabulate absolute magnitudes for a variety of filters, including ground-based (MKO) J and H, and HST F125W and F140W, as well as Spitzer Channels 1 and 2 ([3.6] and [4.5] μm). We calculated a χ² value based on absolute [4.5] μm flux density using the Spitzer Ch2 photometry and our distance estimate, as well as up to five photometric colors: J − [4.5], H − [4.5], [F125W] − [4.5], [F140W] − [4.5], and [3.6] − [4.5],

$$\begin{eqnarray} \chi ^2&=&\frac{({\rm Abs}[4.5]_{{\rm obs}}-{\rm Abs}[4.5]_{{\rm model}})^2}{(5/\ln (10) \sigma _{D}/D)^2+\sigma ([4.5])_{{\rm obs}})^2}\nonumber \\ &&+\sum \limits _i\frac{(({\rm mag}_i-[4.5])_{{\rm obs}}-({\rm mag}_i-[4.5])_{{\rm model}})^2}{\sigma ({\rm mag}_i)^2+\sigma ([4.5])_{{\rm obs}}^2},\quad\quad \end{eqnarray} \tag{ 3 }$$

where D is the distance to the source, Abs[4.5] = [4.5] − 5 × log (D/10 pc) is the absolute 4.5 μm magnitude, and mag_i is the magnitude in the relevant band. The minimum χ² values for each source were determined through the interpolated (mass, age) grid with (0.1 Gyr < Age < 10 Gyr, 5 < Mass < 80 M_Jup) for the BT-Settl models, yielding the model parameters in Table 9. For the coldest Y dwarfs, the data suggest T_eff < 400 K and in these cases we used a coarser grid of BT-Settl models, sampling (300 K < T_eff < 400 K and 3.0 < log g < 5.5) for an assumed radius of 1 R_Jup, and where log g is the log of the surface gravity. For the Morley models, we interpolated in a (T_eff, log g) grid for discrete values of f_sed. The solution spaces for each source, Log(χ²) as a function of model parameters, are shown in Figures 25 and 26. Tables 9 and 10 give the fitted values for each source with their associated uncertainties derived from a Monte Carlo analysis in which the distances and photometric values were varied according to their nominal uncertainties. For the cold BT Settl cases, the uncertainties reflect the coarseness of the grid, not the observational uncertainties. The tables include values of radius and log g from the appropriate evolutionary tracks, as well as the χ² of the fits. Table 10 also includes the differences in the derived values of T_eff, mass, and age between the Morley and BT-Settl models.

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The BT-Settl models (Figure 25) show a valley of preferred values in the (mass, age) plane with quite good fits (χ² < 10 with 2–3 degrees of freedom, dof) for some of the sources, with a median value of χ² = 22 for 2–3 dof. For the coolest sources (i.e., WISE 1828+2650 (⩾Y2), WISE 1541−2250 (Y0.5), and WISE 2209+2711 (Y0:)), the fits converge on T_eff = 350 K and log g = 4.5 with χ² > 400. Figure 27(a)–(o) shows the best-fitting BT-Settl models. Generally, the BT-Settl solutions have a broad range of masses from 12 to 28 M_Jup, with an average of 20 ± 6 M_Jup, and ages from 3.4 to 8.8 Gyr, with an average of 7 ± 2 Gyr. The Y dwarfs have lower masses and temperatures than the T dwarfs—15 versus 25 M_Jup, and 390 K versus 580 K. Figure 28 shows the range in temperature for the late T and Y dwarfs derived from the two sets of models.

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Overall, the Morley models fit the data less well, with a median value of χ² (with 2 or 3 dof) of 47 compared to 22 for the BT-Settl models. These models have uniformly high surface gravities, log g ∼ 5, at the high end of the input grid and thus yield higher masses than BT-Settl cases, ∼30 M_Jup in many cases. In fact, if the model grid is allowed to extend to log g = 5.5, then the masses approach 60 M_Jup with ages of 15 Gyr, which does not seem reasonable. The difficult sources to fit with the BT-Settl models (e.g., WISE 1828+2650 and WISE 1541−2250) also have high χ² values with the Morley models. The derived effective temperatures in the two sets of models (Morley versus BT-Settl) are similar, with the Morley models being 80 K warmer (Figure 28).

As noted in Beichman et al. (2013), WISE 1828+2650 resists simple modeling due to the large disparity between the short and long wavelength magnitudes with H − [4.5] = 8.1 mag. While the 3–5 μm data alone yield a good fit to a BT-Settl model (T_eff = 440 K, log g = 4.5), such a model fails by ∼3 mag to fit the shorter wavelength data. Similarly, fitting only the short-wavelength data yields a BT-Settl model (T_eff = 300, log g = 4.5) that fails to reproduce the longer wavelength observations by comparable amounts. Adding extinction due to a very thick cloud layer with the absorption properties of "interstellar grains" suppresses the near-IR bands relative to the longer wavelengths and results in a model (T_eff = 474 K, log g = 4.6, A_V = 19 mag) with a significantly better χ² = 202 (3 dof) than the model without extinction, χ² = 3700 (4 dof). Adding a cloud layer also improved the BT-Settl fit for WISE 2209+2711 (T_eff = 420 K, log g = 4.8, A_V = 15 mag) with a χ² = 123 (2 dof). Figure 27 shows these extinct models as dotted lines for these two objects. Some previously unmodeled aspect of atmospheric physics or evolutionary status that results in a strongly absorbing cloud layer may prove necessary to understanding these objects.

It is worth noting that the poor model fits for the coldest sources are not improved by invoking a binary brown dwarf system. Fits to objects of common age but disparate masses did not show any improvement relative to the single object solutions. The juxtaposition of two unrelated sources is not a palatable solution either, because there is no evidence for one stationary and one moving object in the imaging data. Similar to the putative Y dwarf companion to WD 0806−661 (Luhman et al. 2012), objects like WISE 1828+2650 and WISE 1541−2250 must be underluminous at short wavelengths (or overluminous in the long-wavelength bands) due to some as yet poorly understood aspect of these very cold atmospheres.

Figure 29 compares the data with a number of models in two color–magnitude diagrams, J versus J−H and [4.5] versus [3.6] − [4.5]. Deviations in both color spaces are apparent with the BT-Settl models (orange, dot-dashed) and the Morley models bracketing most of the objects in Spitzer/WISE bands. The BT-Settl models tend to be ∼1 mag bluer at a given absolute magnitude than is observed, or 1–2 mag underluminous than observed at a given [3.6] − [4.5] color. Three varieties of Morley models are shown—one cloud-free, one with sulfide clouds (f_sed = 5; Table 10), and one incorporating water clouds (C. V. Morley et al., in preparation). The Morley models tend to be 1 mag redder than the observations at a given absolute magnitude, or 1 mag overluminous at a given color. Taken as a group, the Morley and BT-Settl models straddle the observations, but few of the models can be taken as providing a good fit, particularly for the less luminous, colder cases. There is a much wider divergence between the models and the observations in the JH color–magnitude diagram. These figures also include models incorporating water vapor clouds (C. V. Morley et al., in preparation). The BT-Settl models provide a good fit to the J−H colors and absolute magnitudes for the warmer objects, whereas the Morley objects do a better job on the colder objects at these wavelengths. WISE 1828+2650 stands out as extremely red in J−H and is poorly fitted in any of the models.

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Finally, there are a number of conclusions to be drawn from this discussion. The BT and Morley models provide reasonable fits to the properties of the warmer T dwarfs, with the cloud-free BT-Settl models providing the best representation of the absolute magnitudes and colors. However, the coldest objects are difficult to fit and thus properties such as mass and age are quite uncertain. In some cases masses ∼10–15 M_Jup are close to the range inferred for the objects ("planets") found to be orbiting nearby stars, but precise determinations of masses and other properties may simply be impossible using broad photometric bands. Even determining an effective temperature using a bolometric luminosity (Dupuy & Kraus 2013) requires a bolometric correction that is model dependent and, as we have seen, quite uncertain. High-resolution spectroscopy with James Webb Space Telescope (JWST) across the 1–10 μm band would yield unambiguous information on surface gravity and composition and would greatly improve our understanding of these objects. In addition, anchoring these models with a few sources with known ages and masses is absolutely critical. This can be accomplished by studying brown dwarfs in binary systems or investigating objects with higher mass companions of known ages (e.g., the potential Y dwarf companion to the white dwarf WD 0806−661; Luhman et al. 2012).

We have no direct indication of the ages of our sample. The BT-Settl and Morley models are consistent with higher surface gravity, higher mass, and thus older ages of a few Gyr or more. As independent age estimates are important because of the mass–age degeneracy, we use the kinematic information to make a crude estimate of the ages of these stars. The tangential velocity of each object comes from its proper motion and distance: v_tan = 4.74μ/Π km s⁻¹, where μ is the total proper motion and Π the parallax (Smart 1977). For the 12 objects with distance uncertainty less than 15%, the average value of v_tan is 34 km s⁻¹ with a dispersion of 24 km s⁻¹, which falls within the distribution of tangential velocities measured for nearby (<20 pc) L and T brown dwarfs (Faherty et al. 2009). There are, however, significant outliers in this distribution. WISE 0313+7807 has a remarkably small proper motion, 110 mas yr⁻¹, and thus a very small v_tan = 3 ± 1 km s⁻¹. At the other extreme, WISE 0410+1502 and WISE 2209+2711 have v_tan = 72 ± 4 km s⁻¹ and 59 ± 4 km s⁻¹, respectively. WISE 0335+4310 has the most extreme proper motion, v_tan = 78 ± 10 km s⁻¹.

These tangential velocities are consistent with M7-T9 objects studied by Faherty et al. (2009), suggesting that the extreme T and Y dwarfs studied here are drawn from the same kinematic population. For their 20 pc sample, Faherty et al. (2009) suggested ages of 2 to 4 Gyr for the objects with v_tan < 100 km s⁻¹. An object with a high v_tan like WISE 0335+4310 might be somewhat older, up to 8 Gyr. Faherty et al. (2009) suggested that a subset of their sample with low proper motions were younger than the average, perhaps <1 Gyr. Thus, the object with the lowest v_tan, WISE 0313+7807, might be younger than the other sources. Yet its BT-Settl model age is 9 Gyr and a low S/N spectrum of WISE 0313+7807 (Kirkpatrick et al. 2011) does not reveal any obvious peculiarities. The BT-Settl ages are all around 4 to 9 Gyr and thus consistent with the ages suggested by the kinematics.

Without radial velocity (RV) information it is impossible to rule definitively on the association of any of these objects with nearby clusters. Beichman et al. (2013) described a search in RV space for V_z = ±100 km s⁻¹ to look for potential associations with nearby, young clusters¹⁶ (Zuckerman & Song 2004). With one exception, none of the sample show a plausible kinematic membership with nearby clusters. If WISE 1804+3117 were to have V_z ∼ −20 km s⁻¹, an association with Tucanae/Horologium would be possible, but since the age of this object from model fitting is ∼5 Gyr, an association with this 30 Myr old cluster would be problematical.

The incidence of planetary-mass, field brown dwarfs is small. Within 10 pc, the RECONS database (Henry et al. 2006) shows 376 objects in 259 systems as of 2012. Of these objects, 248 are M stars, 16 are T8-T9.5 objects (Kirkpatrick et al. 2012), and 11 are Y dwarfs. Thus extremely low-mass objects represent just 7% of the local population. For the best-fitting BT-Settl models (Table 9) there are only five Y dwarfs with masses <15 M_Jup. Whereas these mass estimates are obviously speculative and model dependent, it is clear that objects with masses less than ∼15 M_Jup form only a small percentage of the local population. The ratio of local (<10 pc) M dwarfs (75 < M < 600) M_Jup to low-mass brown dwarfs (5 < M < 15) M_Jup in logarithmic mass units, N(M₁ → M₂)/log (M₁/M₂), is large ∼10:1 with an obviously large uncertainty due to the uncertain mass estimates. Kirkpatrick et al. (2012) cite a similar ratio, 6:1, from their volume limited brown dwarf sample. Evidently the star formation processes responsible for populating the local solar neighborhood did not produce large numbers of <15 M_Jup objects. This same effect is seen in young clusters where the ratio of stars to brown dwarfs is more precisely estimated to be ∼6:1 (Anderson et al. 2008 and references therein).

It is interesting to note that objects with <15 M_Jup also appear to be difficult to create in the protostellar environments. RV studies find that massive objects are rare in the inner reaches of planetary systems, with objects >5 M_Jup accounting for fewer than 79 out of 882, or 9%, presently cataloged planets within 10 AU of their host stars (Cumming et al. 2008; Howard et al. 2012). There are only 26 10 M_Jup objects out of 882, or just 3%. Here we have ignored the differences between M sin(i) and M, which statistically reduces the number of low-mass objects. Imaging surveys targeting the outer reaches of nearby A-F stars as well as lower mass M stars are beginning to either find objects of ∼5–10 M_Jup or set limits on their occurrence. These coronagraphic studies are typically sensitive to 5–20 M_Jup objects with ages <1 Gyr and located at orbital distances of tens to a few hundreds of AU. Apart from dramatic examples like HR8799, Fomalhaut, and β Pictoris, the success rate of these surveys has been limited, typically a few percent. Around A stars Vigan et al. (2012) find the occurrence rate of a "planet" in the (3–14 M_Jup, 5–320 AU) range is 5.9%–18% (1σ), nominally a factor of two higher than the incidence of a "brown dwarf" in the (15–75 M_Jup, 5–320 AU) range. Nielsen et al. (2013) find the occurrence rate <20% for (>4 M_Jup, 59 and 460 AU) at 95% confidence, and <10% (>10 M_Jup, 38–650 AU). They conclude by noting that "fewer than 10% of B and A stars can have an analog to the HR 8799 b (7 M_Jup, 68 AU) planet at 95% confidence." Around M stars, Montet et al. (2014) find an occurrence rate of 6.5% ± 3.0% for companions in the (1–13 M_Jup, 1–20 AU) range.

Imaging studies are in their infancy, with significant advances in sensitivity and angular resolution likely coming in few years with the Gemini Planet Imager (Macintosh et al. 2012) and P1640 (Oppenheimer et al. 2012). The improvements in contrast and sensitivity will increase the completeness of imaging surveys in terms of their mass limit. Improvements in Inner Working Angle will increase survey completeness for as yet unexplored orbital separations and may thus find many more "super-Jupiters."

With these (uncertain) mass estimates in hand we can speculate as to the formation mechanism of these free floating planetary-mass objects. Observational evidence suggests at least two methods for brown dwarf formation: star-like formation from fragmentation of a molecular cloud (e.g., Bate et al. 2003), possibly aided by turbulence (Padoan & Nordlund 2004), and protostellar disk fragmentation (Boss 2000; Stamatellos et al. 2007; Stamatellos & Whitworth 2009). Huard et al. (2006) and André et al. (2012) have both discovered proto-brown dwarf cores, indicating a star-like formation mechanism for at least some brown dwarfs. Young brown dwarfs have a similar disk fraction to young stars (Luhman et al. 2007) and show the same scaling between mass and accretion rate as stars, $\dot{M} \propto M^2$ (Muzerolle et al. 2003, 2005; Mohanty et al. 2005), again suggesting a common formation mechanism for stars and brown dwarfs. On the disk fragmentation side, Thies & Kroupa (2007) argue that there is a discontinuity in the IMF at the hydrogen-burning limit if unresolved binaries are taken into account, implying that brown dwarfs form differently from stars. Turbulent fragmentation has trouble explaining low-mass binaries—brown dwarf–brown dwarf pairs have not been observed in the numbers predicted (Reggiani & Meyer 2011), indicating that a different formation mechanism may be at work.

There are numerous reasons why both molecular cloud fragmentation and disk fragmentation produce fewer brown dwarfs than stars. While the opacity-limited minimum mass of fragments (either disk-born or cloud-born) is only 1–10 M_Jup (Low & Lynden-Bell 1976; Larson 2005; Whitworth & Stamatellos 2006), such fragments typically accrete mass and become stars (Bate et al. 2003; Kratter et al. 2010) given typical masses for molecular cloud cores and onset times for protostellar collapse (Myers 2009). Vorobyov (2013) argues that the probability of fragment survival in gravitationally unstable disks is low, as inward migration and subsequent ejection of fragments is efficient. Vorobyov also shows that fragment survival requires that the instability must happen in the T-Tauri phase of disk evolution, rather than the embedded phase, yet the necessary conditions for T-Tauri disk fragmentation may occur only rarely. The median disk/star mass ratio of Class II YSOs inferred from dust continuum observations is only 0.9% (Andrews & Williams 2005, 2007). Even when gas is observed directly, as in the deuterated H₂ (HD) observations of TW Hydrae (Bergin et al. 2013), the masses inferred are almost always less than the 0.1M_disk/M_* threshold required for disk fragmentation (Rafikov 2005).

If molecular cloud formation and disk fragmentation are both unlikely, it makes sense to consider whether core accretion—the planet formation process in which a solid core eventually grows large enough to hydrodynamically accrete gas from a disk—might form low-mass field brown dwarfs. Numerical simulations by Ford et al. (2001) show that 30% of the interactions between two giant planets near the stability boundary result in ejection, while microlensing measurements by Sumi et al. (2011) reveal a population of possibly unbound 1 M_Jup planetary-mass objects in the galactic bulge. Mordasini et al. (2012) find that the planet mass produced by core accretion falls off dramatically for M > 3 M_Jup in disks with M < 0.06 M_☉, which would explain the dearth of high-mass planets and field brown dwarfs. Yet Veras & Raymond (2012) argue that planet–planet scattering alone cannot explain the large number of unbound planets discovered by Sumi et al., who estimate two free-floating planets per solar-type star. Veras & Raymond instead suggest that other mechanisms for forming free floaters must be at work.

The likely formation mechanism for the free-floating objects presented here depends sensitively on their mass and velocity dispersion. In most cases the masses inferred from the BT-Settl and Morley models are consistent with either disk fragmentation or star-like formation, as most of the objects are above the rolloff in the planetary-mass function predicted by Mordasini et al. (2012). For the lowest mass Y dwarfs, such as WISE 1828+2650 (∼5–10 M_Jup), core accretion followed by ejection from a planetary system might be the more favored mechanism as such low-mass objects are at or below the opacity-limited minimum mass. However, core accretion near the star (where formation of a massive planet is favorable) followed by planet–planet scattering produces objects with a high velocity dispersion. Our objects have tangential velocities consistent with stars in the solar neighborhood and inconsistent with an origin in nearby young clusters (i.e., <100 pc and <100 Myr (Section 5.2)), implying that core accretion and ejection from a close-packed planetary system is unlikely. Both star-like formation and disk fragmentation followed by the ejection of partially contracted clumps (Basu & Vorobyov 2012) produce objects with low velocity dispersion, as observed for young brown dwarfs in Cha I (Joergens & Guenther 2001). Distinguishing between star-like formation and disk instability is difficult, as both mechanisms are consistent with the observed IMF (Hennebelle & Chabrier 2009; Stamatellos & Whitworth 2009). In a review of brown dwarf observations to date, Luhman et al. (2007) conclude that star-like formation is the most likely origin for low-mass free floaters, whereas Bate (2012) argues that star-like formation, disk fragmentation, and ejection of collapsing cores from molecular clouds probably operate together. The existence of mass solutions that are typical for star-like formation or disk fragmentation, combined with the low velocity dispersion, suggests that our objects are brown dwarfs rather than free-floating planets.