PARALLAXES AND PROPER MOTIONS OF ULTRACOOL BROWN DWARFS OF SPECTRAL TYPES Y AND LATE T

Determining accurate distances to brown dwarfs is important for a number of reasons. First, distance is a vital quantity in establishing not only the space density of these objects, but also the luminosity function which can then be used to test models of star formation at the lowest masses. Second, distances allow the spectra of brown dwarfs to be placed on an absolute flux scale to provide more quantitative checks of atmospheric models. Third, distances for the nearest objects allow us to construct a more complete view of our own solar neighborhood, allowing us to directly visualize the relative importance of brown dwarfs in the Galactic context. Sometimes, distance determinations produce results wholly unanticipated. For example, the J-band overluminosity of the T4.5 dwarf 2MASS J05591914−1404488 (Figure 2 of Dahn et al. 2002) was unexpected despite its location near the J-band bump at the L/T transition (e.g., Looper et al. 2008), a feature thought to be associated with decreasing cloudiness (Marley et al. 2010). It has been suggested, however, that the overluminosity is due to the presence of an unresolved binary (Burgasser et al. 2002; Dupuy & Liu 2012). Similarly unexpected was the recent determination that young, field L dwarfs are often significantly underluminous for their spectral types at near-infrared magnitudes (Faherty et al. 2012).

Some of the earliest parallax determinations for brown dwarfs were by Dahn et al. (2002), Tinney et al. (2003), Vrba et al. (2004), once surveys such as the Two Micron All-Sky Survey (2MASS; Skrutskie et al. 2006), the Sloan Digital Sky Survey (SDSS; York et al. 2000), and the Deep Near-infrared Survey of the southern sky (Epchtein et al. 1997) began to identify L and T dwarfs in large numbers. More recently, parallax programs by groups such as Marocco et al. (2010) and Dupuy & Liu (2012) have pushed astrometry measurements to the latest T spectral subclasses. With the discovery of Y dwarfs from WISE (Cushing et al. 2011; Kirkpatrick et al. 2012) we are now pushing these measurements to even colder temperatures (Beichman et al. 2012). In this paper, we present distance and/or proper motion measurements for an additional eight Y dwarfs, along with three nearby late-T dwarfs, and present the first tangential velocity measurements for Y dwarfs.

Our set of objects includes all known Y dwarfs for which we have imaging data at a sufficient number of epochs for parallax and proper motion estimation. The exception is WISE 1828+2650, presented separately by Beichman et al. (2012). In addition, we have included three late T dwarfs from an investigation of the low-mass end of the substellar mass function within 8 pc of the Sun (Kirkpatrick et al. 2011). The complete sample is listed in the observing log shown in Table 1.

Each of these objects has been observed at two or three epochs by the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) and at least four more epochs of imaging observations by the IRAC instrument (Fazio et al. 2004) on the Spitzer Space Telescope (Spitzer; Werner et al. 2004), the WFC3 instrument (Straughn et al. 2011) of the Hubble Space Telescope (HST), and various ground-based observatories. The observatories and instruments used are listed in the footnote of Table 1, and further details are given by Kirkpatrick et al. (2011, 2012).

Astrometric information was extracted from the observed images at the various epochs using the standard maximum likelihood technique in which a point-spread function (PSF) is fit to each observed source profile. The technique was essentially the same as used in 2MASS, the details of which are given by Cutri et al. (2003), except that the source extraction results presented here were made using co-added images rather than focal-plane images. The positional uncertainties were estimated using an error model which includes the effects of instrumental and sky background noise and PSF uncertainty. The PSF and its associated uncertainty map were estimated for each image individually using a set of bright stars in the field, the median number for which was 14. Since the co-added images were Nyquist sampled or better, sinc interpolation was appropriate during PSF estimation and subsequent profile fitting to the data.

In order to minimize the systematic effects of focal-plane distortion and plate scale and rotation errors, our astrometry is based on relative positions using a reference star (or set of reference stars) in the vicinity of the object. For most objects we were able to find a reference star within ∼10'' common to all images except for those of WISE, due to the lower sensitivity of the latter. In order to incorporate the WISE data it has therefore been necessary to include bright reference stars which in general were much more widely separated from the brown dwarf (up to ∼100''). Most of these were taken from the 2MASS Point Source Catalog (Cutri et al. 2003). In order not to let these stars significantly compromise the astrometry measurements from the more sensitive images with close reference stars, we used a hybrid scheme in which the bright stars were treated as secondary references, bootstrapped to the close reference stars using the images in which they were in common.

The procedure is based on the following measurement model for the observed separation between the brown dwarf and reference star:

$$\begin{eqnarray} \alpha _t - \alpha _{it}^{\rm ref} & = & \alpha _{t}^{\rm BD} - \big(\alpha _i^{\rm cat} + \Delta \alpha _i^{\rm cat} \big) + \nu _t - \nu _{it} \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \delta _t - \delta _{it}^{\rm ref} & = & \delta _t^{\rm BD} - \big(\delta _i^{\rm cat} + \Delta \delta _i^{\rm cat} \big) + \nu ^{\prime }_t - \nu ^{\prime }_{it}, \end{eqnarray} \tag{ 2 }$$

where α_t, δ_t and α^ref_it, δ_it^ref represent the extracted positions of the brown dwarf and ith reference star, respectively, estimated from the image at epoch t based on the nominal position calibration of that image; α^cat_i, δ_i^cat represent the catalog position of the reference star, and Δα^cat_i, Δδ_i^cat represent errors in the catalog position; ν_t, ν'_t represent the estimation errors for the brown dwarf, and ν_it, ν'_it represent the estimation errors for the reference star. These estimation errors include the effects of random measurement noise on the source extraction as well as the residual effects of focal-plane distortion in the position differences. We assume that they can all be described by zero-mean Gaussian random processes.

If we further assume that the Δα^cat_i, Δδ_i^cat are described a priori by zero-mean Gaussian random processes with standard deviations substantially larger than the extraction uncertainties of the reference stars, then an optimal estimate of the brown dwarf position can be obtained from

$$\begin{eqnarray} \hat{\alpha}_{t}^{\rm BD} & = & \alpha _t + \frac{1}{N_t} \sum _{i\in {\cal R}(t)} \big(\alpha _i^{\rm cat} - \alpha _{it}^{\rm ref} \big) \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \hat{\delta}_{t}^{\rm BD} & = & \delta _t + \frac{1}{N_t} \sum _{i\in {\cal R}(t)} \big(\delta _i^{\rm cat} - \delta _{it}^{\rm ref} \big), \end{eqnarray} \tag{ 4 }$$

where ${\cal R}(t)$ is the set of detected reference stars in the image at epoch t, and N_t is the number of stars in the set.

The resulting estimates are included in Table 1 in the form of offsets from the nominal position of the brown dwarf at each epoch, and the set of reference stars used is given in Table 2. After having obtained $\hat{\alpha}_{t}^{\rm BD}$ and $\hat{\delta}_{t}^{\rm BD}$, the individual reference star catalog errors can then be estimated using

$$\begin{eqnarray} \hat{\Delta \alpha}_{i}^{\rm cat} & = & {-}\alpha _{i}^{\rm cat} + \frac{1}{N_i}\sum _{t\in {\cal E}(i)} \big(\hat{\alpha}_{t}^{\rm BD} + \alpha _{it}^{\rm ref} - \alpha _t \big) \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} \hat{\Delta \delta}_{i}^{\rm cat} & = & {-}\delta _{i}^{\rm cat} + \frac{1}{N_i}\sum _{t\in {\cal E}(i)} \, \big(\,\hat{\delta}_{t}^{\rm BD} + \delta _{it}^{\rm ref} - \,\delta _t \big), \end{eqnarray} \tag{ 6 }$$

where ${\cal E}(i)$ is the set of all epochs for which the ith reference star is detected in the corresponding image, and N_i is the number of epochs in the set.

These values can be applied as corrections to the catalog positions of the reference stars, enabling a corresponding time series of estimated brown dwarf positions to be obtained separately for each individual reference star via Equations (1) and (2). The scatter in these estimates then provides a check on the assumptions regarding systematic effects such as focal-plane distortion and possible small proper motions of the reference stars themselves. We have included the effect of this scatter in the final quoted error bars in Table 1.

The measurement model incorporated the effects of parallax and linear proper motion, with appropriate correction for the Earth-trailing orbit in the case of Spitzer observations. The equations used (Kirkpatrick et al. 2011) were as follows:

$$\begin{eqnarray} \cos \delta _1 (\alpha _i-\alpha _1) & = & \Delta \alpha + \mu _\alpha (t_i-t_1) + \pi _{\rm trig} {\bm R}_i \cdot \hat{W} \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \delta _i-\delta _1 & = & \Delta \delta +\mu _\delta (t_i-t_1) - \pi _{\rm trig} {\bm R}_i \cdot \hat{N}, \end{eqnarray} \tag{ 8 }$$

where t_i is the observation time [yr] of the ith astrometric measurement, and R_i is the vector position of the observer relative to the Sun in celestial coordinates and astronomical units. $\hat{N}$ and $\hat{W}$ are unit vectors pointing north and west from the position of the source. R_i is the position of the Earth for 2MASS, SDSS, WISE, and HST observations; for Spitzer observations, R_i is the position of the spacecraft. The observed positional difference on the left-hand side is in arcseconds, the parameters Δα and Δδ are in arcseconds, the proper motion μ_α and μ_δ are in arcseconds yr⁻¹, and the parallax π_trig is in arcseconds.

Maximum likelihood estimates, based on the assumption of Gaussian measurement noise, were made of five parameters: the R.A. and decl. position offsets of the source at a specified reference time, the R.A. and decl. rates of proper motion, and the parallax. The uncertainties were derived using the standard procedure for maximum likelihood estimation (Whalen 1971) using the positional uncertainties quoted in Table 3. The resulting estimates of proper motion and parallax and their associated uncertainties are given in Table 3, and the model fits with respect to the astrometry measurements are presented in Figures 1–3. The chi-squared values, χ², for the parameter fits in Table 3 are, for the most part, close to the number of degrees of freedom, N_df, indicating reasonably good modeling of position uncertainties. Formally, the probability of exceeding χ² given N_df has a median value 0.29.

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The parallaxes that we present are, strictly speaking, relative parallaxes since no correction has been made for the small parallaxes and proper motions of the reference stars, most of which are relatively nearby. However, the expected correction for such effects is only ∼2 mas (Dupuy & Liu 2012) which is at least an order of magnitude smaller than our typical astrometric uncertainties listed in Table 3, so in this error regime the distinction between relative and absolute parallaxes is unimportant.

In order to check to what extent our parallax and proper motion estimates may have been affected by systematic effects of focal-plane distortion not properly modeled by the statistical assumptions of the previous section, we have compared the rms residuals of the above fits (obtained using multiple reference stars) with those obtained using a single reference star for each brown dwarf, and found that there was no significant difference. This suggests that whatever residual focal-plane distortion errors exist, they are smaller than the random errors of source extraction.

We have converted our maximum likelihood estimates of parallax into most probable estimates of distance taking into account both the parallax measurements themselves and prior information. The latter includes an assumption that our objects are spatially distributed in a statistically uniform manner. Formally, that would imply that parallax values are distributed a priori as P(π)∝π⁻⁴; the singularity at zero would then lead to difficulties in estimating the a posteriori most probable π. Even though the zero parallax can be excluded on physical grounds, there is still a bias toward small values such that for S/N < 4, maximum likelihood parallax estimates become insignificant (Lutz & Kelker 1973). Fortunately there is additional prior information to alleviate this problem; small parallaxes (i.e., large distances) can be excluded if they are inconsistent with the observed proper motion based on an assumed velocity dispersion of the objects being studied (Thorstensen 2003).

With these considerations in mind, our estimates of distance, d, are based on the following assumptions:

• 1.  
  Our maximum likelihood parallax values, π_ML, are distributed as Gaussians with standard deviation σ_π.
• 2.  
  Our objects are distributed spatially in a statistically uniform way, so that the a priori probability density distribution of d is proportional to d².
• 3.  
  The distribution of tangential velocities of Y dwarfs in the solar neighborhood can be described by a Gaussian random process with mean and standard deviation $\bar{V}$ and σ_V, respectively; we assume the values $\bar{V}=30$ km s⁻¹ and σ_V = 20 km s⁻¹ respectively, representative of previous observations of T dwarfs (Faherty et al. 2009).

We then obtain the most probable distance, $\hat{d}$, by maximizing the conditional probability density P(d|π_ML, μ_ML), which from Bayes' rule can be expressed by

$$\begin{eqnarray} P(d|\pi _{\rm ML},\mu _{\rm ML}) &\propto & d^2 \exp \big(-(\mu _{\rm ML}d - \bar{V})^2/\big(2\sigma _V^2 \big)\big) \nonumber\\ && \times \exp \big(-(\pi _{\rm ML} - 1/d)^2/\big(2\sigma _\pi ^2 \big)\big), \end{eqnarray} \tag{ 9 }$$

where μ_ML represents the magnitude of our maximum likelihood estimate of proper motion. Our distance estimates are presented in Column 9 of Table 3. The error bars correspond to the 0.159 and 0.841 points of the cumulative distribution with respect to P(d|π_ML, μ_ML).

As is evident from Table 1, our observations represent a mixed bag in terms of telescopes (and hence spatial resolution) and time sampling since they were not specifically designed for astrometry, but rather for follow-up photometry of brown dwarfs detected by WISE. The quality of the observations was quite varied, and not always with sufficient pixel subsampling for the estimation of the high-quality PSFs necessary for astrometry. In the case of Spitzer, for example, each observation consisted of a set of only five dithered images.

In addition, the time sampling of the parallactic cadence is of key importance in the estimation of parallax. The ideal sampling involves observations at solar elongation angles of ±90°, and this is achieved by WISE, albeit with large position errors (typically ∼01–03). These elongation angles are critical for an object on the ecliptic and less important at high ecliptic latitudes. For the parallax measurements described here, the worst example of poor sampling was WISE 1541−2250 for which all of the non-WISE observations were in one quadrant of solar elongation angle (see Column 8 of Table 1), so it is not surprising that the observations did not yield a significant parallax measurement. The previous measurement, corresponding to an estimated distance range of 2.2–4.1 pc (Kirkpatrick et al. 2011), was based on even fewer observations and furthermore used position estimates for which the PSF errors were somewhat underestimated. Our present result of >6 pc therefore supercedes that estimate, but this object should clearly be revisited once a more optimal sampling of the parallactic ellipse has been obtained. By and large, however, there is a good correlation between the sampling cadence and the quality of the parallax estimate; future observations will be optimized both for image quality and cadence.

Nevertheless, significant parallaxes (S/N > 3) have been obtained for five of the eight Y dwarfs and all three of the T dwarfs, thus providing distance estimates. Also, we have combined the latter with our proper motion estimates to yield tangential velocities, V_tan. Of course, our estimated values, $\hat{d}$ and $\hat{V}_{\rm tan}$, are somewhat dependent on the assumed prior distribution of V_tan in Equation (9), and the assumed similarity to the T dwarf distribution may not be valid if the Y dwarfs represent a significantly older population. In order to assess the sensitivity to this assumption, the distance estimates were repeated using a σ_V of 100 km s⁻¹. It was found that for a parallax significance S/N > 4, the increase in σ_V led to no more than a 20% change (always in the positive direction) in $\hat{d}$ and hence $\hat{V}_{\rm tan}$. For lower values of S/N, $\hat{V}_{\rm tan}$ becomes biased toward the a priori value, $\bar{V}$, in Equation (9). Thus in Table 3 we quote $\hat{V}_{\rm tan}$ values only for S/N > 4. Similarly, for S/N < 4 the reliability of our distance estimates is dependent on the validity of the assumptions regarding the a priori distribution of V_tan.

On this basis we obtained significant values of V_tan for two of our Y dwarfs; both are <100 km s⁻¹, suggesting membership in the thin disk population (Dupuy & Liu 2012). Similar analysis techniques, both in terms of the source extraction and parallax estimation, were used by Wright et al. (2012) to estimate the distance to the T8.5 object WISE 1118+3125, inferred (with the aid of its observed common proper motion) to be a member of the ξ UMa system, with excellent agreement with the known distance of that system.

The distance estimates for the present sample, all of which are ≳ 3 pc, have enabled the estimation of absolute magnitudes. These indicate that luminosities plummet at the T/Y boundary (Kirkpatrick et al. 2012) as illustrated by Figures 4 and 5, which represent updated versions of the absolute magnitude versus spectral-type plots from the latter work. The steep decrease may at least partially account for the apparent scatter in absolute magnitudes of objects of the same spectral type, since in the Y0 regime an error of half a spectral type apparently corresponds to more than a magnitude difference in luminosity. More data will be required to make a definitive statement, however.

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The absolute magnitudes also provide valuable guidance for models in the ultra-cool regime. To this end we have compared our observational results with model-based and empirical predictions using plots of absolute magnitude as a function of color, as shown in Figure 6. The M_J versus J – H plot in the upper panel shows that the Y dwarfs continue the trend set by the L and T dwarfs based on the parallax observations of Dupuy & Liu (2012). A key feature is the turnover in the blueward progression of the color at M_J ∼ 16, at considerably redder J – H than predicted by cloud-free models (Saumon & Marley 2008) as illustrated by the solid curve. Such behavior is also apparent in the color–magnitude diagrams for cloud-free models presented by Leggett et al. (2010). The dotted/dashed curves in Figure 6 represent models incorporating the effect of clouds containing various amounts of Cr, MnS, Na₂S, ZnS, and KCl condensates (Morley et al. 2012), as indicated by the sedimentation efficiency parameter, f_sed; lower values correspond to optically thicker clouds. It is apparent that these models can account at least partly for the relative redness of some of the J – H colors, but they predict a blueward hook for temperatures below 400 K, which does not appear to be matched by the observations. Perhaps some of the scatter in J – H colors in Figure 6 might be explained in terms of a patchy cloud model; it is also possible that the inclusion of water clouds might improve consistency with the observations.

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Figure 6 does show reasonable consistency between observations and models based on IRAC colors, i.e., M_[3.6] and M_[4.5] as a function of the [3.6]–[4.5] color. The only major discrepancy is that WISE 1828+2650, whose effective temperature is believed to be ∼300 K, falls at a location more indicative of 500 K on these plots.

We thank C. Morley for providing the results of model calculations and also the referee for very helpful comments. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. Support for this work was provided by NASA through an award issued to programs 70062 and 80109 by JPL/Caltech. This work is also based in part on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program 12330, support for which was provided by NASA through a grant from the Space Telescope Science Institute. This paper also includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile. This research has made use of the NASA/IPAC Infrared Science Archive (IRSA), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration, and also the SIMBAD database, operated at CDS, Strasbourg, France.