# Trigonometric Parallaxes of Two T Dwarfs with Keck and ShaneAO Astrometry

Jayke Nguyen  Lawrence Livermore National Laboratory, 7000 East Ave,
Livermore, CA 94550, USA  Center for Astrophysics and Space Sciences,
Department of Physics, University of California, San Diego, 9500 Gilman Dr.,
La Jolla, CA 92093, USA  S. Mark Ammons  Lawrence Livermore National
Laboratory, 7000 East Ave, Livermore, CA 94550, USA  Kaitlin Dennison
Stanford University, Department of Aeronautics and Astronautics, Durand
Building, 496 Lomita Mall, Stanford, CA 94305-4035, USA  E. Victor Garcia  kWh
Analytics, 609 Mission Street, Fl 2, San Francisco, CA 94105, USA  Jessica R.
Lu Astronomy Department, University of California, Berkeley, CA 94720, USA
Stephen McMillan  Minnesota Institute for Astrophysics, University of
Minnesota, 116 Church Street SE, Minneapolis, MN 55455, USA  Maissa Salama
Institute for Astronomy, University of Hawai’i at Mānoa, Hilo, HI 96720-2700,
USA

(Received June 14, 2021; Revised July 22, 2021; Accepted July 26, 2021)

###### Abstract

We present trigonometric parallax and proper motion measurements for two
T-type brown dwarfs. We derive our measurements from infrared laser guide star
adaptive optics observations spanning five years from the ShaneAO/SHARCS and
NIRC2/medium-cam instruments on the Shane and Keck telescopes, respectively.
To improve our astrometric precision, we measure and apply a distortion
correction to our fields for both instruments. We also transform the Keck and
ShaneAO astrometric reference frames onto the ICRS using five-parameter
parallax and proper motion solutions for background reference stars from Gaia
DR2. Fitting for parallax and proper motion, we measure parallaxes of $73.5\pm
9.2\text{\,}\mathrm{mas}$ and $70.1\pm 6.7\text{\,}\mathrm{mas}$ for
WISEJ19010703+47181688 (WISE1901) and WISEJ21543294+59421370 (WISE2154),
respectively. We utilize Monte Carlo methods to estimate the error in our
sparse field methods, taking into account overfitting and differential
atmospheric refraction. Comparing to previous measurements in the literature,
all of our parallax and proper motion values fall within $2\sigma$ of the
published measurements, and 4 of 6 measurements are within $1\sigma$. These
data are among the first parallax measurements of these T dwarfs and serve as
precise measurements for calibrating stellar formation models. These two
objects are the first results of an ongoing survey of T dwarfs with Keck/NIRC2
and the Shane Adaptive Optics system at Lick Observatory.

Astrometry, Brown dwarfs, Parallax, Proper motions, T dwarfs, Laser guide star

††journal: PASP††software: astropy (Astropy Collaboration et al., 2013, 2018),
astroscrappy (McCully et al., 2018), Python (Van Rossum & Drake Jr, 1995),
StarFinder (Diolaiti et al., 2000), Starlink (Currie et al., 2014)

tablenum SIXtablenum

##  1 Introduction

A fundamental parameter of a stellar object is its distance from Earth.
Historically, astronomers have constrained the distances of nearby stars with
a variety of techniques, including the trigonometric parallax. Parallax
measurements of stars can in turn be used to derive absolute magnitudes,
allowing stars to be placed on color-magnitude diagrams and compared to
theoretical star formation tracks. Brown dwarfs are substellar objects that
were first theoretically devised as the lowest mass products of stellar
formation processes (Kumar, 1962; Hayashi & Nakano, 1963). They straddle the
boundary between fusion-powered M dwarfs and giant planets. Brown dwarfs,
particularly T and Y dwarfs, remain elusive in observational surveys due to
their low luminosities at visible wavelengths.

The advent of all-sky infrared surveys like the Sloan Digital Sky Survey
(SDSS), Two Micron All Sky Survey (2MASS), Wide-field Infrared Survey Explorer
(WISE), and UKIRT Infrared Deep Sky Survey (UKIDSS) has led to the discovery
of hundreds of brown dwarfs in our stellar neighborhood (Eisenstein et al.,
2011; Skrutskie et al., 2006; Kirkpatrick et al., 2011; Lawrence et al.,
2007). However, only a small percentage of these objects have trigonometric
parallax measurements. The dedicated astrometric survey Gaia (Gaia
Collaboration et al., 2016, 2018) has derived parallax and proper motions for
millions of stars (Lindegren, L. et al., 2018) but Gaia’s astrometric
precision suffers for faint T dwarfs that fall below its visible band
sensitivity limit. Therefore, targeted observations at infrared wavelengths
are required to derive parallaxes for the faintest of T dwarfs.

Current ground based laser guide star adaptive optics (LGSAO) systems provide
relative astrometric precision of milliarcseconds or less (Lu et al., 2010;
Ammons et al., 2014; Neichel et al., 2014) which can be used to derive
trigonometric parallax of field brown dwarfs at similar levels of precision
(Ammons et al., 2016; Garcia et al., 2017). The main drawback of LGSAO systems
is that they are limited in their field of view to the anisokinetic angle of
the site, typically $20\text{\,}\arcsec$ to $40\text{\,}\arcsec$, in infrared
wavelengths. Observing a single object only requires a few minutes of
observation time per epoch, and many objects may be observed in a single night
even with limited time. Nonetheless, this method of astrometry is a precise
and reliable way to measure parallax and proper motion of faint T dwarfs.

We have selected a sample of 63 T dwarfs from the SDSS, 2MASS, WISE, and
UKIDSS infrared surveys with nearby tip/tilt stars of brightness $R<17$ and at
least 3 background reference stars in a $33\text{\,}\arcsec$ diameter field of
view. These brown dwarfs are currently being monitored astrometrically with
the LGSAO systems at the 3-m Shane telescope at Lick Observatory (Gavel et
al., 2014) and the 10-m Keck II telescope at Keck Observatory (Wizinowich et
al., 2000; Wizinowich, 2006; Max et al., 1995).

In this paper we present parallax and proper motion constraints for WISE1901
and WISE2154, the first two objects in our survey that have accumulated at
least six epochs. We make use of a novel astrometric calibration method that
works in sparse star fields. To validate our results, we compare our
trigonometric parallax values to existing literature values (Theissen, 2018;
Kirkpatrick et al., 2020).

Section 2 describes the sample selection and survey strategy. Section 3
outlines the data reduction process for Shane and Keck data, including plate
scale measurements for SHARCS and NIRC2. Section 4 outlines stellar position
finding using Starfinder, the alignment procedure, and the final parallax and
proper motion fitting routine. Section 5 presents our error analysis using M92
data and Monte Carlo methods. Section 6 presents our measurements of the
position, proper motion, and parallax for WISE1901 and WISE2154. Lastly,
Section 7 discusses our methodology and results in the context of astrometry
and presents future prospects using our sparse-reference-star method.

##  2 Sample and Survey Construction

###  2.1 Laser Guide Star Adaptive Optics

Using LGSAO at infrared wavelengths is advantageous for parallax measurements
for one primary reason: the decreased point spread function (PSF) width
compared to seeing-limited PSFs increases the signal-to-noise ratio (SNR) of
detection against a bright infrared background. When faint stellar sources
such as brown dwarfs are imaged without adaptive optics, astrometric errors
due to SNR may be higher than the atmospheric differential tip/tilt jitter
(DTTJ). With AO, these terms are reduced and, depending on the geometry of the
reference stars constellation, astrometric errors can hit the systematic noise
floor of the instrument within minutes of exposure time.

As mentioned previously, a limitation of LGSAO systems is the field of view as
set by the anisokinetic angle, typically $20\text{\,}\arcsec$ \-
$40\text{\,}\arcsec$ at infrared wavelengths. This limited field of view
occasionally fails to provide an adequate number of background reference
stars. For this reason, the larger survey is limited to L and T dwarfs that
happen to have at least three reference stars within this patch as well as a
tip/tilt star for LGSAO guiding within the vicinity. Due to the dependence of
astrometric DTTJ error on stellar separation, nearby reference stars are more
stable than distant references, and only a few reference stars are needed to
establish a local reference frame for position and rotation. In addition, all
of our reference stars have positions and proper motions constrained in the
ICRS system by the Gaia mission, which makes it possible to place our LGSAO
observations from multiple observatories onto the same astrometric reference
frame. We test the precision of this method and the implications for the
errors on our parallax and proper motion with a Monte Carlo experiment in M92
in Section 5.2.

###  2.2 Target Selection Criteria

The larger sample was selected from the SDSS, 2MASS, WISE, and UKIDSS brown
dwarf surveys. The criteria were: (1) spectral type later than L1; (2) at
least three background reference stars with $R<20$ in the USNO A2.0 catalog
(Monet, 1998) within a diameter of $33\text{\,}\arcsec$ (within at least
$0.5\text{\,}\arcsec$ of the edge, to ensure the PSF halo is entirely within
the frame); and (3) a $R<17$ tip/tilt star within $60\text{\,}\arcsec$ of the
brown dwarf. The two targets presented here (see Table 1), all selected from
the WISE survey, are the two in our larger sample with the largest number of
observations.

Since the laser guide star was only used during bright time at both
observatories, the observing cadence is roughly monthly with eight observing
runs allocated per year. Lick Observatory is prone to poorer weather during
the winter and spring months, so observations were typically only successful
during the summer and early autumn time frames. This cadence presents a
challenge for parallax measurements which ideally are sampled evenly
throughout the year. To mitigate this effect, we make use of Keck LGS data as
well which is less affected by seasonal weather. In addition, we obtained
observations over a longer time baseline than is typical for parallax
measurements (4 to 5 years, rather than the standard 1 or 2) to improve the
SNR on the parallax and proper motion parameters.

##  3 Data

Table 1: Brown Dwarf Type and Photometric Data

Object | SpT | J-H | H-Ks | J-Ks | J | H | W1-W2 | W2
---|---|---|---|---|---|---|---|---
WISE1901 | T5(3) | $0.39\pm 0.12^{(3)}$ | $-0.17\pm 0.07^{(3)}$ | … | $15.86\pm 0.07^{(3)}$ | … | … | …
WISE2154 | T5(1) | … | … | … | $15.66^{(1)*}$ | $15.77\pm 0.17^{(2)}$ | $2.14\pm 0.08^{(2)}$ | $13.51\pm 0.03^{(2)}$

Note. — References. (1) Looper et al. (2007), (2) Kirkpatrick et al. (2012),
(3) Burgasser et al. (2004)

∗ 2MASS J band

Table 2: Gaia Reference Stars

Label | J2000 RA | J2000 Dec | PM RA | PM Dec | Parallax | Gaia Source ID
---|---|---|---|---|---|---
| hr | deg | mas/yr | mas/yr | mas |
WISE1901
Shane Reference Stars
1901S0 | 19 01 07.012 | +47 18 16.772 |  -0.196 $\pm$ 0.190 | -3.984 $\pm$ 0.184 | 0.318 $\pm$ 0.093 | 2131382365365031040
1901S2 | 19 01 06.219 | +47 18 01.525 | -1.335 $\pm$ 0.239 | -3.908 $\pm$ 0.234 | 0.736 $\pm$ 0.116 | 2131383125572944768
1901S3 | 19 01 06.022 | +47 17 59.638 | -1.113 $\pm$ 0.038 |  -6.649 $\pm$ 0.039 | 1.566 $\pm$ 0.019 | 2131383125576327552
Keck Reference Stars
1901K1 | 19 01 07.012 | +47 18 16.772 |  -0.196 $\pm$ 0.190 | -3.984 $\pm$ 0.184 | 0.318 $\pm$ 0.093 | 2131382365365031040
1901K2 | 19 01 06.817 | +47 18 03.618 |  -0.685 $\pm$ 0.662 | -4.982 $\pm$ 0.632 | 0.394 $\pm$ 0.322 | 2131382365365026048
1901K3 | 19 01 06.219 | +47 18 01.525 | -1.335 $\pm$ 0.239 | -3.908 $\pm$ 0.234 | 0.736 $\pm$ 0.116 | 2131383125572944768
WISE2154
Shane Reference Stars
2154S1 | 21 54 32.709 | +59 42 03.752 | -2.603 $\pm$ 0.439 | -2.699 $\pm$ 0.425 |  -0.670 $\pm$ 0.205 | 2202738436624981376
2154S2 | 21 54 31.314 | +59 42 01.835 | -1.803 $\pm$ 1.294 | -1.857 $\pm$ 0.961 |  -0.773 $\pm$ 0.497 | 2202738436624980608
2154S3 | 21 54 30.856 | +59 42 10.874 | -2.132 $\pm$ 0.773 | -2.841 $\pm$ 0.631 | -0.425 $\pm$ 0.343 | 2202739192539229056
Keck Reference Stars
2154K1 | 21 54 32.709 | +59 42 03.752 | -2.603 $\pm$ 0.439 | -2.699 $\pm$ 0.425 |  -0.670 $\pm$ 0.205 | 2202738436624981376
2154K2 | 21 54 31.314 | +59 42 01.835 | -1.803 $\pm$ 1.294 | -1.857 $\pm$ 0.961 |  -0.773 $\pm$ 0.497 | 2202738436624980608
2154K3 | 21 54 30.856 | +59 42 10.874 | -2.132 $\pm$ 0.773 | -2.841 $\pm$ 0.631 | -0.425 $\pm$ 0.343 | 2202739192539229056

Note. — Only WISE1901 uses different reference stars than our Shane data. For
consistency between targets, matching reference stars between Shane and Keck
data are duplicated in the table. Negative parallaxes are technically
unphysical, but allowed in Gaia data (Luri et al., 2018). In our methodology,
we treat negative parallaxes as zero.

###  3.1 Shane Data

Figure 1: Example $1020\text{\,}\mathrm{px}$ by $1020\text{\,}\mathrm{px}$
cropped frame from 2014 May 6 of processed M92 data from ShaneAO. This frame
has been processed to remove bad pixels and a circular crop on the science
field is applied. The circle has a radius of $510\text{\,}\mathrm{px}$ and the
sensor has a plate scale of
$32.812\text{\,}\mathrm{mas}\text{\,}{\mathrm{px}}^{-1}$,
$32.806\text{\,}\mathrm{mas}\text{\,}{\mathrm{px}}^{-1}$ in $x$ and $y$
respectively.

We observed brown dwarf targets using the SHARCS camera with ShaneAO on the
Shane telescope from 2014 November 8 to 2019 August 17. The SHARCS camera uses
a circular subsection of a cooled, engineering grade Teledyne HAWAII-2RG chip
about 1020 pixels in diameter, with a $33\text{\,}\arcsec$ field of view. All
ShaneAO brown dwarf science frames were taken in H band and have a total
exposure time of $120\text{\,}\mathrm{s}$. All observing sequences are
undithered and we attempt to return the field to the same pixel in-between
epochs by using the same offsets from the tip/tilt star. Within epochs we see
$\sim$$1\text{\,}\mathrm{px}$ of jitter due to errors in telescope pointing
over time. Between epochs, we see $\sim$$10\text{\,}\mathrm{px}$ of
unintentional shifts due to stellar proper motions and variations in
mechanical tip/tilt mirror response, such as actuator hysteresis. An example
of a SHARCS M92 frame is shown in Figure 1. The flat-field images were
constructed by median combining twilight flats when available and dome flats
otherwise. Dark frames were also subtracted using frames matching the exposure
time of our science targets. The Gaia reference stars used for aligning Shane
data are found on Table 2. A list of targets and the number of frames used are
listed on Tables 3 and 4.

For all stellar positions estimated from SHARCS data, we apply a fourth order
polynomial distortion map derived from M92 data taken on 2014 August 12. The
resulting distortion map is shown in Figure 2. We directly map matching stars
in SHARCS M92 observations to Hubble ACS M92 observations (Yelda et al.,
2010). A gnomonic projection is not necessary since we do not transform into a
spherical coordinate system at this step. The distortion map utilizes 205
stars and fits only the nonlinear components of the combined optical and
physical sensor distortion. The distortion map is well defined over the
circular subsection, but is less accurate near the edges due to a declining
density of well-detected stars.

Figure 2: Vector field of the SHARCS distortion map across the science field
subset of the image sensor in absolute pixel units. The pixel coordinates
range from 0 to 2048 in $x$ and $y$ but only a circular subsection is used.
The circle is centered at roughly
$($1108\text{\,}\mathrm{px}$,$769\text{\,}\mathrm{px}$)$ with a radius of
$510\text{\,}\mathrm{px}$. Note that the distortion map is large in magnitude
and poorly constrained at the edge of the bounding circle since there are few
stars used in the fit near the edge.

All of our Shane data, except our M92 data, were taken with the laser guide
star. Although ShaneAO was initially near-diffraction-limited in 2014,
starting in 2016 the LGS return flux declined, resulting in a much broader PSF
and decreased precision. To combat this, we implement a full width half
maximum (FWHM) cut, removing data above the 75th percentile. As mentioned
previously, seeing conditions varied significantly and were generally fair,
but some data are affected by poor seeing conditions.

###  3.2 Keck Data

Figure 3: Example $1024\text{\,}\mathrm{px}$ by $1024\text{\,}\mathrm{px}$
frame of WISE2154 from Keck taken on 2014 September 8. The fainter unlabeled
stars with low SNR are not visible in Shane images and are thus unsuitable
reference stars. We measure the plate scale of NIRC2 medium cam in $x$ and $y$
to be $19.885\text{\,}\mathrm{mas}\text{\,}{\mathrm{px}}^{-1}$ and
$19.788\text{\,}\mathrm{mas}\text{\,}{\mathrm{px}}^{-1}$, respectively, pre-
bench realignment and a plate scale of
$19.849\text{\,}\mathrm{mas}\text{\,}{\mathrm{px}}^{-1}$ post-bench
realignment, where the bench realignment occurred on 2015 April 13. The
reference stars and brown dwarf are labeled “Ref” and “BD” in the image. Note
that NIRC2 has very few bad pixels as compared to SHARCS. The field was also
positioned to ensure our data are not impacted by the crack in the bottom left
of the sensor.

We obtained images of our brown dwarf target fields using the NIRC2 medium
camera with the LGSAO system at the 10-m Keck II Observatory. We use data from
2014 August 6, 2014 September 8, 2015 June 4, and 2015 June 30. NIRC2 medium
camera has a field of view of $20\text{\,}\arcsec$ with an Aladin-3
$1024\times 1024$ pixel array. All data was taken with a
$60\text{\,}\mathrm{s}$ exposure time in the Ks band. Like the Shane data,
Keck frames were not dithered. Within epochs we see
$\sim$$0.5\text{\,}\mathrm{px}$ of jitter due to errors in telescope pointing
over time. Between epochs, we see $\sim$$3\text{\,}\mathrm{px}$ of
unintentional shifts due to stellar proper motions and variations in
mechanical tip/tilt mirror response, such as actuator hysteresis. Images are
flat fielded using dome flats or twilight flats and dark subtracted using
standard techniques. An example of a NIRC2 brown dwarf science frame is shown
in Figure 3. The NIRC2 sensor has very few bad pixels and did not require any
bad pixel treatment. Compared to SHARCS data, the diffraction-limited NIRC2
data are far more precise and seeing conditions were excellent on all four
epochs. The Gaia reference stars used for aligning Keck data are found on
Table 2. A list of targets and number of frames used are listed on Tables 3
and 4.

To create a distortion map for the Keck data, we fit a sixth-order polynomial
to 919 stars in M92 data from 2015 April 4 using the medium camera
configuration of NIRC2 (Figure 4). However, on 2015 April 13, a bench
realignment of the Keck II telescope optics was performed, correcting a y-axis
PSF elongation, changing the optical distortion pattern on the sensor. Our
first two epochs occur before this date, and our last two epochs occur after.
We were not able to collect M92 data post-bench realignment but the a
distortion map was measured before and after this date using the narrow camera
on NIRC2 (Yelda et al., 2010; Service et al., 2016). The pre- distortion map
is a bivariate b-spline fit, while the post- distortion map is a sixth-order
polynomial fit. Changing cameras is effectively a change in the optical system
and therefore the narrow camera distortion map cannot be directly applied to
data taken on the medium camera. However, using the difference between the
pre- and post- distortion map we measure the change in distortion after the
bench realignment and apply this change to our pre-bench realignment
distortion map, effectively creating a distortion map for post-bench
realignment data on medium camera. The difference distortion map is shown in
Figure 5 and is visually a good match to Service et al. (2016).

Figure 4: NIRC2 medium camera distortion map across the
$$20\text{\,}\arcsec$\times$20\text{\,}\arcsec$$ field of view, fit using a
sixth-order polynomial. Since the medium camera on NIRC2 is not often used,
the distortion map was not previously measured and had to be directly measured
from M92 observations from 2015 April 4. Figure 5: NIRC2 distortion map
difference between pre- and post- bench realignment across the whole sensor.
The arrow scale is increased by a factor of two relative to Figure 4. The
bench realignment corrected this, resulting in the difference distortion map
showing an elongation of the x component and compression the y component of
the distortion correction.

###  3.3 SHARCS Plate Scale Measurements

Figure 6: Residuals of the SHARCS plate scale linear transform fit. The
standard deviations of the residuals in $x$ and $y$ are
$(\sigma_{x},\sigma_{y})=($0.213\text{\,}\mathrm{px}$,$0.172\text{\,}\mathrm{px}$)$.
The distortion map was applied to stellar positions measured from M92 data
collected from the Shane telescope on 2014 August 12 and compared to Hubble
ACS astrometric measurements (Yelda et al., 2010)

Measurements of the plate scale were made by cross referencing distortion-
corrected M92 data against distortion-corrected Hubble ACS data (Yelda et al.,
2010). Since we compare two projected systems, a gnomonic projection is not
needed and we may fit a linear transformation between the two systems
directly. Using observations of M92 collected on 2014 August 12, we use 19
reference stars to measure the plate scale by fitting a five-parameter linear
transformation (x/y translation, rotation, and x/y magnification) to the
values. The resulting plate scale is
$32.812\text{\,}\mathrm{mas}\text{\,}{\mathrm{px}}^{-1}$,
$32.806\text{\,}\mathrm{mas}\text{\,}{\mathrm{px}}^{-1}$ in $x$ and $y$
respectively, which we use as a fixed value in all transformations. Figure 6
shows the residuals from that linear transformation. The standard deviations
of the residuals in $x$ and $y$ are
$(\sigma_{x},\sigma_{y})=($0.213\text{\,}\mathrm{px}$,$0.172\text{\,}\mathrm{px}$)$.

###  3.4 NIRC2 Plate Scale Measurements

Figure 7: Residuals of the pre-bench realignment NIRC2 plate scale fit using
M92 data from 2014 August 5. The standard deviations of the residuals in $x$
and $y$ are
$(\sigma_{x},\sigma_{y})=($0.129\text{\,}\mathrm{px}$,$0.217\text{\,}\mathrm{px}$)$.
We fit magnification in $x$ and $y$ since we expect an optical defect that
preferentially elongates the PSF in the $x$ direction (Service et al., 2016).
Figure 8: Residuals of the post-bench realignment NIRC2 plate scale fit. The
standard deviations of the residuals in $x$ and $y$ are
$(\sigma_{x},\sigma_{y})=($0.167\text{\,}\mathrm{px}$,$0.219\text{\,}\mathrm{px}$)$.
We fit a single magnification to the plate scale since after the bench
realignment the PSFs we obtain should now be very close to square (Service et
al., 2016). Fits to the two dimensional plate scale post-bench realignment
data show a less than a $0.01\%$ difference between $x$ and $y$, confirming
that a single plate scale measurement for both dimensions is sufficient.

The process for measuring the NIRC2 plate scale is almost exactly the same
procedure as outlined in Section 3.3. However, we know that prior to the bench
realignment there is an expected y-axis elongation in NIRC2. Pre-bench
realignment data is fit with a x/y translation, rotation, and x/y
magnification, while post-bench realignment data is fit with a x/y
translation, rotation, and single magnification. Additionally, we apply the
difference distortion map to the post-bench realignment data. For the pre-
bench realignment we use 22 data points and measure an $x$ and $y$ plate scale
of $19.885\text{\,}\mathrm{mas}\text{\,}{\mathrm{px}}^{-1}$ and
$19.788\text{\,}\mathrm{mas}\text{\,}{\mathrm{px}}^{-1}$, respectively. For
post-bench realignment data we use 31 data points and measure a single plate
scale of $19.849\text{\,}\mathrm{mas}\text{\,}{\mathrm{px}}^{-1}$. Figures 7
and 8 shows the residuals of the pre- and post- bench realignment,
respectively. The standard deviations of the pre- and post- bench realignment
residuals in $x$ and $y$ are
$(\sigma_{x},\sigma_{y})=($0.129\text{\,}\mathrm{px}$,$0.217\text{\,}\mathrm{px}$)$
and
$(\sigma_{x},\sigma_{y})=($0.167\text{\,}\mathrm{px}$,$0.219\text{\,}\mathrm{px}$)$,
respectively.

Table 3: List of M92 Shane and Keck Observations Date | Start Time | Instrument | Band | No. Frames | Integration Time
---|---|---|---|---|---
UTC | UTC |  |  |  | sec
2014-04-15 | 12:03:37.598 | SHARCS, Shane | H | 2 | 15
2014-05-07 | 09:37:33.111 | SHARCS, Shane | H | 9 | 15
2014-08-12 | 04:43:33.644 | SHARCS, Shane | H | 1 | 15
2014-08-05 | 08:20:36.24 | NIRC2, Keck | Ks | 1 | 10
2015-04-03 | 12:35:21.91 | NIRC2, Keck | Ks | 1 | 30

##  4 Astrometry

###  4.1 Overview

We develop an coordinate transformation procedure to unite astrometry from
both Shane and Keck onto the same astrometric system, the International
Celestial Reference System (ICRS). All steps utilize positions measured from
three reference stars in the field. As a first step, we align frames within
epochs to the epoch mean with two parameter translation fits. Next, we place
the multiple epochs onto the same reference frame using three parameter
translation and rotation fits. We then derive a World Coordinate System (WCS)
transform onto the ICRS using Gaia positions for reference stars. Lastly, we
fit the five-parameter parallax and proper motion solutions to the brown dwarf
positions on the ICRS. Astrometry derived from Shane and Keck are aligned
separately until the final parallax and proper motion fit at which point they
are both on the ICRS.

Figure 9: An output template PSF of Shane WISE2154 data from StarFinder. The
template PSF is stacked with other objects in the frame, smoothed, and
normalized. Stacking all of the objects in the frame help minimize the bias
introduced by bad pixel replacement.

###  4.2 Stellar Position Finding

We use StarFinder (Diolaiti et al., 2000) to find the stellar position of each
object in our science field. In each frame, a template PSF
$30\text{\,}\mathrm{px}$ wide is estimated using all of the objects in the
frame. Figure 9 shows an example of a template PSF used to find stellar
positions of objects in the frame. Each object used to create the template PSF
is interpolated, allowing us to center with sub-pixel accuracy. However, the
interpolation methods require a clean image free of bad pixels to determine
the stellar position properly. Therefore, we use pixel value replacement
methods to fix the bad pixels in the image. We use astroscrappy, a Python
implementation of the L.A. Cosmic algorithm (McCully et al., 2018; van Dokkum,
2001) in addition to sigma clipping to replace bad pixels with the median
value of their neighbors. Since all of our targets are relatively isolated, we
also cropped the relevant objects in each frame and created a mosaiced image
that would run through StarFinder faster. The initial crop parameters are used
to recover the actual stellar positions on the frame. In the final stack of
translated PSFs, we mask bad pixels, take the median pixel values across the
entire image stack, and smooth with a variable box size median filter to
create the master PSF for the frame. Each master PSF has at least three
stacked sources per frame. This master PSF is then correlated with all the
objects in the frame and the stellar position is determined by aligning the
master PSF to each object.

Immediately after finding the stellar position, we apply the SHARCS or NIRC2
distortion map to the data. Each stellar position found in each frame is
distortion-corrected.

###  4.3 Within-Epoch Alignment

Within a given epoch, we have several frames from which we derive a single set
of stellar position coordinates for all the unique objects in the frame.
First, we take the mean stellar position of each object in the field,
averaging the distortion-corrected stellar position of each object across
frames, and use the averaged stellar position object to create a “mean frame”.
A simple two parameter translation is fit (Equation 1) and applied to each
frame to shift the stellar positions as close as possible to the mean frame. A
translation is sufficient as we expect telescope tracking errors to be small.
To handle outliers, we iteratively shrink a circle centered on the average
stellar position of each object in the frame, starting at
$100\text{\,}\mathrm{mas}$ in radius and shrinking to
$20\text{\,}\mathrm{mas}$ over $1\text{\,}\mathrm{mas}$ intervals, rejecting
any frames with stellar positions outside of that radius. Any time a frame is
rejected, the mean frame is remeasured and refitted. The set of stellar
positions measured taking the average stellar positions of the translated
frames is our final measurement for the epoch. We repeat this procedure for
each epoch, producing a single final mean frame per epoch.

$\begin{bmatrix}x\\\ y\end{bmatrix}+\begin{bmatrix}x_{0}\\\
y_{0}\end{bmatrix}=\begin{bmatrix}x_{f}\\\ y_{f}\end{bmatrix}$ (1)

###  4.4 Epoch-to-Epoch Alignment

Using the resulting final mean frame from each epoch, we align the new set of
epoch frames using a similar process as Section 4.3. Given the large temporal
difference between epochs, we expect that there are much larger systematic
errors that result from telescope pointing error, changes in the adaptive
optics system, and differing atmospheric parameters in each observation. To
account for small rotations in the fields between epochs, we fit an additional
rotational parameter (Equation 2). Additionally, since we expect the brown
dwarf to have high proper motion, the brown dwarf is excluded from the
alignment fit but included when the fit is applied. The brown dwarf is easily
identified as the object that moves significantly between epochs relative to
the other objects.

$\begin{bmatrix}\cos{\theta}&-\sin{\theta}\\\
\sin{\theta}&\cos{\theta}\end{bmatrix}\begin{bmatrix}x\\\
y\end{bmatrix}+\begin{bmatrix}x_{0}\\\
y_{0}\end{bmatrix}=\begin{bmatrix}x_{f}\\\ y_{f}\end{bmatrix}$ (2)

###  4.5 Epoch-to-Gaia Alignment

Lastly, we fit a gnomonic projection WCS transform to convert our epoch
aligned data to the ICRS. The fit parameters in the WCS transform are CRVAL1
(translation), CRVAL2 (translation), and $\theta$ (rotation). CRPIX1 and
CRPIX2 are assumed to be in the middle of the image, while the $x$ and $y$
plate scale, $k_{x}$ and $k_{y}$ (Equation 3), are measured independently from
M92 data. At minimum, we require that we have three reference stars for an
accurate fit to Gaia. Knowing the relative shape of the background stars and
the plate scale of both SHARCS and NIRC2, we are able to find the ICRS
coordinates of the background stars in Gaia DR2. After removing the parallax
and proper motion of the reference stars, we fit our data to the Gaia
coordinates with a WCS transform. The resulting output yields the coordinates
of the brown dwarf at each epoch in right ascension and declination in the
ICRS.

$\begin{bmatrix}\cos{\theta}&-\sin{\theta}\\\
\sin{\theta}&\cos{\theta}\end{bmatrix}\begin{bmatrix}k_{x}&0\\\ 0&k_{y}\\\
\end{bmatrix}\begin{bmatrix}x\\\ y\end{bmatrix}+\begin{bmatrix}x_{0}\\\
y_{0}\end{bmatrix}=\begin{bmatrix}x_{f}\\\ y_{f}\end{bmatrix}$ (3)

###  4.6 Parallax Fit

The parallax and proper motion equation is given by Kirkpatrick et al. (2011)
and Urban & Seidelmann (2013):

$\cos{\delta_{1}}(\alpha_{i}-\alpha_{1})=\Delta_{\alpha}+\mu_{\alpha}(t_{i}-t_{1})+\pi\vec{R_{i}}\cdot\hat{W}$
(4)

$(\delta_{i}-\delta_{1})=\Delta_{\delta}+\mu_{\delta}(t_{i}-t_{1})-\pi\vec{R_{i}}\cdot\hat{N}$
(5)

Where the north-pointing and and west-pointing vectors are defined as:

$\hat{N}=-\hat{x}\cos{\alpha}\sin{\delta}-\hat{y}\sin{\alpha}\sin{\delta}+\hat{z}\cos{\delta}$
$\hat{W}=\hat{x}\sin{\alpha}-\hat{y}\cos{\alpha}$

$\alpha$ is the right ascension, $\delta$ is the declination,
$\Delta_{\alpha}$ is the right ascension offset, $\Delta_{\delta}$ is the
declination offset, $\mu_{\alpha}$ is the right ascension proper motion,
$\mu_{\delta}$ is the declination proper motion, $\vec{R}$ is the Earth’s
position vector, $t$ is the time, and $\pi$ is the parallax. Subscript $1$
denotes the initial epoch at time $t_{1}$ and subscript $i$ denotes a
particular epoch at time $t_{i}$.

From the brown dwarf positions that are mapped to the ICRS, we fit Equations 4
and 5 to the data. As a result, we get a measurement on the parallax and
proper motion of the brown dwarf target.

It is also important to note that Shane and Keck data are aligned separately
until the final parallax and proper motion fit is performed. Only then are
both the Keck and Shane data in the same ICRS coordinate system.

##  5 Error and Monte Carlo Analysis

###  5.1 Positional Error

As a first order estimate of our positional error, we compute the standard
deviation in $x$ and $y$ of the reference star residuals after the epoch-to-
epoch alignment step. In this computation we include residuals from all
epochs, combining both Shane and Keck data. The resulting errors are:
($6.733\text{\,}\mathrm{mas}$, $7.510\text{\,}\mathrm{mas}$) and
($5.338\text{\,}\mathrm{mas}$, $2.963\text{\,}\mathrm{mas}$) for WISE1901 and
WISE2154 respectively. This calculation is a simple estimate of the random
portion of the positional error and will capture effects from varying PSF FWHM
widths, low SNR, and errors due to bad pixel correction. To determine the true
systematic error, a more rigorous analysis of our sparse-reference star
method, capturing effects from overfitting as well as contributions from
differential atmospheric refraction, needs to be performed.

###  5.2 Systematic Error Due to Overfitting

With only a few reference stars per target, we risk overfitting our data if we
use transformations with too many parameters. With three reference stars we
have three parameters to fit and six data points measured per frame (each
reference star has two data points, $x$ and $y$). Any additional fitting terms
(e.g. magnification, skew) in the transformations could result in additional
error due to overfitting. To model this, in our positional errors, we add an
extra term in quadrature that captures the systematic overfitting error of our
alignment process due to only using three reference stars.

We estimate this systematic overfitting component by simulating sparse-field
parallax and proper motion measurements with M92 cluster data taken over
multiple epochs. Unlike our targets, M92 has many bright stars in the field,
all with effectively zero parallax and high SNR. For any random selection of
three “reference” stars in M92 and one mock “brown dwarf” we would expect to
derive close to zero relative motion over two successive epochs using our
sparse-reference star method for alignment. Any residuals we do detect would
be mostly due to overfitting.

We start with two M92 frames acquired with ShaneAO on 2014 April 14 and nine
frames on 2014 May 6, from which stellar positions have been measured with
StarFinder. For each of 1820 Monte Carlo iterations, we randomly pick
combinations of four stars. These combinations are analogous to having three
reference stars and one virtual brown dwarf. We perform our alignment and
fitting routine using the reference stars for alignment, stopping at the
epoch-to-epoch step. By excluding the epoch-to-Gaia alignment, we ensure that
this simulation accounts for sources of systematic error such as jitter
between frames and unintentional dithering between epochs. We then measure the
change in position of the mock brown dwarf between the two epochs. We then
compute the standard deviation of these positional changes for all 1820 Monte
Carlo iterations. We divide this standard deviation of the residuals by a
factor of $\sqrt{2}$ as we are using a difference between two epochs to
estimate the error for a single epoch. The resulting single-axis systematic
errors from overfitting are ($\sigma_{x}$,$\sigma_{y}$) =
($8.989\text{\,}\mathrm{mas}$, $6.852\text{\,}\mathrm{mas}$) when using three
reference stars.

###  5.3 Differential Atmospheric Refraction

Light from stars passing through the atmosphere is refracted by different
amounts depending on the zenith angle at which it is observed and the spectrum
of the star. This results in a difference in separation between two stars in
the field due to the atmosphere, the differential atmospheric refraction
(DAR). This phenomena is generally not a large effect, but becomes important
at milliarcsecond level precision.

For each target, we compute the achromatic and chromatic differential
atmospheric refraction (ADAR, CDAR) corrections for all frames according to
the model defined by Gubler & Tytler (1998). The ADAR and CDAR are calculated
using ambient temperature, pressure, humidity, and zenith angle data measured
at the beginning of each frame. The strength of the DAR is largely controlled
by the zenith angle, but other atmospheric parameters still play a role. We
utilize the Starlink function sla_refco (Currie et al., 2014) to calculate the
coefficients for calculating the ADAR and CDAR. Additionally, using the
function AltAz from astropy (Astropy Collaboration et al., 2013) and the Gaia
coordinates, we determine the zenith angle of each star in the frame based on
the observation time and the geographic coordinates of the Shane and Keck
telescopes. We calculate an “effective wavelength” ($\lambda_{\text{eff}}$)
using Equation 8 from (Gubler & Tytler, 1998), integrating over H band. For
the reference stars, we assume a temperature of $7000\text{\,}\mathrm{K}$ as
they are likely to be field FG dwarfs (the resulting DAR corrections are
relatively insensitive to this assumption). To calculate the effective
wavelength for the brown dwarf we directly integrate spectra from 2MASS
0559-14, a T5 dwarf, using data from McLean et al. (2003). This calculation
yields effective wavelengths of $1.620\text{\,}\mathrm{\SIUnitSymbolMicro m}$
and $1.635\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for the brown dwarf and
reference star, respectively, in SHARCS H band. Similarly, we calculate
$2.110\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$2.134\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for the brown dwarf and
reference star, respectively, in NIRC2 Ks band. Note that at the L-T
transition, the color reversal makes it possible for the brown dwarf to have a
shorter effective wavelength than the reference star in H and Ks bands (Dupuy
& Liu, 2012).

For the Shane observations the standard deviations of the set of ADAR values
are $2.074\text{\,}\mathrm{mas}$, $1.735\text{\,}\mathrm{mas}$ and the
standard deviations of the set of CDAR values are
$0.389\text{\,}\mathrm{mas}$, $0.144\text{\,}\mathrm{mas}$ for WISE1901 and
WISE2154, respectively. Similarly, for the Keck observations we measure the
standard deviation of ADAR values of our targets to be
$1.923\text{\,}\mathrm{mas}$ and $1.527\text{\,}\mathrm{mas}$ and the standard
deviation of the CDAR values of our targets to be $0.251\text{\,}\mathrm{mas}$
and $0.008\text{\,}\mathrm{mas}$ for WISE1901 and WISE2154, respectively. The
perturbative effects of both ADAR and CDAR on our positions are small relative
to our positional error bars and so we do not account for ADAR and CDAR
explicitly in our position measurements. However, we add these terms in
quadrature to our existing error estimation.

###  5.4 Monte Carlo Analysis

To derive errors bars on our parallax and proper motion measurements, we
perform a Monte Carlo simulation of our coordinate transformation, alignment
scheme, and parallax and proper motion fitting procedure. This MC analysis
uses our best estimate of the total error on the brown dwarf position as
described below.

$\sigma^{2}_{\text{tot}}=\sigma^{2}_{\text{pos}}+\sigma^{2}_{\text{sys}}+\sigma^{2}_{\text{ADAR}}+\sigma^{2}_{\text{CDAR}}$
(6)

Adding the positional error from the brown dwarf data, the systematic
overfitting error from M92 data, and the error of ADAR and CDAR in quadrature
(Equation 6), we measure total errors. For the Shane data, we measure errors
($\sigma_{x}$, $\sigma_{y}$) of ($11.428\text{\,}\mathrm{mas}$,
$10.383\text{\,}\mathrm{mas}$) for WISE1901 and
($10.599\text{\,}\mathrm{mas}$, $7.666\text{\,}\mathrm{mas}$) for WISE2154.
For the Keck data, we measure errors ($\sigma_{x}$, $\sigma_{y}$), of
($11.397\text{\,}\mathrm{mas}$, $10.350\text{\,}\mathrm{mas}$) for WISE1901
and ($10.566\text{\,}\mathrm{mas}$, $7.620\text{\,}\mathrm{mas}$) for
WISE2154.

For the Monte Carlo analysis we perturb the position of each stellar object
(including the brown dwarf) in the frame by sampling Gaussian distributions
with width, $\sigma_{\text{tot}}$, equal to our previously determined total
error. For each Monte Carlo iteration, we follow the same alignment and
parallax fitting procedure as is performed for the unperturbed data. The
positional perturbation is injected at the beginning of the within-epoch
alignment step. Sampling our distribution 10000 times yields Gaussian
posterior distributions for parallax and proper motion for each of our brown
dwarf targets. We then measure the error on each of our model parameters by
calculating the standard deviation of the posterior functions.

##  6 Results

Table 4: List of Shane and Keck Observations

Date | Start Time | Instrument | Band | No. Frames | Integration Time | Mean FWHM
---|---|---|---|---|---|---
UTC | UTC |  |  |  | seconds | px
WISE1901
2015-06-10 | 09:51:04.636 | SHARCS, Shane | H | 2 | 120 | 13.423
2015-08-08 | 08:49:16.143 | SHARCS, Shane | H | 6 | 120 | 14.677
2015-08-30 | 06:40:27.708 | SHARCS, Shane | H | 10 | 120 | 15.199
2015-10-04 | 04:14:26.476 | SHARCS, Shane | H | 4 | 120 | 14.953
2018-08-26 | 05:42:32.297 | SHARCS, Shane | H | 6 | 120 | 13.164
2019-07-21 | 06:48:22.497 | SHARCS, Shane | H | 7 | 120 | 15.952
2019-08-17 | 07:24:33.809 | SHARCS, Shane | H | 9 | 120 | 13.284
2014-08-06 | 10:11:55.01 | NIRC2, Keck | Ks | 4 | 120 | 2.962
2014-09-08 | 05:59:49.44 | NIRC2, Keck | Ks | 8 | 120 | 3.124
2015-06-04 | 08:57:52.14 | NIRC2, Keck | Ks | 4 | 120 | 3.056
2015-06-30 | 08:10:40.09 | NIRC2, Keck | Ks | 4 | 120 | 3.723
WISE2154
2014-11-08 | 09:29:18.811 | SHARCS, Shane | H | 12 | 120 | 11.863
2015-08-08 | 09:29:18.811 | SHARCS, Shane | H | 10 | 120 | 15.190
2015-08-30 | 09:06:54.778 | SHARCS, Shane | H | 12 | 120 | 15.131
2015-10-04 | 05:17:06.431 | SHARCS, Shane | H | 6 | 120 | 15.070
2016-06-20 | 10:59:51.252 | SHARCS, Shane | H | 18 | 120 | 13.203
2016-09-15 | 05:49:23.903 | SHARCS, Shane | H | 8 | 120 | 14.747
2017-08-05 | 10:49:42.324 | SHARCS, Shane | H | 13 | 120 | 14.539
2018-08-26 | 09:05:23.968 | SHARCS, Shane | H | 3 | 120 | 16.170
2014-08-06 | 10:26:45.81 | NIRC2, Keck | Ks | 3 | 120 | 5.553
2014-09-08 | 08:25:23.85 | NIRC2, Keck | Ks | 11 | 120 | 4.743

Note. — All of our Keck data was taken in the span of about ten months. Shane
data was taken over the course of five years until ShaneAO temporarily went
offline in 2019 due to low laser guide star return. Furthermore, Shane data
was typically obtained around the same time of the year due to poorer weather
in the winter and spring months.

###  6.1 WISE1901

Figure 10: Eight epochs of residuals from the alignment of frames within-epoch
for WISE1901 Shane data. We see relatively good dispersion and the points are
mostly aligned. The third object was our faintest source, as a result, it has
larger residuals than the rest of the objects. The standard deviations for
these residuals, in units of Shane pixels, are: ($\sigma_{x}$,$\sigma_{y}$) =
($0.128\text{\,}\mathrm{px}$, $0.115\text{\,}\mathrm{px}$),
($0.095\text{\,}\mathrm{px}$, $0.130\text{\,}\mathrm{px}$),
($0.179\text{\,}\mathrm{px}$, $0.141\text{\,}\mathrm{px}$),
($0.108\text{\,}\mathrm{px}$, $0.127\text{\,}\mathrm{px}$) for the four
objects in the frame, in order. Figure 11: Four epochs of residuals from the
alignment of frames within-epoch for WISE1901 Keck data. Keck data was much
better resolved than Shane data, even for the faint object. The standard
deviations for these residuals, in units of Keck pixels, are:
($\sigma_{x}$,$\sigma_{y}$) = ($0.068\text{\,}\mathrm{px}$,
$0.100\text{\,}\mathrm{px}$), ($0.066\text{\,}\mathrm{px}$,
$0.049\text{\,}\mathrm{px}$), ($0.028\text{\,}\mathrm{px}$,
$0.054\text{\,}\mathrm{px}$), ($0.036\text{\,}\mathrm{px}$,
$0.048\text{\,}\mathrm{px}$) for the four objects in the frame, in order.
Figure 12: Residuals from Shane and Keck data of epoch-to-epoch alignment for
WISE1901 in units of Shane pixels. Shane and Keck are not on the ICRS yet and
their fits are independent one another. The eight Shane epoch residuals are
tightly clustered when aligning between epochs. The Keck residuals shows
clustering for the pairs of data taken in 2014 and 2015, but are overall still
tightly clustered. Across all objects, we measure overall standard deviations
of ($\sigma_{x}$,$\sigma_{y}$) = ($7.433\text{\,}\mathrm{mas}$,
$9.291\text{\,}\mathrm{mas}$) for the Shane data and
($\sigma_{x}$,$\sigma_{y}$) = ($5.104\text{\,}\mathrm{mas}$,
$1.573\text{\,}\mathrm{mas}$) for the Keck data. Figure 13: Residuals from
WISE1901 epoch-to-Gaia alignment for both Shane and Keck. The final alignment
of the Shane data to Gaia shows somewhat large residuals, but the large number
of epochs helps smooth out the final fit. Across all objects, we measure
overall standard deviations of ($\sigma_{x}$,$\sigma_{y}$) =
($11.056\text{\,}\mathrm{mas}$, $12.144\text{\,}\mathrm{mas}$) for the Shane
data and ($\sigma_{x}$,$\sigma_{y}$) = ($11.023\text{\,}\mathrm{mas}$,
$3.731\text{\,}\mathrm{mas}$) for the Keck data. Figure 14: Final parallax and
proper motion fit for WISE1901. Note that the data was generally taken around
the same time of year every year. The residuals have a standard deviation of
($\sigma_{x}$,$\sigma_{y}$) = ($10.129\text{\,}\mathrm{mas}$,
$9.711\text{\,}\mathrm{mas}$). The reduced chi-squared of the fit is
$\chi^{2}_{\nu}=1.152$.

The WISE1901 field as seen by Shane and Keck do not share the same set of
reference stars. In particular, there is a very bright star (1901S3) in the
Shane frame that is just outside the field in the Keck frame. For the Shane
data we include this bright star as an astrometric reference while for the
Keck data we include a fainter nearby object (1901K2) instead. The very bright
star is near the edge of the frame and a nearby fainter star also influences
the PSF, resulting in poorer positional precision. The Shane position
measurement error was therefore significantly higher for this target. For
WISE1901, we measure $\pi=$73.5\pm 9.2\text{\,}\mathrm{mas}$$,
$\mu_{\alpha}=$117.9\pm 2.9\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$,
and $\mu_{\delta}=$403.4\pm
2.2\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$. Figures 10 and 11 show
the within-epoch residuals, Figure 12 show the epoch-to-epoch residuals,
Figure 13 shows the epoch-to-Gaia residuals, and Figure 14 shows the parallax
and proper motion fit. The overall residuals for the Figure 14 are clustered
with no clear bias, but shows the low precision due to the bright reference
star in the Shane data. Compared to Kirkpatrick et al. (2020), our parallax is
within $0.7\sigma$ of their measured value of $\pi=$67.3\pm
3.4\text{\,}\mathrm{mas}$$. Their proper motion values of
$\mu_{\alpha}=$122.9\pm 1.4\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$
and $\mu_{\delta}=$405.7\pm
1.4\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$ are within $1.7\sigma$
and $1.0\sigma$ of our values, respectively. The proper motion in right
ascension for WISE1901 is the largest deviation in our data set, but is still
within a $2\sigma$. The chance of this result occurring is small, but not too
large to be totally improbable. Therefore, we still find our results to be
consistent with Kirkpatrick et al. (2020).

###  6.2 WISE2154

Figure 15: Eight epochs of residuals from the alignment of frames within-epoch for WISE2154 Shane data. This data is our cleanest set of data. The standard deviations for these residuals, in units of Shane pixels, are: ($\sigma_{x}$,$\sigma_{y}$) = ($0.104\text{\,}\mathrm{px}$, $0.123\text{\,}\mathrm{px}$), ($0.082\text{\,}\mathrm{px}$, $0.126\text{\,}\mathrm{px}$), ($0.105\text{\,}\mathrm{px}$, $0.134\text{\,}\mathrm{px}$), ($0.112\text{\,}\mathrm{px}$, $0.154\text{\,}\mathrm{px}$) the four objects in the frame, in order. Figure 16: Two epochs of residuals from the alignment of frames within-epoch for WISE2154 Keck data. The standard deviations for these residuals, in units of Keck pixels, are: ($\sigma_{x}$,$\sigma_{y}$) = ($0.034\text{\,}\mathrm{px}$, $0.039\text{\,}\mathrm{px}$), ($0.019\text{\,}\mathrm{px}$, $0.024\text{\,}\mathrm{px}$), ($0.028\text{\,}\mathrm{px}$, $0.038\text{\,}\mathrm{px}$), ($0.025\text{\,}\mathrm{px}$, $0.031\text{\,}\mathrm{px}$) for the four objects in the frame, in order. Figure 17: Residuals from Shane and Keck data of epoch-to-epoch alignment for WISE2154 in units of Shane pixels. Shane and Keck are not on the ICRS yet and their fits are independent one another. Across all objects, we measure overall standard deviations of ($\sigma_{x}$,$\sigma_{y}$) = ($5.952\text{\,}\mathrm{mas}$, $3.314\text{\,}\mathrm{mas}$) for the Shane data and ($\sigma_{x}$,$\sigma_{y}$) = ($0.936\text{\,}\mathrm{mas}$, $0.948\text{\,}\mathrm{mas}$) for the Keck data. Figure 18: Residuals from WISE2154 epoch-to-Gaia alignment for both Shane and Keck. Shane epoch-to-Gaia fits are slightly offset from the reference stars. Across all objects, we measure overall standard deviations of ($\sigma_{x}$,$\sigma_{y}$) = ($16.546\text{\,}\mathrm{mas}$, $7.805\text{\,}\mathrm{mas}$) for the Shane data and ($\sigma_{x}$,$\sigma_{y}$) = ($2.055\text{\,}\mathrm{mas}$, $4.679\text{\,}\mathrm{mas}$) for the Keck data. Figure 19: Final parallax and proper motion fit for WISE2154. This target has has the smallest residuals of all of our targets. In particular, WISE2154 is well sampled throughout the periodic motion of the brown dwarf. The residuals have a standard deviation of ($\sigma_{x}$,$\sigma_{y}$) = ($6.860\text{\,}\mathrm{mas}$, $6.316\text{\,}\mathrm{mas}$). The reduced chi-squared of the fit is $\chi^{2}_{\nu}=1.338$. Table 5: Final parallax and proper motion Fit Measurements Target | RA | Dec | PM RA | PM Dec | Parallax | $\chi^{2}_{\nu}$
---|---|---|---|---|---|---
at 2015.5 | deg | deg | mas/yr | mas/yr | mas |
WISE1901 | $285.2759846(33)$ | $47.3056997(22)$ | $\phantom{-}117.9\pm 2.9$ | $\phantom{-}403.4\pm 2.2$ | $73.5\pm 9.2\phantom{0}$ | $1.152$
WISE2154 | $328.6367953(34)$ | $59.7032306(22)$ | $-166.7\pm 2.9$ | $-463.1\pm 3.5$ | $70.1\pm 6.7\phantom{0}$ | $1.338$

We present a total of seven Shane epochs and two Keck epochs for WISE2154. In
both the Shane and Keck data, we utilize the same set of three reference
stars. For WISE2154, we measure $\pi=$70.1\pm 6.7\text{\,}\mathrm{mas}$$,
$\mu_{\alpha}=$-166.7\pm
2.9\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$, and
$\mu_{\delta}=$-463.1\pm
3.5\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$. Figures 15 and 16 show
the within-epoch residuals, Figure 17 show the epoch-to-epoch residuals,
Figure 18 shows the epoch-to-Gaia residuals, and Figure 19 shows the parallax
and proper motion fit. The parallax and proper motion fit for this target has
small, unbiased residuals and more even temporal sampling compared to the
previous target. Kirkpatrick et al. (2020) measures $\pi=$71.0\pm
2.3\text{\,}\mathrm{mas}$$, $\mu_{\alpha}=$-164.9\pm
0.7\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$,
$\mu_{\delta}=$-465.0\pm
0.7\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$, corresponding to within
$0.1\sigma$, $0.6\sigma$, and $0.5\sigma$ of our results, respectively. The
Kirkpatrick et al. (2020) measurements are all consistent with ours, to within
$1\sigma$.

##  7 Conclusion

In this paper we present parallax and proper motion measurements of two T
dwarfs using combined observations from the Shane and Keck telescopes spanning
five years. Results are summarized in Table 5. All observations utilized laser
guide star adaptive optics systems on these telescopes. We utilize a Monte
Carlo simulation of our sparse-reference star procedure to quantify our
parallax and proper motion errors, using our best estimates of both the random
and systematic portions of the astrometric errors. We constrain the parallax
and proper motion components of WISE1901 to be $\pi=$73.5\pm
9.2\text{\,}\mathrm{mas}$$, $\mu_{\alpha}=$117.9\pm
2.9\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$, $\mu_{\delta}=$403.4\pm
2.2\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$. For WISE2154, we
measure $\pi=$70.1\pm 6.7\text{\,}\mathrm{mas}$$, $\mu_{\alpha}=$-166.7\pm
2.9\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$,
$\mu_{\delta}=$-463.1\pm
3.5\text{\,}\mathrm{mas}\text{\,}{\mathrm{yr}}^{-1}$$. All our results are
consistent within $2\sigma$, with 4 of 6 values within $1\sigma$, of
previously published five-parameter astrometric solutions in the literature.

This study leverages five-parameter parallax and proper motion solutions for
our reference stars from Gaia DR2. Use of Gaia data allow us to place our
measurements on the ICRS astrometric reference frame as well as mitigate
errors due to unknown motions of background reference stars. For these
reasons, we are able to constrain the parallax and proper motions of these two
brown dwarfs using only three Gaia reference stars per target.

These two targets are the first of a larger sample of T dwarfs being
astrometrically monitored. With the data analysis pipeline established, future
publications will include the remaining targets once a minimum of six epochs
per target are reached. Distances to the objects and luminosities may be
derived for these data, and these measurements may be used as part of larger
samples to calibrate the lowest-mass end of stellar formation models.

##  8 Acknowledgements

We acknowledge all members of the ShaneAO commissioning team for assistance in
obtaining M92 images as well as Sasha Safonova and Don Pham for assistance
with data reduction. This work performed under the auspices of the U.S.
Department of Energy by Lawrence Livermore National Laboratory under Contract
DE-AC52-07NA27344 with release number LLNL-JRNL-821003. Some of the data
presented herein were obtained at the W. M. Keck Observatory, which is
operated as a scientific partnership among the California Institute of
Technology, the University of California and the National Aeronautics and
Space Administration. The Observatory was made possible by the generous
financial support of the W. M. Keck Foundation. The authors wish to recognize
and acknowledge the very significant cultural role and reverence that the
summit of Maunakea has always had within the indigenous Hawaiian community. We
are most fortunate to have the opportunity to conduct observations from this
mountain. 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. This research made use of
Astropy,111http://www.astropy.org a community-developed core Python package
for Astronomy (Astropy Collaboration et al., 2013, 2018). The Starlink
software (Currie et al., 2014) is currently supported by the East Asian
Observatory.

## References

  * Ammons et al. (2014) Ammons, S. M., Neichel, B., Lu, J., et al. 2014, in Adaptive Optics Systems IV, ed. E. Marchetti, L. M. Close, & J.-P. Véran, Vol. 9148, International Society for Optics and Photonics (SPIE), 553 – 561, doi: 10.1117/12.2057233
  * Ammons et al. (2016) Ammons, S. M., Garcia, E. V., Salama, M., et al. 2016, in Adaptive Optics Systems V, ed. E. Marchetti, L. M. Close, & J.-P. Véran, Vol. 9909, International Society for Optics and Photonics (SPIE), 1698 – 1706, doi: 10.1117/12.2233775
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Burgasser et al. (2004) Burgasser, A. J., McElwain, M. W., Kirkpatrick, J. D., et al. 2004, The Astronomical Journal, 127, 2856, doi: 10.1086/383549
  * Currie et al. (2014) Currie, M. J., Berry, D. S., Jenness, T., et al. 2014, in Astronomical Society of the Pacific Conference Series, Vol. 485, Astronomical Data Analysis Software and Systems XXIII, ed. N. Manset & P. Forshay, 391
  * Diolaiti et al. (2000) Diolaiti, E., Bendinelli, O., Bonaccini, D., et al. 2000, in Adaptive Optical Systems Technology, ed. P. L. Wizinowich, Vol. 4007, International Society for Optics and Photonics (SPIE), 879 – 888, doi: 10.1117/12.390377
  * Dupuy & Liu (2012) Dupuy, T. J., & Liu, M. C. 2012, The Astrophysical Journal Supplement Series, 201, 19, doi: 10.1088/0067-0049/201/2/19
  * Eisenstein et al. (2011) Eisenstein, D. J., Weinberg, D. H., Agol, E., et al. 2011, The Astronomical Journal, 142, 72, doi: 10.1088/0004-6256/142/3/72
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1, doi: 10.1051/0004-6361/201833051
  * Garcia et al. (2017) Garcia, E. V., Ammons, S. M., Salama, M., et al. 2017, The Astrophysical Journal, 846, 97, doi: 10.3847/1538-4357/aa844f
  * Gavel et al. (2014) Gavel, D., Kupke, R., Dillon, D., et al. 2014, in Adaptive Optics Systems IV, ed. E. Marchetti, L. M. Close, & J.-P. Véran, Vol. 9148, International Society for Optics and Photonics (SPIE), 38 – 48, doi: 10.1117/12.2055256
  * Gubler & Tytler (1998) Gubler, J., & Tytler, D. 1998, Publications of the Astronomical Society of the Pacific, 110, 738, doi: 10.1086/316172
  * Hayashi & Nakano (1963) Hayashi, C., & Nakano, T. 1963, Progress of Theoretical Physics, 30, 460, doi: 10.1143/PTP.30.460
  * Kirkpatrick et al. (2011) Kirkpatrick, J. D., Cushing, M. C., Gelino, C. R., et al. 2011, The Astrophysical Journal Supplement Series, 197, 19, doi: 10.1088/0067-0049/197/2/19
  * Kirkpatrick et al. (2012) Kirkpatrick, J. D., Gelino, C. R., Cushing, M. C., et al. 2012, The Astrophysical Journal, 753, 156, doi: 10.1088/0004-637x/753/2/156
  * Kirkpatrick et al. (2020) Kirkpatrick, J. D., Gelino, C. R., Faherty, J. K., et al. 2020, The Field Substellar Mass Function Based on the Full-sky 20-pc Census of 525 L, T, and Y Dwarfs.  https://arxiv.org/abs/2011.11616
  * Kumar (1962) Kumar, S. S. 1962, AJ, 67, 579, doi: 10.1086/108658
  * Lawrence et al. (2007) Lawrence, A., Warren, S. J., Almaini, O., et al. 2007, Monthly Notices of the Royal Astronomical Society, 379, 1599, doi: 10.1111/j.1365-2966.2007.12040.x
  * Lindegren, L. et al. (2018) Lindegren, L., Hernández, J., Bombrun, A., et al. 2018, A&A, 616, A2, doi: 10.1051/0004-6361/201832727
  * Looper et al. (2007) Looper, D. L., Kirkpatrick, J. D., & Burgasser, A. J. 2007, The Astronomical Journal, 134, 1162, doi: 10.1086/520645
  * Lu et al. (2010) Lu, J. R., Ghez, A. M., Yelda, S., et al. 2010, in Adaptive Optics Systems II, ed. B. L. Ellerbroek, M. Hart, N. Hubin, & P. L. Wizinowich, Vol. 7736, International Society for Optics and Photonics (SPIE), 588 – 594, doi: 10.1117/12.859737
  * Luri et al. (2018) Luri, X., Brown, A. G. A., Sarro, L. M., et al. 2018, A&A, 616, A9, doi: 10.1051/0004-6361/201832964
  * Max et al. (1995) Max, C. E., Gavel, D. T., & Olivier, S. S. 1995, in Adaptive Optical Systems and Applications, ed. R. K. Tyson & R. Q. Fugate, Vol. 2534, International Society for Optics and Photonics (SPIE), 412 – 422, doi: 10.1117/12.217760
  * McCully et al. (2018) McCully, C., Crawford, S., Kovacs, G., et al. 2018, astropy/astroscrappy: v1.0.5 Zenodo Release, v1.0.5, Zenodo, doi: 10.5281/zenodo.1482019
  * McLean et al. (2003) McLean, I. S., McGovern, M. R., Burgasser, A. J., et al. 2003, The Astrophysical Journal, 596, 561, doi: 10.1086/377636
  * Monet (1998) Monet, D. G. 1998, Amer. Astron. Soc. Meeting, 193, 120
  * Neichel et al. (2014) Neichel, B., Lu, J. R., Rigaut, F., et al. 2014, Monthly Notices of the Royal Astronomical Society, 445, 500, doi: 10.1093/mnras/stu1766
  * Service et al. (2016) Service, M., Lu, J. R., Campbell, R., et al. 2016, Publications of the Astronomical Society of the Pacific, 128, 095004, doi: 10.1088/1538-3873/128/967/095004
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, The Astronomical Journal, 131, 1163, doi: 10.1086/498708
  * Theissen (2018) Theissen, C. A. 2018, The Astrophysical Journal, 862, 173, doi: 10.3847/1538-4357/aaccfa
  * Urban & Seidelmann (2013) Urban, S. E., & Seidelmann, P. K. 2013, Explanatory Supplement to the Astronomical Almanac, 3rd Edition (Mill Valley, California: University Science Books), 261–263
  * van Dokkum (2001) van Dokkum, P. G. 2001, Publications of the Astronomical Society of the Pacific, 113, 1420, doi: 10.1086/323894
  * Van Rossum & Drake Jr (1995) Van Rossum, G., & Drake Jr, F. L. 1995, Python reference manual (Centrum voor Wiskunde en Informatica Amsterdam)
  * Wizinowich et al. (2000) Wizinowich, P., Acton, D. S., Shelton, C., et al. 2000, Publications of the Astronomical Society of the Pacific, 112, 315, doi: 10.1086/316543
  * Wizinowich (2006) Wizinowich, P. L. 2006, in ICO20: Optical Design and Fabrication, ed. J. Breckinridge & Y. Wang, Vol. 6034, International Society for Optics and Photonics (SPIE), 1 – 9, doi: 10.1117/12.668084
  * Yelda et al. (2010) Yelda, S., Lu, J. R., Ghez, A. M., et al. 2010, The Astrophysical Journal, 725, 331, doi: 10.1088/0004-637x/725/1/331