Parallaxes of Cool Objects with WISE: Filling in for Gaia

This paper uses the multi-epoch astrometry from the Wide-field Infrared Survey Explorer (WISE) to demonstrate a method to measure proper motions and trigonometric parallaxes with precisions of $\sim$4 mas yr$^{-1}$ and $\sim$7 mas, respectively, for low-mass stars and brown dwarfs. This method relies on WISE single exposures (Level 1b frames) and a Markov Chain Monte Carlo method. The limitations of Gaia in observing very low-mass stars and brown dwarfs are discussed, and it is shown that WISE will be able to measure astrometry past the 95% completeness limit and magnitude limit of Gaia (L, T, and Y dwarfs fainter than $G\approx19$ and $G=21$, respectively). This method is applied to WISE data of 20 nearby ($\lesssim17$ pc) dwarfs with spectral types between M6-Y2 and previously measured trigonometric parallaxes. Also provided are WISE astrometric measurements for 23 additional low-mass dwarfs with spectral types between M6-T7 and estimated photometric distances $<17$ pc. Only nine of these objects contain parallaxes within Gaia Data Release 2.


INTRODUCTION
With the imminent release of proper motions and trigonometric parallax measurements for over a billion sources from the Gaia satellite (Perryman et al. 2001), it is important to understand what objects will not be included in the final catalog. Theissen et al. (2017) investigated the Gaia shortfall and found that Gaia will be limited in its ability to observe ultracool dwarfs (spectraltypes later than mid-L) at distances 10 pc due to its relatively blue bandpass.
A number of projects are aimed at measuring trigonometric parallaxes of these ultracool dwarfs (e.g., Dupuy & Liu 2012;Beichman et al. 2013;Faherty et al. 2012;Kirkpatrick et al. 2014;Dupuy et al. 2016;Skinner et al. 2016;Smart et al. 2017b). However, these projects rely on either: 1) numerous epochs of ground-based and space-based observations, using facilities such as the Spitzer Space Telescope (Werner et al. 2004); or 2) survey data spanning multiple epochs, such as the Digitized Sky Survey (DSS), the Wide-field Infrared Survey Explorer (WISE ; Wright et al. 2010), or the Two Micron All-Sky Survey (2MASS; Skrutskie et al. 2006).
Here, I outline a method to measure proper motions and trigonometric parallaxes of nearby ( 20 pc), ultracool objects using publicly available WISE data. In Section 2, I discuss the properties and limitations of Gaia and WISE. I describe my method for measuring proper motions and parallaxes in Section 3. I make comparisons between my method and previous literature measurements for 17 nearby, low-mass dwarfs in Section 3.1. In Section 3.2, I provide new measurements for ten nearby, ultracool dwarfs. I discuss the utility of my method for the immediate future in Section 4.

GAIA AND WISE LIMITATIONS
Gaia is currently conducting the largest astrometric mission to date, with an expected yield of over 1 billion sources with measured proper motions and precise trigonometric parallaxes ( 16 µas for low-mass stars with V 14; Perryman et al. 2001). Theissen et al. (2017) quantified the shortfall of ultracool objects within Gaia Data Release 1 (Gaia Collaboration et al. 2016b,a), using matches between the Late-Type Extension to the Motion Verified Red Stars catalog (LaTE-MoVeRS) and Gaia DR1. They found that Gaia is severely limited in its ability to observe spectral types later than ∼L5 farther than ∼10 pc.
I reevaluated the Gaia shortfall using the LaTE-MoVeRS sample by comparing the fraction of LaTE-MoVeRS sources with a counterpart found within Gaia DR1 as a function of W 2 magnitude (4.6 µm) and Sloan Digital Sky Survey (SDSS; York et al. 2000) i − z color (Figure 1). The fraction of matches typically drops below ∼30% for sources later than ∼L4 with W 2 11. This is similar to the fraction of known L and T dwarfs matched to Gaia DR1 in the study of Smart et al. (2017a), ∼ 24% matched; 321 out of 1317 with G < 23. Figure 1 also shows the total WISE positional uncertainty for a single frame (i.e., σ 2 α + σ 2 δ ) as a function of W 2 magnitude using the original MoVeRS catalog (Theissen et al. 2016). The W 2-absolute magnitude ranges for M, L, and T dwarfs (Filippazzo et al. 2015), and Y dwarfs (Tinney et al. 2014) are indicated with gray shaded regions. The approximate Gaia limiting magnitude is denoted with the red dotted line (W 2 = 11), indicating that WISE can provide astrometric measurements of ultracool objects past the limits of Gaia.

PARALLAXES USING WISE MULTI-EPOCH DATA
The all-sky observations made by WISE are ideal for astrometric studies because they span multiple epochs, most separated by ∼ 6 months. The original WISE mission surveyed the entire sky in four bands, 3.4, 4.6, 12, and 22 µm (hereafter W 1, W 2, W 3, and W 4). This original mission lasted from December 2009 to August 2010, after which the cryogen was depleted, and WISE observed in W 1, W 2, and W 3 until September (3-band survey; ∼30% of the sky 1 ). WISE continued to observe in W 1 and W 2 as part of the Near-Earth Object Wide-field Infrared Survey Explorer (NE-OWISE; Mainzer et al. 2011) mission. In December 2013, WISE was reactivated to continue surveying the entire sky in W 1 and W 2 as part of the NEOWISE-Reactiviation (NEOWISE-R; Mainzer et al. 2014) mission. The NEOWISE-R mission is currently ongoing. The combined WISE dataset contains 7 epochs for every source, with a cadence of ∼6 months and a time baseline of ∼6.5 years. Each single epoch has 12-13 7.7 second exposures in W 1 and W 2 2 , and possibly more exposures depending on depth of coverage for a given line-of-sight.
The survey strategy of WISE was to observe fields close to 90 • Solar elongation, which places observed objects close to their maximum parallax factors ). Many studies have computed parallaxes using WISE data, combined with either higherpositional precision observations (e.g., Spitzer, Keck), and/or data providing a longer time baseline (e.g., DSS, 2MASS), for nearby objects (e.g., Beichman et al. 2013;Luhman 2013;Kirkpatrick et al. 2014;Scholz 2014). However, the numerous epochs of current WISE data now allow relatively precise ( 10 mas) parallax measurements to be made without the need for further data.
An illustration of the parallax method described here is shown in Figure 2 for 2MASS J02550357−4700509, a nearby (∼5 pc) L8 dwarf (Martín et al. 1999;Patten et al. 2006;Kirkpatrick et al. 2008;Faherty et al. 2012). First, all Level 1b (L1b) source catalogs (i.e., All-  The number on each bin indicates the total number of LaTE-MoVeRS sources within that bin, and the color of the bin corresponds to the fraction of stars with matches in Gaia DR1 (black colored text indicates a fraction 0.7). The fraction of Gaia DR1 matches drops below ∼30% for all spectral types later than ∼L4 with W 2 11 (area enclosed with red-dotted lines). Approximate spectral types from Schmidt et al. (2015) are listed on the top. Bottom: 2-d histogram of total WISE positional uncertainty (single frames) versus W 2 magnitude for the MoVeRS catalog. Bin areas are 0.1 mag × 5 mas. Also shown are the MW 2 ranges (gray shaded areas) for M, L, and T dwarfs taken from Filippazzo et al. (2015), and the range for Y dwarfs from Tinney et al. (2014). The astrometric precision hits a floor of ∼50 mas for relatively bright sources. Y dwarfs typically have a single-band measurement (W 2), with low signal-to-noise, which will push them to higher positional uncertainties (> 300 mas). The approximate limit for Gaia is indicated by the red dotted line (W 2 = 11).
Sky, 3-band, NEOWISE Post-Cryo, and NEOWISE-R) are queried for objects within 30 of the expected position of 2MASS J02550357−4700509. L1b source catalogs contain sources extracted from each single exposure 3 . Sources were grouped by epoch, demarcated by 6 month periods starting 91 days after the mean modified Julian date (MJD) of the first epoch (shown as dotted lines in the top and middle panels of Figure 2). Next, the uncertainty weighted average position for each epoch is determined using the WISE reported astrometry position values (α, δ, σ α , σ δ ) and a 3-σ clip to remove outliers. Additionally, the observing epoch time is selected to be the average MJD for each epoch, over a period that may span ∼1-10 days.
Uncertainties (σ α and σ δ ) were computed using the weighted positional uncertainty per epoch, as illustrated in the inset figure of the bottom panel of Figure 2 and given by, where N is the number of frames within the given epoch. The astrometric solution was computed from, where the subscript 0 denotes the first epoch, and the subscript i denotes each subsequent epoch. P α,δ represents the parallax factors (van Dekamp 1967) given by (Green 1985), where X, Y, Z are the components of the barycentric position vector of the Earth obtained from the JPL DE430 solar system ephemeris. These equations were solved using a Markov Chain Monte Carlo (MCMC) routine built on the emcee code (Foreman-Mackey et al. 2013), assuming normally distributed parameters and uniform priors.
3.1. Comparison to Literature Astrometric Measurements I applied the MCMC routine described in the previous section to 17 known, nearby, low-mass objects with generally well-determined parallaxes (<15% uncertainty) sourced from the literature. Sources were chosen to cover a range of spectral types, distances, and W 2 magnitudes. I solved for the astrometric solutions, and Table 1 reports the median values, and the uncertainties are estimated as the largest deviation between the median values and the 16th and 84th percentile values.    c Measurements made using only NEOWISE(-R) data.  Figure 3. Figure 4 shows the residuals between the parallax value derived using WISE and the highest precision literature parallax, as a function of W 2 magnitude. The astrometric precision severely deteriorates for sources fainter than W 2 ≈ 14, setting the approximate limit for the where this method is valid.
The sources beyond 1-σ include 2MASS J2322−3133 and WISEP J1506+7027. 2MASS J2322−3133 is a relatively distant (17.1 ± 1.6 pc), L0 dwarf (Reid et al. 2008;Faherty et al. 2012), indicating the approximate distance limit at this W 2 magnitude. The precision of this method will have a strong dependence on W 2 magnitude, and can be assessed in the future with a larger control sample.
The second outlier, WISEP J1506+7027, has a > 2-σ discrepant parallax measurement from the value reported in Marsh et al. (2013). However, WISEP J1506+7027 is an outlier in the Marsh et al. (2013) absolute magnitude-spectral type diagrams (see Marsh et al. 2013 Figures 4 and 5). The distance computed here of 5.2 ± 0.2 pc places WISEP J1506+7027 on the In principle, this method can be applied to any source bright enough to be extracted within a single WISE L1b frame. Saturated photometry may cause an issue with centroiding. Crowded fields also pose a challenge due to multiple objects within each search radius, however, with proper source selection in each epoch's catalog, any nearby source detected by WISE can have its parallax measured. It is unlikely that robust parallaxes ( 15% uncertainty) can be measured farther than ∼17 pc using WISE alone, assuming an average parallax precision of 9 mas, however, this distance limit is highly dependent on W 2 magnitude. Only the first six objects listed in Table 1 are contained within Gaia DR1, which is roughly consistent with the W 2 11 limit discussed in Section 2.
The computed parallax values are relative measurements, not absolute, since the positional solution for WISE is derived from both moving and non-moving sources. Without knowing the bulk movement of the calibration objects, absolute measurements cannot be made. Assuming a correction of ∼2 mas from relative to absolute using the modeling results reported in Dupuy & Kraus (2013), any correction will be smaller than the uncertainty in this method, so the reported values are   Marsh et al. (2013). The blue dashed line shows the approximate distance limit (right y-axis) as a function of W 2 magnitude for 15% uncertainties using this method.
likely consistent with absolute parallaxes within the uncertainties.

New and Significantly Improved Astrometric Measurements
There are many known low-mass objects estimated to be within 17 pc based on spectro-photometric parallax relationships, that have either no trigonometric parallax measurement (254 within the Winters et al. 2015 southern hemisphere 25 pc sample alone), or measurements with large uncertainties (>20%). Here, I investigate ten such cases, sourced from the literature to cover a range of spectral types, distances, and W 2 magnitudes, three of which are listed in Gaia DR1 (Table 2). In Figure 5, I show the astrometric solution to one of these cases, SDSS J141624.08+134826.7, an unusually blue L6 dwarf with an estimated distance of ∼8 pc (Bowler et al. 2010;Schmidt et al. 2010a;Scholz 2010). The sources in Gaia DR1 should have more precise measurements in the near future. The remainder will require additional astrometric observations to more precisely determine their parallaxes.
Using the sources from Tables 1 and 2, I computed a 2nd order polynomial fit to the parallax uncertainties divided by 15% (the approximate parallax limit where 15% uncertainties can be achieved) as a function of W 2 magnitude. The fit is shown in Figure 3 as the blue dashed line corresponding the the y-axis on the right. Bright sources (W 2 8) can potentially have their parallaxes measured out to distances of ∼25 pc, with sources at the Gaia magnitude limit (W 2 ≈ 11) requiring distances within ∼17 pc. These limits will be validated in the future with a larger control sample.

DISCUSSION
The technique presented here has the potential to find new, nearby, ultracool objects, and measure relatively accurate parallaxes without the need for follow-up observations. This is particularly important as Spitzer is expected to be retired in 2018. Its replacement, the James Webb Space Telescope (JWST ; Gardner et al. 2006), while sensitive to these faint dwarfs, is an unlikely facility for a dedicated parallax program.
As discussed here and in Theissen et al. (2017); Smart et al. (2017a), Gaia will not provide parallaxes for most of the lowest mass stars and brown dwarfs, leaving only ground-based programs. The WISE method described here is a useful alternative for the nearest ( 17 pc) ultracool objects, and a follow-up paper will report measurements for all ultracool dwarfs estimated to be within this distance. C.A.T would like to give his sincerest thanks to Adam Burgasser, Julie Skinner, Aurora Kesseli, and Andrew West for providing comments which greatly improved this manuscript. C.A.T. would also like to thank Julie Skinner for helpful discussion leading up to this manuscript. This material is based upon work supported by the National Aeronautics and Space Administration under Grant No. NNX16AF47G issued through the Astrophysics Data Analysis Program.
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.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www. cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https:// www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

SDSS-IV is managed by the Astrophysical Research
This research made use of Astropy, a communitydeveloped core Python package for Astronomy (Astropy Collaboration et al. 2013). Plots in this publication were made using Matplotlib (Hunter 2007). This research has made use of the SIMBAD database, operated at    CDS, Strasbourg, France. This research has made use of NASA's Astrophysics Data System. This research has also made use of the VizieR catalogue access tool, CDS, Strasbourg, France (Wenger et al. 2000).