section. Unfortunately, such handling is subject to numerical instabilities in
the fitting process leading to often erroneous coefficients. For the case of a
mosaic it is still true that there is a native raw coordinate system for each
device that is identical to the $(x,y)$ system in the previous section. This
case benefits from a new coordinate system intermediate between $(x,y)$ and
$(\xi,\eta)$ that we will call the monolith coordinates denoted by the
subscript $m$. A well-constructed mosaic camera is one where all of the
imaging devices lie in the focal plane of the instrument. In such cases, the
raw coordinates of each device is related to the other devices by a
translation and a rotation. The following treatment explicitly ignores any
out-of-plane tilt a detector might have. The choice of the monolithic
coordinate system is arbitrary, so for convenience we choose one of the
devices to be co-joined with the monolithic system. It is also useful to
choose this one device to be the one that contains or is at least closest to
the optical axis. In this coordinate system the position of the optical axis
is defined to coincide with the origin of the monolithic system. The
transformation from the $i^{\rm{th}}$ raw device to monolith coordinates is
thus,

$\displaystyle x_{m}^{i}$ $\displaystyle=$
$\displaystyle((A^{i}_{1,1}(x-x_{c}^{i})+A^{i}_{1,2}(y-y_{c}^{i}))/d$ (A8)
$\displaystyle y_{m}^{i}$ $\displaystyle=$
$\displaystyle((A^{i}_{2,1}(x-x_{c}^{i})+A^{i}_{2,2}(y-y_{c}^{i}))/d$ (A9)

where $A$ is a rotation matrix related to a rotation angle $\theta$ by

$\displaystyle A=\left({\begin{array}[]{cc}cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}}\right)$ (A12)

Note that the elements of $A$ are often referred to as the “CD matrix” found
in FITS headers for WCS (world coordinate system) conversions (Greisen &
Calabretta, 2002; Calabretta & Greisen, 2002). The values for $(x_{0},y_{0})$
are often found in the CRPIXn keywords. The choice of the reference chip leads
naturally to $\theta=0$ for that image. With this construction, the
$x_{0}^{i}$ values are the raw coordinates of the optical axis for each image.
While it is useful and convenient to impose no rotation between the monolith
system and the anchor chip, there are cases were a global rotation is useful.
The rotation for the anchor chip will impart that global rotation if applied
at the right step. In our IDL software, astcvt.pro, this global rotation is
supported.

###  A.4 Inverting an astrometric solution

The equations above make it very easy to go from position $(x,y)$ to sky-plane
coordinates, either $(\alpha,\delta)$ or $(\xi,\eta)$. However, it is often
very handy to be able to go in the other direction. That is what astcvt.pro
does. Note that the convertion between $(\alpha,\delta)$ and $(\xi,\eta)$ in
either direction is very simple and is not discussed here. If the solution is
linear the conversion is deterministic and exact via an analytic expression.
If the solution is non-linear the conversion requires an iterative numerical
solution. The routine astcvt will do this inversion for any input solution.

###  A.5 FITS keywords

All of the keywords in the Buie system start with AST. The variables
expressions shown refer back to the quantities defined above. The units of the
variables are provided in brackets.

ASTINFO (boolean) – Flag to indicate this header contains the astrometric
information in this format.

ASTXCEN (float,$x_{0}$,[pixels]) – X pixel location of the optical axis.

ASTYCEN (float,$x_{0}$,[pixels]) – Y pixel location of the optical axis.

ASTPROT (float,[radians]) – Chip pre-rotation angle.

ASTRNORM (double,$d$,[pixels]) – Normalization factor.

ASTTRA (double,$\alpha_{0}$,[HMS string]) – Right ascension of the tangent
point. It is not required, but highly desireable that this point be coincident
with ($x_{0},y_{0}$).

ASTTDEC (double,$\delta_{0}$,[DMS string]) – Declination of the tangent point.
It is not required, but highly desireable that this point be coincident with
($x_{0},y_{0}$).

The following three-keyword set is given for every term in the basis set. The
end of the keyword is a numeric digit, starting at 1 and increasing for as
many terms as are required for the mapping. The order of the terms is
arbitrary.

ASTTNi (string) – Name of the term. The names indicate the term and the basis
set to be used. There are three basis sets supported by astterms.pro: simple
polynomial, Chebyshev degree 1 or degree 2 polynomials, and Legendre
polynomials (M=0). The recognized term names are CONST, X, Y, XX, YY, XY, XXX,
YYY, XYY, XXY, XXXX, YYYY, XYYY, XXYY, XXXY, XXXXX, YYYYY, XYYYY, XXYYY,
XXXYY, XXXXY, or all of the preceeding with a T prepended for degree 1
Chebychev polynomials, a U prepended for degree 2 Chebychev polynomials, or P
prepended for Legendre polynomials. The order of the term is indicated by the
repetition of X or Y. For example, the $x^{3}$ term is denoted with XXX.

ASTCXi (double, $a_{i}$, arcsec) – The transformation coefficients from
$(x,y)$ to $\xi$. The units are in arcsec raised to the degree of the term.

ASTCEi (double, $b_{i}$, arcsec) – The transformation coefficients from
$(x,y)$ to $\eta$. The units are in arcsec raised to the degree of the term.

###  A.6 Useful Buie IDL library tools

The following software tools are in direct support of handling operations
related to astrometry. The top level tools are the ones most often used to
build applications to extract or use astrometry. The low-level routines are
usually just used internally in the library. The mid-level routines are
somewhere in between the other two categories.

####  A.6.1 Top-level routines

astcvt – Convert from one astrometric coordinate system to another given an
astrometric description of the image. The coordinate systems supported are raw
pixel positions in the image (xy), detector mosaic monolithic coordinates
(XY), tangent plane coordinates in radians (sn), tangent plane coordinates in
arcsec (SN), and equatorial celestial coordinates (rd).

astinfo – Read (or write) astrometric information from (to) the header.

mkastinfo – Create a synthetic astrometric description. This is intended only
for creating linear descriptions.

####  A.6.2 Mid-level routines

astchi1 – Compute the goodness of fit for a solution based on some rotation
and offset. This is used internally to help solve for inter-chip rotations in
a multi-detector camera.

####  A.6.3 Low-level routines

asteval – Evaluate an astrometric polynomial function given the independent
variables $(x,y)$, a list of terms, and the polynomial coeffiecients.

astrd2sn – Convert from $(\alpha,\delta)$ to $(\xi,\eta)$. This routine is
called by astcvt but is useful on its own.

astrd2xy – Convert from $(\alpha,\delta)$ to $(x,y)$. This routine works only
for linear transformations. astcvt is almost always preferred.

astsn2rd – Convert from $(\xi,\eta)$ to $(\alpha,\delta)$. This routine is
called by astcvt but is useful on its own.

astsn2xy – Convert from $(\xi,\eta)$ to $(x,y)$. This routine is called by
astcvt and is rarely called directly.

astsolve.pro – This routine performs the least-squares fit for an astrometric
solution. The inputs are correlated lists of pixel coordinates $(x,y)$ and
tangent plane coordinates $(\xi,\eta)$ along with a list of fitting terms to
use. This tool performs a robust fit, automatically culling discrepant
positions based on the noise statistics in the fit. Considerable work is
required before-hand to set up the correlated lists but this tool does the
basic work of the fit. One example of an interactive tool for solving for
astrometry that feeds this solver is astrom.pro. This one example is a heavily
used tool for other projects but the scale of the data reduction process for
the New Horizons search was such that it was more efficient to write dedicated
and very non-general solving tools.

astterms – Evaluate the independent vectors to match an astrometric polynomial
function.

astxy2rd – Convert from $(x,y)$ to $(\alpha,\delta)$. astcvt is usually used
rather than this tool.

astxy2sn – Convert from $(x,y)$ to $(\xi,\eta)$. This routine is called by
astcvt and is rarely called on its own.

##  Appendix B Details of key Buie IDL library routines

The following routines were particularly useful in the data processing and
results presented in this paper. Many more were also used but these are either
the essential computational engines or contain interesting and perhaps non-
obvious algorithmic concepts.

###  B.1 basphote.pro

Aperture photometry summation and centroiding is provided by basphote.pro.
This routine and its use has been discussed in Buie & Bus (1992) and Buie
(1996). Alternate versions are available in C and were also ported to IRAF
many decades ago. The object aperture integral is the sum of all pixels
weighted by the exact area of overlap between the pixel and the circular
aperture. Sky signal is determined from a sample of pixels whose centers lie
between the inner and outer radii of the sky annulus. The robust mean of the
sample (robomean.pro) is computed and the sky mean and the sky noise is used
to remove sky signal from the object aperture summation and to propagate into
the final noise estimation. Detector gain, exposure time, and detector read-
noise can also be provided for more accurate and consistent flux and noise
computations. During the measurement, the input position is treated as a rough
location. The image is scanned for the maximum within the distance of the
object aperture radius. From this position, an initial centroid is calculated
without subtracting sky. This first centroid is then used to place the sky
annulus, determine sky, then compute the object aperture flux and final
centroid.

###  B.2 dewarp.pro

This tool was created in a collaborative project with a Harvey Mudd College
clinic team. The basic function of this tool is to map image data from one
grid to another. The input is an astronomical image and an image descriptor
that contains an astrometric solution for the image. Another input is a second
astrometric solution for an image to be constructed. The two solutions are
used to map flux in the input image to the correct location in the output
image. The images are resampled, guided by the astrometric solution, using an
interpolator that is equivalent to a sinc interpolation.

This tool had many uses in our project. It can be used to map one image onto
another eliminating any distortion, sampling, or rotation from the resampled
image and then replicating any distortion in the destination image. This type
of operation is particularly valuable when building a deep stack of the sky
that matches a single image in preparation for subtracting that template
without sampling the original image.

For the ground-based searches it was more advantageous to define a synthetic,
linear image that all search data mapped to for a particular field. In this
case, all images are resampled but the result is strictly linear and
registered. Given that the image transformation is always done with full
astrometric knowledge, the resampled images are just as valid for astrometric
measurements of discovered objects as the originals.

###  B.3 findsrc.pro

Automatic source detection is important element of getting astrometry and
photometry from a digital sky image. This tool is similar in function to the
DAOPHOT tool, SExtractor (Stetson, 1987; Bertin & Arnouts, 1996). However, the
internal algorithmic workings of this program are quite a bit different and
its output can have very different characteristics. In particular, the number
of false positives is usually much lower.

The first step in this tool is a scan of the image to find pixels that exceed
a given threshold SIGTHRESH relative to the noise in the background of the
image. For each pixel, four windows on the background are computed. These
windows extend away from the pixel in the cardinal directions of the image
($\pm x$ and $\pm y$). The length of each window is controlled by the WINDOW
keyword and defaults to 6 pixels. The windows are displaced by GAP pixels from
the target pixel with a default of 2 pixels. This value is normally set to a
value similar to the FWHM of the PSF in the image. From these windows, four
sky signal values are computed with an associated standard deviation. If the
pixel signal exceeds the threshold above the sky in 3 or 4 directions, that
pixel is flagged. The result of this step is a detection array with a binary
value for the pixel exceeding the threshold or not. Note that an additional
check is made to restrict the detected pixels to have a signal level less than
the saturation limit imposed by the MAXPHOTSIG keyword.

The next step is to collapse the detection array. This is needed since many
pixels from the same source can be flagged as detections. This step is
performed by an internal rountine, collapse. The detection image is eroded so
that only one pixel is left flagged per source and that pixel is at the local
maximum in the image. The local maximum are further constrained to be at least
GAP pixels apart. The erosion logic is designed to also collapse large regions
from saturated stars and minimize the number of pixels flagged as detected.
This step is reasonably effective but local variations along bleed trail from
a saturated source can have local maxima that will confuse this tool and lead
to a string of detections along the trail.

The eroded detection image is then a map of source locations. Aperture
photometry is computed with basphote.pro for each flagged pixel in this map.
The radius of the object aperture defaults to GAP but can be altered with the
OBJRAD keyword if desired. The sky annulus for a local sky computation is set
to objrad+1 and objrad+5. Those sources with sensible values for FWHM and
instrumental magnitude are kept. Also, the SNR of the source must also be
above the value of SIGTHRESH. During the photometry calculation, a centroid
location is computed as well.

At the end, the centroid position, FWHM, flux, magnitude, magnitude error,
SNR, sky and sky noise are returned to the caller or possible saved to a file
for later processing. This routine is reasonably fast with the most expensive
part being the computation of the aperture photometry and centroids.

###  B.4 frmdxdy.pro

One of the most important tasks during the determination of an astrometric
solution is the correlation of two lists of positions. In this case, we need
to take a list of star positions, usually in $(x,y)$ coordinates and determine
the correspondence into a list of star catalog positions, usually in
$(\alpha,\delta)$. The algorithm used here was inspired by a method developed
by Bruce Koehn at Lowell Observatory for the LONEOS project. Approximate
information about the image scale and orientation is used to convert the
catalog positions to estimated pixels positions. This tool is extremely
resilient against differences in photometric depth between the two lists.
Rough knowledge of the plate scale is needed. The position shift between the
two lists can be large with no penalty on speed of execution or quality of
result. The rotation of the field, however, must be known very well. The
requirement on this knowledge gets more stringent as the field-of-view
increases. This limitation can be ameliorated somewhat by restricting the
initial determination of shift to a subset of a large image. The presence of
uncharacterized non-linearities in the image also limits the ability to find a
good shift but images subsets help here too.

###  B.5 frmdxyr.pro

This routine is an attempt to fix the problem of needing to know an exact
rotation angle for frmdxdy. Rotation is scanned over a range and a binary
search is run to find the best rotation angle that gives a sensible list
correlations. This tool can be very time-intensive to run if the rotation
range is large and is really only a stop-gap measure when there is small
rotation error in the datasets. This tool is completely useless for the class
of problem where the rotation is not known at all. For most astronomical data,
one usually has a pretty good idea of the information for these routines.

###  B.6 jitterk.pro

PSF-fitting to an image to get the position and flux of a source requires a
good understanding or characterization of the PSF. For stellar data taken with
stellar guiding, knowing the PSF is sufficient and is also easy to determine
from the data (see psfstack). For solar system observations, the sources of
interest are moving. The situation is even more complex for Hubble Space
Telescope data where parallax in introduced.

This tool was specifically written to support HST data reductions though it
can be usefully used on ground-based data as well. The input is a time-history
of the motion of an object relative to the pixel coordinates during an
exposure. In the case of the HST search data for this project, we track the
KBO. As a consequence, the stars are trailed by the target motion and parallax
induced by the telescope’s motion around the Earth. In principle, one can also
add in the measured jitter of HST pointing as measured with its fine-guidance
sensors but this level of detail was never needed.

At first glance, most of the trailed HST images look like linear trails. In
practice, this visual appearance is mis-leading. The parallactic motion can
still be along a line but sometimes, the rate of motion is slower at the start
of the exposure than at the end or vice versa. If the images are fitted as
lines and the line center used, there can be an effective temporal shift in
the derived astrometry. This tool builds a time-history of the pointing that
is used to create a convolution kernel that can be applied to the PSF prior to
the fitting process. This ensures that the temporal registration between
smeared and un-smeared targets is preserved.

###  B.7 lowess.pro

This tool is a LOcally-WEighted Scatterplot Smoother. It provides a means to
draw a smooth curve through data to follow the trend in the data even in the
presence of noise without imposing a smoothing or fitting function.

###  B.8 ois.pro

This tool was originally written by Miller et al. (2008) as a port of the
Alard & Lupton (1998) optimal image subtraction (OIS) software into IDL. That
tool was very slow and not very practical to use. Our version was optimized
during a Harvey Mudd College clinic project and runs at essentially the same
speed as the original C-code. An additional enhancement was added by Miller et
al. (2008) to encode a $\delta$-function basis set for the PSF manipulations
rather than the nested gaussian approach from Alard & Lupton (1998). During
our extensive work on the ground-based data for this project, we found the
$\delta$-function basis set, coupled with very precise astrometric
registration eliminated all need for spatially variable convolution kernels in
the OIS process and that feature of the software was never used.

###  B.9 robomean.pro

Numerous operations in our data require the computation of a mean from a
sample of data. Most real data unfortunately are affected by samples that do
not fit the underlying noise distribution of the quantity being measured. This
tool is designed to provide a robust mean from a sample is is probably the
most heavily used routine from the Buie library. The goal is to iteratively
process the data until all of the surviving points are consistent with the
noise distribution in the surviving points. A threshold can be set on the
filtering and is normally set to 3$\sigma$. Convergence depends on getting
convergence in the value of the mean, standard deviation, skew, and kurtosis
consistent with the errors on these four quantities as points are (possibly)
removed. This is used for measuring sky brightness around a source for
position and flux measurements as well as just about any other place where a
mean is desired.

###  B.10 psfstack.pro

Many operations require a PSF for an image. This routine determines a
numerical PSF by stacking many sources from the images to get an average PSF.
An input list of sources is provided and they are stacked in descending order
of SNR down to a limit that defaults to 25. Sources that are not good PSFs
(such as a double-star blend) are excluded.

###  B.11 skyfit.pro

The degree of image crowding in the image made it difficult to measure the
signal level due to the light from the sky. A typical method is to collect a
large sample of pixels across the image and then compute a robust mean,
removing unusual outliers caused by such things as pixels illuminated by
stars. This method requires the sample to be dominated by actual sky
measurements. Levels of contamination up to 10-20% are easily removed but when
that fraction rises too high our usual tool failed and returned an estimate of
the sky that is systematically and significantly too bright.

This new routine includes robust estimation and simultaneous fitting of a low-
order 2-D polynomial to the sky background with one important enhancement. An
examination of a histogram of the random image pixels shows an asymmetric
distribution. The appearance of this histogram in Fig. B.11 reveals what looks
like a normal distribution for signal at or below the mode of the
distribution. Signals higher than the mode are contaminated by faint star
signals that are just barely brighter than the sky and thus add flux to the
apparent distribution of signal from the sky. If there were no stars, the
distribution will always look like a symmetric Gaussian but as the number of
star-illuminated pixels increases the distribution becomes more and more
distended on the upper half. Pixels below the mode are minimally affected by
background star light.

Sky signal example. The plotted histogram shows the residuals from a 2-d fit
to the sky in a single image. The thick orange curve is a lowess curve of the
histogram and the peak is used as an estimate of the mode. The vertical green
lines mark the mode and the $-1\sigma$ point on the low side of the
distribution. The gray shaded region shows the approximate Gaussian
representation of the sky signal and noise.

In detail, the program first selects 20,000 random pixels from the image (2k
by 4k array size) representing 0.2% of the pixels. A first-order 2-D
polynomial is then fitted to these pixels. Those pixels whose residuals are
more than 3$\sigma$ from the mean residual are rejected. If there are new
rejected pixels, the fitting process is repeated. When there are no additional
pixels removed with up to 10 passes, the 2-D fitting is deemed complete. The
mode of the surviving pixels is computed and then the 1/e half-width is
measured from a smoothed curve based on the data using lowess.pro applied to
data falling below the mode. The true sky signal is assumed to be normally
distributed and we use the mode and 1/e half-width to approximate the sky
signal and its uncertainty. In the example demonstrated in Fig. B.11, 23% of
the samples were excluded by the initial robust sky fit and the plotted
histogram is of the surviving samples. The mode from this histogram is then
added to the constant term of the 2-d polynomial fit and the width of the grey
shaded region is taken to represent the sky noise in the image. In the
histogram shown, the portion above the gaussian represents pixels that are
contaminated by faint starlight. The fraction thus contaminated are too
numerous to be excluded by typical outlier rejection algorithms. Without this
correction to the sky fit, we would systematically fail to detect the faintest
sources which happen to be the most likely to be of interest in our search.
This approach still contains a small systematic bias to a higher sky level but
this bias is significantly smaller than the sky noise level and was thus
neglected. Note that zero in this plot is effectively the naive mean of the
sky computed without sufficient regard to the residual star contamination. In
this case, the correction to the sky level is nearly 200 DN which is roughly a
3-$\sigma$ shift.

###  B.12 stacker.pro

Building image templates or stacks is done with this program. A set of images
can be added up and a robust mean is estimated from the stack at each pixel.
This tool also supports shifting the images to get a coherent stack on a
object moving with respect to the pixel coordinates.

##  Appendix C Ice Hunters – Citizen Science Participation

The rise of citizen science projects has been a consequence of attempts to
harness the interest and attention of a large number of initially untrained
people to tackle a large and either tedious or difficult to code task,
generally in the form of sorting, classification, or searching through data.
The search for KBOs in these crowded stellar fields was seen to be an
interesting case.

For this project, all the data were pre-processed as described in the main
section of this paper. The basic product of these calculations were difference
images where the stars are (mostly) subtracted and moving objects are seen as
white and black pairs. We provided annotated training images to acquaint the
participants with what to look for. After successful training, access to live
data was granted.

The images being scanned were much larger (2k by 4k) than could be displayed
given the technology at the time. Rather than rely on image navigation driven
by the user, we chose to present sub-images from the larger images. The size
of the sub-array was dictated by the least capable graphical system to be used
– in this case we were limited by the capabilities of the first version of the
Apple iPad and a sub-array size of 424 x 424. To eliminate missing objects on
the edge, we built-in a 10% overlap between adjacent sub-arrays. At the time
of the sub-array creation, a fixed stretch of the image was used – scaled to
$-3$ to $+8\sigma$ of the sky signal and the result was saved as a PNG-format
image.

Breaking up the images resulted in a very large number of files. These files
had to be loaded on a cloud server which was hosted on Amazon Web Services
(AWS). The tool for uploading data to the server was GUI-based and designed
for small batches of files. The transfer process was extremely time-consuming
and would often fail if too many files were attempted at once. This problem
proved to be a nearly fatal bottleneck for this project.

The system was designed to present images to a participant for examination.
Performing a complete search meant that each sub-image was viewed and
processed by up to 16 different people with a small fraction seen as many as
40 times. The list of images was stepped through as each load request came in.
Our design allowed for very fast processing of the data given enough
participants. Once an image was viewed enough times, it was removed from the
candidate list for processing. The goal during processing was to ensure that
there were always images in the queue waiting for processing. In practice it
was difficult to load images to AWS faster than they could be processed by the
citizen scientists.

This method of processing created a very large number of false positive
detections. The total number of flagged transients was 728,954 and we had to
apply some automated filtering steps to reduce the work load. Any transient
seen $\geq$3 times was considered worthy of a confirmation effort. This
threshold excluded 61% of the flagged transients. An additional 13% were
excluded on the basis of visual examination of the candidate. The surviving
candidates were visually graded into 3 different classes: a likely KBO (124,
0.02%), a likely main-belt asteroid (16,011, 2.2%), and a likely variable star
(79,383, 11%). An additional 93,734 (13%) candidates could not be excluded or
placed into any of these three classes and simply left marked as “unknown” and
not considered further.

The final performance of this system versus our normal processing methods was
the same for KBO detections. The citizen science approach was far more
effective at finding and recording all of the other category of time variable
elements contained in the data for essentially the same amount of effort. The
citizen science approach would also be very effective if the data were
populated with simulated objects for the purposes of characterizing the
observational biases.

The following is a list of all those that participated as an “Ice Hunter:” T.
Abbate, T. Abrahamson, M. Adam, J. Adamson, A. Agbedor, B. Alexander, U.
Allan, X. Alldredge, A. M. H. Allen, D. Alton, M. P. Alves, V. S. Alves, O.
Amerongen, W. L. Anderson, J. Andrews, S. Andrews, M. Anstett, K. Applebaum,
J. Arrington, E. T. F. Ashton, W. Astl, L. G. Atkins, K. Bädker, B. Badnaruk,
E. Baeten, R. C. Bailey, R. Balick, J. L. Ball, B. E. Barnett, J. S. Bartel,
C. Bartholomew, P. Baumgaertner, A. Becker, S. Becker, É. Bedard, I. Beer, J.
Begg, J. Begonja, P. Bel, S. Berry, J. Bevc, V. A. Bezugly, G. Bianchini, M.
Biel, B. Bigelow, M. Billiani, J. Bishop, M. C. Blanaru, S. Blystone, G.
Boden, T. Boesch, R. Bohnes, C. Boland, F. Bomeny, J. Borck, D. Borncamp, M.
Boschat, G. P. A. Boscher, M. A. Boutilier, S. Boyd, N. B. Bradford, S.
Bratcher, P. Brayshaw, A. Brinkmann, C. Brittain, D. Brogan, R. Brooks, D.
Browne, T. Brydon, D. Burluraux, C. Butler, P. Caado, N. G. Cairns, F.
Callebout, D. Cameron, P. Campbell, J. Campos, J. Candeias, A. Caracciolo, S.
Carlino, A. Carpenter, C. Carrascal, J. Carruthers, U. Carsenty, F. Castro, L.
D. Chadwick, E. Chaghafi, B. Chamberlain, K. Champion, C. Chandler, J. K. H.
Cheng, D. Chestnov, C. Chinery, J. Chisholm, A. CIvinskas, A. Clark, D.
Cleaver, M. Cline, S. Clingenpeel, B. Collins, E. Conseil, S. Corfield, R.
Costello, R. Costin, M. Cotton, C. Craddock, M. Crosio, J. A. Cross, B.
Crossland, C. Crowe, S. Cruze, R. Damerell-Moss, J. Danker, M. Darcy, G.
David, J. M. A. Dawey, V. P. De A Barreto, C. De Grandis, D. de Vries, S. de
Witt, K. Dekker, S. Delille, J. Demers, A. Dether, K. Devine, W. A. Dias, S.
Diecks, E. Dimitrova, K. M. Doll, P. Dowden, P. N. Duggan, A. Dumitriu, L. M.
Duncan, G. Dungworth, P. Dunlea, D. Dzierwa, A. Edwards, C. Elidoro, P.
Ellinger, L. J. Elsnau, M. Elustondo, Y. A. Emery, H. F. Evans, J. Evans, S.
Evdokimova, A. Eve, A. Fakahany, A. Farmer, P. K. Farmer, K. Farrelly, P.
Fazekas, A. Felsenstein, N. Fequet, S. Finney, T. Fisher, M. S. Fletcher, R.
Fletcher, K. Flippo, J. P. B. Fonseca, C. Foote, K. Fox, D. C. Fromm, S.
Fronczak, R. Fuchs, H. Fukushi, M. Furskog, M. Gaebel, R. Gagliano, D. Gash,
S. Ghegoiu, B. Gilbert, J. Gilbertson, R. Gill, A. Gillis, R. Glover, V.
Gonano, S. Gonzaga, S. J. Goodman, A. Gorton, L. Goznell, H. Grady, T. Green,
P. Greutmann, C. Griffing, P. Gronowski, R. Guerreiro, E. Gutiérrez Gómez, D.
Gütler, C. Guttau, P. Gwenn, L. Gyarmati, L. Hackl, T. Hager, E. Hall, J.
Haller, T. Hantel, J. Harkin, A. Harlander, C. Hartel, E. Hartmeijer, R. F.
Harvey, A. Hatwood, R. Heikes, H. Heilman, F. Helk, E. Helmersen, R.
Hendricks, S. Hennessey, B. Hensley, B. Herentrey, M. Herr, J. Herridge, J. F.
Higgins, L. Higgins, S. Hignell, T. Hodge, T. Hoffman, T. Hollerung, T.
Horvath, D. Hrdlicka, M. Huefner, A. E. Huerta-Martin, P. Hughes, M.-T. Hui,
C. Hunt, C. Hutchinson, N. Ihle, B. Imp, S. Ivanchenko, K. Izakovic, P.
Jackson, W. Jammer, J. D. Jamulla, N. Janssen, A. Jaureguiberry, D. Jaworski,
P. Jennings, R. Jewell, W. Johnson, L. F. Jones, S. L. Kaczmarek, A.
Kaczmarowski, M. Kaluzny, N. Kaminski, M. Kamp, K. Kanegae, S. Karge, A. Karl,
D. Karthikeyan, A. Kastengren, S. Kavalli, N. Kaylor, B. Kelly, C. Kendall, A.
Kennedy, B. Kerr, S. Kidner, M. Kidulich, R. King, B. Kirschenmann, J. H.
Klouwen, K. Klug, K. Koester, M. Kolling, R. Korber, P. Kostrzewa, R. Kracht,
H. Krawczyk, J. W. Krawczyk, R. Krikken, P. Król, H. Kumar, M. Kumrucu-
Lohmiller, P. Kyriakides, D. J. Lactin, A. Lamperti, J. Lander, M. Larson, F.
Laurentiu, C. Le Garff, P. Le Gendre, K. Le Tourneau, P. Ledin, D. V.
Leegwater, N. Lenke, P. A. Leon, A. V. Levin, D. Lindberg, J. Lipinski, C.
Lloyd, V. P. Loeffler III, R. Lopez-Fabrega, E. Luchinat, S. Luers, J. Lummus,
P. J. Lunn, A. Lysiak, C. Macmillan, A. Macumber, R. Madala, K. L. P. Madsen,
S. Magee, R. Maherjinqiu, C. Mangili, A. Mankevich, B. Manning, D. Marion, A.
Martin, D. Martinez, J. A. Martins, M. H. Massuda, B. Matter, C. Mays, I.
Mazouzi, K. Mazowiecka, M. T. Mazzucato, R. McClure, B. McDaniel, P. McGarry,
G. D. McKee, D. McMillan, R. Mellor, P. Mellors, J. Merc, S. Mercer, K.
Miller, A. Mimeev, G. Mitchell, J. L. Moe, G. Mohr, B. B. Mölders, R. M.
Moldoveanu, E. Moody, C. Mooney, C. Moreno-Martinez, C. Morford, A. Morgan, D.
Morgan, K. Morrill, V. Mottino, B. Moyant, W. Muehlisch, E. Müller, T.
Myllymaki, S. Napier, J. Naujok, C. Neely, M. Negus, D. Nelson, S. Nembrini,
J. Nethercott, B. Newgrosh, C. Newman, O. Nica, J. Nicholson, M. Nicholson, P.
Nicholson, M. Nissinen, J. Nordnes, A. G. Norris, O. North, P. Northrop, K.
Nuber, M.-C. Nuta, G. O’Callaghan, A. Oikkonen, R. Olejarski, R. Orsval, J.
Ostler, K. Ozga, N. N. Paklin, A. Pandey, M. Pani, P. Pantoni, E. R. Pearsall,
J. K. Pedersen, J. Peixoto, S. Pemberton, C. Perello, P. R. G. Perlmutter, E.
Perozzi, B. Perry, D. Pettitt, M. Phillips, K. Pidgley, N. Piekarska, R.
Pierce, F. Piroddu, T. Playle, M. Plewinska, L. Pöffel, S. Pogrebenko, T.
Pollard, C. W. Poole, A. Post, D. L. M. Preston, T. Pulokas, J. Purontakanen,
E. Pusch, S. Pyne, J. Quinn, E. Quiroga-RodrÌguez, H. Raab, C. A. Radford, V.
M. Rahimi, C. E. Ravasi, I. Regan, F. Reisch, M. Renz, B. Replogle, J.
Richmond, E. Rike, Q. Ringsak, F. Ringwald, A. E. M. Rojas, J. P. Ronsen, K.
Roovers, J. Ros, J. Rossman, S. L. Roubroeks, T. Rounds, G. Roynon, C.
Ruedinger, M. Ruh, N. Rutledge, G. F. Salinas, O. Sanislav, H. Sankala, L.
Saracino, M. Savels, J. A. Sayle, T. Schäfer, U. Scheuss, M. Schindler, S.
Schmalz, K. Schmida, T. Scholze, P. Schulz, L. Schupper, J. Scott, M. Scott,
J. Sejpka, G. Selig, B. M. Shaw, G. Shearwater, L. Sheldon, J. Shelton, A.
Sheppard, A. Shirvanian, M. Shockley, K. L. Siddall, M. J. Sijens, P. Silva,
D. J. Simister, M. Simmons, D. Skillman, R. Slivka, I. A. Smith, M. Smith, R.
Smith, J. Snyder, B. Sobczuk, L. L. Somsikova, M. Souders, M. Spanjaard, A.
Springer, T. Stallone, M. Stano, D. Stelter, R. W. Stephan, R. Stevens, P. D.
Stewart, A. D. Stoca, J. Stuart, J. Sue, P. S. Summerlin, S. R. Taylor, M.
Tepper, R. Tertnes, J. Thebarge, P. Thomaidis, J. Thornton, D. Tomic, J.
Tompkins, T. Tormanen, A. Torså, G. Triltzsch, D. Truog II, D. Tukendorf, T.
Turaev, I. Uhl, C. Unger, C. van Boven, R. Van Der Hilst, P. Van Grijfland, M.
Veillette, R. H. B. Velthuis, P. Verdelis, M. Verga, G. Verhoeven, A.
Vertinskis, P. Vigil, D. Völkel, T. Vorley, T. Vorobjov, V. Vorochilov, L. L.
W. Wah, E. Walravens, D. Ward, S. Ward, J. D. Warner, J. Wartenberg, G.
Washbourn, P. Waterfield, S. Wegert, A. Wegner-Kaminski, D. Weilant, D.
Westbrook, J. Wheat, N. Wheeldon, D. Whitfield, A. Wille, M. V. Winkle, G.
Wollenhaupt, D. Woodhams, A. Wrobel, C. Yandle, J. Zeddam, D. R. Zeigler, L.
Zinke, B. Zinn, and I. Zormpa.

Subaru, Magellan, HST, CFHT, Gemini

## References

  * Alard & Lupton (1998) Alard, C., & Lupton, R. H. 1998, ApJ, 503, 325, doi: 10.1086/305984
  * Bernstein et al. (2004) Bernstein, G. M., Trilling, D. E., Allen, R. L., et al. 2004, The Astronomical Journal, 128, 1364, doi: 10.1086/422919
  * Bertin & Arnouts (1996) Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393, doi: 10.1051/aas:1996164
  * Boulade et al. (2003) Boulade, O., Charlot, X., Abbon, P., et al. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4841, Proc. SPIE, ed. M. Iye & A. F. M. Moorwood, 72–81, doi: 10.1117/12.459890
  * Brown et al. (2018) Brown, A. G. A., Vallenari, A., Prusti, T., et al. 2018, Astronomy & Astrophysics, 616, A1, doi: 10.1051/0004-6361/201833051
  * Buie (1996) Buie, M. W. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 135
  * Buie & Bus (1992) Buie, M. W., & Bus, S. J. 1992, Icarus, 100, 288, doi: 10.1016/0019-1035(92)90101-C
  * Buie et al. (2020) Buie, M. W., Porter, S. B., Tamblyn, P., et al. 2020, AJ, 159, 130, doi: 10.3847/1538-3881/ab6ced
  * Calabretta & Greisen (2002) Calabretta, M. R., & Greisen, E. W. 2002, A&A, 395, 1077, doi: 10.1051/0004-6361:20021327
  * Dressel (2019) Dressel, L. 2019, Wide Field Camera 3 Instrument Handbook, Version 12.0 (Space Telescope Science Institute)
  * Dressler et al. (2006) Dressler, A., Hare, T., Bigelow, B. C., & Osip, D. J. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6269, Proc. SPIE, 62690F, doi: 10.1117/12.670573
  * Elliot et al. (2005) Elliot, J. L., Kern, S. D., Clancy, K. B., et al. 2005, AJ, 129, 1117, doi: 10.1086/427395
  * Farnocchia et al. (2016) Farnocchia, D., Chesley, S. R., Micheli, M., et al. 2016, Icarus, 266, 279, doi: 10.1016/j.icarus.2015.10.035
  * Fraser et al. (2014) Fraser, W. C., Brown, M. E., Morbidelli, A., Parker, A., & Batygin, K. 2014, ApJ, 782, 100, doi: 10.1088/0004-637X/782/2/100
  * Fraser et al. (2008) Fraser, W. C., Kavelaars, J. J., Holman, M. J., et al. 2008, Icarus, 195, 827, doi: 10.1016/j.icarus.2008.01.014
  * Fuentes et al. (2010) Fuentes, C. I., Holman, M. J., Trilling, D. E., & Protopapas, P. 2010, ApJ, 722, 1290, doi: 10.1088/0004-637X/722/2/1290
  * Gaia Collaboration et al. (2016a) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016a, A&A, 595, A1, doi: 10.1051/0004-6361/201629272
  * Gaia Collaboration et al. (2016b) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2016b, A&A, 595, A2, doi: 10.1051/0004-6361/201629512
  * Gennaro et al. (2018) Gennaro, M., Anderson, J., Baggett, S., et al. 2018, WFC3 Data Handbook Version 4.0 (Space Telescope Science Institute)
  * Gonzaga et al. (2012) Gonzaga, S., Hack, W., Fruchter, A., & Mack, J. 2012, The DrizzlePac Handbook (Space Telescope Science Institute)
  * Greisen & Calabretta (2002) Greisen, E. W., & Calabretta, M. R. 2002, A&A, 395, 1061, doi: 10.1051/0004-6361:20021326
  * Gwyn (2014) Gwyn, S. D. J. 2014, Journal of Instrumentation, 9, C04003, doi: 10.1088/1748-0221/9/04/C04003
  * Kavelaars et al. (2021) Kavelaars, J. J., Petit, J.-M., Gladman, B., et al. 2021, ApJ, 920, L28, doi: 10.3847/2041-8213/ac2c72
  * Kozhurina-Platais et al. (2016) Kozhurina-Platais, V., Mackenty, J., Golimovski, D., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9904, Proc. SPIE, 99046I, doi: 10.1117/12.2233793
  * Krist et al. (2011) Krist, J. E., Hook, R. N., & Stoehr, F. 2011, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8127, Proc. SPIE, 81270J, doi: 10.1117/12.892762
  * Lindegren et al. (2016) Lindegren, L., Lammers, U., Bastian, U., et al. 2016, A&A, 595, A4, doi: 10.1051/0004-6361/201628714
  * Lindegren et al. (2018) Lindegren, L., Hernández, J., Bombrun, A., et al. 2018, Astronomy & Astrophysics, 616, A2, doi: 10.1051/0004-6361/201832727
  * McLeod et al. (2015) McLeod, B., Geary, J., Conroy, M., et al. 2015, PASP, 127, 366, doi: 10.1086/680687
  * Miller et al. (2008) Miller, J. P., Pennypacker, C. R., & White, G. L. 2008, PASP, 120, 449, doi: 10.1086/588258
  * Miyazaki et al. (2002) Miyazaki, S., Komiyama, Y., Sekiguchi, M., et al. 2002, PASJ, 54, 833, doi: 10.1093/pasj/54.6.833
  * Miyazaki et al. (2018) Miyazaki, S., Komiyama, Y., Kawanomoto, S., et al. 2018, PASJ, 70, S1, doi: 10.1093/pasj/psx063
  * Monet et al. (2003) Monet, D. G., Levine, S. E., Canzian, B., et al. 2003, AJ, 125, 984, doi: 10.1086/345888
  * Parker & Kavelaars (2010) Parker, A. H., & Kavelaars, J. J. 2010, Icarus, 209, 766, doi: 10.1016/j.icarus.2010.04.018
  * Petit et al. (2011) Petit, J. M., Kavelaars, J. J., Gladman, B. J., et al. 2011, AJ, 142, 131, doi: 10.1088/0004-6256/142/4/131
  * Porter et al. (2022) Porter, S. B., Spencer, J. R., Verbiscer, A., et al. 2022, , 3, 23, doi: 10.3847/PSJ/ac3491
  * Smart & Green (1977) Smart, W. M., & Green, E. b. R. M. 1977, Textbook on Spherical Astronomy (Cambridge University Press)
  * Spencer et al. (2003) Spencer, J., Buie, M., Young, L., Guo, Y., & Stern, A. 2003, Earth Moon and Planets, 92, 483, doi: 10.1023/B:MOON.0000031963.58573.97
  * Stern (2008) Stern, S. A. 2008, Space Sci. Rev., 140, 3, doi: 10.1007/s11214-007-9295-y
  * Stetson (1987) Stetson, P. B. 1987, PASP, 99, 191, doi: 10.1086/131977
  * Verbiscer et al. (2019) Verbiscer, A. J., Porter, S., Benecchi, S. D., et al. 2019, AJ, 158, 123, doi: 10.3847/1538-3881/ab3211
  * Verbiscer et al. (2022) Verbiscer, A. J., Helfenstein, P., Porter, S. B., et al. 2022, , 3, 95, doi: 10.3847/PSJ/ac63a6
  * Zacharias et al. (2012) Zacharias, N., Finch, C. T., Girard, T. M., et al. 2012, VizieR Online Data Catalog, I/322A
  * Zacharias et al. (2015) Zacharias, N., Finch, C., Subasavage, J., et al. 2015, VizieR Online Data Catalog, I/329