11institutetext: Charles University, Faculty of Mathematics and Physics,
Institute of Astronomy, V Holešovičkách 2, 180 00 Prague, Czech Republic
11email<EMAIL_ADDRESS>22institutetext: Astronomical
Institute, Academy of Sciences of the Czech Republic, Fričova 1, 251 65
Ondřejov, Czech Republic 33institutetext: Institute of Astronomy of V. N.
Karazin Kharkiv National University, Sumska Str. 35, Kharkiv 61022, Ukraine
44institutetext: Faculty of Physics, Weizmann Institute of Science, 234 Herzl
St., Rehovot 7610001, Israel 55institutetext: E. Kharadze Georgian National
Astrophysical Observatory, Abastumani, Georgia 66institutetext: Institute of
Astronomy and NAO, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee
Blvd., BG-1784 Sofia, Bulgaria 77institutetext: Blue Mountains Observatory, 94
Rawson Pde. Leura, NSW 2780, Australia 88institutetext: Samtskhe-Javakheti
State University, Rustaveli Street 113, Akhaltsikhe 0080, Georgia
99institutetext: Institute for Astronomy, University of Edinburgh, Royal
Observatory, Edinburgh, EH9 3HJ, UK 1010institutetext: Astronomy Research
Center, Research Institute of Basic Sciences, Seoul National University, 1
Gwanak-ro, Gwanak-gu, Seoul 08826, Korea 1111institutetext: Modra Observatory,
Department of Astronomy, Physics of the Earth, and Meteorology, FMPI UK,
Bratislava SK-84248, Slovakia 1212institutetext: Astronomical Observatory
Institute, Faculty of Physics, Adam Mickiewicz University, Słoneczna 36,
60-286 Poznań, Poland 1313institutetext: Crimean Astrophysical Observatory,
Simeiz, Crimea, Ukraine 1414institutetext: Université Côte d’Azur,
Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France
1515institutetext: Ulugh Beg Astronomical Institute, Astronomicheskaya Str.
33, Tashkent 100052, Uzbekistan 1616institutetext: Carl Sagan Center at the
SETI Institute, 189 Bernardo Av., Mountain View, CA 94043, USA
1717institutetext: Sugarloaf Mountain Observatory, South Deerfield, MA, USA

# Secular change in the spin states of asteroids due to radiation and
gravitation torques. New detections and updates of the YORP effect

J. Ďurech 11 D. Vokrouhlický 11 P. Pravec 22 Yu. Krugly 331212 D. Polishook 44
J. Hanuš 11 F. Marchis 1616 A. Rożek 99 C. Snodgrass 99 L. Alegre 99 Z.
Donchev 66 Sh. A. Ehgamberdiev 1515 P. Fatka 22 N. M. Gaftonyuk 1313 A. Galád
1111 K. Hornoch 22 R. Ya. Inasaridze 5588 E. Khalouei 1010 H. Kučáková 22 P.
Kušnirák 22 J. Oey 77 D. P. Pray 1717 A. Sergeev 331414 I. Slyusarev 33

(Received ?; accepted ?)

###### Abstract

Context. The rotation state of small asteroids is affected in the long term by
perturbing torques of gravitational and radiative origin (the YORP effect).
The former can be detected by a change in the spin-axis orientation in the
inertial space; the latter manifests itself by a quadratic increase in the
rotation phase.

Aims. Direct observational evidence of the YORP effect is the primary goal of
our work. This includes both the YORP detection for new objects and an
improvement in the accuracy of previously known detections.

Methods. We carried out photometric observations of five near-Earth asteroids:
(1862) Apollo, (2100) Ra-Shalom, (85989) 1999 JD6, (138852) 2000 WN10, and
(161989) Cacus. Then we applied the light-curve inversion method to all
available data to determine the spin state and a convex shape model for each
of the five studied asteroids. The YORP effect was modeled as a linear change
of the rotation frequency $\upsilon\equiv\mathrm{d}\omega/\mathrm{d}t$. In the
case of (2100) Ra-Shalom, the analysis required that the spin-axis precession
due to the solar gravitational torque also be included.

Results. We obtained two new detections of the YORP effect: (i)
$\upsilon=(2.9\pm 2.0)\times 10^{-9}\,\text{rad}\,\text{d}^{-2}$ for (2100)
Ra-Shalom, and (ii) $\upsilon=(5.5\pm 0.7)\times
10^{-8}\,\text{rad}\,\text{d}^{-2}$ for (138852) 2000 WN10. The analysis of
Ra-Shalom also reveals a precession of the spin axis with a precession
constant $\alpha\sim 3000\,\arcsec$ yr-1. This is the first such detection
from Earth-bound photometric data. For the other two asteroids, we improved
the accuracy of the previously reported YORP detection: (i) $\upsilon=(4.94\pm
0.09)\times 10^{-8}\,\text{rad}\,\text{d}^{-2}$ for (1862) Apollo, and (ii)
$\upsilon=(1.86\pm 0.09)\times 10^{-8}\,\text{rad}\,\text{d}^{-2}$ for
(161989) Cacus. With this value, Apollo has the most precisely determined YORP
effect so far. Despite the recent report of a detected YORP effect for (85989)
1999 JD6, we show that the model without YORP cannot be rejected
statistically. Therefore, the detection of the YORP effect for this asteroid
requires future observations. In several of our targets, the currently
available observations do not provide enough constraints on the shape model
(even at large scales) to compute the theoretical YORP effect with sufficient
precision. Nevertheless, the interpretation of the detected signal as the YORP
effect is fairly plausible. The spin-axis precession constant of Ra-Shalom
determined from observations matches the theoretically expected value.

Conclusions. The total number of asteroids with a YORP detection has increased
to 12. In all cases, the rotation frequency increases in time. The analysis of
a rich photometric data set of irregularly shaped asteroids may require
inclusion of spin-axis precession in future studies.

###### Key Words.:

Minor planets, asteroids: general, Methods: data analysis, Techniques:
photometric

##  1 Introduction

Photometric observations of asteroids can be used to determine their rotation
state (in most cases, a unique rotation period and spin-axis direction) and
shape. In contrast to the shape, which is most often only a convex
approximation due to the limited information content of the disk-integrated
light curves, the spin state can be determined very precisely. This precision
increases with increasing number of available observations. The ability to
accurately constrain the direction of the spin axis principally stems from a
number of independent viewing geometries, defined by the observer (usually
Earth), the asteroid, and the Sun, that are represented by the data. In the
case of a few exceptional near-Earth asteroids, a sufficient number of
observations may be achieved in one or two apparitions (Monteiro et al., 2020;
Kwiatkowski et al., 2021). More often, only data accumulated over years or
decades eventually help to determine the spin-axis direction to within a few
degrees of accuracy. The precision of the rotation-period determination mainly
depends on the time span covered by observations. With light curves that are
observed over several decades and a typical rotation period of some hours, the
rotation-period precision can reach about $0.01$ s.

The simplest lowest-energy rotation model, characterized by a fixed direction
of the spin axis in the inertial space and by a constant rotation frequency,
is adequate to fit the observations of most asteroids. This model is about as
sophisticated as the Keplerian description of the asteroid heliocentric motion
on a fixed ellipse, however. As data spanning sufficiently long intervals of
time become available and help determine the spin state very accurately, the
basic model may need generalization by effects of even tiny torques,
especially if they accumulate over time into a strong perturbation. The
relevant torques originate (i) in the solar gravitational field and (ii) in
the solar radiation.

The first torque (i) is a well-known effect that has been empirically known
and was later theoretically analyzed by astronomers for centuries in the case
of the rotation of Earth (the difference is only that two centers, the Sun and
the Moon, act together to produce the resulting lunisolar precession of the
Earth axis). The explanation of the effect was first given by Isaac Newton
(e.g., Chandrasekhar, 1995, Sec. 23) and was later mathematically mastered by
Jean d’Alembert (D’Alembert, 1749). While it is obvious for the Earth, it took
centuries before the solar-induced gravitational precession was discussed in
the context of asteroid rotation. The pioneering works in this respect came
from the Uppsala school during the 1990s (e.g. Skoglöv et al., 1996; Skoglöv,
1997, 1998; Skoglöv & Erikson, 2002). Still, in spite of significant
improvements in mathematical modeling, the effect was elusive to direct
detection. Only when very precise measurements from the Dawn spacecraft
visiting (4) Vesta became available was the asteroidal precession
observationally determined (e.g., Konopliv et al., 2014). However, the
accuracy was not impressive, to the point that some consider the Vesta case
inconclusive (see, e.g., Archinal et al., 2018). The reason is that the effect
is rather small for a roundish shape like that of Vesta, only $\simeq
0.28^{\circ}$ cy-1, and Dawn stayed near Vesta for only about a year. The
detection of asteroidal spin precession using much less precise Earth-bound
observations can only be successful when the effect at a properly chosen
target is much stronger than was found for Vesta. Ideally, this would require
a slowly rotating, highly irregular body on a near-Earth orbit for which the
effect may increase to a fraction of a degree in a year. Still, a good-quality
data set spanning a long period of time is needed. The first hint that it
might be possible to detect the asteroidal spin precession has recently been
given by Ďurech et al. (2022b) for (1620) Geographos and (1685) Toro. However,
the first convincing detection awaited the current paper. For the first time,
we provide the detection of the spin-axis precession for (2100) Ra-Shalom.
This object is nearly an optimum target for this goal: it resides on an Aten
orbit, it rotates slowly, it has an irregular shape and obliquity away from
the unfavorable values of $0^{\circ}$, $90^{\circ}$ and $180^{\circ}$, and the
available numerous photometric data span a long interval of $44$ years. The
details are given in Sec. 3.2.

The second torque (ii) is a novel process without relevance for planets,
satellites, or even large asteroids. It has to do with radiation pressure that
is imparted on small, irregularly shaped asteroids, and it consists of two
components: a smaller torque due to directly reflected sunlight in the optical
band, and a more important component due to the recoil of the thermally
emitted radiation by the asteroid itself. The analysis of this phenomenon has
undergone an impressive revival in the last two decades after the pioneering
work of Rubincam (2000). Rubincam also coined the acronym YORP, which stands
for Yarkovsky-O’Keefe-Radzievski-Paddack effect. Many of the important
applications of YORP in planetary science can be found in the review by
Vokrouhlický et al. (2015). While the gravitational torque only affects the
direction of the spin axis, the YORP effect triggers secular perturbation of
both the spin-axis direction and the spin frequency. Both are tiny effects, so
that it is not surprising that the direct detection of YORP proved to be a
tricky task. Vokrouhlický et al. (2004) pointed out that the only foreseeable
way to detect the YORP effect is the secular change in the rotation frequency,
which fortuitously produces a quadratic perturbation in the rotation phase.
This is directly observable with photometric data. Taylor et al. (2007); Lowry
et al. (2007); Kaasalainen et al. (2007) indeed followed this method and
obtained the first YORP detections for the small near-Earth asteroids (54509)
YORP and (1862) Apollo. Since then, the observations of a handful of other
asteroids allowed us to detect the YORP torque directly, but the sample still
remains very limited. Most importantly, the available results are still
insufficient to solve the quantitative challenges posed by the theory of the
YORP effect (see Vokrouhlický et al., 2015, for a review and detailed
discussion). This also motivated our work: We aimed to increase the sample of
asteroids with YORP detections, to determine their physical properties, and to
compare the observed YORP with the theory.

Building partly on our previous research and also reporting on new targets, we
present evidence for the YORP effect in four near-Earth asteroids: (i) An
improved accuracy of the YORP effect detection for the two bodies (1862)
Apollo and (161989) Cacus, and (ii) a determination of the YORP effect for the
first time for another two bodies, (2100) Ra-Shalom and (138852) 2000 WN10.
Several aspects related to these detections are notable. The YORP effect
detection in (1862) Apollo now has the highest signal-to-noise ratio, with a
very good prospect of further improvements. In the case of (138852) 2000 WN10,
the YORP effect is detected for a prograde-rotating body for the first time.
Finally, (2100) Ra-Shalom represents a difficult analysis in which the spin-
axis precession is detected for the first time together with the YORP effect
on a body with the lowest rotation frequency so far. Additionally, we analyzed
available data for asteroid (85989) 1999 JD6, for which the YORP effect has
previously been reported by Tian et al. (2022). However, the addition of new
observations to the data set of this object shows that the YORP effect has not
yet been detected at a statistically significant level. Nevertheless, we find
it interesting that while a no-YORP solution is statistically admissible at a
level of about one sigma for (85989) 1999 JD6, most solutions require the
rotational frequency to be decelerated. If this is confirmed in the future,
this would be a breakthrough case in the context of the YORP effect theory
because all YORP detections so far have shown an acceleration of the rotation
frequency (see Sect. 4).

Figure 1: Bootstrap distribution of the pole direction in the ecliptic
longitude $\lambda$ and latitude $\beta$ (top), the YORP parameter $\upsilon$,
and the rotation period $P_{0}$ (bottom) for asteroid (1862) Apollo (the
latter corresponding to epoch JD0 2452276.0). The color-coding corresponds to
the density of points. The black cross marks the best-fit solution based on
the full original (not bootstrapped) data set.

The paper is organized as follows. In Sec. 2 we describe the available
observation data sets, the reconstructed models, and the uncertainty in the
rotation parameters obtained using our light-curve inversion method. We
compare these values with the theoretical prediction in Sec. 3, and we discuss
the broader context of our results in Sec. 4.

##  2 New photometric observations and model reconstruction

This section describes new photometric observations and model reconstruction
for each asteroid in our sample. The final model is clearly based not only on
our new data, but also uses previously published observations whenever
available. The observatories contributing to our campaign are listed in Table
2. Observations carried out with the Danish Telescope at La Silla (DK154) and
the Ondřejov Observatory telescope (D65) were calibrated, the other photometry
was relative without all-sky calibration. Observing times were converted from
the reported UTC into TDB.

The new observations are summarized in the tables in the appendix. For
simplicity, we only list individual nights, not individual light curves when
more of them (due to observations in different filters or calibration issues)
were observed in a single night with the same telescope.

To reconstruct the physical model, we used the light-curve inversion method of
Kaasalainen et al. (2001) and Kaasalainen & Torppa (2001), in which the
angular frequency $\omega$ evolves linearly in time as
$\omega=\omega_{0}+\dot{\omega}t$, where the rate
$\upsilon\equiv\dot{\omega}=\text{d}\omega/\text{d}t$ and the initial value
$\omega_{0}$ are free parameters of the optimization (a comment on the
validity of this approach can be found in Sec. 4). In the following text, we
call $\upsilon$ a YORP parameter, although the connection between the change
in the rotation rate detected from light curves and the YORP effect is
discussed below in Sect. 3. Instead of $\omega_{0}$, we report the sidereal
rotation period $P_{0}=2\pi/\omega_{0}$. Because the rotation period evolves
with time, it is necessary to also report the epoch JD0, for which $P_{0}$ is
given. In our previous works (e.g., Ďurech et al., 2008, 2018, 2022b), we set
JD0 to the epoch of the first observation. However, this causes a correlation
between $\upsilon$ and $P_{0}$, a shorter initial rotation period (at the
beginning of the observing data set) and a smaller YORP produce about the same
evolution in the rotation phase angle as a longer initial period that evolves
faster due to a larger YORP parameter $\upsilon$. The rotation phase angle
$\varphi$ evolves over time $t$ from some initial value $\varphi_{0}$ as

$\varphi(t)=\varphi_{0}+\frac{2\pi}{P_{0}}\left(t-\text{JD}_{0}\right)+\frac{1}{2}\upsilon\left(t-\text{JD}_{0}\right)^{2}\,.$
(1)

With JD0 at the beginning of the observing data set, the second term is always
positive, so that by decreasing $P_{0}$, the phase $\varphi$ increases for all
observations, and this can partly be balanced by decreasing $\upsilon$, from
which a positive correlation between $P_{0}$ and $\upsilon$ arises. However,
when JD0 is somewhere in the middle of the observing interval, the second term
is negative for half of the observations, and a small change in $P_{0}$ leads
to an increase in $\varphi$ for half of the observations and a decrease for
the remaining half, which cannot be compensated for by changing $\upsilon$. To
avoid a correlation between $\upsilon$ and $P_{0}$, we now set JD0 somewhere
close to the center of the interval covered by observations.

Figure 2: Convex shape model of asteroid (1862) Apollo shown from the
equatorial level (left and center, $90^{\circ}$ apart) and pole-on (right).
Figure 3: Example light curves of (1862) Apollo. The epoch of observation is
shown by the label at the top of each panel. The blue points are the observed
data, and the red curve is our best model including YORP. The geometry of the
observation is described by the aspect angle $\theta$, the solar aspect angle
$\theta_{0}$, and the solar phase angle $\alpha$.

To estimate the uncertainties in the spin and YORP parameters, we created 3000
bootstrap data sets and repeated the inversion. The bootstrap sample was a
random selection of the same number of light curves; in each selected light
curve, we bootstrapped its points. The parameter uncertainties reported below
are the standard deviations of the parameter distributions.

We also included the precession effect of the spin axis due to solar torque
for asteroids (2100) Ra-Shalom and (85989) 1999 JD6, for which this effect is
strong enough to be taken into account (see Ďurech et al., 2022b, for
details).

###  2.1 (1862) Apollo

Apollo was one of the two asteroids, together with (54509) YORP, for which the
YORP effect was detected for the first time (Kaasalainen et al., 2007; Lowry
et al., 2007; Taylor et al., 2007). Ďurech et al. (2008) published an update
of the original detection and derived a YORP acceleration of $(5.5\pm
1.2)\times 10^{-8}\text{rad}\,\text{d}^{-2}$ from 27 years of observations
from 1980–2007.

We observed Apollo during five more apparitions in 2014, 2017, 2019, 2021, and
2023. We also used other published observations from 2014 (Warner, 2014) and
2021 (Warner & Stephens, 2022) that are available in the ALCDEF
database111https://alcdef.org (Warner et al., 2009). The full data set now
consists of 87 light curves covering the time 1980–2023. The new light curves
are listed in Table 3. The observations taken with DK154 and D65 were
absolutely calibrated in the Johnson-Cousins VR photometric system. We
determined a color index $(V-R)=0.428\pm 0.010$ mag, which is compatible with
its Q-type spectral classification.

The new model we derived has the following spin parameters: an ecliptic
longitude $\lambda=(61.1\pm 4.6)^{\circ}$, an ecliptic latitude
$\beta=(-71.2\pm 2.2)^{\circ}$, a sidereal rotation period
$P_{0}=(3.065\,422\,6\pm 0.000\,000\,2)$ h (given at JD0 2452276.0 epoch), and
a secular change in the rotation rate $\upsilon=(4.94\pm 0.09)\times
10^{-8}\,\text{rad}\,\text{d}^{-2}$. The uncertainties of these spin
parameters were estimated from the distribution of the bootstrap solutions
shown in Fig. 1. The shape model is shown in Fig. 2. The difference in the
rotation phase accumulated over 43 years between a constant-period model with
$P_{0}$ and our best-fit model is $\sim 340^{\circ}$, almost one full
rotation. In Fig. 3 we show the match between the data and model for the six
new light curves as an example.

Figure 4: Color map of the reduced $\chi^{2}$ values of the light-curve fit
for asteroid Ra-Shalom for different values of the precession rate $\alpha$
and the YORP strength $\upsilon$. The red point marks the lowest $\chi^{2}$
value, the inner yellow contour is the estimated $1\sigma$ uncertainty limit,
and the outer white contour is a $3\sigma$ boundary. Figure 5: Convex shape
model of asteroid (2100) Ra-Shalom shown from the equatorial level (left and
center, $90^{\circ}$ apart) and pole-on (right).

###  2.2 (2100) Ra-Shalom

Previous photometry data of Ra-Shalom were analyzed by Ďurech et al. (2012a)
and later by Ďurech et al. (2018). In both cases, the data set was found to be
too short to reveal the YORP effect, and the constant period model was found
adequate to match the available observations. The best Ďurech et al. (2018)
could find was that the $3\sigma$ uncertainty interval of the YORP parameter,
namely $-1.0\times 10^{-8}<\upsilon<1.5\times
10^{-8}\,\text{rad}\,\text{d}^{-2}$, was slightly asymmetric with respect to
the origin. This finding indicated a possible positive $\upsilon$, which we
aimed to detect here by adding new observations. The new photometric
observations from 2019 and 2022 are listed in Table LABEL:tab:aspect_2100.
Part of these observations was conducted with the $28^{\prime\prime}$
telescope at the Wise Observatory (Brosch et al., 2015). We also included data
published by Warner & Stephens (2020a, 2023) in our analysis.

We also used light-curve observations from a citizen science project
spearheaded by the SETI Institute, which connects telescope owners of the
eVscope and eQuinox models developed by the France-based company, Unistellar.
These digital, automated, and compact telescopes, boasting an 11.3 cm mirror,
were tailored for people with very little background in astronomy.
Consequently, the telescope is seamlessly operated using a smartphone through
a dedicated mobile app.

An observing campaign focusing on Ra-Shalom commenced in August 2022. A myriad
of citizen astronomers collated and submitted their data, covering a nearly
two-month span. We selected a subset of data with adequate photometric
quality, which is detailed in Table 5. Nonetheless, these light curves were
not incorporated into the final modeling because their photometric quality was
inferior to that of our primary data set. This discrepancy can be attributed
to the small aperture of the telescope and to the fact that the data typically
encapsulated limited segments of the complete rotation phase (approximately
one to two hours relative to the almost 20-hour rotation period of Ra-Shalom).
We still used this as a chance to corroborate the data reliability from the
citizen project by juxtaposing it against our model predictions. This endeavor
underscores the potential of the Unistellar burgeoning telescope network
(comprising nearly 2000 potential contributors of various scientific
modalities backed by Unistellar and its partner, the SETI Institute) for the
photometry of pertinent targets, especially near-Earth asteroids (NEAs). For
Ra-Shalom, a comprehensive data set from experienced professional and amateur
astronomers was at our disposal, rendering the Unistellar observations
superfluous for the model reconstruction. This burgeoning citizen astronomer
network harbors significant potential for the future, however, particularly in
capturing photometric data for any sufficiently luminous NEA (refer to the
instance of (7335) 1989 JA, Lambert et al., 2023).

Figure 6: Example light curves of (2100) Ra-Shalom. The blue points show
observed data, and the red curve shows our best model with YORP and
precession. The last two light curves are examples of Unistellar eVscope data
that were not used for the modeling.

The first photometric observations of Ra-Shalom were taken in 1978. The
interval covered by available observations therefore spans 44 years, with 190
light curves in total. Being a near-Earth asteroid with the semimajor axis
$a=0.832$ au, an eccentricity $e=0.437$, and a rotation period of about 19.82
h, the solar gravitational torque causes a potentially non-negligible
precession of the Ra-Shalom rotation axis. In order to justify this extension
of the model, along the lines developed in Ďurech et al. (2022b), we first
theoretically estimated the corresponding perturbation of the rotation pole
direction. The model without YORP has a pole direction
$(286.4^{\circ},-56.4^{\circ})$, a rotation period $P=19.820\,040$ h, and the
dynamical ellipticity of the shape model is $\Delta=0.232$ (assuming a uniform
density distribution). The formal uncertainty in the pole direction is only a
few degrees. The precession constant $\alpha$ (see Eq. (1) in Ďurech et al.,
2022b, for its definition) may be theoretically estimated with these data. We
obtained $\alpha=2430\,^{\prime\prime}\,\text{yr}^{-1}$, which corresponds to
a shift of $27^{\circ}$ on the precession cone, or $16^{\circ}$ in $\lambda$
and $-4^{\circ}$ in $\beta$ over 44 years. As a result, the pole precession is
large enough and must be considered when interpreting the available light
curves.

Following the approach of Ďurech et al. (2022b), we thus added the precession
constant $\alpha$, along with the YORP strength $\upsilon$, as a second free
parameter in the model. We scanned the values of $\upsilon$ and $\alpha$ on a
grid to determine whether the fit level depends on these two parameters.
Although the parameter $\alpha$ has to be positive, we formally tested values
between $-5000$ and $5000\,^{\prime\prime}\,\text{yr}^{-1}$. The YORP
parameter $\upsilon$ was let free to range the interval $-5$ and $5\times
10^{-9}\,\text{rad}\,\text{d}^{-2}$. The result is shown in Fig. 4. The best
solution with the lowest root mean square (RMS) residuals is for
$\alpha=3000\,^{\prime\prime}\,\text{yr}^{-1}$ and $\upsilon=2.9\times
10^{-9}\,\text{rad}\,\text{d}^{-2}$. We determined the uncertainty contours in
the same way as in Vokrouhlický et al. (2011) or Polishook (2014), namely by
defining an appropriate level of $\chi^{2}$ with respect to the minimum
$\chi^{2}$. The $3\sigma$ interval defined in this way still covers the
$\upsilon=0$ value, but the $1\sigma$ interval is about $\pm 2\times
10^{-9}\,\text{rad}\,\text{d}^{-2}$, which is lower than the value of
$\upsilon$ itself. Moreover, these $\chi^{2}$-based error intervals are larger
than the uncertainties estimated by bootstrapping (see Appendix A in Ďurech et
al., 2022b). The bootstrap $1\sigma$ error of the YORP parameter for
$\alpha=3000^{\prime\prime}\,\text{yr}^{-1}$ is $\delta\upsilon=8\times
10^{-10}\,\text{rad}\,\text{d}^{-2}$. Thus, our conservative conclusion is
that the YORP is detected at the $2\sigma$ level and the precession due to
solar gravitation torque at the $1\sigma$ level.

The best model has a pole $\lambda_{0}=(257\pm 4)^{\circ}$, $\beta_{0}=(-52\pm
2)^{\circ}$, $P_{0}=(19.820\,072\pm 0.000\,008$) h (both for JD0 2443763.0),
and $\Delta=0.259$. With this updated $\Delta$ value, the theoretical
precession constant $\alpha$ is now $2720\,^{\prime\prime}\,\text{yr}^{-1}$.
This agrees excellently with the formally best value obtained from
observations, given the uncertainty and simplification of the convex shape
model. The final shape model is shown in Fig. 5, and its synthetic light
curves are shown in Fig. 6.

Figure 7: Convex shape model of asteroid (85989) 1999 JD6 shown from the
equatorial level (left and center, $90^{\circ}$ apart) and pole-on (right).

Figure 8: Bootstrap distribution of the pole direction in the ecliptic
longitude $\lambda$ and latitude $\beta$ (top) and the YORP parameter
$\upsilon$ and the rotation period $P_{0}$ (bottom) for asteroid (85989) 1999
JD6. The color-coding corresponds to the density of points. The cross marks
the solution based on the original (not bootstrapped) data set. Figure 9:
Color map of the reduced $\chi^{2}$ values of the light-curve fit for asteroid
(85989) 1999 JD6 for different values of the precession rate $\alpha$ and the
YORP strength $\upsilon$. The inner yellow contour is the estimated $1\sigma$
uncertainty limit, and the outer white contour is a $3\sigma$ boundary.

###  2.3 (85989) 1999 JD6

Marshall et al. (2015) and later Marshall (2017) combined Goldstone and
Arecibo radar observations obtained during the close approach in 2015 with
optical light curves and reconstructed a physical model of this asteroid. The
shape model of 1999 JD6 is a contact binary, with two connected lobes and
dimensions $3.0\times 1.2\times 1.0\,$km. The fortuitous geometry of the radar
observations allowed Marshall (2017) to determine the pole direction very
accurately, namely $(\lambda,\beta)=(220.3^{\circ},-73.43^{\circ})$, with an
uncertainty of $0.25^{\circ}$. The sidereal rotation period was
$7.664\,346\,4\pm 0.000\,005\,6$ h, and these authors provided an upper limit
on the YORP value $1.6\times 10^{-6}\,\text{deg}\,\text{d}^{-2}$, which
corresponds to $|\upsilon|<2.8\times 10^{-8}\,\text{rad}\,\text{d}^{-2}$.

Recently, Tian et al. (2022) analyzed the optical light curves spanning a
20-year time interval (between 2000 and 2020). From this data set, they
reported the YORP detection for this asteroid. In particular, their model has
a rotation period $7.667\,749\pm 0.000\,009$ h (for JD 2451728.0) and a
rotation pole direction $\lambda=(232\pm 2)^{\circ}$, $\beta=(-59\pm
1)^{\circ}$. In addition, their analysis required a rotation acceleration
$\upsilon=(2.4\pm 0.3)\times 10^{-8}\,\text{rad}\,\text{d}^{-2}$ ($1\sigma$
error), which Tian et al. (2022) interpreted as the YORP effect.

To reconstruct our independent shape and spin model, we used a similar light-
curve data set as Tian et al. (2022), but extended it by additional
observations taken in May and June 2023. Altogether, our data extend over ten
apparitions from 1999 to 2023 (see Table LABEL:tab:aspect_85989). To our
surprise, we obtained a different result than the previous two studies. Our
best model is shown in Fig 7, has a pole direction $(260^{\circ},-60^{\circ})$
and a period $P=7.664\,354$ h. When allowing for a nonzero $\upsilon$, the
best value is $\upsilon=-5.0\times 10^{-9}\,\text{rad}\,\text{d}^{-2}$. The
bootstrap estimates of the errors in the pole direction, period, and the YORP
parameter are shown in Fig. 8. The $1\sigma$ uncertainties are $4.8^{\circ}$
in $\lambda$, $4.0^{\circ}$ in $\beta$, $1.5\times 10^{-6}$ h in $P$, and
$4.1\times 10^{-9}\,\text{rad}\,\text{d}^{-2}$ in $\upsilon$. Therefore, even
if the formally best fit and bootstrap samples suggest a negative YORP
acceleration of about $-5.0\times 10^{-9}\,\text{rad}\,\text{d}^{-2}$, its
uncertainty is at about the same level, and zero YORP is still consistent with
the current data set.

Figure 10: Example light curves of (85989) 1999 JD6. The blue points show
observed data, the red curve shows our best model without YORP, and the dashed
curve shows the best model with the spin and YORP parameters fixed at values
given by Tian et al. (2022) (clearly not matching the available observations).

As to the difference with respect to the radar study, we note that our spin-
axis direction does not agree with the precise value determined by Marshall
(2017). Fixing the pole direction at their value still gives an acceptable
fit, but the $\chi^{2}$ value increases by about 5%. When we treat all data
sets as relative light curves, the pole direction shifts to
$(252^{\circ},-67^{\circ})$, which is closer to the radar pole, but still
outside the error intervals. The reason for this discrepancy is not known to
us. We speculate that it might be related to our model being convex, while the
real shape of 1999 JD6 is bilobed, as was clearly revealed by radar delay-
Doppler images.

The convex shape model shown in Fig. 7 has a dynamical flattening of
$\Delta=0.40$. With a semimajor axis 0.883 au, an excentricity 0.633, and an
inclination $17.06^{\circ}$, the theoretically estimated solar precession
constant is $\alpha=2130\,^{\prime\prime}\,\text{yr}^{-1}$. Over 24 years of
observations, this corresponds to a shift of almost $13^{\circ}$ on the
precession cone, which is $7.3^{\circ}$ in $\lambda$ and $-2.6^{\circ}$ in
$\beta$. These values are comparable to the pole uncertainties, so that the
solar precession effect should also be included in the modeling because it has
a non-negligible effect, and we wondered whether this omission might not
contribute to the difference with respect to the results of Marshall (2017).

We proceeded similarly as in the case of asteroid Ra-Shalom: We probed values
of $\alpha$ between $-5000$ and $5000\,^{\prime\prime}\,\text{yr}^{-1}$, and
the YORP parameter value $\upsilon$ was tested in between $-3$ and $3\times
10^{-8}\,\text{rad}\,\text{d}^{-2}$. The result is shown in Fig. 9. The
formally best solutions with the lowest RMS residuals are obtained for
$\alpha<0$, which is unrealistic. However, the estimated $3\sigma$ uncertainty
interval is so wide that it includes all tested values of $\alpha$. This means
that our data set is not sensitive to $\alpha$, and we currently cannot
constrain it from observations. For any value of the precession constant, the
constant period, namely $\upsilon=0$, is still consistent with the data.
However, as in the case of the bootstrap samples, the uncertainty interval is
not symmetric around zero here either, and negative values of $\upsilon$ are
preferred.

###  2.4 (138852) 2000 WN10

The data set we used for the shape reconstruction consisted of light curves
from 2008 and 2009 obtained by Skiff et al. (2012), from 2015 by Warner (2016)
downloaded from the Asteroid Lightcurve Database222https://alcdef.org (Warner
et al., 2009), and our new observations listed in Table
LABEL:tab:aspect_138852.

The light-curve inversion led to a strong YORP detection. The best spin
solution has the following parameters: $\lambda=(318\pm 4)^{\circ}$,
$\beta=(60\pm 8)^{\circ}$, $P=4.463\,667\,7\pm 0.000\,000\,6$ h, (for JD
2457023.5), and $\upsilon=(5.5\pm 0.7)\times
10^{-8}\,\text{rad}\,\text{d}^{-2}$. The corresponding best-fit shape model of
(138852) 2000 WN10 is shown in Fig. 11. The bootstrap distribution of the
poles and rotation parameters is shown in Fig 12.

In addition to the best-fit solution described above, there was another local
minimum in $\chi^{2}$ for the sidereal rotation $P=4.465\,939\,1$ h. However,
this solution was significantly worse, as it provided the same fit with
nonzero YORP as the best period above without YORP. The pole direction was not
unique, likely because of the limited geometry; there were also possible
solutions with the pole within $\pm 10^{\circ}$ around the direction
$\beta=90^{\circ}$. Some of them are also shown in the upper plot in Fig. 12.
However, for this spin-axis direction, a YORP value between 5 and $6\times
10^{-8}\,\text{rad}\,\text{d}^{-2}$ also provides a significantly better fit
than a constant period model.

Figure 11: Convex shape model of asteroid (138852) 2000 WN10 shown from the
equatorial level (left and center, $90^{\circ}$ apart) and pole-on (right).

Figure 12: Bootstrap distribution of the pole direction in the ecliptic
longitude $\lambda$ and latitude $\beta$ (top), the YORP parameter $\upsilon$,
and the rotation period $P_{0}$ (bottom) for asteroid (138852) 2000 WN10 (the
latter at epoch JD0 2457022.5). The color-coding corresponds to the density of
points, and the cross marks the solution based on the original (not
bootstrapped) data set. Pole ecliptic latitudes $\beta$ higher than
$90^{\circ}$ correspond to pole directions $\beta-90^{\circ}$,
$\lambda-180^{\circ}$. Figure 13: Example light curves of (138852) 2000 WN10.
The blue points show the observed data, the red curve shows our best model
with YORP, and the dashed curve shows the best model without YORP.

###  2.5  (161989) Cacus

The previous model based on observations from 1978–2016 was published by
Ďurech et al. (2018). Its spin parameters were $\lambda=(254\pm 5)^{\circ}$,
$\beta=(-62\pm 2)^{\circ}$, and $\upsilon=(1.9\pm 0.3)\times
10^{-8}\,\text{rad}\,\text{d}^{-2}$. We updated the model by adding new light
curves from 2022 that were observed with TRAPPIST and by Panfichi & Pajuelo
(2023) (see Table 8).

The new model (Fig. 14) is essentially the same as the old one, with the same
spin parameters, but smaller uncertainties: The pole direction $(252\pm
3^{\circ},-63\pm 2^{\circ})$, period $P=3.755\,052\,7$ (for JD0 2451544.5),
and the YORP value $\upsilon=1.86\times 10^{-8}$ rad d-2.

However, in contrast to the old model by Ďurech et al. (2018), the YORP
detection now does not depend on the isolated data from 1978. Even without the
two light curves by Degewij et al. (1978) and Schuster et al. (1979), the
best-fit YORP value is $\upsilon=1.6\times
10^{-8}\,\text{rad}\,\text{d}^{-2}$. The bootstrap results are shown in Fig.
15. The plot of the period versus YORP shows a core of points and diagonal
tails. This pattern corresponds to the unusual time distribution of the
photometric observations. Two light curves were taken in 1978, and the
remaining part of the data set (another 33 light curves) consists of
observations taken between 2003 and 2022. The outlying diagonal points
correspond to bootstrap samples that by chance do not contain light curves
from 1978. There were 381 such cases out of 3000. For them, the zeropoint lies
even before the first observation, so that there is a strong correlation
between the period and YORP. On the other hand, the bulk of the points in the
center contains observations from 1978, so that the correlation is much
weaker. The formal standard deviations for the whole bootstrap sample are
$0.000\,000\,6$ h and $1.3\times 10^{-9}\,\text{rad}\,\text{d}^{-2}$. For a
limited subsample including 1978 data, it is $0.000\,000\,3$ h and $8.5\times
10^{-10}\,\text{rad}\,\text{d}^{-2}$.

Figure 14: Convex shape model of asteroid (161989) Cacus shown from the
equatorial level (left and center, $90^{\circ}$ apart) and pole-on (right).

Figure 15: Bootstrap distribution of the pole direction in the ecliptic
longitude $\lambda$ and latitude $\beta$ (top), the YORP parameter $\upsilon$,
and the rotation period $P_{0}$ (bottom) for asteroid (161989) Cacus (the
$P_{0}$ value is given at epoch JD0 2451544.5). The color-coding corresponds
to the density of points, and the cross marks the solution based on the
original (not bootstrapped) data set.

##  3 Justification of the rotation rate change using a theoretical model

The secular change in the sidereal rotation rate, expressed by the parameter
$\upsilon$, has been empirically derived in Sec. 2, without a particular
theoretical interpretation. This approach, namely including $\upsilon$ into
the light-curve inversion method as a free, solved-for parameter, has been
found to be very efficient in most previous studies. However, it ideally
requires a subsequent step of connecting its value to a particular physical
phenomenon. Spontaneous particle ejections (Scheeres et al., 2020) or
micrometeoroid impacts (Wiegert, 2015) may be examples of these candidate
processes. We took a different standpoint here by adopting a link of the
observationally determined $\upsilon$ values to the YORP effect (Vokrouhlický
et al., 2015).

To set up this connection, we must evaluate $\upsilon_{\rm model}$ using a
numerical model. As always, the latter depends on many physical parameters
${\bf p}$, thus $\upsilon_{\rm model}({\bf p})$. We may take the liberty of
adjusting ${\bf p}$ to certain optimum values ${\bf p}_{\star}$ to match
$\upsilon_{\rm model}({\bf p}_{\star})\simeq\upsilon$, but the point is to
ensure that ${\bf p}_{\star}$ has reasonable (expected) values. Unfortunately,
the YORP is still slightly more complex: (i) Some of the parameters are well-
constrained observationally or may be estimated using an educated guess (e.g.,
the rotation state, the large-scale convex shape, the size, the surface
thermal inertia, and the bulk density, as an example of the latter group), but
(ii) others are not easily accessible from ground-based observations (e.g.,
shape nonconvexities and small-scale surface irregularities). The predicted
$\upsilon_{\rm model}$ depends on (ii), however, which may represent a
significant perturbation in specific cases (e.g., Statler, 2009; Breiter et
al., 2009). The specific physical phenomena that were considered to
quantitatively test the YORP-dependence on these hidden parameters had to do
with the lateral heat conduction in small-scale surface irregularities (e.g.,
Golubov & Krugly, 2012; Ševeček et al., 2015; Golubov & Lipatova, 2022), or
with the anisotropy of the thermal emission (”thermal beaming”) and mutual
irradiation of the surface facets (e.g., Rozitis & Green, 2012, 2013a). We did
not evaluate the contribution of (ii) here, and leave this part to future
studies of specific targets. We only modeled (i). To do this we used our well-
tested numerical approach that we presented in Čapek & Vokrouhlický (2004,
2005). In general terms, the model solves the 1D heat diffusion problem
independently for each of the surface facets, with nonlinear Robin boundary
conditions at the surface and deep interior below the facet. The time domain
of the solution spans one revolution about the Sun, and the rotation period is
slightly adjusted (typically within its uncertainty limits) to have an integer
number of rotation cycles in one revolution period (in this way, the
periodicity of the solution is imposed). The discretization steps in the space
and time domains are chosen to satisfy the von Neumann criterion, and the
iterations are repeated until a subdegree tolerance for all surface facets is
satisfied. After reaching convergence, we evaluated the total thermal torque
and acceleration along the heliocentric revolution using the appropriate sum
over all surface facets. More details are provided in the references mentioned
above.

For the parameters ${\bf p}$, we used the nominal (best-fit) rotation period,
pole orientation, and shape model determined from the light-curve inversion in
Sec. 2. Noting the strong dependence of the YORP effect on the asteroid shape,
we chose an additional ten variant solutions of the spin state and shape, all
providing a statistically acceptable fit to the light curves. We ran the model
for all of them and report (i) the median and (ii) the range of the
$\upsilon_{\rm model}$ values. The other physical parameters, such as the size
and bulk density, were either taken from previously published studies,
constrained by the observed Yarkovsky effect, or assumed. Their values are
individually discussed in the next sections. We note that $\upsilon_{\rm
model}$ does not depend on the surface thermal inertia in the simple model of
the YORP effect we use here (see Čapek & Vokrouhlický (2004) for numerical
evidence and Nesvorný & Vokrouhlický (2007); Breiter & Michalska (2008) for
analytical proof). Therefore, we do not need to specify this parameter when we
report the $\upsilon_{\rm model}$.

Even though our model caveat consists of the absence of the observationally
hidden parameters, we further justify it by predicting the thermal
acceleration in heliocentric motion (the Yarkovsky effect). All five NEAs
analyzed in this paper have a reliably measured secular drift of the semimajor
axis by now. Similarly to the case of the steady increase in the sidereal
rotation rate, we assume that the origin of this perturbation is due to the
thermal accelerations in the heliocentric motion (the Yarkovsky effect). An
important justification then arises from the common prediction of the YORP and
Yarkovsky strength based on our model. Importantly, the Yarkovsky (orbital)
part is somewhat less dependent on the details of the asteroid shape and
surface roughness (our runs for the best-fit model and the sample of ten
variant models usually provides a Yarkovsky effect prediction within $5-15$%
of the median value). As a result, it can provide a reasonable constraint on
the bulk density. The value of the secular drift of the semimajor axis
$\mathrm{d}a/\mathrm{d}t$ derived within the orbit determination process has
been calculated in several independent studies published over the past decade
or more (see, e.g., Chesley et al., 2008; Nugent et al., 2012; Farnocchia et
al., 2013; Vokrouhlický et al., 2015; Chesley et al., 2016; Tardioli et al.,
2017; Del Vigna et al., 2018; Greenberg et al., 2020, listed here sequentially
in time). As soon as the available astrometric information was accurate and
abundant, the $\mathrm{d}a/\mathrm{d}t$ solution across these different
sources converged to a single value. We used the most recently updated
solution provided by the JPL Horizons website.333https://ssd.jpl.nasa.gov/
This is important because the statistically significant value of
$\mathrm{d}a/\mathrm{d}t$ for (161989) Cacus only became available after the
radar astrometry was taken in September 2022.

###  3.1 (1862) Apollo

While discovered already in April 1932 (see Wolf & Reinmuth, 1932, becoming
later a namesake of its orbital class among the near-Earth asteroids), this
body shares the fate of several other NEAs by being lost for many decades and
rediscovered only in 1973. A fortuitous pair of very close encounters with
Earth in November 1980 and April 1982 offered the opportunity to take a
sequence of multiwavelength observations: (i) visible photometry (Harris et
al., 1987), (ii) infrared (Lebofsky et al., 1981), and (iii) even radar
observations (Goldstein et al., 1981; Ostro et al., 2002). This wealth of data
allowed an early determination of many physical parameters (although some
needed slight corrections later on), making Apollo exclusive even in its
category. The parameters included the rotation state and pole orientation
(e.g., Harris et al., 1987), a size between $1.2$ and $1.6$ km, a geometric
albedo of $0.2$ to $0.26$ (e.g., Harris, 1998; Ostro et al., 2002), and a
Q-type spectral classification (Tholen, 1984; Bus & Binzel, 2002). Apollo was
recognized to be one of the first targets for a detection of the Yarkovsky
effect (Vokrouhlický et al., 2005b) (see even previous evidence of
nongravitational perturbations of its heliocentric motion by Ziolkowski (1983)
and Yeomans (1991), who suspected unseen cometary activity). The actual
detection of the Yarkovsky effect was reported in 2008 (see Chesley et al.,
2008), even preceded by the detection of the YORP effect (see Kaasalainen et
al., 2007; Ďurech et al., 2008). Apollo has been found to have an unusually
small satellite in its class (Ostro et al., 2005), which was uniquely detected
in the radar observations taken in October and November 2005. Because it is so
very small, not much is known about it, but it cannot influence our model
determination of either the Yarkovsky or the YORP effects. Apollo was also
taken as an exemplary case to show the possible role of planetary close
encounters in explaining asteroidal Q-type spectra, which are similar to
ordinary chondrite analogs (e.g., Nesvorný et al., 2010). In summary, not much
is known about (1862) Apollo, but this body continues to be an inspiration for
interesting concepts of asteroidal science.

Figure 16:  Predicted values of the Yarkovsky effect and density for asteroid
(1862) Apollo. Bottom panel: Predicted semimajor axis drift
$\mathrm{d}a/\mathrm{d}t$ (ordinate) due to the Yarkovsky effect from our
model for three different values of the bulk density ($2.5$, $2.75$, and $3$ g
cm-3; see the labels and the middle curve for $2.75$ g cm-3) are shown by
solid curves. The abscissa is the surface thermal inertia in SI units (J m-2
s-0.5 K-1). We assumed the rotation state and shape model from the light-curve
inversion in Sec. 2.1, and an effective size of $1.55$ km. The grayscale
horizontal region shows the value $-(1.94\pm 0.16)\times 10^{-4}$ au My-1 from
the orbit determination. The grayscale vertical region shows the range of the
best-fit surface thermal inertia value $140^{+140}_{-100}$ J m-2 s-0.5 K-1(see
Rozitis & Green, 2012). Top panel: Model-predicted bulk density to match the
observed value of the semimajor axis drift shown by the grayscale region. The
solid line in the middle shows the exact correspondence surrounded by a map of
the sigma interval of the Yarkovsky drift. A nominal effective size of $1.55$
km is used. If this value were higher or lower, the density solution would
shift in the direction indicated by the arrows (preserving the $\rho_{\rm
b}\,D$ value). The analysis of ten different shape models of (1862) Apollo
reveals a variation of $\pm 12$% about the median value used in the figure.
Effects of small-scale surface roughness, if important, would shift the
solution toward a lower value by typically $10-30$% (see Rozitis & Green,
2012).

One such step has been undertaken by Rozitis et al. (2013), who combined
information about the detected Yarkovsky and YORP effects and interpreted them
using a single thermophysical model of this asteroid (a goal that has been
imagined by Čapek & Vokrouhlický, 2004, see their Sec. 5, but not achieved
before). We adopted their solution for an Apollo size $D=1.55\pm 0.07$ km, a
geometric albedo $p_{V}=0.20\pm 0.02$, a surface thermal inertia
$\Gamma=140^{+140}_{-100}$ J m-2 s-0.5 K-1, and a bulk density $\rho_{\rm
b}=2.85^{+0.48}_{-0.68}$ g cm-3. In a way, our work is a mere repetition of
the Rozitis et al. (2013) calculation, except that we now have updated and
more accurate values of both YORP and Yarkovsky effects. As for the latter,
the current astrometric data set provides $\mathrm{d}a/\mathrm{d}t=-(1.94\pm
0.16)\times 10^{-4}$ au My-1, while the value of Rozitis et al. (2013) was
higher by about $10$%. Similarly, our new value for the YORP effect (Sec. 2.1)
is about $10$% smaller and much more accurate than the value $(5.5\pm
1.2)\times 10^{-8}$ rad d-2 from Ďurech et al. (2008) considered by Rozitis et
al. (2013).

Figure 16 shows the comparison between the predicted Yarkovsky drift
$\mathrm{d}a/\mathrm{d}t$ from our model and the observed value (the shape
variants produce scatter of roughly $\pm 10$% about the median value used in
Fig. 16 in addition to to the bulk density dependence). Assuming a bulk
density of $2.75$ g cm-3, which we nominally used in the simulation, the
predicted semimajor axis drift matches the observed value very well. Small
adjustments of size, surface roughness, and/or thermal inertia may further
help to match any value in the uncertainty interval of
$\mathrm{d}a/\mathrm{d}t$. The same model predicts a median $\upsilon_{\rm
model}\simeq 5.0\times 10^{-8}$ rad d-2 for the $2.75$ g cm-3 bulk density,
and a range between $1.80\times 10^{-8}$ rad d-2 and $7.25\times 10^{-8}$ rad
d-2. This is fairly consistent with the observed value $\upsilon\simeq
4.94\times 10^{-8}$ rad d-2 (Sec. 2.1). In addition to the global shape
dependence, further differences may readily be explained by the thermal
beaming of a rough surface (Rozitis & Green, 2012), and/or simply a slightly
larger body size.

###  3.2 (2100) Ra-Shalom

Ra-Shalom has received a wealth of infrared observations, both from the ground
and from space (e.g., Harris et al., 1998; Delbó et al., 2003; Trilling et
al., 2010; Usui et al., 2011). In spite of the persisting uncertainty, the
results converge to a size of $2-2.8$ km, a geometric albedo $0.1-0.18$, and
an unusually high value of the surface thermal inertia (possibly $\Gamma\simeq
1000$ J m-2 s-0.5 K-1). We adopted the results of Shepard et al. (2008), who
combined various data sets, including radar observations, to obtain the most
complete picture of this asteroid. Their preferred size was $D=2.3\pm 0.2$ km
and a geometric albedo $p_{V}=0.13\pm 0.03$. Shepard et al. (2008) also
analyzed a composite visible-to-infrared spectrum of Ra-Shalom and found that
it was similar to spectra of CV3 meteorites. Based on this similarity, they
argued that a K-type classification would be the best match for this asteroid
(refining the previous C- or Xc-type classification444Admittedly though, there
is an uncertainty in the Ra-Shalom classification, since Binzel et al. (2019)
found its spectrum neutral to blueish and classified it as a B-type object.
The slow rotation makes it unclear whether this observation speaks for the
whole body or a particular surface feature. More data are clearly needed in
this respect.). Combining the measured grain density of CV3 meteorites and the
estimated $\sim 30$% porosity of C-class asteroids, Shepard et al. (2008)
suggested a bulk density of $\rho_{\rm b}=2.4\pm 0.6$ g cm-3. Finally, these
authors also interpreted their radar measurements and thermal observations as
providing evidence for a coarse or rocky surface with only a thin or
unimportant regolith layer.

Figure 17:  Predicted values of the Yarkovsky effect and density for asteroid
(2100) Ra-Shalom. Bottom panel: Predicted semimajor axis drift
$\mathrm{d}a/\mathrm{d}t$ (ordinate) due to the Yarkovsky effect from our
model for three different values of the bulk density ($2$, $2.5$, and $3$ g
cm-3; see the labels) are shown by solid curves. The abscissa is the surface
thermal inertia in SI units (J m-2 s-0.5 K-1). We assumed the rotation state
and shape model from the light-curve inversion in Sec. 2.2 and an effective
size of $2.3$ km. The grayscale horizontal region shows the value $-(2.01\pm
0.43)\times 10^{-4}$ au My-1 from the orbit determination. The vertical dashed
line indicates the suggested surface high-thermal inertia of $\simeq 1000$ J
m-2 s-0.5 K-1. Top panel: Model-predicted bulk density to match the observed
value of the semimajor axis drift shown by the grayscale region. The solid
line in the middle shows the exact correspondence surrounded by a map of the
sigma interval of the Yarkovsky drift. A nominal effective size of $2.3$ km is
used. If this value were higher or lower, the density solution would shift in
the direction indicated by the arrows (preserving the $\rho_{\rm b}\,D$
value). The analysis of ten different shape models of (2100) Ra-Shalom reveals
a variation of $\pm 10$% about the median value used in the figure. Effects of
small-scale surface roughness, if important, would shift the solution toward a
lower value by typically $10-30$% (see Rozitis & Green, 2012). Figure 18:
Rotation pole evolution of (2100) Ra-Shalom due to the solar torque and its
comparison with the simple Colombo top model. The coordinates on the axes are
defined in the orbital plane, namely (i) the obliquity $\epsilon$ (or
$\cos\epsilon$) on the right and left ordinates, and (ii) the longitude $\phi$
reckoned from the direction $90^{\circ}$ away from the orbital node (located
at $\phi=-90^{\circ}$; e.g., Breiter et al., 2005). The numerically propagated
evolution of the Ra-Shalom pole over the $44$ yr time span is shown by the
solid black line. The gray lines are solutions of the simple Colombo top
model. For most of the obliquity values, they are basically simple straight
lines of approximately constant $\epsilon$. The exception is the Cassini
resonant zone at $\epsilon\simeq 90^{\circ}$ (with the separatrix shown by the
red curves, and the stationary Cassini state 2 shown by the red dot). Unlike
the case of asteroid (433) Eros, whose spin axis librates in the resonant zone
(Vokrouhlický et al., 2005a), the evolution of the Ra-Shalom pole is simpler
and only consists of nearly regular precession. The zoom in the right panel
shows that the change in obliquity is minimum, but conforms to the Colombo top
solution.

The large size and favorable orbit of Ra-Shalom, belonging to what Milani et
al. (1989) classified as the Toro orbital group residing near or inside the
mean motion resonances with the Earth, offered numerous opportunities to
observe this asteroid with radar. It belongs to the record-holders in the
number of radar observations at different apparitions (in this case, seven
observations between 1981 and 2022). Their accuracy and long time-base
resulted in a high-quality orbit determination that allowed a firm detection
of the Yarkovsky effect (see already Vokrouhlický et al., 2005b).
Interestingly, the related drift of the semimajor axis
$\mathrm{d}a/\mathrm{d}t=-(2.01\pm 0.43)\times 10^{-4}$ au My-1 is well
matched by our model, adopting a bulk density in between $2$ and $3$ g cm-3
and the suggested high surface thermal inertia (Fig. 17). These two parameters
are correlated: If the thermal inertia were lower, the required bulk density
would also be lower. Additionally, the unaccounted-for effects of the surface
roughness in our model may slightly decrease the bulk density solution, as
indicated by the arrow in the top panel of Fig. 17.

Ďurech et al. (2012b) and Ďurech et al. (2018) repeatedly sought the YORP
effect in the available photometric data of that date, but were only able to
place an upper limit on the value. In this paper, we finally determined the
value from a series of light-curve data extended to the last year. In contrast
to the conclusion in Ďurech et al. (2018), who speculated that Ra-Shalom might
be the first case for which YORP will be found to decelerate the rotation
rate, we found a weakly accelerating signal of $\upsilon\simeq 2.9\times
10^{-9}$ rad d-2 (Sec. 2.2). This is even lower than the value found for
asteroid (1685) Toro, even though Toro (i) is larger and (ii) has a larger
semimajor axis. This comparison confirms the unusual weakness of the YORP
effect for Ra-Shalom.

Assuming a bulk density of $2.5$ g cm-3, fine-tuned to match the Yarkovsky
effect, our simulation predicts a median $\upsilon_{\rm model}\simeq-2.1\times
10^{-8}$ rad d-2. Not only is the absolute value of $\upsilon_{\rm model}$
higher than the detected $\upsilon$, but its sign is opposite (similarly to
what has been found in Ďurech et al., 2018). Clearly, the nominal convex shape
model, which just has a low resolution at large scales, cannot capture
important features of the YORP effect. The full range of predicted
$\upsilon_{\rm model}$ values from the sample of $11$ equivalent models is
$-0.15\times 10^{-8}$ rad d-2 to $-6.1\times 10^{-8}$ rad d-2. This is still
negative, but some of the extreme values are now closer to the detected
$\upsilon$.

A possibly important clue to the solution is the evidence of a significant
surface roughness (Shepard et al., 2008). This could promote a significant
contribution of the fine-scale surface effects, such as mutual facet
irradiation and/or transverse heat conduction, of which at least the latter
preferentially contributes by an acceleration component in YORP (e.g., Golubov
& Krugly, 2012). The situation of (2100) Ra-Shalom may therefore be
reminiscent of (25143) Itokawa, for which the early analyses, making use of
the large-scale resolution shape models of this asteroid, consistently
predicted a negative $\upsilon$ value (e.g., Vokrouhlický et al., 2004;
Scheeres et al., 2007; Breiter et al., 2009). However, after enough
observations were finally available, the detected $\upsilon$ value proved to
be positive (Lowry et al., 2014), which implied a spin-up state of Itokawa.
Lowry et al. (2014), following an earlier theoretical concept formulated by
Scheeres & Gaskell (2008) (see also Breiter et al., 2009), suggested that the
perturbing effect was due to a difference in the bulk density of the two main
shape features, notably the quasi-ellipsoidal lobes in contact. However, the
difference required to explain the overall effect appeared rather large.
Ševeček et al. (2015) proposed an alternative, and perhaps more natural,
explanation based on the novel concept of lateral heat conduction on small-
scale surface features discovered by Golubov & Krugly (2012). We hypothesize
that the same model may explain the disparity in the observed and modeled
$\upsilon$ values for Ra-Shalom. However, we leave a detailed study for the
future.

We now briefly return to the observationally revealed precession of the Ra-
Shalom rotation pole (Sec. 2.2). We found that the best-fit shape model
corresponds to the dynamical flattening value $\Delta\simeq 0.26$, which
provides a theoretical value for the precession constant $\alpha\simeq 2720$ ″
yr-1 (which is fully compatible with the best-fit value of this parameter;
Fig. 4). From the simplest possible point of view, in which the heliocentric
orbit would be fixed in space, the pole would precess with an angular speed of
$-\alpha\cos\epsilon\simeq 2522$ ″ yr-1, where $\epsilon\simeq 158^{\circ}$ is
the rotation pole obliquity, about the direction of the orbital angular
momentum (therefore normal to the orbital plane; e.g., Bertotti et al., 2003).
The situation becomes more complicated when the orbital plane evolves in time
due to planetary perturbations. In the case of Ra-Shalom, the essence of the
generalization may be described using a model that is still very simple, in
which the orbital inclination remains approximately constant ($I\simeq
15.7^{\circ}$), but the longitude of node precesses in inertial space about
the sum of the planetary orbital angular momenta with a frequency
$s\simeq-22.98$ ″ yr-1. The dynamics of the rotation pole within this setup is
described by the Colombo top model (e.g., Colombo, 1966; Haponiak et al.,
2020), whose implementation for asteroids can be found, for instance, in
Vokrouhlický et al. (2006). We refer to these references for more details and
note here that the Colombo top model has two fundamental parameters: (i) the
orbit inclination $I$, and (ii) the ratio of frequencies
$\kappa=\alpha/(2\sigma)\simeq-59$. The complexity of the pole evolution is
determined by the relation of $\kappa$ to a critical parameter
$\kappa_{\star}=-\frac{1}{2}\left(\sin^{2/3}I+\cos^{2/3}I\right)^{3/2}\simeq-0.823$.
The most complicated situation occurs when $\kappa$ is just slightly smaller
than $\kappa_{\star}$. In our situation, $\kappa$ is much smaller than
$\kappa_{\star}$, and this confines the complexity to a narrow obliquity slab
near $90^{\circ}$, which restores the expected regularity at high and low
values of $\epsilon$. The latter corresponds to the case of Ra-Shalom.

To verify the conclusion, we numerically integrated the evolution of the Ra-
Shalom rotation pole over the $44$ yr time span between the first and last
available light-curve observations (September 1978 to September 2022).555We
used the symplectic numerical integrator described in Breiter et al. (2005)
implemented in the widely used orbital dynamics package swift (e.g.,
http://www.boulder.swri.edu/~hal/swift.html). The results are summarized in
Fig. 18. The $44$ yr track of the pole evolution is shown by the solid black
segment, and the gray lines show solutions of the Colombo top model (the
proximity of the true evolution of the Ra-Shalom pole from our numerical
simulation to the Colombo top solution starting at nearby initial conditions
shows that the latter is a fairly good but not complete approximation). The
existence of the resonant zone at about $90^{\circ}$ obliquity perturbs the
Ra-Shalom obliquity only very slightly, producing a negligible variation
during the time span of interest. The principal dynamical effect consists of a
$\simeq 31^{\circ}$ drift in longitudinal angle $\phi$ associated with the
obliquity $\epsilon$ (right panel in Fig. 18). The true angular distance
$\delta$ between the Ra-Shalom pole direction in 1978 and in 2022 in space is
smaller, however. We easily find that
$\cos\delta=1-2\sin^{2}\epsilon\sin^{2}\phi/2$, and therefore, $\delta\simeq
11.5^{\circ}$. Because the inclination of the Ra-Shalom orbit is rather small,
most of this effect is projected into a change in ecliptic longitude of the
pole, while a small part, about $4^{\circ}$, is the change in ecliptic
latitude.

Figure 19:  Predicted values of the Yarkovsky effect and the density for
asteroid (85989) 1999 JD6. Bottom panel: Predicted semimajor axis drift
$\mathrm{d}a/\mathrm{d}t$ (ordinate) due to the Yarkovsky effect from our
model for three different values of the bulk density ($1.5$, $2$, and $2.5$ g
cm-3; see the labels) shown by the solid curves; the abscissa is the surface
thermal inertia in SI units (J m-2 s-0.5 K-1). We assumed the rotation state
and shape model from the light-curve inversion in Sec. 2.3 and an effective
size of $1.45$ km. The grayscale horizontal region shows the value $-(4.26\pm
0.98)\times 10^{-4}$ au My-1 from the orbit determination. Top panel: Model-
predicted bulk density to match the observed value of the semimajor axis drift
shown by the grayscale region. The solid line in the middle shows the exact
correspondence surrounded by a map of the sigma interval of the Yarkovsky
drift. The nominal effective size of $1$ km is used. If this value were lower
or higher, the density solution would shift in the direction indicated by the
arrows (preserving the $\rho_{\rm b}\,D$ value). The analysis of ten different
shape models of (85989) 1999 JD6 reveals a variation of $\pm 10$% about the
median value used in the figure.

###  3.3 (85989) 1999 JD6

The most complex study of this asteroid was presented by Marshall (2017), who
combined several sets of multiwavelength data from visible photometry and
infrared observations to very detailed radar sensing during its 2015
apparition. The results revealed the highly elongated shape of a contact
binary with a volume equivalent size of $D=1.45\pm 0.14$ km and a surface
thermal inertia $\Gamma\simeq 280$ J m-2 s-0.5 K-1(see also Campins et al.,
2009). The radar data were complemented with light-curve observations between
1999 and 2015, which allowed determining the sidereal rotation period to
$7.664\,346\,4\pm 0.000\,005\,6$ hr. The abundant radar data in 2015 and their
fortuitous viewing geometry discussed in Marshall (2017) resulted in an
unusually accurate determination of the asteroid rotation pole, namely
$(\lambda,\beta)=(220.3^{\circ},-73.43^{\circ})$, with only a fraction of a
degree of uncertainty. This implies an obliquity of nearly $180^{\circ}$.
Marshall (2017) also searched for the YORP signal in the data available until
2015, but he set only an upper limit $|\upsilon|<2.6\times 10^{-8}$ rad d-2.
The recent analysis of the Yarkovsky effect with Gaia DR3 precise astrometric
measurements included by Dziadura et al. (2023) provided
$\mathrm{d}a/\mathrm{d}t=-(4.26\pm 0.98)\times 10^{-4}$ au My-1 for this
asteroid. The taxonomic classification of (85989) 1999 JD6 is not clearly
constrained, with suggestions of K class Thomas et al. (2014), X class Carry
et al. (2016), and L class Binzel et al. (2019), which means that only some of
the primitive classes are excluded. From BVR photometry, Polishook & Brosch
(2008) concluded that RaShalom belongs to the K or S class.

Recently, Tian et al. (2022) analyzed a larger set of photometric observations
taken between 2000 and 2020 and claimed to have detected the YORP effect with
$\upsilon=(2.4\pm 0.3)\times 10^{-8}$ rad d-2. However, the results we
presented in Sec. 2.3 do not confirm their finding. Instead, we obtained
$\upsilon=-(5\pm 4)\times 10^{-9}$ rad d-2, with a zero value still
statistically acceptable. This is a tighter constraint than was found by
Marshall (2017), but it does not yet prove the YORP signal. The reason for the
difference between our results and that in Tian et al. (2022) eludes us. The
best we could have done is to double-check our solution, which is compatible
for the YORP nondetection with that of Marshall (2017).

We also used our rotation state and convex shape model to estimate the
expected value of the semimajor axis secular drift (the Yarkovsky effect) and
the sidereal rotation rate secular change (the YORP effect). We assumed a $2$
g cm-3 bulk density and ran simulations for a wide range of surface thermal
inertia. The results are shown in Fig. 19. The predicted semimajor axis drift
is generally consistent with the value determined by Dziadura et al. (2023),
but if the low thermal inertia of the surface holds, the bulk density also
needs to be low ($\simeq 1.3-1.7$ g cm-3). The intrinsic bulk density might be
higher because by using the convex-hull model, we overestimated the volume
that is occupied by matter for this contact binary object.

As expected, the formal median value $\upsilon_{\rm model}\simeq 2.7\times
10^{-7}$ rad d-2 obtained with our nominal shape model and a bulk density $2$
g cm-3 is significantly different from the upper limit $|\upsilon|\leq
10^{-8}$ rad d-2 (with even a preference for the negative values). However,
the full range of $\upsilon_{\rm model}$ from our $11$ shape/spin variant
solutions extends from $-3.5\times 10^{-7}$ rad d-2 to $3.7\times 10^{-7}$ rad
d-2, indicating that the YORP effect cannot be accurately predicted from the
simple convex models and available photometric observations.

###  3.4 (138852) 2000 WN10

The population of the Earth coorbitals is an interesting transient subclass of
NEAs with a characteristic dynamical lifetime of some tens of thousands of
years (e.g., Morais & Morbidelli, 2002). Coorbitals, or objects that are
orbitally very close to them, may offer a unique possibility among NEAs to be
observable every year for a decade or more. Because these annual Earth
encounters may be close, even very small NEAs may be observed. In spite of the
slight drawback that they typically repeat a similar observing geometry,
coorbitals are particularly suitable in situations when a series of
observations seeks to determine the cumulative effect of a weak perturbation.
This is the case of the Yarkovsky and YORP effects. For this reason,
Vokrouhlický et al. (2005b) considered the detection of the Yarkovsky effect
for a handful of small coorbital asteroids, including (54509) YORP (for which
the Yarkovsky effect was swiftly reported by Chesley et al., 2008) and
(138852) 2000 WN10 (for which the Yarkovsky effect was reported by
Vokrouhlický et al., 2015, due its larger size and unfortunate absence of the
radar astrometry). Even more importantly, (54509) YORP was one of the first
two objects for which the YORP effect was detected (Taylor et al., 2007; Lowry
et al., 2007). The YORP detection for (138852) 2000 WN10 proved to be harder
and had to wait until the present paper. This is due to its larger size and
intrinsically weaker YORP strength.666The rotation rate acceleration of
(54509) YORP reads $\upsilon\simeq 350\times 10^{-8}$ rad d-2. If we were to
plainly scale this value for about three times larger and about twice denser
(138852) 2000 WN10, we may expect the YORP strength of about $\upsilon\simeq
350/2/3^{2}\times 10^{-8}\simeq 20\times 10^{-8}$ rad d-2 (note that both
asteroids have about the same orbit and both have extreme values of the
obliquity). Yet, the observed $\upsilon$ is four times smaller (Sec. 2.4),
perhaps due to a more regular shape.

To compare the observed $\mathrm{d}a/\mathrm{d}t$ and $\upsilon$ (from Sec.
2.4) with the theoretical values, we need to know or assume several physical
parameters of the asteroid. Because 2000 WN10 is small and lacks radar
observations, the information about this asteroid is unfortunately sparse. The
light-curve inversion provides complete information about the spin state and a
rough (convex) shape model. Based on broadband photometry, Ieva et al. (2018)
found (138852) 2000 WN10 to be compatible with the S-type group of asteroids.
Perna et al. (2018) conducted a spectroscopic survey of small NEAs and
included (138852) 2000 WN10 in their program. The visible spectrum corresponds
to the Sq taxonomic type (similarly Binzel et al., 2019, found it to be a
Q-type object). Adopting a mean geometric albedo of $0.24$ for this group,
they argued for a rather small size of $\simeq 250$ m. However, without
infrared observations, the albedo and size values are simply an educated
guess. If we were to assume the mean albedo value of the NEA population
$0.15$, the size would recalibrate to $\simeq 320$ m. At this moment, we adopt
$D\simeq 300$ m for our thermal model. In practice, we rescale unconstrained
dimensions of the shape model from Sec. 2.4 such that its volume is equivalent
to a sphere with a diameter of $300$ m. Finally, we need to adopt some value
of (i) the surface thermal inertia $\Gamma$ and (ii) the bulk density
$\rho_{\rm b}$. In this case, we ran a series of simulations for different
$\Gamma$, sufficiently sampling the interval of values from thermophysical
modeling of small NEAs (see, e.g., Delbó et al., 2015). For the bulk density,
we used $2$ g cm-3. The results for different $\rho_{\rm b}$ values were
readily obtained with the $\mathrm{d}a/\mathrm{d}t\propto\rho_{\rm b}^{-1}$
relation.

Figure 20:  Predicted values of the Yarkovsky effect and density for asteroid
(138852) 2000 WN10. Bottom panel: Predicted semimajor axis drift
$\mathrm{d}a/\mathrm{d}t$ (ordinate) due to the Yarkovsky effect from our
model for three different values of the bulk density ($1.5$, $2$, and $2.5$ g
cm-3; see the labels) are shown by the solid curves; the abscissa is the
surface thermal inertia in SI units (J m-2 s-0.5 K-1). We assumed the rotation
state and shape model from the light-curve inversion in Sec. 2.4 and an
effective size of $300$ m. The grayscale horizontal region shows the value
$(17.8\pm 2.9)\times 10^{-4}$ au My-1 from the orbit determination. Top panel:
Model-predicted bulk density to match the observed value of the semimajor axis
drift shown by the grayscale region. The solid line in the middle shows the
exact correspondence surrounded by a map of the sigma interval of the
Yarkovsky drift. The nominal effective size of $300$ m is used. If this value
were higher or lower, the density solution would shift in the direction
indicated by the arrows (preserving the $\rho_{\rm b}\,D$ value). The analysis
of ten different shape models of (138852) 2000 WN10 reveals a variation of
$\pm 3$% about the median value used in the figure. The effects of small-scale
surface roughness, if important, would shift the solution toward a lower value
by typically $10-30$% (see Rozitis & Green, 2012).

Figure 20 shows our theoretical predictions for the Yarkovsky semimajor axis
drift as compared to the observed value $\mathrm{d}a/\mathrm{d}t=(17.8\pm
2.9)\times 10^{-4}$ au My-1 (the variation in the predicted values due to
different shape models is small in this case, only $\pm 3$%). The plausible
range $1.7$ to $2.4$ g cm-3 for the bulk density is fully acceptable, but is
shifted to slightly lower values than those of the small Q-type asteroid
(6489) Golevka, as also determined by the Yarkovsky effect (see Chesley et
al., 2003). At the same time, the required thermal inertia in between $150$
and $600$ J m-2 s-0.5 K-1 favorably compares to what is expected from limited
cases of small S-type NEAs with good visible and infrared observations (Delbó
et al., 2015). There is clearly room for further adjustments due to small
changes in the effective size $D$ and the effects of small-scale roughness
(the arrows in the upper panel schematically illustrate their influence).

Figure 21:  Predicted values of the Yarkovsky effect and density for asteroid
(161989) Cacus. Bottom panel: Predicted semimajor axis drift
$\mathrm{d}a/\mathrm{d}t$ (ordinate) due to the Yarkovsky effect from our
model for three different values of the bulk density ($1.5$, $2$ and $2.5$ g
cm-3; see the labels) are shown by the solid curves; the abscissa is the
surface thermal inertia in SI units (J m-2 s-0.5 K-1). We assumed the rotation
state and shape model from the light-curve inversion in Sec. 2.5 and an
effective size of $1$ km. The grayscale horizontal region shows the value
$-(4.45\pm 0.78)\times 10^{-4}$ au My-1 from the orbit determination. The
grayscale vertical region shows the range of the acceptable surface thermal
inertia value $200$ to $2000$ J m-2 s-0.5 K-1and the best-fitting core $500$
to $800$ J m-2 s-0.5 K-1(see Ďurech et al., 2018). Top panel: Model-predicted
bulk density to match the observed value of the semimajor axis drift shown by
the grayscale region. The solid line in the middle shows the exact
correspondence surrounded by a map of the sigma interval of the Yarkovsky
drift. The nominal effective size of $1$ km is used. If this value were higher
or lower, the density solution would shift in the direction indicated by the
arrows (preserving the $\rho_{\rm b}\,D$ value). The analysis of ten different
shape models of (161989) Cacus reveals a variation of $\pm 3$% about the
median value used in the figure. The effects of small-scale surface roughness,
if important, would shift the solution toward a lower value by typically
$10-30$% (see Rozitis & Green, 2012).

The same simulations that provided satisfactory results for the Yarkovsky
effect were also used to compute the YORP effect. The light-curve data
analysis resulted in a low and positive value $\upsilon\simeq 5.5\times
10^{-8}$ rad d-2, indicating that the rotation rate of 2000 WN10 increases
slowly. However, using the obtained shape model, the bulk density of $2$ g
cm-3, and the effective size of $300$ m, our simplified nominal model provides
$\upsilon_{\rm model}\simeq-8.4\times 10^{-8}$ rad d-2. This is a similar
mismatch as for Ra-Shalom, presented in Sec. 3.2. Here, however, we find a
much stronger variation of the predicted $\upsilon_{\rm model}$ values
computed for $11$ possible solutions from light-curve fitting: $-15\times
10^{-7}$ rad d-2 to $26\times 10^{-7}$ rad d-2. Clearly, the sample of
resulting convex shape models cannot predict the YORP effect accurately. This
likely originates from two effects: (i) the very low obliquity of $\simeq
10^{\circ}$ means that the polar flattening of the shape model from the light-
curve inversion is only poorly resolved, and (ii) the coorbital nature of the
2000 WN10 orbit inherently limits the viewing geometry for an Earth-bound
observed for subsequent seasons (see discussion in Taylor et al., 2007).
Because of the low obliquity, the solar aspect angle of our observations has
been always limited to between 82 and $88^{\circ}$. The shape-modeling
procedure clearly interpolates the surface even in the zone that was poorly
sampled by the observations. This produces a huge missing piece of
information, however, that would be needed to determine $\upsilon_{\rm
model}$. At this moment, we must satisfy ourselves by observing that
$\upsilon$ and $\upsilon_{\rm model}$ are generally on the same order of
magnitude (or at least within the expected limits).

###  3.5 (161989) Cacus

This asteroid was discovered during its close encounter with the Earth in
February 1978, but it was subsequently lost until its rediscovery in January
2003 (the close encounter in September 1997 was missed because the position of
Cacus was at a low declination on the southern sky). Nevertheless, the 1978
apparition is important by providing an astrometric tie point, but mainly by
providing the early photometric observations (Degewij et al., 1978; Schuster
et al., 1979). Cacus has been regularly observed since 2003, and both
astrometry and photometry were accumulated. The radar observations taken from
the Goldstone DSS-14 dish in August and September 2022 have significantly
increased the orbit accuracy. A low-accuracy detection of the YORP effect was
reported by Ďurech et al. (2018) and was improved in this paper by adding
photometric observations taken in 2022. The post-radar orbit also allows us to
constrain the Yarkovsky effect fairly well, that is, the semimajor axis
secular drift of $\mathrm{d}a/\mathrm{d}t=-(4.45\pm 0.78)\times 10^{-4}$ au
My-1. Cacus was also fortuitously observed by the WISE spacecraft in its
cryogenic phase (Mainzer et al., 2011), which allowed Ďurech et al. (2018) to
determine its effective size $D=1.0\pm 0.2$ km and the most likely range of
the surface thermal inertia $\Gamma=500-800$ J m-2 s-0.5 K-1. Finally, Thomas
et al. (2014) and Binzel et al. (2019) reported spectroscopic observations,
classifying Cacus consistently as a Q- or S-class object. Overall, in a quite
short interval of time, Cacus has been elevated to the rank of NEAs with
fairly complete information.

We used the best-fit spin state and shape model found in Sec. 2.5, together
with the assumed effective size of $1$ km, to compute the Yarkovsky effect for
a plausible range of surface thermal inertia and bulk density values. The
results are shown in Fig. 21. The observed semimajor axis drift
$\mathrm{d}a/\mathrm{d}t=-(4.45\pm 0.78)\times 10^{-4}$ au My-1 is well
compatible with a bulk density between $1.7$ and $2.4$ g cm-3 (Fig. 21).
Because the taxonomic class is similar to (1862) Apollo and (138852) 2000
WN10, the inferred bulk density for Cacus appears to be acceptable (a slightly
smaller size and/or lower thermal inertia may push the value higher and close
to the value inferred for Apollo or Golevka).

As above, we were less successful when we tried to match the observed
acceleration of the rotation rate $\upsilon\simeq 1.94\times 10^{-8}$ rad d-2
(Sec. 2.5). Using the nominal model as above for the Yarkovsky effect (and
specifically, $2$ g cm-3 bulk density), we obtained a median $\upsilon_{\rm
model}\simeq-2\times 10^{-8}$ rad d-2 and a full range between $-20\times
10^{-8}$ rad d-2 and $18\times 10^{-8}$ rad d-2 for a sample of ten variant
shape or spin solutions. Similarly to the case of 2000 WN10, we suspect that
the overall uncertainty in the shape model plays a fundamental role. The
obliquity of Cacus is basically $180^{\circ}$, implying that the spin axis
stretch of the model, as well as other details of the shape model along this
direction, are very loosely constrained. This means that the result strongly
depends on model details, even at medium and large scales. At this moment, the
data simply do not provide us with an accurate enough model to firmly predict
the $\upsilon_{\rm model}$ value.

##  4 Discussion and conclusions

Table 1:  YORP detections as of August 2023. The first segment of 12 objects
corresponds to highly reliable cases. The two additional objects at the end of
the table list candidate cases for which additional observations in the future
will likely result in a YORP detection.

Asteroid |  $\upsilon$ | $\upsilon/\delta\upsilon$ | $C_{Y}$ | $P$ | $D$ | $\epsilon$ | Reference
---|---|---|---|---|---|---|---
|  | [$10^{-8}\,\text{rad}\,\text{d}^{-2}$] |  |  | [h] | [km] | [deg] |
– YORP detections –
1620 | Geographos | $1.14\pm 0.03$ | 38.0 | $0.013$ | $\phantom{15}5.2233360$ | $\phantom{2}2.56$ | 154 | (1)
1685 | Toro | $0.33\pm 0.03$ | 11.0 | $0.0080$ | 10.197826 | 3.5 | 160 | (1)
1862 | Apollo | $4.94\pm 0.09$ | 54.9 | $0.025$ | $\phantom{15}3.0654226$ | $\phantom{2}1.55$ | 163 | This work
2100 | Ra-Shalom | $0.29\pm 0.2\phantom{9}$ | $\phantom{5}1.5$ | $0.0011$ | 19.820072 | 2.3 | 158 | This work
3103 | Eger | $1.1\phantom{9}\pm 0.5\phantom{9}$ | $\phantom{5}2.2$ | $0.0077$ | $\phantom{5}5.710156$ | $\phantom{2}1.78$ | 176 | (2)
10115 | 1992 SK | $8.3\phantom{9}\pm 0.6\phantom{9}$ | 13.8 | $0.014$ | $\phantom{5}7.320232$ | 1.0 | 161 | (1)
25143 | Itokawa | $3.54\pm 0.38$ | $\phantom{5}9.3$ | $0.00072$ | 12.132371 | $\phantom{1}0.32$ | 178 | (3)
54509 | YORP | $350\pm 35\phantom{0}$ | 10.0 | $0.0053$ | $\phantom{555}0.20290020$ | $\phantom{15}0.114$ | 173 | (4,5)
68346 | 2001 KZ66 | $8.43\pm 0.69$ | 12.2 | $0.013$ | $\phantom{1}4.985997$ | $\phantom{15}0.797$ | 159 | (6)
101955 | Bennu | $6.34\pm 0.91$ | $\phantom{5}7.0$ | $0.0010$ | $\phantom{1}4.296057$ | $\phantom{15}0.490$ | 178 | (7)
138852 | 2000 WN10 | $5.5\phantom{9}\pm 0.7\phantom{9}$ | $\phantom{5}7.9$ | $0.00056$ | $\phantom{51}4.4636677$ | 0.3 | $\phantom{1}10$ | This work
161989 | Cacus | $1.86\pm 0.09$ | 20.7 | $0.0027$ | $\phantom{55}3.7550527$ | 1.0 | 179 | This work
– Possible, but weak YORP signal –
85989 | 1999 JD6 | $-0.5\pm 0.4$ | $\phantom{55}1.25$ | $0.00083$ | $\phantom{1}7.664354$ | 1.53 | 160 | This work
85990 | 1999 JV6 | $\phantom{-}3.1\pm 2.4$ | $\phantom{5}1.3$ | $0.00069$ | $\phantom{1}6.536787$ | $\phantom{5}0.442$ | 174 | (8)

777The first two columns provide the number and designation of the asteroid.
The third column gives the change in the secular rotation-rate $\omega$,
$\upsilon=d\omega/dt$, which is empirically determined from the observations,
and the fourth column gives the statistical significance of $\upsilon$. The
fifth column lists the parameter $C_{Y}$ introduced in Eq. (1) of Rozitis &
Green (2013b) (see also Rossi et al., 2009). Columns 5 to 8 give the rotation
period $P$ (at the epoch of detection), the size $D$, and the obliquity
$\epsilon$. The last column provides the source.

(1) Ďurech et al. (2022b), (2) Ďurech et al. (2018), (3) Lowry et al. (2014),
(4) Taylor et al. (2007), (5) Lowry et al. (2007), (6) Zegmott et al. (2021),
(7) Hergenrother et al. (2019), (8) Rożek et al. (2019a)

It has become tradition over the past years to conclude a paper that reported
a new detection of the YORP effect by recalling the previous detections and
noting that in all cases, the rotation rate increases (i.e., $\upsilon$ is
positive for all asteroids). We continue this tradition. For brevity, we
recall Table 3 of Tian et al. (2022), who listed the relevant last YORP
detections. Our work represents two modifications: (i) an extension by two new
detections for (2100) Ra-Shalom and (138852) 2000 WN10, but (ii) also one
retraction, at least as far as the situation stands now, namely a nondetection
of YORP for (85989) 1999 JD6. Taken together, the YORP effect has been
detected for 12 asteroids, and the rotation rate has indeed been found to
increase in all cases.888Ra-Shalom spins up contrary to expectations from
Ďurech et al. (2018). Another YORP-detection candidate for which previous
analysis predicted the rotation rate to slow down is (1917) Cuyo (see Rożek et
al., 2019b). In this case, it would also have the first YORP detection for an
Amor-class NEA (i.e., perihelion higher than 1 au). Based on our new results,
we provide a list of YORP detections in Table 1. In addition to the spin
parameters and the size, the table also lists the parameter $C_{Y}$, which is
a nondimensional coefficient introduced by Rozitis & Green (2013b) (see also
Rossi et al., 2009) to describe the YORP strength for a particular asteroid.
The $C_{Y}$ parameter is the $\upsilon$ parameter normalized for the semimajor
axis $a$, the eccentricity $e$, the size $D$, and the density $\rho$ (we used
the same density of $2500$ kg cm-3 according to the formula

$C_{Y}=\upsilon\,\frac{a^{2}\sqrt{1-e^{2}}\,\rho\,D^{2}}{G_{1}}\,,$ (2)

where $G_{1}\approx 6.4\times 10^{16}$ kg m s-2 is a modified constant of
solar radiation ($2/\pi$ times the solar radiation pressure per unit area at a
unit distance $1\times 10^{17}$ kg m s-2). The values of $C_{Y}$ allowed us to
directly compare the YORP strength after removing the effects of the size and
heliocentric distance. Apollo has the highest value of $C_{Y}=0.025$, which is
also likely the reason for the good agreement of its theoretical value of
$\upsilon_{\mathrm{model}}$ with the observed value and why it is stable with
respect to bootstrap shape variants. For the other three asteroids we modeled
in this work, Ra-Shalom, 2000 WN10, and Cacus, $C_{Y}$ is smaller by at least
one order of magnitude. This means that the detected YORP is weak and thus is
more sensitive to the shape details. The $\upsilon_{\mathrm{model}}$ values
are therefore spread much more widely for these three cases.

While not an exception from the rule, we may mention for the sake of interest
that (138852) 2000 WN10 is the first case in which the YORP effect has been
determined for a prograde-rotating asteroid. This is expected, however,
because the theory does not expect any difference between the prograde- and
retrograde-rotating cases. Retrograde rotators predominate in the NEA
population and represent a fraction of about two-thirds of the total. This is
confirmed by direct observations of their rotation pole (e.g., La Spina et
al., 2004; Kryszczyńska et al., 2007), but it is also fairly well reflected in
the statistics of the Yarkovsky detections (e.g., Farnocchia et al., 2013;
Tardioli et al., 2017; Greenberg et al., 2020). In this respect, it seems a
small portion that the YORP effect is detected for only one prograde-rotator
in a sample of 12 objects (if YORP acts equally on prograde- and retrograde-
rotating bodies). Using a binomial distribution, we find that the probability
that this occurs by chance is $\simeq 5$%. As far as we know, no apriori
selection bias exists to preferentially detect the YORP effect for retrograde-
rotating NEAs. As a result, the situation may still indicate that the sample
of NEAs for which the YORP effect was detected is still too small to draw bold
conclusions.

Following the same line of argument, the binomial distribution may indicate
the likelihood that none of the YORP detections revealed negative $\upsilon$
if the model were not to give a preference to its sign. Simple algebra
provides a probability of only $0.05$% that 12 cases have the same sign of
$\upsilon$ by chance alone. This would all become more realistic if the model
were to give a preference to spinning up, rather than spinning down, by YORP.
Golubov & Krugly (2012) discovered this hidden element in the theory by
accounting for the lateral heat conduction in the small-scale surface features
of the asteroid (an interesting attempt for a combined model of the YORP
effect may be found in Golubov & Scheeres, 2019). If the still uncertain
fraction to accelerate the rotation rate is about $70-80$%, the chance of
missing a deceleration case out of 12 trials would increase to
$\simeq(1.5-7)$%. Here again, we conclude that the sample of asteroids in
which the YORP effect has been detected is small. This motivates further
efforts to search for asteroids for which the YORP effect may be detected. Of
particular interest may be a future follow-up of (85989) 1999 JD6. If the
tendency from our solution in Sec. 2.3 is confirmed, it may become the first
asteroid for which the detected YORP effect decelerates the rotation rate.

Our new detections add new insight into the properties of the sample of
objects for which the YORP effect has been determined in several respects. For
instance, $\upsilon=(2.9\pm 2.0)\times 10^{-9}\,\text{rad}\,\text{d}^{-2}$ for
(2100) Ra-Shalom is the lowest value detected so far. This confirms that it
might be possible to extend YORP detections to large asteroids, eventually
even beyond the near-Earth asteroid group if enough data are available in the
future. Moreover, Ra-Shalom has the longest rotation period for which the YORP
effect has been detected so far. At the same time, the Ra-Shalom YORP
detection came together with the detection of the spin-axis precession. In
this respect, we note that the detection of the YORP effect for an asteroid in
a tumbling state is yet another novelty. An indication of such detection can
be found in the analysis of the rotation state of the very small asteroid 2012
TC4 (see Lee et al., 2021). The accurately monitored rotation state of (99942)
Apophis (e.g. Pravec et al., 2014; Lee et al., 2022; Ďurech et al., 2022a)
might indicate that this is a very suitable candidate, especially if high-
quality data were obtained from both space-borne and ground-based observations
before its very close approach to the Earth in April 2029.

The accuracy of the observationally determined YORP strength for (1862)
Apollo, expressed by a value $\upsilon/\delta\upsilon\simeq 55$ for the
signal-to-noise ratio, is superior to all currently known cases (followed with
not much worse solution for (1620) Geographos; see Ďurech et al., 2022b).
Future observations, maybe within the next decade or two, will certainly
continue to improve the solution. Additionally, for asteroids as large as
(1862) Apollo, these observations do not require large instruments: $0.5-1$ m
scale telescopes may be suitable. As an example, if a good-quality light-curve
observation of (1862) Apollo is obtained during its close approach in October
2030, the signal-to-noise ratio of the YORP detection would increase to
$\simeq 70-75$. The question arises, however, whether this is even needed and
what the scientific justification would be.

A straightforward answer is provided by continuing a high-accuracy theoretical
modeling work along the line of Rozitis et al. (2013). If the theoretical
prediction of the Yarkovsky and YORP strength could still be improved,
spectacular constraints might be derived on parameters such as the bulk
density (see Farnocchia et al., 2021, where only the Yarkovsky side was
considered). In cases like (2100) Ra-Shalom (Sec. 2.2), (1620) Geographos,
and/or (1685) Toro (Ďurech et al., 2022b), further observations are clearly
motivated to better decorrelate the YORP effect (thus also to improve its
accuracy) from the effect of the regular precession of the spin axis. The
observationally determined value of the precession constant may also
contribute significantly to the consistency of the whole approach and might
even help to answer interesting scientific questions (e.g., improve the shape
model or suggest a possible inhomogeneity of the density distribution). One
additional aspect of the YORP detection methods that have been used so far may
be clarified in view of the prospects for further significant improvements in
the detection accuracy.

Figure 22:  Nondimensional factor $\zeta$ (on the ordinate) related to the
periodic part of the rotation frequency change $\Delta\omega$ (see the text),
expressed as a function of the heliocentric distance (at the abscissa). The
sense of motion is clarified by the arrows, which imply that the upper branch
of the curve corresponds to the motion from perihelion to aphelion, and vice
versa for the lower branch of the curve. This prediction corresponds to the
specific case of (1862) Apollo; the secular part with the $\upsilon$ value
determined in Sec. 2.1 has been subtracted from the rotation-rate $\omega(t)$
evolution and used for the normalization of the $\zeta$ function. The high
eccentricity ($\simeq 0.56$) of the Apollo orbit implies that the amplitude of
the effect is not too small. The red symbols indicate the epochs along the
orbits for which light-curve observations are available. Some of the
observations were taken during close encounters on extreme values of phase
angle exceeding $90^{\circ}$ at heliocentric distances smaller than $1$ au
(indicated by the vertical dashed line). However, Apollo is large enough to
also be easily observable near the aphelion of its orbit at a heliocentric
distance of $2.29$ au (see also Table 3).

The baseline model with the constant sidereal rotation frequency
$\omega(t)=\omega_{0}$ has always been generalized in the simplest way, namely
using the linear model $\omega(t)=\omega_{0}+\upsilon t$ so far (implying that
$\omega_{0}$ only holds at some arbitrary epoch $t=0$). However, when the
asteroid orbit is highly eccentric, all this is slightly more general because
the YORP torque depends at any moment along the orbits on the instantaneous
heliocentric distance, on the orientation of the body frame with respect to
the heliocentric position vector, and on other factors (e.g., Rubincam, 2000;
Čapek & Vokrouhlický, 2004). The empirical factor $\upsilon$ in the formula
above is only the result of averaging the YORP effect over one revolution
about the Sun. As a result, the correct model for the rotation frequency
evolution in time should read $\omega(t)=\omega_{0}+\upsilon
t+\Delta\omega(t)$, where $\Delta\omega(t)$ is a periodic function with the
baseline periodicity $T$ of the asteroid revolution about the Sun. We may
rearrange the formula to provide the variation in the rotation rate, so that
it reads $\omega(t)-\omega_{0}=\upsilon\left[t+T\,\zeta\left(t\right)\right]$,
where we introduced a nondimensional factor
$\zeta(t)=\Delta\omega(t)/(T\,\upsilon)$. Figure 22 shows $\zeta$ as a
function of heliocentric distance rather than time, which is possible because
the heliocentric motion on a fixed ellipse is periodic (here we clearly may
neglect planetary perturbations). We note that the variation has a non-
negligible amplitude of $\simeq 0.3$. However, the principal point is that it
does not accumulate in time. At maximum, its positive value lasts half of the
orbital period $T/2$, and the variation in the rotation frequency is
$\simeq\upsilon\,\zeta_{\rm max}\,T/2$, producing an accumulated phase shift
of $\simeq\upsilon\,\zeta_{\rm max}\,T^{2}/4$. For (1862) Apollo, this amounts
to $\simeq 0.1^{\circ}$. By repeating the same calculation for (101955) Bennu,
we found that the periodic part of the YORP signal only has an amplitude of
$\simeq 0.006^{\circ}$ (because Bennu has a lower eccentricity and shorter
orbital period). This is likely too small even for the very accurate
observations of the OSIRIS-REx spacecraft mission.

We may conclude that in the case of large asteroids, such as Apollo, with
currently reachable YORP detection, we may only rely on the secular change in
the rotation rate $\upsilon$, neglecting variations in the instantaneous
radiation torques along the orbit. The situation would be reversed in a
hypothetical experiment, when an extremely strong torque would affect the
rotation of a very small body and the available observations would cover a
timescale comparable to or even shorter than the revolution period $T$ about
the Sun. The future may perhaps offer this interesting possibility.

###### Acknowledgements.

This work has been supported by the Grant Agency of the Czech Republic, grants
20-04431S and 23-04946S. The work at Modra was supported by the Slovak Grant
Agency for Science VEGA, Grant 1/0530/22. YK thanks for supporting the
European Federation of Academies of Sciences and Humanities (grants ALLEA
EFDS-FL1-18). AR and CS acknowledge support from the UK Science and Technology
Facilities Council. This project has received funding from the European
Union’s Horizon 2020 research and innovation programme under grant agreement
No 870403 (NEOROCKS). DP is thankful for the Wise Observatory staff.

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##  Appendix A New photometric observations

The aspect tables below list the asteroid distance from the Sun $r$ and from
the Earth $\Delta$, the solar phase angle $\alpha$, the geocentric ecliptic
coordinates of the asteroid $(\lambda,\beta)$, and the observatory at which
the data were taken (see Table 2) or a reference to the original publication.

Table 2:  Observatories and telescopes.

Abbreviation | Telescope/Observatory | Telescope aperture [cm] | MPC code
---|---|---|---
D65 | Ondřejov Observatory, Czech Republic | 65 | 557
DK154 | Danish Telescope, ESO, La Silla, Chile | 154 | W74
Rozhen | Rozhen Observatory, Bulgaria | 200 | 071
Simeiz | Simeiz Observatory, Crimea, Ukraine | 100 | 094
Abastumani | Abastumani Observatory, Georgia | 70 | 119
Maidanak | Maidanak Observatory, Uzbekistan | 150 | 188
Kharkiv | Chuguyiv Observatory, Kharkiv, Ukraine | 70 | 121
Modra | Modra Observatory, Slovakia | 60 | 118
PROMPT | Cerro Tololo Inter-American Observatory, Chile | 41 |
SM | Sugarloaf Mountain Observatory, South Deerfield, MA, USA | 50 |
Wi | Wise Observatory, Israel | 71 | 097
BMO | Blue Mountains Observatory, Australia | 35 | Q68
TS | TRAPPIST South, ESO, La Silla, Chile | 60 | I40

999The table lists telescopes used in our work.

Table 3:  Aspect data for the new observations of (1862) Apollo.

Date | $r$ | $\Delta$ | $\alpha\phantom{g}$ | $\lambda$ | $\beta$ | Observatory or
---|---|---|---|---|---|---
| [au] | [au] | [deg] | [deg] | [deg] | Reference
2014 03 20.8 | 1.784 | 0.793 | 4.7 | 182.1 | $8.2$ | D65
2014 03 25.2 | 1.757 | 0.766 | 5.3 | 179.6 | $8.1$ | (1)
2014 03 26.2 | 1.751 | 0.760 | 5.8 | 179.0 | $8.0$ | (1)
2014 03 27.3 | 1.744 | 0.754 | 6.4 | 178.3 | $8.0$ | (1)
2014 03 28.9 | 1.734 | 0.747 | 7.6 | 177.4 | $7.9$ | D65
2014 03 30.1 | 1.726 | 0.742 | 8.4 | 176.7 | $7.8$ | PROMPT
2014 04 03.1 | 1.700 | 0.727 | 11.7 | 174.2 | $7.6$ | SM
2014 12 28.7 | 1.679 | 0.707 | 7.8 | 97.1 | $13.3$ | D65
2014 12 29.7 | 1.686 | 0.714 | 7.7 | 96.5 | $13.3$ | D65
2017 01 21.1 | 2.166 | 1.196 | 5.8 | 126.8 | $11.6$ | DK154
2017 01 22.3 | 2.170 | 1.198 | 5.5 | 126.3 | $11.6$ | DK154
2019 02 09.2 | 2.292 | 1.318 | 5.2 | 146.0 | $10.6$ | DK154
2019 02 11.3 | 2.291 | 1.315 | 4.7 | 145.1 | $10.6$ | DK154
2019 02 13.3 | 2.290 | 1.313 | 4.5 | 144.3 | $10.6$ | DK154
2021 12 11.1 | 1.151 | 0.326 | 52.0 | 12.3 | $10.9$ | (2)
2021 12 12.2 | 1.161 | 0.336 | 51.0 | 14.1 | $10.9$ | (2)
2023 02 17.3 | 1.787 | 1.098 | 29.3 | 210.1 | $5.9$ | DK154
2023 02 23.3 | 1.750 | 1.003 | 28.3 | 211.0 | $6.0$ | DK154
2023 02 25.3 | 1.738 | 0.972 | 27.9 | 211.3 | $6.1$ | DK154

(1) Warner (2014); (2) Warner & Stephens (2022); PROMPT observation by Joe
Pollock

Table 4:  Aspect data for the new observations of (2100) Ra-Shalom. Date | $r$ | $\Delta$ | $\alpha\phantom{g}$ | $\lambda$ | $\beta$ | Observatory or
---|---|---|---|---|---|---
| [au] | [au] | [deg] | [deg] | [deg] | Reference
2019 07 25.0 | 1.122 | 0.449 | 64.9 | 29.9 | $18.3$ | Wi
2019 07 26.0 | 1.125 | 0.444 | 64.4 | 30.0 | $18.2$ | Wi
2019 07 26.9 | 1.129 | 0.440 | 64.0 | 30.1 | $18.1$ | Wi
2019 07 27.0 | 1.129 | 0.440 | 63.9 | 30.1 | $18.1$ | D65
2019 07 32.0 | 1.145 | 0.414 | 61.5 | 30.4 | $17.3$ | Wi
2019 08 04.0 | 1.154 | 0.398 | 59.9 | 30.4 | $16.7$ | Wi
2019 08 05.0 | 1.157 | 0.392 | 59.4 | 30.4 | $16.5$ | D65
2019 08 08.0 | 1.164 | 0.376 | 57.7 | 30.2 | $15.9$ | Wi
2019 08 09.0 | 1.166 | 0.370 | 57.1 | 30.2 | $15.7$ | Wi
2019 08 09.0 | 1.167 | 0.370 | 57.0 | 30.2 | $15.7$ | D65
2019 08 16.4 | 1.181 | 0.327 | 52.0 | 29.0 | $13.8$ | (1)
2019 08 17.4 | 1.182 | 0.321 | 51.2 | 28.7 | $13.5$ | (1)
2019 08 19.4 | 1.185 | 0.310 | 49.5 | 28.1 | $12.8$ | (1)
2019 08 20.4 | 1.187 | 0.304 | 48.6 | 27.8 | $12.5$ | (1)
2019 08 21.4 | 1.188 | 0.298 | 47.8 | 27.4 | $12.1$ | (1)
2019 08 22.4 | 1.189 | 0.292 | 46.8 | 27.0 | $11.7$ | (1)
2019 08 23.4 | 1.190 | 0.287 | 45.8 | 26.6 | $11.3$ | (1)
2019 08 23.9 | 1.190 | 0.284 | 45.3 | 26.3 | $11.1$ | Wi
2019 08 24.1 | 1.191 | 0.283 | 45.2 | 26.2 | $11.1$ | D65
2019 08 25.0 | 1.191 | 0.278 | 44.2 | 25.8 | $10.7$ | Wi
2019 08 26.0 | 1.192 | 0.272 | 43.1 | 25.2 | $10.2$ | D65
2019 09 11.6 | 1.192 | 0.196 | 17.4 | 9.0 | $-1.4$ | BMO
2019 09 22.4 | 1.178 | 0.180 | 12.8 | 350.5 | $-12.7$ | BMO
2019 09 23.5 | 1.176 | 0.180 | 15.2 | 348.5 | $-13.8$ | BMO
2019 09 24.5 | 1.175 | 0.181 | 17.2 | 346.8 | $-14.7$ | BMO
2019 09 25.5 | 1.173 | 0.182 | 19.5 | 344.9 | $-15.7$ | BMO
2019 09 26.4 | 1.171 | 0.183 | 21.6 | 343.1 | $-16.5$ | BMO
2019 09 28.6 | 1.166 | 0.187 | 26.5 | 339.2 | $-18.4$ | BMO
2022 07 18.0 | 1.030 | 0.329 | 78.4 | 33.4 | $34.3$ | D65
2022 07 19.0 | 1.036 | 0.325 | 77.5 | 32.9 | $34.3$ | D65
2022 07 20.0 | 1.041 | 0.322 | 76.7 | 32.5 | $34.3$ | D65
2022 07 22.0 | 1.052 | 0.315 | 74.9 | 31.6 | $34.3$ | D65
2022 07 24.1 | 1.062 | 0.307 | 73.0 | 30.7 | $34.3$ | D65
2022 07 27.0 | 1.076 | 0.296 | 70.3 | 29.1 | $34.3$ | D65
2022 07 28.0 | 1.081 | 0.293 | 69.4 | 28.6 | $34.3$ | D65
2022 08 03.0 | 1.107 | 0.269 | 63.3 | 24.5 | $34.2$ | D65
2022 08 04.0 | 1.111 | 0.265 | 62.1 | 23.6 | $34.1$ | D65
2022 08 04.9 | 1.115 | 0.261 | 61.1 | 22.9 | $34.1$ | D65
2022 08 08.0 | 1.126 | 0.249 | 57.4 | 20.0 | $33.8$ | D65
2022 08 16.1 | 1.152 | 0.218 | 46.0 | 10.1 | $32.3$ | D65
2022 08 16.4 | 1.153 | 0.217 | 45.6 | 9.7 | $32.2$ | (2)
2022 08 16.9 | 1.154 | 0.215 | 44.7 | 8.8 | $32.0$ | D65
2022 08 17.4 | 1.156 | 0.214 | 43.9 | 8.1 | $31.8$ | (2)
2022 08 17.9 | 1.157 | 0.212 | 43.1 | 7.4 | $31.6$ | D65
2022 08 18.4 | 1.158 | 0.211 | 42.3 | 6.6 | $31.5$ | (2)
2022 08 19.0 | 1.160 | 0.209 | 41.1 | 5.6 | $31.2$ | D65
2022 08 19.3 | 1.160 | 0.208 | 40.8 | 5.2 | $31.1$ | (2)
2022 08 20.4 | 1.163 | 0.204 | 38.7 | 3.3 | $30.6$ | (2)
2022 08 21.3 | 1.165 | 0.202 | 37.2 | 1.8 | $30.2$ | (2)
2022 08 22.3 | 1.168 | 0.199 | 35.3 | 360.0 | $29.6$ | (2)
2022 08 23.3 | 1.170 | 0.197 | 33.5 | 358.2 | $29.0$ | (2)
2022 08 25.2 | 1.174 | 0.193 | 30.0 | 354.8 | $27.7$ | (2)
2022 08 26.0 | 1.175 | 0.192 | 28.6 | 353.3 | $27.2$ | D65
2022 08 29.0 | 1.181 | 0.188 | 23.2 | 347.7 | $24.7$ | D65
2022 08 29.9 | 1.182 | 0.188 | 21.6 | 345.8 | $23.8$ | D65
2022 09 06.9 | 1.192 | 0.193 | 16.5 | 331.6 | $15.3$ | D65
Table 4: continued.

(1) Warner & Stephens (2020a); (2) Warner & Stephens (2023)

Table 5:  Aspect data for Unistellar observations of (2100) Ra-Shalom. Date | $r$ | $\Delta$ | $\alpha\phantom{g}$ | $\lambda$ | $\beta$ | Observer
---|---|---|---|---|---|---
| [au] | [au] | [deg] | [deg] | [deg] |
2022 08 15.0 | 1.149 | 0.222 | 47.7 | 11.6 | $32.6$ | M. Billiani
2022 08 18.7 | 1.159 | 0.210 | 41.7 | 6.1 | $31.3$ | K. Fukui
2022 08 20.0 | 1.162 | 0.206 | 39.5 | 4.0 | $30.8$ | B. Guillet
2022 08 20.9 | 1.164 | 0.203 | 37.8 | 2.4 | $30.3$ | P. Tikkanen
2022 08 21.0 | 1.165 | 0.203 | 37.8 | 2.4 | $30.3$ | P. Kuossari
2022 08 24.9 | 1.173 | 0.194 | 30.6 | 355.4 | $28.0$ | anonymous
2022 08 25.9 | 1.175 | 0.192 | 28.7 | 353.5 | $27.2$ | P. Kuossari
2022 08 25.9 | 1.175 | 0.192 | 28.6 | 353.4 | $27.2$ | S. Price
2022 08 25.9 | 1.175 | 0.192 | 28.7 | 353.5 | $27.2$ | anonymous
2022 08 27.0 | 1.177 | 0.191 | 26.7 | 351.4 | $26.4$ | B. Guillet
2022 08 27.9 | 1.179 | 0.190 | 25.1 | 349.7 | $25.6$ | P. Kuossari
2022 08 27.2 | 1.178 | 0.190 | 26.3 | 350.9 | $26.2$ | S. Kardel
2022 08 28.5 | 1.180 | 0.189 | 24.0 | 348.5 | $25.1$ | M. Shimizu
2022 08 28.8 | 1.180 | 0.189 | 23.4 | 347.9 | $24.8$ | O. Clerget
2022 08 29.0 | 1.181 | 0.188 | 23.1 | 347.6 | $24.6$ | M. Lauvernier
2022 08 28.7 | 1.180 | 0.189 | 23.7 | 348.2 | $24.9$ | K. Fukui
2022 08 29.2 | 1.181 | 0.188 | 22.8 | 347.2 | $24.5$ | M. Loose
2022 08 29.6 | 1.182 | 0.188 | 22.1 | 346.4 | $24.1$ | K. Oura
2022 08 31.7 | 1.185 | 0.187 | 18.9 | 342.4 | $22.0$ | I. Chairman
2022 09 01.7 | 1.186 | 0.187 | 17.8 | 340.6 | $21.0$ | I. Chairman
2022 09 01.2 | 1.185 | 0.187 | 18.4 | 341.6 | $21.6$ | E. Hickok
2022 09 01.8 | 1.186 | 0.188 | 17.7 | 340.4 | $20.9$ | O. Clerget
2022 09 01.1 | 1.185 | 0.187 | 18.5 | 341.7 | $21.6$ | S. Will
2022 09 01.1 | 1.185 | 0.187 | 18.5 | 341.7 | $21.6$ | J. Randolph
2022 09 01.9 | 1.186 | 0.188 | 17.6 | 340.3 | $20.8$ | D. Martin
2022 09 02.9 | 1.187 | 0.188 | 16.8 | 338.5 | $19.8$ | P. Kuossari
2022 09 03.0 | 1.188 | 0.188 | 16.6 | 338.2 | $19.6$ | A. Schmidt
2022 09 02.1 | 1.187 | 0.188 | 17.4 | 339.9 | $20.6$ | G. Simard
2022 09 03.8 | 1.189 | 0.189 | 16.2 | 336.8 | $18.7$ | P. Kuossari
2022 09 03.9 | 1.189 | 0.189 | 16.2 | 336.7 | $18.7$ | A. Katterfeld
2022 09 03.9 | 1.189 | 0.189 | 16.2 | 336.7 | $18.7$ | O. Clerget
2022 09 03.6 | 1.188 | 0.189 | 16.3 | 337.2 | $19.0$ | W. Yue
2022 09 08.9 | 1.193 | 0.197 | 18.1 | 328.6 | $13.1$ | P. Tikkanen
2022 09 10.9 | 1.194 | 0.203 | 20.5 | 325.7 | $10.9$ | Y. Lorand
2022 09 11.8 | 1.195 | 0.206 | 21.8 | 324.4 | $9.9$ | O. Clerget
2022 09 11.9 | 1.195 | 0.206 | 21.8 | 324.4 | $9.9$ | D. Martin
Table 6:  Aspect data for observations of (85989) 1999 JD6. Date | $r$ | $\Delta$ | $\alpha\phantom{g}$ | $\lambda$ | $\beta$ | Observatory or
---|---|---|---|---|---|---
| [au] | [au] | [deg] | [deg] | [deg] | Reference
1999 05 22.0 | 1.289 | 0.426 | 41.8 | 231.5 | $57.7$ | D65
1999 05 24.0 | 1.299 | 0.434 | 41.0 | 228.5 | $56.2$ | D65
1999 05 24.9 | 1.304 | 0.439 | 40.7 | 227.2 | $55.5$ | D65
1999 06 13.0 | 1.384 | 0.557 | 39.2 | 213.4 | $40.6$ | D65
2000 07 06.4 | 1.321 | 0.370 | 29.8 | 261.4 | $33.9$ | (7)
2000 07 07.4 | 1.316 | 0.368 | 30.6 | 259.9 | $33.5$ | (7)
2000 07 09.4 | 1.306 | 0.366 | 32.4 | 257.0 | $32.6$ | (7)
2000 07 10.3 | 1.301 | 0.364 | 33.3 | 255.5 | $32.1$ | (7)
2004 05 15.0 | 1.303 | 0.448 | 41.3 | 261.9 | $53.7$ | (1)
2004 05 16.0 | 1.308 | 0.449 | 40.7 | 260.2 | $53.7$ | (1)
2004 05 17.0 | 1.313 | 0.450 | 40.1 | 258.4 | $53.7$ | (1)
2004 05 23.0 | 1.341 | 0.459 | 36.9 | 248.1 | $52.5$ | (1)
2004 05 27.9 | 1.361 | 0.472 | 35.2 | 240.6 | $50.5$ | (1)
2004 06 05.4 | 1.391 | 0.505 | 34.4 | 230.4 | $45.8$ | (8)
2004 06 06.5 | 1.394 | 0.510 | 34.4 | 229.4 | $45.1$ | (8)
2004 06 09.5 | 1.402 | 0.526 | 34.7 | 226.8 | $43.2$ | (8)
2004 06 11.4 | 1.407 | 0.537 | 35.0 | 225.4 | $42.0$ | (8)
2004 06 12.4 | 1.410 | 0.543 | 35.2 | 224.8 | $41.3$ | (8)
2005 07 08.9 | 1.261 | 0.293 | 29.7 | 272.1 | $35.4$ | (1)
2005 07 10.0 | 1.255 | 0.289 | 30.6 | 270.1 | $35.2$ | (1)
2005 07 14.9 | 1.223 | 0.272 | 36.2 | 259.9 | $33.3$ | Modra
2005 07 15.9 | 1.217 | 0.269 | 37.5 | 257.9 | $32.7$ | Modra
2014 05 20.4 | 1.394 | 0.533 | 36.0 | 275.9 | $42.7$ | (2)
2014 05 21.4 | 1.397 | 0.530 | 35.4 | 274.9 | $42.9$ | (2)
2014 05 22.3 | 1.399 | 0.528 | 34.9 | 273.9 | $43.1$ | (2)
2015 06 07.4 | 1.336 | 0.575 | 45.1 | 323.1 | $22.0$ | (3)
2015 06 08.4 | 1.332 | 0.563 | 45.0 | 323.4 | $22.2$ | (3)
2015 06 11.4 | 1.317 | 0.526 | 44.9 | 324.2 | $22.8$ | (3)
2015 06 12.4 | 1.312 | 0.514 | 44.9 | 324.5 | $23.0$ | (3)
2015 06 14.4 | 1.302 | 0.490 | 44.8 | 325.1 | $23.4$ | (3)
2015 06 15.4 | 1.297 | 0.477 | 44.8 | 325.4 | $23.6$ | (3)
2018 06 01.2 | 1.181 | 0.518 | 58.8 | 167.6 | $41.2$ | (4)
2018 06 02.3 | 1.188 | 0.527 | 58.2 | 168.9 | $40.6$ | (4)
2018 06 03.3 | 1.195 | 0.536 | 57.6 | 170.1 | $39.9$ | (4)
2018 06 04.3 | 1.202 | 0.546 | 57.0 | 171.3 | $39.3$ | (4)
2019 06 03.4 | 1.437 | 0.522 | 29.3 | 269.2 | $41.1$ | (5)
2019 06 04.4 | 1.438 | 0.520 | 28.9 | 268.0 | $41.1$ | (5)
2019 06 05.4 | 1.439 | 0.518 | 28.5 | 266.8 | $41.1$ | (5)
2019 06 06.4 | 1.439 | 0.516 | 28.2 | 265.6 | $41.1$ | (5)
2019 06 07.4 | 1.440 | 0.514 | 28.0 | 264.4 | $41.0$ | (5)
2020 06 18.4 | 1.231 | 0.445 | 51.6 | 337.8 | $20.0$ | (6)
2020 06 19.4 | 1.224 | 0.433 | 51.9 | 338.6 | $20.1$ | (6)
2020 06 20.4 | 1.217 | 0.420 | 52.3 | 339.3 | $20.2$ | (6)
2020 06 21.4 | 1.211 | 0.408 | 52.6 | 340.1 | $20.3$ | (6)
2020 06 22.4 | 1.204 | 0.395 | 53.0 | 341.0 | $20.5$ | (6)
2020 06 23.4 | 1.197 | 0.383 | 53.5 | 341.8 | $20.6$ | (6)
2023 05 10.0 | 1.057 | 0.335 | 72.7 | 135.7 | $67.6$ | D65
2023 05 15.9 | 1.110 | 0.368 | 65.0 | 156.7 | $61.9$ | D65
2023 06 25.0 | 1.356 | 0.729 | 47.5 | 195.2 | $30.0$ | DK154
Table 6: continued.

(1) Polishook & Brosch (2008); (2) Warner (2014); (3) Warner (2015); (4)
Warner (2018); (5) Warner & Stephens (2019); (6) Warner & Stephens (2020b);
(7) observed by Brandon Bozek, other information lost; (8) observed by Luke
Dundon, other information lost

Table 7:  Aspect data for observations of (138852) 2000 WN10. Date | $r$ | $\Delta$ | $\alpha\phantom{g}$ | $\lambda$ | $\beta$ | Observatory
---|---|---|---|---|---|---
| [au] | [au] | [deg] | [deg] | [deg] |
2008 11 27.8 | 1.172 | 0.190 | 11.8 | 53.5 | $6.9$ | Rozhen
2009 11 18.0 | 1.130 | 0.146 | 13.4 | 63.6 | $-13.2$ | Simeiz
2009 11 22.9 | 1.150 | 0.163 | 4.7 | 56.0 | $-2.8$ | Simeiz
2009 11 26.9 | 1.165 | 0.183 | 11.8 | 51.2 | $4.0$ | Simeiz
2010 11 13.0 | 1.104 | 0.135 | 30.1 | 69.7 | $-28.6$ | Abastumani
2010 11 25.8 | 1.157 | 0.174 | 11.9 | 49.2 | $0.5$ | Maidanak
2010 11 26.8 | 1.160 | 0.179 | 13.7 | 48.2 | $2.1$ | Maidanak
2010 12 07.7 | 1.199 | 0.254 | 29.5 | 41.1 | $14.4$ | Maidanak
2011 11 19.9 | 1.129 | 0.145 | 12.3 | 53.3 | $-13.7$ | D65
2011 11 20.7 | 1.133 | 0.148 | 11.2 | 52.1 | $-11.6$ | Abastumani
2011 11 22.8 | 1.141 | 0.157 | 10.9 | 49.4 | $-6.9$ | Abastumani
2011 11 22.9 | 1.142 | 0.157 | 10.9 | 49.4 | $-6.8$ | D65
2011 11 23.9 | 1.146 | 0.162 | 11.7 | 48.2 | $-4.7$ | Kharkiv
2011 11 24.9 | 1.149 | 0.167 | 12.9 | 47.2 | $-2.8$ | Kharkiv
2011 11 26.0 | 1.154 | 0.173 | 14.6 | 46.1 | $-0.8$ | Abastumani
2012 11 19.2 | 1.127 | 0.144 | 14.8 | 49.3 | $-15.0$ | DK154
2012 11 22.8 | 1.142 | 0.160 | 14.6 | 45.1 | $-6.7$ | Abastumani
2012 11 23.8 | 1.145 | 0.165 | 15.5 | 44.2 | $-4.8$ | Abastumani
2013 10 30.4 | 1.032 | 0.157 | 71.4 | 103.2 | $-64.4$ | DK154
2013 10 31.3 | 1.037 | 0.153 | 69.2 | 98.8 | $-63.5$ | DK154
2013 11 05.3 | 1.061 | 0.136 | 56.1 | 77.6 | $-56.0$ | DK154
2013 11 06.3 | 1.066 | 0.133 | 53.1 | 74.0 | $-53.8$ | DK154
2013 11 07.3 | 1.070 | 0.131 | 50.0 | 70.5 | $-51.5$ | DK154
2013 11 08.3 | 1.075 | 0.130 | 46.7 | 67.3 | $-48.9$ | DK154
2013 11 09.3 | 1.080 | 0.128 | 43.5 | 64.4 | $-46.2$ | DK154
2013 11 11.3 | 1.089 | 0.127 | 37.0 | 59.2 | $-40.4$ | DK154
2013 11 22.8 | 1.138 | 0.159 | 17.6 | 41.8 | $-8.5$ | Abastumani
2013 11 23.3 | 1.140 | 0.161 | 17.9 | 41.5 | $-7.5$ | DK154
2014 11 17.2 | 1.111 | 0.135 | 24.0 | 43.9 | $-25.1$ | DK154
2014 12 16.1 | 1.216 | 0.333 | 39.9 | 33.2 | $16.1$ | DK154
2015 11 14.4 | 1.094 | 0.128 | 33.6 | 42.8 | $-36.8$ | DK154
2015 11 15.3 | 1.098 | 0.130 | 31.2 | 41.5 | $-33.6$ | DK154
2015 11 16.3 | 1.102 | 0.132 | 29.1 | 40.3 | $-30.4$ | DK154
2016 10 23.3 | 0.986 | 0.188 | 87.2 | 153.9 | $-75.3$ | DK154
2016 12 01.1 | 1.162 | 0.219 | 33.1 | 29.1 | $2.5$ | DK154
2016 12 07.1 | 1.184 | 0.267 | 37.3 | 29.1 | $8.8$ | DK154
2017 11 21.1 | 1.119 | 0.157 | 31.1 | 27.0 | $-17.7$ | DK154
2017 11 24.1 | 1.131 | 0.173 | 31.3 | 26.6 | $-10.8$ | DK154
2017 11 25.2 | 1.135 | 0.180 | 31.7 | 26.5 | $-8.5$ | DK154
2018 11 29.2 | 1.147 | 0.207 | 36.0 | 23.6 | $-2.9$ | DK154
2019 11 04.3 | 1.032 | 0.145 | 70.0 | 15.0 | $-76.4$ | DK154
2019 11 05.0 | 1.035 | 0.142 | 68.3 | 15.5 | $-74.3$ | DK154
2019 11 26.1 | 1.131 | 0.187 | 36.8 | 20.8 | $-9.5$ | DK154
2019 11 27.1 | 1.135 | 0.194 | 37.0 | 21.0 | $-7.7$ | DK154
2019 11 28.0 | 1.139 | 0.200 | 37.3 | 21.2 | $-6.0$ | DK154
2019 12 02.0 | 1.154 | 0.229 | 38.8 | 22.1 | $0.0$ | DK154
2020 11 18.1 | 1.097 | 0.153 | 41.9 | 15.4 | $-27.8$ | DK154
2020 11 21.2 | 1.111 | 0.167 | 39.7 | 16.6 | $-19.6$ | DK154
2021 11 28.1 | 1.135 | 0.212 | 41.5 | 16.4 | $-6.8$ | DK154
2022 11 21.2 | 1.102 | 0.174 | 45.6 | 9.2 | $-22.2$ | DK154
Table 7: continued. Table 8:  Aspect data for the new observations of (161989)
Cacus.

Date | $r$ | $\Delta$ | $\alpha\phantom{g}$ | $\lambda$ | $\beta$ | Observatory or
---|---|---|---|---|---|---
| [au] | [au] | [deg] | [deg] | [deg] | Reference
2022 01 28.1 | 1.286 | 0.439 | 39.0 | 114.6 | $-54.2$ | TS
2022 01 29.1 | 1.284 | 0.435 | 39.0 | 113.6 | $-53.7$ | TS
2022 01 30.1 | 1.282 | 0.432 | 39.1 | 112.6 | $-53.2$ | TS
2022 02 15.1 | 1.242 | 0.391 | 42.2 | 100.8 | $-40.3$ | TS
2022 02 16.1 | 1.239 | 0.389 | 42.5 | 100.4 | $-39.4$ | TS
2022 02 17.1 | 1.236 | 0.388 | 42.9 | 99.9 | $-38.3$ | TS
2022 08 28.0 | 1.012 | 0.068 | 86.6 | 64.8 | $26.0$ | (1)
2022 08 30.0 | 1.019 | 0.061 | 79.6 | 59.1 | $13.6$ | (1)
2022 08 31.0 | 1.023 | 0.058 | 75.5 | 56.0 | $6.2$ | (1)
2022 08 31.1 | 1.023 | 0.058 | 75.2 | 55.7 | $5.6$ | (1)