# Col-OSSOS: The Two Types of Kuiper Belt Surfaces

Wesley C. Fraser Herzberg Astronomy and Astrophysics Research Centre, National
Research Council, 5071 W. Saanich Rd. Victoria, BC, V9E 2E7, Canada
Department of Physics and Astronomy, University of Victoria, Elliott Building,
3800 Finnerty Road, Victoria, BC V8P 5C2, Canada  Rosemary E. Pike Center for
Astrophysics — Harvard & Smithsonian; 60 Garden Street, Cambridge, MA, 02138,
USA  Michaël Marsset European Southern Observatory, Alonso de Córdova 3107,
Santiago, Chile  Department of Earth, Atmospheric and Planetary Sciences, MIT,
77 Massachusetts Avenue, Cambridge, MA 02139, USA  Megan E. Schwamb
Astrophysics Research Centre, Queen’s University Belfast, Belfast, County
Antrim BT7 1NN, UK  Michele T. Bannister School of Physical and Chemical
Sciences — Te Kura Matū University of Canterbury, Private Bag 4800,
Christchurch 8140, New Zealand  Laura Buchanan Astrophysics Research Centre,
Queen’s University Belfast, Belfast, County Antrim BT7 1NN, UK  JJ Kavelaars
Herzberg Astronomy and Astrophysics Research Centre, National Research
Council, 5071 W. Saanich Rd. Victoria, BC, V9E 2E7, Canada  Department of
Physics and Astronomy, University of Victoria, Elliott Building, 3800 Finnerty
Road, Victoria, BC V8P 5C2, Canada  Susan D. Benecchi Planetary Science
Institute, 1700 East Fort Lowell, Suite 106, Tucson, AZ 85719, USA  Nicole J.
Tan School of Physical and Chemical Sciences — Te Kura Matū University of
Canterbury, Private Bag 4800, Christchurch 8140, New Zealand  Nuno Peixinho
Instituto de Astrofísica e Ciências do Espaço, Universidade de Coimbra,
PT3040-004 Coimbra, Portugal  Stephen D. J. Gwyn Herzberg Astronomy and
Astrophysics Research Centre, National Research Council, 5071 W. Saanich Rd.
Victoria, BC, V9E 2E7, Canada  Mike Alexandersen Center for Astrophysics —
Harvard & Smithsonian; 60 Garden Street, Cambridge, MA, 02138, USA  Ying-Tung
Chen Institute of Astronomy and Astrophysics, Academia Sinica; 11F of AS/NTU
Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617,
Taiwan, R.O.C  Brett Gladman Department of Physics and Astronomy, University
of British Columbia, Vancouver, BC, Canada  Kathryn Volk Lunar and Planetary
Laboratory, The University of Arizona, 1629 E University Blvd, Tucson, AZ
85721 Wesley C. Fraser<EMAIL_ADDRESS>
###### Abstract

The Colours of the Outer Solar System Origins Survey (Col-OSSOS) has gathered
high quality, near-simultaneous (g-r) and (r-J) colours of 92 Kuiper Belt
Objects (KBOs) with (u-g) and (r-z) gathered for some. We present the current
state of the survey and data analysis. Recognizing that the optical colours of
most icy bodies broadly follow the reddening curve, we present a new
projection of the optical-NIR colours, which rectifies the main non-linear
features in the optical-NIR along the ordinates. We find evidence for a
bifurcation in the projected colours which presents itself as a diagonal empty
region in the optical-NIR. A reanalysis of past colour surveys reveals the
same bifurcation. We interpret this as evidence for two separate surface
classes: the BrightIR class spans the full range of optical colours and
broadly follows the reddening curve, while the FaintIR objects are limited in
optical colour, and are less bright in the NIR than the BrightIR objects. We
present a two class model. Objects in each class consist of a mix of separate
blue and red materials, and span a broad range in colour. Spectra are modelled
as linear optical and NIR spectra with different slopes, that intersect at
some transition wavelength. The underlying spectral properties of the two
classes fully reproduce the observed structures in the UV-optical-NIR colour
space ($0.4\lesssim\lambda\lesssim 1.4\mbox{ $\mu$m}$), including the
bifurcation observed in the Col-OSSOS and H/WTSOSS datasets, the tendency for
cold classical KBOs to have lower (r-z) colours than excited objects, and the
well known bimodal optical colour distribution.

††facilities: Gemini-North(GMOS, NIRI), CFHT(MegaCam), Subaru (Suprime-
cam)††software: numpy (Harris et al., 2020), matplotlib (Hunter, 2007),
TRailed Imaging in Python, (Fraser et al., 2016), astropy (Astropy
Collaboration et al., 2013, 2018),
nirlin555https://staff.gemini.edu/~astephens/niri/nirlin/nirlin.py, SExtractor
(Bertin & Arnouts, 1996), SExtractor in Python (Barbary, 2016), synphot (STScI
Development Team, 2013)

##  1 Introduction

The colours of Kuiper Belt Objects (KBOs) have long been a topic of interest
to the Solar System science community, having been used to reveal a range of
compositional and cosmogonic properties about those bodies. Broadband colours
have been used to provide compositional inference when targets are too faint
for detailed spectroscopy. For KBOs, colours are often as informative as
spectroscopy, as shortward of the $1.5\mbox{ $\mu$m}$ water-ice absorption
feature, most small KBOs exhibit nearly featureless spectra, that are well
described by a red optical slope (Fornasier et al., 2009) and a less-red NIR
slope (Barucci et al., 2011), with a smooth transition from one to the other
(e.g., Gourgeot et al., 2015). Where known absorptions exist, specific
compositional information for certain materials can be gleaned with well
chosen filters (e.g., Trujillo et al., 2011).

Arguably more useful than compositional information, large samples of
broadband colours have also been used for taxonomic purposes. For example,
optical colours combined with H-band photometry have been useful in
identifying faint Haumea family members (Snodgrass et al., 2010). In recent
years, multiple efforts have been made to generate a taxonomy of KBO surfaces
from their broadband optical-NIR colours (Dalle Ore et al., 2013; Fraser &
Brown, 2012; Peixinho et al., 2015; Pike et al., 2017; Schwamb et al., 2019).
Thus far, no consensus has been found regarding the true number of main KBO
compositional classes, with as few as three classes being proposed (e.g.,
Fraser & Brown, 2012; Pike et al., 2017) and as many as 10 (Dalle Ore et al.,
2013). The count of KBO taxons has important cosmogonic implications, as it is
believed that taxon membership is a consequence of compositional structure
that occurred while objects resided in the protoplanetesimal disk. It is
believed that the taxons reflect the level of radial compositional
stratification that occurred between formation and the dispersal of the
protoplanetesimal disk.

The Colours of the Outer Solar System Origins Survey (Col-OSSOS) was designed
around the experiences of past optical and NIR colour surveys (Schwamb et al.,
2019). The spectrophotometric observations of this program were designed to be
sensitive to the optical and NIR spectral slopes, so as to detect any
potential taxons that may reveal themselves in these wavelength regions
(Tegler & Romanishin, 2003; Delsanti et al., 2006; Peixinho et al., 2012;
Fraser & Brown, 2012). Additional UV and optical bands were acquired for some
targets as telescope time permitted. With a goal of homogenous signal-to-noise
(SNR) in all bands, and for all targets, the Col-OSSOS sample was designed
with a focus on providing confident taxonomic classification for KBOs. Here we
report the current state of the Col-OSSOS survey, presenting the grJ colours
of 92 targets acquired at the Gemini-North Telescope. We also present the
first results of the u-band observations that were acquired by an accompanying
program on the Canada-France-Hawaii Telescope.

We present a new colour projection scheme that is motivated by the spectral
behaviour of the bulk of icy bodies to fall near the so-called reddening
curve, or the curve of constant spectral slope through colour-space. This
projection into axes with distance along, and perpendicularly away from the
reddening curve, rectifies the non-linear structures seen in the optical-NIR
colour space of KBOs, for the Col-OSSOS dataset and others. With this
reprojection we find a bifurcation in the optical-NIR colours of KBOs, with
significant overlap of the optical and NIR colours of each class. The
bifurcation is apparent for colours which include a NIR filter, with
wavelength $\lambda\gtrsim 0.8\mbox{ $\mu$m}$. These findings are similar to
those of the Hubble/WFC3 Test of Surfaces in the Outer Solar System (H/WTSOSS,
Fraser & Brown, 2012), and a reanalysis of those data shows the same
bifurcation. These results argue for only two surface classes for small KBOs
(absolute magnitudes $H_{r}\gtrsim 5$). One class is dominated mainly by Cold
Classical KBOs (CCKBOs) with very red optical, and less red NIR colours. The
other class has optical colours that span the full range exhibited by KBOs,
but for objects that are equally as red as the first class, have much higher
NIR reflectivities. Further, we argue that the now well-known bifurcation in
optical colours is not due to one spectral class being found with only neutral
to moderately red optical colours, and the other with very red colours.
Rather, the class that the majority of excited KBOs belongs to spans the full
optical colour range.

In Section 2 we present a summary of the Col-OSSOS survey design, target
selection, observations, and data reductions. In Section 3 we present the
colours of Col-OSSOS targets, along with a new projection technique that helps
reveal structures in the multi-dimensional UV-optical-NIR colour space by
rectifying those colours along and perpendicular to the reddening line. We use
that projection to search for evidence of separate classes of KBOs in the
optical and NIR range of KBO colours. In Section 4, independent of the Col-
OSSOS dataset, we re-consider other large optical and NIR colour datasets to
increase the wavelength range and sample size in our analysis. Those results
corroborate what we find with the Col-OSSOS dataset. In Section 5 we present a
simple spectral model that accounts for the bulk behaviours of the different
colours datasets we have considered. In Section 6 we present a synthesis or
our results, and show how they fit within past compositional interpretations.
We discuss the cosmogonic significance of our results, and finish with
concluding remarks in Section 7.

##  2 Observational Data

We make use of the Col-OSSOS dataset. This consists mainly of observations in
the Sloan u, g, and r, filters, and the Mauna Kea J filter, which were
acquired with the Gemini Multi-Object Spectrograph (GMOS; Hook et al., 2004)
and Near Infrared Imager (NIRI; Hodapp et al., 2003) at Gemini-North, and the
MegaPrime imager (Boulade et al., 2003) on the Canada-France-Hawaii Telescope.
A small number of targets received supplemental observations with the Suprime-
Cam imager (Miyazaki et al., 2002) on the Subaru Telescope. The data we
present here include measurements presented previously (Fraser et al., 2017;
Pike et al., 2017; Schwamb et al., 2019; Marsset et al., 2019, 2020; Fraser et
al., 2021), but are updated to reflect improvements in the reduction
pipelines. Full details of the observing program, data reductions, and colour
analysis can be found in Schwamb et al. (2019), though we summarize the
procedures here. We also list specific changes and improvements to the
pipelines and analysis procedures since last publication (Schwamb et al.,
2019).

The nominal Col-OSSOS sample are those 98 objects found in the E, H, L, O, S,
and T blocks of the Outer Solar System Origins Survey (OSSOS, Bannister et
al., 2018), with discovery triplet brightness r$<23.6$ mags. The E, S, and T
blocks have 6, 1, and 2 targets that are yet to be observed at the time of
writing. A summary of the field details is available in Table 1.

The standard observing procedure for a given target was to image the target in
a r-g-J-g-r pattern, with g, and r observations acquired with GMOS, and the J
observations acquired with NIRI. Exposure times were 300 s exposure for GMOS
images, and 120 s for individual NIRI frames, unless otherwise noted (see
Table 2). The number of exposures acquired in each filter was tuned to match
the target’s discovery brightness, and designed to achieve a signal to noise
greater than 20 in J, and 25 in g, and r, assuming pessimistic KBO colours of
(g-r)=1.1 and (r-J)=1.2. If circumstance allowed it, GMOS images in z-band
were acquired in an ad-hoc approach, either before or after, or bracketing the
nominal sequence.

When available, observations in r and u where acquired at CFHT, simultaneously
with the Gemini sequence. These observations usually used an r-u-r pattern,
with $\sim 14$ u-band images of 320 s, with two r-band bracketing exposures of
300 s. Though some sequences acquired in 2014 did not have bracketing r-band
images. A couple sequences also included g-band observations. Here we present
only measurements from the u-band data, and have not made use of the r-band
imagery. A full list of all exposures considered here are listed in Table 2.

###  2.1 Reductions

GMOS imagery was first flat fielded and de-biased in the standard way using
flats produced as part of Gemini’s day calibrations program. Rough photometric
calibrations were made using in-image stellar sources, with magnitudes
uncorrected for differences between the GMOS and catalog photometric systems.
Linear colour terms were measured, and applied to correct the rough image
zeropoints, to place them in the catalog calibration magnitude system. Source
magnitudes were measured using the Trailed Image Photometry in Python package
(TRIPPy; Fraser et al., 2017) with pill apertures accounting for the source’s
rate of motion, and radius of 1.2 times the Full Width at Half Maximum (FWHM)
of nearby stellar sources. Point-spread functions (PSFs) were generated using
in-image point sources, from which aperture corrections were measured,
enabling correction to the circular apertures with radius $4\times$ FWHM used
on the calibration stars.

After bad-pixel masking and cosmic ray rejection, NIRI images were flattened,
and had dark-current removed using sky-frames created from a rolling 15-frame
window for each target NIRI sequence. The images of a science sequence were
aligned using in-field point-sources, at the sidereal rate, as well as the
rate of motion of the science target. Two mean stacks consisting of the first
and second halves of the images in the sequence were produced. PSFs were
generated from the sidereal imagery, from which FWHMs were measured. TRIPPy
pill apertures of radius $1.2\times$ FWHM were used, and aperture corrections
applied to match the $4\times$ FWHM were used on the calibration stars.

The Subaru photometry we report here is an updated reduction of the values
previously reported in Pike et al. (2017). We highlight a change to the
reductions. Specifically, sources were more closely scrutinized to better
exclude non-point-sources from the list of sources used to evaluate image
PSFs. Unfortunately, in the original analysis some PSF reference sources were
non-stellar (eg. galaxy cores), resulting in inflated PSFs, and therefore,
inflated TRIPPy apertures.

NIRI photometric calibrations were made from standard star sequences acquired
before and after the standard r-g-J-g-r sequence. Each standard sequence
consisted of 9 frames each consisting of 6 1.5 s coadds, and were reduced in
the same way as the science frames, but utilizing a single sky frame for the
full sequence. During analysis, an inconsistency was discovered in measured
calibration magnitudes from night to night. The source was traced to non-
symmetric variations in the PSF, presumably due to the primary mirror
aberrations not settling during the short calibration sequences. The result
was an asymmetric PSF whose profile was still well described by a Moffat
profile along any given radial line (from centre outwards) but whose width
varied with angle around source peak. The inconsistent calibration photometry
was measured using a circular aperture scaled off the FWHM measured from the
average radial profile of the star, resulting in large portions of the
calibrator source flux falling outside the aperture in regions where the PSF
was particularly wide. The solution was to measure an azimuthally varying
Moffat profile, in 8 equal pie segments around the source. The adopted FWHM
was then twice the widest measured Half-Width at Half Maximum from the
separate segments. Reported zeropoints are the median values extracted from
all nine images of a sequence. By scaling apertures in this way, the image-to-
image zeropoints were reflected correctly, enabling calibrations accurate to
$\sim 3\%$. See Appendix A for a discussion of the precision of this
technique.

The MegaPrime u-band images were debiased, flattened, and photometrically
calibrated to the uCFHT bandpass following the MegaPipe procedures (Gwyn,
2008). All calibrated u-band images of a science sequence were then stacked at
sidereal and non-sidereal rates using the same procedures as used for the
J-band data. PSFs and aperture radii were evaluated from the sidereal stacks,
and photometry of the science target was measured from the non-sidereal stack.
We report photometry of only cases of clear source detection. No effort was
made to measure upper limits of ambiguous- or non-detections. Analysis of
u-band data was completed only for some data taken in semesters 2014B through
2016A. A more robust analysis including upper limits and all outstanding
u-band Col-OSSOS data – the majority of available u-band data – will be the
focus of a future work.

With photometry in hand, colours were measured in the same way as previously
reported. That is, each object was assumed to exhibit a linear lightcurve
during the 1.5-4 hour Gemini sequences. A linear lightcurve, and colours with
respect to r was fit to the optical data reported in the GMOS passbands. The
fitted line was then extrapolated to the J and u-band epochs to measure (u-g)
and (r-J) colour values. We point the reader to Schwamb et al. (2019) for
further details of this fitting procedure.

Finally, using the colour terms evaluated during our optical data calibrations
(see Appendix B) and from Gwyn (2008), colours were evaluated in the Sloan
Digital Sky Survey (SDSS Fukugita et al., 1996; Padmanabhan et al., 2008) (for
the u-band photometry) and the Panoramic Survey Telescope and Rapid Response
System Pan-STARRS PS1 colour bands (Tonry et al., 2012), with appropriate
propagation of uncertainties. Colours of our targets in the Gemini/CFHT, SDSS,
and PS1 colour systems are reported in Table 5. The updated colours derived
from Subaru observations are presented in the SDSS colour system in Table 4.

A number of notable changes and improvements have been made to the Col-OSSOS
pipeline. These changes are:

  1. 1.

All optical photometric calibrations are against the Pan-STARRS1 archive only.
We no longer make use of calibrations against the SDSS archive.

  2. 2.

J-band calibration source PSF profiles were measured using the azimuthally
varying moffat profile technique described above.

  3. 3.

Reported J-band uncertainties now include a 0.03 magnitude (0.02 in prior
reductions) contribution due to the precision by which we can measure
zeropoints, added in quadrature with contributions from other noise sources.

  4. 4.

A bug was fixed in the exposure times of NIRI imagery that was caused by
alteration of the _EXPTIME_ header keyword as part of the Gemini _nirlin_
procedure.

  5. 5.

We utilized the background measurement routine in _SExtractor in Python_
package (Barbary, 2016), with a 128x128 box, to remove the background in each
quadrant of each NIRI frame, in advance of the _cleanir_
111https://www.gemini.edu/instrumentation/niri/data-reduction artifact removal
and cosmic ray rejection routines. This resulted in a drastic improvement in
residual background structure compared to using the default _SExtractor_
background estimation for the whole image.

  6. 6.

The linear lightcurve fitting routing was altered to weight each measurement’s
contributions to the overall fit likelihood according to the uncertainty of
each measurement.

Special consideration was made for observations in the 2016B semester. The
observing plan was to make use of stars GD246 and G-129 as NIR calibration
sources. It was readily apparent from our calibration data that GD246 was
variable. Fortunately, two other stars happened to fall in the field of view,
providing an extra calibration source. A sequence of observations were
acquired in 8 different epochs in 2019B at twilight during which GD246, and
calibration sources F108, and SA114-750 were acquired, enabling brightness
calibration; the two stars near GD246 have $J=11.879\pm 0.005$ and $J=15.03\pm
0.03$, respectively. Across 12 separate epochs, there was clear variability
for the fainter source, but not for the brighter.

In addition to the unexpected variability of GD246, an error was made in the
observing plan, causing observations to be acquired of a similarly bright star
at 30’ higher RA than for G-129. This affected the Col-OSSOS sequences of 7
targets from the S and T blocks. Unfortunately this star has a faint secondary
within the wings of the primary, and about 5% the brightness of the primary.
Removal of the secondary was a relatively simple task by making a PSF from the
primary, and manually removing the secondary from the look up table.
Subtraction residuals from the secondary were roughly 0.1% of the primary’s
flux. At five separate epochs, GD246 and this incorrect star were observed in
the same Col-OSSOS sequence, enabling a flux measurement of $J=14.56\pm 0.03$.
No variability was detected for the primary within the precision of the
brightness measurement. With brightness measurements of the two alternate
calibration sources – the incorrectly pointed star, and the star below GD246 –
in hand, and confirmation of non-variability for both sources, photometric
calibration of the 2016B data was performed as normal, and accounted for the
precision of the brightness measurements of the two alternate calibrations
sources.

A full list of observations utilized in this paper are reported in Table 2.
The raw Gemini data files are available through the Gemini Observatory Archive
(https://archive.gemini.edu). The reduced imagery are available for download
at the Canadian Astronomy Data Centre: doi: _to be updated at publication._

##  3 Colours of Col-OSSOS

In Figure 1 we show the full (g-r) and (r-J) measurements of the Col-OSSOS
sample, which consists of 110 individual measurements and 92 unique objects.
In this figure, we highlight two broad dynamical groups: the cold classical
Kuiper Belt objects (CCKBOs), and the dynamically excited objects. For the
former, we consider two different definitions. Classically, CCKBOs have been
defined as those with low inclinations, $i<6^{\circ}$ and high perihelia
$q>37$ AU, and semi-major axes bounded by the 3:2 and 2:1 mean-motion
resonances, $40<a<48$ AU. We note that the exact maximum inclination has
varied slightly (eg. Brown, 2001; Elliot et al., 2005; Fraser & Brown, 2012),
though slight variations in the adopted value have little bearing on our
results. Like in past papers, we also include in this group objects in the 7:4
and 5:3 resonances with $i<6^{\circ}$. For a more modern definition, we
consider objects’ free inclinations – the inclination vector after the forced
component is removed (Volk & Malhotra, 2017; Van Laerhoven et al., 2019; Huang
et al., 2022), $i_{\textrm{free}}<4^{\circ}$, and a more restrictive semi-
major axis range $42<a<48$ to select the least disturbed sample of classical
KBOs, which are those most likely to have formed in-situ. Through out this
manuscript, we discuss each subset separately where the distinction between
these two definitions of cold classical object is important. Finally, we take
as the dynamically excited sample, all those objects that are not considered
CCKBOs by either definition.

Figure 1 presents some of the features of the optical-NIR KBO colour space
that past surveys have shown, including the tendency for CCKBOs to be
optically red, and for most objects to fall to the lower-right of the
reddening curve, or curve of constant spectral slope in colour-space. This
sample also shows a bimodality in the optical colour distribution as has been
reported numerous times in the past (for recent examples, see Tegler et al.,
2016; Fraser & Brown, 2012; Peixinho et al., 2015; Marsset et al., 2019).
Hartigan’s DIP test of bimodality (Hartigan & Hartigan, 1985) implies that the
full sample has 3.5% probability of being drawn from a unimodal distribution.
That probability of unimodality changes to near 100% when excluding the
CCKBOs. Interestingly, the optical plane clustering test (FOP test; Fraser &
Brown, 2012) does not reveal any statistically significant bifurcation in the
(g-r) and (r-J) colour space. We note, however, an intriguing region with a
dearth of sources. This region can be roughly described as a diagonal region
redward in both the optical and NIR colours of $(g-r)\sim 0.75$ and $(r-J)\sim
1.5$.

Figure 1:  Left: The (g-r) and (r-J) colours of Col-OSSOS targets. Repeat
measurements for targets (6 objects) have been connected by dashed lines. The
reddening curve is shown by the solid black line. Right: The projection values
$PC^{1}$ and $PC^{2}$ of the (g-r) and (r-J) colours. In this projection, the
reddening curve is a horizontal line with $PC^{2}=0$. Grey hashes show lines
of constant (g-r) or (r-J) colour. The horizontal line at $PC^{2}=-0.13$ marks
the value below which all FaintIR objects are found. These mostly consist of
CCKBOs. The bottom and side panels show histograms of the full dataset (black)
and of the objects in the BrightIR (blue, bottom panel) and FaintIR classes
(orange, bottom panel). In both panels, red points are CCKBOs defined by their
$i_{\textrm{free}}<4^{\circ}$, yellow and red points together are CCKBOs by
the historical definition based on $i$ (see Section 2), and black points are
all those excited objects outside the CC region. Objects identified as
belonging to the class of _blue binary_ KBOs (Fraser et al., 2021) are
outlined with cyan halos.

To analyze the grJ colour space, we introduce a new approach inspired by the
recognition that the colours of icy bodies tend to roughly follow the
reddening curve – the curve of equal spectral slope in all colours (see Figure
1 for an example). This is true of Jupiter Trojans (see Szabó et al., 2007,
for example), KBOs, and Centaurs (Peixinho et al., 2015), the bulk of which
have optical colours usually within a few tenths of a magnitude of the
reddening curve (we present the optical colours of Trojans, KBOs, and Centaurs
in Figures 2, 3). The majority of small bodies in each of these populations
exhibit spectra dominated by the so-called _optical gap_ feature associated
with C-H bonds in the organic materials (see Grundy et al., 2020; Seccull et
al., 2021, for recent discussions). The tendency for the spectra of these
bodies to follow the reddening curve is true to a lesser extent in the near-
UV, and NIR where albedos tend to deviate away from the reddening curve. This
deviation is most dramatic in the NIR, though these populations still exhibit
correlated optical and NIR colours (Emery et al., 2011; Fraser & Brown, 2012).
The non-linear shape of the reddening curve is a common feature of colour-
space that can be utilized in interpretation of observed colours.

Figure 2:  The optical colours of Jupiter Trojans reported in the Sloan
Digital Sky Survey Moving Object Catalog v4 (Ivezić et al., 2001) with
$\Delta(g-r)<0.04$ and $\Delta(r-i)<0.04$. The datapoints generally scatter
around and follow the reddening curve which is shown by the solid line.
Errorbars are omitted for clarity. Semi-transparent points are used to show
overlapping table values. The shading is a kernel density estimate to
highlight the densest regions of the data.

Figure 3: Left: The optical colours of Kuiper Belt Objects and Centaurs
reported in Peixinho et al. (2015). The datapoints generally scatter around
and follow the reddening curve which is shown by the solid line. Right: The
projection values $PC^{1}$ and $PC^{2}$ of the (V-R) and (R-I) colours shown
in the left panel. Only the colours of objects with colour uncertainty
$\Delta(V-R)<0.1$, $\Delta(R-I)$, and $H_{\textrm{R}}>4$ are plotted. The
limit on absolute magnitude is made to avoid the few dwarf planets who have
surfaces dominated by their own rheologies, and therefore have colours that
are outliers on this plot. Red points are CCKBOs defined by their
$i_{\textrm{free}}$, yellow and red points together are CCKBOs by the
historical definition based on $i$, and black points are all those excited
KBOs outside the CC region.

As such, we adopt a projection of a 2-dimensional colour space onto a new set
of basis vectors that removes the reddening curve trends seen in the colours
samples. Specifically, we adopt two basis vectors that are the instantaneously
parallel and perpendicular vectors along the reddening curve, with projection
values that we label $PC^{1}$ and $PC^{2}$, respectively. We use subscripts to
denote which filters are considered. $PC^{1}$ and $PC^{2}$ are respectively,
the line integral along the reddening curve from some predefined reference
value, and the distance from the reddening curve along a vector perpendicular
to the reddening curve. In this way, negative values of
$PC^{2}_{\textrm{grJ}}$ reflect objects with concave spectra, or lower
spectral slope in (r-J) than in (g-r), and positive values of
$PC^{2}_{\textrm{grJ}}$ reflect convex spectra. We choose as the origin the
Solar (g-r) and (r-J) colours (see Appendix C) for $PC^{1}_{\textrm{grJ}}=0$,
or when put in terms of spectral slope, a value $s=0\mbox{ \%/100
$n$m}$222Spectral slope, $s$, is defined as increase in reddening per 100 nm,
with respect to the V-band ($\sim 550$ $n$m). eg., a spectral slope of 50
%/100 $n$m is 50% more reflective at 650 $n$m than at 550 $n$m.. For a visual
demonstration of this projection, see Appendix Figure 13. The projection
values are tabulated in Table 5.

This new projection is similar in concept to projections derived from
Principle Component Analysis. Though, rather than adopting an algorithm that
finds the _linear_ projection that maximizes the data variance along a single
basis vector, our approach is agnostic of the dataset in question. Rather this
projection is physically driven by the observation that in general, bulk
colour samples of icy bodies (Jupiter Trojans, comet nuclei, centaurs, KBOs,
etc.) tend toward the reddening curve, with usually only moderate spectral
deviations away from it.

We present the projection of the main Col-OSSOS data, (g-r) and (r-J), into
the PC space in Figure 1. From this figure, the primary advantage of this
projection is apparent - the non-linear structures in colour space that follow
the reddening curve are nearly rectified parallel to the ordinates of the new
basis. For example, the diagonal region that is absent of objects in grJ
colour space now appears as an obvious bifurcation in $PC^{2}_{\textrm{grJ}}$.
Hartigan’s DIP test of bimodality (Hartigan & Hartigan, 1985) suggests that
this apparent bifurcation has a 20% chance of occurring randomly. The gap
between the two peaks is only $\sim 0.1$ mags, and so we consider smaller
uncertainty subsets of the (g-r) and (r-J) sample. The probability that
apparent bifurcation is random chance is $<2\%$ for uncertainty values in both
colours of 0.06 to 0.1 mags. For example, with uncertainties of
$\Delta(g-r)\leq 0.1\mbox{ and }\Delta(r-J)\leq 0.1$ (79 of 110 g-r and r-J
colour pairs), the probability is 2%. The probability reduces even further
still, to 0.0015% for uncertainties in both colours $<0.09$ mags. The sample
with uncertainty $<0.06$ mags is too small for meaningful inference.
Considering a sample that rejects the most uncertain points provides a better
representation of the significance of the bifurcation in
$PC^{2}_{\textrm{grJ}}$ because the observed gap between the two inferred
populations is only $\sim 0.1$ magnitudes. Thus, including objects with larger
uncertainties will only mask the bifurcation. We find no evidence for further
bifurcations in the colour, or projected colour spaces. That is, we only have
evidence for 2 separate groups in the (g-r) and (r-J) colour space. A value of
$PC^{2}_{\textrm{grJ}}=-0.13$ provides a relatively clean division into two
classes. We label these two classes, BrightIR and FaintIR, as one class is
notably brighter in the NIR with respect to their optical colours than is the
other class.

In the Col-OSSOS sample, the objects with $PC^{2}_{\textrm{grJ}}>-0.13$ are
almost entirely all dynamically excited objects. Of the 52 objects (57 unique
measurements) in that region, only 6 are CCKBOs, and only 3 by the
$i_{\textrm{free}}$ definition. A higher concentration of CCKBOs are FaintIR
objects ($PC^{2}_{\textrm{grJ}}<-0.13$). Of the 40 with
$PC^{2}_{\textrm{grJ}}<-0.13$ (excluding 2013 JK64, see below), 22 are
dynamically excited, and the rest are CCKBOs; $PC^{2}_{\textrm{grJ}}<-0.13$ is
where the majority of CCKBOs are found.

Intriguingly, the majority of objects with $PC^{2}_{\textrm{grJ}}<-0.13$ also
have $PC^{1}_{\textrm{grJ}}>0.4$. Only 6 objects have values
$PC^{2}_{\textrm{grJ}}<-0.13$ and $PC^{1}_{\textrm{grJ}}<0.4$. Five of those 6
objects have similar colours, with $(g-r)\sim 0.75$ and $(r-J)\sim 1.2$
($PC^{1}_{\textrm{grJ}}\sim 0.2$, $PC^{2}_{\textrm{grJ}}\sim-0.2$).
Examination of their colours reveals that those 6 objects are closer to the
objects with $PC^{2}_{\textrm{grJ}}>-0.13$, suggesting that they are a tail of
the BrightIR class rather than belonging to the FaintIR class with lower
$PC^{2}_{\textrm{grJ}}<-0.13$ and $PC^{1}_{\textrm{grJ}}>0.4$. This is true of
those 6 objects in both the colour and the projection spaces. All six are also
dynamically excited KBOS. We suggest that these 6 objects are members of the
BrightIR class, and that a two-dimensional boundary exists between the two
classes of KBOs: FaintIR has $PC^{1}_{\textrm{grJ}}\gtrsim 0.4$ and
$PC^{2}_{\textrm{grJ}}<-0.13$, and BrightIR objects fall outside this region.

Using the above two dimensional boundary between BrightIR and FaintIR objects,
of the 6 BrightIR CCKBOs, three objects with $i_{\textrm{free}}$ too inflated
to be considered CCKBOs by that definition are so-called _blue binaries_
(Fraser et al., 2017). These objects, identified by their less red optical
colours, and 100% binary rate, are theorized to be push-out survivors from
more interior regions of the Solar System – though some difficulties with this
interpretation have been pointed out (Nesvorny et al., 2022). We consider
these interesting objects in the Discussion Section.

Our results suggest that the dynamically excited KBOs occupy a different
optical-NIR colour space than do the CCKBOs. To test the statistical
significance of the apparent differences in distribution of
$PC^{2}_{\textrm{grJ}}$ values between the CCKBOs and the dynamically excited
KBOs, we turn to the non-parametric, 2-sample Anderson-Darling (AD) statistic,
which we calibrate with bootstrap sampling with repeated samples allowed
(Fraser & Brown, 2012). From this we find that the probability that the CCKBOs
and excited KBOs share the same $PC^{2}_{\textrm{grJ}}$ distribution is
essentially 0; in 10,000 evaluations of the bootstrapped AD statistic, not one
evaluation produced a value as extremal as the observed sample. That is, the
CCKBOs – by either definition – span an entirely different distribution of
$PC^{2}_{\textrm{grJ}}$ values than does the full excited sample. We highlight
that the application of the AD statistic directly to the (r-J) colour
distributions of the two dynamical samples results in a probability of 3%.
Thus, by projecting into PC space, the apparent bifurcation of the colour
distributions is maximally separate, and significantly more rectilinear.

The result that the majority of CCKBOs occupy a colour space of bluer NIR
colours than similarly optically coloured excited KBOs is similar to the
result found by Pike et al. (2017). From Col-OSSOS data, they found that many
CCKBOs have bluer (r-z) colours than their excited counterparts. We revisit
that result here, and present (g-r) vs. (r-z) along with the reddening curve
projection of those colours in Figure 4. The colours we present here use a
slightly enhanced reduction of the same data in Pike et al. (2017).
Unfortunately, a number of the colours reported in that paper were influenced
by poor PSF star selection – itself caused by the highly undersampled PSF in
much of the raw imagery – which resulted in poorly estimated aperture
corrections. We have corrected this issue during a reanalysis of the data.
Despite this error, the conclusions drawn in Pike et al. (2017) are not
influenced: many CCKBOs occupy a unique region of (g-r) and (r-z) colour
space. This is apparent from the distribution of $PC^{2}_{\textrm{grz}}$
values; all the dynamically excited KBOs for which we have (r-z) measurements
have $PC^{2}_{\textrm{grz}}>-0.13$, and 5 of 10 CCKBO measurements fall below
this value.

Figure 4: Left: The (g-r) and (r-z) colours of Col-OSSOS targets. Repeat
measurements for targets have been connected by dashed lines. Right: The
projection values $PC^{1}$ and $PC^{2}$ of the (g-r) and (r-z) colours shown
in the left panel. Red points are CCKBOs defined by their $i_{\textrm{free}}$,
yellow and red points together are CCKBOs by the historical definition based
on $i$, and black points are all those excited KBOs outside the CC region. The
horizontal line at $PC_{\textrm{2}}=-0.13$ marks the value below which all
FaintIR objects are found. These mainly consist of CCKBOs. The colour scheme
follows that of Figure 1.

Application of the AD statistic to the $PC^{2}_{\textrm{grz}}$ distribution
demonstrates that the probability that the CC and non-CCKBO samples are drawn
from the same parent distribution is only 0.05%. We draw the same conclusions
as Pike et al. (2017); like in the grJ space, the colours of CCKBOs are
differently distributed than their excited counterparts in the grz space as
well. Unlike in the grJ colour space, however, no statistically significant
bifurcation in $PC^{2}_{\textrm{grz}}$ is apparent. This is likely a sample
size effect as Col-OSSOS has only 26 objects (and two repeated) with (g-r) and
(r-z) colours. Similarly, the lack of dynamically excited objects with
$PC^{2}_{\textrm{grz}}<-0.13$ is probably just a sample size effect; from the
grJ sample, we might expect one such excited object with these colours when we
find zero.

We highlight that due to spectral behaviour differences between (r-z) and
(r-J), the value of $PC^{2}_{\textrm{grJ}}=-0.13$ that separates BrightIR from
FaintIR may not be the same for $PC^{2}_{\textrm{grz}}$. The small grz sample
however, precludes searching for variations.

In the grz colour space, no observed objects are found with intermediate
colours, (g-r) $\sim 0.8$. The DIP test, however, demonstrates that in
projected grz the distribution of objects with $PC^{2}_{\textrm{grz}}>-0.13$
has a probability of unimodality of 73%. Moreover, a number of objects are
found with those (g-r) colours in the larger grJ sample, implying that the
apparent lack of objects in that region is merely a result of the smaller grz
sample, and not a real feature of the colour distribution.

An intriguing result is found when one considers the (r-J) colours of those
objects in the grz sample. All objects with $PC^{1}_{\textrm{grz}}>0.75$ and
$PC^{2}_{\textrm{grz}}>-0.13$ have $PC^{2}_{\textrm{grJ}}<-0.13$. It seems
that just because an object is found to have $PC^{2}<-0.13$ in one NIR band,
does not necessarily mean that it will have $PC^{2}<-0.13$ in a different NIR
band. We will discuss this result within the context of a spectral model in
Section 5.

We highlight the 5:2 resonant object, 2013 JK64 (OSSOS designation o3o11).
This object exhibits significant variations in its (r-J) colour (see Figure
1). The $\sim 0.5$ magnitude variation in is so extreme as to result in
confident but different classification into BrightIR at one visit, and FaintIR
in the next. It is notable that JK64 does not exhibit intermediate colours
between the BrightIR and FaintIR classes, but instead the measured values fall
on either side of the $PC^{2}=-0.13$ value leading to an ambiguous
classification for this interesting target. This is a similar result as that
of CCKBO, 2013 UN15 (o3l63) which exhibits variable (r-J) and (r-z) colours,
resulting in ambiguous classification from the (r-z) colour alone. We consider
these targets along with KBOs that are known to be spectrally variable in the
Discussion Section.

Finally, we consider the (u-g) colours sample, which we present alongside
their projection in Figure 5. Unlike in the NIR bands, the single CCKBO in the
(u-g) sample does not stand out, but rather, falls in the same range of (u-g)
colours as the excited KBOs that possess similar (g-r) colours. This may be a
result of the low SNR (u-g) colours compared to the (r-z) and (r-J), though it
may be that the distribution of $PC^{2}$ is not bimodal in the NUV.

For objects with (g-r)$<0.75$, the range of (u-g) colours is compressed, with
$1.25\lesssim(u-g)\lesssim 1.8$ compared to the redder objects with
$1.2\lesssim(u-g)\lesssim 2.5$. This is most likely a result of the lower
optical albedos of the less red KBOs (Lacerda et al., 2014). That is, because
the albedos are lower, there simply is not as large a range of physically
allowable (larger than zero) u-band albedos, while still having red (u-g)
colours. This will be the topic of a future paper in which a compositional
interpretation of the Col-OSSOS sample is presented. In summary, in the NUV-
optical space, we find no evidence that the CCKBOs standout with a different
colour distribution than the non-CCKBOs possessing similar optical colours.

Figure 5: Left: The (u-g) and (g-r) colours of Col-OSSOS targets. Right: The
projection values $PC^{1}$ and $PC^{2}$ of the (g-r) and (u-g) colours shown
in the left panel. Cold classical CCKBOs are marked in red. In these colours,
CCKBOs cannot be differentiated from excited KBOs, to the SNR limits of the
dataset. The colour scheme follows that of Figure 1; red points are CCKBOs
defined by $i_{\textrm{free}}$, and black are KBOs outside the CC region.

##  4 Published Colour Datasets

In the following section, we consider other optical and NIR colour datasets.
In particular, we analyze how the KBO colour distributions in other filters
appears in the reddening curve projection space.

###  4.1 H/WTSOSS

We turn our attention to the observations of the Hubble/Wide Field Camera 3
Test of Surfaces of the Outer Solar System (H/WTSOSS; Fraser & Brown, 2012).
Like Col-OSSOS, H/WTSOSS was an optical-NIR photometric survey of a large
sample of KBOs, but of a target sample with unknown discovery and tracking
biases and no limiting magnitude. Unlike in Fraser & Brown (2012) we do not
limit ourselves to only those data with colour uncertainty less than 0.1
magnitudes in (F606w-F814w) and (F814w-F139m), but consider the entire dataset
in our projection analysis.

We present a projection into PC colour space of the (F606w-F814w) and
(F814w-F139m) colours from the H/WTSOSS sample independent of the Col-OSSOS
(g-r) and (r-J) in Figure 6. This sample reveals a similar bifurcation in
$PC^{2}$ and at a similar value, $PC^{2}=-0.13$, as in the Col-OSSOS
sample333For ease of reading, we omit the filter label subscripts when
discussing the projection of the H/WTSOSS data.. When considering only those
objects with uncertainty of less than 0.1 magnitudes in each colour, we find
an 8% chance of a bifurcation in $PC^{2}$, as evaluated with the DIP test. We
point out that this bifurcation is the same as the diagonal bifurcation in
optical-NIR that has already been found in the H/WTSOSS sample, albeit
revealed with a much simpler analysis than the more complicated multi-
dimensional clustering methods previously used (Fraser & Brown, 2012).

Figure 6: Left: The (F606w-F814w) and (F814w-F139m) colours of Kuiper Belt
Objects and Centaurs reported in (Fraser & Brown, 2012). The reddening curve
which is shown by the solid line. Colours are in the STMAG system. The left
and bottom panels show histograms of the colours. Right: The projection values
$PC^{1}$ and $PC^{2}$ of the (F606w-F814w) and (F814w-F139m) colours shown in
the left panel. CCKBOs are marked in red. The left and bottom panels show
histograms of $PC^{1}$ and $PC^{2}$. The horizontal line at $PC^{2}=-0.13$
marks the value below which all FaintIR objects are found. These mainly
consist of CCKBOs. The colour scheme follows that of Figure 1; red points are
CCKBOs defined by $i_{\textrm{free}}$, yellow and red points are those based
on $i$, and black are KBOs outside the CC region. The _blue binary_ 2003 HG57
is outlined with a cyan halo. All histograms show only objects that have
uncertainties less than 0.1 magnitudes in both colours to avoid inclusion of
uncertain measurements that may mask the presence of the bifurcation in
$PC^{2}$.

Dynamically, the H/WTSOSS sample shows a very similar result as that of the
Col-OSSOS sample. That is, objects with $PC^{2}>-0.13$ in the H/WTSOSS filters
are predominantly dynamically excited objects: only 3 of the 18 H/WTSOSS
CCKBOs have $PC^{2}>-0.13$. Further, like the Col-OSSOS objects, the H/WTSOSS
objects with $PC^{2}>-0.13$ span the full optical or $PC^{1}$ colour range.

In general, the H/WTSOSS and Col-OSSOS datasets present the same result: in
the optical-NIR colour space there is a bifurcation in $PC^{2}$ above and
below a value of $\sim-0.13$, and CCKBOs are predominantly found with
$PC^{2}<-0.13$, while most excited KBOs have $PC^{2}>-0.13$. That the same
systematic colour-dynamical divisions occur in both datasets is a
demonstration that the apparent structures in projected colours are a robust
feature of the KBO population, and not a consequence of filter or target
selection.

We note that no bifurcation in $PC^{2}$ was detected in the (F606w-F814w) and
(F139m-F153m) colour space. That is to say, the two groups of objects found
above and below $PC^{2}=-0.13$ fully overlap in the (F139m-F153m) colour.
F153m samples the centre of the $1.5\mbox{ $\mu$m}$ water-ice absorption band.
The lack of bifurcation implies that the BrightIR and FaintIR objects share
similar water-ice absorption depths, and differ only in the behaviour of their
optical-gap absorptions which dominate at shorter wavelengths.

###  4.2 Optical Spectrophotometric Datasets

The results we have presented thus far have demonstrated that in the NIR, the
CCKBOs occupy a different region of colour space than do most dynamically
excited KBOs. This is apparent at wavelengths, $\lambda\geq 0.9\mbox{
$\mu$m}$. We wish to test if such a bifurcation is present at shorter bands.
To that end, we make use of the Peixinho et al. (2015) optical colours
dataset. This is a large and reliable multidimensional dataset of optical
colours derived from Johnson BVRI photometry that was extracted from prior
published optical colours. Importantly, they reject all colour measurements
spanning timespans larger than 1.5 hours, in an attempt to minimize the
influence of lightcurve effects on the reported colours. The main weakness of
this colours sample is the lack of simultaneity in more than two bands. It is
common that reported colours were gathered at different epochs. It should be
noted however, that only a small fraction of KBOs have exhibited optical
spectral variability (see Fraser et al., 2015, and Table 2), and as such, the
Peixinho dataset is currently the best available for our purposes.

In our analysis of the Peixinho dataset, we projected 4 different combinations
of colours (B-V), (B-R), (V-R), (V-I), and (R-I). The most populous subset of
that dataset is (V-R) and (R-I), the projection of which is presented in
Figure 3. We also considered subsets with cuts in uncertainty (0.05 mags, 0.1
mags, etc.) and cuts in absolute magnitude. In all cases, only hints of
bifurcation in $PC^{2}$ were found. This is most prominent in the colours
involving I-band. No statistically significant bifurcation was found, however.
The distribution of $PC^{2}$ appears to be unimodal when considering only
optical colours. This is true regardless of dynamical population cuts. While
the CCKBOs all possess red optical colours, and therefore have high $PC^{1}$
values, the do not differentiate from the excited objects in $PC^{2}$. It
seems plausible that the FaintIR and BrightIR classes do differentiate in
colours involving the I-band, though the two classes appear to be closer
together in $PC^{2}$ compared to longer wavelengths, and so this signal is not
detectable in the available colour datasets. Combining these results with
those from Col-OSSOS, we conclude that the BrightIR and FaintIR classes as
defined above, predominantly overlap in the ugr/BVRI colours space. That is,
both classes can be found tracking the reddening curve, and exhibit nearly
linear spectra through the optical range. Any spectral differences between the
two classes only become readily apparent at wavelengths $\lambda\gtrsim
0.9\mbox{ $\mu$m}$.

##  5 Spectral Interpretation

In this section, we consider the properties of KBO spectra that could account
for the structures we have detected in the UV-optical-NIR colour
distributions. Generally, there are five properties of the UV-optical-NIR
colour space that we are interested in understanding:

  1. 1.

The tendency to follow the reddening curve in the ugr/BVRI filter range

  2. 2.

The bifurcation in the optical colour distributions (eg. (g-r) (B-R), etc.)

  3. 3.

Deviations away from the reddening curve to less-red colours at NIR
wavelengths ($\lambda\gtrsim 0.8\mbox{ $\mu$m}$)

  4. 4.

The separation of two groups in an optical-NIR colour space (e.g., grJ, grz,
or the filters utilized by H/WTSOSS), separated at a $PC^{2}\sim-0.13$

  5. 5.

The fact that not all objects consistently fall above or below
$PC^{2}\sim-0.13$ across all of the band passes considered.

In the most general sense, the spectra of KBOs can be described by red, linear
optical spectra, that turnover to nearly linear NIR spectra with less-red
slopes (Cruikshank et al., 1998; Barucci et al., 2011; Gourgeot et al., 2015;
Grundy et al., 2020; Seccull et al., 2021). Observed exceptions to this rule
include wavelength regions influenced by water-ice (Barkume et al., 2008),
objects 2003 AZ84 and 2004 EW95 which appear to have spectra closer to C-type
asteroids than KBOs (Fornasier et al., 2004; Seccull et al., 2018), and the
spectra of the largest KBOs, which exhibit spectra dominated by select ices
that are too volatile to survive on the surfaces of smaller KBOs (Brown et
al., 2011, 2012). To that end, we adopt a simple spectral model with which to
interpret the colour distributions we present above. A model spectrum is
defined by a linear optical spectrum and a linear NIR spectrum, with the two
spectra meeting at some transition wavelength. Free parameters are the optical
spectral slope, $s_{opt}$, the NIR spectral slope, $s_{ir}$, and a transition
wavelength, $w$.

The distribution of (g-r) and (r-J) colours of Col-OSSOS targets shows a lot
of similarity to the distribution of colours of the H/WTSOSS dataset. That is,
both show evidence for two separate groups of objects occupying unique ranges
of colour. Those classes also show a continuum of colours, with correlated
optical and NIR colours. Inspired by the similarity between the two datasets,
we turn to the mixing model put forth by Fraser & Brown (2012) to model the
colours and correlated optical-NIR colours of H/WTSOSS targets. In that model,
the members of a compositional class are made up of a mixture of two main
material components. The colours of a class then fall along a continuum of
colour with the continuum’s end-points matching the colours of the materials,
with the specific colours of an object determined by the mixture of those two
materials. Fraser & Brown (2012) found that within that framework, the colours
of the two materials that made up a class were consistent with one material
being mainly silicate in nature (on the blue end of the class’s continuum) and
the other component being a water-rich organic material (on the red end of the
continuum).

Using the mixture model approach, we model the two colour groups detected in
the Col-OSSOS grJ colour space with two separate classes as seen in Figure 7.
These two mixture model curves have been chosen in a similar fashion to those
proposed in Fraser & Brown (2012): both classes share the neutral-coloured
material component, with each red material chosen to describe _only_ the
features of the Col-OSSOS grJ distribution. These two mixture model curves are
presented in Figure 7. We emphasize that these mixture models are not a fit to
observed colours, but merely are a representative model. Though we do point
out that with only 6 parameters (two colours, (g-r) and (r-J), for each
material component), 72 of the 110 observed (g-r) and (r-J) colours are
$1-\sigma$ consistent with the chosen mixing models, and 91 of 110 are
$2-\sigma$ consistent. We deem the chosen model as a reasonable representation
of the grJ colour distribution.

Figure 7:  The Col-OSSOS dataset plotted with the mixing model representation
(red dashed curves in the top panel). Simulated spectra were those randomly
drawn that were consistent with the (g-r) and (r-J) colours of the mixing
models. Other colours were evaluated from those spectra. The resultant colours
for mixing model classes 1 and 2 are shown by the orange and blue points
respectively.

From the two mixture models, syntehtic spectra were sampled. This was done by
randomly drawing model spectral parameters $(s_{opt},s_{ir},w)$, and
calculating their (g-r) and (r-J) colours, collecting those that are within
$0.04$444This value is chosen to approximate the slight apparent deviations
away from non-linearity exhibited by KBOs in the optical, but is relatively
arbitrary, as a range of similar values produce equally satisfactory results.
magnitudes of the mixing model curves. The only restriction was made on the
NIR spectral slope, requiring $s_{ir}\geq 0$, as few to no KBOs exhibit
negative spectral slopes in the grzJ spectral range (e.g. Barucci et al.,
2011). In this way, we create representative model spectra that broadly
reflect the known spectral behaviours of KBOs, and match the (g-r) and (r-J)
colours of both surface classes in the Col-OSSOS dataset. The sample of
synthetic KBO colours is shown in Figure 7.

As the above modelling approach produces simple spectra, we can also draw
colours using other filters, such as z, or VRI. The interesting result of the
above modelling is apparent when we consider the predicted (r-z) colours of
the model spectra. Specifically, we see that the BrightIR objects – the
objects with redder (r-J) colours – also have redder (r-z) colours with a
spread in (r-z) that falls along a relatively narrow strip. This is a result
of the fact that to be consistent with the grJ colours of the redder class
requires that $s_{ir}$ be only marginally bluer than $s_{opt}$. FaintIR
objects – the class with bluer (r-J) colours – exhibit a larger spread of
colours in (r-z). This is because of the degeneracy between spectral
parameters $s_{ir}$ and $w$ which can result in a specific (r-J) colour from a
large spread of slope and transition wavelength values, thus resulting in a
broad range in (r-z) for a single (r-J) value. Moreover, the (r-z) colours of
this class tend to pile-up at values $(r-z)\sim 0.55$. This occurs when the
transition wavelength, $w$, is shorter than the blue edge of the z filter,
$w\lesssim 0.83\mbox{ $\mu$m}$. The BrightIR class however, has (r-z) values
that pile-up near the reddening line. This has the effect of creating a
bimodal (r-z) colour distribution for optically red $(g-r)\gtrsim 0.85$ KBOs
(see Figure 8). This is consistent with the observed (r-z) colour
distribution, which also appears bimodal for red KBOs (Pike et al., 2017), and
implies that the spectra of many CCKBOs have a transition wavelength,
$w<0.83\mbox{ $\mu$m}$ (Seccull et al., 2021).

Figure 8:  Histograms of (r-z) and (g-r) colour samples from our mixing model.
Orange and blue points represent FaintIR and BrightIR objects, respectively,
with the black curve showing the overall sample histograms. The histograms
were weighted to have an equal number of objects in each class, which is
approximately true of the observed sample. The absolute values in this figure
are meaningless. The (r-z) histogram is only of optically red KBOs with
(g-r)$>0.85$ which show a bimodality in (r-z) that is apparent in both the
simulations and in the observations. Neither the simulated or observed (r-z)
distribution is bimodal for objects with (g-r)$<0.85)$.

In the framework of only two mixing models, the optical colour distributions
of both compositional classes almost fully overlap one another. To account for
the bimodal optical colour distribution therefore requires a non-uniform
distribution of objects along each of the mixing model curves. That is,
uniform sampling in the spectral model parameters, or uniform sampling along
the model curves themselves results in a unimodal optical colour distribution.
From the Col-OSSOS grzJ colour distribution (or the H/WTSOSS sample) however,
it is clear that each class is not spread uniformly in colour space. Rather,
BrightIR objects appear to cluster at $(g-r)\sim 0.63$, or spectral slope
values $s_{opt}=13\%/100\mbox{ $n$m}$, while FaintIR objects cluster at redder
optical colours, $(g-r)\sim 0.86$ or $s_{opt}=30\%/100\mbox{ $n$m}$. Inspired
by this observation we adopt a trivial modelling approach. Specifically, when
sampling synthetic KBO spectra parameters, we randomly sample $s_{opt}$ from
two Gaussian curves, one for each class. We use median values of $s_{opt}=30$
and $13\%/100\mbox{ $n$m}$, with standard deviations of $4$ and $6\%/100\mbox{
$n$m}$, for the FaintIR and BrightIR classes, respectively. The BrightIR class
requires a wider width to reflect the broader range of optical colours those
objects occupy. $s_{ir}$ and $w$ were sampled uniformly. As seen in Figure 8,
when sampled in this way, the model optical colour distribution (e.g. $(g-r)$,
$(V-R)$, etc. ) is bimodal. Notably, by virtue of the relatively wide widths
of the Gaussians from which $s$ were sampled, the resultant model colours
still span the full optical-NIR range of both the BrightIR and FaintIR
classes.

We highlight that the double Gaussian sampling of $s$ is required only to
broadly describe the distribution of sources along each mixing line, and hence
the bimodal optical colour distribution. The bimodal (r-z) distribution for
optically red KBOs is not a result of the Gaussian sampling, but rather, is
entirely driven by the location of the wavelengths of the z-band filter, and
the overlap between the two mixing model curves. A bimodal distribution in
(r-z) does result even when fictitious colours are sampled uniformly along the
mixing model curves.

To summarize, the four main model assumptions are:

  * •

Two classes of KBO with the colours of each described by geometric mixing
models, with the neutral-coloured end-member material being shared between
both classes

  * •

Simple linear optical and NIR spectra with different slopes, and a transition
wavelength

  * •

NIR spectra do not have negative spectral slopes

  * •

Sources are found spread across a mixing colour curve, but tend towards an
optical spectral slope specific to each class.

This model reproduces nearly all features of the distribution of optical-NIR
colours of KBOs, though it is not without weakness. Specifically, the model
breaks down at the shortest and longest wavelengths. That is, the model
produces spectra that are too red in the (u-g) colour (Figure 7). This argues
for deviations away from linearity at wavelengths $\lambda\lesssim 0.4\mbox{
$\mu$m}$, or the approximate blue edges of the B and g-bands. Such deviations
have been detected in some spectra, such as the spectrum of Pholus which shows
a slightly convex spectrum that is bluer at $\lambda\sim 0.4\mbox{ $\mu$m}$
than at longer optical wavelengths (Cruikshank et al., 1998). Further weakness
is shown for passbands beyond J. In particular, our model results in objects
from the (r-J) red model compositional class that are too red in the Hubble
passbands (F814w-F139m). To rectify this while preserving the (r-z)
distribution requires the reddest NIR spectra to turnover to nearly neutral
slopes at wavelengths beyond the J-band, $\lambda\gtrsim 1.2\mbox{ $\mu$m}$.
An example of such a turnover can be seen in the spectrum of Arrokoth (Grundy
et al., 2020).

We emphasize that the model we present here is not an attempt to reflect some
of the quantitative properties of the observed or intrinsic colour
distributions. For example, the model makes no robust attempt to match the
distribution of colours along each mixing model curve, but merely adopts
simple Gaussian distributions in optical spectral slope. Moreover, no attempt
has been made to match the observed balance of objects between the two
compositional classes. Other potential improvements include a more realistic
spectral model that includes features such as an optical-NIR transition region
and not just a single wavelength, consideration for water-ice absorption, and
adjustments to account for a broader wavelength range where the assumption of
linear NIR spectra breaks down. These and other considerations will be the
topic of a future manuscript.

##  6 Discussion

We have presented a new interpretation of KBO colours, in which two
populations, which we label as BrightIR and FaintIR occupy different optical-
NIR colour regions. Specifically, both classes span the full optical colour
range exhibited by KBOs, but are separate in NIR colours: FaintIR is found at
moderate NIR colours while BrightIR has redder NIR colours just shy of the
reddening curve. From the data spanning the NUV-optical-NIR range, a clean
separation of the BrightIR and FaintIR classes is not apparent in the ugr, or
BVRI band-passses, as both classes span an overlapping range in those colours.
The separation is only apparent for longer passes, z, J, F139m etc. This is
supported by the results of the DIP test of unimodality. In the 8 different
colour pairs we used in our search (3 from Col-OSSOS, 1 from H/WTSOSS, and 4
from the Peixinho dataset), only the projection of the colour pairs: (g-r) and
(r-J); and the (F606w-F814w) and (F814w-F139m) have low probabilities of
unimodality, at 0.0015% and 8% respectively. To test the overall significance
of this result, we performed a numerical experiment whereby we simulated 8
separate uniform colour distributions of length matching those of the real
datasets, and asked if across those 8 samples, if any set had DIP-test
probabilities less than each of the observed values. That is, one sample of
the 8 with probability less than $0.0015\%$, and one with less than $8\%$. In
3,000 iterations of this experiment, only 26 met this threshold. Therefore, we
conclude that after accounting for the fact that bifurcations were searched
for in multiple datasets, the detected bifurcation remains statistically
significant.

We do acknowledge that our analysis has an air of probability hacking. That
is, the bifurcations in each of the colour sets we analyzed were only
significant once the maximum measurement uncertainty of each sample was
lowered below a specific value. We feel justified in this approach however, as
the separations between the two groups detected in each sample were of order
the uncertainty threshold below which the statistics become significant. Put
another way, when the data were typically more uncertain than the apparent
separation of the two underlying groups, the presence of the FaintIR and
BrightIR classes was masked. We emphasize that the presence of two groups was
detected in three entirely unrelated colour datasets, and across many
different unrelated optical and NIR colours. Despite this weakness in our
analysis, the detection of the BrightIR and FaintIR classes is statistically
significant, and robust.

The work we present here is not the first time a bifurcation of KBOs into two
populations has been published. It has long been apparent that the optical
colour distribution of small ($H\gtrsim 6$) KBOs is bifurcated (see Fraser &
Brown, 2012; Peixinho et al., 2012, 2015; Tegler et al., 2016; Marsset et al.,
2019, for recent examples). This can also be seen in the (g-r) colours of Col-
OSSOS (see Figure 1). The bifurcation in the optical colour distribution has
been interpreted as evidence for the presence of two distinct populations of
KBOs, separated with optical spectral slopes smaller and larger than $s\sim
20\%/100\mbox{ $n$m}$. These populations are now colloquially referred to as
_Less Red_ (LR) and _Very Red_ (VR). Tegler et al. (2016) pointed out that the
LR and VR colour distribution tails must slightly overlap, hence why published
significance of bifurcation values rarely exceeds 2-$\sigma$, even for colours
samples counting in the hundreds. The H/WTSOSS sample, and the data presented
in Schwamb et al. (2019) demonstrated that a cleaner bifurcation is apparent
in optical-NIR colour space. Generally, past works have concluded that the LR
population spans a spectral slope range $0\lesssim s\lesssim 25\%/100\mbox{
$n$m}$, and $20\lesssim s\lesssim 50\%/100\mbox{ $n$m}$ for the VR population.

We and other teams have argued in the past for three spectro-dynamical
classes. That is, alongside the LR, and VR KBOs, the CCKBOs make a third class
(Fraser & Brown, 2012; Pike et al., 2017). Here, we argue that this
interpretation is incorrect. Rather, we have found evidence for only two
spectral classes that significantly overlap in optical colours. The bimodality
in optical colour of excited KBOs and Centaurs is a result of the distribution
of each spectral class along their mixing lines, with a preference for one
class to be found at more neutral optical colours than the other. Though we
emphasize that both classes have members with optical colours $0.7<(g-r)<1.0$.
As the BrightIR and FaintIR class have an overlap that spans much of the
optical colour range of KBOs, a statistically robust bifurcation will never be
found with optical colours alone. This subtle transformation of the modern
interpretation of the KBO colours distribution has cosmogonic implications
that are less than subtle, and will be discussed below.

We find no evidence of any further bifurcation beyond the two classes. This is
true of the Col-OSSOS, H/WTSOSS, and Peixinho samples, and subsets thereof. At
most, only two groups are apparent in all of the colour datasets we have
considered, which span the UV-optical-NIR range. And notably, the bulk
properties of all observed colours across all colours datasets can be modelled
with only two classes.

The classification scheme we present here is a two class system, and is
different than the three (or more) taxon systems previously put forth. In most
cases, the CCKBOs were assigned their own taxon. This was motivated by their
distinct physical properties including binarity (Noll et al., 2008), size
distribution (Fraser et al., 2014), high albedos (Brucker et al., 2009), and
unique colours (Pike et al., 2017) compared to the excited KBOs. The optical
bimodality of the excited sample was then interpreted as evidence for at least
two additional taxons. We have shown here that once the true shape of the
FaintIR class is revealed, no additional bifurcation in the optical-NIR colour
distribution of the excited KBOs is apparent. Rather, we find evidence for
only two main taxa.

Our assertion of the presence of only two classes for the bulk of small KBOs
comes with two predictions about KBOs that must hold true:

  * •

The albedos of VR KBOs that belong to the BrightIR class will be as equally
low as the LR objects

  * •

The size distribution of excited KBOs from the FaintIR class will match the
size distribution of red cold classical KBOs, and inversely, BrightIR objects
in the cold classical range will have a size distribution that matches that of
dynamically excited BrightIR objects.

The former prediction arises from the observation that LR objects (less red
optical colours, $s\lesssim 20\%/100\mbox{ $n$m}$) typically have lower
albedos (Lacerda et al., 2014; Fraser & Brown, 2012). It seems that CCKBOs
have systematically higher albedos than at least the LR KBOs, with most having
albedos $\rho\gtrsim 0.1$ (Brucker et al., 2009; Lacerda et al., 2014).
Considering our two group classification then, it follows that all FaintIR
objects will possess equally high albedos. VR objects (red optical spectral
slopes, $s\gtrsim 20\%/100\mbox{ $n$m}$) with high albedos can be found in all
excited classes, consistent with this idea. Similarly, the LR objects have low
albedos $\rho\lesssim 0.1$, with only a few exceptions. It follows that all
BrightIR KBOs will have lower albedos. The observed albedos are compatible
with this prediction, as optically red KBOs with low-albedos have been
detected in the excited KBO populations. Alternatively, it is possible that
optically red BrightIR bodies may have albedos like those of FaintIR objects,
though this would imply that BrightIR objects have a large range of albedos
alongside the large range of optical colours.

The second prediction arises from the observation that the low-inclination
component of the Kuiper Belt possesses a different size distribution than the
excited bodies (Bernstein et al., 2004; Fraser & Kavelaars, 2009; Fuentes et
al., 2009; Petit et al., 2011; Fraser et al., 2014). While the low and high-
inclination components are mixtures of bodies from BrightIR and FaintIR
objects, the CCKBOs (or the low-i component) are predominantly FaintIR
members. This implies that the steep size distribution of the low-i component
is a reflection of the size distribution of FaintIR bodies, and that BrightIR
objects have a shallower size distribution for bodies larger than the break
diameter $D\gtrsim 100$ km. The prediction then is that FaintIR objects found
on excited orbits will have a steep size distribution like that of the CCKBOs.
The contrary possibility, that of differing size distributions for different
dynamically separated classes of FaintIR object, would be very hard to rectify
in our current understanding of the formation of the Kuiper Belt, and would
challenge the two group classification scheme we advocate for here.

###  6.1 Spectral Shapes of BrightIR and FaintIR Objects

The classification scheme we present here results in two classes with distinct
colour distributions, and therefore distinct spectral shapes. These
differences can be summarized as lower NIR reflectance for FaintIR objects
compared to BrightIR objects with the same optical colours. Though we
highlight that there are BrightIR objects with equally low NIR reflectance as
FaintIR bodies, but only with bluer optical colours. This is visually
demonstrated in Figure 9, where we plot relative reflectances of Col-OSSOS
targets. These reflectances are derived from their observed mean ugrzJ
photometry. It can be seen that FaintIR objects typically have lower relative
reflectances for wavelengths $\lambda\gtrsim 0.9\mbox{ $\mu$m}$ compared to
BrightIR objects with $PC^{1}_{\textrm{grJ}}>0.4$. The BrightIR objects with
$PC^{1}_{\textrm{grJ}}<0.4$ have NIR reflectances fully overlap that of the
FaintIR objects.

Figure 9: The range of reflectivities for Col-OSSOS targets for BrightIR and
FaintIR objects. The BrightIR objects have been divided into those objects
with $PC^{1}$ less than and greater than $0.4$ for presentation purposes only;
no evidence for any further subdivisions beyond BrightIR and FaintIR has been
detected. Relative reflectance values were derived from the observed ugrzJ
colours and adopting Solar colours (u-g)=1.43, (g-r)=0.44, (r-z)=0.18, and
(r-J)=0.98 (see Appendix C). Spectra are normalized to unity in the r-band.
Uncertainties in the observed colours were not considered when generating the
range of spectra, only the nominal measured values.

Pike et al. (2017) speculated that the differences between BrightIR and
FaintIR objects were due to an absorption feature that influences the z-band
reflectance of CCKBOs that is not present to the same extent on excited KBOs.
Our finding that the bifurcation is apparent at even longer wavelengths
disfavours the absorption band hypothesis, as the purported absorption band
must span $0.9\lesssim\lambda\lesssim 1.4\mbox{ $\mu$m}$ so as to account for
the separation of the two classes at these wavelengths. Such an absorption is
extremely wide compared to most solid-state absorptions, and thus, this
hypothesis seems unlikely.

The high quality spectra of CCKBOs reported by Seccull et al. (2021) support
our interpretation. That is, they find no evidence for a specific absorption
feature influencing the $\gtrsim 0.9\mbox{ $\mu$m}$ region. The same is true
of the reflectance spectrum of 2014 MU69 Arrokoth. This small CCKBO (mean
diameter $\sim 40~{}km$) has colours that reveal it to be a FaintIR member,
and does not exhibit any notable absorptions in the
$0.9\lesssim\lambda\lesssim 1.5\mbox{ $\mu$m}$ range (Grundy et al., 2020).
Rather, like most small and excited KBOs, CCKBOs exhibit spectra that can be
characterized as near-linear in the optical, that roll over to shallower
slopes in the NIR. It happens that FaintIR objects, which dominate the CCKBOs,
exhibit a roll-over at shorter wavelengths than their excited counterparts
(e.g., Barucci et al., 2011). Like the excited KBOs, the full spectral
behaviour of CCKBOs from $0.3\lesssim\lambda\lesssim 1.4\mbox{ $\mu$m}$
appears to be dominated almost solely by the optical-gap feature of organic
rich materials.

It seems that the apparent spectral differences between the BrightIR and
FaintIR classes are driven by differences in the overall shapes of their
optical-gap absorptions. The optical-gap is a broad feature that is itself a
sum of overlapping absorptions driven by the varied organic materials present
on the surfaces of a body. It follows then that organic materials present on
KBOs differ between the two classes. This is likely caused by many factors,
including but not limited to the budget of non-organic contaminants within the
molecular structure of the organic materials (e.g., Izawa et al., 2014), the
duration of short-lived volatiles on their surfaces (Brown et al., 2011), and
the amount of irradiation driven de-hydrogenation that each class has
experienced (Brunetto et al., 2006). Specific compositional interpretations
will require observations at wavelengths $0.1\lesssim\lambda\lesssim 0.4\mbox{
$\mu$m}$, and at $\lambda\gtrsim 3\mbox{ $\mu$m}$ where many organic materials
exhibit diagnostic absorption features.

###  6.2 The Orbital Distributions and Origins of FaintIR and BrightIR Objects

We now turn our attention to the orbital element distributions of the FaintIR
and BrightIR objects. In Figure 10 we present the orbital elements of both
classes. No significant difference is detected in the eccentricity or
inclination distributions of FaintIR and BrightIR CCKBOs. This is also true of
the eccentricity distribution of the excited objects in both classes.
Interestingly, the AD test suggests that the probability that the semi-major
axis distributions of the BrightIR and FaintIR CCKBOs share the same parent
distribution, is only 5%. This is true of both the H/WTSOSS and Col-OSSOS
datasets. We caution though that the former has unknown observational biases,
and the latter remains incomplete. Therefore, this result while intriguing,
should be considered tentative at best.

Figure 10: Free inclination, inclination, and eccentricity vs. semi-major axes
of the objects from Col-OSSOS and H/WTSOSS. Those two surveys are the only two
from which reliable classification into BrightIR and FaintIR can be made. The
left column shows the cold classical region, while the right column shows the
broader extended Kuiper belt.

The only immediately apparent appreciable difference in orbital element
distributions rests within the inclination distributions of the excited
objects. Specifically, the highest inclination objects are predominantly found
in the BrightIR class. Marsset et al. (2019) demonstrated that the inclination
distributions of the VR and LR KBO populations were statistically different
throughout the observed excited TNO region. That is, a distinct decrease in
density of VR objects occurs for inclinations greater than $i\sim 21^{\circ}$.
We find a similar result here, but rephrase that result in terms of FaintIR
and BrightIR. Only 1 of the 12 objects with $i\geq 21^{\circ}$ are FaintIR
members. This is a similar ratio to the ratio of VR to LR objects in the
Marsset sample: of the 44 objects that can be reliably classed as either VR or
LR, only 5 are VR. While many of the VR objects are also FaintIR members, the
correspondence is not 1-to-1. In the Col-OSSOS (g-r) and (r-J) sample, 9 of
the 44 VR objects ((g-r)$>0.78$) actually belong to the BrightIR class. We
draw the reader’s attention to Marsset et al. (in prep; to be updated at
redline stage) where the colour-inclination correlation is revisited in the
Bright/FaintIR taxonomy we present here. We also point out Pike et al. (in
prep; to be updated at redline stage) who present an analysis of the
distribution of BrightIR and FaintIR bodies in the Col-OSSOS resonant
populations.

The preponderance of FaintIR objects found in the cold classical region (by
either definition), and their reduced occurrence at high inclinations argues
for different dynamical origins of these two populations. It is generally
accepted that the origins of the chemo-dynamical classes of object are
primordial, being the result of compositional differences in the
protoplanetesimal disk from which KBOs originated. This interpretation is
favoured over the possibility of the different classes forming as a result of
some evolutionary process that occurred after disk dispersal and emplacement
in the Kuiper Belt. The evidence against the latter idea is numerous, but
rests mainly on the fact that both BrightIR and FaintIR objects are found in
various ratios across all dynamical classes and regions of the Kuiper Belt.
Thus, it is hard to construct an evolutionary mechanism that selectively
alters the colours of only some KBOs, that otherwise have experienced the same
dynamical, thermal, and collisional histories as dynamically nearby bodies.
This selective mechanism must also preserve the nearly identical colours of
components in binary systems (Benecchi et al., 2009). Rather, it seems most
likely that objects in the protoplanetesimal belt earned their spectral types
before dispersal, probably as a result of processes that alter the
compositional makeups of planetesimal bodies according to their locales (e.g.
Brown et al., 2011).

The above reasoning has led to recent efforts to use the observed distribution
of compositional classes to infer properties of the protoplanetesimal disk,
given a dynamical model (see Gladman & Volk, 2021, for a review). From the
properties of a subset of BrightIR bodies found in the cold classical region –
the so-called _blue binaries_ , Fraser et al. (2017) inferred that some
BrightIR objects must have originated only a handful of AU interior to the
current cold classical region where the majority of FaintIR objects are found
(also see Fraser et al., 2021). From a simple exponential disk model, Schwamb
et al. (2019) used the ratio of LR and VR excited bodies to infer that the LR
bodies formed approximately between $\sim 30$ and $\sim 39$ AU, though this
was made under the assumption that KBOs consist of three separate classes.
From this work we can now appreciate that many of the red dynamically excited
KBOs are in fact just the reddest members of the FaintIR class, and therefore
must have formed alongside the rest of the FaintIR bodies.

To date the most comprehensive analysis of KBO formation locations is the work
of Nesvorný et al. (2020). They consider only two classes of object, the LR
and VR objects, with CCKBOs being considered members of the latter. From the
ratios of LR to VR objects found in various orbital classes, they conclude
that the LR objects must have formed interior 40 AU. Such a formation location
for the LR objects naturally explains why that class has higher inclinations
than the VR class, as the latter, by virtue of their formation locations,
experienced weaker excitation during dispersal.

Considering the results we present here, one cannot simply apply a cut in
optical colour alone to properly divide a sample of KBOs into their intrinsic
classes. From the Col-OSSOS grJ sample, 48 or, 52% of the 91 objects with
unambiguous BrightIR/FaintIR classification would belong to the LR class
((g-r)$<0.78$). These are the KBOs that are considered to have formed in
regions closer than they currently are. Of the 91 objects, 58, or 63% belong
to the BrightIR class, suggesting that many more objects formed closer to the
Sun than would be implied by a simple cut in optical colour. The difference is
even more stark when considering only the 64 excited bodies; 44 would be
classed as LR but 51 belong to the BrightIR class. Therefore, when using only
optical colour alone to classify objects, one underestimates the fraction of
excited objects that formed at closer regions by $\sim 15\%$. While this
correction may seem small, the observed fraction of excited objects that
belong to the BrightIR class is $0.79\pm 0.06$, which is $2-\sigma$ discrepant
from the equivalent value inferred when considering the LR/VR classification.

We highlight that a few objects with FaintIR colour measurements are found on
highly excited orbits. Namely, 2004 PB112 and 2013 JK64 belong to the 27:4 and
5:2 mean-motion resonance respectively, and 2014 UA225 is found on a detached
orbit. Intriguingly, these three objects have relatively low inclinations,
$i\leq 15^{\circ}$ (UA225 has $i=3.57^{\circ}$). It seems likely that the
dynamical pathways each of these objects experienced to place them on their
current orbits will provide significant constraint on the formation locations
of FaintIR bodies, and of the migration history of Neptune.

We also see a few BrightIR objects amongst the CCKBOs. Many of these bodies
are found near some of the prominent mean-motion resonances. Huang et al.
(2022) demonstrated that objects in mean-motion resonances can experience
large inclination variations. We speculate that some of these BrightIR CCKBOs
were originally loosely captured resonant bodies that experienced a downward
diffusion to low-i before being dropped from the resonance.

In the scenario discussed above, in which BrightIR bodies formed closer to the
Sun than the FaintIR bodies, the formation of the _blue binaries_ remains a
challenge. These bodies tend to have inflated $i_{\textrm{free}}$ compared to
most other low inclination classical KBOs (Fraser et al., 2021). Some _blue
binaries_ , are members of the CCKBOs based on $i_{\textrm{free}}$. Take for
example 2003 HG57, which has $i_{\textrm{free}}=2.1^{\circ}$. In principle the
slightly inflated $i_{\textrm{free}}$ values of the _blue binaries_ are
consistent with the idea that these objects have been pushed out to the cold
classical region. Though, as pointed out in Nesvorny et al. (2022) when
considering detailed migration histories, the push-out scenario would still
produce more single objects than binary systems, and so the origins of these
peculiar objects remain unclear. It seems clear that like the excited FaintIR
bodies, the _blue binaries_ are bound to provide some significant cosmogonic
insights.

We point out that objects 2013 UN15, 2013 JK64, and 2013 SA100 have highly
variable surface colours, with variations much larger than the uncertainties
in their colours. Similarly large colour variations have been detected on
objects 1998 SM165, 2005 TV189, and 2001 PK47 (Fraser et al., 2015). All 6 of
these targets are unambiguously dynamically excited, with inclinations
$i>8^{\circ}$. Within the dynamically excited sample, the prevalence of highly
spectrally variable targets is $\sim 5\%$, and so the lack of detection of
similar variability in the Col-OSSOS CCKBO sample is not particularly
meaningful. Of these objects, JK64 and PK47, both have ambiguous
BrightIR/FaintIR classifications, with each exhibiting statistically
significantly different colours with class consistency that changes from epoch
to epoch. These two objects are of particular interest as understanding the
full extent of their spectral variability may provide insight regarding the
true nature of the difference between BrightIR and FaintIR surfaces.

##  7 Conclusions

Here we have presented the full colours dataset of the Colours of the Outer
Solar System Origins Survey. This includes a sample of 98 objects with (g-r)
and (r-J) colours, a number of which that have also received near-simultaneous
(u-g) and (r-z) measurements. Using a technique which projects a two
dimensional colour space onto the reddening curve, we find a bifurcation in
the observed (g-r) and (r-J) colours, into two spectral classes. The BrightIR
class closely follows the reddening curve, while the FaintIR class has lower
reflectivity at NIR wavelengths compared to BrightIR objects of similar
optical colour. The presence of this bifurcation is confirmed with the
H/WTSOSS dataset, the (g-r) and (r-z) colours from Col-OSSOS, and the optical
colours from Peixinho et al. (2015). Available optical colour datasets reveal
that this bifurcation is not as apparent at wavelengths $\lambda\lesssim
0.8\mbox{ $\mu$m}$, resulting in unimodal optical colour distributions, to the
precision of available datasets.

We present a simple model that can fully account for all of the observed
structures in the optical-NIR colour space. A KBO spectrum is modelled with a
simple linear optical and linear NIR spectrum, that intersect at some
transition wavelength. Parameter values are chosen to match two mixing model
curves which follow the main trends seen in the the grJ colour space, and
account for nearly 80% of the observed colours. In this way, the model
reproduces:

  * •

The separation of two groups in an optical-NIR colour space

  * •

The bifurcation in the optical colour distributions

  * •

Deviations away from the reddening curve to less-red colours at NIR
wavelengths ($\lambda\gtrsim 0.85\mbox{ $\mu$m}$)

  * •

The fact that not all objects fall above or below $PC^{2}\sim-0.13$ across all
band passes considered.

The bimodal optical colour distribution requires that objects are not
uniformly distributed along the mixing model curves, but rather cluster around
optical spectral slopes of $s=13$ and $30\%/100\mbox{ $n$m}$, for the BrightIR
and FaintIR classes, respectively.

In summary, we find that all the major properties of the UV-optical-NIR colour
distribution can be accounted for with a model consisting of only two
compositional classes. This model predicts that the size distribution of
dynamically excited FaintIR objects will match that of the red cold classical
KBOs. And conversely, the size distribution of BrightIR objects in the cold
classical region will match that of the dynamically excited BrightIR objects.
Testing such a prediction will be possible with the completion of the (g-r)
and (r-J) colour measurements of the outstanding ST and E-block Col-OSSOS
targets.

We also argue that when making use of KBO population statistics to constrain
cosmogonic models, one cannot simply apply a selection in optical colour to
separate the objects into the two compositional classes. Such a cut will
overestimate the fraction of objects that formed inside the transition
distance that hypothetically divided the two classes. Rather a selection in
optical-NIR colour space is required. Cosmogonic inference based on population
statistics drawn from optical colours alone should be considered with an
appropriate amount of caution.

.  Field Label Area R.A. Dec. Ec. Lon. Ec. Lat. N(r$<23.6$) Complete
($\circ^{2}$) ($\circ$) ($\circ$) ($\circ$) ($\circ$) E 21 14:15:28.89
-12:32:28.5 215:50:40.3 +00:59:02.5 32 N O 21 15:58:01.35 -12:19:54.2
239:57:37.5 +07:59:03.1 13 Y** — The centaur, 2013 JC64 - o3o01 has drifted
into the galactic plane, and is not retrievable in ground-based data. O-block
should only be considered complete for orbits with semi-major axes beyond
Neptune. H 21 01:35:14.39 +13:28:25.3 26:58:37.1 +03:18:09.0 22 Y L 20
00:52:55.81 +03:43:49.1 13:37:29.5 -01:47:10.0 18 Y S 10.827 00:30:08.35
+06:00:09.5 09:17:17.9 +02:31:33.3 11 N T 10.827 00:35:08.35 +04:02:04.5
09:39:27.8 +00:13:35.4 6 N

Table 1: Summary of OSSOS Survey Fields

Note. — Listed coordinates are the central coordinates of each field, and in
the J2000 epoch. For further details, see Bannister et al. (2018).

.  Designation MPC OSSOS Header Exposure Filter MJD Gemini Mag. Zeropoint PS
Mag. Exp. Time (s) 2014 UE225 o4h50 Col3N01 N20140825S0353.fits rG0303
56894.511300 $22.852\pm 0.019$ $28.254\pm 0.007$ $22.899\pm 0.019$ 200 2014
UE225 o4h50 Col3N01 N20140825S0376.fits rG0303 56894.562150 $22.867\pm 0.019$
$28.259\pm 0.007$ $22.915\pm 0.019$ 200 2014 UE225 o4h50 Col3N01
N20140825S0354.fits gG0301 56894.514650 $23.873\pm 0.030$ $28.178\pm 0.009$
$23.832\pm 0.030$ 200 2014 UE225 o4h50 Col3N01 N20140825S0355.fits gG0301
56894.517910 $23.831\pm 0.028$ $28.163\pm 0.008$ $23.790\pm 0.028$ 200 2014
UE225 o4h50 Col3N01 N20140825S0375.fits gG0301 56894.558820 $23.819\pm 0.028$
$28.175\pm 0.009$ $23.778\pm 0.028$ 200 2014 UE225 o4h50 Col3N01
Col3N01_0.fits J 56894.540820 $20.967\pm 0.086$ $23.888\pm 0.020$ – 720 2014
UE225 o4h50 Col3N01 Col3N01_1.fits J 56894.550450 $21.227\pm 0.100$ $23.888\pm
0.020$ – 840

Table 2: Observations

Note. — Example observations for 2014 UE225. A full list of observations is
available in the online version.

.  Object (g-r) (g-r) $\Delta$(g-r) (r-z) (r-z) $\Delta$(r-z) (r-J) (r-J)
$\Delta$(r-J) (u-g) (u-g) $\Delta$(u-g) Bracket Second Simult. OSSOS MPC PS
SDSS PS SDSS PS SDSS PS SDSS r Half u Sun – – 0.453 – – 0.091 – – 1.020 – –
1.430 – - T - o3e01 2002 GG166 0.503 0.591 0.013 – – – 1.493 1.496 0.050 1.823
1.732 0.047 T T F o3e02 2013 GH137 0.609 0.713 0.032 – – – 1.753 1.757 0.097
1.423 1.314 0.145 T T T o3e04 2013 GJ137 0.512 0.601 0.023 – – – 1.690 1.693
0.097 – – – T T - o3e04 2013 GJ137 0.529 0.621 0.030 – – – 1.654 1.657 0.086 –
– – T T - o3e04 2013 GJ137 0.663 0.776 0.082 – – – – – – – – – F F - o3e05
2013 GW136 0.618 0.724 0.019 – – – 1.692 1.697 0.065 1.415 1.305 0.092 T T F
o3e09 2013 GY136 0.435 0.512 0.010 0.385 0.394 0.012 – – – – – – T T - o3e09
2013 GY136 0.436 0.513 0.018 – – – 1.189 1.191 0.109 1.443 1.364 0.093 T T T
o3e11 2013 GZ136 0.609 0.714 0.020 – – – 1.604 1.608 0.122 – – – T T - o3e16
2013 GS137 0.865 1.010 0.022 – – – 1.713 1.720 0.080 – – – T T - o3e19 2013
GR136 0.612 0.717 0.026 – – – 1.462 1.466 0.102 – – – T T - o3e20PD 2001 FK185
0.713 0.833 0.033 – – – 1.768 1.773 0.077 1.750 1.624 0.293 T T T o3e21 2013
GQ137 0.763 0.892 0.022 – – – 1.868 1.874 0.062 2.034 1.899 0.301 T T T o3e22
2013 GN137 0.902 1.053 0.010 – – – 1.734 1.742 0.070 1.912 1.753 0.230 F T T
o3e23PD 2001 FO185 0.717 0.838 0.080 – – – – – – – – – F F - o3e23PD 2001
FO185 0.737 0.861 0.022 – – – 1.861 1.866 0.068 – – – T T - o3e23PD 2001 FO185
0.840 0.981 0.071 – – – – – – – – – F F - o3e27PD 2004 EU95 0.830 0.969 0.023
– – – 1.806 1.813 0.064 – – – T T - o3e28 2013 GX137 0.842 0.983 0.028 – – –
1.452 1.460 0.081 – – – T T - o3e29 2013 GO137 0.657 0.768 0.027 – – – 1.723
1.727 0.059 2.389 2.273 0.966 T T T o3e30PD 2013 EM149 0.820 0.958 0.021 – – –
1.644 1.651 0.060 2.662 2.518 0.966 T T T o3e31 2013 GT137 0.891 1.040 0.038 –
– – 1.780 1.788 0.086 – – – T T - o3e34PD 2013 GF138 0.919 1.073 0.026 – – –
1.700 1.708 0.065 – – – T T - o3e35 2013 GP137 0.807 0.943 0.033 – – – 1.261
1.268 0.100 – – – T T - o3e37PD 2004 HJ79 0.816 0.953 0.020 – – – 1.588 1.595
0.068 – – – T T - o3e39 2013 GP136 0.657 0.769 0.020 – – – 1.628 1.633 0.066 –
– – T T - o3e43 2013 GV137 0.789 0.922 0.022 – – – 1.269 1.275 0.124 – – – T T
- o3e43 2013 GV137 0.810 0.947 0.060 – – – – – – – – – F F - o3e44 2013 GG138
0.933 1.090 0.031 – – – 1.845 1.854 0.075 – – – T T - o3e45 2013 GQ136 0.862
1.006 0.055 0.446 0.483 0.057 – – – – – – F F - o3e45 2013 GQ136 0.923 1.078
0.025 – – – – – – – – – T T - o3e49 2013 HR156 0.498 0.586 0.027 – – – 1.360
1.363 0.112 – – – T T - o3e51 2013 GM137 0.508 0.597 0.041 – – – 1.191 1.194
0.129 – – – T T - o3e54 2013 GW137 0.813 0.949 0.023 – – – – – – – – – T T -
o3e55 2013 GX136 0.627 0.734 0.023 – – – 1.641 1.645 0.071 – – – T T - o3l01
2013 UR15 0.569 0.667 0.023 – – – 1.637 1.641 0.093 1.594 1.492 0.229 T T T
o3l05 2013 UJ17 0.582 0.682 0.036 – – – – – – – – – T T - o3l06PD 2001 QF331
0.710 0.830 0.025 – – – 1.574 1.579 0.074 1.530 1.404 0.162 T T T o3l09 2013
US15 0.900 1.050 0.015 – – – 1.482 1.491 0.065 1.667 1.508 0.228 T T T o3l13PD
2003 SR317 0.551 0.646 0.012 – – – 1.356 1.359 0.065 – – – T T - o3l15 2013
SZ99 0.504 0.592 0.019 – – – 1.535 1.538 0.083 1.785 1.695 0.241 T T T o3l18
2010 RE188 0.577 0.677 0.015 – – – 1.426 1.429 0.079 – – – T T - o3l32 2013
SP99 0.837 0.977 0.020 – – – 1.598 1.606 0.067 – – – T T - o3l39 2016 BP81
0.486 0.572 0.026 – – – 1.570 1.572 0.086 – – – T T - o3l43 2013 UL15 0.766
0.895 0.031 – – – 1.476 1.482 0.076 2.219 2.083 0.276 T T T o3l46 2013 UP15
0.756 0.884 0.020 – – – 1.896 1.902 0.087 – – – T T - o3l50 2013 UO15 0.817
0.955 0.019 – – – 1.698 1.705 0.055 – – – T T - o3l57 2013 UM15 0.898 1.048
0.010 – – – 1.721 1.729 0.068 1.787 1.628 0.228 F T F o3l60 2006 QF181 0.769
0.898 0.025 – – – 1.533 1.539 0.072 1.839 1.704 0.270 T T F o3l63 2013 UN15
0.900 1.050 0.027 0.446 0.486 0.090 1.316 1.324 0.110 – – – T T - o3l63 2013
UN15 0.931 1.087 0.038 0.704 0.746 0.053 1.866 1.875 0.057 – – – T T - o3l63
2013 UN15 0.935 1.091 0.032 – – – – – – – – – T T - o3l69 2013 UX18 0.760
0.888 0.010 – – – 1.649 1.655 0.093 1.353 1.219 0.200 F T T o3l76 2013 SQ99
0.831 0.971 0.023 0.444 0.478 0.030 1.689 1.696 0.072 2.359 2.212 0.320 T T T
o3l77 2013 UQ15 0.401 0.473 0.032 – – – 0.940 0.942 0.120 1.634 1.560 0.263 T
T F o3l79 2013 SA100 0.544 0.638 0.020 0.381 0.396 0.025 1.675 1.678 0.100
1.659 1.562 0.144 T T F o3l79 2013 SA100 0.562 0.659 0.020 0.372 0.388 0.015
1.401 1.405 0.054 – – – T T - o3o09 2013 JB65 0.619 0.725 0.018 – – – 1.097
1.101 0.090 1.623 1.513 0.177 T T T o3o11 2013 JK64 0.770 0.900 0.022 – – –
1.911 1.917 0.061 2.071 1.935 0.173 T T T o3o11 2013 JK64 0.904 1.055 0.057 –
– – 1.853 1.861 0.101 – – – F F - o3o14 2013 JO64 0.466 0.548 0.027 – – –
1.149 1.152 0.092 – – – T T - o3o15 2013 JD65 0.675 0.790 0.030 – – – 1.877
1.881 0.087 1.835 1.715 0.966 T T T o3o18 2013 JE64 0.592 0.694 0.032 0.493
0.512 0.035 – – – – – – T T - o3o18 2013 JE64 0.816 0.954 0.120 – – – 1.465
1.472 0.247 – – – F T - o3o20PD 2007 JF43 0.838 0.978 0.011 – – – 1.946 1.953
0.048 – – – T T - o3o21 2013 JR65 0.380 0.450 0.010 – – – 1.592 1.593 0.073 –
– – F T - o3o21 2013 JR65 0.485 0.570 0.035 0.429 0.440 0.043 – – – – – – T T
- o3o27 2013 JJ65 0.922 1.076 0.027 – – – 1.686 1.695 0.075 – – – T T - o3o28
2013 JN65 0.492 0.578 0.013 – – – 1.631 1.634 0.062 – – – T T - o3o29 2013
JL64 0.581 0.681 0.025 – – – 1.424 1.428 0.132 1.457 1.353 0.167 T T T o3o34
2013 JH64 0.595 0.697 0.019 – – – 1.466 1.470 0.079 1.788 1.682 0.130 T T T
o3o51 2013 JX67 0.595 0.697 0.016 – – – 1.395 1.399 0.062 – – – T T - o4h01
2014 UJ225 0.535 0.627 0.012 – – – 1.235 1.238 0.105 1.863 1.767 0.122 T T T
o4h03 2014 UQ229 0.801 0.936 0.018 – – – 1.989 1.996 0.062 2.215 2.073 0.147 T
T T o4h05 2014 UX229 0.557 0.653 0.013 – – – 1.460 1.463 0.090 1.506 1.407
0.052 T T T o4h07 2010 TJ182 0.476 0.559 0.018 – – – 1.338 1.340 0.064 1.558
1.472 0.063 T T T o4h09 2014 UV228 0.504 0.592 0.021 – – – 1.458 1.461 0.060
1.836 1.745 0.216 T T T o4h11 2014 UO229 0.622 0.728 0.022 – – – 1.158 1.162
0.078 1.710 1.599 0.172 T T T o4h13 2014 UD229 0.590 0.691 0.016 – – – 1.300
1.304 0.076 1.532 1.426 0.197 T T T o4h14 2014 US229 0.535 0.628 0.020 – – –
1.415 1.418 0.068 1.534 1.438 0.124 T T T o4h18 2014 UX228 0.422 0.497 0.022 –
– – 1.486 1.488 0.060 1.713 1.635 0.098 T T T o4h18 2014 UX228 0.469 0.552
0.017 0.513 0.523 0.024 – – – – – – T T - o4h19 2014 UK225 0.837 0.978 0.017
0.628 0.664 0.022 1.670 1.677 0.055 2.064 1.917 0.168 T T T o4h20 2014 UL225
0.388 0.458 0.023 0.235 0.240 0.036 – – – – – – T T - o4h20 2014 UL225 0.473
0.556 0.030 – – – 0.767 0.769 0.128 1.444 1.358 0.173 T T F o4h29 2014 UH225
0.452 0.532 0.017 0.338 0.348 0.037 1.626 1.628 0.056 1.756 1.673 0.143 T T T
o4h31 2014 UM225 0.677 0.792 0.014 – – – 1.524 1.529 0.061 – – – T T - o4h39
2007 TC434 0.572 0.670 0.015 – – – 1.495 1.499 0.063 – – – T T - o4h45 2014
UD225 0.608 0.712 0.016 – – – 1.247 1.251 0.087 2.069 1.961 0.173 T T F o4h48
2001 RY143 0.764 0.892 0.027 – – – 1.884 1.890 0.074 2.119 1.984 0.267 T T T
o4h50 2014 UE225 0.891 1.040 0.017 – – – 1.809 1.817 0.067 2.220 2.063 0.215 T
T T o4h67PD 2006 QP180 0.814 0.950 0.082 – – – 2.062 2.069 0.099 – – – T T -
o4h69PD 1995 QY9 0.629 0.737 0.021 – – – 1.460 1.464 0.057 – – – T T - o4h70
2014 UF228 0.520 0.611 0.024 – – – 1.377 1.380 0.072 – – – T T - o4h75 2014
UN228 0.497 0.585 0.040 – – – 1.264 1.267 0.130 – – – T T - o4h75 2014 UN228
0.506 0.595 0.010 – – – 1.757 1.760 0.103 – – – F T - o4h75 2014 UN228 0.531
0.623 0.064 – – – 1.348 1.351 0.194 – – – T T - o4h76PD 2001 RX143 0.721 0.843
0.038 – – – 1.309 1.315 0.109 – – – T T - o5s01 2015 RK277 0.447 0.527 0.015 –
– – 1.352 1.355 0.052 – – – T T - o5s05 2015 RV245 0.572 0.670 0.018 0.404
0.421 0.035 1.398 1.402 0.088 – – – T T - o5s06 2015 RW245 0.583 0.683 0.020
0.377 0.394 0.037 1.539 1.543 0.075 – – – T T - o5s07 2015 RU277 0.494 0.581
0.023 – – – 1.251 1.254 0.094 – – – T T - o5s07 2015 RU277 0.533 0.625 0.028 –
– – 1.229 1.232 0.093 – – – T T - o5s16 2004 PB112 0.636 0.744 0.044 0.449
0.470 0.057 – – – – – – F F - o5s16 2004 PB112 0.705 0.824 0.022 – – – 1.417
1.422 0.059 – – – T T - o5s32 2015 RJ277 0.544 0.638 0.020 – – – 1.172 1.175
0.108 – – – T T - o5s36 2015 RB281 0.654 0.766 0.039 – – – 1.471 1.475 0.095 –
– – T T - o5s36 2015 RB281 0.860 1.004 0.038 – – – 1.514 1.522 0.104 – – – T T
- o5s45 2015 RG277 0.814 0.950 0.018 – – – 1.635 1.642 0.061 – – – T T - o5s52
2015 RU278 0.518 0.609 0.030 – – – 1.361 1.364 0.126 – – – T T - o5s68 2015
RR245 0.654 0.765 0.012 0.466 0.488 0.018 1.183 1.188 0.083 – – – T T - o5t04
2015 RU245 0.757 0.884 0.018 0.563 0.592 0.028 1.565 1.572 0.066 – – – T T -
o5t09 2014 UA225 0.814 0.950 0.018 0.581 0.615 0.019 1.458 1.464 0.060 – – – T
T - o5t11PD 2001 QE298 0.714 0.834 0.018 0.598 0.624 0.027 1.488 1.494 0.063 –
– – T T - o5t31 2015 RT245 0.808 0.944 0.047 0.456 0.489 0.054 1.578 1.585
0.122 – – – F F -

Table 3: Col-OSSOS Colours

Note. — Colours are reported for u, g, r, and z observations in both the PS
and SDSS filter systems. Each separate epoch for a target is reported.
Magnitudes are in AB magnitudes for all but the J-band which is in the Vega
magnitude system. Each row is marked according to if the sequence used in
deriving that colour had bracketing r-band observations, if that sequence was
missing one half of the GMOS observations, and if the reported u-band
observations were acquired simultaneously with the Gemini observations. OSSOS
and Minor Planet Center (MPC) designations are shown.

.  Object (r-i) $\Delta$(r-i) (r-z) $\Delta$(r-z) OSSOS MPC SDSS SDSS o4h50
2014 UE225 0.468 0.024 0.753 0.031 o4h01 2014 UJ225 – – 0.555 0.058 o4h45 2014
UD225 – – 0.409 0.089 o4h31 2014 UM225 – – 0.360 0.100 o3l39 2016 BP81 – –
0.499 0.084 o3l77 2013 UQ15 – – 0.159 0.069 o3l63 2013 UN15 – – 0.595 0.077
o3l09 2013 US15 – – 0.663 0.041 o3l01 2013 UR15 0.392 0.035 0.424 0.055 o3l43
2013 UL15 0.414 0.027 0.658 0.036 o3l46 2013 UP15 – – 0.404 0.117 o3l06PD 2001
QF331 0.469 0.027 0.737 0.032 o3l15 2013 SZ99 – – 0.415 0.084 o3l79 2013 SA100
– – 0.500 0.048 o3l76 2013 SQ99 – – 0.738 0.071

Table 4: Subaru Colours

Note. — Updated colours of Col-OSSOS targets acquired at Subaru in the 2014B
semester during simultaneous observations with the Gemini telescope. For more
details, see .

.  Object $PC^{1}_{grz}$ $\Delta PC^{1}_{grz}$ $PC^{2}_{grz}$ $\Delta
PC^{2}_{grz}$ $PC^{1}_{grJ}$ $\Delta PC^{1}_{grJ}$ $PC^{2}_{grJ}$ $\Delta
PC^{2}_{grJ}$ Sun – 0.0 – 0.0 – 0.0 – 0.0 – o4h50 2014 UE225 – – – – 1.067
0.060 -0.192 0.060 o4h01 2014 UJ225 – – – – 0.298 0.105 -0.093 0.105 o4h45
2014 UD225 – – – – 0.337 0.087 -0.169 0.087 o4h20 2014 UL225 – – – – -0.188
0.129 -0.137 0.129 o3l43 2013 UL15 – – – – 0.653 0.078 -0.251 0.078 o3l18 2010
RE188 – – – – 0.501 0.076 -0.074 0.076 o3l39 2016 BP81 – – – – 0.591 0.077
0.079 0.077 o3l77 2013 UQ15 – – – – -0.035 0.120 -0.022 0.120 o3l63 2013 UN15
0.705 0.067 -0.248 0.071 0.557 0.128 -0.459 0.128 o3l57 2013 UM15 – – – –
0.991 0.064 -0.253 0.064 o3l09 2013 US15 – – – – 0.748 0.072 -0.384 0.072
o3l01 2013 UR15 – – – – 0.691 0.082 0.025 0.082 o3l69 2013 UX18 – – – – 0.817
0.087 -0.162 0.087 o3l46 2013 UP15 – – – – 1.031 0.069 -0.017 0.069 o3l06PD
2001 QF331 – – – – 0.715 0.070 -0.149 0.070 o3l32 2013 SP99 – – – – 0.823
0.067 -0.263 0.067 o3l13PD 2003 SR317 – – – – 0.425 0.062 -0.071 0.062 o3l15
2013 SZ99 – – – – 0.569 0.075 0.047 0.075 o3l79 2013 SA100 0.317 0.020 -0.017
0.015 0.712 0.086 0.067 0.086 o3l76 2013 SQ99 0.627 0.022 -0.204 0.023 0.909
0.068 -0.209 0.068 o3l60 2006 QF181 – – – – 0.712 0.074 -0.228 0.074 o3l79
2013 SA100 0.322 0.013 -0.039 0.016 0.472 0.051 -0.067 0.051 o4h29 2014 UH225
0.216 0.030 0.041 0.021 0.624 0.047 0.139 0.047 o4h19 2014 UK225 0.760 0.014
-0.063 0.018 0.894 0.052 -0.226 0.052 o3l63 2013 UN15 0.902 0.033 -0.055 0.045
1.151 0.049 -0.188 0.049 o4h31 2014 UM225 – – – – 0.646 0.059 -0.138 0.059
o3l50 2013 UO15 – – – – 0.906 0.051 -0.191 0.051 o4h39 2007 TC434 – – – –
0.565 0.060 -0.040 0.060 o4h03 2014 UQ229 – – – – 1.137 0.046 0.003 0.046
o4h05 2014 UX229 – – – – 0.526 0.084 -0.039 0.084 o4h07 2010 TJ182 – – – –
0.375 0.060 0.002 0.060 o4h09 2014 UV228 – – – – 0.497 0.055 0.018 0.055 o4h11
2014 UO229 – – – – 0.251 0.079 -0.212 0.079 o4h13 2014 UD229 – – – – 0.384
0.076 -0.132 0.076 o4h14 2014 US229 – – – – 0.472 0.065 -0.033 0.065 o4h18
2014 UX228 – – – – 0.489 0.054 0.115 0.054 o4h48 2001 RY143 – – – – 1.026
0.060 -0.031 0.060 o3e01 2002 GG166 – – – – 0.529 0.046 0.032 0.046 o3e05 2013
GW136 – – – – 0.770 0.056 0.001 0.056 o3e09 2013 GY136 – – – – 0.219 0.106
0.000 0.106 o3e22 2013 GN137 – – – – 1.007 0.066 -0.249 0.066 o3e29 2013 GO137
– – – – 0.817 0.052 -0.022 0.052 o3e30PD 2013 EM149 – – – – 0.856 0.059 -0.223
0.059 o3e39 2013 GP136 – – – – 0.735 0.059 -0.070 0.059 o3e44 2013 GG138 – – –
– 1.137 0.065 -0.205 0.065 o3o11 2013 JK64 – – – – 1.114 0.086 -0.174 0.086
o3o14 2013 JO64 – – – – 0.192 0.090 -0.042 0.090 o3o15 2013 JD65 – – – – 0.959
0.070 0.045 0.070 o3o20PD 2007 JF43 – – – – 1.134 0.037 -0.055 0.037 o3o27
2013 JJ65 – – – – 0.980 0.074 -0.296 0.074 o3o34 2013 JH64 – – – – 0.549 0.076
-0.077 0.076 o3e02 2013 GH137 – – – – 0.814 0.081 0.041 0.081 o3e04 2013 GJ137
– – – – – – – – o3e16 2013 GS137 – – – – 0.956 0.075 -0.228 0.075 o3e20PD 2001
FK185 – – – – 0.894 0.066 -0.052 0.066 o3e21 2013 GQ137 – – – – 1.012 0.050
-0.041 0.050 o3e23PD 2001 FO185 – – – – – – – – o3e23PD 2001 FO185 – – – – – –
– – o3e35 2013 GP137 – – – – 0.447 0.110 -0.381 0.110 o3e37PD 2004 HJ79 – – –
– 0.799 0.068 -0.248 0.068 o3e43 2013 GV137 – – – – – – – – o3e55 2013 GX136 –
– – – 0.728 0.063 -0.033 0.063 o3o09 2013 JB65 – – – – 0.192 0.093 -0.226
0.093 o3o11 2013 JK64 – – – – 1.054 0.048 -0.020 0.048 o3o18 2013 JE64 – – – –
0.671 0.257 -0.309 0.257 o3o21 2013 JR65 – – – – 0.561 0.062 0.199 0.062 o3o28
2013 JN65 – – – – 0.646 0.053 0.100 0.053 o3o29 2013 JL64 – – – – 0.501 0.125
-0.078 0.125 o5t09 2014 UA225 0.707 0.016 -0.085 0.015 0.660 0.065 -0.309
0.065 o5s68 2015 RR245 0.472 0.014 -0.054 0.012 0.293 0.086 -0.239 0.086 o5s06
2015 RW245 0.342 0.030 -0.054 0.023 0.610 0.070 -0.034 0.070 o5t04 2015 RU245
0.637 0.019 -0.061 0.021 0.735 0.064 -0.200 0.064 o5t11PD 2001 QE298 0.620
0.018 -0.004 0.020 0.632 0.063 -0.191 0.063 o5t31 2015 RT245 0.612 0.041
-0.178 0.040 0.783 0.122 -0.246 0.122 o5s05 2015 RV245 0.357 0.027 -0.027
0.022 0.472 0.084 -0.078 0.084 o5s16 2004 PB112 0.444 0.045 -0.051 0.038 – – –
– o5s45 2015 RG277 – – – – 0.841 0.058 -0.221 0.058 o5s01 2015 RK277 – – – –
0.380 0.049 0.040 0.049 o5s32 2015 RJ277 – – – – 0.241 0.109 -0.122 0.109
o3l63 2013 UN15 – – – – – – – – o3l05 2013 UJ17 – – – – – – – – o3e09 2013
GY136 0.240 0.009 0.084 0.008 – – – – o4h18 2014 UX228 0.362 0.017 0.129 0.015
– – – – o3e11 2013 GZ136 – – – – 0.685 0.110 -0.033 0.110 o3e19 2013 GR136 – –
– – 0.553 0.097 -0.097 0.097 o3e23PD 2001 FO185 – – – – 0.989 0.055 -0.021
0.055 o3e27PD 2004 EU95 – – – – 1.012 0.055 -0.139 0.055 o3e28 2013 GX137 – –
– – 0.674 0.087 -0.340 0.087 o3e31 2013 GT137 – – – – 1.041 0.078 -0.210 0.078
o3e34PD 2013 GF138 – – – – 0.989 0.063 -0.285 0.063 o3e43 2013 GV137 – – – –
0.442 0.134 -0.359 0.134 o3e45 2013 GQ136 0.663 0.050 -0.223 0.044 – – – –
o3e49 2013 HR156 – – – – 0.407 0.106 -0.013 0.106 o3e45 2013 GQ136 – – – – – –
– – o3e51 2013 GM137 – – – – 0.246 0.127 -0.077 0.127 o3e54 2013 GW137 – – – –
– – – – o5s07 2015 RU277 – – – – 0.298 0.090 -0.044 0.090 o5s07 2015 RU277 – –
– – 0.293 0.094 -0.093 0.094 o5s16 2004 PB112 – – – – 0.557 0.060 -0.212 0.060
o5s36 2015 RB281 – – – – 0.584 0.093 -0.138 0.093 o5s36 2015 RB281 – – – –
0.754 0.110 -0.329 0.110 o3e04 2013 GJ137 – – – – 0.685 0.074 0.072 0.074
o3e04 2013 GJ137 – – – – 0.708 0.081 0.107 0.081 o3o51 2013 JX67 – – – – 0.481
0.061 -0.104 0.061 o3o21 2013 JR65 0.309 0.032 0.064 0.028 – – – – o3o18 2013
JE64 0.441 0.026 0.015 0.024 – – – – o4h67PD 2006 QP180 – – – – 1.199 0.069
0.045 0.069 o4h69PD 1995 QY9 – – – – 0.561 0.054 -0.116 0.054 o4h70 2014 UF228
– – – – 0.429 0.068 -0.031 0.068 o4h75 2014 UN228 – – – – 0.313 0.126 -0.044
0.126 o4h75 2014 UN228 – – – – 0.407 0.184 -0.052 0.184 o4h75 2014 UN228 – – –
– 0.761 0.084 0.144 0.084 o4h76PD 2001 RX143 – – – – 0.455 0.116 -0.271 0.116
o4h20 2014 UL225 0.087 0.029 0.047 0.020 – – – – o5s52 2015 RU278 – – – –
0.416 0.120 -0.035 0.120

Table 5: PC Values

Note. — Projection values for the grz and grJ colour spaces, calculated from
the colours in the SDSS system that are tabulated in Table 3.

The authors wish to recognize and acknowledge the very significant cultural
role and reverence that the summit of Maunakea has always had within the
indigenous Hawaiian community. We are most fortunate to have the opportunity
to conduct observations from this mountain. This research is based on
observations from the Large and Long Program GN-2014B-LP-1, GN-2015A-LP-1,
GN-2015B-LP-1, GN-2016A-LP-1, GN-2016B-LP-1, GN-2017A-LP-1, GN-2018A-Q-118,
GN-2018A-Q-223, and GN-2020B-Q-127 obtained at the international Gemini
Observatory, a program of NSF’s NOIRLab, which is managed by the Association
of Universities for Research in Astronomy (AURA) under a cooperative agreement
with the National Science Foundation. on behalf of the Gemini Observatory
partnership: the National Science Foundation (United States), National
Research Council (Canada), Agencia Nacional de Investigación y Desarrollo
(Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina),
Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and
Korea Astronomy and Space Science Institute (Republic of Korea). This research
is also based on observations obtained with MegaPrime/MegaCam, a joint project
of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is
operated by the National Research Council (NRC) of Canada, the Institut
National des Science de l’Univers of the Centre National de la Recherche
Scientifique (CNRS) of France, and the University of Hawaii. The observations
at the Canada-France-Hawaii Telescope were performed with care and respect
from the summit of Maunakea which is a significant cultural and historic site.
Finally, this research is based in part on data collected at the Subaru
Telescope, which is operated by the National Astronomical Observatory of
Japan. This research used the facilities of the Canadian Astronomy Data Centre
operated by the National Research Council of Canada with the support of the
Canadian Space Agency. This work also made use of the Gemini Observatory
Archive, NASA’s Astrophysics Data System Bibliographic Services, and the JPL
HORIZONS web interface (https://ssd.jpl.nasa.gov/horizons.cgi). This work made
use of the the Pan-STARRS1 Surveys (PS1) and the PS1 public science archive,
which 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. REP acknowledges funding from NASA Emerging Worlds
grant 80NSSC21K0376.

##  Appendix A Photometry of Highly Azimuthally Variable Point Sources

Here we present a new technique developed to accurately measure the photometry
of the stellar calibration imagery acquired as part of every NIR observation.
These sequences were a series of 10 frames acquired with integrations of a few
to $\sim 6$ s, with dithers of a few tens of arcseconds between each frame. An
issue with inconsistent zeropoints was found when studying the calibration
frames in depth. This has been attributed to the highly variable shape of the
stellar images acquired during these sequences. When measuring zeropoints from
these images, the standard method of measuring of measuring flux in a large
aperture, $r=4$ FWHM – was adopted. The FWHM itself was determined from the
radially averaged stellar profile. Such a large aperture radius is usually
enough to encompass $\sim 99\%$ of a point source flux, and provide stable
absolute calibrations. It was found however, that the resultant zeropoints
would vary by a couple tenths of a magnitude, a factor of $\sim 5$ more than
expected. This variation disappeared however, when unusually large apertures
were considered, hinting that oddly, some large, and variable fraction of the
stellar flux was being excluded from the normally large apertures.

The solution to the highly variable zeropoints was discovered when examining
the residuals of subtracting the radially averaged moffat profile from the
stellar images. See Figure 11. Large asymmetries containing substantial
portions of the stellar flux were present, including a significant portion of
flux in the profile wings. These asymmetries inspired the new technique. The
stellar image is broken up into $N$ equal sized wedges, that pivot around the
source centroid. In each wedge, a moffat profile is fit to the pixels in that
wedge. Linear interpolation is used to generate functions dependent on the
azimuthal angle $\theta$ that replace the usual moffat parameter,
$\alpha=\alpha\left(\theta\right)$ with a constant $\beta$. We adopted $N=6$
as a good compromise between risks of overfitting, and quality of the
modelling of the profile wings.

At the cost of potentially worsening the subtraction residuals near the core
of the profile, the azimuthally variable profile resulted in much better
modelling of the profile wings. We found that by adopting an aperture radius
$r=4$ FWHM where the FWHM is the maximum determined from each wedge, the
resultant zeropoint scatter was substantially reduced, with a scatter of order
$\sim 0.1$ magnitudes. The adopted zeropoint was taken as the median of
zeropoints of a set of calibration frames, after rejecting the two most
discrepant points. Accordingly, we adopt a zeropoint uncertainty of 0.03
magnitudes for each sequence.

While successful at reducing calibration uncertainties to acceptable levels,
the above method still leaves significant room for improvement; from
poissonian consideration alone, sub-1% calibration quality should be
achievable. We speculate that the calibration issues we (and likely many other
groups) experienced is due to temporal variations in the telescope image shape
that do not average out sufficiently in short exposures. The obvious solution
is to use significantly longer exposure times for the calibration frames, that
in itself requires a set of fainter sources than those typically used. Further
efforts will be required to determine appropriate solutions. We would like to
raise a point of caution in considering published science results that make
use of the typical type of calibration images like we gathered, but with
standard photometry calibration techniques.

Figure 11:  Cutouts around a typical NIRI calibration star. Left: original
image. Middle: Subtraction of an average moffat profile. Right: Subtraction of
the azimuthally variable moffat profile. The residual flux in the wings are
improved through use of the azimuthally variable profile.

##  Appendix B Colour Terms between the Gemini, Pan-STAARS, and SDSS Systems

Colour terms between the Gemini and Pan-STAARS griz filter sets were
calculated in the same way as presented in (Schwamb et al., 2019). Stars were
selected as those sources in the Pan-STAARS release 1 dataset with difference
between Kron and aperture magnitudes of less than 0.05 magnitudes, and
magnitudes $17<r<22.5$, $g<22.5$, $z>17.0$ and colours $0.3<(g-r)<1.2$. Linear
colour terms were fit to all measurements of catalog sources across all GMOS
data acquired prior to publication of this manuscript. Terms were calculated
separately for the E2V and Hamamatsu detectors. The difference between
instrumental magnitudes and (g-r) colours for the catalog stars in the images
acquired with the Hamamatsu detectors are shown in Figure 12. The resultant
linear colour terms are:

$\displaystyle g_{\textrm{G}}$ $\displaystyle=$ $\displaystyle
g_{\textrm{PS}}+0.019(\pm 0.005)\times(g-r)_{\textrm{PS}}$ (B1) $\displaystyle
r_{\textrm{G}}$ $\displaystyle=$ $\displaystyle r_{\textrm{PS}}-0.047(\pm
0.004)\times(g-r)_{\textrm{PS}}$ (B2) $\displaystyle i_{\textrm{G}}$
$\displaystyle=$ $\displaystyle i_{\textrm{PS}}-0.198(\pm
0.016)\times(g-r)_{\textrm{PS}}$ (B3) $\displaystyle z_{\textrm{G}}$
$\displaystyle=$ $\displaystyle z_{\textrm{PS}}-0.096(\pm
0.012)\times(g-r)_{\textrm{PS}}$ (B4)

for the Hamamatsu GMOS detectors, and

$\displaystyle g_{\textrm{G}}$ $\displaystyle=$ $\displaystyle
g_{\textrm{PS}}+0.046(\pm 0.003)\times(g-r)_{\textrm{PS}}$ (B5) $\displaystyle
r_{\textrm{G}}$ $\displaystyle=$ $\displaystyle r_{\textrm{PS}}-0.053(\pm
0.002)\times(g-r)_{\textrm{PS}}$ (B6) $\displaystyle i_{\textrm{G}}$
$\displaystyle=$ $\displaystyle i_{\textrm{PS}}-0.12(\pm
0.03)\times(g-r)_{\textrm{PS}}$ (B7) $\displaystyle z_{\textrm{G}}$
$\displaystyle=$ $\displaystyle z_{\textrm{PS}}-0.118(\pm
0.007)\times(g-r)_{\textrm{PS}}$ (B8)

for the E2V GMOS detectors.

Figure 12: GMOS Instrumental magnitudes vs. (g-r) colours of stars in the Pan-
STAARS v1 dataset. Red lines show the linear colour terms derived from those
measurements.

##  Appendix C The Reddening Curve Projection

Here we provide a visual demonstration of the reddening curve (Fig. 13). The
$PC^{1}$ and $PC^{2}$ basis vectors each take on the instantaneous tangential
and perpendicular vectors of the reddening line, respectively, with projection
values increasing with increasing red colours. The software written to perform
this non-linear projection is available at
https://github.com/fraserw/projection (Will be available at time of
submission).

Figure 13: Visual demonstration of the reddening curve projection. The dashed
grid shows curves of constant $PC^{1}$ and $PC^{2}$ in intervals of 0.1.
Objects with positive and negative values of $PC^{2}$ have convex and concave
spectra, and are found above and below the reddening curve, respectively. The
point values show the extremal projection values.

In this work, we adopt Solar Colours of (u-g)=1.43, (g-r)=0.44, (r-z)=0.18,
(r-J)=0.98, (F606w-F814w)=-0.47, (F814w-F139m)=-1.24, (B-V)=0.66, (V-R)=0.36,
and (R-I)=0.35. The bandpasses u, g, r, and z, are AB magnitudes in the SDSS
system. The J bandpass is specifically the MaunaKea Observatories J-band
filter666http://www.gemini.edu/sciops/instruments/niri/filters/ in Vega
magnitudes. The F606w, F814w, and F139m filters are references to the Hubble
Space Telescope STMAG “flight” system (Fraser & Brown, 2012). We adopt Vega
magnitudes for the B, V, R, and I bandpasses. For B, V, R, I, J, and the HST
STMAG system, Solar colours were computed using the _synphot_ package, with
the specific bandpasses of each filter, and the _sun_reference_001.fits_ data
file provided as part of the _synphot data files_.

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