11institutetext: V. N. Karazin Kharkiv National University, 4 Svobody Sq.,
Kharkiv, 61022, Ukraine
11email<EMAIL_ADDRESS>22institutetext: Crimean Astrophysical
Observatory, Nauchny, Crimea 33institutetext: Astronomical Institute of the
Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
44institutetext: Main Astronomical Observatory of the National Academy of
Sciences of Ukraine, 27 Zabolotnoho Str., 03143 Kyiv, Ukraine 55institutetext:
Taras Shevchenko National University of Kyiv, Astronomical Observatory,
Ukraine 66institutetext: ICAMER Observatory of NASU, 27 Zabolotnoho Str.,
Kyiv, 03143, Ukraine 77institutetext: Université Côte d’Azur, Observatoire de
la Côte d’Azur, CNRS, Laboratoire Lagrange, France 88institutetext: Keldysh
Institute of Applied Mathematics, RAS, Miusskaya Sq. 4, Moscow 125047, Russia

# Small solar system objects on highly inclined orbits

Surface colours and lifetimes

T. Hromakina 11 I. Belskaya 11 Yu. Krugly 11 V. Rumyantsev 22 O. Golubov 11 I.
Kyrylenko 11 O. Ivanova 334455 S. Velichko 11 I. Izvekova 66 A. Sergeyev 77 1
1 I. Slyusarev 11 I. Molotov 88

(—; —)

###### Abstract

Context. Less than one percent of the discovered small solar system objects
have highly inclined orbits ($i>60^{\circ}$), and revolve around the Sun on
near-polar or retrograde orbits. The origin and evolutionary history of these
objects are not yet clear.

Aims. In this work we study the surface properties and orbital dynamics of
selected high-inclination objects.

Methods. BVRI photometric observations were performed in 2019-2020 using the
2.0m telescope at the Terskol Observatory and the 2.6m telescope at the
Crimean Astrophysical Observatory. Additionally, we searched for high-
inclination objects in the Sloan Digital Sky Survey and Pan-STARRS. The
dynamics of the selected objects was studied using numerical simulations.

Results. We obtained new photometric observations of six high-inclination
objects (468861) 2013 LU28, (517717) 2015 KZ120, 2020 EP, A/2019 U5
(A/PanSTARRS), C/2018 DO4 (Lemmon), and C/2019 O3 (Palomar). All of the
objects have similar $B-V$, $V-R$, $R-I$ colours, which are close to those of
moderately red TNOs and grey Centaurs. The photometric data that were
extracted from the all-sky surveys also correspond to moderately red surfaces
of high-inclination objects. No signs of ultra-red material on the surface of
high-inclination asteroids were found, which supports the results of previous
works. The comet C/2018 DO4 (Lemmon) revealed some complex morphology with
structures that could be associated with particles that were ejected from the
cometary nucleus. Its value of the parameter $Af\rho$ is around 100 cm for the
aperture size of 6 000 km. The value of $Af\rho$ for the hyperbolic comet
C/2019 O3 (Palomar) is much larger, and is in the range from 2 000 to 3 700 cm
for the aperture sizes from 25 000 km to 60 000 km. For objects 2013 LU28,
2015 KZ120, and 2020 EP we estimated future and past lifetimes on their
orbits. It appears that the orbits of considered objects are strongly chaotic,
and with the available accuracy of the orbital elements no reliable
predictions can be made about their distant past or future. The lifetimes of
high-inclination objects turned out to be highly non-sensitive to the
precision of the orbital elements, and to the Yarkovsky orbital drift.

###### Key Words.:

Minor planets, asteroids: general – Comets: general Kuiper belt: general –
Oort Cloud – techniques: photometric – Surveys

##  1 Introduction

The vast majority of solar system objects have prograde orbits with low
inclinations. According to the Minor Planet
Center111https://minorplanetcenter.net (MPC), only about 250 asteroids on
near-polar ($i$ around 90∘) or retrograde ($i$¿90∘) orbits have been
discovered to date. The majority of high-inclination objects belong to
Centaurs, trans-Neptunian objects (TNOs), and distant objects on cometary
orbits. The remaining $\sim$10 % are near-Earth objects (NEOs). To date no
retrograde main belt asteroids have been discovered. On the other hand, about
a half of long-period comets (including those on parabolic and hyperbolic
orbits) that may have their origin in the Oort cloud have retrograde orbits.
Based on their inclination distribution, these comets are also designated as
nearly isotropic comets (Levison, 1996).

Jewitt (2005) introduced a dynamical class called the damocloids, point-source
objects on cometary orbits that do not show signs of cometary activity and
have a Tisserand parameter $T_{J}<2$. Based on this definition, most of the
high-inclination objects can be classified as damocloids.

The origin of very high-inclination objects is not yet clear and remains a
matter of debate. Different explanations have been proposed for the presence
of such objects in the present-day solar system. A number of studies have been
performed to investigate the origin and evolution of such objects by means of
the numerical integration of their orbits. According to Gladman et al. (2009),
such extreme TNOs can be outliers from the already explored reservoirs like
the Kuiper belt and/or the outer Oort cloud. Conversely, they could also come
from a currently unobserved reservoir, which can also supply the Halley-type
comets (Gladman et al., 2009). Brasser et al. (2012) and Volk & Malhotra
(2013) showed that the Kuiper belt is a very unlikely source of retrograde
centaurs. Chen et al. (2016) suggested the existence of a mostly undiscovered
population of extreme objects ($q>10$ AU, $a<100$ AU, and $i>60^{\circ}$) that
occupy a common plane and have the same origin. Another possible explanation
is the existence of an extended scattered disk created by a hypothetical yet
undiscovered big planet (Batygin & Brown, 2016). The recent paper by Namouni &
Morais (2020) suggests that high-inclination Centaurs were captured from the
interstellar medium. This conclusion is based on their finding, to a high
level of probability, for high-inclination Centaurs to be on near-polar orbits
4.5 Gyr in the past. However, this result was rebutted by Morbidelli et al.
(2020).

Retrograde near-Earth asteroids, according to model by Greenstreet et al.
(2012), are produced by the main belt, with the majority of objects
originating from 3:1 resonance with Jupiter. Another evolutionary pathway that
produces retrograde NEOs was proposed by de la Fuente Marcos et al. (2015).
They suggest that it is the 9:4 mean motion resonance with Jupiter that
triggers the orbital flip and, as a result, production of retrograde objects.

The results of numerical integration show that most retrograde objects are
unstable and have short life spans of less than ¡1 Myr (de la Fuente Marcos &
de la Fuente Marcos, 2014; Li et al., 2019; Kankiewicz & Włodarczyk, 2016).
Kankiewicz & Włodarczyk (2016) also pointed out that the influence of the
Yarkovsky effect can potentially be important, especially on Myr timescales.

Mean motion resonances with planets play an important role in the evolution of
particular objects and in the dynamical configuration of a planetary system as
a whole (Morbidelli, 2002; Ferraz-Mello et al., 2006; Beaugé et al., 2003).
Interestingly, Namouni & Morais (2015) showed that retrograde objects are more
easily captured in mean motion resonances than prograde ones. Various authors
looked for resonances among high-inclination and retrograde objects. Wiegert
et al. (2017) identified that object 2015 BZ509 is the first retrograde co-
orbital asteroid of Jupiter. Connors & Wiegert (2018) found that asteroid 2007
VW266 is in 13/14 retrograde mean motion resonance. A retrograde TNO 2011 KT19
was found to be in 7/9 resonance with Neptune by Morais & Namouni (2017).
Finally, a recent survey of resonances among retrograde asteroids was
performed by Li et al. (2019). They identified 15 retrograde objects that are
currently captured in resonances with giant planets, and 30 retrograde objects
that will be captured in resonances in the future.

Given the rarity and faintness of high-inclination objects, only for a few of
them were some physical properties determined. Jewitt (2005, 2015) found that
damocloids show a lack of extremely red surfaces. Licandro et al. (2018)
obtained spectral data for bodies on cometary orbits. They observed several
objects on retrograde and near-polar orbits and classified them as D type
based on their visible spectra. Pinilla-Alonso et al. (2013) obtained spectra
and photometric data for two retrograde objects 2008 YB3 and 2005 VD. They
concluded that both objects have moderately red surfaces. The authors also
investigated the dynamical evolution of these two bodies by integrating their
orbits for $10^{8}$ years into the past. They found that the dynamical
evolution of 2005 VD is dominated by a Kozai resonance with Jupiter (Kozai,
1962), while that of 2008 YB3 is dominated by close encounters with Jupiter
and Saturn. They concluded that 2005 VD comes from the Oort cloud, whereas
2008 YB3 originated in the trans-Neptunian region.

Here we present the first results of our observational programme dedicated to
the investigation of high-inclination small solar system objects. This work
contains new photometric observations of six objects. We also made an attempt
to enlarge statistics on the visible colours of high-inclination bodies by
using available data from multi-band sky surveys. The obtained data on visible
colours were compared with the published data on high-inclination objects and
other dynamical classes. Finally, we simulated orbits of observed objects to
estimate their lifetimes and to investigate the possible influence of the
Yarkovsky effect.

##  2 Photometric observations and data reduction

Photometric observations were carried out at the 2.6m Shain Telescope at the
Crimean Astrophysical Observatory (CrAO) and at the 2.0m Zeiss-2000 telescope
at the Peak Terskol observatory in 2019-2020. Shain telescope is equipped with
a 2048$\times$2048 FLI PL-4240 CCD camera. The field of view is
14.4$\times$14.4 arcmin and the pixel scale is 0.84”/pxl with a 2$\times$2
binning. The Zeiss telescope is equipped with a 2048$\times$2048 FLI PL-4301
CCD camera that covers a field of view of 10.7$\times$10.7 arcmin. The pixel
scale is 0.62”/pxl with a 2$\times$2 binning that was applied during the
observations. The typical seeing at CrAO ranged from 1.9 to 2.4 arcsec, and
the typical seeing at the Terskol Observatory was in the range from 3.0 to 3.7
arcsec.

Image reduction was done in the standard way that included subtraction of the
dark images and flat-field correction. Aperture photometry was performed with
the use of the photutils Python package (Bradley et al., 2019). For objects
that do not show signs of cometary activity, the most appropriate aperture
size was determined from the full width at half maximum (FWHM) values of
neighbouring stars in the image. Several aperture sizes were used for objects
with cometary activity. Sky subtraction was performed using an annular
aperture around each object’s photocentre.

The absolute calibration was made using the solar-type stars in the image
field. The magnitudes of the selected stars were taken from the Pan-STARRS
catalogue. For the images that were obtained at CrAO and Terskol, all suitable
stars in the image were used. For the Pan-STARRS images, due to their large
field of view, we did not use all stars in the frame. Instead, we only
considered the stars in a 0.15 arcmin radius around the target. On average,
this radius corresponded to about 20-50 stars per image.

Our observations were performed in the standard Johnson–Cousins photometric
system in BVRI broad-band filters. The exposure times of individual images
were from 90 to 180 s, depending on the brightness of the object. Typical
errors for individual frames were 0.03-0.05 mag in B filter, 0.02-0.04 in R
and V filters, and 0.06-0.10 mag in I filter.

##  3 Results

Only a small number of high-inclination objects were bright enough to be
observable on small ground-based telescopes during the observational runs in
2019-2020. In total, we obtained photometric data for six objects, including
three objects on retrograde orbits (2013 LU28, C/2018 DO4 (Lemmon), and A/2019
U5) and three objects on near-polar orbits (2015 KZ120, C/2019 O3 (Palomar),
and 2020 EP).

Our list of potentially observable targets did not initially include active
comets as one of the main aims of the programme was to assess the surface
properties of high-inclination objects. However, two of the observed targets
appeared to be comets. For both objects the preliminary reports on the
possible cometary activity appeared during the observation planning phase and
the corresponding publications in Minor Planet Electronic Circular were
released soon after our observations. The rest of the bodies do not show any
signs of cometary activity in our data. Table 1 presents the observational
circumstances, such as heliocentric ($r$) and geocentric ($\Delta$) distances
and phase angles ($\alpha$) together with measured and reduced magnitudes.

Table 1: Geometrical circumstances and magnitudes of the observed objects

Name | UT date | r (au) |  $\Delta$ (au) |  $\alpha$ (deg) | Apparent mag.∗ | Reduced mag. | Filter | Observatory
---|---|---|---|---|---|---|---|---
(468861) 2013 LU28 | 2019 08 01.80 | 13.06 | 13.29 | 4.3 | 21.02$\pm$0.06 | 9.82 | B | CrAO
|  |  |  |  | 20.18$\pm$0.04 | 8.98 | V | CrAO
|  |  |  |  | 19.56$\pm$0.08 | 8.36 | R | CrAO
|  |  |  |  | 19.46$\pm$0.09 | 8.26 | I | CrAO
| 2019 08 31.77 | 12.95 | 13.42 | 3.9 | 20.92$\pm$0.06 | 9.72 | B | CrAO
|  |  |  |  | 20.09$\pm$0.04 | 8.89 | V | CrAO
|  |  |  |  | 19.52$\pm$0.08 | 8.32 | R | CrAO
|  |  |  |  | 18.85$\pm$0.09 | 7.65 | I | CrAO
| 2020 03 22.05 | 12.20 | 11.66 | 4.0 | 20.64$\pm$0.12 | 9.44 | B | Terskol
|  |  |  |  | 19.72$\pm$0.02 | 8.95 | V | Terskol
|  |  |  |  | 19.20 $\pm$0.02 | 8.43 | R | Terskol
(517717) 2015 KZ120 | 2019 08 01.84 | 8.41 | 8.26 | 6.9 | 20.83$\pm$0.03 | 11.62 | B | CrAO
|  |  |  |  | 19.89$\pm$0.03 | 10.68 | V | CrAO
|  |  |  |  | 19.47$\pm$0.04 | 10.26 | R | CrAO
|  |  |  |  | 18.92$\pm$0.06 | 9.71 | I | CrAO
| 2019 08 30.81 | 8.42 | 8.39 | 6.9 | 20.74$\pm$0.04 | 11.49 | B | CrAO
|  |  |  |  | 19.96$\pm$0.01 | 10.71 | V | CrAO
|  |  |  |  | 19.47$\pm$0.02 | 10.22 | R | CrAO
|  |  |  |  | 18.85$\pm$0.05 | 9.60 | I | CrAO
2020 EP | 2020 04 23.97 | 2.55 | 1.78 | 17.6 | 20.95$\pm$0.05 | 17.67 | B | CrAO
|  |  |  |  | 20.12$\pm$0.05 | 16.84 | V | CrAO
|  |  |  |  | 19.61$\pm$0.04 | 16.33 | R | CrAO
|  |  |  |  | 19.19$\pm$0.09 | 15.91 | I | CrAO
A/2019 U5 | 2020 07 19.91 | 8.67 | 8.32 | 6.4 | 19.79$\pm$0.05 | 10.50 | B | CrAO
|  |  |  |  | 19.00$\pm$0.05 | 9.71 | V | CrAO
|  |  |  |  | 18.59$\pm$0.05 | 9.30 | R | CrAO
|  |  |  |  | 18.04$\pm$0.05 | 8.75 | I | CrAO
| 2020 07 27.96 | 8.64 | 8.28 | 6.4 | 19.73$\pm$0.04 | 10.46 | B | Terskol
|  |  |  |  | 19.05$\pm$0.04 | 9.78 | V | Terskol
|  |  |  |  | 18.63$\pm$0.04 | 9.36 | R | Terskol
|  |  |  |  | 18.12$\pm$0.04 | 8.85 | I | Terskol
| 2020 08 14.94 | 8.53 | 8.23 | 6.6 | 19.74$\pm$0.04 | 10.51 | B | CrAO
|  |  |  |  | 18.95$\pm$0.03 | 9.72 | V | CrAO
|  |  |  |  | 18.52$\pm$0.03 | 9.29 | R | CrAO
|  |  |  |  | 17.97$\pm$0.03 | 8.74 | I | CrAO
C/2018 DO4 | 2019 10 01.05 | 2.45 | 1.99 | 23.4 | 14.05$\pm$0.08 | 10.61 | B | CrAO
|  |  |  |  | 13.13$\pm$0.03 | 9.69 | V | CrAO
|  |  |  |  | 12.70$\pm$0.03 | 9.26 | R | CrAO
|  |  |  |  | 11.99$\pm$0.07 | 8.55 | I | CrAO
| 2019 10 03.06 | 2.45 | 1.97 | 23.0 | 13.34$\pm$0.04 | 9.92 | R | Terskol
| 2019 10 05.08 | 2.46 | 1.93 | 22.5 | 14.22 $\pm$0.06 | 10.84 | B | Terskol
|  |  |  |  | 13.32 $\pm$0.04 | 9.94 | V | Terskol
|  |  |  |  | 12.88 $\pm$0.04 | 9.50 | R | Terskol
C/2019 O3 | 2020 07 19.92 | 8.92 | 8.27 | 5.2 | 18.30$\pm$0.03 | 8.71 | B | CrAO
|  |  |  |  | 17.54$\pm$0.03 | 7.95 | V | CrAO
|  |  |  |  | 17.11$\pm$0.03 | 7.52 | R | CrAO
|  |  |  |  | 16.69 $\pm$0.09 | 7.10 | I | CrAO
| 2020 08 14.90 | 8.90 | 8.32 | 5.5 | 18.28$\pm$0.03 | 8.93 | B | CrAO
|  |  |  |  | 17.54$\pm$0.03 | 8.19 | V | CrAO
|  |  |  |  | 17.14$\pm$0.03 | 7.79 | R | CrAO
|  |  |  |  | 16.76$\pm$0.03 | 7.41 | I | CrAO

∗ The magnitudes of C/2018 DO4 and C/2019 O3 were calculated for the 10 000 km
and 15 000 km apertures, respectively.

###  3.1 (468861) 2013 LU28

(468861) 2013 LU28 (previously designated as 2014 LJ9 and 2015 KB157) is a TNO
on a retrograde orbit ($i=125.4^{\circ}$) with a very high eccentricity
$e=0.95$. The semi-major axis of its orbit is $a=187.1$ au, and perihelion and
aphelion distances are $q=8.7$ au and $Q=365.5$ au, respectively.

The photometric observations were carried out in BVRI filters at CrAO during
two nights in August 2019 and in BVR filters at Terskol observatory on March
21, 2020. No colour variation was found for the different dates. The obtained
average surface colours ($B-V=0.83\pm 0.04$, $V-R=0.57\pm 0.04$, and
$R-I=0.68\pm 0.05$) imply a moderately red surface of 2013 LU28. The
differences in reduced magnitude values measured at different dates are within
the estimated brightness errors.

The absolute magnitude was calculated using the formalism by Bowell et al.
(1989) and assuming $G=0.15$. Then the visible absolute magnitude for 2013
LU28 is $H_{V}=8.6\pm 0.1$ mag, which is fainter than the value adopted in MPC
(8.2 mag). In order to estimate the diameter of the object, we assume a low to
moderate albedo in 0.04-0.1 range. In this case the diameter is 80-127 km.

###  3.2 (517717) 2015 KZ120

2015 KZ120 is a Centaur on a near-polar ($i=85^{\circ}$) and highly eccentric
($e=0.82$) orbit with $a=46.7$ au, $q=8.4$ au, and $Q=85.0$ au.

Our observations were carried out during two nights in August 2019 at CrAO in
BVRI photometric filters. The measured surface colours $B-V=0.78\pm 0.05$,
$V-R=0.49\pm 0.03$, and $R-I=0.61\pm 0.06$ suggest that the object has a
moderately red surface.

No significant variation is found in the values of reduced magnitude. The
calculated visible absolute magnitude is $H_{V}=10.2\pm 0.1$ mag, which is
similar to that reported in MPC (10.0 mag). The estimated diameter (again
assuming the albedo in 0.04-0.1 range) is 39-61 km.

###  3.3 2020 EP

2020 EP is a recently discovered object on a near-polar ($i=76^{\circ}$) and
very eccentric orbit ($e=0.76$). Its semi-major axis is $a=10.5$ au, the
perihelion distance is $q=2.5$ au, and the aphelion distance $Q=18.5$ au.

2020 EP was observed during one night on April 23, 2020, at CrAO observatory
in BVRI filters. The object is on a cometary orbit and at the time of
observation it was close to its perihelion at a heliocentric distance $r=2.55$
au. However, we found no signs of cometary activity for 2020 EP. The measured
surface colours correspond to a moderately red surface ($B-V=0.83\pm 0.09$,
$V-R=0.51\pm 0.06$, and $R-I=0.42\pm 0.10$).

The calculated visible absolute magnitude is $H_{V}=15.9\pm 0.1$ mag and is
similar to the value in MPC (16.0 mag). Using this value of absolute
magnitude, the estimated diameter of the object is rather small, only 3-4 km.

###  3.4 A/2019 U5 (A/PanSTARRS)

A/2019 U5 (A/PanSTARRS) (hereafter A/2019 U5) is a recently discovered object
on a hyperbolic cometary orbit with $i=113.5^{\circ}$, $e=1.002$, and $q=3.6$
au. With the current orbital elements A/2019 U5 will be reaching its
perihelion in March 2023. To date, no cometary activity has been detected for
this object, thus it is currently classified as a hyperbolic asteroid. Very
few such asteroids have been discovered to date, and the majority of them
eventually showed signs of cometary activity.

We observed A/2019 U5 during three nights in the BVRI filters. On July 19 and
August 14, 2020, the object was observed at CrAO, and on July 27, 2020, at
Terskol observatory. The obtained colour indices are similar and correspond to
a moderately red surface. The average colours for three nights are
$B-V=0.73\pm 0.06$, $V-R=0.42\pm 0.06$, and $R-I=0.55\pm 0.06$. The calculated
reduced magnitudes are very similar for all of our observations.

The estimated absolute magnitude is $H_{V}=9.3\pm 0.1$ mag (compared to
$H_{V}=9.68\pm 0.68$ in the JPL Small-Body
Database222https://ssd.jpl.nasa.gov/sbdb.cgi), and the corresponding diameter
is 60-95 km.

###  3.5 C/2018 DO4 (Lemmon)

C/2018 DO4 (Lemmon) (hereafter C/2018 DO4) is a retrograde comet
($i=160^{\circ}$) with $e=0.91$, $a=25.9$ au, $q=2.41$ au, and $Q=49.3$ au.
The comet reached perihelion on August 18, 2018 and was observed post-
perihelion at CrAO on October 02, 2019 and at the Terskol observatory on
October 03 and 05, 2019. A cometary coma is clearly seen in all of the
obtained images.

Colours of the comet were calculated using different apertures that correspond
to a linear size range of about $\sim$5 000-20 000 km. The obtained colours
are shown in Table 2. Additionally, in Fig. 1 we present the B-V colours that
were obtained at different apertures. No significant changes of colours are
detected in the range from $\sim$5 000 to 20 000 km.

Table 2: Colours of the comet C/2018 DO4 obtained on October 03 and 05, 2019. Date |  $\rho$, km | B-V | V-R | V-I
---|---|---|---|---
Oct 02 |  $\sim$5 000 | 0.93$\pm$0.04 | 0.42$\pm$0.04 | 1.14$\pm$0.08
|  $\sim$10 000 | 0.92$\pm$0.04 | 0.43$\pm$0.04 | 1.14$\pm$0.08
|  $\sim$15 000 | 0.89$\pm$0.04 | 0.43$\pm$0.04 | 1.15$\pm$0.08
|  $\sim$20 000 | 0.90$\pm$0.04 | 0.47$\pm$0.04 | 1.27$\pm$0.08
Oct 05 |  $\sim$5 000 | 0.91$\pm$0.07 | 0.44$\pm$0.05 |
|  $\sim$10 000 | 0.92$\pm$0.07 | 0.41$\pm$0.05 |
|  $\sim$15 000 | 0.85$\pm$0.07 | 0.44$\pm$0.05 |
|  $\sim$20 000 | 0.86$\pm$0.07 | 0.45$\pm$0.05 |
Figure 1: B-V colours of the comet C/2018 DO4 at different aperture sizes.

The level of dust production for a comet can be quantified by means of the
parameter Af$\rho$ (A’Hearn et al., 1984). It contains the value of the
geometric albedo A; the radius of the photometric aperture $\rho$ in cm; and
the factor f, which is the total scattering cross-section of the grains within
this aperture. This parameter can be estimated as

$Af\rho=\frac{4r^{2}\Delta^{2}10^{0.4(m_{sun}-m_{c})}}{\rho},$ (1)

where $\Delta$ is the geocentric distance in au, $r$ is the heliocentric
distance in cm, and $m_{sun}$ and $m_{c}$ are the apparent magnitudes of the
Sun and the comet, respectively. We used the solar colours from Holmberg et
al. (2006). In Fig. 2 we show the values of Af$\rho$ that were calculated for
different filters on October 02 and 05, 2019. The decrease in Af$\rho$ with
the distance from the nucleus most likely can be explained by the
fragmentation of the particles as the fragmentation causes a decrease in the
particle scattering cross section. We see a noticeable difference in the
radial profiles for the B and I filters. Since the light scattering
characteristics of particles change quickly when the particle size is
comparable with the wavelength, this confirms the predominance of submicron-
and micron-sized particles in the cometary coma. The values of Af$\rho$ in the
R filter for the 6 000 km aperture is 97$\pm$3 cm and 103$\pm$5 cm for
observations on October 02 and 05, 2019, respectively, which suggest that the
level of dust production did not change significantly.

Figure 2: Values of Af$\rho$ for C/2018 DO4 (Lemmon) at different apertures.
The top x-axis shows the angular aperture size. The bottom x-axis shows the
linear aperture size. The filled symbols correspond to observations at CrAO
made on October 01 and the open symbols correspond to observations at the
Terskol observatory made on October 03.

In order to reveal low-contrast structures in the coma of comet C/2018 DO4 the
images were treated with digital filters. We applied a combination of four
numerical techniques: rotational gradient (Larson & Sekanina, 1984), 1/$\rho$
profile, azimuthal average, and renormalization (Samarasinha & Larson, 2014).
The first two techniques allow us to remove the bright background from a
cometary coma and to highlight the low-contrast features. Division by the
azimuthal average is very good at separating brighter broad jets. We applied
each technique to the individual frames and to the composite images in order
to evaluate whether the revealed features are real or not. We also studied the
change in the jet structure due to the shift of the comet’s optocentre. Even
in the case of significant displacement of the optocentre by 6 pixels, the
detected structures were preserved for all images. Previously this technique
was used to pick out structures in several comets with good results (e.g.
Ivanova et al., 2018, 2019; Picazzio et al., 2019).

Figure 3 shows the direct B, V, R, and I images of the comet in relative
intensity obtained on October 02, 2019. Figure 4 shows the direct R images of
the comet obtained on October 02 and 05, 2019. The relative intensities of the
adjacent contours of the isophots differ by a factor of $\sqrt{2}$. The
observed coma is strongly asymmetric, and most probably was formed due to
activity of an isolated area on the cometary nucleus. In processed images we
revealed two strong structures: a bright near nucleus region (A) and a tail
(B), which is elongated along the negative velocity vector in the direction of
the coma. The position angle of the tail, measured anticlockwise from the
north through the east, is $75^{\circ}$.

In CrAO images we can see two isolated structures ($a$, $b$) placed along the
tail at distances of 20 000–60 000 km from the optocentre (Fig. 3); structure
$a$ can also be seen in the Terskol images (Fig. 4). These structures are most
pronounced in the images processed by the rotational gradient method because
this method is very sensitive to faint structures in cometary coma. The
rotational gradient technique highlights the structures that are distributed
radially from the nucleus. In general, these structures are formed by the
effects of the nucleus rotation on the released particles. The separated
structures were observed for all observation sets from October 02 to 05, 2019.
Following Farnham (2009), we define these features as arcs or shells. Most
likely the visible separation into such shells is a result of the nucleus
rotation and the projection of a particle flow that was ejected from the
nucleus. The collimated appearance of a tail and structures, which are similar
in all the filters that were used, favours the dust-dominated nature of the
comet. A more detailed analysis and modelling of the morphology of the comet
will be addressed in a future work.

Figure 3: Relative intensity images of the comet C/2018 DO4 (Lemmon) in the B,
V, R, and I filters obtained on October 02, 2019, at CrAO. (a) Direct images
of the comet with the isophots differing by a factor $\sqrt{2}$; (b) Image
processed by a rotational gradient method (Larson & Sekanina, 1984); (c) and
(d) Images to which the azimuthal average and the division by 1/$\rho$ profile
methods were applied, respectively (Samarasinha & Larson, 2014). The arrows
show the direction to the Sun ($\odot$), north (N), east (E), and the negative
velocity vector of the comet projected onto the sky plane (V). Negative
distance is in the solar direction. Figure 4: Relative intensity images of the
comet C/2018 DO4 (Lemmon) in the R filter obtained on October 03, 2019 (top),
and on October 05, 2019 (bottom), at Terskol Observatory. (a) Direct images of
the comet with the isophots differing by a factor $\sqrt{2}$; (b) Image
processed by the rotational gradient method (Larson & Sekanina, 1984); (c)
Image to which the azimuthal average method was applied (Samarasinha & Larson,
2014). The arrows show the directions to the Sun ($\odot$), north (N), east
(E), and the negative velocity vector of the comet projected onto the sky
plane (V). Negative distance is in the solar direction.

###  3.6 C/2019 O3 (Palomar)

C/2019 O3 (Palomar) (hereafter C/2019 O3) is another object on a near-polar
($i=89.8^{\circ}$) hyperbolic ($e=1.002$) orbit with $q=8.8$ au. It is
classified as a comet as cometary activity has been detected. The object will
reach its perihelion in March 2021.

We observed the comet C/2019 O3 during two nights on July 19 and August 14,
2020, at CrAO in the BVRI filters. The calculated colours that correspond to
the aperture size of 15 000 km are $B-V=0.75\pm 0.03$, $V-R=0.47\pm 0.03$, and
$V-I=0.94\pm 0.09$. Table 3 contains the surface colours that were measured at
different apertures.

Table 3: Colours of the comet C/2019 O3 obtained on July 19 and August 14, 2020. Date |  $\rho$, km | B-V | V-R | V-I
---|---|---|---|---
Jul 19 |  $\sim$30 000 | 0.76$\pm$0.03 | 0.44$\pm$0.04 | 0.85$\pm$0.08
|  $\sim$40 000 | 0.80$\pm$0.03 | 0.41$\pm$0.04 | 0.85$\pm$0.09
|  $\sim$50 000 | 0.81$\pm$0.03 | 0.43$\pm$0.04 | 085$\pm$0.09
|  $\sim$60 000 | 0.78$\pm$0.03 | 0.43$\pm$0.04 | 0.95$\pm$0.09
Aug 14 |  $\sim$30 000 | 0.74$\pm$0.03 | 0.41$\pm$0.04 | 0.78$\pm$0.08
|  $\sim$40 000 | 0.72$\pm$0.03 | 0.45$\pm$0.04 | 0.82$\pm$0.09
|  $\sim$50 000 | 0.73$\pm$0.03 | 0.43$\pm$0.04 | 086$\pm$0.09
|  $\sim$60 000 | 0.72$\pm$0.03 | 0.43$\pm$0.04 | 0.93$\pm$0.09

The calculated value of Af$\rho$ for observations in July and August, 2020,
are similar within the uncertainties, thus we provide here the average values.
In the R filter and for the aperture size of about 25 000 km the value of
$Af{\rho}$ is $3654\pm 77$ cm, which is significantly larger than that for the
comet C/2018 DO4. In Fig. 5 we show the mean values of $Af\rho$ in the BVRI
filters calculated with various apertures.

As for the comet C/2018 DO4 we obtained a noticeable difference in the
Af$\rho$ profiles for the B and I filters. Again, this confirms the
predominance of submicron- and micron-sized particles in the cometary coma of
C/2019 O3.

Figure 5: Average values of the parameter Af$\rho$ for the comet C/2019 O3
observed on July 19 and August 14, 2020. The top x-axis shows the angular
aperture size in arcseconds and the bottom x-axis shows the linear aperture
size (in km).

The cometary activity was detected in our images when the object was at
$r=8.9$ au from the Sun. This implies that the activity mechanism must be
different from water ice sublimation, which is only possible at a distance
from the Sun of up to $3-4$ au (Meech & Svoren, 2004). At larger distances
other mechanisms are considered, such as sublimation of CO and/or CO2,
transition between amorphous and crystalline phases of water ice, or annealing
of the amorphous water ice (Meech et al., 2009; Prialnik & Bar-Nun, 1992; Luu,
1993).

##  4 Search for high-inclination objects in multi-band sky surveys

In addition to our observations, we performed a search for high-inclination
objects in databases. In particular, we used data from the Sloan Digital Sky
Survey (SDSS) and Pan-STARRS surveys. For the SDSS we used data from a
catalogue and for the Pan-STARRS we worked directly with the images. The SDSS
and Pan-STARRS surveys both used Sloan photometric filters. Thus, in order to
compare the obtained colours with those in the Johnson–Cousins photometric
system we used the transformation equations from Kostov & Bonev (2018).

###  4.1 SDSS catalogue

The Sloan survey provides astrometric and photometric observations of a large
variety of astronomical objects, including small solar system bodies. The
photometric observations were conducted using the 2.5m wide-angle telescope at
the Apache Point Observatory (New Mexico, United States) (York et al., 2000).
The camera consists of 30 2048$\times$2048 photometric CCDs that are mosaicked
on the telescope’s focal plane, which allows a field of view of $3^{\circ}$
(Gunn et al., 1998).

The SDSS camera obtained quasi-simultaneous ugriz images with a time interval
of approximately 72 s between each exposure, and the largest temporal baseline
is about 300s between the r and g filters. The selection of retrograde
asteroids was carried out from the SDSS SSO catalogue (Sergeyev & Carry,
2020), which contains more that 1M measurements of solar system objects
observed in 1998-2009. It was prepared by analysing the moving objects that
were extracted from the DR16 and the Stripe82 SDSS photometry catalogues. To
distinguish known asteroids, the cone-search queries of SkyBoT service were
used (Berthier et al., 2006, 2016).

We found only four high-inclination objects with multiple observations:
Centaur 2005 VD, TNOs 2005 OE and 2012 DR30, and a NEO 2012 MS4. Given the
rather large magnitude errors for the individual objects, we consider here
only the average colours of the found objects: B-V=0.86$\pm$0.18,
V-R=0.53$\pm$0.11, and R-I=0.40$\pm$0.15. Thus, although the use of SDSS data
did not help to increase the statistics on colours of highly inclined objects,
we can conclude that the calculated mean colours are consistent with
moderately red surfaces.

###  4.2 Pan-STARRS images

We used the data obtained by the Pan-STARRS survey telescope (Tonry et al.,
2012) during the first 18 months from March 2010 to May 2014. The Pan-STARRS
survey was conducted using the 1.8m telescope of the Haleakala observatory
(Hawaii, United States). The telescope is equipped with a 1.4 gigapixel CCD
camera GPC1 that covers a field of view of $\sim 7\times 7^{\circ}$. The grizy
filter system used by the survey is similar to the Sloan photometric system
(Tonry et al., 2012). In this work we only used the data obtained in the g, r,
and i filters. The search for the objects that are present on Pan-STARRS
images was performed using the Solar System Object Image Search (Gwyn et al.,
2012).

We found quasi-simultaneous observations for three TNOs: (336756) 2010 NV1,
(342842) 2008 YB3, and 2012 DR30. The measured colours for all found TNOs
correspond to moderately red surfaces consistent with the previously measured
colours for these objects (see Table 4). Similarly to SDSS, we found very few
high-inclination objects in the Pan-STARRS images. It appears that the study
of these rare and faint objects using all-sky survey data is not very
fruitful, and that a dedicated programme is necessary.

##  5 Analysis of surface colours for high-inclination asteroids

Our new observations of four high-inclination asteroids showed that all of
these extreme objects are moderately red. Similarly, the colours of several
other high-inclination objects that were found in the databases suggest that
they also have moderately red surfaces. In Table 4 we collect all the
available colours for high-inclination objects. Previously the colour indices
were measured for 21 high-inclination asteroids. Our new observations increase
the number of objects with known colours, but the number of studied high-
inclination objects remains very small (around 10% of the discovered objects).
The table contains orbital elements ($i,a,e$); albedos and diameters, if
available; absolute magnitudes from MPC; and surface colours in the
Johnson–Cousins photometric system. As can be seen, all of the studied high-
inclination objects have surface colours from neutral to moderately red, which
confirms the conclusion by Jewitt (2015) on the absence of extremely red
surfaces among these objects.

Such surface colours are inherent for different dynamical classes of solar
system objects. Figure 6 shows the colour-colour diagram of high-inclination
objects together with other dynamical types, such as Jupiter trojans,
Centaurs, TNOs, and comets. As can be seen, the high-inclination objects
(including those that were observed in this work) do not show a wide variety
of colours, unlike TNOs or Centaurs. They fall in a rather restricted area
intersecting with grey Centaurs, moderately red TNOs, and Jupiter trojans.

Figure 6: Colour-colour diagram comparing different dynamical classes of small
solar system objects, such as Jupiter family comets (JFC) (Lamy & Toth, 2009),
long-period comets (LPC) (Solontoi et al., 2012), Jupiter trojans (Emery et
al., 2011), TNOs, and Centaurs (Belskaya et al., 2015) with objects on highly
inclined orbits. The black symbol shows the solar colours.

To further explore the similarity between these populations, we compare the
average spectra of trojans (Emery et al., 2011), grey Centaurs, and moderately
red TNOs (Merlin et al., 2017) with that of high-inclination objects. To do
this we converted the average colours and the corresponding standard deviation
into intensities and subtracted the Sun colours (Fig. 7). The mean colours of
high-inclination objects have the best match with grey Centaurs, but also
overlap with those of moderately red TNOs and to a lesser degree with those of
Jupiter trojans.

There is a tendency for ultra-red material to be mostly present among low-
inclination objects (Tegler & Romanishin, 2000; Trujillo & Brown, 2002;
Doressoundiram et al., 2005; Gulbis et al., 2006; Fulchignoni et al., 2008;
Peixinho et al., 2008). As Fig. 6 shows, there is a clear absence of very red
surfaces among polar and retrograde objects, even though they are present on
some TNOs and Centaurs. Different scenarios are proposed in order to explain
this absence in comets (Jewitt, 2002) and damocloids Jewitt (2005, 2015). One
is the size-dependent effects of collisional resurfacing as measured
damocloids are smaller than Centaurs and KBOs. This scenario was considered
the least plausible by Jewitt (2005). It is not supported by the new
observations of relatively large high-inclination objects (or damocloids), as
can be seen in Table 4.

Another explanation is the thermal dissociation or the instability of ultra-
red matter at higher temperatures Jewitt (2005); Lamy & Toth (2009). It is
suggested that higher temperatures are uncharacteristic for the complex
organics associated with the ultra-red matter (Dalle Ore et al., 2015; Jewitt,
2002). What somewhat decreases the likelihood of such scenario is the
discovery of very red objects that come very close to the Sun and retain their
very red surfaces. For example, the objects on cometary orbit 2016 ND21
($q=3.76$ au, $e=0.56$, $a=8.46$ au) was observed at heliocentric distance
$r=3.8$ au and was found to have an extremely red surface colours similar to
those of very red TNOs and Centaurs (Hromakina et al., 2019). However, this
object’s orbit is highly eccentric, and the body spends a lot of time in the
outer part of the solar system, and may not have spent enough time at close
distances to lose its redness.

Finally, the blanketing of the primordial red surfaces created by outgassing
activity may be responsible for the absence of ultra-red surfaces (Jewitt,
2002; Grundy, 2009). This scenario, although very plausible, has unresolved
issues. In particular, the discovery of a wide variety of colours among active
comets, including ultra-red ones (Lamy & Toth, 2009; Bauer et al., 2003),
requires a better understanding of the blanketing mechanism. It appears that
the existence of activity does not necessarily result in the creation of a
mantle that covers the ultra-red matter on the surface (Lamy & Toth, 2009;
Bauer et al., 2003), and other effects may prevent the dust from settling on
the surface. It is possible that such the mantle would appear only after a
prolonged activity.

Table 4: Orbital and physical properties of high-inclination objects

Object | i | e | a, au |  H1, mag |  A2 | D, km | B-V | V-R | R-I |  Ref3
---|---|---|---|---|---|---|---|---|---|---
(330759) 2008 SO218 | 170.3 | 0.56 | 8.12 | 12.8 | 0.08 | 13.5 |  0.89$\pm$0.03 |  0.55$\pm$0.02 |  0.50$\pm$0.04 | J15
(336756) 2010 NV1 | 140.7 | 0.97 | 281.2 | 10.4 | 0.04 | 52.2 |  0.79$\pm$0.03 |  0.53$\pm$0.02 |  0.39$\pm$0.02 | J15
|  |  |  |  |  |  |  0.74$\pm$0.03 |  0.50$\pm$0.02 |  | T16
|  |  |  |  |  |  |  0.87$\pm$0.10 |  0.53$\pm$0.10 |  | This work
(342842) 2008 YB3 | 105.1 | 0.44 | 11.62 | 9.3 | 0.05 | 79.0 |  0.82$\pm$0.01 |  0.46$\pm$0.01 |  | S10
|  |  |  |  |  |  |  |  0.50$\pm$0.06 |  0.5$\pm$0.08 | P13
|  |  |  |  |  |  |  0.80$\pm$0.01 |  0.46$\pm$0.01 |  0.95$\pm$0.01 | H12
|  |  |  |  |  |  |  0.74$\pm$0.02 |  0.51$\pm$0.02 |  | J15
|  |  |  |  |  |  |  0.77$\pm$0.02 |  0.46$\pm$0.02 |  | T16
|  |  |  |  |  |  |  0.81$\pm$0.06 |  0.49$\pm$0.06 |  | This work
(418993) 2009 MS9 | 68.0 | 0.97 | 377.9 | 9.8 |  |  |  0.84$\pm$0.04 |  0.52$\pm$0.04 |  | J15
(434620) 2005 VD | 172.9 | 0.25 | 6.67 | 14.1 |  |  |  0.60$\pm$0.17 |  0.45$\pm$0.11 |  | P13
(468861) 2013 LU28 | 125.4 | 0.95 | 187.10 | 8.2 |  |  |  0.83$\pm$0.04 |  0.57$\pm$0.04 |  0.68$\pm$0.05 | This work
|  |  |  |  |  |  |  0.84$\pm$0.07 |  0.62$\pm$0.09 |  0.42$\pm$0.09 | This work
(517717) 2015 KZ120 | 85.5 | 0.82 | 46.73 | 10.0 |  |  |  0.78$\pm$0.05 |  0.49$\pm$0.03 |  0.61$\pm$0.06 | This work
|  |  |  |  |  |  |  0.93$\pm$0.05 |  0.43$\pm$0.05 |  0.54$\pm$0.07 | This work
1999 LE31 | 151.8 | 0.47 | 8.14 | 12.5 |  |  |  0.74$\pm$0.07 |  0.51$\pm$0.05 |  0.49$\pm$0.05 | J05
|  |  |  |  |  |  |  0.75$\pm$0.02 |  0.45$\pm$0.02 |  0.54$\pm$0.03 | J05
2000 HE46 | 158.6 | 0.9 | 23.7 | 14.8 |  |  |  0.87$\pm$0.06 |  0.55$\pm$0.07 |  0.40$\pm$0.06 | J05
2002 RP120 | 119.1 | 0.95 | 53.71 | 12.3 | 0.10 | 14.6 |  0.83$\pm$0.03 |  0.47$\pm$0.03 |  0.52$\pm$0.03 | J05
2008 KV42 | 103.4 | 0.49 | 41.68 | 8.8 |  |  |  0.82$\pm$0.09 |  0.47$\pm$0.06 |  0.42$\pm$0.06 | H12
|  |  |  |  |  |  |  |  0.47$\pm$0.06 |  0.42$\pm$0.06 | S10
2010 BK118 | 143.9 | 0.98 | 376.1 | 10.2 | 0.06 | 52.0 |  0.76$\pm$0.03 |  0.57$\pm$0.03 |  0.51$\pm$0.03 | J15
2010 OM101 | 118.8 | 0.92 | 26.1 | 17.0 | 0.04 | 3.2 |  0.81$\pm$0.04 |  0.53$\pm$0.04 |  | J15
2010 OR1 | 143.9 | 0.92 | 26.9 | 16.2 | 0.11 | 2.3 |  0.76$\pm$0.03 |  0.52$\pm$0.09 |  0.48$\pm$0.04 | J15
|  |  |  |  |  |  |  0.81$\pm$0.04 |  0.51$\pm$0.09 |  0.43$\pm$0.02 | J15
2010 WG9 | 70.4 | 0.65 | 53.2 | 8.3 | 0.07 | 112.7 |  |  |  0.52$\pm$0.02 | R13
|  |  |  |  |  |  |  0.73$\pm$0.05 |  0.37$\pm$0.04 |  | J15
2012 DR30 | 78.0 | 0.98 | 1445.3 | 7.1 |  |  |  0.65$\pm$0.04 |  0.56$\pm$0.03 |  0.42$\pm$0.04 | K13
|  |  |  |  |  |  |  0.81$\pm$0.05 |  0.49$\pm$0.05 |  0.49$\pm$0.07 | This work
2012 YO6 | 106.9 | 0.48 | 6.3 | 14.7 |  |  |  |  |  0.34$\pm$0.05 | J15
2013 NS11 | 130.3 | 0.79 | 12.6 | 13.6 | 0.03 | 15.2 |  0.74$\pm$0.02 |  0.56$\pm$0.04 |  0.52$\pm$0.08 | J15
2013 YG48 | 61.3 | 0.75 | 8.18 | 17.2 |  |  |  0.77$\pm$0.02 |  0.55$\pm$0.02 |  | J15
A/2019 U5 | 113.5 | 1.002 | – | 9.7 |  |  |  0.75$\pm$0.04 |  0.42$\pm$0.04 |  0.54$\pm$0.05 | This work
|  |  |  |  |  |  |  0.68 $\pm$0.05 |  0.42$\pm$0.05 |  0.51$\pm$0.05 | This work
2020 EP | 76.4 | 0.76 | 10.50 | 15.9 |  |  |  0.83$\pm$0.09 |  0.51$\pm$0.06 |  0.42$\pm$0.10 | This work

1 Absolute magnitudes are taken from MPC.

2 Albedo and diameters are from Licandro et al. (2016); Fernández et al.
(2005); Nugent et al. (2016); Bauer et al. (2013).

3 H12: Hainaut et al. (2012), J05: Jewitt (2005), J15: Jewitt (2015), K13:
Kiss et al. (2013), P13: Pinilla-Alonso et al. (2013), R13: Rabinowitz et al.
(2013), S10: Sheppard (2010), T16: Tegler et al. (2016).

Figure 7: Mean colours of high-inclination objects (black circles) together
with mean spectra of Jupiter trojans (green) from Emery et al. (2011), grey
Centaurs (blue), and moderately red TNOs (red) from Merlin et al. (2017).

As many of the high-inclination objects reside on cometary orbits, there is a
question of whether these objects are indeed comets with as-yet-undetected
cometary activity. We checked how close to the Sun these bodies were observed.
In Fig. 8 (upper panel) we show the absolute magnitude of high-inclination
asteroids and the heliocentric distances at which the corresponding objects
were observed. Most of the considered objects were observed only once, and the
value of $r$ corresponds to the day of observation. When multiple observations
were available, we chose the one that corresponds to the smaller value of $r$.
The lower panel of Fig. 8 shows a distribution of perihelion distances at the
time of observations for both high-inclination asteroids and comets. From the
histogram we can see that the two populations have similar distribution up to
about 10 au (the only exception being a larger number of asteroids with $q$
around 3 au). Thus, the figure shows that for the majority of the observed
high-inclination asteroids, we cannot assume that the cometary activity was
not detected due to the larger distance from the Sun. If such asteroids are in
fact dormant or dead comets, the absence of ultra-red matter can be associated
with its blanketing, as we already discussed in this section. However, there
is a clear trend for smaller asteroids to be detected closer to the Sun, which
may be caused by the observational bias.

Figure 8: Upper panel: Absolute magnitude vs. heliocentric distance at the
time of observation for high-inclination asteroids. Lower panel: Number of
discovered high-inclination comets and asteroids observed at various
perihelion distances.

##  6 Lifetimes of high-inclination objects

###  6.1 Simulation of orbit lifetimes

We simulated the orbital dynamics of the considered asteroids into the past
and into the future, to estimate the time when they entered their retrograde
orbits and when they will leave them. The simulations were conducted using the
REBOUND package.333REBOUND code is freely available at
http://github.com/hannorein/rebound As a compromise between speed and
accuracy, we used the WHFast integrator, included the perturbations from the
four Jovian planets, and excluded the terrestrial planets. For each asteroid
we created 96 clones with the normal distribution within the error ellipsoid
of the orbital elements. The orbital elements and the covariation matrix of
their errors were taken from JPL HORIZONS database.444JPL HORIZONS database is
available at https://ssd.jpl.nasa.gov/horizons.cgi

The results for the fraction of the surviving clones as a function of time are
shown in Fig. 9. The figure shows that there is no sharp moment when most
clones leave the system: the population of surviving clones decays with time
almost exponentially. This happens because the orbits of the considered
objects are strongly chaotic, and the available precision of the orbital
elements is insufficient to accurately predict their orbits, but can merely
describe them statistically. In the vicinity of the nominal orbits there are
some with very short and very long lifetimes, and each clone is randomly
assigned a lifetime. In probabilistic terms, the nearly exponential decay
implies that the probability for a clone to escape remains nearly constant
with time. Deviations from the exponential decay in Fig. 9 include a small
plateau for 2013 LU28 at small times (corresponding to relative stability of
its present-day orbit) and a strong tails for 2015 KZ120 and 2020 EP at large
times (corresponding to the existence of more stable orbits in close vicinity
to the nominal orbit).

Figure 9: Fraction of surviving clones of asteroids 2013 LU28 (top), 2015
KZ120 (middle), and 2020 EP (bottom) as a function of time. The red and blue
lines correspond to the forward and backward integration of orbits,
respectively. The dashed lines show the exponential least-squares fits.

The median lifetimes of the clones are given in Table 5. The median lifetime
is easier to compute than the mean lifetime or its higher-order moments as it
is less sensitive to the few longest surviving particles, and its computation
does not require waiting until all the particles leave the system. The
statistical errors on the median lifetimes are computed, as explained in
Appendix A, based on the assumption of the exponential decay of the number of
surviving particles with time.

We believe that the estimated future lifetimes of the clones are a good
statistical indicator of the expected future lifetime of the real asteroid,
whereas the past lifetimes of the clones do not necessary indicate the real
time passed by an asteroid on its orbit. The overwhelming majority of such
simulated clones come from interstellar space, a highly improbable process due
to the very low density of interstellar asteroids. Ignoring such clones and
considering only the ones coming from the Kuiper belt or the main asteroid
belt could significantly alter the lifetime distribution, but would be very
hard to compute because of the extremely small number of such clones. A
Bayesian analysis of the most plausible ways to enter the existing orbits is
needed to estimate the past lifetimes more accurately.

Table 5: Median lifetimes of asteroid clones simulated into the future and into the past. The computation of errors is explained in Appendix A. | Lifetime | Lifetime
---|---|---
Asteroid | in the past, Myr | in the future, Myr
(468861) 2013 LU28 | $-11.1\pm 1.6$ | $9.2\pm 1.4$
(517717) 2015 KZ120 | $-16.1\pm 2.4$ | $16.3\pm 2.4$
2020 EP | $-0.32\pm 0.05$ | $0.23\pm 0.03$

Lifetimes of 2013 LU28 can be compared with the calculations by Kankiewicz &
Włodarczyk (2017). Assigning errors in accordance with our Appendix A, we can
reword Kankiewicz & Włodarczyk (2017) by saying that the lifetime in the past
is $-8.9\pm 1.3$ Myr with the Yakovsky effect and $-9.2\pm 1.3$ for purely
gravitational interactions; instead, into the future the lifetimes are $8.4\pm
1.2$ Myr and $6.9\pm 1.0$ Myr with and without Yarkovsky, respectively.
Although in all four cases the times are somewhat shorter than our estimates,
the differences all fall within $1.5\sigma$, making them statistically
insignificant.

###  6.2 Influence of the Yarkovsky effect

The results with and without the Yarkovsky effect in Kankiewicz & Włodarczyk
(2017) seem statistically indistinguishable in most cases. The relative
deviations in the median lifetimes with and without the Yarkovsky effect are
shown in Fig. 10. Along the horizontal axis of this figure we plot $\Delta
a/a$, where $a$ is the semi-major axis of an asteroid, and $\Delta a$ is
determined as the minimum distance to a planet’s orbit as measured from the
asteroid’s pericentre, apocentre, or semi-major axis (e.g. if the pericentre
is 0.5 au from Saturn, the apocentre 0.1 au from Neptune, and the semi-major
axis 0.2 au from Uranus, then $\Delta a=0.1$ au). The parameter $\Delta a$
could serve as a crude indicator of the effect of a small change in orbit on
the asteroid’s dynamics. If a small perturbation of the asteroid’s orbit
allows or forbids the intersection of certain planetary orbits or puts the
asteroid into a 1:1 resonance with a planet, then it can dramatically change
the lifetime of the asteroid’s orbit. Otherwise small perturbations such as
the Yarkovsky effect might be less significant.

We see in Fig. 10 that for $\Delta a/a>0.05$ most points lie within the
$1\sigma$ area, and all points lie within the $2\sigma$ area (in yellow). Only
for the smallest $\Delta a/a$ do some points go beyond $2\sigma$. Therefore,
we conclude that for most asteroids the Yarkovsky effect changes the results
on the level of the shot noise, thus it is indistinguishable from a random
perturber. The few exceptions correspond to the orbits whose semi-major axis,
aphelion, or perihelion are close to the orbit of one of the planets.

Figure 10: Relative change of the median escape times for asteroids due to the
Yarkovsky effect. Vertical coordinates are obtained by comparing computations
with and without YORP for each asteroid from Kankiewicz & Włodarczyk (2016)
and Kankiewicz & Włodarczyk (2018). Colour-coding distinguishes between the
two datasets, as well as integration into the past or the future. The
horizontal axis is the relative distance to the closest planet from the
asteroid’s pericentre, apocentre, or semi-major axis (see text for details).
The 1$\sigma$ and 2$\sigma$ areas for the shot noise are shaded in yellow.

###  6.3 Influence of the orbit uncertainty

In Fig. 11 we study how the distribution of asteroid clones over the lifetimes
depends on the precision of the asteroid’s orbit. We artificially multiply or
divide errors on the orbital elements by some factor, and find that the
lifetime distribution remains almost unaffected, even if the error is
multiplied by 100 or divided by $10^{10}$. For 2020 EP we also try to shrink
the error ellipsoid by the factor of 100, and then shift the median of the
orbital element distribution to a random point within the old error ellipsoid
(orange lines in the bottom panel), again with no significant change in the
lifetime distribution.

Figure 11: Distribution of asteroid clones over ejection times for the
backward integration of asteroids (468861) 2013 LU28 (top) and 2020 EP
(bottom) for artificially magnified, shrunk, or shifted error ellipsoids.

In particular, this result implies that although the orbit of 2020 EP is very
uncertain (the observation arc is only 54 days), the estimated lifetime will
not change even if the accuracy of the orbital elements is improved by a
significant factor.

This stability of the distributions seems less striking if we note that the
Lyapunov times in the considered region of the solar system are typically much
shorter than the orbit lifetimes. For example, for (468861) 2013 LU28
Kankiewicz & Włodarczyk (2018) find the Lyapunov time to be 0.104 Myr, and the
lifetime 6.9 Myr. The 66 times discrepancy between them translates into the
error increment during the time of integration by the factor of $e^{66}\approx
10^{29}$. This factor rapidly turns a good knowledge of initial conditions
into total unpredictability, and equates initial errors of $10^{-2}$ and
$10^{-10}$. To have a chance of precisely predicting the orbit of 2013 LU28
throughout its entire lifetime, we need the initial relative error in the
orbital elements of the order of $10^{-29}$. Only then is there a chance of
obtaining in Figure 11 a sharp drop at a certain instance of time instead of a
dull exponent. Still, even with such an unrealistic accuracy of the initial
conditions, for the precise prediction of the escape we will need an equally
unrealistic accuracy, for example of the orbit integrator and planet masses.
Even the Yarkovsky effect, although it gets lost in errors at the current
accuracy of computations, will be crucial to know precisely if the other
errors are eliminated.

##  7 Summary and conclusions

We have presented new photometric observations for six high-inclination
objects. All of the observed targets have moderately red surfaces, which seem
to be typical for high-inclination asteroids. Two of the targets in this work
(C/2018 DO4 and C/2019 O3) appeared to be comets. The level of dust production
for comets was estimated by the means of parameter Af$\rho$. For C/2018 DO4
this value is around 100 cm at the aperture of about 6 000 km. The study of
morphology for the comet C/2018 DO4 using different filtering techniques
revealed some complex structures that may be associated with particles that
were ejected from the cometary nucleus. The measured Af$\rho$ values for the
comet C/2019 O3 are much larger than those for the comet C/2018 DO4 and are in
a range from about 2 000 to 3 700 cm at apertures from 30 000 to 60 000 km.

From the search in all-sky surveys, we were able to find some additional
photometric data: four asteroids were found in the SDSS catalogue, and three
in the Pan-STARRS images. However, due to the large error bars and scarcity of
the found objects, it did not help us to increase the statistics of high-
inclination objects with knows surface properties. Nonetheless, we can
conclude that the mean colours of the found objects correspond to moderately
red surfaces.

No confident detection of ultra-red objects were found in this work, which
supports the results of Jewitt (2005, 2015). The average colours of high-
inclination objects are very similar to those of grey Centaurs and moderately
red TNOs, and to a lesser extent to those of Jupiter trojans.

For three high-inclination bodies, we simulated the orbital dynamics backward
and forward in time, and estimated the moment when these objects entered and
will leave their unstable orbits. The lifetimes in the forward and backward
integrations both appeared strikingly insensitive to selection of the orbital
elements of the asteroid within their error margins, and to the value of the
errors themselves. For example, increasing the error by a factor of 100 or
decreasing it by a factor of $10^{10}$ produced virtually no change in the
computed lifetime distribution of asteroid (468861) 2013 LU28. Evaluation of
the shot noise of the median lifetime estimate allowed us to re-assess the
previous claims (Kankiewicz & Włodarczyk, 2017, 2018) about the influence of
the Yarkovsky effect on the asteroid lifetime, and in most cases we found its
influence to be below the threshold of statistical significance. It could
imply that in such a strongly chaotic region of the solar system it is
virtually impossible to conduct any reliable predictions of orbits into the
distant past or future, but only very crude statistical estimates for
ensembles of drastically different closely positioned orbits. These
statistical estimates might be very robust with respect to small perturbations
(such as the errors on the orbital elements or the Yarkovsky effect), but
their predictive value is very limited.

In conclusion, our knowledge about high-inclination objects remains quite
limited. Thus, we are planning to continue the programme dedicated to the
investigations of these rare solar system objects.

###### Acknowledgements.

OI thanks the Slovak Academy of Sciences grant VEGA 2/0023/18 and the Slovak
Research and Development Agency under the Contract no. APVV-19-0072.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS acknowledges support and resources from the
Center for High-Performance Computing at the University of Utah. The SDSS web
site is www.sdss.org. SDSS is managed by the Astrophysical Research Consortium
for the Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science, Carnegie
Mellon University, the Chilean Participation Group, the French Participation
Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica
de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley
National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-
Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für
Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik
(MPE), National Astronomical Observatories of China, New Mexico State
University, New York University, University of Notre Dame, Observatório
Nacional / MCTI, The Ohio State University, Pennsylvania State University,
Shanghai Astronomical Observatory, United Kingdom Participation Group,
Universidad Nacional Autónoma de México, University of Arizona, University of
Colorado Boulder, University of Oxford, University of Portsmouth, University
of Utah, University of Virginia, University of Washington, University of
Wisconsin, Vanderbilt University, and Yale University.
The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been
made possible through contributions by the Institute for Astronomy, the
University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society
and its participating institutes, the Max Planck Institute for Astronomy,
Heidelberg and the Max Planck Institute for Extraterrestrial Physics,
Garching, The Johns Hopkins University, Durham University, the University of
Edinburgh, the Queen’s University Belfast, the Harvard-Smithsonian Center for
Astrophysics, the Las Cumbres Observatory Global Telescope Network
Incorporated, the National Central University of Taiwan, the Space Telescope
Science Institute, the National Aeronautics and Space Administration under
Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA
Science Mission Directorate, the National Science Foundation Grant No.
AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the
Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.

## References

  * A’Hearn et al. (1984) A’Hearn, M. F., Schleicher, D. G., Millis, R. L., Feldman, P. D., & Thompson, D. T. 1984, AJ, 89, 579
  * Batygin & Brown (2016) Batygin, K. & Brown, M. E. 2016, ApJ, 833, L3
  * Bauer et al. (2003) Bauer, J. M., Fernández, Y. R., & Meech, K. J. 2003, PASP, 115, 981
  * Bauer et al. (2013) Bauer, J. M., Grav, T., Blauvelt, E., et al. 2013, ApJ, 773, 22
  * Beaugé et al. (2003) Beaugé, C., Ferraz-Mello, S., & Michtchenko, T. A. 2003, ApJ, 593, 1124
  * Belskaya et al. (2015) Belskaya, I. N., Barucci, M. A., Fulchignoni, M., & Dovgopol, A. N. 2015, Icarus, 250, 482
  * Berthier et al. (2016) Berthier, J., Carry, B., Vachier, F., Eggl, S., & Santerne, A. 2016, MNRAS, 458, 3394
  * Berthier et al. (2006) Berthier, J., Vachier, F., Thuillot, W., et al. 2006, in Astronomical Society of the Pacific Conference Series, Vol. 351, Astronomical Data Analysis Software and Systems XV, ed. C. Gabriel, C. Arviset, D. Ponz, & S. Enrique, 367–+
  * Bowell et al. (1989) Bowell, E., Hapke, B., Domingue, D., et al. 1989, in Asteroids II, ed. R. P. Binzel, T. Gehrels, & M. S. Matthews, 524–556
  * Bradley et al. (2019) Bradley, L., Sipőcz, B., Robitaille, T., et al. 2019, astropy/photutils: v0.6
  * Brasser et al. (2012) Brasser, R., Duncan, M. J., Levison, H. F., Schwamb, M. E., & Brown, M. E. 2012, Icarus, 217, 1
  * Chen et al. (2016) Chen, Y.-T., Lin, H. W., Holman, M. J., et al. 2016, ApJ, 827, L24
  * Connors & Wiegert (2018) Connors, M. & Wiegert, P. 2018, Planet. Space Sci., 151, 71
  * Dalle Ore et al. (2015) Dalle Ore, C. M., Barucci, M. A., Emery, J. P., et al. 2015, Icarus, 252, 311
  * de la Fuente Marcos & de la Fuente Marcos (2014) de la Fuente Marcos, C. & de la Fuente Marcos, R. 2014, Ap&SS, 352, 409
  * de la Fuente Marcos et al. (2015) de la Fuente Marcos, C., de la Fuente Marcos, R., & Aarseth, S. J. 2015, MNRAS, 446, 1867
  * Doressoundiram et al. (2005) Doressoundiram, A., Peixinho, N., Doucet, C., et al. 2005, Icarus, 174, 90
  * Emery et al. (2011) Emery, J. P., Burr, D. M., & Cruikshank, D. P. 2011, AJ, 141, 25
  * Farnham (2009) Farnham, T. L. 2009, Planet. Space Sci., 57, 1192
  * Fernández et al. (2005) Fernández, Y. R., Jewitt, D. C., & Sheppard, S. S. 2005, AJ, 130, 308
  * Ferraz-Mello et al. (2006) Ferraz-Mello, S., Micchtchenko, T. A., & Beaugé, C. 2006, Regular motions in extra-solar planetary systems, ed. B. A. Steves, A. J. Maciejewski, & M. Hendry, 255
  * Fulchignoni et al. (2008) Fulchignoni, M., Belskaya, I., Barucci, M. A., de Sanctis, M. C., & Doressoundiram, A. 2008, Transneptunian Object Taxonomy, ed. M. A. Barucci, H. Boehnhardt, D. P. Cruikshank, A. Morbidelli, & R. Dotson, 181
  * Gladman et al. (2009) Gladman, B., Kavelaars, J., Petit, J. M., et al. 2009, ApJ, 697, L91
  * Greenstreet et al. (2012) Greenstreet, S., Gladman, B., Ngo, H., Granvik, M., & Larson, S. 2012, ApJ, 749, L39
  * Grundy (2009) Grundy, W. M. 2009, Icarus, 199, 560
  * Gulbis et al. (2006) Gulbis, A. A. S., Elliot, J. L., & Kane, J. F. 2006, Icarus, 183, 168
  * Gunn et al. (1998) Gunn, J. E., Carr, M., Rockosi, C., et al. 1998, AJ, 116, 3040
  * Gwyn et al. (2012) Gwyn, S. D. J., Hill, N., & Kavelaars, J. J. 2012, PASP, 124, 579
  * Hainaut et al. (2012) Hainaut, O. R., Boehnhardt, H., & Protopapa, S. 2012, A&A, 546, A115
  * Holmberg et al. (2006) Holmberg, J., Flynn, C., & Portinari, L. 2006, MNRAS, 367, 449
  * Hromakina et al. (2019) Hromakina, T., Velichko, S., Belskaya, I., Krugly, Y., & Sergeev, A. 2019, in EPSC-DPS Joint Meeting 2019, Vol. 2019, EPSC–DPS2019–1216
  * Ivanova et al. (2019) Ivanova, O., Luk’yanyk, I., Kolokolova, L., et al. 2019, A&A, 626, A26
  * Ivanova et al. (2018) Ivanova, O. V., Picazzio, E., Luk’yanyk, I. V., Cavichia, O., & Andrievsky, S. M. 2018, Planet. Space Sci., 157, 34
  * Jewitt (2005) Jewitt, D. 2005, AJ, 129, 530
  * Jewitt (2015) Jewitt, D. 2015, AJ, 150, 201
  * Jewitt (2002) Jewitt, D. C. 2002, AJ, 123, 1039
  * Kankiewicz & Włodarczyk (2016) Kankiewicz, P. & Włodarczyk, I. 2016, in 37th Meeting of the Polish Astronomical Society, ed. A. Różańska & M. Bejger, Vol. 3, 286–289
  * Kankiewicz & Włodarczyk (2017) Kankiewicz, P. & Włodarczyk, I. 2017, MNRAS, 468, 4143
  * Kankiewicz & Włodarczyk (2018) Kankiewicz, P. & Włodarczyk, I. 2018, Planet. Space Sci., 154, 72
  * Kiss et al. (2013) Kiss, C., Szabó, G., Horner, J., et al. 2013, A&A, 555, A3
  * Kostov & Bonev (2018) Kostov, A. & Bonev, T. 2018, Bulgarian Astronomical Journal, 28, 3
  * Kozai (1962) Kozai, Y. 1962, AJ, 67, 591
  * Lamy & Toth (2009) Lamy, P. & Toth, I. 2009, Icarus, 201, 674
  * Larson & Sekanina (1984) Larson, S. M. & Sekanina, Z. 1984, AJ, 89, 571
  * Levison (1996) Levison, H. F. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 107, Completing the Inventory of the Solar System, ed. T. Rettig & J. M. Hahn, 173–191
  * Li et al. (2019) Li, M., Huang, Y., & Gong, S. 2019, A&A, 630, A60
  * Licandro et al. (2016) Licandro, J., Alí-Lagoa, V., Tancredi, G., & Fernández, Y. 2016, A&A, 585, A9
  * Licandro et al. (2018) Licandro, J., Popescu, M., de León, J., et al. 2018, A&A, 618, A170
  * Luu (1993) Luu, J. X. 1993, PASP, 105, 946
  * Meech et al. (2009) Meech, K. J., Pittichová, J., Bar-Nun, A., et al. 2009, Icarus, 201, 719
  * Meech & Svoren (2004) Meech, K. J. & Svoren, J. 2004, Using cometary activity to trace the physical and chemical evolution of cometary nuclei, ed. M. C. Festou, H. U. Keller, & H. A. Weaver, 317
  * Merlin et al. (2017) Merlin, F., Hromakina, T., Perna, D., Hong, M. J., & Alvarez-Candal, A. 2017, A&A, 604, A86
  * Morais & Namouni (2017) Morais, M. H. M. & Namouni, F. 2017, MNRAS, 472, L1
  * Morbidelli (2002) Morbidelli, A. 2002, Modern celestial mechanics : aspects of solar system dynamics
  * Morbidelli et al. (2020) Morbidelli, A., Batygin, K., Brasser, R., & Raymond, S. N. 2020, Monthly Notices of the Royal Astronomical Society: Letters, 497, L46
  * Namouni & Morais (2015) Namouni, F. & Morais, M. H. M. 2015, MNRAS, 446, 1998
  * Namouni & Morais (2020) Namouni, F. & Morais, M. H. M. 2020, Monthly Notices of the Royal Astronomical Society, 494, 2191
  * Nugent et al. (2016) Nugent, C. R., Mainzer, A., Bauer, J., et al. 2016, AJ, 152, 63
  * Peixinho et al. (2008) Peixinho, N., Lacerda, P., & Jewitt, D. 2008, AJ, 136, 1837
  * Picazzio et al. (2019) Picazzio, E., Luk’yanyk, I. V., Ivanova, O. V., et al. 2019, Icarus, 319, 58
  * Pinilla-Alonso et al. (2013) Pinilla-Alonso, N., Alvarez-Candal, A., Melita, M. D., et al. 2013, A&A, 550, A13
  * Prialnik & Bar-Nun (1992) Prialnik, D. & Bar-Nun, A. 1992, A&A, 258, L9
  * Rabinowitz et al. (2013) Rabinowitz, D., Schwamb, M. E., Hadjiyska, E., Tourtellotte, S., & Rojo, P. 2013, AJ, 146, 17
  * Samarasinha & Larson (2014) Samarasinha, N. H. & Larson, S. M. 2014, Icarus, 239, 168
  * Sergeyev & Carry (2020) Sergeyev, A. V. & Carry, B. 2020, in preprint
  * Sheppard (2010) Sheppard, S. S. 2010, AJ, 139, 1394
  * Solontoi et al. (2012) Solontoi, M., Ivezić, Ž., Jurić, M., et al. 2012, Icarus, 218, 571
  * Tegler & Romanishin (2000) Tegler, S. C. & Romanishin, W. 2000, Nature, 407, 979
  * Tegler et al. (2016) Tegler, S. C., Romanishin, W., Consolmagno, G. J., & J., S. 2016, AJ, 152, 210
  * Tonry et al. (2012) Tonry, J. L., Stubbs, C. W., Lykke, K. R., et al. 2012, ApJ, 750, 99
  * Trujillo & Brown (2002) Trujillo, C. A. & Brown, M. E. 2002, ApJ, 566, L125
  * Volk & Malhotra (2013) Volk, K. & Malhotra, R. 2013, Icarus, 224, 66
  * Wiegert et al. (2017) Wiegert, P., Connors, M., & Veillet, C. 2017, Nature, 543, 687
  * York et al. (2000) York, D. G., Adelman, J., Anderson, John E., J., et al. 2000, AJ, 120, 1579

##  Appendix A Estimate of the error in the median orbit lifetime

Here we show how we estimate the error on the median orbit lifetime. This
estimate is useful not merely to establish confidence limits and compare our
results to the predecessors, but also allows us to make more general
conclusions regarding, for example, the statistical insignificance of the
lifetime alteration by the Yarkovsky effect for most retrograde asteroids.

Consider a probability density function $f(t)=dP/dt$ of asteroid clones over
lifetimes $t$. Let $m$ be the median value of this probability distribution,
so that the probability of $t$ being smaller than $m$ is equal to one-half:

$P(t<m)=\int_{0}^{m}f(t)dt=\frac{1}{2}.$ (2)

The probability density function and its median value are illustrated in
Figure 12 with red lines.

$f(t)$$t$$\delta t$$\delta N$$m_{N}$$m$ Figure 12: Illustration of the sample
median value. Theoretical distribution and its median value $m$ are in red.
The sample points and the sample median value $m_{N}$ are in blue.

Assume we are sampling the distribution over lifetimes for $N$ clones (small
blue circles in Fig. 12). The number of clones below the theoretical median
value $m$ is governed by the binomial distribution and may deviate from
$\frac{N}{2}$ by some value $\delta N$. If $N$ is large, $\delta N$ is well
approximated by the normal distribution, with the mean squared deviation

$\sigma_{\delta N}=\sqrt{\frac{1}{2}\cdot\frac{1}{2}\cdot
N}=\frac{\sqrt{N}}{2}.$ (3)

When the sample median value $m_{N}$ is computed, it will be shifted with
respect to $m$ by some value $\delta t=m_{N}-m$, such that the region between
$m$ and $m_{N}$ accommodates $\delta N$ points. If $\delta t$ is small, the
number of points between $m$ and $m_{N}$ can be approximated as the
mathematical expectation of the number of points within the rectangle $\delta
t\times f(m)$ (blue rectangle in Figure 12):

$\delta N\approx Nf(m)\delta t.$ (4)

Therefore, the distribution of $\delta t$ is proportional to the distribution
of $\delta N$, and is thus also a normal distribution. Its mean squared
deviation follows from Eq. (3) and Eq. (4):

$\sigma_{\delta t}=\frac{1}{2\sqrt{N}f(m)}.$ (5)

For the strongly chaotic orbits of asteroid clones, we expect to have a more
or less stable decay rate over time, with the probability density function
approximately described by the decay exponent:

$f(t)=\frac{1}{t_{0}}\exp\left(-\frac{t}{t_{0}}\right).$ (6)

This assumption is justified by Fig. 9, where we see that the number of
surviving clones is well approximated by an exponent. As the derivative of the
number of surviving clones over time is proportional to the probability
density function $f(t)$, the latter is thus also approximately exponential.

Then the median of such an exponential distribution is

$m=t_{0}\log{2}.$ (7)

The error on the median as determined from Eq. (5) is

$\sigma_{m}=\frac{t_{0}}{\sqrt{N}}.$ (8)

Finally, the relative error is

$\varepsilon_{m}=\frac{\sigma_{m}}{m}=\frac{1}{\sqrt{N}\log{2}}.$ (9)

This equation allows us to estimate the error on the median lifetime due to
the shot noise, determined from the simulation of $N$ asteroid clones.