# No Evidence for Orbital Clustering in the Extreme Trans-Neptunian Objects

K. J. Napier Department of Physics, University of Michigan, Ann Arbor, MI
48109, USA  D. W. Gerdes Department of Physics, University of Michigan, Ann
Arbor, MI 48109, USA  Department of Astronomy, University of Michigan, Ann
Arbor, MI 48109, USA  Hsing Wen Lin ( 林省文) Department of Physics, University
of Michigan, Ann Arbor, MI 48109, USA  S. J. Hamilton NSF Graduate Research
Fellow  Department of Physics, University of Michigan, Ann Arbor, MI 48109,
USA  G. M. Bernstein Department of Physics and Astronomy, University of
Pennsylvania, Philadelphia, PA 19104, USA  P. H. Bernardinelli Department of
Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
T. M. C. Abbott  Cerro Tololo Inter-American Observatory, NSF’s National
Optical-Infrared Astronomy Research Laboratory, Casilla 603, La Serena, Chile
M. Aguena  Departamento de Física Matemática, Instituto de Física,
Universidade de São Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil
Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino
77, Rio de Janeiro, RJ - 20921-400, Brazil  J. Annis  Fermi National
Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA  S. Avila
Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049
Madrid, Spain  D. Bacon  Institute of Cosmology and Gravitation, University of
Portsmouth, Portsmouth, PO1 3FX, UK  E. Bertin  CNRS, UMR 7095, Institut
d’Astrophysique de Paris, F-75014, Paris, France  Sorbonne Universités, UPMC
Univ Paris 06, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris,
France  D. Brooks  Department of Physics & Astronomy, University College
London, Gower Street, London, WC1E 6BT, UK  D. L. Burke  Kavli Institute for
Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University,
Stanford, CA 94305, USA  SLAC National Accelerator Laboratory, Menlo Park, CA
94025, USA  A. Carnero Rosell  Instituto de Astrofisica de Canarias, E-38205
La Laguna, Tenerife, Spain  Universidad de La Laguna, Dpto. Astrofísica,
E-38206 La Laguna, Tenerife, Spain  M. Carrasco Kind  Department of Astronomy,
University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL
61801, USA  National Center for Supercomputing Applications, 1205 West Clark
St., Urbana, IL 61801, USA  J. Carretero  Institut de Física d’Altes Energies
(IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193
Bellaterra (Barcelona) Spain  M. Costanzi  INAF-Osservatorio Astronomico di
Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy  Institute for
Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy  L. N.
da Costa  Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal.
José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil  Observatório
Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
J. De Vicente  Centro de Investigaciones Energéticas, Medioambientales y
Tecnológicas (CIEMAT), Madrid, Spain  H. T. Diehl  Fermi National Accelerator
Laboratory, P. O. Box 500, Batavia, IL 60510, USA  P. Doel  Department of
Physics & Astronomy, University College London, Gower Street, London, WC1E
6BT, UK  S. Everett  Santa Cruz Institute for Particle Physics, Santa Cruz, CA
95064, USA  I. Ferrero  Institute of Theoretical Astrophysics, University of
Oslo. P.O. Box 1029 Blindern, NO-0315 Oslo, Norway  P. Fosalba  Institut
d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain  Institute of
Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193
Barcelona, Spain  J. García-Bellido  Instituto de Fisica Teorica UAM/CSIC,
Universidad Autonoma de Madrid, 28049 Madrid, Spain  D. Gruen  Department of
Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA
Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,
Stanford University, Stanford, CA 94305, USA  SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA  R. A. Gruendl  Department of Astronomy,
University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL
61801, USA  National Center for Supercomputing Applications, 1205 West Clark
St., Urbana, IL 61801, USA  G. Gutierrez  Fermi National Accelerator
Laboratory, P. O. Box 500, Batavia, IL 60510, USA  D. L. Hollowood  Santa Cruz
Institute for Particle Physics, Santa Cruz, CA 95064, USA  K. Honscheid
Center for Cosmology and Astro-Particle Physics, The Ohio State University,
Columbus, OH 43210, USA  Department of Physics, The Ohio State University,
Columbus, OH 43210, USA  B. Hoyle  Faculty of Physics, Ludwig-Maximilians-
Universität, Scheinerstr. 1, 81679 Munich, Germany  Max Planck Institute for
Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany
Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität
München, Scheinerstr. 1, 81679 München, Germany  D. J. James  Center for
Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138,
USA  S. Kent  Fermi National Accelerator Laboratory, P. O. Box 500, Batavia,
IL 60510, USA  Kavli Institute for Cosmological Physics, University of
Chicago, Chicago, IL 60637, USA  K. Kuehn  Australian Astronomical Optics,
Macquarie University, North Ryde, NSW 2113, Australia  Lowell Observatory,
1400 Mars Hill Rd, Flagstaff, AZ 86001, USA  N. Kuropatkin  Fermi National
Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA  M. A. G. Maia
Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José Cristino
77, Rio de Janeiro, RJ - 20921-400, Brazil  Observatório Nacional, Rua Gal.
José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil  F. Menanteau
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W.
Green Street, Urbana, IL 61801, USA  National Center for Supercomputing
Applications, 1205 West Clark St., Urbana, IL 61801, USA  R. Miquel
Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain
Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science
and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain  R. Morgan
Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison,
1150 University Avenue Madison, WI 53706-1390  A. Palmese  Fermi National
Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA  Kavli Institute
for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA  F.
Paz-Chinchón  Institute of Astronomy, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK  National Center for Supercomputing Applications, 1205
West Clark St., Urbana, IL 61801, USA  A. A. Plazas  Department of
Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ
08544, USA  E. Sanchez  Centro de Investigaciones Energéticas,
Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain  V. Scarpine  Fermi
National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA  S.
Serrano  Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona,
Spain  Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can
Magrans, s/n, 08193 Barcelona, Spain  I. Sevilla-Noarbe  Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid,
Spain  M. Smith  School of Physics and Astronomy, University of Southampton,
Southampton, SO17 1BJ, UK  E. Suchyta  Computer Science and Mathematics
Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831  M. E. C. Swanson
National Center for Supercomputing Applications, 1205 West Clark St., Urbana,
IL 61801, USA  C. To  Department of Physics, Stanford University, 382 Via
Pueblo Mall, Stanford, CA 94305, USA  Kavli Institute for Particle
Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA
94305, USA  SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
A. R. Walker  Cerro Tololo Inter-American Observatory, NSF’s National Optical-
Infrared Astronomy Research Laboratory, Casilla 603, La Serena, Chile  R.D.
Wilkinson  Department of Physics and Astronomy, Pevensey Building, University
of Sussex, Brighton, BN1 9QH, UK

(Received 17 October 2020; Revised 7 January 2021; Accepted 9 February 2021)

###### Abstract

The apparent clustering in longitude of perihelion $\varpi$ and ascending node
$\Omega$ of extreme trans-Neptunian objects (ETNOs) has been attributed to the
gravitational effects of an unseen 5-10 Earth-mass planet in the outer solar
system. To investigate how selection bias may contribute to this clustering,
we consider 14 ETNOs discovered by the Dark Energy Survey, the Outer Solar
System Origins Survey, and the survey of Sheppard and Trujillo. Using each
survey’s published pointing history, depth, and TNO tracking selections, we
calculate the joint probability that these objects are consistent with an
underlying parent population with uniform distributions in $\varpi$ and
$\Omega$. We find that the mean scaled longitude of perihelion and orbital
poles of the detected ETNOs are consistent with a uniform population at a
level between $17\%$ and $94\%$, and thus conclude that this sample provides
no evidence for angular clustering.

Solar system (1528), Planetary science (1255), Trans-Neptunian objects (1705)

††journal: The Planetary Science Journal

##  1 Introduction

The apparent clustering in longitude of perihelion $\varpi$ and ascending node
$\Omega$ of solar system bodies known as extreme trans-Neptunian objects
(ETNOs) motivated the hypothesis that the solar system contains a 5-10 Earth-
mass planet (Planet X/Planet 9) at 400-800 times Earth’s distance from the sun
(Trujillo & Sheppard, 2014; Batygin & Brown, 2016; Batygin et al., 2019). Some
have proposed even more exotic sources of the apparent clustering, such as
gravitational perturbations from a primordial black hole captured into orbit
around the sun (Scholtz & Unwin, 2020).

While there is no universally accepted definition for the ETNOs, recent
literature has emphasized objects with semi-major axis $a\gtrsim 230$ au and
perihelion $q>30$ au. Because ETNOs follow highly elliptical orbits, and their
brightness decreases like $1/r^{4}$, they are almost always discovered within
a few decades of perihelion. Moreover, telescopic surveys observe a limited
area of the sky, at particular times of year, to a limited depth. These
effects result in significant selection bias. The six ETNOs considered in the
Batygin & Brown (2016) (BB16) analysis were discovered in an assortment of
surveys with unknown or unpublished selection functions, making it difficult
to establish that the observed angular clustering was indeed of physical
origin.

More recent surveys have carefully characterized their selection functions and
applied these tools to small samples of new ETNOs. The Outer Solar System
Origins Survey (OSSOS, Bannister et al. (2016)) analyzed the bias present in
the discovery of 8 objects they detected with $a>150$ au and $q>30$ au
(Shankman et al., 2017). They found that their detected objects were
consistent with a uniform underlying population in $\varpi$ and $\Omega$.
Bernardinelli et al. (2020a) analyzed samples of 3-7 variously-defined ETNOs
discovered by the Dark Energy Survey (DES, DES Collaboration (2005); Dark
Energy Survey Collaboration et al. (2016)), and also found the data consistent
with angular isotropy.

Brown & Batygin (2019) (BB19) attempted to reverse-engineer the survey bias in
the entire then-known population of 14 ETNOs using a sampling method (Brown,
2017) on all TNOs known to the Minor Planet Center. In contrast to the
individual survey-level analyses described above, BB19 concluded that the
observed clustering is highly likely to be a physical effect, and they argued
that the best explanation remains a massive distant planet.

While no single survey has discovered enough ETNOs to reach a statistically
compelling conclusion, a stronger statement becomes possible when data from
multiple surveys are combined. According to the criteria above, there are 14
ETNOs (Table 1) detected by three independent surveys with characterized
selection functions, all published since BB16. Using the published pointing
history, depth, and TNO tracking selections for DES (5 objects) (Khain et al.,
2020; Bernardinelli et al., 2020b), OSSOS (5 objects) (Bannister et al.,
2018), and the survey of Sheppard & Trujillo (2016) (ST) (4 objects), we
calculate the joint probability that these objects are consistent with the
null hypothesis: an underlying population distributed uniformly in the
longitudes $\varpi$ and $\Omega$. If the purported clustering is indeed a
physical effect, we would expect it to remain consistent with the data in this
larger, independent sample when selection functions are modeled.

Object |  $a$ (au) | $e$ |  $i$ ($\degree$) |  $q$ (au) |  $\omega$($\degree$) |  $\Omega$ ($\degree$) | H (mag) | Survey
---|---|---|---|---|---|---|---|---
2015 BP519 | 448.8 | 0.92 | 54.1 | 35.2 | 348.1 | 135.2 | 4.4 | DES
2013 SL102 | 314.3 | 0.88 | 6.5 | 38.1 | 265.5 | 94.7 | 7.1 | DES
2013 RA109 | 462.4 | 0.90 | 12.4 | 46.0 | 263.0 | 104.8 | 6.2 | DES
2014 WB556 | 289.1 | 0.85 | 24.2 | 42.5 | 234.6 | 114.9 | 7.3 | DES
2016 SG58 | 233.0 | 0.85 | 13.2 | 35.1 | 296.3 | 119.0 | 7.5 | DES
2013 SY99 | 733.1 | 0.93 | 4.2 | 50.1 | 32.2 | 29.5 | 6.7 | OSSOS
2015 RX245 | 426.4 | 0.89 | 12.1 | 45.7 | 65.1 | 8.6 | 6.2 | OSSOS
2015 GT50 | 311.4 | 0.88 | 8.8 | 38.5 | 129.0 | 46.1 | 8.5 | OSSOS
2015 KG163 | 679.7 | 0.94 | 14.0 | 40.5 | 32.1 | 219.1 | 8.2 | OSSOS
uo5m93 | 283.0 | 0.86 | 6.8 | 39.5 | 43.3 | 165.9 | 8.8 | OSSOS
2013 FT28 | 295.4 | 0.85 | 17.4 | 43.4 | 40.7 | 217.7 | 6.7 | ST
2014 SR349 | 296.6 | 0.84 | 18.0 | 47.7 | 341.2 | 34.9 | 6.7 | ST
2015 TG387 | 1101.3 | 0.94 | 11.7 | 65.1 | 118.0 | 301.0 | 5.5 | ST
2014 FE72 | 1559.5 | 0.98 | 20.6 | 36.2 | 133.9 | 336.8 | 6.1 | ST
2012 VP113* | 262.7 | 0.69 | 24.1 | 80.5 | 293.8 | 90.8 | 4.0 | ST
2013 RF98* | 363.6 | 0.90 | 29.5 | 36.1 | 311.8 | 67.6 | 8.7 | DES SN
Table 1:  Barycentric orbital elements of the ETNOs used in our analysis. All
reported values are at the epoch JD $2459000.5$ (except for uo5m93 whose
elements are for the epoch JD $2457163.82647$). DES SN indicates discovery in
the DES supernova fields. *Note that in order to maintain an independent
sample from BB16 we do not include 2013 RF98 or 2012 VP113 in our main
analysis. We discuss their effects separately.

##  2 Methods

The three surveys we consider have very different designs and scientific
goals, and consequently quite different ETNO selection functions. This is
readily apparent from their survey footprints, shown in Figure 1. DES, which
was on-sky between 2012 and 2019, used the Dark Energy Camera (DECam, Flaugher
et al. (2015)) on the 4-meter Blanco telescope at CTIO to carry out an
extragalactic survey designed to measure cosmological parameters. It consisted
of two interwoven surveys. In the 30 sq. deg. supernova survey, ten separate
fields were visited approximately weekly in the $griz$ bands during the six
months per year that DES was in operation. In the 5000 sq. deg. wide survey,
each field was imaged a total of 10 times at a sparse temporal cadence in each
of the $grizY$ bands over the duration of the survey. The wide survey reached
a limiting $r$-band magnitude of $\approx 23.5$. DES had limited near-ecliptic
coverage centered near ecliptic longitude of 0, and a large off-ecliptic
footprint that made it particularly sensitive to high-inclination objects. For
our main analysis, we consider only the ETNOs detected in the DES wide survey,
and treat the supernova fields separately. The OSSOS survey (2013-2017), by
contrast, was optimized to detect and track TNOs in eight $\sim 20$ sq. deg.
blocks distributed along the ecliptic. This survey used the 3.6-meter Canada-
France-Hawaii Telescope and reached a limiting $r$-band magnitude of
24.1–25.2. Finally, the ST survey (2007-2015) used the Blanco, Subaru, Large
Binocular, and Magellan telescopes to cover 1080 sq. deg. at an average
distance of 13 deg. from the ecliptic to a depth of approximately $VR\sim 25$.
This survey aimed to detect the most distant objects: ETNOs and Inner Oort-
Cloud (IOC) objects such as Sedna. Therefore, only those candidates with an
estimated heliocentric distance greater than 50 au were selected for followup
and tracking.

The most complete way to account for survey bias in the discovery of the solar
system objects is to use a survey simulator (Petit et al., 2011; Lawler et
al., 2018). In essence, a survey simulator simulates detections of a model
population of solar system bodies by using a survey’s pointing history, depth,
and tracking criteria. This allows for the computation of a survey’s selection
function for a given population, which enables us to account for bias, and
therefore understand the true underlying populations. While it gives a
reasonable approximation, the technique employed in BB19 cannot fully
substitute for actually simulating each survey to calculate its selection
function. Since the known ETNOs were discovered by a variety of surveys, the
task of developing an appropriate simulator is nontrivial. Our simulator
(FastSSim) is highly parametric, requiring only the few pieces of information
common among all well-characterized surveys: pointing history, limiting
magnitudes, and followup criteria.111The tools for the FastSSim algorithm have
now been compiled into an open-source Python package SpaceRocks. It is under
active development at https://github.com/kjnapier/spacerocks. The basic flow
of the simulator is as follows:

  1. 1.

Map a survey’s published pointing history to a HEALPix222It is not important
that we used a HEALPix mapping. We could have used any mapping onto the
sphere. grid, as in Figure 1.

  2. 2.

Generate a distribution of fake objects at a single epoch.

  3. 3.

Calculate the objects’ HEALPix pixels and apparent magnitudes.

  4. 4.

Determine which fake objects fall in a survey’s footprint.

  5. 5.

Make cuts according to the survey’s limiting magnitudes and followup criteria.

Note that this simulation method makes several approximations. We compute the
sky coordinates of our objects at objects at a single epoch, we use a single
color and limiting magnitude for each survey field, we do not consider CCD-
level detections (so we do not account for complications such as chip gaps),
and we employ a step-function detection criterion (so we do not model survey
cadence or linking efficiency). We use a single HEALPix pixel for each survey
pointing. We have chosen the pixel scales for each telescope as follows:
Blanco uses NSIDE of 64 (except for the DES supernova fields, for which we use
NSIDE of 1024), and the Magellan, Large Binocular, and Subaru telescopes use
NSIDE of 128. These assumptions ignore the time history of the surveys, as
well as apparent motion of the objects. FastSSim works well for this
application because the objects move slowly, the telescopes have large fields
of view, and the sensitivity does not have much spatial variation.

Figure 1: A HEALPix mapping of the currently-released sky coverage of the
three major TNO surveys of this generation. The surveys by OSSOS and Sheppard
and Trujillo hug the ecliptic plane (plotted in red), while DES, designed as a
cosmological survey, has a much more expansive footprint.

We acquired the non-DES survey pointings and limiting magnitudes from Sheppard
& Trujillo (2016), Sheppard et al. (2019), and Bannister et al. (2018). We
choose each HEALPix pixel size to most closely match the field of view of the
telescope used. This does not allow for a perfect mapping between pointings
and pixels, but it turns out to be sufficient for our needs. In fact, we find
that FastSSim performs remarkably well in cross-checks against both our full
chip-level DES simulator (Hamilton, 2019), and the OSSOS survey simulator
described in Petit et al. (2011) (see Figure 2). While FastSSim misses some of
the fine details of the selection functions, the small sample of ETNOs and the
approximate nature of this analysis make such fine details unimportant to our
overall conclusion. Given the success of these cross-checks, we are confident
in extending its use to characterize the survey of Sheppard and Trujillo.

Figure 2: OSSOS selection functions for 1,615 detections from the nominal
population described in §3 in the angles $\omega$, $\Omega$, and $\varpi$, as
calculated by FastSSim (red) and the OSSOS/CFEPS survey simulator (black). The
distributions have the same general shape.

To simulate the surveys we randomly generate ETNOs in accordance with a
nominal scattered disk model (specified in §3) until each survey has
accumulated $10^{5}$ detections, to allow for high-resolution characterization
of a survey’s sensitivity in $\varpi$ and $\Omega$. This typically requires
the generation of approximately $10^{10}$ fakes, so our set of simulated
objects spans the parameter space of the ETNOs. We consider an object to be
detected if it is in one of the survey’s HEALPix pixels, is brighter than the
pixel’s limiting magnitude, and has a perihelion distance $q\geq$ 30 au. For
the survey of Sheppard and Trujillo, we satisfy a tracking criterion specified
in Sheppard et al. (2016) by requiring an object to have a heliocentric
distance of at least 50 au at the time of detection.

As a quantitative example of the effectiveness of FastSSim, Figure 2 shows a
comparison with the CFEPS/OSSOS simulator. For this test each simulator uses
the population model defined in §3. Using Kuiper’s test, we find that the
distributions of the 1,615 data points calculated by FastSSim are
statistically indistinguishable from those computed using the CFEPS/OSSOS
survey simulator. Thus in order to distinguish the two simulators one would
need $>1,615$ ETNOs—well above the quantity discovered by OSSOS. We achieve
similar results in quantitative comparisons against the distributions computed
using the DES survey simulators (see Figure 5.1 in Hamilton (2019) and Figure
2 in Bernardinelli et al. (2020a)).

In Figure 3 we plot a Gaussian kernel density estimate of each simulator’s
detections in the $(x,y)$ and $(p,q)$ spaces ($x$, $y$, $p$, and $q$ are
defined in §4). Since the distributions appear to be in good agreement we
believe that our selection functions, which are derived directly from the
surveys’ pointing histories, depths, and TNO tracking strategies, more
faithfully model their respective surveys than than those inferred indirectly
in BB19 (see their Figure 4). Furthermore, the p-values we calculate using the
CFEPS/OSSOS survey simulator do not significantly differ from those we
calculate using FastSSim.

Figure 3: The black contours (which follow a linear scale) are Gaussian kernel
density estimates of $10^{6}$ iterations of sampling 5 points from the OSSOS
biases calculated using the CFEPS/OSSOS survey simulator (because OSSOS
discovered 5 ETNOs), and plotting the mean $(x,y)$ and $(p,q)$ positions (see
§4 for definitions of $x$, $y$, $p$, and $q$). The red contours represent the
same statistic, calculated using FastSSim. The blue points are the mean
positions of the 5 ETNOs discovered by OSSOS. Our simulator reproduces the
results of the OSSOS simulator with better fidelity than the heuristic method
of BB19 (see their Figure 4).

##  3 Scattered Disk Model

To test the dependence of our analysis on the choice of scattered disk model,
we simulated models with various distributions of semi-major axis ($a$),
eccentricity ($e$), inclination ($i$), and absolute magnitude ($H$), while
keeping the orbital angles $\Omega$ and $\omega$ (and thus $\varpi$) uniform
from $0\degree$ to $360\degree$. We tested manifold permutations with the
parameter distributions: $N(a)\propto a^{\zeta}$ with
$a\in[230\,\text{au},1600\,\text{au}]$ and $\zeta\in[0.5,1.0]$, uniform
$i\in[0\degree,60\degree]$, Brown distribution $i$ (Brown (2001)) with a
variety of widths ranging from 5$\degree$ to 25$\degree$, and $N(H)\propto
10^{H\zeta}$ with $H\in[4,10]$ and $\zeta\in[0.6,0.9]$.

We found that our conclusions were not significantly affected by the variation
of the model parameters. Our results are also robust to changes in pericenter
distribution (see the Appendix for the distribution of orbital elements for
populations with $q>30,35$, and $38$ au). (Shankman et al., 2017) found
similar resilience to changes in scattered disk model. Noting the weak
dependence of the outcome of our simulations on the choice of model, we
proceed using the following scattered disk model:

  * –

$a$ follows a single power-law distribution such that $N(a)\propto a^{0.7}$,
where $a\in[230\,\text{au},1600\,\text{au}]$

  * –

$e$ is distributed uniformly $\in[0.69,0.999]$

  * –

$i$ follows a Brown distribution such that
$N(i)\propto\sin(i)\exp{\left[-\frac{(i-\mu_{i})^{2}}{2\sigma_{i}^{2}}\right]}$
with $\mu_{i}=0\degree$ and $\sigma_{i}=15\degree$

  * –

$H$ follows a single power-law distribution such that $N(H)\propto 10^{0.8H}$,
where $H\in[4,10]$

  * –

Perihelion distance $q>30\,\text{au}$

These model parameters produce posteriors in $a$, $e$, $q$, $i$, and $H$ that
appear to be in reasonable agreement with the real ETNO detections by each
survey. See the Appendix for histograms of the posteriors in each of these
variables, overlaid with a rug plot of each survey’s real detections.

##  4 Analysis and Results

Performing a clustering analysis in the variables $\varpi$ and $\Omega$ is
complicated, as the two are strongly correlated. We proceed by working in the
orthogonal $\\{x,y,p,q\\}$ basis discussed in BB19 (importantly, $\varpi$ and
$\Omega$ are linearly independent in this basis). Note that these vectors are
not normalized, but instead have their lengths modulated by eccentricity and
inclination. The coordinates are defined as follows.

$\Gamma=1-\sqrt{1-e^{2}}\qquad\qquad Z=\sqrt{1-e^{2}}\left[1-\cos(i)\right]$
(1) $x=\sqrt{2\Gamma}\cos(\varpi)\qquad\qquad y=\sqrt{2\Gamma}\sin(\varpi)$
(2) $p=\sqrt{2Z}\cos(\Omega)\qquad\qquad q=\sqrt{2Z}\sin(\Omega)$ (3)

Note that $\Gamma$ and $Z$ have been scaled by a factor of
$\sqrt{GM_{\odot}a}$ from their traditional forms, since the semi-major axis
is not relevant to this argument. Figure 4 shows our calculated selection
functions in the $xy$ and $pq$-planes.

Figure 4: Kernel density estimates of each survey’s selection function in the
canonical $xy$-space (top row) and $pq$-space (bottom row). The contours
represent simulated detections (the contours scale linearly, and darker
contours are more densely populated), while the blue dots represent the ETNOs
detected by each survey. The outlier in both DES panels is the object 2015
BP519 (Becker et al., 2018).

For the sake of comparison, we used the method presented in BB19 to test the
consistency of each survey’s detected ETNOs with its selection function. We
first perform $10^{6}$ iterations sampling from our simulated detections a set
of objects whose cardinality is equal to that of the set of real ETNOs
detected by the given survey. We then take the average $\\{x,y,p,q\\}$
position of each sample, and use these values to construct a 4-dimensional
histogram. We display a Gaussian kernel density estimation of these data in
the $xy$ and $pq$-planes in Figure 5.

Figure 5: Kernel density estimates of the mean $(x,y)$ and $(p,q)$ position of
$10^{6}$ samples of ETNOs drawn from the PDFs shown in Figure 4. The number of
objects in each sample corresponds to the number of ETNOs detected by the
given survey. The contours represent the samples (the contours scale linearly,
and darker contours are more densely populated), while the red dots represent
the mean position of the ETNOs detected by each survey.

We perform a Gaussian kernel density estimation on our mean-sampled histograms
to obtain a probability distribution function (PDF). Next we draw N samples
from our simulated data (where N is the number of ETNOs actually detected by
the survey), find the mean $\\{x,y,p,q\\}$ position, and evaluate our PDF at
that position. We repeat this $10^{5}$ times to construct a likelihood
function. Next we compute this value for the ETNOs actually discovered by the
survey. To calculate the probability of a survey detecting the ETNOs it
actually detected (as opposed to some other set of ETNOs), we find the
fraction of the $10^{5}$ sample likelihood values that the survey’s actual
likelihood value exceeds. Rounded to the nearest $1\%$, this probability for
each survey is as follows: ${\mathcal{P}_{DES}}\sim 0.06$,
${\mathcal{P}_{OSSOS}}\sim 0.53$, and ${\mathcal{P}_{ST}}\sim 0.59$.

The joint probability of N surveys detecting objects with given probabilities
(or some less likely set of values) can be calculated as the volume under the
surface of constant product of probabilities in the domain of the
N-dimensional unit hypercube, given by

$\mathcal{P}_{joint}=P\sum_{k=0}^{N}(-1)^{k}\frac{\log(P)^{k}}{k!}$ (4)

where $P\equiv\prod_{k}\mathcal{P}_{k}$. In our case,
$k\in\\{\text{DES},\text{OSSOS},\text{ST}\\}$. Using Equation 4, we calculate
the joint probability to be $24\%$.

With such a small sample size, this work is sensitive to outliers and the
definition of the ETNOs itself. The high-inclination object 2015 BP519 is
among the most dynamically anomalous objects in the solar system (Becker et
al., 2018), and we cannot discount the possibility that it is of a different
dynamical origin than the other ETNOs. If we redo our analysis without 2015
BP519, $\mathcal{P}_{DES}$ increases to $84\%$, and thus $\mathcal{P}_{joint}$
increases to $\sim 85\%$. 2014 FE72 has an extremely large semi-major
axis—roughly four standard deviations above the mean of the ETNOs considered
in this work. Its large semi-major axis carries it deep into the IOC region,
where interactions with galactic tides make its secular relationship with a
putative Planet X/Planet 9 less certain. If we exclude 2014 FE72,
$\mathcal{P}_{\text{ST}}$ increases to $88\%$ and thus
$\mathcal{P}_{\text{joint}}$ increases to $31\%$. If we include 2012 VP113,
$\mathcal{P}_{\text{ST}}$ increases to $60\%$, and
$\mathcal{P}_{\text{joint}}$ remains $24\%$. We also address the fact that the
clustering by a putative Planet X/Planet 9 should be more robust in the sample
of ETNOs with $q>40$ au, since these objects avoid strong perturbations by
Neptune. If we restrict our ETNOs to these 8 objects,
$\mathcal{P}_{\text{joint}}$ increases to $94\%$. Finally, we analyze the
subset of objects which are either stable or metastable in the presence of the
putative Planet X/Planet 9 (Batygin et al., 2019): 2015 TG387, 2013 SY99, 2015
RX245, 2014 SR349, 2012 VP113, 2013 RA109, and 2013 FT28. For this subset
$\mathcal{P}_{\text{joint}}=82\%$.

For the sake of completeness, we also use a more traditional sampling method
to determine the significance of the clustering of ETNOs. We begin by
performing a Gaussian kernel density estimate on each survey’s posterior
distributions. We then perform $10^{5}$ iterations in which we randomly draw
$N$ points from each survey’s posterior distribution (where $N$ is the number
of ETNOs detected by the survey) and multiply each of the $N$ probabilities
together to calculate a likelihood. Finally, we calculate the same metric for
each survey’s actual detections and compare the value to the distribution of
our samples. As before, the probability for each survey is the fraction of the
$10^{5}$ sample likelihood values that the survey’s actual likelihood value
exceeds. Rounded to the nearest $1\%$, the probability for each survey is as
follows: ${\mathcal{P}_{DES}}\sim 0.06$, ${\mathcal{P}_{OSSOS}}\sim 0.41$, and
${\mathcal{P}_{ST}}\sim 0.43$. The joint probability is thus $17\%$.

For a more physically intuitive representation of the survey bias, refer to
Figure 6. Here the radial quantity represents the barycentric distance, and
the azimuthal quantity represents true longitude (the true anomaly +
$\varpi$). The edge of the black circle is at 30 au. The white regions
represent the combined surveys’ sensitivity (brighter regions correspond to
higher sensitivity), weighted by the number of real ETNO detections. The red
dots represent the real ETNOs at the epoch of discovery. The observations are
in good agreement with the combined selection function, qualitatively
confirming the conclusions of our formal statistical analysis performed on
canonical variables.

Figure 6: Combined ETNO selection function for all three surveys. The radial
quantity is the ETNO’s barycentric distance, and the azimuthal quantity is
true longitude. The edge of the black circle is at 30 au. The white regions
represent the combined surveys’ sensitivity (brighter regions correspond to
higher sensitivity), weighted by the number of real ETNO detections. The red
dots represent the real ETNOs at the epoch of discovery. The outer ring is
caused by the 50 au tracking criterion imposed by ST.

###  4.1 DES Supernova Fields

The ETNO 2013 RF98 was discovered in the deep DES supernova fields (DES SN
hereafter). Since the DES SN fields are so small, they suffer from severe
selection bias. Additionally, since their observing cadence and depth
($\sim$24.5 in the $r$-band) are significantly different than the wide survey,
they need to be treated independently. We generated 1,829 simulated detections
in the DES SN fields (since the fields are so small, it is computationally
prohibitive to generate $10^{5}$ synthetic detections as we do for DES, OSSOS,
and ST) from the population model defined in §3. We show the posteriors in
$\\{a,e,i,H,\Omega,\varpi\\}$ in Figures 8, 9, 10, 11, 12, 13, and 14. Figure
7 shows a kernel density estimate of the detections in $\\{x,y,p,q\\}$ space.
In all parameters, 2013 RF98 appears to be a rather ordinary detection for the
DES SN fields.

Figure 7: Kernel density estimates of the DES SN selection function in the
canonical $xy$-space (left) and $pq$-space (right). The contours represent
simulated detections (the contours scale linearly, and darker contours are
more densely populated), while the blue dots represent 2013 RF98.

Since there is only one data point here, we can just numerically integrate to
find $\mathcal{P}_{\text{DESSN}}=0.33$ (i.e. a p-value of 0.33). Treating DES
SN as its own survey, we may use Equation 4 to calculate the 4-survey joint
probability to find $\mathcal{P}_{\text{joint}}=25\%$.

##  5 Discussion and Conclusions

We use quantified selection bias calculations on all the ETNOs discovered by
the three most productive ETNO surveys, each with with a quite different
survey strategy and selection function, to test the consistency of the ETNOs
with a uniform underlying distribution. Given a joint probability between
$17\%$ and $94\%$ (i.e. a $p$-value between 0.17 and 0.94), we conclude that
the sample of ETNOs from well-characterized surveys is fully consistent with
an underlying parent population with uniform distributions in the longitudes
$\varpi$ and $\Omega$. Our result differs drastically from the corresponding
value in BB19 of $0.2\%$. Closer inspection sheds some light on the apparent
discrepancy. If we examine only the overlapping set of ETNOs used this work
and in BB19 (2015 BP519, 2013 RF98, 2013 SY99, 2015 RX245, 2015 GT50, 2015
KG163, 2013 FT28, 2014 SR349, and 2014 FE72), $\mathcal{P}_{\text{joint}}$
drops to $<0.005$. This indicates an expected issue: small number statistics
are sensitive to fluctuations. For example, when BB19 performed their analysis
a small but important set of the ETNOs had not yet been reported to the MPC.
As a concrete demonstration of the importance of the omission of a few ETNOs
from BB19, consider DES. Of the five ETNOs discovered by the DES wide survey,
BB19 included only 2015 BP519. From Figure 4 it is clear that this object
lands in an extremely low-probability region. This drives down
$\mathcal{P}_{\text{joint}}$, and thus gives a satisfactory answer as to why
the result of this work differs so significantly from that of BB19.

It is important to note that our work does not explicitly rule out Planet
X/Planet 9; its dynamical effects are not yet well enough defined to falsify
its existence with current data. This work also does not analyze whether some
form of clustering could be consistent with the 14 ETNOs we consider. For
example, the ETNOs could happen to be clustered precisely where current
surveys have looked. In that case, a survey with coverage orthogonal to the
regions shown in Figure 6 would find far fewer ETNOs than expected. Various
realizations of Planet X/Planet 9 predict clustering of various widths,
modalities, and libration amplitudes and frequencies; we do not test for
consistency with any of these distributions. Instead, we have shown that given
the current set of ETNOs from well-characterized surveys, there is no evidence
to rule out the null hypothesis. Increasing the sample of ETNOs with ongoing
and future surveys with different selection functions such as the Deep
Ecliptic Exploration Project (DEEP) (Trilling et al., 2019) and the Legacy
Survey of Space and Time (LSST) at the Vera Rubin Observatory (Schwamb et al.,
2018) will allow for more restrictive results. Despite other lines of indirect
evidence for Planet X/Planet 9, in the absence of clear evidence for
clustering of the ETNOs the argument becomes much weaker. Future studies
should consider other mechanisms capable of giving the outer solar system its
observed structure, while preserving a uniform distribution of ETNOs in the
longitudes $\Omega$ and $\varpi$.

This material is based upon work supported by the National Aeronautics and
Space Administration under Grant No. NNX17AF21G issued through the SSO
Planetary Astronomy Program and by the National Science Foundation under Grant
No. AST-2009096. Funding for the DES Projects has been provided by the U.S.
Department of Energy, the U.S. National Science Foundation, the Ministry of
Science and Education of Spain, the Science and Technology Facilities Council
of the United Kingdom, the Higher Education Funding Council for England, the
National Center for Supercomputing Applications at the University of Illinois
at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the
University of Chicago, the Center for Cosmology and Astro-Particle Physics at
the Ohio State University, the Mitchell Institute for Fundamental Physics and
Astronomy at Texas A&M University, Financiadora de Estudos e Projetos,
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro,
Conselho Nacional de Desenvolvimento Científico e Tecnológico and the
Ministério da Ciência, Tecnologia e Inovação, the Deutsche
Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy
Survey. The Collaborating Institutions are Argonne National Laboratory, the
University of California at Santa Cruz, the University of Cambridge, Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the
University of Chicago, University College London, the DES-Brazil Consortium,
the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH)
Zürich, Fermi National Accelerator Laboratory, the University of Illinois at
Urbana-Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the
Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory,
the Ludwig-Maximilians Universität München and the associated Excellence
Cluster Universe, the University of Michigan, NFS’s NOIRLab, the University of
Nottingham, The Ohio State University, the University of Pennsylvania, the
University of Portsmouth, SLAC National Accelerator Laboratory, Stanford
University, the University of Sussex, Texas A&M University, and the OzDES
Membership Consortium. Based in part on observations at Cerro Tololo Inter-
American Observatory at NSF’s NOIRLab (NOIRLab Prop. ID 2012B-0001; PI: J.
Frieman), which is managed by the Association of Universities for Research in
Astronomy (AURA) under a cooperative agreement with the National Science
Foundation. The DES data management system is supported by the National
Science Foundation under Grant Numbers AST-1138766 and AST-1536171. The DES
participants from Spanish institutions are partially supported by MICINN under
grants ESP2017-89838, PGC2018-094773, PGC2018-102021, SEV-2016-0588,
SEV-2016-0597, and MDM-2015-0509, some of which include ERDF funds from the
European Union. IFAE is partially funded by the CERCA program of the
Generalitat de Catalunya. Research leading to these results has received
funding from the European Research Council under the European Union’s Seventh
Framework Program (FP7/2007-2013) including ERC grant agreements 240672,
291329, and 306478. We acknowledge support from the Brazilian Instituto
Nacional de Ciência e Tecnologia (INCT) do e-Universo (CNPq grant
465376/2014-2). This manuscript has been authored by Fermi Research Alliance,
LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy,
Office of Science, Office of High Energy Physics.

## References

  * Bannister et al. (2016) Bannister, M. T., Kavelaars, J. J., Petit, J.-M., et al. 2016, AJ, 152, 70, doi: 10.3847/0004-6256/152/3/70
  * Bannister et al. (2018) Bannister, M. T., Gladman, B. J., Kavelaars, J. J., et al. 2018, ApJS, 236, 18, doi: 10.3847/1538-4365/aab77a
  * Batygin et al. (2019) Batygin, K., Adams, F. C., Brown, M. E., & Becker, J. C. 2019, Physics Reports, 805, 1 , doi: https://doi.org/10.1016/j.physrep.2019.01.009
  * Batygin & Brown (2016) Batygin, K., & Brown, M. E. 2016, AJ, 151, 22, doi: 10.3847/0004-6256/151/2/22
  * Becker et al. (2018) Becker, J. C., Khain, T., Hamilton, S. J., et al. 2018, The Astronomical Journal, 156, 81, doi: 10.3847/1538-3881/aad042
  * Bernardinelli et al. (2020a) Bernardinelli, P. H., Bernstein, G. M., Sako, M., et al. 2020a, The Planetary Science Journal, 1, 28, doi: 10.3847/PSJ/ab9d80
  * Bernardinelli et al. (2020b) —. 2020b, ApJS, 247, 32, doi: 10.3847/1538-4365/ab6bd8
  * Brown (2001) Brown, M. E. 2001, The Astronomical Journal, 121, 2804.  http://stacks.iop.org/1538-3881/121/i=5/a=2804
  * Brown (2017) —. 2017, The Astronomical Journal, 154, doi: 10.3847/1538-3881/aa79f4
  * Brown & Batygin (2019) Brown, M. E., & Batygin, K. 2019, The Astronomical Journal, 157, 62, doi: 10.3847/1538-3881/aaf051
  * Dark Energy Survey Collaboration et al. (2016) Dark Energy Survey Collaboration, Abbott, T., Abdalla, F. B., et al. 2016, MNRAS, 460, 1270, doi: 10.1093/mnras/stw641
  * DES Collaboration (2005) DES Collaboration. 2005, ArXiv e-prints.  https://arxiv.org/abs/astro-ph/0510346
  * Flaugher et al. (2015) Flaugher, B., Diehl, H. T., Honscheid, K., et al. 2015, AJ, 150, 150, doi: 10.1088/0004-6256/150/5/150
  * Hamilton (2019) Hamilton, S. 2019, PhD thesis, University of Michigan
  * Khain et al. (2020) Khain, T., Becker, J. C., Lin, H. W., et al. 2020, AJ, 159, 133, doi: 10.3847/1538-3881/ab7002
  * Lawler et al. (2018) Lawler, S. M., Kavelaars, J., Alexandersen, M., et al. 2018, ArXiv e-prints.  https://arxiv.org/abs/1802.00460
  * 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
  * Scholtz & Unwin (2020) Scholtz, J., & Unwin, J. 2020, Phys. Rev. Lett., 125, 051103, doi: 10.1103/PhysRevLett.125.051103
  * Schwamb et al. (2018) Schwamb, M. E., Jones, R. L., Chesley, S. R., et al. 2018, arXiv e-prints, arXiv:1802.01783.  https://arxiv.org/abs/1802.01783
  * Shankman et al. (2017) Shankman, C., Kavelaars, J. J., Bannister, M. T., et al. 2017, AJ, 154, 50, doi: 10.3847/1538-3881/aa7aed
  * Sheppard & Trujillo (2016) Sheppard, S. S., & Trujillo, C. 2016, AJ, 152, 221, doi: 10.3847/1538-3881/152/6/221
  * Sheppard et al. (2016) Sheppard, S. S., Trujillo, C., & Tholen, D. J. 2016, ApJ, 825, L13, doi: 10.3847/2041-8205/825/1/L13
  * Sheppard et al. (2019) Sheppard, S. S., Trujillo, C. A., Tholen, D. J., & Kaib, N. 2019, The Astronomical Journal, 157, 139, doi: 10.3847/1538-3881/ab0895
  * Trilling et al. (2019) Trilling, D., Gerdes, D., Trujillo, C., et al. 2019, in EPSC-DPS Joint Meeting 2019, Vol. 2019, EPSC–DPS2019–395
  * Trujillo & Sheppard (2014) Trujillo, C. A., & Sheppard, S. S. 2014, Nature, 507, 471, doi: 10.1038/nature13156

Figure 8: Posterior pericenter distance distributions of simulated detections.
The red triangles are the real ETNO detections by each survey. Note that DES
has partially overlapping data points at $q\approx 35$ au. Figure 9: Posterior
semi-major axis distributions of simulated detections. The red triangles are
the real ETNO detections by each survey. Note that ST has partially
overlapping data points at $a\approx 296$ au. The grey, red, and blue
histograms correspond to cuts with $q>30,35$ and $38$ au, respectively. Figure
10: Posterior eccentricity distributions of simulated detections. The red
triangles are the real ETNO detections by each survey. Note that DES has
overlapping data points at $e=0.85$. The grey, red, and blue histograms
correspond to cuts with $q>30,35$ and $38$ au, respectively. Figure 11:
Posterior inclination distributions of simulated detections. The red triangles
represent the real ETNO detections by each survey. The grey, red, and blue
histograms correspond to cuts with $q>30,35$ and $38$ au, respectively. Figure
12: Posterior absolute magnitude distributions of simulated detections. The
red triangles represent the real ETNO detections by each survey. Note that ST
has overlapping data points at $H=6.7$. The grey, red, and blue histograms
correspond to cuts with $q>30,35$ and $38$ au, respectively. Figure 13:
Posterior longitude of ascending node distributions of simulated detections.
The red triangles represent the real ETNO detections by each survey. The grey,
red, and blue histograms correspond to cuts with $q>30,35$ and $38$ au,
respectively. Figure 14: Posterior longitude of pericenter distributions of
simulated detections. The red triangles represent the real ETNO detections by
each survey. The grey, red, and blue histograms correspond to cuts with
$q>30,35$ and $38$ au, respectively.