# A New Blind Asteroid Detection Scheme

Nathan Golovich Lawrence Livermore National Laboratory, 7000 East Avenue,
Livermore, CA 94550, USA  Noah Lifset Lawrence Livermore National Laboratory,
7000 East Avenue, Livermore, CA 94550, USA  Robert Armstrong Lawrence
Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550, USA
Eric Green Lawrence Livermore National Laboratory, 7000 East Avenue,
Livermore, CA 94550, USA  Michael D. Schneider Lawrence Livermore National
Laboratory, 7000 East Avenue, Livermore, CA 94550, USA  Roger Pearce  Lawrence
Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550, USA
Center for Applied Scientific Computing, 7000 East Avenue, Livermore, CA
94550, USA

###### Abstract

As astronomical photometric surveys continue to tile the sky repeatedly, the
potential to push detection thresholds to fainter limits increases; however,
traditional digital-tracking methods cannot achieve this efficiently beyond
time scales where motion is approximately linear. In this paper we prototype
an optimal detection scheme that samples under a user defined prior on a
parameterization of the motion space, maps these sampled trajectories to the
data space, and computes an optimal signal-matched filter for computing the
signal to noise ratio of trial trajectories. We demonstrate the capability of
this method on a small test data set from the Dark Energy Camera. We recover
the majority of asteroids expected to appear and also discover hundreds of new
asteroids with only a few hours of observations. We conclude by exploring the
potential for extending this scheme to larger data sets that cover larger
areas of the sky over longer time baselines.

Computational methods, Small solar system bodies, Asteroids, Near-Earth
objects

††facilities: CTIO(Blanco/DECam)

##  1 Introduction

The vast majority of objects that exist within the solid angle of an
astronomical exposure are undetectable above the noise of the image. Typical
detection schemes employ a point spread function (PSF) or other convolution-
based detection method and define “detections” as fluctuations above some
threshold, typically measured in units of the signal-to-noise ratio (SNR). The
number of false positives grows as this threshold is lowered such that a
threshold below $\text{SNR}=5$ is rare and below $\text{SNR}=3$ is not
typically useful for large, modern cameras. The most straightforward way to
improve detection sensitivity is to increase the exposure time, but moving
objects exhibit trailing losses and this does not allow for the removal of
cosmic rays and other imprinted correlated noise that appear differently in
each exposure and add over time. A common method for increasing detection
sensitivity is taking multiple exposures and stacking the images and
performing object detection on the median stack of the images. This reduces
the noise proportional to the square root of the number of exposures (assuming
equal exposure times and similar, uncorrelated noise in each), and thus allows
for fainter detection thresholds. This method works well with modern cameras
which have small read noise and dark current. Faint stars and galaxies are
frequently observed for hours using this method.

Methods to stack the signal of moving objects are more challenging in
practice. Large scale asteroid surveys such as the Catalina Sky Survey (CSS;
Larson et al., 1998), the Panoramic Survey Telescope and Rapid Response System
(Chambers et al., 2016, Pan-STARRS;[), and the Zwicky Transient Facility (ZTF;
Masci et al., 2019) do not detect moving objects systematically with such
methods. Instead, each of these surveys implement a similar method where
objects are detected above an SNR threshold in single-epochs before they are
linked together into orbital tracks and monitored until a confident orbital
determination is made. The Pan-STARRS Moving Object Processing System (MOPS;
Denneau et al., 2013) is an example method, which is also forming the basis
for the method used by the upcoming Legacy Survey of Space and Time (LSST,
formerly the Large Synoptic Survey Telescope; Ivezić et al., 2019), which will
utilize a MOPS-like framework and report tracklets to the Minor Planet Center
(MCP:111https://minorplanetcenter.net/data Williams et al., 2009).

On the other hand, special surveys for moving objects have been carried out in
order to go deeper than these single-epoch methods allow. It is informative to
first consider the angular rate of motion of various classes of solar system
objects of interest. The furthest comets, asteroids and Kuiper Belt objects
(KBOs) move slowly enough to permit (without trailing losses) exposures of up
to $\sim$20 minutes; whereas inner solar-system asteroids in the main-belt
(MBAs) and near-Earth objects (NEOs) permit exposures of $\sim$2 minutes and
as short as $\sim$1 second, respectively (Shao et al., 2014; Heinze et al.,
2015). In order to overcome this motion, “digital tracking” methods have been
developed. Digital tracking refers to the stacking of consecutive short
exposures (short enough to avoid substantial trailing losses) along moving
trajectories in order to add the signal of a moving object. This gives the
benefits of tracking the motion of a moving object and taking a long exposure,
with the added benefit of simpler removal of stationary objects through
standard difference imaging. Furthermore, by not assuming the motion of the
objects of interest, the intrinsic scatter in angular motion between moving
objects may be probed generically by stacking along a fine grid in angular
motion space and detecting objects on “shifted” median stacks (due to this
methodology, this technique is also often known colloquially as “shift-and-
stack”).

Tyson et al. (1992) first applied this method to search for KBOs, which
resulted in a null detection but demonstrated the capability to search for
moving objects down to faint limits without dealing with trailing losses.
Later, Cochran et al. (1995) demonstrated the power of this method, by
discovering a large population of KBOs with $m_{V}\leq 28$ using the Hubble
Space Telescope (HST). Bernstein et al. (2004) also used HST to conduct a
search for trans-Neptunian objects (TNOs) complete at the 50%-level down to
$m_{606W}\leq 29.2$. More recently, Shao et al. (2014) and Zhai et al. (2014)
used this method for fast moving NEOs, and Heinze et al. (2015) used it to
search for MBAs. In the last year, authors have implemented faster searches
with graphical processing units (GPUs) and massively parallel analyses capable
of searching a finer grid and stacking a deeper stack of short exposures (Zhai
et al., 2019; Whidden et al., 2019). Additionally, digital tracking has been
used to estimate more precise astrometry, which is vital for fast moving
objects for accurate orbit determination (Zhai et al., 2018). Most recently,
Heinze et al. (2019) has used it to conduct a complete search for MBAs down to
$m_{V}=25$.

In each of these studies, a linear assumption was made in order to avoid
smearing the signal, which is an inherent cap on the usefulness of this
technique. In this paper, we will describe a method that relaxes this
requirement and enables a fully blind search for moving objects within survey
data. A fully implemented version of our method could bring wide-area, long
time-domain searches into play for digital tracking methods. We develop this
method with an eye toward LSST (Ivezić et al., 2019), which will probe
2$\times 10^{5}$ square degrees to a $\text{SNR}=5$ depth of $m(r)=27.5$ by
stacking $\sim 1000$ images per pointing in six filters (ugrizy). It will
likely detect hundreds of thousands of asteroids over the ten-year survey with
single-epoch detection methods. Our method is suitable for detecting asteroids
in wide field surveys well below the single-epoch detection limit while also
recovering valuable orbital and astrometric information simultaneously. In §2
we describe our test data from the Dark Energy Camera (DECam; Flaugher et al.,
2015) mounted on the 4m Blanco Telescope at Cerro-Tololo Inter-American
Observatory (CTIO) in Chile. In §3 we describe our detection method. We
analyze our test data and present our results in §4. Finally, in §5 we offer a
discussion of future extensions to this method and contextualize our simple
test here in this larger picture.

##  2 Data

###  2.1 Observations

We obtained $\sim 3$ hours of observations on the Blanco Telescope at CTIO
with DECam. These were spread over two consecutive nights (2018 December 22
and 23) during a 90 minute gap in the observability of the scheduled
observations (Program 2018A-0273; PALS, PI William Dawson). The moon was full
on 2018 December 21, so the nights were in bright time. In each night, we
observed a single pointing toward the ecliptic plane, separated by a pro-grade
translation of 1$\arcdeg$ day-1 along the ecliptic. Each night consisted of a
$\sim 90$ minutes stack of 40 second exposures in a single filter. We dithered
randomly by selecting pointings from $\mathcal{N}(\mu_{RA},\sigma_{RA}^{2})$
and $\mathcal{N}(\mu_{DEC.},\sigma_{DEC}^{2})$, where $\mu_{RA}$ and
$\mu_{DEC}$ are listed in Table 1 and $\sigma_{RA}=\sigma_{DEC}=2\arcmin$. The
pointings over the two nights are shown also in Figure 1.

Table 1: Summary of observations Date | Filter | Exposures | $\mu_{RA}$ | $\mu_{DEC}$ | Median Seeing
---|---|---|---|---|---
|  | Number | ∘ | ∘ | $\arcsec$
2018 Dec 22 | r | 85 | 135.00 | 17.04 | 0.97
2018 Dec 23 | r | 82 | 136.01 | 16.77 | 0.87
Figure 1: The data cube depth (number of frames) in R.A. and Dec. To generate
the data cube, we cut the entire region into “patches” with the LSST data
management pipeline, followed by creation of a master pixel grid. All analyses
occurred on this grid.

###  2.2 Data Reduction

We obtained our exposures after standard processing by the DECam Community
Pipeline (Valdes et al., 2014). We then used the LSST software
package222https://pipelines.lsst.io (Bosch et al., 2019) to produce difference
images. This can be split into three different pieces: making calibrated
single epoch catalogs and images, creating a static template image, and
subtracting the template from the single epoch images to produce difference
images.

The LSST stack single epoch processing includes: the masking of cosmic rays,
measuring the PSF, detecting objects, deblending and measuring individual
sources and calibrating the astrometry and photometry. A detailed explanation
of each of these steps is explained in the HSC data release papers (Aihara et
al., 2018, 2019). For calibration purposes we used the Pan-STARRS catalog
(Flewelling et al., 2016).

To create the template images we first interpolated the data onto a tangent
plane projection centered on the average pointing. The total observed area was
divided into a regular grid with each grid $4000\times 4000$ pixels or $\sim
17\arcmin$ on a side (see Figure 1). For the templates we selected the best
seeing images which were those with $\text{seeing}<1.2\arcsec$. For each grid
the static sky is constructed following the procedure outlined in (Aihara et
al., 2018, 2019). Briefly, the procedure builds a two-sigma clipped coadd to
construct a static image of the sky. We subtract this coadd from each
individual image and identify variable sources. Those detections that are
truly variable will only appear in a small subset of the single epoch visits.
The variable sources are then masked and a coadd image is created by taking
the mean of all the images.

We use the Alard-Lupton algorithm (Alard & Lupton, 1998) as implemented in the
LSST stack to create difference images. This procedure estimates a convolution
kernel which, when convolved with the template, matches the PSF of the
template with that of the science image by minimizing the mean squared
difference between the template and science image. The Alard-Lupton procedure
uses linear basis functions, with potentially spatially-varying linear
coefficients, to model the matching kernel which can flexibly account for
spatially-varying differences in PSFs between the two images. The algorithm
has the advantage that it does not require direct measurement of the images’
PSFs. Instead it only needs to model the differential matching kernel in order
to obtain an optimal subtraction.

After examining the difference images, we found that there remained a large
number of artifacts due to problems in the difference imaging algorithm. A
majority of these objects were bright stars, but even fainter stars caused
problems for our analysis. We tried to remove these issues by masking out
known objects. We selected objects from PAN-STARS with $m_{r}<20$ and masked
them using the procedure described in Coupon et al. (2018). For each star the
mask was composed of a magnitude-dependent circle for the star and a rectangle
for the bleed trail. We visually tuned the size of the circle and rectangle to
match our data. We split the data into two class above and below 14th
magnitude. The size of the radii in arcseconds are:

${\rm radius}=\begin{cases}400\,e^{-m_{r}/3.8},&\text{for}\ 14<m_{r}<20\\\
400\,e^{-m_{r}/4.1},&\text{for}\ m_{r}<14.\end{cases}$ (1)

The length and width of the rectangles are 1.5$\times$0.15 times the circles
diameter and for $14<m_{r}<20$ and 6$\times$0.3 times the diameter for
$m_{r}<14$. The longer rectangles were necessary to remove some long bleed
trails for bright stars.

##  3 Methods

A typical shift and stack detection strategy is carried out by sampling a fine
grid of angular rates and stacking multiple frames along this proper motion to
produce a shifted median stack image, which is then used to detect point
sources that are moving at the matched rate of the shifted stack. This is the
optimal search methodology when a linear assumption can be safely made
regarding the motion of the objects of interest and the grid is fine enough to
ensure all point sources are detected as such. This is indeed the best method
for analyzing our data in hand. However, we seek a more generic search method.
If we imagine instead that the data cover a wide area of sky and are observed
over a large time-baseline (e.g., from an LSST-like survey), no longer can the
flux from a given asteroid that intersects many images be efficiently co-added
in a manner like shift-and-stack since the linear assumption is no longer
valid. Since a given set of orbits are not parallel trajectories through the
data volume, multiple objects are not detectable simultaneously. This fact is
what limits shift-and-stack methods from being useful beyond the time where
non-linear motion occurs, but it also means that a search that moves beyond
this time scale must be of a fundamentally different design. In this section
we build the idea of sampling individual trajectories through a survey and
combining the requisite pixels to gather all of the expected flux for an
individual trajectory. This method, while not optimal for our data set, still
provides relatively complete results (see §4) in a straight forward manner.
Ultimately, we will extend the methodology and code infrastructure built here
for larger and wider surveys.

###  3.1 Prior Generation

Still thinking generically, we sample from a set of priors that parameterize
the motion through the data. For example, we could randomly sample Keplerian
orbital elements (allowing $e>1$) if we aimed to search for objects on
hyperbolic orbits, or we could sample randomly from elliptical orbits with
perihelia smaller than 1.3 AU to tailor our search to NEOs. The task is then
to compute the intersection of a given survey with a randomly sampled
trajectory and determine if the randomly selected set of pixels constitutes a
detection. Note that this is not an easy task. The number of trajectories
needed to fill a wide area and long time baseline survey is exceedingly large.
However, the individual computations needed for each sampled trajectory are
rather small.

For our data at hand, we can sample linear trajectories. Each is defined by
four parameters: proper motions and intercepts in on-sky angular coordinates,
respectively. The proper motions are selected from a prior distribution while
the intercepts are chosen randomly from a uniform distribution that spans
beyond the extent of our data volume such that rays that enter the cube on the
side are allowed. For a given trajectory, the position was selected as a
random valid pixel within the data cube. This was done to ensure that at least
one image was properly intersected by the trajectory. The proper motions were
selected separately from two different probability distribution functions–one
for motion parallel to the ecliptic and one for motion perpendicular to the
ecliptic. After these were selected, they were converted to equatorial and
then to pixel coordinates.

We simulated both MPC and NEOs from Granvik et al. (2018) to generate prior
distributions for the proper motions. The simulated NEO population provided a
more accurate distribution for NEOs since only a small, biased fraction of the
MPC database contains NEOs. We randomly simulated orbits up our data space
selecting those that were within the field of view (FOV) of DECam at the time
of our observations. We chose to calculate proper motion priors in ecliptic
coordinates because asteroid motion is predominantly along the ecliptic, and
this allowed us to create the sharpest difference between the orthogonal
components of the proper motion. The computed proper motion distributions are
in Figure 2.

Figure 2: Proper motion distribution used for trajectory priors. This
distribution was based on simulating all known objects in the MPC database
observable in our survey. It was smoothed using a boxcar filter of width
$5\arcsec/hr$ and $9\arcsec/hr$ in latitude and longitude respectively.

###  3.2 Random trajectories to pixels

In this subsection, we describe the manner in which random trajectories
interact with the data in memory. This is a simple approach to efficiently
accessing the relevant data in the case where it may all be loaded into random
access memory (RAM). We will first describe the approach and then expand on
possible extensions to it that could yield the requisite speed-up to make the
method more applicable to larger data sets.

We utilized the Flash cluster at Lawrence Livermore National Laboratory (LLNL)
employing 2x Xeon E5-2670 v3 (12 cores each) CPUs and 256 GB of SSD memory.
This is important to avoid having to access the data through the file system,
which adds substantial latency to the queries. This also begins to approximate
the requirements for larger survey searches, which will need to be distributed
over many nodes of a large high-performance computing (HPC) system. This will
add additional challenges such as node to node communication bandwidth and
communication cross-linking, which we will discuss in more detail later in
this subsection and in §5.

In general, any time-domain survey data may be thought of as filling a three-
dimensional space (two spatial and one time), where the units of the two-
dimensional spatial domain are pixels covering the two-sphere and the time
domain is discretized at irregular steps corresponding to the time of each
exposure in the survey. In the linear regime this space may be simplified to a
“cube” with square pixels and a short time axis. This assumption allows data
collision and lookup indexing to be fast and relatively simple, but is
inaccurate over wider surveys and longer periods of time – this is the same
challenge that makes traditional shift-and-stack fail beyond a short time
regime.

In our case, the survey data set is constructed into this conceptual cube by
composing the data space as a three-dimensional stack of layers, with each
layer itself being constructed from a series of “tiles” that come from the
LSST data processing stack utilized in §2.2. Tiles contain the “real” surface
of the data, i.e. the portions which exists somewhere on disk in a FITS file,
and each FITS file in the data set is represented as a virtual tile in memory.
The FITS files also contain information on the position and size in this
global linear coordinate system and that information is used for positioning
the corresponding tile within each layer.

The global linear coordinate space is partitioned at each level of the data
cube hierarchy. At the top-level, the cube’s two-dimensional projection is the
minimum bounding box containing all of the tiles in the reduced data. The
layers of the cube can be thought of as the intersection of the cube with
parallel planes spaced at the time of each exposure (or perhaps thick planes
with thickness corresponding to the exposure time). Each layer shares the same
two-dimensional coordinate space (with small differences corresponding to
dithering) and differ only in time. Layers also contain information about the
length of the exposure represented by the tiles on that layer, so that the
flatted two-dimensional footprint of a trajectory’s intersection with the
layer can be accurately calculated (this is important for using a proper
signal-matched filter later). Tiles have a local coordinate space equal to
that of the data array in the FITS file they represent.

Once a random trajectory is sampled, we propagate it through the data cube. At
each time where an image exists in the cube, we calculate the trajectory–data
cube intersection (in the linear assumption this is trivial), and we pull in
the relevant pixel data for the SNR calculation. When looking up pixel data,
each level in the cube hierarchy is queried to access the portion of the
coordinate stored before moving to the remainder of the query to subordinate
levels, if applicable, until the pixel values are fully resolved. All such
queries are done by giving the cube object a point in three-dimensional space.
First, we determine if the requested position is within the bounding box of
the layer and if so, we then determine which layer was requested and pass the
remaining two-dimensional query to that layer. We then determine whether or
not that point falls within a tile of the layer or within a chip gap or star
mask. If the pixel queried exists and contains data, the query is passed to
that tile. Once the query reaches the tile, the code retrieves the requested
data from its associated FITS file. This is done by renormalizing the
requested point to the local coordinate space of the tile and indexing into
the data array segment of its associated FITS file to retrieve the requested
pixel. A specially encoded NaN value is returned if the requested point cannot
be further resolved to a real pixel at any point in this process (e.g., this
is the case if the pixel lands in a chip gap or star mask).

In the interest of performance, we did not use any existing FITS library to
perform the retrieval of pixel data. Instead, we use Linux’s mmap (or memory
map) feature to “map” the primary data array of the FITS file to a 1D array in
memory. This provided a number of challenges to overcome but also proved
significantly more performant than using, e.g.,
CFITSIO333https://heasarc.gsfc.nasa.gov/fitsio/ for this task. The main hurdle
for our method was that FITS files store data records in (archaic) spans of
2880 bytes while the memory page size on a typical modern compute sysetm is
4096 bytes, so it is unlikely that the start of a FITS data record will be
page-aligned on a modern computer. Linux’s mmap only allows mappings that
begin from file offsets that are multiples of the system page size. To work
around this, we instead map the offset of the system memory page that contains
the start of the data record we are interested in reading and note the
difference between the two offsets when performing file accesses. FITS files
may also store their data values in different forms of endianness, so we
perform an endian swap when reading the pixel data if the endianness of the
FITS file does not match that of the system.

###  3.3 Number of trajectories

Here we estimate the number of required trajectories to satisfactorily cover
the space. Following Heinze et al. (2015), the grid space needed to fully
sample the proper motion parameters is given by:

$\Delta=\frac{\sqrt{2}b_{max}}{t_{int}},$ (2)

where $b_{max}$ is the maximum allowable blur ($\sim 1$ PSF) and $t_{int}$ is
the total integration time of the stack (about 100 minutes in our case). Our
prior has a radius of $\sim 50\arcsec$ s-1 in two-dimensional proper motion
(Figure 2), so a rough estimate of the required number of trajectories that
must be sampled is given by the two-dimensional proper motion domain divided
by the square of Eq. 2 multiplied by the number of PSFs in a DECam FOV. This
procedure gives $\sim 300\times 10^{9}$. Computing detections for this number
of rays took roughly six days on ten nodes of the Flash computing system. In
§5 we will discuss improvements to our code that can vastly increase the
sampling rate, even with larger data loads and more complex trajectories.

###  3.4 Signal-to-noise Ratio Calculations

Once the requisite pixels are pulled from the data, the SNR is calculated with
a weighted sum using a two-dimensional Gaussian profile convolved with a line
segment equal to the length of the expected streak, which we can compute from
the sampled proper motion for each trial trajectory (note that for most of the
trajectories this is very short, and the convolution reduces to the Gaussian
PSF). The flux and noise in each pixel were summed across all images that a
given source intersected. Within one image intersection, we measure the flux
$F$ by calculating a weighted sum over the pixels $(p)$ inside the streak:

$F=\sum_{i}{w_{i}*p_{i}}.$ (3)

The weights $(w)$ are determined from the profile, which is the convolution of
a line segment with a two-dimensional Gaussian. The weights are normalized
such that their sum equals the effective number of pixels in the streak:

$n_{eff}=\sum_{i}w_{i}.$ (4)

The effective number of pixels is calculated by integrating the flux profile:

$n_{eff}^{-1}=\int{dxdy\,P(x,y)^{2}}.$ (5)

Specifically, we consider a streak with length $L$ oriented along the x-axis,
width $\sigma_{y}$, flux per unit length $l_{0}$, and center (0,0), convolved
with a symmetric bivariate Gaussian PSF of width $\sigma_{\pi}$:

$P(x,y)=\frac{l_{0}}{2L}\frac{1}{\sqrt{2\pi(\sigma_{\pi}^{2}+\sigma_{y}^{2})}}\exp\left(\frac{-y^{2}/2}{\sigma_{\pi}^{2}+\sigma_{y}^{2}}\right)\,\times\\\
\left(\operatorname{erf}{\left(\frac{L-2x}{2\sqrt{2\sigma_{\pi}^{2}}}\right)}+\operatorname{erf}{\left(\frac{L+2x}{2\sqrt{2\sigma_{\pi}^{2}}}\right)}\right).$
(6)

Assuming that the object is unresolved, $\sigma_{\pi}>>\sigma_{y}$, this
simplifies accordingly:

$P(x,y)=\frac{l_{0}}{2L}\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp{\left({\frac{-y^{2}}{2\sigma^{2}}}\right)}\,\times\\\
\left(\operatorname{erf}{\left(\frac{L-2x}{2\sqrt{2\sigma^{2}}}\right)}+\operatorname{erf}{\left(\frac{L+2x}{2\sqrt{2\sigma^{2}}}\right)}\right),$
(7)

where we have defined $\sigma\equiv\sigma_{\pi}$.

We choose an area surrounding the streak to sum over that balances capturing
as much signal as possible with limiting computation time. We use twice the
full-width-at-half-maximum of the point spread function ($2\sigma$) as a
limiting radius away from the streak’s ridge to integrate to, which
encompasses over 99 percent of the signal.

Sources of noise include sky background Poisson noise, read noise, dark
current, and Poisson noise from the signal itself. The total noise $N$
includes each of these but is dominated by the sources of Poisson noise, which
add in quadrature. Thus we have:

$SNR=\frac{F}{\sqrt{F+N}}.$ (8)

The background noise is calculated as the standard deviation of pixel values
in the difference image. The expected value per pixel will then be the square
of this number, and thus we get the total background signal will be:

$N=\sigma_{\rm pixel}^{2}\,n_{\rm eff}.$ (9)

All together we have:

$SNR=\frac{F}{\sqrt{F+n_{\rm eff}\,\sigma_{\rm pixel}^{2}}},$ (10)

where

$F=\frac{\sum_{i}{(l_{i}\,p_{i})}\,\sum_{i}{l_{i}}}{\sum_{i}{l_{i}^{2}}},$
(11)

This is derived from equations 2, 3, and 4, where $l_{i}$ is the formula for
the profile and the sums are taken over pixels, which is an approximation for
the integral.

###  3.5 Cuts and Outlier Rejection

Before calculating the overall SNR, some initial filtering is required to deal
with potential image artifacts intersected at a small number of exposures. To
start, we store a pseudo “light curve” for each trajectory, which is the local
SNR on each image the trajectory intersects. Even a few outliers in the light
curve can dominate the overall computed SNR for faint asteroids, which are
often well below single-epoch detection thresholds, so we apply some outlier
rejection methods. We start with a rolling-median sigma clip, and then also
apply an overall sigma clip (as opposed to a rolling-median) to trajectories
that only intersect 20 or fewer images. We then calculate the total SNR for
the remaining unclipped images. We require at least five images intersected
and $SNR>5$. After this point, a trajectory is considered a candidate and we
store its requisite information: proper motions and intercepts in pixel
$(x,y)$ coordinates, overall SNR, and frames intersected. Many of these
candidates are false positives or duplicates that are later filtered out.

Next, we apply a filter that removes duplicate detections of the same source.
This is a frequent occurrence, especially for bright asteroids because a
trajectory that only slightly intersects the deposited flux can quickly
surpass the SNR threshold along only a portion of the true trajectory. This
filter removes the vast majority of the candidate list. We use a k-d tree to
find nearest neighbors for all the trajectories in the four dimensional space
of $(x,y)$ position and proper motion (the proper motions are converted to
spatial coordinates by multiplying by the total integration time of the data
for each night). Using this four-distance metric, we consider all trajectories
within $\epsilon=15\arcsec$ of other duplicates. To be explicit, we require:

$\sqrt{\left(|\vec{x}_{0,i}-\vec{x}_{0,j}|\right)^{2}+\left(\Delta
t\,|\vec{\pi}_{i}-\vec{\pi}_{j}|\right)^{2}}<\epsilon,$ (12)

where $\vec{x}_{0,ij}$ are the position of two trajectories at the first
exposure, $\Delta t$ is the time between the last and first exposure, and
$\vec{\pi}_{ij}$ are the proper motions.

We keep the trajectory with the highest SNR. We chose $\epsilon=15\arcsec$ as
a balance between leaving in too many duplicates and cutting out trajectories
that were not duplicates but simply close to each other; 15$\arcsec$ is small
enough such that no two known asteroids from the MPC database that are
expected to appear in our data are within this range of each other (see Figure
3). It is still likely that some duplicates pass this cut. We address these
later.

Figure 3: The number of asteroids removed from MPC catalog in our survey as a
function of the size of the duplicate cut. No two asteroids are within
20$\arcsec$ in this catalog; there are likely some faint asteroids removed
with a cut this large. We use a 15$\arcsec$ duplicate cut in the first step.

We refine each trajectory to more properly align with the source using a Monte
Carlo method. This is done by selecting a small postage stamp cutout around
each trajectory’s intersection with an image. We then sample $10^{6}$ trial
trajectories within these postage stamps centered on the original detection
trajectory (specifically, a bi-variate Gaussian of 3$\arcsec$.6 hr-1 and
2$\arcsec$.6 width along the proper motion and position axes, respectively).
We retain the highest SNR trajectory of the sample as the new, refined
trajectory. We then apply a second duplicate filter setting
$\epsilon=8\arcsec$. Any duplicates that passed the first cut will get closer
to each other, and their source, though the refinement process. Thus, the
second duplicate process can use a more stringent duplicate cut to
successfully filter out almost all remaining duplicates.

Next, we apply a series of filters on the median stack image. This median
image is calculated by taking the median across the postage stamp cutouts for
each candidate. In this median image, a trajectory that perfectly aligns with
a source would appear as a centered single PSF (or short streak for fast
moving sources). First we calculate the SNR of this median stack using a
weighted sum, and we require $SNR_{med}>7$. Next we apply two filters intended
to cut out false positives that are a result of artifacts that survived the
difference imaging procedure (typically near bright star masks or the edge of
the DECam FOV). First we calculate the noise of the median stack image as the
sigma clipped standard deviation of pixel values, and require that it is less
than 10 ADU. Next we take all the pixel values outside the source profile and
calculate the $\chi^{2}$:

$\chi^{2}=\frac{1}{N_{pix}}\sum{\frac{p_{i}^{2}}{N_{i}^{2}}},$ (13)

and require it to be less than 1.5.

Next, we apply a “near-hit” filter that attempts to cut out any trajectory
that only partially overlaps with a real source in the data. These are a
consequence of under sampling the trajectory priors, especially in the wings
of the proper motion prior. We pause here to note that our primary goal is not
to optimally detect asteroids in this trial field. We are more interested in
developing a framework that may be properly scaled to larger data sets, and we
are using this small test survey to being that development. We will return to
this discussion in §5. The near-hit filter is carried out by analyzing the
distribution of local SNR values in the light curve. For a good hit, most of
the local SNR values would distribute around a mean value due to Poisson noise
in the flux and the sky background as well as intrinsic variability due to
rotation of the asteroid. For a near-hit, the portion of the trajectory that
misses the true asteroid motion would exhibit near-zero SNR. To differentiate
between these two, we rank-order the local SNR values into an array and
compared the standard deviation of the highest values of the distribution
(fourth quartile) to the standard deviation of the lowest (first through third
quartile). We required that the fourth quartile variance be less than the
first through third quartile variance, thus requiring that most of the high
value local SNRs are clumped around a value indicative of a detection on all
of the frames that intersect the trajectory. We demonstrate asteroids that
both pass and fail this cut in Figure 4. In the case where all values are
clumped around zero, and this filter is passed, the overall SNR would likely
be below our threshold in the first place.

Figure 4: Example distributions of Local SNR values of frames hit for
detection trajectories that are bad (left panel) and good (right panel)
approximations of the true trajectory. The near hit filter compares the STD of
the first through third quartile (blue bins) with the STD of the fourth
quartile (orange bins). The near-hit mostly has local SNRs around 0 and some
reaching higher corresponding to when the trail trajectory overlaps the flux
from the asteroid, and so fails the test. The right panel shows a good hit,
which has mostly non-zero local SNRs and passes.

Finally, we also apply a similar filter where we take each trajectory and
randomly split its light curve in two. We then require that each half
individually must reach the SNR cut of $\frac{5}{\sqrt{2}}$ to check that flux
is rougly split between two random halves of the lightcurve.

We then visually inspect each result to remove any remaining false positives
or potential duplicates. In figure 5 we demonstrate the overall process as
well as the number of detections after each stage for each night.

Figure 5: The flow chart for our filtering process along with numbers of
detections for each night after every stage. The “Initial Filters” consist of
just the median image filters, while the “Final Filters” consist of median
image and light curve filters.

##  4 Results

###  4.1 Injection and completeness

During data reduction, we injected fake sources into the data to allow us to
measure completeness. Roughly 1700 sources were injected each night with
uniform distributions of proper motions up to 360$\arcsec$ $\text{hr}^{-1}$
and apparent r-band magnitudes between 18 and 28. We measured completeness by
calculating the percentage of injected sources that were detected with respect
to both magnitude and proper motion, where the latter is heavily affected by
proper motion priors and refinement.

We determined which detections corresponded to injected sources using the same
four-distance metric from Eq. 12. For each detection, we found the closest
injected source, and if it was within 20$\arcsec$, we considered it a match.
We performed a similar analysis on known asteroids in the MPC database. We
took all MPC asteroids expected to be within our observations and matched them
with a larger minimum four-distance of $70\arcsec$. This was larger to
accommodate errors in MPC orbital elements and potential orbit perturbations
since last observation. We chose these two four-distance values empirically
based on distributions of minimum four-distances, where there was a very clear
demarcation between matched and un-matched sources (see Figure 6).

Figure 6: The distribution of four-distances from each detection to the
nearest intended matching target (MPC objects on the left panel and injected
sources on the right). There is a clear demarcation for each population; the
red dotted line shows where we have chosen to separate matches from unmatched
objects. These distributions are for night two only and has been cropped to
show the small distances. Night one has similar distributions.

In figures 7 and 8, we show completeness along with our proper motion priors.
The 5 sigma, single-epoch detection limit was $m_{r}<21.9$ and $m_{r}<21.8$
for nights one and two, respectively. We determined the 5 sigma cuts using the
DECam exposure time calculator with a sky background estimated using the model
described in Krisciunas & Schaefer (1991), which is derived based on Mie
scattering of aerosols and Rayleigh scattering. The Mie scattering term is an
empirical fit at large angles though Schaefer (1991) suggests it is valid to
within roughly 10∘ of the moon, which is closer than our fields for each
night. The lack of completeness below the red dashed line is suggestive of an
incomplete search of the priors, which we mentioned was evident from the large
number of near-hits. We again refer readers to §5 on this point, as our main
goal of this analysis is to work toward an analysis framework that is
scaleable to larger data sets on HPC resources. Indeed, the detections above
the red dashed lines in Figures 7 and 8 suggest that the method is working.

In Figure 9, we show completeness specifically for known MPC asteroids. We
successfully detect nearly all bright known asteroids and the majority of MPC
objects brighter than $m_{V}=21$. These figures display only the detections
that are matched with known MPC asteroids, while a substantial fraction of our
detections are new.

Figure 7: Night one injected source recovery. The red dotted line represents
the 5 sigma limiting magnitude for single epoch detection (the limit drops
minimally over the x-axis shown). The proper motion priors shown below heavily
influence completeness with respect to proper motion. The detection of fast
moving objects is a result of imperfect initial detection with possible
adjustment with refinement. Figure 8: Same as Figure 7 but for night two.
Figure 9: _Left:_ night one recovery of known MPC asteroids. The magnitude
distribution of known MPC asteroids in the FOV is shown in solid blue. Those
that were successfully recovered are shown in orange. We matched our
detections with the known MPC asteroids using the four-distance metric on MPC
asteroid positions propagated to the relevant epoch. _Right:_ Same for night
two, as with the larger number of false positives, we find that night two was
also less sensitive to MPC known asteroids.

###  4.2 Detections

We detected a total of 487 asteroids across both nights, of which 301 are
recovered MPC objects. The detections for each night are shown in Figure 10
overlaid on the proper motion prior in log-space to emphasize the tails of the
distribution. Two objects detected only in night one where detected in the
tails of this prior. A thumbnail of the median stack along the detection
trajectory is also presented for each detection for nights one and two in
Figures 11 and 12, respectively.

In total, we detected 64 asteroids on both night one and night two. We
selected these 64 asteroids by comparing detections across nights and
enforcing the same flux was measured and the two detections occurred within
$50\arcsec$ of the extrapolated position from the first night detections to
the second night. All detections have been reported to the MPC. We present
them in Tables 2 and 3. We explored using the rotational reflex velocity (RRV)
method to determine these asteroids distances (Heinze & Metchev, 2015; Lin et
al., 2016); however, because our observations were not taken near opposition,
this method does not provide a one-to-one mapping to distance.

Figure 10: Proper motion of detected asteroids for both nights. The grey scale
background shows the proper motion priors used in sampling trajectories. The
occasional alignment of dots from each night is the result of detecting the
same object across both nights.

##  5 Discussion

###  5.1 HPC Considerations

We have alluded to future use cases and extensions for our constructed
pipeline throughout this paper, so in the spirit of full specificity, we will
discuss the reasons for adopting such a pipeline, the benefits of it, and also
the challenges that are unique to this framework.

Between several surveys (past, present, and future) the vast majority of the
sky will be observed repeatedly over several decades. Regardless of flux,
photons from a large fraction of all asteroids in the Solar System have been
deposited in these exposures on hundreds if not thousands of occasions – the
vast majority of which may be too faint to detect in single-epoch detection
schemes, thus making the embedded information useless for most asteroid
detection and linking methods. Accessing this information and combining it to
detect more asteroids in a given survey is our fundamental goal. In §3, we
laid out the basic idea behind the method. The key features of our method are
sampling random trajectories under a prior distribution followed by a finely
tuned mapping and gather operation on the data space. The gather operation
quickly becomes the bottleneck because a single orbit touches numerous files,
which must be accessed separately before the simple requisite computation
(e.g., PSF convolution, weighted sum, etc.) can be made. We note here that
this challenge is an impediment that traditional shift-and-stack methods do
not suffer from because the underlying assumption of linearity ensures that
parallel trajectories may always be tested simultaneously via detection on the
shifted median stack. Here, we have relaxed the linear assumption and address
the I/O challenge head on.

In this paper, to get around this challenge, we used a compute system that was
capable of loading all of the requisite data into RAM. We thus avoided latency
in accessing the files repeatedly over a file server. In the case where the
data load is much larger, this will no longer be possible. A simple mmap
implementation becomes much less efficient in the manner we carried out. We
are now developing a batched method, which accepts the latency of multiple
reads from the file server and instead spends RAM on simultaneous propagation
of a larger number of trajectories through time. We have found comparable
trajectory sampling rates per node but an ability to scale to arbitrarily
large numbers of compute nodes. A follow-on paper will analyze the new
implementation performance on a similar data set before we graduate to
sampling non-linear trajectories and larger data volumes over longer time
domains (Golovich et al. in preparation).

Unfortunately, many of the newest HPC clusters (and more scheduled in the
future) do not enable the communication bandwidth to make such a search
tractable on large surveys. We must address these issues in order to follow
our approach to larger data volumes. As the data volumes grow, the separation
of requisite pixels for a given SNR computation grows as well. That is, pixels
distributed around an HPC system’s RAM are needed to be combined. In HPC
parlance, our code has poor _locality_. Furthermore, since nearly any number
of distinct trajectories touch each pixel, there is a high data access to
computation ratio. This tends to add latency in a traditional HPC computation
for data fetches. Lumsdaine et al. (2006) describe these types of
computational issues for the case of graph algorithms on large data sets,
which share many challenges with our vision of a blind asteroid detection
pipeline.

Longer time domains also present a new challenge. That is, the search space
grows rapidly: a fractional change in orbital elements results in a larger
projected, on-sky spatial separation over time. In fact, for large surveys
(e.g., Pan-STARRS, ZTF, LSST), the number of possible unique trajectories
through the data space becomes exceedingly large.

Despite this search volume problem, the potential to detect extremely faint
asteroids remains, so we now explore some of the requirements and challenges
posed. Given the immense search space, the most likely use case for a code
that can extract detections from a whole survey is in densely searching niche
orbital regimes of particular interest such as Earth Trojan asteroids (e.g.,
Connors et al., 2011) or interstellar objects (e.g., Meech et al., 2017,
although we should note that the search space for these is even more immense
given the unbounded eccentricity). Under a narrow set of priors for the motion
of specific types of objects, an efficient sampling, mapping, and gathering of
relevant pixel data from an entire survey could lead to new discoveries.
Expanding the discovery space requires optimization of each step (sampling,
mapping physical trajectories to the pixel data, and gathering that data). We
have shown in this study the importance of optimized loading of data into RAM.
In larger surveys, loading all the data into RAM becomes impossible on smaller
compute systems, necessitating parallel distributed memory methods, which are
foreign to most astronomers. Cutting-edge compute systems have upwards of
several petabytes of aggregate RAM, enough to hold even very large surveys;
however, accessing this data across many hundreds of compute nodes brings
about additional challenges. For one, because an individual orbit may touch
hundreds of images across the survey yet those images are distributed through
the RAM, there is very little localization. In order to efficiently access the
small number of pixels needed to compute the SNR, e.g., an immense amount of
cross-talk between compute nodes querying other nodes for their data. These
issues will be explored in our future studies.

###  5.2 Summary

In this paper we motivated and developed a prototype for a new type of blind
search for asteroids in astronomical survey data. We provided a proof of
concept analysis on a small three hour observation with DECam detecting
blindly numerous asteroids fainter than the single epoch depth would permit.
We achieve this by parametrizing the motion of asteroids of interest and
randomly sampling from prior probability density functions for the motion
model and mapping those trajectories to the data space and computing the
optimal signal-matched filter for detection. We present the results of our
detection pipeline on our test data, finding 487 asteroids with 301 of these
matching to known asteroids in the MPC database and 186 being new discoveries.
These detections were made despite challenging observing conditions made
within $20\arcdeg$ of a nearly full moon. We thoroughly discuss the prospects
for such a code in larger surveys and explore the HPC requirements for present
and future asteroid surveys.

Figure 11: night one detections. Figure 12: night two detections.

Table 2: Night One Detections  MPC Designation | $m_{r}$ | R.A. | Dec. |  $\pi_{\alpha}\cos{\delta}$aa$\alpha$ and $\delta$ refer to right ascension and declination, respectively. | $\pi_{\delta}$ | Observation Time bbObservation time refers to the time at which the R.A. and Dec. values were observed. These were the first exposure the detected trajectory intersected.
---|---|---|---|---|---|---
| Magnitude | ∘ | ∘ | $\arcsec\,hr^{-1}$ | $\arcsec\,hr^{-1}$ | MJD
| 20.21 | 134.79317 | 17.34582 | 57.431 | 16.524 | 58475.300347
5512 | 14.48 | 134.02066 | 17.10905 | 49.997 | 4.529 | 58475.300347
| 20.75 | 134.29134 | 17.63576 | 56.773 | -8.453 | 58475.301169
326927 | 18.31 | 135.53799 | 17.57403 | 41.795 | -5.965 | 58475.301169
422173 | 18.92 | 134.0232 | 17.14441 | 65.643 | 39.024 | 58475.300347
377387 | 19.2 | 134.71131 | 16.80628 | 58.3 | 28.746 | 58475.300347
| 19.77 | 134.27197 | 16.42143 | 56.311 | -4.806 | 58475.301169
| 19.61 | 135.2303 | 17.9343 | 35.45 | 60.322 | 58475.300347
| 20.19 | 135.09889 | 16.94402 | 66.378 | -7.589 | 58475.300347
| 19.05 | 135.34779 | 16.84297 | 34.476 | -3.1 | 58475.300347
| 18.0 | 134.62291 | 16.47335 | 11.164 | 221.666 | 58475.300347
501015 | 21.06 | 135.26071 | 17.70396 | 66.039 | 16.29 | 58475.300347
100878 | 17.99 | 135.14364 | 16.62999 | 38.803 | 26.496 | 58475.300347
| 20.33 | 134.82584 | 16.46962 | 77.055 | 6.383 | 58475.300347
| 20.36 | 134.67973 | 17.80584 | 48.188 | 26.118 | 58475.300347
229604 | 18.85 | 135.74152 | 17.25404 | 55.572 | 1.447 | 58475.300347
| 19.51 | 135.45125 | 16.83472 | 31.218 | 5.512 | 58475.301169
369867 | 19.02 | 135.46566 | 16.95659 | 64.4 | 33.97 | 58475.300347
501117 | 20.73 | 134.42677 | 16.95386 | 68.706 | 15.689 | 58475.300347
| 20.76 | 135.37694 | 17.89672 | 48.122 | -6.48 | 58475.301169
242209 | 18.01 | 135.12496 | 16.23311 | 75.061 | 9.547 | 58475.301169
497263 | 19.31 | 134.78628 | 17.06208 | 30.148 | 20.783 | 58475.300347
42019 | 17.32 | 135.57697 | 17.02326 | 55.307 | 17.87 | 58475.300347
125679 | 17.26 | 134.48391 | 17.87131 | 61.07 | 48.136 | 58475.300347
563034 | 18.83 | 135.41812 | 17.29607 | 52.998 | 49.363 | 58475.300347
62327 | 17.86 | 135.19413 | 17.99334 | 55.54 | 27.205 | 58475.300347
783819 | 19.49 | 135.28325 | 16.90288 | 50.448 | -13.525 | 58475.300347
99083 | 18.1 | 135.94501 | 16.56385 | 55.344 | -16.92 | 58475.301169
559253 | 20.09 | 135.72067 | 16.58669 | 76.843 | 4.673 | 58475.301169
315998 | 19.73 | 135.56097 | 16.99877 | 43.285 | 21.197 | 58475.300347
636698 | 19.84 | 133.87031 | 17.0173 | 50.014 | 37.303 | 58475.301968
180367 | 18.64 | 135.75742 | 16.67427 | 55.399 | 30.834 | 58475.300347
199168 | 18.51 | 135.82092 | 17.68091 | 64.178 | -12.744 | 58475.300347
287521 | 18.28 | 135.61687 | 17.63634 | 40.531 | 45.061 | 58475.300347
320504 | 18.58 | 134.23352 | 16.64174 | 46.233 | -3.2 | 58475.300347
545217 | 19.87 | 134.74767 | 17.43255 | 70.063 | -10.462 | 58475.300347
166030 | 18.21 | 135.72696 | 17.19883 | 43.37 | 21.078 | 58475.300347
496976 | 19.68 | 134.77711 | 17.69075 | 33.327 | 23.594 | 58475.300347
110244 | 18.02 | 134.92272 | 17.62696 | 57.894 | 23.569 | 58475.300347
575004 | 20.22 | 135.1702 | 16.49363 | 68.63 | 26.687 | 58475.301968
214124 | 17.69 | 135.79507 | 16.93669 | 30.802 | 42.509 | 58475.300347
350748 | 17.89 | 135.40597 | 17.87755 | 61.462 | -40.19 | 58475.300347
| 19.65 | 135.70601 | 17.40023 | 48.279 | 13.637 | 58475.301968
307572 | 19.43 | 135.39642 | 17.39503 | 44.701 | 20.88 | 58475.300347
77485 | 17.42 | 135.61702 | 16.26229 | 44.789 | 23.292 | 58475.301169
321863 | 18.96 | 135.68024 | 17.80152 | 64.584 | 35.323 | 58475.300347
| 20.09 | 135.75291 | 16.95639 | 70.368 | -31.471 | 58475.300347
| 20.14 | 135.79085 | 16.79599 | 49.26 | 19.26 | 58475.300347
| 20.29 | 135.17341 | 17.07839 | 66.86 | 5.108 | 58475.301169
507022 | 19.66 | 134.43732 | 17.21644 | 65.624 | 7.78 | 58475.300347
48143 | 17.07 | 135.02526 | 17.83662 | 34.815 | 51.106 | 58475.300347
102405 | 18.16 | 134.77825 | 18.03703 | 54.663 | 31.115 | 58475.300347
162069 | 18.79 | 134.68747 | 16.88823 | 71.141 | 27.857 | 58475.300347
| 19.96 | 134.27535 | 16.95371 | 56.777 | 20.66 | 58475.300347
506183 | 20.74 | 134.61118 | 17.42175 | 65.341 | 19.609 | 58475.300347
254574 | 18.55 | 135.79797 | 17.14525 | 40.31 | 24.242 | 58475.300347
107655 | 20.34 | 134.59861 | 16.16852 | 56.539 | -1.181 | 58475.300347
173983 | 17.54 | 134.91364 | 17.69367 | 34.551 | 36.223 | 58475.300347
655789 | 19.69 | 135.4815 | 16.84174 | 59.223 | 42.707 | 58475.300347
367784 | 18.99 | 134.78362 | 17.65727 | 59.545 | 49.518 | 58475.300347
724316 | 20.47 | 134.76377 | 16.83589 | 45.08 | 7.517 | 58475.300347
190276 | 19.56 | 135.78347 | 17.03281 | 68.821 | 16.402 | 58475.300347
522895 | 19.31 | 135.8193 | 17.59754 | 40.039 | 23.076 | 58475.300347
745999 | 18.38 | 134.47392 | 17.58256 | 47.03 | 8.892 | 58475.301169
| 20.63 | 135.39451 | 16.85289 | 62.923 | 4.46 | 58475.300347
474046 | 19.7 | 134.59682 | 17.4886 | 56.689 | 15.055 | 58475.300347
| 21.06 | 134.59864 | 17.24698 | 36.805 | 15.124 | 58475.301169
429278 | 19.87 | 135.71753 | 17.18038 | 48.23 | 28.087 | 58475.300347
170608 | 18.78 | 135.31984 | 17.33802 | 53.01 | 14.767 | 58475.302766
61818 | 17.66 | 134.93991 | 17.67428 | 59.961 | 48.089 | 58475.300347
| 19.52 | 134.74307 | 16.2332 | 41.346 | 43.186 | 58475.300347
214666 | 17.1 | 135.56074 | 16.50028 | 34.317 | 68.267 | 58475.300347
125851 | 18.4 | 134.8533 | 16.95914 | 54.672 | 35.107 | 58475.300347
| 19.45 | 134.9119 | 17.41068 | 48.654 | 26.384 | 58475.301169
120792 | 18.61 | 135.43493 | 17.32942 | 127.374 | 29.077 | 58475.352257
474942 | 19.74 | 135.37887 | 16.29108 | 50.546 | 31.295 | 58475.302766
343950 | 19.12 | 135.28006 | 16.78169 | 49.698 | 39.056 | 58475.300347
373174 | 19.75 | 135.92121 | 17.04615 | 26.347 | 28.044 | 58475.300347
| 20.38 | 135.87684 | 16.88091 | 44.222 | 23.526 | 58475.300347
273784 | 19.77 | 134.04942 | 17.18428 | 59.407 | 13.064 | 58475.300347
190380 | 19.62 | 135.36075 | 16.11803 | 30.317 | 19.31 | 58475.300347
| 19.93 | 134.55301 | 16.34607 | 50.397 | 29.196 | 58475.300347
| 20.29 | 134.97073 | 17.45837 | 38.634 | 40.252 | 58475.300347
| 20.49 | 135.27444 | 17.38082 | 58.443 | -9.882 | 58475.300347
509252 | 20.61 | 135.85848 | 16.91407 | 56.211 | 17.845 | 58475.301169
350990 | 18.34 | 135.65633 | 16.96134 | 32.189 | 32.944 | 58475.300347
22308 | 17.25 | 136.1153 | 17.00435 | 32.55 | 42.667 | 58475.300347
456011 | 19.32 | 135.09549 | 17.39107 | 52.49 | -7.736 | 58475.300347
166530 | 19.93 | 135.79449 | 17.00166 | 24.684 | 21.931 | 58475.300347
| 19.83 | 135.66423 | 16.56798 | 52.686 | 13.334 | 58475.301169
| 19.86 | 135.43679 | 17.77679 | 56.434 | -12.992 | 58475.300347
266184 | 18.76 | 134.91723 | 17.22933 | 54.472 | 29.606 | 58475.300347
| 18.45 | 134.08582 | 16.68713 | 54.288 | 52.081 | 58475.300347
508023 | 19.66 | 135.26764 | 17.84768 | 53.882 | 7.841 | 58475.300347
| 19.64 | 134.78689 | 17.34378 | 49.751 | -0.234 | 58475.300347
| 19.21 | 134.82559 | 16.5333 | 59.508 | -5.465 | 58475.300347
178084 | 19.36 | 135.84415 | 16.56622 | 54.598 | 17.176 | 58475.300347
46883 | 16.7 | 135.15826 | 17.73524 | 61.521 | 34.949 | 58475.301169
| 20.83 | 134.10219 | 17.19007 | 55.722 | -4.338 | 58475.300347
| 19.94 | 135.7474 | 16.80471 | 31.926 | 15.469 | 58475.300347
250392 | 18.1 | 134.25628 | 16.49468 | 55.088 | -12.042 | 58475.300347
9470 | 15.69 | 135.69139 | 17.5841 | 50.502 | 10.037 | 58475.300347
| 19.98 | 135.08763 | 17.79337 | 54.468 | -2.128 | 58475.300347
5769 | 16.55 | 134.91065 | 17.49583 | 55.945 | 17.863 | 58475.300347
286186 | 18.88 | 135.29339 | 17.40232 | 53.641 | 16.063 | 58475.300347
| 21.01 | 134.8931 | 18.00769 | 1.749 | -3.132 | 58475.301968
| 19.27 | 134.0956 | 16.93541 | 76.241 | -13.748 | 58475.300347
17642 | 17.04 | 135.23332 | 17.75534 | 62.437 | 33.548 | 58475.300347
| 18.83 | 135.32308 | 17.94623 | 35.372 | 76.723 | 58475.300347
220307 | 18.17 | 136.09491 | 16.98661 | 36.44 | -4.09 | 58475.300347
| 17.21 | 135.54982 | 17.02742 | 46.318 | 19.634 | 58475.300347
| 20.82 | 135.59494 | 17.30234 | 61.552 | 9.878 | 58475.300347
300231 | 18.92 | 135.28969 | 16.74339 | 57.323 | 8.737 | 58475.300347
| 20.08 | 135.87517 | 16.59475 | 59.324 | -24.808 | 58475.301169
| 20.28 | 135.12598 | 17.20985 | 50.76 | 30.722 | 58475.300347
| 20.29 | 135.39359 | 16.3298 | 43.382 | 12.798 | 58475.300347
278673 | 19.63 | 134.79528 | 16.03467 | 50.989 | 17.352 | 58475.301169
188669 | 19.53 | 135.40357 | 18.02587 | 55.071 | 22.158 | 58475.300347
16776 | 15.68 | 134.57762 | 16.06845 | 33.971 | 12.496 | 58475.302766
| 20.1 | 134.0741 | 16.60081 | 40.609 | 29.66 | 58475.302766
| 20.82 | 135.34161 | 16.52361 | -1.294 | 3.586 | 58475.300347
261394 | 20.37 | 135.60673 | 16.52606 | 49.005 | 36.634 | 58475.300347
| 19.4 | 135.94739 | 17.5561 | 41.569 | 42.232 | 58475.300347
| 20.44 | 134.97392 | 17.00893 | 28.638 | 24.57 | 58475.300347
77266 | 17.31 | 135.12389 | 16.70643 | 40.632 | 19.616 | 58475.300347
542227 | 20.58 | 135.18455 | 17.41052 | 57.822 | 30.258 | 58475.301968
| 20.77 | 135.23566 | 17.24622 | 70.585 | 7.51 | 58475.301169
165159 | 18.23 | 134.37372 | 17.14882 | 63.925 | 47.412 | 58475.300347
| 20.03 | 134.51699 | 17.70129 | 63.985 | -39.287 | 58475.301169
| 20.36 | 134.81407 | 17.63194 | 46.492 | 0.594 | 58475.300347
474027 | 18.87 | 134.40392 | 16.64714 | 52.944 | 42.167 | 58475.301968
418129 | 18.19 | 134.87483 | 17.88348 | 39.931 | -18.148 | 58475.300347
6538 | 17.25 | 134.7705 | 17.12793 | 66.588 | 15.804 | 58475.300347
| 20.6 | 134.48707 | 17.02795 | 60.734 | 31.702 | 58475.300347
| 19.36 | 135.37545 | 16.18635 | 45.609 | 35.316 | 58475.300347
260492 | 17.37 | 135.49424 | 16.2389 | 40.585 | -18.641 | 58475.300347
| 19.7 | 135.34271 | 16.45115 | 51.413 | -9.634 | 58475.300347
4443 | 15.51 | 134.47003 | 16.55552 | 60.06 | 39.683 | 58475.300347
488781 | 20.44 | 134.31472 | 16.90327 | 59.865 | 29.275 | 58475.300347
103530 | 18.55 | 135.73 | 17.06041 | 54.742 | 14.324 | 58475.300347
119376 | 17.49 | 135.3276 | 16.49491 | 41.236 | 1.555 | 58475.300347
361051 | 18.16 | 134.62338 | 17.21597 | 33.166 | 48.899 | 58475.300347
| 20.78 | 135.31916 | 16.59105 | 54.508 | 4.007 | 58475.301169
461884 | 19.89 | 135.03417 | 17.74525 | 60.311 | 24.088 | 58475.301169
| 20.66 | 135.09864 | 17.62182 | 37.968 | -9.104 | 58475.300347
| 20.45 | 134.91923 | 17.66643 | 72.066 | 20.606 | 58475.300347
539738 | 18.64 | 134.89698 | 17.12403 | 41.935 | 43.034 | 58475.300347
| 20.66 | 133.94558 | 17.13093 | 58.313 | 22.471 | 58475.300347
| 19.91 | 135.1765 | 17.64411 | 35.298 | 27.18 | 58475.300347
64015 | 17.44 | 134.87733 | 17.90454 | 54.156 | 12.55 | 58475.301169
8978 | 16.0 | 135.14704 | 17.89014 | 65.675 | 26.978 | 58475.300347
423840 | 19.85 | 134.46646 | 16.46711 | 71.194 | 29.585 | 58475.300347
| 20.22 | 134.60573 | 17.11111 | 44.773 | 34.639 | 58475.300347
147773 | 19.53 | 134.80763 | 17.06768 | 63.33 | 19.264 | 58475.300347
| 20.75 | 135.46048 | 16.25968 | 57.169 | 19.055 | 58475.301169
661526 | 19.93 | 135.77553 | 16.96214 | 57.715 | 1.17 | 58475.301169
358445 | 19.35 | 135.23542 | 16.63116 | 47.188 | 11.246 | 58475.300347
24761 | 15.67 | 135.30375 | 17.65341 | 56.99 | -18.551 | 58475.300347
126301 | 17.2 | 134.16922 | 17.22408 | 40.864 | 21.326 | 58475.300347
| 20.56 | 134.37262 | 17.00279 | 78.943 | -1.699 | 58475.300347
| 20.61 | 135.50253 | 17.51833 | 82.856 | -17.575 | 58475.300347
| 17.81 | 135.91577 | 16.68033 | 85.789 | -28.674 | 58475.300347
376906 | 18.83 | 134.75587 | 16.76423 | 83.931 | -52.499 | 58475.312384
| 19.88 | 135.55198 | 16.8105 | 43.921 | 9.85 | 58475.300347
183688 | 18.34 | 135.57588 | 16.29158 | 58.884 | 24.35 | 58475.300347
287010 | 18.24 | 134.90166 | 17.88768 | 53.569 | -3.884 | 58475.300347
| 20.19 | 135.13612 | 17.81496 | 52.154 | 26.939 | 58475.300347
| 20.17 | 135.3928 | 16.60968 | 43.422 | 16.607 | 58475.300347
602145 | 19.84 | 134.38636 | 16.55467 | 58.335 | 28.688 | 58475.300347
| 20.73 | 134.83023 | 18.0493 | 38.884 | -54.864 | 58475.300347
| 19.95 | 134.73546 | 16.37383 | 54.449 | 3.078 | 58475.300347
| 18.77 | 135.76502 | 17.00668 | 54.048 | 16.711 | 58475.300347
255691 | 19.85 | 134.25897 | 16.61406 | 71.802 | 19.865 | 58475.300347
294239 | 19.51 | 134.39383 | 16.54091 | 60.448 | 14.357 | 58475.300347
| 18.58 | 134.57582 | 16.88552 | 51.52 | -22.064 | 58475.300347
46837 | 17.9 | 135.46872 | 17.51203 | 67.994 | 21.485 | 58475.300347
| 20.15 | 134.31936 | 16.90309 | 68.194 | 26.618 | 58475.300347
| 20.4 | 134.5761 | 17.55046 | 59.958 | -0.076 | 58475.300347
581272 | 18.83 | 135.13438 | 16.20795 | 55.833 | -15.7 | 58475.301169
177858 | 19.63 | 134.60746 | 17.35789 | 60.952 | 10.915 | 58475.300347
150968 | 18.05 | 134.51746 | 17.735 | 50.031 | 18.05 | 58475.301169
18668 | 17.46 | 134.78653 | 16.97594 | 51.23 | 22.698 | 58475.301169
217833 | 19.8 | 135.83334 | 16.68884 | 61.384 | 27.943 | 58475.300347
663839 | 20.47 | 135.7013 | 17.22348 | 60.729 | -3.262 | 58475.300347
| 20.87 | 135.61127 | 17.78009 | 60.101 | -15.649 | 58475.300347
417541 | 18.99 | 134.55183 | 17.44795 | 59.349 | 29.804 | 58475.300347
| 19.38 | 135.57666 | 16.70859 | 43.738 | 30.967 | 58475.300347
| 20.74 | 134.94554 | 17.95619 | 44.185 | 28.296 | 58475.300347
214491 | 19.52 | 134.43224 | 16.85051 | 56.798 | 19.332 | 58475.300347
| 20.37 | 135.2321 | 16.33634 | 39.604 | 20.264 | 58475.300347
260657 | 19.25 | 135.03231 | 17.36967 | 55.331 | 15.451 | 58475.300347
| 20.5 | 134.76348 | 16.10563 | 77.295 | -12.722 | 58475.300347
552209 | 19.43 | 134.60418 | 16.19545 | 57.644 | 5.886 | 58475.300347
27019 | 15.41 | 134.09866 | 17.39066 | 52.487 | 13.752 | 58475.300347
| 19.99 | 134.52439 | 17.04456 | 51.731 | 12.791 | 58475.300347
198537 | 18.37 | 134.27491 | 16.53696 | 62.309 | 7.164 | 58475.300347
| 19.72 | 135.91934 | 17.32968 | 71.835 | -0.475 | 58475.300347
| 20.35 | 135.35328 | 17.17408 | 51.372 | 24.916 | 58475.300347
| 17.75 | 135.362 | 16.11228 | 67.688 | -0.616 | 58475.300347
205682 | 17.99 | 136.1518 | 16.90252 | 46.979 | -14.569 | 58475.301169
| 20.32 | 135.19088 | 17.63654 | 53.788 | 4.147 | 58475.300347
| 20.7 | 134.11844 | 17.13383 | 73.308 | -7.337 | 58475.300347
677931 | 18.88 | 135.46847 | 17.37129 | 39.337 | 47.97 | 58475.300347
262895 | 17.92 | 134.8744 | 17.51558 | 44.239 | -3.668 | 58475.300347
521280 | 20.02 | 135.64237 | 17.05015 | 49.957 | 14.994 | 58475.300347
| 17.43 | 135.02616 | 17.09924 | 62.297 | 14.274 | 58475.300347
677518 | 19.1 | 134.51064 | 17.13048 | 47.823 | 25.56 | 58475.300347
534631 | 20.29 | 134.26493 | 17.19806 | 63.942 | 16.999 | 58475.300347
9889 | 16.25 | 135.67274 | 17.36914 | 61.378 | 23.71 | 58475.300347
579476 | 19.58 | 134.56231 | 16.97512 | 27.566 | 26.953 | 58475.301968
| 18.63 | 134.62099 | 16.93608 | 21.91 | 33.829 | 58475.300347
427745 | 19.54 | 134.76143 | 16.15773 | 70.677 | -4.054 | 58475.300347
| 21.02 | 134.85982 | 17.50874 | 59.171 | -2.113 | 58475.301968
| 19.78 | 134.83798 | 17.27431 | 33.599 | 25.769 | 58475.300347
513029 | 18.79 | 135.00649 | 16.34023 | 60.03 | -3.812 | 58475.300347
| 19.72 | 135.12797 | 17.79551 | 57.816 | 17.039 | 58475.300347
230642 | 18.52 | 135.44288 | 16.3727 | 68.2 | 5.422 | 58475.301169
| 20.57 | 135.25256 | 17.27381 | 49.12 | 17.485 | 58475.300347
| 19.8 | 135.93276 | 16.62426 | 72.478 | -14.119 | 58475.300347
391646 | 19.7 | 134.94115 | 16.85069 | 53.38 | 17.932 | 58475.300347
256708 | 17.67 | 134.74153 | 17.02885 | 18.433 | 25.373 | 58475.300347
525327 | 19.26 | 134.82291 | 17.70847 | 63.129 | -7.61 | 58475.300347
430234 | 18.66 | 134.99828 | 17.53782 | 76.034 | -13.327 | 58475.300347
673147 | 19.63 | 135.8769 | 16.82667 | 64.148 | 2.995 | 58475.300347
130163 | 19.15 | 134.96891 | 16.84801 | 59.807 | 25.06 | 58475.300347
| 19.74 | 134.84806 | 16.93507 | 55.956 | 31.91 | 58475.300347
538702 | 19.58 | 135.55464 | 16.4501 | 56.813 | 23.288 | 58475.300347
415273 | 19.81 | 135.8166 | 17.55665 | 51.653 | 10.472 | 58475.300347
148710 | 18.53 | 134.72185 | 16.12625 | 64.92 | 13.165 | 58475.300347
| 19.85 | 135.58038 | 16.53938 | 49.747 | 13.061 | 58475.300347
| 18.35 | 135.51165 | 17.31583 | 49.308 | -9.374 | 58475.300347
| 19.51 | 134.79216 | 17.22858 | 49.39 | 33.797 | 58475.300347
| 20.82 | 134.09934 | 17.1773 | 60.706 | -2.61 | 58475.300347
467952 | 20.12 | 134.57006 | 17.41323 | 60.037 | 30.337 | 58475.301169
| 20.14 | 134.51405 | 16.12569 | 49.237 | 5.602 | 58475.302766
| 20.37 | 135.76149 | 17.19018 | 53.882 | 9.738 | 58475.300347
498850 | 19.19 | 135.54189 | 16.95697 | 46.745 | 24.606 | 58475.301968
621143 | 19.84 | 134.96609 | 16.8278 | 65.547 | 27.659 | 58475.300347
| 17.8 | 134.82962 | 17.84133 | 56.016 | 14.378 | 58475.300347
| 19.66 | 135.2296 | 16.53891 | 53.674 | 20.596 | 58475.300347
| 20.35 | 134.10112 | 17.13622 | 56.182 | 13.122 | 58475.300347
| 20.54 | 135.03577 | 17.10945 | 62.049 | -5.566 | 58475.301169
| 18.41 | 135.43409 | 17.3294 | 46.147 | 17.276 | 58475.345914
600245 | 20.22 | 134.64215 | 16.98929 | 57.779 | 13.284 | 58475.300347
318146 | 19.28 | 134.52753 | 17.43151 | 63.579 | 3.269 | 58475.300347
95582 | 17.56 | 134.12376 | 16.6841 | 53.13 | 38.164 | 58475.300347
44014 | 16.97 | 135.91186 | 17.26817 | 41.452 | 23.414 | 58475.300347
| 20.77 | 135.05198 | 16.69015 | 36.056 | 23.54 | 58475.301968
77420 | 16.21 | 135.21516 | 16.4483 | 28.509 | -19.984 | 58475.302766
| 20.38 | 134.65146 | 17.70069 | 46.886 | 34.067 | 58475.300347
463401 | 19.72 | 134.70812 | 17.87152 | 43.682 | 36.565 | 58475.300347
| 20.42 | 135.24513 | 17.59055 | 57.772 | 5.267 | 58475.300347
472892 | 19.92 | 135.66265 | 16.61926 | 55.977 | 29.243 | 58475.300347
| 19.24 | 135.55943 | 17.17658 | 49.82 | 28.397 | 58475.300347
255808 | 17.76 | 135.54513 | 17.41992 | 39.632 | 33.862 | 58475.300347
| 20.69 | 134.93874 | 18.00672 | 43.501 | 24.703 | 58475.300347
216107 | 20.21 | 134.59559 | 17.7875 | 66.563 | 21.114 | 58475.300347
291327 | 19.23 | 133.99728 | 17.00698 | 64.214 | 10.35 | 58475.300347
431811 | 19.64 | 135.31303 | 16.75942 | 57.649 | 14.828 | 58475.301169
| 20.17 | 134.64992 | 17.1545 | 45.905 | 75.348 | 58475.300347
674128 | 18.12 | 134.62777 | 16.92916 | 61.382 | -43.229 | 58475.300347
89964 | 17.36 | 135.07826 | 17.18778 | 47.276 | 23.893 | 58475.300347
| 19.12 | 135.24719 | 16.53393 | 34.629 | 18.644 | 58475.300347
| 19.75 | 134.94336 | 17.3722 | 48.585 | 24.016 | 58475.300347
| 19.29 | 134.52739 | 17.42166 | 60.691 | 26.028 | 58475.300347
499477 | 20.02 | 135.57338 | 16.77963 | 63.713 | 39.11 | 58475.300347
| 20.5 | 134.64071 | 16.93852 | 34.507 | 12.726 | 58475.300347
170158 | 18.7 | 134.56489 | 17.10633 | 50.193 | 20.329 | 58475.302766
| 20.13 | 134.53876 | 16.29287 | 26.634 | 35.057 | 58475.300347
| 20.1 | 134.38619 | 16.59145 | 47.522 | 14.101 | 58475.301169
41901 | 16.72 | 134.32881 | 17.61989 | 43.198 | 18.792 | 58475.305961
381522 | 19.38 | 135.49721 | 17.27391 | 60.77 | 7.276 | 58475.300347
100533 | 17.95 | 135.11626 | 16.73943 | 65.071 | 21.946 | 58475.300347
| 19.58 | 134.66398 | 17.37553 | 27.568 | 23.173 | 58475.300347
| 20.27 | 135.53247 | 16.92004 | 53.636 | -2.405 | 58475.301169
352472 | 18.07 | 134.05337 | 17.12481 | 38.357 | 72.414 | 58475.300347
256636 | 18.28 | 135.36548 | 17.93669 | 60.883 | 4.316 | 58475.300347
408797 | 20.0 | 134.49007 | 17.34945 | 79.785 | 17.413 | 58475.300347
| 19.76 | 134.2307 | 16.55724 | 37.475 | 52.787 | 58475.301169
| 20.47 | 133.97791 | 17.15478 | 47.449 | -21.524 | 58475.300347
| 20.31 | 135.3578 | 16.65846 | 52.213 | 29.905 | 58475.300347
351861 | 20.18 | 134.71354 | 17.27571 | 62.581 | 17.24 | 58475.300347
| 18.85 | 134.27187 | 16.81163 | 44.321 | 54.972 | 58475.300347
508599 | 20.03 | 135.01639 | 17.48889 | 75.36 | 8.471 | 58475.300347
| 20.44 | 134.61301 | 17.57297 | 53.024 | 21.593 | 58475.301169
| 20.05 | 135.73037 | 17.10609 | 41.069 | 26.773 | 58475.300347
215060 | 18.99 | 134.82543 | 16.45099 | 52.328 | 21.262 | 58475.300347
| 20.11 | 135.3333 | 16.23553 | 55.839 | 12.704 | 58475.300347
| 19.94 | 134.86455 | 17.59219 | 46.684 | 35.15 | 58475.300347
| 20.31 | 134.91288 | 17.05437 | 28.373 | 24.613 | 58475.300347
| 19.81 | 134.56303 | 16.49217 | 43.829 | 14.022 | 58475.301169
381436 | 20.05 | 135.22639 | 17.61307 | 64.614 | 19.49 | 58475.300347
72997 | 16.69 | 134.86342 | 16.48966 | 51.379 | -7.078 | 58475.300347
107889 | 16.93 | 135.51453 | 17.00693 | 33.066 | 48.546 | 58475.300347
347455 | 19.3 | 135.24629 | 17.80423 | 51.194 | 25.297 | 58475.300347
| 20.59 | 134.27526 | 16.73583 | 32.724 | 0.14 | 58475.300347

Table 3: Night Two Detections  MPC Designation | $m_{r}$ | RA | Dec | $\pi_{RA}\cos{\delta}$ | $\pi_{\delta}$ | Observation Time
---|---|---|---|---|---|---
| magnitudes | ∘ | ∘ | $\arcsec\,hr^{-1}$ | $\arcsec\,hr^{-1}$ | MJD
170608 | 18.63 | 135.22188 | 17.36393 | 55.573 | 15.898 | 58476.300833
576938 | 18.74 | 136.43999 | 15.95802 | 44.19 | 20.596 | 58476.302454
| 20.06 | 135.03027 | 16.6883 | 36.68 | 28.919 | 58476.300833
| 19.86 | 135.8096 | 17.08086 | 32.103 | 19.102 | 58476.300833
277630 | 18.92 | 135.66741 | 16.22197 | 42.883 | -1.184 | 58476.300833
341429 | 18.58 | 135.79427 | 15.79003 | 47.4 | 23.281 | 58476.302454
215444 | 19.65 | 136.99119 | 17.07061 | 63.862 | 33.106 | 58476.301655
522895 | 19.52 | 135.74501 | 17.6383 | 44.641 | 24.624 | 58476.300833
| 19.59 | 135.69084 | 16.83357 | 35.794 | 17.402 | 58476.300833
299971 | 18.07 | 135.65043 | 16.15148 | 56.851 | -9.274 | 58476.300833
7059 | 15.24 | 136.86659 | 17.06729 | 48.982 | 8.442 | 58476.300833
358445 | 19.32 | 135.14534 | 16.64968 | 50.581 | 12.301 | 58476.300833
111700 | 17.36 | 135.93196 | 16.55727 | 42.168 | 29.754 | 58476.300833
350990 | 18.88 | 135.59726 | 17.01963 | 35.084 | 34.42 | 58476.300833
460720 | 19.32 | 135.7289 | 16.15551 | 73.714 | 16.448 | 58476.300833
177937 | 18.81 | 135.68836 | 15.88187 | 54.286 | 13.468 | 58476.300833
67077 | 17.41 | 136.32295 | 17.19308 | 69.773 | 7.776 | 58476.300833
220480 | 19.3 | 136.2989 | 16.11685 | 59.96 | 25.11 | 58476.302454
214124 | 17.65 | 135.74019 | 17.01302 | 36.728 | 43.888 | 58476.300833
330131 | 19.43 | 136.31937 | 15.79908 | 63.384 | 30.625 | 58476.301655
| 20.65 | 136.67627 | 16.49828 | 49.039 | 18.896 | 58476.300833
100878 | 17.78 | 135.07358 | 16.67615 | 44.532 | 27.266 | 58476.300833
10054 | 16.18 | 136.33663 | 16.61992 | 52.22 | 16.78 | 58476.302454
118468 | 17.42 | 136.51153 | 16.62008 | 46.452 | 26.827 | 58476.302454
| 20.27 | 136.34698 | 16.40979 | 69.765 | 5.317 | 58476.302454
198563 | 18.3 | 136.33217 | 16.34901 | 53.015 | 13.626 | 58476.300833
| 18.89 | 135.41844 | 17.29914 | 52.348 | -8.597 | 58476.324873
286186 | 19.03 | 135.19343 | 17.42729 | 56.234 | 15.854 | 58476.308854
| 20.41 | 135.31207 | 16.63987 | 46.599 | 19.39 | 58476.300833
120792 | 17.65 | 135.34902 | 17.37105 | 50.908 | 26.273 | 58476.300833
620113 | 20.35 | 136.33872 | 17.04476 | 77.986 | 18.418 | 58476.300833
| 19.23 | 135.28843 | 16.83695 | 38.147 | -1.303 | 58476.300833
77266 | 17.47 | 135.05051 | 16.73986 | 45.313 | 20.725 | 58476.300833
| 20.07 | 135.47953 | 17.31833 | 65.547 | 10.332 | 58476.300833
99083 | 17.01 | 135.86024 | 16.56101 | 50.211 | 0.515 | 58476.300833
| 19.98 | 135.75424 | 17.04098 | 29.064 | 23.256 | 58476.301655
308315 | 19.91 | 135.33333 | 16.65702 | 63.726 | 9.115 | 58476.300833
12489 | 17.32 | 136.49334 | 16.071 | 45.203 | 14.713 | 58476.300833
| 19.78 | 135.24967 | 16.43344 | 54.615 | -8.489 | 58476.301655
673147 | 19.65 | 135.75638 | 16.83001 | 68.017 | 2.912 | 58476.312106
261394 | 19.88 | 135.48795 | 16.56176 | 52.07 | 13.334 | 58476.301655
410328 | 18.68 | 136.00154 | 16.86384 | 40.932 | 21.29 | 58476.300833
563034 | 18.95 | 135.32085 | 17.38142 | 54.671 | 49.993 | 58476.300833
77422 | 17.33 | 136.19751 | 17.23524 | 48.395 | 9.425 | 58476.300833
504540 | 19.76 | 136.40104 | 17.25066 | 56.621 | 11.898 | 58476.300833
175458 | 16.44 | 136.52869 | 16.5626 | 46.28 | 26.737 | 58476.300833
| 20.21 | 135.76753 | 16.55105 | 63.337 | -23.436 | 58476.300833
272748 | 18.42 | 136.03915 | 16.42488 | 48.578 | 24.275 | 58476.300833
11775 | 17.05 | 136.25558 | 16.4975 | 56.937 | 16.153 | 58476.300833
166603 | 18.37 | 136.61049 | 16.28888 | 57.769 | 15.102 | 58476.302454
601077 | 19.53 | 136.4359 | 17.2971 | 54.517 | -5.072 | 58476.301655
80806 | 17.07 | 135.86608 | 17.7458 | 48.77 | 37.85 | 58476.300833
369867 | 19.16 | 135.34519 | 17.01431 | 66.979 | 34.506 | 58476.300833
263991 | 17.77 | 135.71841 | 15.84891 | 58.631 | 38.516 | 58476.300833
193614 | 17.7 | 136.23192 | 17.71814 | 39.663 | 49.021 | 58476.300833
753413 | 19.79 | 136.6366 | 16.59783 | 29.121 | 2.257 | 58476.300833
| 19.17 | 136.66966 | 16.83054 | 54.65 | 1.386 | 58476.300833
| 19.89 | 136.34304 | 16.71795 | 33.92 | 30.262 | 58476.300833
601979 | 18.24 | 136.14168 | 17.22774 | 55.841 | -9.842 | 58476.300833
233849 | 19.24 | 135.88851 | 16.95816 | 52.537 | 17.359 | 58476.300833
| 19.75 | 136.66689 | 17.45598 | 53.062 | 26.924 | 58476.301655
| 19.39 | 135.46489 | 17.22491 | 52.28 | 28.865 | 58476.300833
| 20.23 | 135.65556 | 17.15474 | 44.422 | 29.509 | 58476.300833
429278 | 19.42 | 135.62698 | 17.22858 | 51.532 | 29.027 | 58476.300833
308359 | 18.63 | 135.80359 | 17.02565 | 81.777 | -20.815 | 58476.300833
| 20.18 | 135.78246 | 16.0388 | 63.717 | 5.868 | 58476.300833
127602 | 17.34 | 136.2752 | 16.21634 | 70.532 | 2.218 | 58476.300833
538702 | 19.71 | 135.44846 | 16.48989 | 59.442 | 23.436 | 58476.300833
532854 | 19.76 | 136.79087 | 16.20461 | 71.127 | -6.962 | 58476.301655
255808 | 17.72 | 135.4741 | 17.47971 | 45.041 | 35.759 | 58476.300833
343388 | 18.5 | 135.70054 | 17.6587 | 68.257 | -11.092 | 58476.300833
656468 | 20.32 | 136.74881 | 17.0585 | 52.987 | 8.986 | 58476.300833
594360 | 20.5 | 136.7577 | 16.44631 | 51.739 | 24.66 | 58476.302454
277377 | 19.67 | 136.32328 | 17.12817 | 62.294 | 22.669 | 58476.300833
498850 | 19.13 | 135.4559 | 16.99877 | 50.57 | 25.117 | 58476.300833
166030 | 18.15 | 135.64987 | 17.23355 | 47.223 | 21.514 | 58476.300833
473267 | 18.87 | 135.81728 | 17.32215 | 42.399 | 49.874 | 58476.300833
| 20.08 | 136.31433 | 16.6182 | 50.547 | -0.079 | 58476.302454
119376 | 17.26 | 135.25554 | 16.49822 | 43.99 | 2.772 | 58476.300833
| 19.73 | 135.61529 | 17.42385 | 51.244 | 14.9 | 58476.300833
431072 | 19.47 | 136.97429 | 17.05782 | 43.853 | 0.713 | 58476.301655
626709 | 19.15 | 135.86705 | 16.40152 | 59.217 | 4.345 | 58476.301655
663839 | 20.13 | 135.58638 | 17.21575 | 63.547 | -3.197 | 58476.300833
510285 | 19.98 | 135.32888 | 16.52914 | 67.827 | 5.166 | 58476.301655
365075 | 18.91 | 136.48842 | 16.15177 | 59.528 | 10.94 | 58476.322488
386506 | 19.27 | 136.09069 | 17.75528 | 46.456 | 27.302 | 58476.300833
16500 | 17.23 | 136.64005 | 16.51016 | 53.986 | 22.626 | 58476.300833
| 19.8 | 135.89591 | 16.90669 | 61.6 | 8.903 | 58476.300833
147181 | 18.21 | 136.27812 | 17.51217 | 65.831 | 17.762 | 58476.300833
324998 | 18.25 | 136.69835 | 16.6882 | 25.028 | -32.699 | 58476.300833
| 19.36 | 136.24379 | 16.13965 | 52.325 | 22.198 | 58476.300833
348902 | 20.05 | 136.93307 | 16.53251 | 39.312 | -20.286 | 58476.302454
491052 | 18.58 | 135.972 | 16.36937 | 55.269 | 32.659 | 58476.300833
42019 | 17.67 | 135.47326 | 17.05354 | 58.027 | 18.158 | 58476.300833
68050 | 18.01 | 136.77643 | 16.49843 | 76.136 | 14.782 | 58476.300833
160942 | 17.51 | 135.98831 | 17.58401 | 71.649 | 5.44 | 58476.300833
9889 | 15.79 | 135.55901 | 17.40975 | 66.005 | 25.11 | 58476.300833
| 20.12 | 135.56822 | 16.58966 | 54.599 | 13.216 | 58476.300833
503442 | 19.55 | 134.97665 | 16.81495 | 59.507 | 5.252 | 58476.300833
343950 | 18.84 | 135.18567 | 16.84944 | 53.914 | 39.654 | 58476.300833
| 20.44 | 136.15398 | 17.74642 | 32.686 | 6.793 | 58476.300833
205682 | 18.07 | 136.06945 | 16.87638 | 52.201 | -12.812 | 58476.300833
| 19.63 | 135.80239 | 17.77756 | -2.331 | 3.362 | 58476.308854
| 20.6 | 136.59895 | 16.5701 | 70.376 | -17.874 | 58476.301655
| 18.86 | 136.24951 | 17.03423 | 67.871 | -12.499 | 58476.300833
211948 | 18.42 | 136.31032 | 17.33449 | 33.743 | 10.382 | 58476.300833
36186 | 16.65 | 136.68077 | 16.10726 | 49.262 | 27.374 | 58476.302454
| 20.36 | 136.17558 | 16.43811 | 57.801 | 17.737 | 58476.300833
294715 | 18.65 | 136.76117 | 16.21089 | 61.698 | -9.374 | 58476.301655
214666 | 17.7 | 135.49641 | 16.61914 | 37.035 | 69.217 | 58476.302454
432453 | 19.17 | 135.68294 | 15.83491 | 47.355 | 15.379 | 58476.300833
| 17.95 | 136.48921 | 16.15486 | 45.267 | -23.018 | 58476.300833
69945 | 18.43 | 135.7733 | 15.97912 | 62.584 | 14.483 | 58476.300833
327173 | 18.5 | 136.99326 | 16.56207 | 5.321 | 31.874 | 58476.302454
| 20.5 | 135.6533 | 16.89193 | 51.873 | 13.039 | 58476.300833
9470 | 15.69 | 135.59699 | 17.59959 | 54.588 | 10.469 | 58476.302454
| 19.45 | 135.93277 | 15.99409 | 51.228 | -27.666 | 58476.300833
215249 | 18.56 | 136.21533 | 16.1978 | 67.227 | 40.871 | 58476.300833
74196 | 17.84 | 136.56215 | 16.23027 | 54.267 | 26.165 | 58476.300833
| 19.48 | 135.29135 | 16.24837 | 48.809 | 36.925 | 58476.300833
| 19.89 | 136.13554 | 16.94882 | 57.722 | 8.942 | 58476.302454
160952 | 18.12 | 136.53272 | 16.7094 | 74.49 | 31.54 | 58476.300833
| 19.42 | 136.61242 | 17.31761 | 34.842 | -25.805 | 58476.300833
| 20.26 | 136.63788 | 16.00316 | 60.001 | -7.2 | 58476.301655
368106 | 18.9 | 136.89205 | 16.46623 | 49.552 | 26.496 | 58476.300833
| 20.27 | 135.29237 | 17.13186 | 49.182 | -30.974 | 58476.301655
| 18.5 | 136.82018 | 16.52299 | 51.701 | -8.705 | 58476.3161
661526 | 20.16 | 135.66744 | 16.96293 | 60.218 | 2.531 | 58476.300833
287521 | 18.31 | 135.54068 | 17.71363 | 43.237 | 44.68 | 58476.300833
230642 | 18.65 | 135.31528 | 16.38053 | 72.082 | 6.854 | 58476.300833
97240 | 16.73 | 136.01248 | 16.00087 | 47.499 | 6.235 | 58476.300833
474942 | 20.2 | 135.28387 | 16.34534 | 54.115 | 33.091 | 58476.300833
| 20.08 | 135.84387 | 16.13753 | 44.925 | 21.29 | 58476.301655
| 19.16 | 136.55238 | 17.0633 | 29.329 | 49.28 | 58476.300833
180367 | 18.35 | 135.65373 | 16.72844 | 60.234 | 32.346 | 58476.300833
356324 | 19.36 | 136.43649 | 16.72724 | 46.695 | 44.309 | 58476.301655
210410 | 19.18 | 136.1321 | 16.63173 | 49.737 | 13.712 | 58476.300833
334595 | 19.05 | 135.61312 | 17.67213 | 57.434 | 17.716 | 58476.300833
| 19.59 | 135.13039 | 16.57398 | 56.332 | 21.038 | 58476.300833
| 20.35 | 136.64475 | 17.03865 | 50.759 | -7.153 | 58476.300833
| 20.27 | 136.11317 | 16.54213 | 49.128 | 20.527 | 58476.300833
| 20.45 | 136.29677 | 17.3111 | 54.077 | 18.58 | 58476.300833
171567 | 19.05 | 135.69274 | 16.19179 | 71.332 | 4.918 | 58476.300833
564072 | 19.46 | 136.26338 | 15.95012 | 61.111 | -16.308 | 58476.302454
196982 | 17.73 | 136.16396 | 16.57139 | 49.359 | 35.86 | 58476.300833
499477 | 20.59 | 135.47242 | 16.82739 | 49.813 | 10.159 | 58476.300833
| 19.5 | 135.39774 | 16.84382 | 35.954 | 6.016 | 58476.300833
103530 | 18.4 | 135.62942 | 17.08498 | 57.766 | 15.491 | 58476.300833
139191 | 18.39 | 136.44185 | 16.54856 | 48.923 | 23.479 | 58476.300833
310387 | 18.42 | 136.48133 | 17.72875 | 39.835 | 8.064 | 58476.301655
77420 | 16.44 | 135.17018 | 16.41232 | 32.634 | -20.563 | 58476.300833
| 20.6 | 136.36311 | 16.18875 | 63.489 | 1.35 | 58476.300833
| 19.35 | 136.12582 | 16.95941 | 45.987 | 74.678 | 58476.300833
| 18.96 | 136.29627 | 15.75668 | 18.519 | 31.871 | 58476.302454
| 19.59 | 135.78388 | 17.30166 | 29.191 | 23.206 | 58476.301655
208987 | 17.96 | 135.78051 | 16.51568 | 41.031 | 22.406 | 58476.300833
| 19.47 | 136.34439 | 16.55707 | 56.927 | 1.451 | 58476.300833
141170 | 18.28 | 135.83898 | 16.36287 | 47.654 | 23.746 | 58476.300833
| 18.16 | 136.50655 | 16.57531 | 59.82 | 5.724 | 58476.300833
210210 | 19.12 | 136.4391 | 15.99642 | 56.034 | 11.7 | 58476.300833
12538 | 16.32 | 136.75815 | 16.15224 | 51.782 | 3.636 | 58476.302454
268038 | 18.65 | 136.91207 | 17.19404 | 60.57 | 2.898 | 58476.300833
18456 | 15.66 | 136.47852 | 17.32713 | 36.5 | 14.688 | 58476.300833
326927 | 18.26 | 135.46241 | 17.5636 | 47.769 | -4.997 | 58476.300833
| 20.28 | 136.48621 | 17.23242 | 60.853 | 30.47 | 58476.301655
118976 | 18.62 | 136.30441 | 17.38439 | 54.625 | 22.288 | 58476.300833
83762 | 16.82 | 136.59937 | 16.05942 | 58.064 | -4.025 | 58476.301655
535650 | 19.39 | 135.77725 | 16.50283 | 42.691 | 34.798 | 58476.300833
| 17.8 | 136.45865 | 16.60754 | 50.533 | 30.287 | 58476.302454
300100 | 20.48 | 135.92023 | 16.89878 | 74.564 | 3.992 | 58476.300833
422252 | 19.81 | 136.49046 | 16.02948 | 49.88 | 30.092 | 58476.300833
525560 | 19.87 | 136.24626 | 17.75111 | 53.109 | 23.285 | 58476.300833
217833 | 19.3 | 135.72047 | 16.73632 | 63.379 | 27.961 | 58476.300833
62841 | 17.66 | 136.67904 | 16.17763 | 61.906 | 10.156 | 58476.300833
546659 | 20.46 | 136.27346 | 16.68588 | 66.792 | 25.52 | 58476.325683
| 20.22 | 135.21984 | 16.59767 | 57.936 | 5.368 | 58476.300833
135382 | 18.37 | 135.71195 | 16.19279 | 62.63 | 15.304 | 58476.300833
302010 | 19.77 | 134.94177 | 16.84369 | 65.979 | 8.244 | 58476.300833
| 19.7 | 136.7685 | 17.13505 | 30.098 | 1.39 | 58476.300833
227354 | 18.58 | 136.56422 | 16.77846 | 60.666 | 15.127 | 58476.302454
| 19.8 | 135.80003 | 16.59761 | 74.196 | -12.827 | 58476.300833
| 20.68 | 135.60756 | 17.34904 | 53.114 | 14.332 | 58476.300833
| 19.35 | 135.19157 | 16.87652 | 54.537 | -13.032 | 58476.300833
300231 | 18.71 | 135.18482 | 16.75723 | 60.001 | 9.76 | 58476.301655
677931 | 18.38 | 135.39582 | 17.45446 | 42.997 | 49.021 | 58476.300833
| 20.9 | 135.69761 | 16.20796 | 60.43 | -8.784 | 58476.300833
77485 | 17.29 | 135.53661 | 16.30218 | 48.007 | 24.638 | 58476.300833
| 19.73 | 135.78325 | 17.3273 | 74.949 | 0.594 | 58476.301655
| 20.39 | 136.59471 | 16.56258 | 38.316 | 9.799 | 58476.301655
| 19.61 | 135.30874 | 16.15239 | 34.6 | 21.38 | 58476.300833
| 19.24 | 135.5803 | 17.50614 | 13.088 | 51.754 | 58476.300833

This work was performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and
was supported by the LLNL-LDRD Program under Projects 17-ERD-120, 19-SI-004,
and 20-ER-025. This work was based on observations obtained at Cerro Tololo
Inter-American Observatory a division of the National Optical Astronomy
Observatories, which is operated by the Association of Universities for
Research in Astronomy, Inc. under cooperative agreement with the National
Science Foundation. We thank the PI of the PALS survey (William Dawson) for
granting us eight hours of observational time over four nights during which
their targets were not observable. We thank Maya Gokhale, Eddie Schlafly, Will
Dawson, and Josh Meyers for useful discussions. We thank the Pan-STARRS team
for their dutiful photometry catalogs for calibration. The Pan-STARRS1 Surveys
(PS1) and the PS1 public science archive have been made possible through
contributions by the Institute for Astronomy, the University of Hawaii, the
Pan-STARRS Project Office, the Max-Planck Society and its participating
institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max
Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins
University, Durham University, the University of Edinburgh, the Queen’s
University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las
Cumbres Observatory Global Telescope Network Incorporated, the National
Central University of Taiwan, the Space Telescope Science Institute, the
National Aeronautics and Space Administration under Grant No. NNX08AR22G
issued through the Planetary Science Division of the NASA Science Mission
Directorate, the National Science Foundation Grant No. AST-1238877, the
University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos
National Laboratory, and the Gordon and Betty Moore Foundation. This research
has made use of NASA’s Astrophysics Data System Bibliographic Services. This
research made use of NumPy (Van Der Walt et al., 2011). This research made use
of Astropy, a community-developed core Python package for Astronomy (Astropy
Collaboration et al., 2018, 2013). This research made use of matplotlib, a
Python library for publication quality graphics (Hunter, 2007). This research
made use of pandas (McKinney, 2010, 2011). This research made use of the
REBOUND integrator package (Rein & Liu, 2012).

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