# Phase Curves of Kuiper Belt Objects, Centaurs, and Jupiter Family Comets
from the ATLAS Survey

Matthew M. Dobson Astrophysics Research Centre, School of Mathematics and
Physics, Queen’s University Belfast, Belfast BT7 1NN, UK  Megan E. Schwamb
Astrophysics Research Centre, School of Mathematics and Physics, Queen’s
University Belfast, Belfast BT7 1NN, UK  Susan D. Benecchi Planetary Science
Institute, 1700 East Fort Lowell Rd., Suite 106, Tucson, AZ 85719, USA  Anne
J. Verbiscer Department of Astronomy, University of Virginia, P.O. Box 400325,
Charlottesville, VA 22904-4325, USA  Alan Fitzsimmons Astrophysics Research
Centre, School of Mathematics and Physics, Queen’s University Belfast, Belfast
BT7 1NN, UK  Luke J. Shingles GSI Helmholtzzentrum für Schwerionenforschung,
Planckstraße 1, 64291 Darmstadt, Germany  Astrophysics Research Centre, School
of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, UK
Larry Denneau University of Hawaii Institute for Astronomy, 2680 Woodlawn Dr.,
Honolulu, HI 96822  A. N. Heinze DiRAC Institute and the Department of
Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195
Ken W. Smith Astrophysics Research Centre, School of Mathematics and Physics,
Queen’s University Belfast, Belfast BT7 1NN, UK  John L. Tonry University of
Hawaii Institute for Astronomy, 2680 Woodlawn Dr., Honolulu, HI 96822  Henry
Weiland University of Hawaii Institute for Astronomy, 2680 Woodlawn Dr.,
Honolulu, HI 96822  David. R. Young Astrophysics Research Centre, School of
Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, UK

###### Abstract

The Kuiper belt objects, the Centaurs, and the Jupiter-family comets form an
evolutionary continuum of small outer Solar System objects, and their study
allows us to gain insight into the history and evolution of the Solar System.
Broadband photometry can be used to measure their phase curves, allowing a
first-order probe into the surface properties of these objects, though limited
telescope time makes measuring accurate phase curves difficult. We make use of
serendipitous broadband photometry from the long-baseline, high-cadence ATLAS
survey to measure the phase curves for a sample of 18 Kuiper belt objects,
Centaurs, and Jupiter-family comets with unprecedentedly large datasets. We
find phase curves with previously reported negative slopes become positive
with increased data and are thus due to insufficient sampling of the phase
curve profile, and not a real physical effect. We search for correlations
between phase curve parameters, finding no strong correlations between any
parameter pair, consistent with the findings of previous studies. We search
for instances of cometary activity in our sample, finding a previously
reported outburst by Echeclus and a new epoch of increased activity by Chiron.
Applying the main belt asteroid $HG_{1}G_{2}$ phase curve model to three
Jupiter-family comets in our sample with large phase angle spans, we find
their slope parameters imply surfaces more consistent with those of
carbonaceous main belt asteroids than silicaceous ones.

††facilities: ATLAS (Maunaloa and Haleakalā telescopes)††software: Astropy
(Astropy Collaboration et al., 2013, 2018), Jupyter Notebook (Kluyver et al.,
2016), lmfit (Newville et al., 2014), math (Van Rossum, 2020), Matplotlib
(Hunter, 2007), Numpy (van der Walt et al., 2011; Harris et al., 2020), os
(Van Rossum, 2020), Pandas (pandas development team, 2020), python
(https://www.python.org), SAOImageDS9 (Joye, 2019), sbpy (Mommert et al.,
2019), SciPy (Virtanen et al., 2020), seaborn (Waskom, 2021)

##  1 Introduction

Kuiper belt objects (KBOs) are small planetesimals residing beyond the orbit
of Neptune and are the remnants of planet formation during the early history
of the Solar System. Over 3700 KBOs111From Minor Planet Center:
https://www.minorplanetcenter.net have been discovered to date, and can be
divided into sub-populations based on their present dynamics (Delsanti &
Jewitt, 2006; Gladman et al., 2008; Kavelaars et al., 2008) and physical
properties (Schaller & Brown, 2007a; Brown, 2008; Brown et al., 2011).
Originating from the Kuiper belt population, the Centaurs are small icy
objects that have diffused inwards onto giant planet-crossing orbits (Levison
& Duncan, 1997; Duncan & Levison, 1997; Volk & Malhotra, 2008). Centaurs are
likely precursors to certain populations of short-period comets, most notably
the Jupiter-family comets (JFCs) (Tiscareno & Malhotra, 2003; Fernández et
al., 2013; Bauer et al., 2013; Sarid et al., 2019), whose orbital dynamics are
governed predominantly by Jupiter’s gravitational influence (Vaghi, 1973;
Carusi & Valsecchi, 1987; Levison & Duncan, 1994; Levison, 1996; Gladman et
al., 2008; Sarid et al., 2019) and whose perihelion distances bring them
sufficiently close to the Sun for water ice to sublimate (Womack et al., 2017
and references therein). Studying these populations of primordial objects can
reveal information about the conditions, dynamics and evolutionary processes
of the Solar System, both in its early history and its present state.

Broadband photometry remains the fastest method of characterizing large
numbers of these objects, as they are generally too faint for spectroscopy.
Such photometry can be used to analyse their phase curves: the change in an
object’s distance-corrected brightness when observed at different solar phase
angles $\alpha$, the angle between the Sun and the observer as viewed from the
object. The phase curve of an atmosphereless object is determined by its
surface reflectivity, itself a complex function of its surface properties.
Thus, analysis of an object’s phase curve can allow an explorative probe into
its overall surface properties (Hapke, 1963, 1966, 1968, 1981, 1984, 1986,
2002, 2008, 2012, 2021; Bowell et al., 1989; Nelson et al., 2000; Belskaya et
al., 2008; Verbiscer & Helfenstein, 1998; Verbiscer et al., 2013).

The large heliocentric distances of KBOs and many Centaurs limit Earth-based
observations of these objects to a narrow range of small phase angles
($\alpha\lesssim 2$ deg), requiring in-situ measurements from spacecraft for
high phase angles. Verbiscer et al. (2019) and Verbiscer et al. (2022)
obtained phase curves of KBOs spanning several dynamical sub-populations by
combining data from both Earth-based observations and _NASA’s_ _New Horizons_
spacecraft, extending the phase angle coverage beyond that achievable from
Earth. The phase curves of the KBOs analysed were found to exhibit
similarities to those of other populations of small Solar System bodies,
including JFCs and giant planet satellites theorized to be captured KBOs
(Verbiscer et al., 2019). Additionally, KBOs with high-albedo surfaces
dominated by volatile ices were found to exhibit shallow phase curve slopes,
with steeper slopes being a feature of objects with lower-albedo surfaces (Li
et al., 2015; Ayala-Loera et al., 2018; Verbiscer et al., 2022). Without such
measurements, the small phase angle range observable from Earth impairs the
ability of surface reflectance models to accurately fit the data. However,
beyond the very small range of phase angles where the opposition surge - the
sudden increase in an object’s brightness as $\alpha\rightarrow 0$ deg - is
significant, Centaur and KBO phase curves typically exhibit approximately
linear profiles across phase angles observable from Earth (Binzel &
Mulholland, 1984; Tholen & Tedesco, 1994; Buie et al., 1997; Sheppard &
Jewitt, 2002; Schaefer & Rabinowitz, 2002; Rabinowitz et al., 2006, 2007;
Sheppard, 2007; Belskaya et al., 2008; Schaefer et al., 2009; Verbiscer et
al., 2013), with a slope quantified by the linear phase coefficient, $\beta$.
Fitting a linear function to the phase curves generated using Earth-based
observations has thus allowed comparison of phase curves between individual
objects and populations.

Rabinowitz et al. (2007) and Schaefer et al. (2009) found phase curves of KBOs
and Centaurs to exhibit approximately linear trends when observed from Earth.
These phase curves ranged from flat to steep in profile, with varying phase
coefficient values between different photometric filters. The largest KBOs
exhibited small phase coefficients ($\beta<0.1$ mag deg-1), suggesting such a
property may be indicative of objects with high-albedo volatile-rich surfaces,
in accordance with both the findings of Verbiscer et al. (2022) and
theoretical models of surface reflectivity. Several Centaurs also exhibited
similarly flat phase curves, implying their surfaces are dominated by organic
compounds. This accords with the findings of Sheppard & Jewitt (2002),
laboratory measurements of both high- and low-albedo materials (Shkuratov et
al., 2002) and the improbability of volatile retention by Centaurs due to
their smaller heliocentric distances. Schaefer et al. (2009) reported
correlations between an object’s opposition surge with its color index and
geometric albedo. Both Schaefer et al. (2009) and Verbiscer et al. (2019) find
a distinction between the scattering properties of outer Solar System
satellites and those of the Centaur and KBO populations. The studies of
Alvarez-Candal et al. (2016); Ayala-Loera et al. (2018); Alvarez-Candal et al.
(2019) report several phase coefficients for Centaurs and KBOs whose values
were significantly negative, implying unphysical dimming at small phase angles
which cannot be explained by surface reflectivity models. These studies also
reported a strong correlation between the relative phase coefficient between
filters (defined by Alvarez-Candal et al. (2019) as
$\Delta\beta\equiv\beta_{V}-\beta_{R}$) and an object’s color index
($H_{V}-H_{R}$), in addition to a correlation between relative phase
coefficient and albedo for objects with absolute magnitudes $H_{V}\geq 4.5$.
Kokotanekova et al. (2017) and Kokotanekova et al. (2018) measured shallow
phase coefficients of $\beta<0.1$ mag deg-1 for 14 JFCs, and reported a
possible correlation of increasing geometric albedo with steeper phase
coefficient.

Phase curve measurements require high-cadence observations over a long
baseline to sample the full range of phase angles visible from Earth. Such
ideal measurements are difficult to achieve in practice, due to the limited
availability of telescope time, in addition to the challenges of ground-based
observations. Combining data of an object from multiple telescopes at
different sites, as well as from literature studies, can help augment
otherwise sparse datasets (Alvarez-Candal et al., 2016; Ayala-Loera et al.,
2018; Alvarez-Candal et al., 2019). However, this could introduce systematic
errors into the dataset due to heterogeneous data reduction pipelines and
differing broadband filter wavelength ranges, as well as potential long
timespans between observations. To sample the full range while ensuring all
data come from a homogeneous photometric system and data reduction pipeline
would require observations with a telescope that performs regular-cadence
observations of that object. Such an opportunity is afforded by the Asteroid
Terrestrial-impact Last Alert System (ATLAS) survey (Tonry et al., 2018a, b),
which has accumulated large datasets of serendipitous observations of many
KBOs, Centaurs, and JFCs.

In this paper, we make use of the high-cadence, long baseline photometry
obtained by the ATLAS survey to generate phase curves for a sample of KBOs,
Centaurs, and JFCs visible to the ATLAS telescopes in two broadband filters.
With the advantage of a larger dataset than previous studies across a single
baseline, we measure the phase coefficients of these objects, search for
correlations between these and other object parameters, and look for outbursts
of cometary activity across the survey’s near-continuous, 6-year observation
baseline. This paper is structured as follows. In section 2, we describe the
ATLAS survey. Section 3 details the sample of objects selected for this study.
In section 4, we present details about the photometry obtained by and used in
this study. Section 5 details the data analysis methods, while the results
from this analysis and our discussion of these are presented in Sections 6.
Finally, in Section 7, we draw our conclusions from our analysis.

##  2 ATLAS

The data used in this study originate from observations performed by the ATLAS
wide-field survey (Tonry et al., 2018a), whose primary mission is to detect
near-Earth asteroids that may pose an impact hazard to Earth. Since its first
observations in 2016, ATLAS has serendipitously observed a sample of bright
KBOs, Centaurs, and JFCs, with the long baseline and high cadence of
observations providing ample sampling of the phase curves of these objects.
This eliminates the need to include additional data from external sources,
thereby reducing systematic errors that could arise from combining data
obtained via heterogeneous methods of image processing and data analysis.
ATLAS’s long baseline has been previously utilized by studies analysing other
populations of small Solar System objects, such as asteroids (Erasmus et al.,
2020; Mahlke et al., 2021) and Jupiter Trojans (McNeill et al., 2021).

ATLAS presently consists of four 0.5-m Schmidt telescopes, two of which are
located at Hawai’i (sites at Haleakalā and Maunaloa), with another two
installed in 2022 at sites in Chile and South Africa. The ATLAS telescopes
regularly observe the sky outside $60$ deg of the Sun to a limiting magnitude
of approximately $19.5$ mag in two non-standard wide-band filters - cyan (c,
spanning 420-650 nm) and orange (o, spanning 560-820 nm). When weather
conditions permit, four 30s exposures are taken per pointing over a 1 hour
interval. Each exposure covers 28.9 deg2 field of view (Tonry et al., 2018a,
b), with the two telescopes of ATLAS installed at the time of this study
covering the entire accessible sky over approximately 2 days. Further details
of the ATLAS system and data reduction pipeline are described in Tonry et al.
(2018a, b) and Smith et al. (2020). For this study, we make use of data from
the two ATLAS telescopes at the Hawai’ian sites of Haleakalā and Maunaloa, as
the southern hemisphere telescopes had only recently been installed at the
time of this study.

##  3 Sample of Objects

We select objects from the KBO, Centaur and JFC populations that may have been
observable by ATLAS, downloading the 2021 catalog of objects from JPL
Horizons222https://ssd.jpl.nasa.gov/ - retrieved on 2021 March 25. These
objects are selected according to the following criteria:

  1. 1.

The semimajor axis of each object must lie beyond the aphelion of Jupiter
($a>5.5$ au).

  2. 2.

For ${\geq}1$ month during the ATLAS baseline, the object must be brighter
than $19.5$ mag (ATLAS limiting magnitude) to ensure detection.

  3. 3.

The objects’ predicted positional uncertainties had to be
${<}1^{\prime\prime}$ (equating to ${\sim}$0.5 pixels on ATLAS images), to
ensure proper identification on the ATLAS images;

  4. 4.

The ATLAS observations of each object had to span ${>}$50% of the total
observable phase angle range during the time that ATLAS has been observing, to
ensure adequate sampling of at least half the observable span of the phase
curve;

  5. 5.

Each object had to have at least 50 observations recorded in both c and o
filters, to ensure adequate sampling.

Upon selecting our sample of objects, we subsequently divide them into their
dynamical classes. We define KBOs as small outer Solar System objects whose
semimajor axes lie beyond that of Neptune ($a>a_{N}$). The KBOs in our sample
come from several dynamically hot sub-populations of the Kuiper belt using the
definitions of Gladman et al. (2008): 3 objects from the resonant population
(Pluto, Orcus, Huya), 1 from the detached TNO population (Eris), and 3 from
the hot classical Kuiper belt (Haumea, Quaoar, Makemake). Furthermore, 6 of
our 7 KBOs, with absolute magnitudes $H_{V}<3$, differ significantly from the
remainder of the Kuiper belt. These Pluto-sized objects have greater masses,
allowing retention of high albedo volatiles on their surfaces, augmenting
their absolute magnitude such that they deviate from the brightness
distribution of the Kuiper belt (Bernstein et al., 2004; Schaller & Brown,
2007b; Brown, 2008, 2012).

At present, there is no agreed upon definition for the Centaur population, nor
is there a consensus distinction between the Centaurs and the related JFC
population (Levison & Duncan, 1997; Gladman et al., 2008; Jewitt, 2009;
Kokotanekova et al., 2017; Chandler et al., 2020; Lilly et al., 2021; Roberts
& Muñoz-Gutiérrez, 2021). For our study, we use a dynamical definition to
separate these populations not based on past histories of cometary activity.
We define and distinguish between the Centaur and JFC populations by their
semimajor axes, perihelia, and the effect of Jupiter on their dynamics, the
latter of which can be quantified by the Tisserand parameter with respect to
Jupiter, $T_{J}$, defined as:

$T_{J}\equiv\frac{a_{J}}{a}+2\cos i\sqrt{\frac{a}{a_{J}}(1-e^{2})}$ (1)

where $a$ and $a_{J}$ are the semimajor axes of the object and Jupiter
respectively, $e$ the object’s eccentricity and $i$ the object’s inclination.
We construct our Centaur definition based on those of Gladman et al. (2008)
and Jewitt (2009), classifying Centaurs as small Solar System objects not in
1:1 mean motion resonance with any planet, on orbits with semimajor axes $a$
and perihelia $q$ between the semimajor axes of Jupiter and Neptune
($a_{J}<a,q<a_{N}$), and Tisserand parameters with respect to Jupiter
$T_{J}>3.05$, this threshold value accounting for Jupiter’s non-zero
eccentricity and the resulting effect on small object dynamics (Gladman et
al., 2008). We accordingly define Jupiter-family comets (JFCs) as small Solar
System objects not in 1:1 mean motion resonance with a planet, whose perihelia
lie interior to the orbit of Jupiter ($q<a_{J}$) and with Tisserand parameters
with respect to Jupiter of $2<T_{J}\leq 3.05$. We classify any objects in our
sample with $a<a_{N}$ that do not satisfy the definitions for Centaurs or JFCs
as _Transition Objects_. Table 1 lists the objects in our sample with
semimajor axes $a<a_{N}$, and their dynamical classification according to the
definitions of Centaurs and JFCs from Gladman et al. (2008), Jewitt (2009) and
Sarid et al. (2019), in addition to their classification in this study.

Table 1: Classification of objects used in this study with semimajor axes $5.5$ au $<a<30.1$ au, according to the definitions of Gladman et al. (2008), Jewitt (2009) and Sarid et al. (2019).  Name | Gladman et al. (2008) | Jewitt (2009) | Sarid et al. (2019) | This study
---|---|---|---|---
(944) Hidalgo | JFC | JFC | Neither | JFC
(2060) 95P/Chiron | Centaur | Centaur | Centaur | Centaur
(10199) Chariklo | Centaur | Centaur | Centaur | Centaur
(37117) Narcissus | JFC | JFC | Neither | JFC
(54598) Bienor | Centaur | Centaur | Centaur | Centaur
(60558) 174P/Echeclus | JFC | Centaur | Centaur | Transition Object
(347449) 2012 TW236 | JFC | JFC | Neither | JFC
(349933) 2009 YF7 | JFC | Centaur | Centaur | Transition Object
(459865) 2013 XZ8 | Centaur | Centaur | Centaur | Centaur
(501585) 2014 QA43 | JFC | JFC | Neither | JFC
2016 ND21 | JFC | JFC | Neither | JFC

Our sample consists of 18 objects (7 KBOs, 4 Centaurs, 5 JFCs, and 2
Transition Objects), whose orbital parameters are listed in Table 3, along
with their estimated V-band absolute magnitudes, and, if known, their
published rotational lightcurve periods and amplitudes. The orbital
distribution of our sample is shown in Figure 1 and its $H$ distribution is
plotted in Figure 2. There are a median of 181 and 463 ATLAS observations per
object in the _c_ and _o_ filters, respectively. Across all objects in this
study, the ATLAS data cover a median of 93% (_c_) and 95% (_o_) of the total
observable phase angle range from Earth across the ATLAS baseline. These
objects span apparent magnitudes $14.2<c<19.8$ and $13.8<o<19.7$ in each ATLAS
photometric filter. The galactic latitudes spanned by our sample of objects
during the ATLAS baseline are shown in Figure 3, showing many objects in our
sample cross into the galactic plane where stellar crowding and the resulting
background flux contamination must be dealt with.

Table 2: Heliocentric orbital elements, absolute magnitudes and rotation
periods of KBOs, Centaurs, JFCs and Transition Objects visible to ATLAS survey
used in this study. $a$ = semimajor axis; $e$ = eccentricity; $i$ =
inclination; $q$ = perihelion; $Q$ = aphelion; $T_{J}$ = Tisserand parameter
(with respect to Jupiter); $H$ = (_V_ -band) absolute magnitude; $P$ =
rotation period; $\Delta M$ = peak-to-peak rotational lightcurve amplitude.

Name | $a$ | $e$ | $i$ | $q$ | $Q$ | $T_{J}$ | $H$ | $P$ | $\Delta M$ | Source for $P$ | Dynamical Class
---|---|---|---|---|---|---|---|---|---|---|---
| (au) |  | (deg) | (au) | (au) |  | (mag) | (hrs) | (mag) | and $\Delta M$ |
(136199) Eris | 67.94 | 0.43 | 43.97 | 38.41 | 97.51 | 4.780 | -1.11 | … | … | … | KBO (detached)
(136108) Haumea | 43.12 | 0.2 | 28.21 | 34.65 | 51.54 | 5.086 | 0.25 | 3.915341 | 0.28 | SS17 | KBO (hot classical)
(38628) Huya | 39.69 | 0.28 | 15.47 | 28.55 | 50.49 | 5.235 | 4.86 | 5.28 | 0.02 $\pm$ 0.01 | T14 | KBO (3:2 resonant)
|  |  |  |  |  |  |  | 4.45 | 0.1 | G16 |
|  |  |  |  |  |  |  | 6.75 | $<0.1$ | O03 |
(136472) Makemake | 45.37 | 0.16 | 28.99 | 37.98 | 52.77 | 5.204 | -0.13 | 22.83 | 0.032 $\pm$ 0.005 | H19 | KBO (hot classical)
|  |  |  |  |  |  |  |  | 0.0286 $\pm$ 0.0016 | HL09 |
(90482) Orcus | 39.14 | 0.23 | 20.59 | 30.21 | 48.06 | 5.129 | 2.3 | 10.5 | ${\sim}0.05$ | G16 | KBO (3:2 resonant)
|  |  |  |  |  |  |  |  | 0.18 $\pm$ 0.08 | R06 |
|  |  |  |  |  |  |  |  | 0.03-0.18 | D10 |
|  |  |  |  |  |  |  |  | 0.06 $\pm$ 0.04 | O11 |
(134340) Pluto | 39.45 | 0.25 | 17.09 | 29.57 | 49.32 | 5.228 | -0.4 | 153.29352 | 0.19 | B18 | KBO (3:2 resonant)
|  |  |  |  |  |  |  |  | (Pluto+Charon) |  |
(50000) Quaoar | 43.66 | 0.04 | 7.99 | 41.85 | 45.35 | 5.844 | 2.51 | 8.8394 | 0.17 $\pm$ 0.04 | O03 | KBO (hot classical)
|  |  |  |  |  |  |  |  | 0.17 | FB10 |
(459865) 2013 XZ8 | 13.42 | 0.37 | 22.53 | 8.42 | 18.41 | 3.140 | 9.69 | … | … | … | Centaur
(54598) Bienor | 16.46 | 0.2 | 20.75 | 13.18 | 19.71 | 3.578 | 7.55 | 9.14 | 0.7 | O02 | Centaur
|  |  |  |  |  |  |  |  | 0.75 $\pm$ 0.09 | O03 |
|  |  |  |  |  |  |  |  | 0.34 $\pm$ 0.08 | R06 |
|  |  |  |  |  |  |  |  | 0.088 $\pm$ 0.008 (2014) | FV17 |
|  |  |  |  |  |  |  |  | 0.082 $\pm$ 0.007 (2015) | FV17 |
|  |  |  |  |  |  |  |  | 0.10 $\pm$ 0.02 (2016) | FV17 |
(10199) Chariklo | 15.84 | 0.17 | 23.35 | 13.17 | 18.52 | 3.485 | 6.54 | 7.004 | 0.11-0.13 | F14 | Centaur
|  |  |  |  |  |  |  |  | 0.06 ± 0.02 | L17 |
95P/(2060) Chiron | 13.69 | 0.38 | 6.93 | 8.52 | 18.87 | 3.365 | 5.84 | 5.918 | 0.088 ± 0.003 | LJ90 | Centaur
|  |  |  |  |  |  |  |  | 0.07 ± 0.02 | D91 |
|  |  |  |  |  |  |  |  | 0.04 ± 0.005 | MB93 |
|  |  |  |  |  |  |  |  | 0.06 ± 0.01 | L97 |
(347449) 2012 TW236 | 6.97 | 0.57 | 11.96 | 3.0 | 10.94 | 2.610 | 12.3 | … | … | … | JFC
(501585) 2014 QA43 | 9.66 | 0.51 | 37.21 | 4.76 | 14.54 | 2.409 | 11.4 | 11.558 | 0.236 | S18 | JFC
2016 ND21 | 8.47 | 0.56 | 21.83 | 3.76 | 13.16 | 2.584 | 12.3 | 17.53 | 0.31 | H19 | JFC
(944) Hidalgo | 5.73 | 0.66 | 42.54 | 1.95 | 9.52 | 2.068 | 10.69 | 10.0630 | 0.48 | W97 | JFC
(37117) Narcissus | 6.88 | 0.55 | 13.8 | 3.07 | 10.68 | 2.616 | 13.29 | … | … | … | JFC
(349933) 2009 YF7 | 12.11 | 0.46 | 30.99 | 6.51 | 17.72 | 2.749 | 10.91 | … | … | … | Transition Object
174P/(60558) Echeclus | 10.71 | 0.46 | 4.34 | 5.81 | 15.60 | 3.031 | 9.17 | 26.802 | 0.24 ± 0.06 (R-filter) | R05 | Transition Object
|  |  |  |  |  |  |  |  | 0.36 ± 0.09 (V-filter) | R05 |

References. — Heliocentric orbital elements and absolute magnitude values
obtained from JPL Horizons. Rotational period/amplitude sources: B18: Benecchi
et al. (2018), SS17: Santos-Sanz et al. (2017), O03: Ortiz et al. (2003),
FB10: Fraser & Brown (2010), LJ90: Luu & Jewitt (1990), D91: Dahlgren et al.
(1991), MB93: Marcialis & Buratti (1993), L97: Lazzaro et al. (1997a), FV17:
Fernández-Valenzuela et al. (2017), O02: Ortiz et al. (2002), D10: Delsanti et
al. (2010), O11: Ortiz et al. (2011), G16: Galiazzo et al. (2016), F14:
Fornasier et al. (2014), L17: Leiva et al. (2017), R05: Rousselot et al.
(2015), HL09: Heinze & de Lahunta (2009), H19: Hromakina et al. (2019), T14:
Thirouin et al. (2014), S18: Safrit (2018), W97: Wisniewski et al. (1997).

Figure 1: Orbital distribution diagram showing semimajor axes ($a$), orbital
eccentricities ($e$) and orbital inclinations ($i$) of the objects used in
this study. Data taken from JPL Horizons. Figure 2: Histogram of _V_ -band
absolute magnitudes from JPL Horizons of the sample of objects with semimajor
axis less than Neptune’s $a<a_{N}$ (solid red) and greater than Neptune’s
$a>a_{N}$ (dashed blue) used in this study. Bin-size = 2 mag. Figure 3: Bar
plot of galactic latitude of observations for each object in ATLAS $c$ (upper
plot) and $o$ (lower plot). The red bar denotes the sky region spanning $\pm
10$ deg from the galactic equator.

##  4 Data and Photometry

We utilize observations made by the two ATLAS telescopes at Haleakalā and
Maunaloa. Data for our sample was obtained using the ATLAS Forced Photometry
Server333https://fallingstar-data.com/forcedphot/ (Shingles et al., 2021),
which fits a point-spread function (PSF) at an object’s position on the image
of an observation as predicted from its orbital ephemeris cataloged by the
Minor Planet Center, calculating the AB magnitude of the flux at that point.
The ATLAS data reduction includes an image subtraction pipeline, which matches
a high signal-to-noise reference image of the static sky (“wallpaper”) to the
location of a given observation’s image, and subtracts the wallpaper flux from
that image, creating a “difference image” containing only the flux not
previously present (Tonry et al., 2018a, b). We chose to use difference images
to reduce the effect of contaminating background stars and galaxies on the
brightness measurements of each object, which can become significant if an
object passes in front of or near a background star on the sky. We select all
ATLAS data up to and including the most recent observations at the time of
selection (2021 June 30), allowing us to obtain sufficient photometry to
generate phase curves for all 18 objects in our sample in both _c_ and _o_
filters. We deem any observation a detection if its apparent magnitude is
brighter than 1) the $5\sigma$ limiting magnitude of the image (to ensure the
observation was of good quality); and 2) the $3\sigma$ upper magnitude limit
derived from the flux uncertainty (ensuring an observed object could be
detected on the image). For every detection that satisfies these criteria, we
correct their Modified Julian Dates (MJDs) for light travel time. The ATLAS
photometry used in this analysis for all objects in our sample are listed in
Table 3.

Table 3: ATLAS photometry for each object.

Name | MJD | Phase | Apparent Magnitude | Apparent Magnitude Uncertainty | Filter | Reduced Magnitude | Flags
---|---|---|---|---|---|---|---
| (light travel time-corrected) | (deg) |  |  |  |  |
10199 Chariklo | 57511.477108 | 3.1056 | 18.516 | 0.17 | c | 6.745 | G
10199 Chariklo | 57891.519318 | 2.689 | 18.771 | 0.151 | o | 6.935 | G
10199 Chariklo | 57895.513409 | 2.5112 | 18.579 | 0.115 | c | 6.749 | G
10199 Chariklo | 57903.490523 | 2.1231 | 19.009 | 0.124 | c | 7.189 | G
10199 Chariklo | 57903.507429 | 2.1222 | 19.243 | 0.129 | c | 7.423 | G

Note. — This table is published in its entirety in the machine-readable
format. A portion is shown here for guidance regarding its form and content.

Note. — Key for flags: G - data used to measure phase curve parameters; C -
cometary activity flagged by detection algorithm; D - data not flagged by
algorithm yet obtained during extended cometary activity; O - opposition
effect.

##  5 Data Analysis

We detail the methods used to fit the phase curves for our sample in both
ATLAS $c$ and $o$ filters, search for rotational lightcurves in the ATLAS
data, detect for epochs of cometary activity, and re-fit the data corrected
for these effects.

###  5.1 Generating Phase Curves

All ATLAS apparent magnitude measurements were transformed to reduced
magnitudes to remove distance effects. The reduced magnitude $M$ is defined
as:

$M=m-5\log_{10}{(r\Delta)}$ (2)

where $m$ is the apparent magnitude, and $r$ and $\Delta$ are the heliocentric
and geocentric distances of the object at the time of each observation,
respectively. Distances and solar phase angles for each observation of a given
object were obtained from JPL Horizons. The uncertainty in reduced magnitude
was taken to be that of the measured apparent magnitude, due to the
comparatively negligible uncertainties of the geocentric and heliocentric
distances. Datapoints with uncertainties larger than 85% of all datapoints
were removed. These observations were likely taken during poor observing
conditions, causing their large uncertainties. Each object’s dataset was then
clipped to remove outlying points beyond 3-sigma from the median reduced
magnitude to exclude measurements from contaminating flux, poor image
subtraction, or poor calibration. All data in a given filter were treated as a
single bin of phase angle unless observations extended to phase angles ${>}20$
deg or ${<}0.1$ deg, where phase curve profiles become non-linear, in which
cases data were split into two or four bins of phase angle, respectively. All
data in each bin were sigma-clipped separately until convergence.

###  5.2 Phase Curve Fitting

Beyond very small phase angles ($\alpha\sim 0.1$ deg, Belskaya et al., 2008)
where the opposition effect is significant, KBO, Centaur, and JFC phase curves
exhibit approximately linear profiles when observed from Earth. We apply a
weighted linear fit to the phase curve of each object in each ATLAS filter,
using the _polyfit_ function from the _NumPy_ Python package (van der Walt et
al., 2011; Harris et al., 2020) to fit a 1st-degree polynomial to the reduced
magnitudes $M(\alpha)$ according to

$M(\alpha)=H+\alpha\beta$ (3)

where $\alpha$ is the solar phase angle of the object at the time of
measurement, $\beta$ is the linear phase coefficient, and $H$ is the absolute
magnitude of the object. We consider only data at phase angles $\alpha\geq
0.1$ deg to remove any influence of an opposition surge on the resulting
$\beta$ value. Each data point was weighted by the inverse of its magnitude
uncertainty. The best fit values of both $\beta$ and $H$ and their associated
uncertainties were computed via Monte Carlo simulation. We generate $10^{5}$
synthetic phase curves with synthetic reduced magnitude measurements
$M_{i,synthetic}$ calculated according to:

$M_{i,synthetic}=M_{i}+{rand[0,\delta M_{i}]}$ (4)

where $rand[0,\delta M_{i}]$ is a random number generated from a normal
distribution centred at zero of standard deviation equal to $\delta M_{i}$,
the uncertainty of each reduced magnitude measurement. The best fit $\beta$
and $H$ values were taken to be the medians of the distributions of all
$10^{5}$ results for each parameter. These distributions tended to be
approximately Gaussian in profile, thus we took the associated uncertainties
of these parameters to be half of the range that includes the central 95% of
their corresponding distributions, yielding $2\sigma$ uncertainty values.
Figure 4 shows an example for the KBO Quaoar of an ATLAS phase curve of our
sample and the associated uncertainty distributions. For objects for which
changes in brightness due to their periodic lightcurve or outbursts of
cometary activity can be identified (applicable to 5 objects in total), we
correct their datasets for these effects (see Sections 5.3 and 5.4 for
lightcurve and cometary activity correction, respectively) and reapply the
phase curve fitting algorithm to the corrected data.

Figure 4: Linear fit to phase curve of (50000) Quaoar in $c$ and $o$ filters,
with 2D histograms showing density plot of error distributions of linear phase
coefficient and absolute magnitude. Best fit function extended to $\alpha=0$
deg for ease of viewing the extrapolated absolute magnitude value and any
potential opposition surge. The complete figure set (18 images) is available
in the online published version of the paper.

###  5.3 Lightcurve Corrections

We aim to correct for the effect of rotational modulation on the phase curves
wherever possible by searching for periodicities corresponding to the period
of the rotational lightcurve of an object for all ATLAS data in our sample.
For most objects in our sample, the uncertainty in the reduced magnitude
measurements exceeds the literature peak-to-peak amplitude of the rotational
lightcurve. This obscures any rotational modulation, thus precluding any
correction of this effect on the phase curve of the object. However, for some
objects in our sample we observe substructure in the residuals to the best-
fitting linear phase curve function, which extends beyond the associated
uncertainties of the brightness measurements, potentially due to rotational
modulation.

To search for periodicities in the lightcurve, we use the Lomb-Scargle
periodogram algorithm (Lomb, 1976; Scargle, 1982) due to the irregular time
spacing of the ATLAS observations. For each filter, the best fit linear phase
curve is subtracted from each dataset. The resulting data, corrected also for
light travel time, were then passed into the Lomb Scargle periodogram function
(_LombScargle_) from the _astropy Python_ package (Astropy Collaboration et
al., 2013, 2018), weighted by the reduced magnitude uncertainty. Frequency
grids were chosen with minimum and maximum rotation frequencies of 0.1 day-1
(240 hr period) and 24 day-1 (1 hr period), with $10^{7}$ periodogram
evaluations between these frequency extremes. Potential rotation periods for
each object in the sample were searched for in the $c$ and $o$ datasets
separately. For a given filter, the maximum peak in the power spectrum was
taken to be the object’s best fitting period value, with its associated
uncertainty equal to half the difference between the period values
corresponding to the closest two stationary points either side the maximum
peak.

We highlight that the best-fit period value is subject to uncertainty due to
aliasing, background noise and uncertainties in the measured magnitudes
(VanderPlas, 2018). A method of quantifying the validity of a given rotation
period is to calculate the false alarm probability of the maximum peak of the
power spectrum - the probability that Gaussian noise could produce a signal of
equal height to a given peak in the power spectrum (VanderPlas, 2018). For the
rotation period value yielded for each filter, we calculated false alarm
probabilities, choosing the Baluev method (Baluev, 2008) for its suitable
accuracy and computational efficiency (VanderPlas, 2018). Only objects whose
periods in both filters came from peaks higher than the 95% false alarm
probability level (${>}2\sigma$ results) were considered to be plausible.

Large photometric uncertainties in the data, plus the generally fewer $c$
filter observations compared to the $o$-filter, could mean that significantly
different best-fit period values are yielded per filter. We set a criterion
that the period values from both filters must match to ${<}0.5$ hours to be
considered valid. For any object which fits this criterion, we check if their
rotation periods match values aliased to Earth’s sidereal period (12, 24, 48,
etc. hours) to ${<}0.1$ hours, to ensure we are not detecting periodicity due
to sampling rate. If so, we check the rotation phase coverage of the data when
phase-folded to the rotation period in each filter. If the phase-folded data
of an object are grouped into clusters with gaps in between, or cover ${<}50$%
of the rotation phase span, we reject that object’s rotation period value.

Three objects in our sample satisfy the above criteria for valid rotation
periods: Pluto, Haumea, and Hidalgo. For these objects, we hence consider only
the rotation period yielded by the filter with more datapoints ($o$). The
ATLAS rotation period for Pluto is consistent with values from literature
(Tholen & Tedesco, 1994; Tholen & Buie, 1997; Sheppard & Jewitt, 2002;
Schaefer et al., 2008; Buie et al., 2010; Benecchi et al., 2018) and, to
within uncertainties, the ATLAS rotation periods equal the half-values of the
literature rotation periods of Haumea (Rabinowitz et al., 2006; Lacerda et
al., 2008; Lellouch et al., 2010; Thirouin et al., 2010; Lockwood et al.,
2014) and Hidalgo (Hanuš et al., 2016; Santos-Sanz et al., 2017). We note that
uncertainties in our reduced magnitude measurements could obscure double-
peaked lightcurves of elongated or irregular-shaped objects, causing them to
appear as single-peaked and the rotation period value yielded by the Lomb-
Scargle algorithm to equate to half the true rotation period. Both Haumea
(Rabinowitz et al., 2006; Lacerda et al., 2008; Lellouch et al., 2010;
Thirouin et al., 2010; Lockwood et al., 2014) and Hidalgo (Ďurech et al.,
2007; Hanuš et al., 2016) exhibit shape elongation. Our measured period values
for these objects are consistent with the half the reported rotation periods,
thus we consider the true rotation period of these two objects to be twice the
value output from the Lomb-Scargle Periodogram. The ATLAS rotation period
values for Pluto, Haumea and Hidalgo, both those yielded by the maximum
periodogram peak and the final chosen rotation period values, are listed in
Table 5.3, in addition to those from literature.

Figure 5: _Upper row:_ Lomb-Scargle periodogram power spectra for 2012 TW236
in the $o$ (orange), $c$ (cyan), and combined $c$ and $o$ (blue) filters. The
black dashed and dotted lines represent the 95% and 99.7% false alarm
probability levels. _Lower row:_ The $o$ (orange, left), $c$ (cyan, centre),
and combined $c$ and $o$ (right) datasets of 2012 TW236, each phase-folded to
the rotation period value corresponding to maximum peak in the power spectrum.
The combined filter dataset shows a clear sinusoidal lightcurve.

Some rotation periods may not appear in individual filter datasets, but may be
discerned when the datasets are combined to augment sampling of any rotational
lightcurve. We repeat our Lomb-Scargle periodogram algorithm on both the ATLAS
$c$ and $o$ filter data combined. We assume that the objects in our sample
exhibit no variability in colour over their rotation periods, and correct the
dataset in each filter for color according to the difference in the median
reduced magnitudes of each filter per object. We consider only Lomb-Scargle
periodogram spectra maxima higher than the 95% false alarm probability level.
For objects which satisfy this criterion, we consider only those where the
data in both filters span the full rotation phase space (ensuring that the
rotation period value is not being driven by one filter only) and where the
standard deviation of the datapoints exceeds twice the median magnitude
uncertainty. If the object’s phase-folded lightcurve exhibits gaps in the
phase-folded lightcurve, and/or the rotation period value lies close to those
aliased to Earth’s sidereal period (12, 24, 48 etc. hours) to ${<}0.1$ hours,
we reject the object’s rotation period value. We recover rotation periods for
Pluto, Haumea, and Hidalgo, all consistent to those found by separate analysis
of each filter. We discover a rotation period value of $5.4749\pm 0.0015$
hours from the Lomb-Scargle periodogram for the JFC 2012 TW236. The Lomb-
Scargle periodogram power spectra and the datasets of 2012 TW236 phase-folded
to the obtained rotation periods are shown in Figure 5. Due to the small size
of this object, it is likely irregularly shaped with a double-peaked
rotational lightcurve, so we consider the rotation period of 2012 TW236 to be
$10.950\pm 0.003$ hours, to our knowledge the first measured value for this
object.

Figure 6: _c_ and _o_ phase curves of Haumea, before (upper plots) and after (lower plots) correction for rotational modulation. Linear best-fit functions plotted as black line, with residuals from the fit displayed beneath each plot. Table 4: Changes in phase coefficient and absolute magnitude in each ATLAS filter for objects whose datasets were corrected for rotational lightcurve.  Name |  $\beta_{c}$ before |  $\beta_{c}$ after |  $\beta_{o}$ before |  $\beta_{o}$ after
---|---|---|---|---
| lightcurve correction | lightcurve correction | lightcurve correction | lightcurve correction
| (mag/deg) | (mag/deg) | (mag/deg) | (mag/deg)
(134340) Pluto | $0.045\pm 0.001$ | $0.047\pm 0.001$ | $0.0460\pm 0.006$ | $0.0444\pm 0.006$
(136108) Haumea | $0.056\pm 0.016$ | $0.090\pm 0.016$ | $0.113\pm 0.010$ | $0.106\pm 0.011$
(944) Hidalgo | $0.0324\pm 0.0003$ | $0.0318\pm 0.0003$ | $0.0301\pm 0.0002$ | $0.0316\pm 0.0001$
(347449) 2012 TW236 | $0.042\pm 0.005$ | $0.049\pm 0.005$ | $0.038\pm 0.008$ | $0.048\pm 0.008$
Name |  $H_{c}$ before |  $H_{c}$ after |  $H_{o}$ before |  $H_{o}$ after
| lightcurve correction | lightcurve correction | lightcurve correction | lightcurve correction
| (mag/deg) | (mag/deg) | (mag/deg) | (mag/deg)
(134340) Pluto | $-0.7305\pm 0.0015$ | $-0.7313\pm 0.0014$ | $-1.1542\pm 0.0007$ | $-1.1534\pm 0.0007$
(136108) Haumea | $0.368\pm 0.014$ | $0.343\pm 0.015$ | $0.1011\pm 0.0097$ | $0.1124\pm 0.0097$
(944) Hidalgo | $11.051\pm 0.009$ | $11.074\pm 0.009$ | $10.776\pm 0.004$ | $10.744\pm 0.004$
(347449) 2012T W236 | $12.549\pm 0.041$ | $12.437\pm 0.042$ | $12.18\pm 0.05$ | $12.12\pm 0.05$
Table 5: Rotation Period obtained from ATLAS datasets in $o$ filter and
combined filters.

Name | $P$ | $P_{final}$ | Filter(s) |  $\Delta M$ $c$ |  $\Delta M$ $o$ | Literature $P$ | Literature $\Delta M$ | Source
---|---|---|---|---|---|---|---|---
| (hr) | (hr) |  | (mag) | (mag) | (hr) | (mag) |
(134340) Pluto | $153.28\pm 0.65$ | $153.28\pm 0.65$ | $o$ | $0.100\pm 0.004$ | $0.080\pm 0.002$ | $153.2328\pm 0.0048$ | $0.120\pm 0.006$ | B18
(136108) Haumea | $1.95767\pm 0.0001$ | $3.91534\pm 0.00019$ | $o$ | $0.19\pm 0.01$ | $0.205\pm 0.004$ | $3.915341\pm 0.000005$ | $0.28\pm 0.02$ | SS17
(944) Hidalgo | $5.031\pm 0.004$ | $10.06\pm 0.009$ | $o$ | $0.101\pm 0.031$ | $0.237\pm 0.014$ | $10.0630\pm 0.0003$ | $0.48$ | W97
(347449) 2012 TW236 | $5.4749\pm 0.0015$ | $10.950\pm 0.003$ |  $c$ and $o$ | $0.444\pm 0.072$ | $0.568\pm 0.116$ | … | … | …
|  |  | combined |  |  |  |  |

References. — Rotational period/amplitude sources: B18: Benecchi et al.
(2018), SS17: Santos-Sanz et al. (2017), W97: Wisniewski et al. (1997).

We phase-fold the datasets of Pluto, Haumea, Hidalgo, and 2012 TW236 in both
filters to their selected rotation period value. We fit a sinusoid of period
equal to the chosen rotation period value to the data using the _lmfit_ Python
package (Newville et al., 2014), weighted by the uncertainty of each
magnitude. This fitted function is subtracted off the reduced magnitude data
to eliminate the rotational modulation, and remove any outlying data in the
corrected dataset to 3-sigma. The linear phase curve model for the object is
refitted again on the corrected data to measure the phase coefficient and
absolute magnitude, corrected for the effect of the object’s rotation on its
axis. The effect of correcting for an object’s rotational lightcurve is
illustrated in Figure 6, showing the significant decrease in spread of the
data about the best fitting linear phase curve function. The root mean square
values of the residuals to the best-fitting linear function decrease from
0.096 mag to 0.066 mag in the $c$ filter and from 0.093 mag to 0.057 mag in
the $o$ filter when the effects of Haumea’s rotational lightcurve is removed.
Table 4 lists the phase coefficient and absolute magnitude values for each
object for which rotational modulation was corrected, before and after
correction. Table 5.3 lists the amplitudes of the best-fitting sinusoid
lightcurve function in both $c$ and $o$, the uncertainty being calculated from
the covariance matrix of the resulting fit. We note that our amplitudes of
rotational modulation differ significantly both between filters and from those
in literature, though the large uncertainties in ATLAS reduced magnitudes are
a possible cause of this.

###  5.4 Cometary Activity Search

The Centaur and JFC populations contain objects known to exhibit cometary
activity (Lowry et al., 2008; Peixinho et al., 2020, and references therein).
It is thus plausible that some objects in our sample may have exhibited
activity, forming a coma which may partly or fully obscure an object’s surface
from direct observation. This would brighten the object’s apparent magnitude,
and could alter an object’s phase curve for a period of time. Given the number
of datapoints per object, we can use the ATLAS phase curves to search for
deviations that may be indicative of possible cometary outgassing or dust
production. To search for cometary outbursts, we select for observations that
were significantly brighter than predicted from the measured phase curve. We
assume that most of an object’s data are taken from epochs of (relative)
quiescence potentially punctuated by shorter epochs of activity or outbursts.
We consider this assumption to be valid as most of our objects exhibit reduced
magnitudes which are consistent in value across separate apparitions. We
subtract the best-fitting phase curve function from the dataset (corrected for
rotational lightcurve if possible), and select observations brighter than
$2\sigma$ from the object’s median brightness across the ATLAS baseline.

Active epochs of Centaurs and JFCs exhibit timescales ranging from weeks to
years (Peixinho et al., 2020, and references therein). Such activity would
manifest as a series of bright observations across consecutive nights. To
detect such cometary activity, we utilize the ability of ATLAS to observe a
given target multiple times per night, and select nights containing at least 3
observations, of which ${\geq}75\%$ have magnitudes brighter than $2\sigma$
from the object’s median brightness. We term such nights of observations
‘bright nights’ henceforth. We further analyse these bright nights to
determine if cometary activity is the source of the brightening. We note that
objects may exhibit activity which do not produce short increases in
brightness, and we would not be able to detect this activity using the
described algorithm. To detect any such lower-level cometary activity, we
consider only observations whose phase-corrected reduced magnitude lie
${<}2\sigma$ from the object’s sigma-clipped median brightness. We consider
each datapoint in turn, and bin all data taken 30 days after each observation.
If at least $14$ observations (the approximate average number of ATLAS
exposures per week) are taken in any given interval, we consider the median
brightness of all data in that interval. If the binned median brightness
exceeds the object’s sigma-clipped median brightness by $0.2$ mag and $1$
standard deviation of the binned data, we consider all data in this interval
to be a long outburst. We remove any data which corresponded to epochs of
cometary activity from the datasets and reapply the phase curve fitting
algorithm to the remaining data. If any instance of cometary activity is found
to extend over time, we remove all data in both filters that falls within the
time range of the activity.

###  5.5 HG1G2 Model

Several objects in our sample were observed across phase angle ranges
comparable to those of main belt asteroids. This allowed us to fit their data
from ATLAS with phase curve functions used for these asteroids. The resulting
best-fit parameters for our sample could then be compared to those of the
asteroid population. We utilize the three-parameter $HG_{1}G_{2}$ phase curve
model developed by Muinonen et al. (2010), where $H$ is the absolute magnitude
of the phase curve, and $G_{1}$ and $G_{2}$ both quantify the slopes of the
phase angle functions that govern the overall phase curve shape, analogous to
the linear phase coefficient $\beta$. We utilize this function as per the
findings of Mahlke et al. (2021) that taxonomic complexes could be
distinguished using the $HG_{1}G_{2}$ function applied to ATLAS data. Mahlke
et al. (2021) also found the related $HG_{12}^{*}$ asteroid phase curve
function (Penttilä et al., 2016) failed to distinguish between complexes for
ATLAS data, thus we do not apply the $HG_{12}^{*}$ function to our data. We
select only phase curves in a given filter to fit with the $HG_{1}G_{2}$ model
which satisfy the following criteria:

  1. 1.

Observations of $\alpha\geq 12$ deg, to ensure we are sampling the section of
the phase curve where curvature becomes significant and the accuracy of linear
fits is reduced.

  2. 2.

Phase curves must contain ${\geq}35$ datapoints at phase angles
$0\leq\alpha\leq 5$ deg, to ensure we have sufficient sampling of the region
where the opposition surge can become significant, which may affect the
profile of the fitted phase curve model.

The $o$ filter phase curves of Narcissus, Hidalgo, and 2014 QA43, and the $c$
filter phase curve of Hidalgo satisfy these criteria. We apply the
$HG_{1}G_{2}$ model to the reduced magnitude values of these datasets using
the _HG1G2_ function from the _Python_ photometry module _sbpy.photometry_
(Mommert et al., 2019). We use a Monte Carlo algorithm to measure the absolute
magnitude and $G_{1}$ and $G_{2}$ parameters, generating $10^{5}$ synthetic
phase curves by varying each reduced magnitude value by a random fraction of
its associated uncertainty (assuming a normal uncertainty distribution) and
fitting the $HG_{1}G_{2}$ model to each synthetic dataset. The final values
for $H$, $G_{1}$, and $G_{2}$ are the median of the resulting distribution of
all $10^{5}$ values of each parameter, with their associated uncertainties
being half of the range between the values which enclose 95% of all values in
the distribution, yielding $2\sigma$ uncertainties. If an object’s data
satisfy the requirements for fitting with the $HG_{1}G_{2}$, we retroactively
use the resulting fit instead of the linear model for eliminating phase angle
dependence when searching for rotation periods and cometary activity in the
full dataset. For the object Narcissus, due to near-identical resulting chi-
square values, we use both the linear and $HG_{1}G_{2}$ fits when searching
for rotation period and cometary activity, finding that the choice of fitted
function has no effect on the results yielded for Narcissus by either
algorithm.

##  6 Results and Discussion

###  6.1 Linear Phase Coefficients - Full Phase Angle Range

We measured the linear phase coefficient $\beta$ and absolute magnitude $H$ in
both ATLAS $c$ and $o$ filters for all 18 objects in our sample across the
full range of phase angles spanned by each dataset (excluding opposition
surges at $\alpha{<}0.1$ deg). Table 6 lists our measured $\beta$ and $H$
values and $2\sigma$ errors for all objects in our sample across the full
phase angle range. Figures for all $c$ and $o$ ATLAS phase curves and the
2-dimensional (2D) histograms of the fitted phase curve parameters are shown
in Figure 4 and its online Figure set. Linear fits applied to the full range
of phase angles spanned by the data were used to correct for variability due
to object rotation and cometary activity for all objects, apart from the $o$
filter datasets of Narcissus and 2014 QA43 and the $c$ and $o$ filter datasets
of Hidalgo which were corrected using the $HG_{1}G_{2}$ function. Four such
objects - 2012 TW236, 2014 QA43, 2016 ND21, and Narcissus - have datasets
which extend to phase angles beyond $\alpha\sim 10$ deg where non-linearity
could become significant, but the small number of objects this applies to
means we consider a linear fit suitable for variability correction. However,
for the remainder of our analysis, including a comparison to the Kuiper belt
and is associated populations, we restrict ourselves to measuring phase
coefficients and absolute magnitudes for a fixed phase angle range of
$0.1\leq\alpha\leq 6$ deg, as detailed in section 6.2.

Table 6: Linear phase coefficients $\beta$ and absolute magnitudes $H$ for all
objects in our sample in both ATLAS broadband filters _c_ and _o_ , measured
over the full range of phase angles for each filter dataset, with associated
$2\sigma$ uncertainties, with phase angle range over which the parameters were
measured in each filter. Parameter values measured across the maximum range of
phase angles in each filter, with data corrected for rotational lightcurves
and cometary activity wherever possible.

Object | $\beta_{c}$ | $H_{c}$ |  $\alpha$ range ($c$ filter) | $N_{c}$ | $\beta_{o}$ | $H_{o}$ |  $\alpha$ range ($o$ filter) | $N_{o}$ | Light | Cometary
---|---|---|---|---|---|---|---|---|---|---
| (mag/deg) | (mag) | (deg) |  | (mag/deg) | (mag) | (deg) |  | curve | activity
|  |  |  |  |  |  |  |  | correction? | correction?
KBOs
(136199) Eris | $0.043\pm 0.088$ | $-1.012\pm 0.042$ | 0.1206-0.6058 | 239 | $0.016\pm 0.043$ | $-1.307\pm 0.018$ | 0.1214-0.6068 | 738 | No | No
(136108) Haumea | $0.089\pm 0.017$ | $0.343\pm 0.014$ | 0.5397-1.1567 | 313 | $0.106\pm 0.01$ | $0.112\pm 0.01$ | 0.544-1.1548 | 1043 | Yes | No
(38628) Huya | $-0.015\pm 0.085$ | $4.974\pm 0.096$ | 0.5515-1.9664 | 68 | $0.058\pm 0.031$ | $4.476\pm 0.04$ | 0.5394-2.0139 | 340 | No | No
(136472) Makemake | $0.005\pm 0.015$ | $0.107\pm 0.013$ | 0.5209-1.1062 | 353 | $0.034\pm 0.009$ | $-0.247\pm 0.008$ | 0.5193-1.1087 | 1088 | No | No
(90482) Orcus | $0.18\pm 0.065$ | $2.238\pm 0.059$ | 0.4008-1.2082 | 271 | $0.07\pm 0.04$ | $2.07\pm 0.037$ | 0.4045-1.2084 | 541 | No | No
(134340) Plutoa | $0.045\pm 0.001$ | $-0.728\pm 0.002$ | 0.1085-1.7313 | 323 | $0.0448\pm 0.0006$ | $-1.1536\pm 0.0007$ | 0.1152-1.7362 | 886 | Yes | No
(50000) Quaoar | $0.029\pm 0.065$ | $2.792\pm 0.057$ | 0.1964-1.341 | 177 | $0.075\pm 0.028$ | $2.221\pm 0.025$ | 0.1925-1.3402 | 590 | No | No
Centaurs
(459865) 2013 XZ8 | $0.026\pm 0.017$ | $9.937\pm 0.087$ | 2.8174-6.8551 | 127 | $0.036\pm 0.011$ | $9.524\pm 0.052$ | 2.1459-6.9036 | 329 | No | No
(54598) Bienor | $0.058\pm 0.025$ | $7.586\pm 0.07$ | 1.3222-4.0222 | 172 | $0.062\pm 0.014$ | $7.248\pm 0.041$ | 1.3103-4.0452 | 431 | No | No
(10199) Chariklo | $0.035\pm 0.019$ | $6.939\pm 0.044$ | 0.1041-3.6314 | 170 | $0.049\pm 0.01$ | $6.477\pm 0.022$ | 0.1535-3.5772 | 454 | No | No
(2060) 95P/Chiron | $0.093\pm 0.011$ | $5.69\pm 0.023$ | 0.2194-3.093 | 234 | $0.097\pm 0.006$ | $5.433\pm 0.011$ | 0.1688-3.1055 | 834 | No | Yes
JFCs
(347449) 2012 TW236 | $0.049\pm 0.005$ | $12.437\pm 0.042$ | 2.0318-17.0886 | 69 | $0.048\pm 0.008$ | $12.123\pm 0.05$ | 1.9453-14.469 | 58 | Yes | No
(501585) 2014 QA43 | $0.063\pm 0.015$ | $11.781\pm 0.131$ | 3.7953-11.9905 | 81 | $0.064\pm 0.005$ | $11.379\pm 0.041$ | 1.2204-11.8886 | 270 | No | Yesb
2016 ND21 | $0.032\pm 0.005$ | $12.635\pm 0.043$ | 0.1533-14.7265 | 91 | $0.048\pm 0.003$ | $12.063\pm 0.027$ | 2.0524-15.0726 | 333 | No | No
(944) Hidalgo | $0.0318\pm 0.0003$ | $11.074\pm 0.009$ | 2.1269-30.5594 | 178 | $0.0316\pm 0.0002$ | $10.744\pm 0.004$ | 1.1296-30.5823 | 730 | Yes | No
(37117) Narcissus | $0.048\pm 0.003$ | $13.573\pm 0.023$ | 2.5505-18.5841 | 69 | $0.044\pm 0.002$ | $13.161\pm 0.021$ | 1.5666-18.3289 | 204 | No | No
Transition Objects
(349933) 2009 YF7 | $0.032\pm 0.018$ | $11.117\pm 0.099$ | 1.9905-8.5739 | 110 | $0.036\pm 0.01$ | $10.747\pm 0.064$ | 1.9827-8.9607 | 285 | No | No
(60558) 174P/Echeclus | $0.083\pm 0.008$ | $9.804\pm 0.039$ | 0.9073-8.6663 | 111 | $0.064\pm 0.003$ | $9.403\pm 0.015$ | 0.383-9.5528 | 363 | No | Yes

^a^afootnotetext: Values are for the Pluto/Charon system, whose components
cannot be resolved at the ATLAS pixel scale.

^b^bfootnotetext: While we do not detect any confirmed instance of cometary
activity for 2014 QA43, we nevertheless remove all data classified by our
activity search algorithm as occurring during an instance of sustained
brightening. It is not clear if this brightening is due to cometary activity
not visible in the ATLAS images or due to changing aspect angle.

###  6.2 Linear Phase Coefficients - Limited Phase Angle Range

The objects in our sample span semimajor axes of $5.7$ au $\leq a\leq$ $68.1$
au, resulting in a wide range of maximum phase angles observable from Earth.
ATLAS observations thus sample different sections of the phase curves
depending on an object’s heliocentric distance. To compare phase curves over a
fixed range of phase angle, we restrict our datasets (corrected for rotational
modulation and cometary activity where possible and excluding data at
$\alpha\leq 0.1$ deg) to include only observations taken at phase angles
$\alpha\leq{6}$ deg, and consider only objects with ${>}50$ datapoints in this
range in both $c$ and $o$. We apply our methods to the resulting phase curves
of 13 objects. The resulting absolute magnitudes and phase coefficients in $c$
and $o$ are listed in Table 7. Comparing the resulting values to those
measured over the maximum phase angle range, we find that the phase
coefficient and absolute magnitude values change beyond $2\sigma$ error bars
only for the Transition Objects Echeclus and 2009 YF7 in the $c$ filter.
Figures for all $c$ and $o$ ATLAS phase curves for data in phase angle range
$0.1\leq\alpha\leq{6}$ are shown in Figure 7 and its online Figure set.

Figure 7: Linear fit to phase curve of (50000) Quaoar in ATLAS $o$ and $c$
filters for data in phase angle range $0.1\leq\alpha\leq 6$ deg. Best fit
function extended to $\alpha=0$ deg for ease of viewing the extrapolated
absolute magnitude value and any potential opposition surge. The complete
figure set (13 images) is available in the online published version of the
paper. Table 7: Linear phase coefficients $\beta$ and absolute magnitudes $H$
for all objects with ${>}50$ data points recorded at phase angles
$0.1\leq\alpha\leq 6$ deg in both ATLAS broadband filters _c_ and _o_ , with
associated $2\sigma$ uncertainties, and number of data points $N$ at
$0.1\leq\alpha\leq 6$ deg in each ATLAS filter.

Object | $\beta_{c}$ | $H_{c}$ | $N_{c}$ | $\beta_{o}$ | $H_{o}$ | $N_{o}$ | Light | Cometary
---|---|---|---|---|---|---|---|---
| (mag/deg) |  | ($\alpha\leq 5$ deg) | (mag/deg) |  | ($\alpha\leq 5$ deg) | curve | activity
|  |  |  |  |  |  | correction? | correction?
KBOs
(136199) Eris | $0.043\pm 0.088$ | $-1.012\pm 0.042$ | 239 | $0.016\pm 0.043$ | $-1.307\pm 0.018$ | 738 | No | No
(136108) Haumea | $0.089\pm 0.017$ | $0.343\pm 0.014$ | 313 | $0.106\pm 0.01$ | $0.112\pm 0.01$ | 1043 | Yes | No
(38628) Huya | $-0.015\pm 0.085$ | $4.974\pm 0.096$ | 68 | $0.058\pm 0.031$ | $4.476\pm 0.040$ | 340 | No | No
(136472) Makemake | $0.005\pm 0.015$ | $0.107\pm 0.013$ | 353 | $0.034\pm 0.009$ | $-0.247\pm 0.008$ | 1088 | No | No
(90482) Orcus | $0.18\pm 0.065$ | $2.238\pm 0.059$ | 271 | $0.07\pm 0.04$ | $2.07\pm 0.037$ | 541 | No | No
(134340) Plutoa | $0.045\pm 0.001$ | $-0.728\pm 0.002$ | 323 | $0.0448\pm 0.0006$ | $-1.1536\pm 0.0007$ | 886 | Yes | No
(50000) Quaoar | $0.029\pm 0.065$ | $2.792\pm 0.057$ | 177 | $0.075\pm 0.028$ | $2.221\pm 0.025$ | 590 | No | No
Centaurs
(459865) 2013 XZ8 | $0.024\pm 0.023$ | $9.948\pm 0.105$ | 85 | $0.031\pm 0.016$ | $9.546\pm 0.069$ | 236 | No | No
(54598) Bienor | $0.058\pm 0.025$ | $7.586\pm 0.07$ | 172 | $0.062\pm 0.014$ | $7.248\pm 0.041$ | 431 | No | No
(10199) Chariklo | $0.035\pm 0.019$ | $6.939\pm 0.044$ | 170 | $0.049\pm 0.01$ | $6.477\pm 0.022$ | 454 | No | No
(2060) 95P/Chiron | $0.093\pm 0.011$ | $5.69\pm 0.023$ | 234 | $0.097\pm 0.006$ | $5.433\pm 0.011$ | 834 | No | Yes
Transition Objects
(60558) Echeclus | $0.094\pm 0.012$ | $9.769\pm 0.048$ | 81 | $0.069\pm 0.006$ | $9.394\pm 0.018$ | 252 | No | Yes
(349933) 2009 YF7 | $0.113\pm 0.033$ | $10.768\pm 0.159$ | 85 | $0.067\pm 0.027$ | $10.611\pm 0.125$ | 137 | No | No

^a^afootnotetext: Values are for the Pluto/Charon system, whose components
cannot be resolved at the ATLAS pixel scale.

We note that our $\beta$ values differ by ${>}2\sigma$ between the ATLAS
filters for many objects. Significant variation of $\beta$ with wavelength has
been previously reported by several multi-filter phase curve studies of
Centaurs and KBOs (Buratti et al., 2003; Rabinowitz et al., 2007; Schaefer et
al., 2009; Alvarez-Candal et al., 2016; Ayala-Loera et al., 2018; Alvarez-
Candal et al., 2019). Figure 8 shows the ATLAS $\beta$ values compared to
those from the literature in the Johnson-Cousins _BVRI_ photometric system,
for which the ATLAS filters $c$ and $o$ exhibit significant overlap in
wavelength range with $B$ and $V$ filters ($c$ filter) and $R$ and $I$ filters
($o$ filter), respectively. Though the ATLAS $\beta$ values often differ in
value from the literature, for most objects they overlap within the $2\sigma$
error bars or lie between the extremes of the range of the literature values
for a given object. Our ATLAS linear phase coefficients are consistent with
those from the literature, as expected, and we attribute any disparity to be
due to different wavelength ranges of the filters of the ATLAS and Johnson-
Cousins photometric systems. We also note that our measured phase coefficient
and absolute magnitudes for Pluto are those of the Pluto-Charon system, the
components of which cannot be resolved at the ATLAS pixel scale.

Figure 9 shows the best-fitting linear phase curve models for all objects in
our sample, divided by dynamical class and photometric filter. The $\beta$
values indicate relatively flat phase curves across our entire sample, with
values ranging from $-0.01$ to $0.18$ mag/deg in $c$, and $0.02$ to $0.106$
mag/deg in $o$. Such flat phase curves have been previously observed for
several JFCs, Centaurs, and Pluto-sized KBOs (Rabinowitz et al., 2006, 2007;
Kokotanekova et al., 2017, 2018; Verbiscer et al., 2022) and are thought to be
indicative of low-albedo organic-rich Centaur and JFC surfaces (Nelson et al.,
2000; Shkuratov et al., 2002; Cruikshank & Dalle Ore, 2003; Rabinowitz et al.,
2007) and high-albedo KBO surfaces covered in granular volatile ices (Brown et
al., 2005; Licandro et al., 2006; Rabinowitz et al., 2006, 2007; Trujillo et
al., 2007). Furthermore, we find that for both the $c$ and $o$ filters, the
phase coefficient values of the Centaurs and Transition Objects all fall in
the same range as those of the KBO population when measured across the same
phase angle range $0.1\leq\alpha\leq 6$ deg. All but one of our sample exhibit
positive phase coefficient slopes in both ATLAS filters. Huya exhibits a
marginally negative phase coefficient in the $c$ filter, as shown in Figure 10
but is consistent within $2\sigma$ of having a flat or positive slope.
Alvarez-Candal et al. (2016); Ayala-Loera et al. (2018); Alvarez-Candal et al.
(2019) reported negative phase coefficients for several objects in their
sample, including Chiron, Eris, Haumea, Huya, and Makemake. $\beta<0$ would
imply an unphysical dimming near opposition. Alvarez-Candal et al. (2016) and
Ayala-Loera et al. (2018) speculated this effect may be due to unidentified
ring systems, epochs of cometary activity, or poorly corrected rotational
modulation. We instead measure phase coefficient values for these objects
which are either positive slopes or consistent with positive slopes to within
$2\sigma$. Furthermore, Huya’s negative $c$ filter phase coefficient is
derived from the second sparsest dataset in our sample of $N_{c}=68$
observations, comparable in size to the $V$ filter dataset from Ayala-Loera et
al. (2018) ($N_{V}=45$) which also yielded a negative phase coefficient value.
This tendency towards positive $\beta$ values with increasing dataset size
suggests negative phase coefficient values are most likely due to small
datasets with large scatter which are unable to sufficiently sample the phase
curve profile, and not a real property of these objects.

Figure 8: _Upper plots:_ Linear phase coefficients in ATLAS $c$ (_left_ ,
black squares) and $o$ (_right_ , black squares), calculated from the phase
angle range $0.1\leq\alpha\leq 6$ deg, with literature values in Johnson-
Cousins broadband filters (blue: $B$, green: $V$, red: $R$, grey: $I$) with
which ATLAS filters exhibit significant overlap. _Lower plots:_ Upper plots
magnified for clarity. AC16: Alvarez-Candal et al. (2016), AL18: Ayala-Loera
et al. (2018), Bu03: Buratti et al. (2003), B97: Buie et al. (1997), B06:
Bagnulo et al. (2006), R05: Rousselot et al. (2005), R06: Rabinowitz et al.
(2006), R07: Rabinowitz et al. (2007), SJ02: Sheppard & Jewitt (2002), SR02:
Schaefer & Rabinowitz (2002).  Figure 9: Best-fit linear phase curves for all
objects in our sample, divided by filter ($c$ filter left, $o$ filter right)
and dynamical classification, spanning the phase angle range of each dataset
in each filter. All phase curves are normalized to 0 mag at opposition
($\alpha=0$ deg) to enable comparison of slope values. Figure 10: Linear fit
to phase curve of (38628) Huya in $c$ and $o$ filters, with 2D histograms
showing density plot of error distributions of linear phase coefficient and
absolute magnitude. Best fit function extended to $\alpha=0$ deg for ease of
viewing any potential opposition surge. Note the nominally negative
$\beta_{c}$ value, whose associated uncertainty nevertheless makes it
consistent with a flat sloped phase curve, consistent with atmosphereless
surface reflectance.

###  6.3 Correlations

We explore potential correlations between phase coefficients $\beta_{c}$ and
$\beta_{o}$, absolute magnitudes $H_{c}$ and $H_{o}$, color index
$H_{c}-H_{o}$, and relative phase coefficients $\beta_{c}-\beta_{o}$ derived
from the ATLAS phase curves. We analyse parameter values measured across the
phase angle range $0.1\leq\alpha\leq 6$ deg to ensure all parameters are
measured over the same region of the phase curve. We make use of the Spearman
rank correlation test to search for monotonic relationships between pairs of
variables, using the _spearmanr_ function from the python module _scipy_
(Virtanen et al., 2020). The _spearmanr_ function calculates two parameters.
The first is the Spearman rank correlation coefficient, $r_{s}$, which
quantifies the strength and direction of any detected monotonic relationship
between two variables; $|r_{s}|\rightarrow 1$ defines a strong correlation,
whereas $r_{s}\approx 0$ implies no correlation. The second parameter is the
null hypothesis probability value of that correlation, or p-value,
$P_{r_{s}}$, which quantifies the probability that any detected correlation is
derived from an inherently uncorrelated distribution of data. We note however
that the Spearman rank correlation test itself does not consider uncertainties
in the analysed variables. To account for this, for every parameter pair
tested, we perform a Monte Carlo simulation on the parameter values. For each
data point, we vary the nominal value within associated uncertainties,
assuming every uncertainty follows a normal distribution with a standard
deviation equal to the nominal value of the uncertainty itself, perform the
rank statistic on the resulting synthetic values, and repeat $10^{5}$ times.
We accumulate all the associated p-values and calculate the percentage of
simulations for which $P_{r_{s}}>0.05$ i.e. we could not rule out an
uncorrelated population to 95% probability. We consider a correlation to exist
if $|r_{s}|\geq 0.5$ and ${<}5\%$ of simulations return $P_{r_{s}}>0.05$. A
correlation that satisfies one of these criteria is considered tentative, and
we reject any correlation if neither is satisfied or $P_{r_{s}}>95\%$. We
apply the Spearman rank correlation test to our full sample and subsamples:
Centaurs/JFCs/Transition Objects; KBOs; dwarf planets ($H\leq 3$); and non-
dwarf planet objects from the KBO, Centaur, JFC, and Transition Object
populations ($H>3$). Figure 11 shows plots of the tested ATLAS parameters for
the full object sample, the Centaurs, JFCs, Transition Objects, the KBOs, the
dwarf planets, and the non-dwarf planet objects in our sample.

Figure 11: Plots of tested parameter pairs derived from ATLAS observations for
our full object sample (first column); Centaurs, JFCs, and Transition Objects
(second column); KBOs (third column); dwarf planets (fourth column); and non-
dwarf planets (last column).

Table 8 lists the parameters from our Spearman rank correlation tests. We find
no strong correlations between $\beta_{c}$ and $H$, $\beta_{o}$ and $H$, and
$\beta_{o}$ and $H_{c}-H_{o}$ for the full sample and all sub-populations.
However, we find a tentative negative correlation between $\beta_{c}$ and
$H_{c}-H_{o}$ for the full sample and all sub-populations as we cannot reject
the null hypothesis to ${>}95\%$ confidence. A tentative negative correlation
between $\beta_{o}$ and $H_{c}-H_{o}$ is also found for the
Centaur/JFC/Transition Object and non-dwarf planet populations. We also find a
tentative negative correlation between $\beta_{c}-\beta_{o}$ and $H_{c}-H_{o}$
for the full sample, the KBO population, and the non-dwarf planet population,
yet no significant correlation between these parameters for the dwarf planet
population, and a tentative correlation that can almost be ruled out to $95\%$
confidence for the Centaur/JFC/Transition Objects sub-population. No strong
correlation is identified for any tested parameter pair. We also find no
significant change in correlation results when applying or correlation search
algorithm to parameter values derived from the full phase angle range.

Table 8: Results of correlation investigation using ATLAS data in phase angle range $0.1\leq\alpha\leq 6$ deg. Spearman rank correlation coefficients $r_{s}$, associated null hypothesis probability values $P_{r_{s}}$, percentage of Monte Carlo simulations where $P_{r_{s}}>0.05$ (i.e. we cannot reject a random distribution to 95% confidence), and number of datapoints analysed for correlation tests between phase curve parameters using ATLAS data.  Parameters | $r_{s}$ | $P_{r_{s}}$ | % Simulations with $P_{r_{s}}>0.05$ | N | Correlation?
---|---|---|---|---|---
Full Sample
$\beta_{c}$ vs $H_{c}$ | 0.214 | 0.482 | 99.985 | 13 | No
$\beta_{o}$ vs $H_{o}$ | 0.159 | 0.603 | 99.99 | 13 | No
$\beta_{c}$ vs $H_{c}-H_{o}$ | -0.747 | 0.003 | 13.931 | 13 | Tentative
$\beta_{o}$ vs $H_{c}-H_{o}$ | -0.297 | 0.325 | 95.694 | 13 | No
$\beta_{c}-\beta_{o}$ vs $H_{c}-H_{o}$ | -0.648 | 0.017 | 52.726 | 13 | Tentative
Centaurs/JFCs/Transition Objects
$\beta_{c}$ vs $H_{c}$ | 0.257 | 0.623 | 100.0 | 6 | No
$\beta_{o}$ vs $H_{o}$ | -0.314 | 0.544 | 99.792 | 6 | No
$\beta_{c}$ vs $H_{c}-H_{o}$ | -0.771 | 0.072 | 63.171 | 6 | Tentative
$\beta_{o}$ vs $H_{c}-H_{o}$ | -0.657 | 0.156 | 73.967 | 6 | Tentative
$\beta_{c}-\beta_{o}$ vs $H_{c}-H_{o}$ | -0.771 | 0.072 | 94.894 | 6 | Tentative
KBOs
$\beta_{c}$ vs $H_{c}$ | -0.286 | 0.535 | 99.94 | 7 | No
$\beta_{o}$ vs $H_{o}$ | 0.643 | 0.119 | 99.538 | 7 | No
$\beta_{c}$ vs $H_{c}-H_{o}$ | -0.75 | 0.052 | 67.321 | 7 | Tentative
$\beta_{o}$ vs $H_{c}-H_{o}$ | -0.036 | 0.939 | 99.975 | 7 | No
$\beta_{c}-\beta_{o}$ vs $H_{c}-H_{o}$ | -0.786 | 0.036 | 72.76 | 7 | Tentative
Dwarf Planets
$\beta_{c}$ vs $H_{c}$ | 0.143 | 0.787 | 99.3 | 6 | No
$\beta_{o}$ vs $H_{o}$ | 0.771 | 0.072 | 98.216 | 6 | No
$\beta_{c}$ vs $H_{c}-H_{o}$ | -0.714 | 0.111 | 68.087 | 6 | Tentative
$\beta_{o}$ vs $H_{c}-H_{o}$ | -0.086 | 0.872 | 99.969 | 6 | No
$\beta_{c}-\beta_{o}$ vs $H_{c}-H_{o}$ | -0.714 | 0.111 | 97.497 | 6 | No
Non-Dwarf Planets
$\beta_{c}$ vs $H_{c}$ | 0.536 | 0.215 | 100.0 | 7 | No
$\beta_{o}$ vs $H_{o}$ | -0.071 | 0.879 | 97.804 | 7 | No
$\beta_{c}$ vs $H_{c}-H_{o}$ | -0.857 | 0.014 | 43.9 | 7 | Tentative
$\beta_{o}$ vs $H_{c}-H_{o}$ | -0.679 | 0.094 | 79.644 | 7 | Tentative
$\beta_{c}-\beta_{o}$ vs $H_{c}-H_{o}$ | -0.857 | 0.014 | 82.876 | 7 | Tentative

We also apply our algorithm to the data of Ayala-Loera et al. (2018), who
reported phase coefficients and absolute magnitudes for 114 objects in
Johnson-Cousins $V$ and $R$ filters, which exhibit overlap in wavelength range
with ATLAS $c$ and $o$. We select only objects from their study with $\beta$
values consistent with positive to within $2\sigma$ and divide their object
sample according to our definitions. Table 9 lists the results from applying
our algorithm to their data. For $\beta$ and $H$, we find tentative negative
correlations for the Centaurs/JFCs/Transition Objects sub-population in both
the $V$ and $R$ filters. For $\beta$ and $H_{V}-H_{R}$, we find no correlation
in V for any sub-population, yet find a tentative positive correlation for the
Centaurs/JFCs/Transition Objects and the dwarf planets sub-population. For
$\beta_{V}-\beta_{R}$ and $H_{V}-H_{R}$, we find highly significant negative
correlations for the full sample, the KBOs, and the non-dwarf planet sub-
populations

Table 9: Results of correlation investigation using data from Ayala-Loera et al. (2018). Spearman rank correlation coefficients $r_{s}$, associated null hypothesis probability values $P_{r_{s}}$, percentage of Monte Carlo simulations where $P_{r_{s}}>0.05$ (i.e. we cannot reject a random distribution to 95% confidence), number of datapoints $N$ analysed for correlation tests between phase curve parameters from Ayala-Loera et al. (2018), and if the parameters exhibit a correlation.  Parameters | $r_{s}$ | $P_{r_{s}}$ | % Simulations with $P_{r_{s}}>0.05$ | N | Correlation?
---|---|---|---|---|---
Full Sample
$\beta_{V}$ vs $H_{V}$ | -0.278 | 0.005 | 18.208 | 100 | No
$\beta_{R}$ vs $H_{R}$ | -0.176 | 0.081 | 77.217 | 99 | No
$\beta_{V}$ vs $H_{V}-H_{R}$ | -0.219 | 0.037 | 81.36 | 91 | No
$\beta_{R}$ vs $H_{V}-H_{R}$ | 0.408 | 0.0 | 10.796 | 91 | No
$\beta_{V}-\beta_{R}$ vs $H_{V}-H_{R}$ | -0.747 | 0.0 | 0.133 | 91 | Yes
Centaurs/JFCs/Transition Objects
$\beta_{V}$ vs $H_{V}$ | -0.508 | 0.026 | 14.087 | 19 | Tentative
$\beta_{R}$ vs $H_{R}$ | -0.521 | 0.022 | 13.153 | 19 | Tentative
$\beta_{V}$ vs $H_{V}-H_{R}$ | -0.449 | 0.054 | 90.363 | 19 | No
$\beta_{R}$ vs $H_{V}-H_{R}$ | -0.573 | 0.010 | 75.758 | 19 | Tentative
$\beta_{V}-\beta_{R}$ vs $H_{V}-H_{R}$ | -0.307 | 0.201 | 84.616 | 19 | No
KBOs
$\beta_{V}$ vs $H_{V}$ | -0.152 | 0.177 | 99.297 | 81 | No
$\beta_{R}$ vs $H_{R}$ | -0.110 | 0.330 | 99.959 | 80 | No
$\beta_{V}$ vs $H_{V}-H_{R}$ | -0.271 | 0.022 | 75.219 | 72 | No
$\beta_{R}$ vs $H_{V}-H_{R}$ | 0.418 | 0.0 | 19.681 | 72 | No
$\beta_{V}-\beta_{R}$ vs $H_{V}-H_{R}$ | -0.802 | 0.0 | 0.248 | 72 | Yes
Dwarf Planets
$\beta_{V}$ vs $H_{V}$ | 0.143 | 0.787 | 100.0 | 6 | No
$\beta_{R}$ vs $H_{R}$ | 0.6 | 0.208 | 99.623 | 6 | No
$\beta_{V}$ vs $H_{V}-H_{R}$ | 0.429 | 0.397 | 86.778 | 6 | No
$\beta_{R}$ vs $H_{V}-H_{R}$ | 0.771 | 0.072 | 86.046 | 6 | Tentative
$\beta_{V}-\beta_{R}$ vs $H_{V}-H_{R}$ | -0.714 | 0.111 | 96.087 | 6 | No
Non-Dwarf Planets
$\beta_{V}$ vs $H_{V}$ | -0.241 | 0.019 | 56.606 | 94 | No
$\beta_{R}$ vs $H_{R}$ | -0.225 | 0.031 | 35.844 | 93 | No
$\beta_{V}$ vs $H_{V}-H_{R}$ | -0.230 | 0.034 | 81.014 | 85 | No
$\beta_{R}$ vs $H_{V}-H_{R}$ | 0.377 | 0.0 | 19.221 | 85 | No
$\beta_{V}-\beta_{R}$ vs $H_{V}-H_{R}$ | -0.745 | 0.0 | 0.28 | 85 | Yes

We find a highly significant negative correlation using data from Ayala-Loera
et al. (2018) for objects with phase coefficient values consistent with
positive to within $2\sigma$. We also find similar negative correlations with
ATLAS data for the full sample, the KBOs, and the non-dwarf planets sub-
populations. However, these correlations are tentative, and an inherently
uncorrelated distribution cannot be ruled out to $95\%$ confidence. Jackknife
re-sampling of the Ayala-Loera et al. (2018) sample, using sub-sample sizes of
$N=13$ (for the full ATLAS sample) and $N=7$ (for both the KBO and non-dwarf
planet sub-populations), yields no correlation for 16.3% of simulations. Thus
the difference in size between our sample and that of Ayala-Loera et al.
(2018) is likely responsible for this discrepancy in correlation certainty.
Furthermore, our ATLAS datasets usually have many more observations than those
of Ayala-Loera et al. (2018), and are taken in filters of different wavelength
ranges. We therefore consider our findings to be consistent with those of
Ayala-Loera et al. (2018) and attribute any difference in correlation
certainty to be due to differing sample sizes, dataset sizes, and filter
wavelength ranges. An increase in the number of objects with phase coefficient
and absolute magnitude measurements from datasets comparable to those afforded
by ATLAS would allow us to potentially recover this negative correlation
between relative phase coefficient - the difference in phase coefficient value
between broadband filters of differing wavelength range - and color index.

Ayala-Loera et al. (2018) reported the strong negative correlation between
relative phase coefficient and color index as persisting even when their
sample was divided by absolute magnitude and semimajor axis. However, their
sample was divided by semimajor axis and absolute magnitude at $a=40$ au and
$H=4.5$, in contrast to our thresholds of $a=30$ au and $H=3$. Dividing our
sample by the sub-population definitions of Ayala-Loera et al. (2018) using
$H$ values calculated from ATLAS data, we find tentative correlations for the
$a<40$ au and $H>4.5$ subpopulations, and no correlation for the $H<4.5$ au
subpopulation, with the $a>40$ au population being too small for an accurate
rank statistic. Applying our algorithm to their sample divided by their
definitions, we also find strong negative correlations between relative phase
coefficient and color index for the $H>4.5$ population and $a>40$ au
population, and tentative negative correlations for the $a<40$ au population
and $H<4.5$ au populations. Dividing their sample by our definitions of sub-
populations, we find no correlation for Centaurs/JFCs/Transition Objects or
the dwarf planets sub-populations, yet a strong correlation for the KBOs and
the non-dwarf planet sub-population which is dominated in number by KBOs. This
difference in observed correlation between populations may point to a
difference in substructure between the surfaces of these objects, likely
caused by an evolutionary effect due to differing insolation. An increased
sample size similar to those of Alvarez-Candal et al. (2016), Ayala-Loera et
al. (2018) and Alvarez-Candal et al. (2019), with precisions of parameter
values comparable to those afforded by the ATLAS datasets, would help to
better recover and study this negative correlation between relative phase
coefficient and color index, which may in fact be a universal property across
different populations of small Solar System objects (Alvarez-Candal et al.,
2022), and confirm the existence of this possible distinction.

###  6.4 Cometary Activity Search

We searched for instances of cometary activity in the ATLAS datasets for all
18 objects in our sample. Most of our sample show no sign of cometary
activity. A small number of objects, namely Chiron, Echeclus, Pluto, and
Quaoar exhibit ‘bright nights’ as defined in Section 5.4, suggesting potential
cometary outbursts or long-term activity. Table 10 lists the bright
observations per object identified by our algorithm as potentially indicative
of cometary activity, either in the form of short outbursts or longer-term
sustained brightening. We discuss each object below:

_Echeclus_ \- we detect three separate instances of bright nights across the
ATLAS baseline, as shown in Figure 12. The first, seen in the $c$ filter on
2018 January 17 (MJD 58135) and the $o$ filter from 2017 December 24 (MJD
58111) to 2018 January 2 (MJD 58120), coincides in time with a previously
reported 2017 December outburst (James, 2018; Kareta et al., 2019). During
this outburst, Echeclus exhibited a fan-shaped coma which remained clearly
visible in the ATLAS images for approximately 1 month, as seen in Figure 13.
This activity seemed to build to a second outburst observed as two consecutive
bright nights, 2018 March 6 (MJD 58183) and 2018 March 8 (MJD 58185). Analysis
of the full width half maximum (FWHM) of the PSF of Echeclus during the night
of 2018 March 6 (the night during which the PSF of Echeclus was more clearly
visible) reveals significant spread compared to those of background stars, as
shown in Figure 14. The PSF FWHM of Echeclus compared to that of the
background stars for this date are listed in Table 11, indicative of a visible
coma. The third instance appears as a single bright night in the $o$ filter on
2018 April 9, however analysis of the ATLAS images reveals Echeclus to lie
close to a poorly-subtracted background star during this time, as seen in
Figure 15, making this a result of background stellar flux contamination, and
not a comet-like outburst.

_Chiron_ \- we detect two separate instances of bright nights, highlighted in
Figure 16. The first is a single bright night at 2016 December 22 (MJD 57744)
observed in $c$; visual inspection reveals Chiron lies close to a poorly-
subtracted background star, as seen in Figure 15, thus this brightening is due
to stellar contamination. The second instance is an increase in brightness
constituting three consecutive bright nights starting at 2021 June 18 (MJD
59383), the first three nights of observation of a new apparition of Chiron.
No coma or tail is visible in the ATLAS images, and Chiron’s flux signal
appears stellar (see Figures 17 and 18 and Table 11), yet this brightness
increase exceeds Chiron’s amplitude of rotational modulation. This new epoch
of increased activity is reported in Dobson et al. (2021a) and Dobson et al.
(2021b). Further analysis of Chiron is beyond the scope of this study, and
will be subject of a later paper.

_Pluto_ \- we detect two bright nights at 2019 July 5 (MJD 58669) and 2020
October 7 (MJD 59129), highlighted in Figure 19, yet neither are caused by
cometary activity. Visual inspection of ATLAS images of each night shows that
Pluto is contaminated by background stars, as seen in Figure 15.

_Quaoar_ \- we detect a single bright night at 2021 March 22 (MJD 59295),
highlighted in Figure 20. We find Quaoar lies close to a poorly-subtracted
background star in all observations from this night, as seen in Figure 15,
contaminating its photometry.

We also detect two further instances of brightening for both Echeclus and 2014
QA43 which may correspond to long-term activity, described below:

_Echeclus_ \- we detect an instance of sustained brightening of 0.2 mag and
${>}1$ standard deviation of the data from the sigma-clipped median ranging
from 2017 November 10 (MJD 58067) to 2018 February 12 (MJD 58161), as
highlighted in Figure 12. This coincides in time with Echeclus’s ${>}2\sigma$
outburst in 2017 December, and a fan-shaped coma is clearly visible in the
ATLAS images during this time, as seen in Figure 13.

_2014 QA43_ \- we detect an instance of brightening ranging in time from 2020
September 16 (MJD 59108) to 2020 December 7 (MJD 59190), spanning a large
fraction of the observations from that apparition, as shown in Figure 21. The
median phase-corrected reduced magnitude of this apparition is brighter than
that of previous years of data by approximately 0.3 mag. Measurements of the
FWHM of 2014 QA43 reveals no significant broadening compared to background
stars, as seen in Figure 22. It remains unclear if this brightening is due to
cometary activity not visible in the images or due to the changing aspect
angle of the object.

In conclusion, only Chiron and Echeclus are found to exhibit definite comet-
like activity. It is not surprising that we find these objects have been
active during the ATLAS survey, as both Chiron (Bus et al., 1988; Luu &
Jewitt, 1990; Dahlgren et al., 1991; Marcialis & Buratti, 1993; Elliot et al.,
1995; Lazzaro et al., 1997b; Silva & Cellone, 2001; Duffard et al., 2002) and
Echeclus (Choi et al., 2006a, b; Jaeger et al., 2011; Rousselot et al., 2016;
James, 2018; Seccull et al., 2019; Kareta et al., 2019) have well-recorded
histories of comet-like activity in the literature. Both these objects have
been classified by previous studies as belonging to the Centaur population,
members of which have been known to exhibit comet-like outbursts. However,
unlike comets, Centaur activity is decoupled from perihelion. The mechanism
responsible for cometary activity on Centaurs currently remains elusive at
present. Jewitt (2009) found that from a population of 23 Centaurs, the median
perihelion of active Centaurs was smaller than that of inactive ones, and
proposed a transition from amorphous to crystalline water ice to be the cause
of Centaur activity. Considering only objects in our sample which satisfy the
Centaur definition of Jewitt (2009), we find that Echeclus and Chiron have
perihelia between those of Jupiter and Saturn, with those of our inactive
Centaurs extending to beyond that of Saturn. Though our sample is too small to
accurately quantify median perihelia, our observations of Centaur activity
appear to be consistent with the findings of Jewitt (2009).

All of the variable (or comet-like activity) observations in the ATLAS dataset
that correspond to instances of potential cometary activity were removed from
the dataset before phase curve analyses were conducted. If any found instance
of possible cometary activity was extended in time, we remove all datapoints
in both filters between the start date and end date of the activity.
Furthermore, any data identified by our algorithm as a bright night yet showed
background star contamination were removed from the dataset. We note that the
number of observations taken during an outburst remains a limiting factor in
detecting such activity. For example, despite occurring during the ATLAS
observation baseline, a previously-recorded outburst of Echeclus in 2016
August (Rousselot et al., 2016) did not cause a ${>}2\sigma$ increase in the
ATLAS data, and observations were too sparse for it to be detected as lower-
level cometary activity. Furthermore, any lower-level cometary activity that
persisted for several consecutive apparitions would likely remain undetected,
thereby reducing the accuracy of any applied phase curve fit. However, our
algorithm has allowed us to detect an instance of cometary activity on Chiron
despite displaying no visible coma or comet-like tail. Such activity would
remain undetected from analysis of ATLAS postage stamps, yet is readily
detectable in our algorithm. Therefore, we present our phase curve fits as the
best analysis that can be carried out within the limits of the dataset.

Table 10: Instances of nights of observation potentially indicative of cometary activity.  Object | Filter | Start Date | End Date | Cometary Activity? | Notes
---|---|---|---|---|---
Echeclus | c | 2018-01-17 (MJD 58135) | 2018-01-17 (MJD 58135) | Short outburst | Fan-shaped coma visible
Echeclus | o | 2017-12-24 (MJD 58111) | 2018-01-02 (MJD 58120) | Short outburst | Fan-shaped coma visible
Echeclus | o | 2018-03-06 (MJD 58183) | 2018-03-06 (MJD 58183) | Short outburst | PSF extension
Echeclus | o | 2021-01-03 (MJD 59217) | 2021-01-03 (MJD 59217) | No | Background stellar flux contamination
Echeclus | o | 2017-11-10 (MJD 58067) | 2018-02-12 (MJD 58161) | Sustained brightening | Fan-shaped coma visible
Chiron | c | 2016-12-22 (MJD 57744) | 2016-12-22 (MJD 57744) | No | Background stellar flux contamination
Chiron | o | 2021-06-18 (MJD 59383) | 2021-06-27 (MJD 59392) | Sustained brightening | No visible activity or PSF extension
|  |  |  |  | yet brightening beyond literature amplitude
|  |  |  |  | and spread of data from previous apparitions
Pluto | c | 2019-07-04 (MJD 58669) | 2019-07-04 (MJD 58669) | No | Background stellar flux contamination
Pluto | o | 2020-10-07 (MJD 59129) | 2020-10-07 (MJD 59129) | No | Background stellar flux contamination
Quaoar | o | 2021-03-22 (MJD 59295) | 2021-03-22 (MJD 59295) | No | Background stellar flux contamination
2014 QA43 | o | 2020-09-16 (MJD 59108) | 2020-12-07 (MJD 59190) | Unknown | No visible coma or No PSF extension
Figure 12: Plots of phase-angle corrected reduced magnitude vs. MJD in ATLAS $c$ (turquoise, upper) and $o$ (orange, upper) and phase curves of reduced magnitude vs. phase angle in ATLAS $c$ (turquoise, lower) and $o$ (orange, lower) for (60558) Echeclus. Red points denote observations taken from nights with ${\geq}3$ observations of which ${75\%}$ are brighter than $2\sigma$ from the object’s sigma-clipped median magnitude. Magenta points denote observations within a 30 day bin whose median magnitude lies 0.2 magnitudes and 1 standard deviation brighter than the object’s sigma-clipped median magnitude. Figure 13:  ATLAS difference (left) and reduced (right) images of Echeclus during its observed cometary outburst. Red crosshairs denote Echeclus’s position on the images. _Upper:_ 2017 December 29 19:31 UTC (MJD 58117.313716) ATLAS $o$ filter images, with fan-shaped coma clearly visible (detected from our short outburst search algorithm). _Middle:_ 2018 January 17 18:09 UTC (MJD 58136.256515) ATLAS $o$ filter images, with fan-shaped coma clearly visible (detected from our longer-term lower-level activity search algorithm). _Lower:_ 2018 March 5 UTC 17:09 (MJD 58183.215153) ATLAS $o$ filter images, showing Echeclus to appear fuzzy compared to background stars, indicating residual coma (detected from our short outburst search algorithm).  Figure 14: PSF FWHM analysis for 2018 March 6 (MJD 58183) ATLAS images of (60558) Echeclus. _Upper row:_ 100x100 pixel inset of 1000x1000 pixel reduced ATLAS images centered on Echeclus. _Lower row:_ Histograms of FWHM of all background stars in each 1000x1000 pixel ATLAS image, with the median stellar FWHM (dashed magenta vertical line) and Echeclus FWHM (solid green vertical line) plotted for reference. In all images, Echeclus displays significant PSF extension. Table 11: PSF FWHM analysis for selected bright nights of Echeclus, Chiron, and 2014 QA43 compared to PSF FWHM of background stars.  Object | UTC Date and Time | MJD | Echeclus PSF FWHM | Stellar PSF FWHM
---|---|---|---|---
|  |  | (arcsec) | (arcsec)
Echeclus | 2018-03-05 17:18 | 58183.215153 | $8.52$ | $4.99\pm 0.68$
Echeclus | 2018-03-05 17:20 | 58183.222678 | $8.22$ | $5.32\pm 0.53$
Echeclus | 2018-03-05 17:30 | 58183.229396 | $8.29$ | $5.19\pm 0.49$
Echeclus | 2018-03-05 17:39 | 58183.23565 | $8.81$ | $5.42\pm 0.47$
Chiron | 2021-06-22 23:07 | 59388.463541 | $3.66$ | $3.70\pm 0.36$
Chiron | 2021-06-22 23:12 | 59388.466769 | $3.79$ | $3.61\pm 0.29$
Chiron | 2021-06-22 23:22 | 59388.474141 | $5.07$ | $3.92\pm 0.33$
Chiron | 2021-06-22 23:48 | 59388.492191 | $4.34$ | $3.85\pm 0.33$
2014 QA43 | 2020-11-11 00:32 | 59164.522729 | $5.06$ | $4.25\pm 0.19$
2014 QA43 | 2020-11-11 01:07 | 59164.546531 | $3.86$ | $4.02\pm 0.21$
2014 QA43 | 2020-11-11 02:12 | 59164.592126 | $5.02$ | $4.00\pm 0.17$
2014 QA43 | 2020-11-11 02:23 | 59164.599976 | $3.25$ | $4.30\pm 0.19$
Figure 15: Collage of ATLAS difference (left) and reduced (right) images of,
in descending order, Pluto (2019-07-04 19:05 UTC, $c$ filter), Pluto
(2020-10-06 14:34 UTC, $o$ filter), Chiron (2016-12-21 15:17 UTC, $c$ filter),
Quaoar (2021-03-21 18:44 UTC, $o$ filter), and Echeclus (2021-01-02 21:56 UTC,
$o$ filter). Red crosshairs denote positions of objects in images. In all
images, the objects lie close to often poorly-subtracted background stars.
Figure 16: Plots of phase-angle corrected reduced magnitude vs. MJD in ATLAS
$c$ (turquoise, upper) and $o$ (orange, upper) and phase curves of reduced
magnitude vs. phase angle in ATLAS $c$ (turquoise, lower) and $o$ (orange,
lower) for (2060) Chiron. Red points denote observations taken from nights
with ${\geq}3$ observations of which ${75\%}$ are brighter than $2\sigma$ from
the object’s median sigma-clipped magnitude. Figure 17: 2021 June 17 23:32
UTCMJD 59383.480799 ATLAS difference image (left) and reduced image (right) of
Chiron during its 2021-2022 active epoch, taken in ATLAS $o$ filter. Red
crosshairs denote Chiron’s position on the images. Chiron’s signal remains
point-like from visual inspection. Figure 18: PSF FWHM analysis for 2021 June
23 (MJD 59388) ATLAS images of (2060) Chiron. _Upper row:_ 100x100 pixel inset
of 1000x1000 pixel reduced ATLAS images centered on (2060) Chiron. _Lower
row:_ Histograms of FWHM of all background stars in each 1000x1000 pixel ATLAS
image, with the median stellar FWHM (dashed magenta vertical line) and (2060)
Chiron FWHM (solid green vertical line) plotted for reference. In all images,
the FWHM of (2060) Chiron remains consistent with the histogram distribution
of the FWHM of background stars, with no significant extension. Figure 19:
Plots of phase-angle corrected reduced magnitude vs. MJD in ATLAS $c$
(turquoise, upper) and $o$ (orange, upper) and phase curves of reduced
magnitude vs. phase angle in ATLAS $c$ (turquoise, lower) and $o$ (orange,
lower) for (134340) Pluto, also corrected for rotational modulation. Red
points denote observations taken from nights with ${\geq}3$ observations of
which ${75\%}$ are brighter than $2\sigma$ from the sigma-clipped object’s
median magnitude. Figure 20: Plots of phase-angle corrected reduced magnitude
vs. MJD in ATLAS $c$ (turquoise, upper) and $o$ (orange, upper) and phase
curves of reduced magnitude vs. phase angle in ATLAS $c$ (turquoise, lower)
and $o$ (orange, lower) for (50000) Quaoar, also corrected for rotational
modulation. Red points denote observations taken from nights with ${\geq}3$
observations of which ${75\%}$ are brighter than $2\sigma$ from the object’s
sigma-clipped median magnitude. Figure 21: Plots of phase-angle corrected
reduced magnitude vs. MJD in ATLAS $c$ (turquoise, upper) and $o$ (orange,
upper) and phase curves of reduced magnitude vs. phase angle in ATLAS $c$
(turquoise, lower) and $o$ (orange, lower) for (501585) 2014 QA43. Red points
denote observations taken from nights with ${\geq}3$ observations of which
${75\%}$ are brighter than $2\sigma$ from the sigma-clipped object’s median
magnitude. Magenta points denote observations within a 30 day bin whose median
magnitude lies 0.2 magnitudes and 1 standard deviation brighter than the
object’s sigma-clipped median magnitude. Figure 22: PSF FWHM analysis for 2020
November 11 (MJD 59164) ATLAS images of (501585) 2014 QA43. _Upper row:_
100x100 pixel inset of 1000x1000 pixel reduced ATLAS images centered on
(501585) 2014 QA43. _Lower row:_ Histograms of FWHM of all background stars in
each 1000x1000 pixel ATLAS image, with the median stellar FWHM (dashed magenta
vertical line) and (501585) 2014 QA43 FWHM (solid green vertical line) plotted
for reference. The PSF of (501585) 2014 QA43 remains consistent with the
histogram distribution of the FWHM of background stars, with no significant
extension.

###  6.5 $HG_{1}G_{2}$ Model

We apply the $HG_{1}G_{2}$ phase curve model to the $c$ filter dataset of
Hidalgo and the $o$ filter phase curve of Hidalgo, Narcissus and 2014 QA43.
The resulting fits to the data are shown in Figure 23 and its online Figure
set. Table 12 lists the absolute magnitudes $H$ and slope parameters $G_{1}$
and $G_{2}$ of these phase curves. Our best-fit slope parameters are
consistent within their uncertainties with monotonic functions of positive
slope, as predicted from surface reflectance models (Muinonen et al., 2010;
Mahlke et al., 2021). Mahlke et al. (2021) utilized serendipitous ATLAS
observations of main belt asteroids to fit their phase curves with the
$HG_{1}G_{2}$ function to determine possible correlations of slope parameters
with main belt asteroid (MBA) taxonomic class. We aim to compare our best-fit
slope parameters with those from the taxonomic classes to infer the surface
types of Hidalgo, Narcissus and 2014 QA43. Figure 24 displays the kernel
density estimation (KDE) plots of $G_{1}$ and $G_{2}$ values of 19708 MBAs as
measured by Mahlke et al. (2021) sorted by taxonomic complex, with our derived
slope parameters for our three JFCs overplotted with their associated error
bars. We include the $c$ filter slope parameter of Hidalgo for completeness,
but for analysis we consider only the slope parameters for our objects in the
$o$ filter as they were derived from more observations. As seen in Figure 24,
the $o$ filter slope parameters of all three fitted JFCs are consistent within
uncertainties with the KDE distributions of all asteroid taxonomic complexes
to $3\sigma$, with greater correspondences to the carbonaceous C-type
complexes. The surfaces of Narcissus and 2014 QA43 remain unclassified, while
Hidalgo has been previously classified as a D-type asteroid (Tholen, 1984;
Belskaya et al., 2017; Geem et al., 2022). Our $G_{1}$ and $G_{2}$ parameters
for Hidalgo are consistent with its previous classification, lying within the
95% confidence interval, albeit just outside the core of the distribution of
the D-type complex.

Figure 23: _Left plot:_ $HG_{1}G_{2}$ model fit to _c_ filter phase curve of (944) Hidalgo. _Right plot:_ 2D histogram of Monte Carlo solutions to slope parameters $G_{1}$ and $G_{2}$ of $HG_{1}G_{2}$ fit. Best fit function extended to $\alpha=0$ deg for ease of viewing the extrapolated absolute magnitude value and any potential opposition surge. The complete figure set (4 images) is available in the online published version of the paper. Table 12: $H$, $G_{1}$, and $G_{2}$ parameters, their associated $2\sigma$ uncertainties, and the number of datapoints $N$ for objects that satisfy criteria for fitting with the $HG_{1}G_{2}$ model. Object | $H$ | $G_{1}$ | $G_{2}$ | N
---|---|---|---|---
Hidalgo (c filter) | $10.135\pm 0.048$ | $-0.017\pm 0.03$ | $0.291\pm 0.008$ | 178
Hidalgo (o filter) | $10.193\pm 0.026$ | $0.355\pm 0.024$ | $0.214\pm 0.006$ | 730
Narcissus (o filter) | $13.044\pm 0.124$ | $0.71\pm 0.216$ | $0.113\pm 0.083$ | 204
2014 QA43 (o filter) | $10.899\pm 0.093$ | $-0.063\pm 0.178$ | $0.393\pm 0.106$ | 270
Figure 24: $G_{1}$,$G_{2}$ KDE distributions of taxonomic complexes as mapped
by Mahlke et al. (2021) in $c$ and $o$ ATLAS filters, with $G_{1}$ and $G_{2}$
parameters of Hidalgo (circle), Narcissus (square), and 2014 QA43 (triangle)
phase curves overplotted. Contours mark KDE levels at which 68.3% ($1\sigma$),
95% ($2\sigma$) and 99.7% ($3\sigma$) of values are encompassed. Uncertainties
for Hidalgo datapoints are smaller than the plot symbol.

##  7 Conclusions

We use serendipitous observations from the long baseline and high cadence
ATLAS survey to generate phase curves for 18 small Solar System objects,
including 7 KBOs, 4 Centaurs, and 5 JFCs. We have measured the linear phase
coefficients for all 18 objects in both wide-band $c$ and $o$ ATLAS filters.
The results are as follows:

  1. 1.

Negative phase coefficients $\beta$ values noted in the literature become more
positive with the larger datasets from ATLAS, and are thus an artifact of
small datasets with large scatter, and not a real physical phenomenon.

  2. 2.

We search for correlations between phase curve parameters using phase
coefficient and absolute magnitude values re-measured across the phase angle
range $0.1\leq\alpha\leq 6$ deg. We find no correlation between the linear
phase coefficients and absolute magnitude for either ATLAS filter. We report
no correlation between phase coefficient and color index in the $o$ filter,
and a tentative negative correlation in the $c$ filter, consistent with the
results of Ayala-Loera et al. (2018) and Alvarez-Candal et al. (2019). We find
a tentative negative correlation between relative phase coefficient and object
color index in the ATLAS $c$ and $o$ filters, consistent with the findings of
Ayala-Loera et al. (2018) and Alvarez-Candal et al. (2019), though an increase
in sample size would help to better resolve this correlation.

  3. 3.

Splitting our sample into sub-populations, no significant correlation is found
between phase coefficient and absolute magnitude in either filter. Tentative
correlations are found in the ATLAS $c$ filter between phase coefficient and
color index for all sub-populations, and in the $o$ filter for the
Centaur/JFCs/Transition Objects and non-dwarf planet sub-populations. We find
a tentative negative correlation between relative phase coefficient and object
surface color for the KBO population, the dwarf planet population, and all
non-dwarf planet objects, consistent with the results of Ayala-Loera et al.
(2018) and Alvarez-Candal et al. (2019). The apparent lack of correlation
exhibited by the Centaurs/JFCs/Transition Objects compared to the KBOs may
indicate a difference in surface properties, but a larger sample size is
needed to confirm this.

  4. 4.

We detect comet-like activity exhibited by Echeclus (corresponding to previous
literature outbursts) and Chiron (corresponding to a newly discovered epoch of
activity reported by Dobson et al. (2021b)), and we also detect potential
cometary activity for 2014 QA43. This complements established methods looking
for extended PSFs and faint coma or tails in images to detect epochs of
cometary activity.

  5. 5.

We apply the $HG_{1}G_{2}$ model to the phase curves of 3 JFCs - Hidalgo,
Narcissus, and 2014 QA43. We find their best-fitting slope parameters to be
more consistent with carbonaceous main belt asteroid taxonomic complexes than
silicaceous ones, indicating low-albedo, carbon-rich surfaces for these
objects.

The long baseline and high observation cadence of ATLAS has yielded larger
photometric datasets per object in our sample than any previous study. Future
surveys utilising larger telescope diameters, such as the Rubin Observatory
Legacy Survey of Space and Time (LSST) by the Vera C. Rubin Observatory (LSST
Science Collaboration et al., 2009; Jurić et al., 2019; Ivezić et al., 2019),
though operating with a lower observation cadence than ATLAS, will help reduce
uncertainties in magnitude measurements, aiding discovery of and correction
for rotational lightcurves. LSST will also significantly increase the number
of objects whose phase curves can be analysed by probing to fainter limiting
magnitudes, allowing increased accuracy for characterizing larger numbers of
the KBO, Centaur, and JFC populations.

##  8 Acknowledgements

This work has made use of data from the Asteroid Terrestrial- impact Last
Alert System (ATLAS) project. The Asteroid Terrestrial- impact Last Alert
System (ATLAS) project is primarily funded to search for near earth asteroids
through NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575; byproducts of
the NEO search include images and catalogs from the survey area. The ATLAS
science products have been made possible through the contributions of the
University of Hawaii Institute for Astronomy, the Queen’s University Belfast,
the Space Telescope Science Institute, the South African Astronomical
Observatory, and the Millennium Institute of Astrophysics (MAS), Chile. MES
and MMD were supported by the UK Science Technology Facilities Council (STFC)
grants ST/V000691/1 and ST/V506990/1, respectively. LJS acknowledges support
by the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (ERC Advanced Grant KILONOVA No. 885281).

This research has made use of data and services provided by the Horizons
system of the Jet Propulsion Laboratory, and has made use of services provided
by NASA’s Astrophysics Data System. All photometric data used in this study is
available from the online ATLAS Forced Photometry Server at
https://fallingstar-data.com/forcedphot/.

The authors thank the researchers Grigori Fedorets, David Jackson, Lucy Dolan,
and Will Bate for useful feedback on the manuscript. The authors thank the
entire ATLAS Collaboration for their productive discussion on this paper. The
authors thank the anonymous reviewers for their feedback that improved the
manuscript of this paper.

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