11institutetext: ESA NEO Coordination Centre, Largo Galileo Galilei, 1, 00044
Frascati, Italy
11email<EMAIL_ADDRESS>11institutetext:  ESA ESRIN / PDO / NEO
Coordination Centre, Largo Galileo Galilei, 1, 00044 Frascati (RM), Italy
11email<EMAIL_ADDRESS>22institutetext: Elecnor Deimos, Via
Giuseppe Verdi, 6, 28060 San Pietro Mosezzo (NO), Italy 33institutetext: RHEA
Systems, Via di Grotte Portella, 28, 00044 Frascati (RM), Italy
44institutetext: Alia Space System Srl, Via San Giuseppe Calasanzio 15, 00044
Frascati (RM), Italy 55institutetext: ESA ESTEC / PDO, Keplerlaan 1, 2201 AZ
Noordwijk, The Netherlands 66institutetext: RHEA Group, Jonckerweg 18, 2201 DZ
Noordwijk, The Netherlands

# An automated procedure for the detection of the Yarkovsky effect and results
from the ESA NEO Coordination Centre††thanks: Full tables 20, 21 and 22 are
only available in electronic form at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/.

Marco Fenucci 1122 Marco Micheli 1133 Francesco Gianotto 1133 Laura Faggioli
1133 Dario Oliviero 1122 Andrea Porru 1144 Regina Rudawska 5566 Juan Luis Cano
11 Luca Conversi 11 Richard Moissl 11

(Received — / Accepted —)

###### Abstract

Context. The measurement of the Yarkovsky effect on near-Earth asteroids
(NEAs) is common practice in orbit determination today, and the number of
detections will increase with the developments of new and more accurate
telescopic surveys. However, the process of finding new detections and
identifying spurious ones is not yet automated, and it often relies on
personal judgment.

Aims. We aim to introduce a more automated procedure that can search for NEA
candidates to measure the Yarkovsky effect, and that can identify spurious
detections.

Methods. The expected semi-major axis drift on an NEA caused by the Yarkovsky
effect was computed with a Monte Carlo method on a statistical model of the
physical parameters of the asteroid that relies on the most recent NEA
population models and data. The expected drift was used to select candidates
in which the Yarkovsky effect might be detected, according to the current
knowledge of their orbit and the length of their observational arc. Then, a
nongravitational acceleration along the transverse direction was estimated
through orbit determination for each candidate. If the detected acceleration
was statistically significant, we performed a statistical test to determine
whether it was compatible with the Yarkovsky effect model. Finally, we
determined the dependence on an isolated tracklet.

Results. Among the known NEAs, our procedure automatically found 348
detections of the Yarkovsky effect that were accepted. The results are overall
compatible with the predicted trend with the the inverse of the diameter, and
the procedure appears to be efficient in identifying and rejecting spurious
detections. This algorithm is now adopted by the ESA NEO Coordination Centre
to periodically update the catalogue of NEAs with a measurable Yarkovsky
effect, and the results are automatically posted on the web portal.

###### Key Words.:

Astrometry and celestial mechanics - Minor planets, asteroids: general -
Methods: statistical

##  1 Introduction

The Yarkovsky effect is a nongravitational force caused by thermal emission.
It affects the dynamics of asteroids whose diameter is smaller than about 40
km (see e.g. Vokrouhlický, 1998; Bottke et al., 2006; Vokrouhlický et al.,
2015), and it mainly manifests as a drift in the semi-major axis of the
object. It is known to cause several phenomena in the dynamics of small bodies
in the Solar System, such as the spreading of asteroid families (see e.g.
Spoto et al., 2015), and the transport of objects from the main belt to the
near-Earth region (see e.g. Granvik et al., 2017). From the point of view of
planetary defence, measuring nongravitational effects acting on near-Earth
asteroids (NEAs) is important for a reliable assessment of the impact risk,
especially for the long-time horizon search of potential impacts (Chesley et
al., 2014; Farnocchia & Chesley, 2014; Spoto et al., 2014; Vokrouhlický et
al., 2015). Additionally, measurements of the Yarkovsky effect can be used to
extrapolate the physical properties of the asteroids (see e.g. Chesley et al.,
2014; Rozitis et al., 2014; Fenucci et al., 2021, 2023), which are key
parameters to know for the design of lander and sample-return spacecraft
missions (Murdoch et al., 2021), or to predict the efficiency of a kinetic
impactor-deflection mission (Cheng et al., 2023; Daly et al., 2023; Thomas et
al., 2023).

Since the first measurement of the Yarkovsky effect on asteroid (6489) Golevka
(Chesley et al., 2003), many detections on NEAs have been found and announced
in a series of related works (Nugent et al., 2012; Farnocchia et al., 2013;
Del Vigna et al., 2018; Greenberg et al., 2020). Today, the information about
the detections is regularly provided by the three main centers for asteroid
orbit computation: the ESA NEO Coordination Centre111https://neo.ssa.esa.int/
(NEOCC), the NEODyS222https://newton.spacedys.com/ service, and the NASA JPL
Solar System Dynamics333https://ssd.jpl.nasa.gov/ (SSD) group. However, the
process of detecting new measurements is not fully automated, and the
identification of spurious detections still relies on manual checks that are
often based on personal judgment. There are several reasons that make a
detection spurious, such as dynamical modeling issues or poor-quality old
astrometry. Some detections may also appear to be incompatible with the
Yarkovsky effect, but they are in fact real and are caused by other poorly
understood phenomena, as in the case of asteroid (523599) 2003 RM (Farnocchia
et al., 2023). Therefore, in some cases it is questionable whether a Yarkovsky
effect has really been detected or not. On the other hand, the new and
improved observational technologies and the high-quality data provided by the
ESA Gaia mission (Tanga et al., 2022) mean that the number of detected
Yarkovsky effects is only expected to increase in the future. This will become
especially true after the beginning of the operational phase of new-generation
ground-based surveys, such as the ESA Flyeye telescope (Conversi et al., 2021)
and the Vera Rubin Observatory (Ivezić et al., 2019), and of the the space-
based infrared telescopes NEO Surveyor by NASA (Mainzer et al., 2021) and
NEOMIR by ESA (Conversi et al., 2023). These new facilities are expected to
increase the number of known asteroids by roughly an order of magnitude (Jones
et al., 2018). This situation calls for the development of new and automated
methods for the detection of the Yarkovsky effect and for the identification
of spurious measurements.

In contrast to previous efforts, we propose a systematic procedure here to
search for candidates for measurements of the Yarkovsky effect on NEAs, and
for the identification of spurious detections. We use a statistical model of
the physical parameters of the asteroids that relies on the most recent NEA
data sets and population models (Granvik et al., 2018; Morbidelli et al.,
2020; Berthier et al., 2023). We first estimate the expected Yarkovsky effect
with a Monte Carlo method. Then, a list of NEA candidates for the detection of
the Yarkovsky effect is determined according to the expected semi-major axis
drift, the observational arc, and the uncertainty of the current orbit. For
each object of the list, a nongravitational force along the transverse
direction is estimated through orbit determination, and only detections with a
good signal-to-noise ratio that are statistically compatible with the
prediction from the physical model are kept. Then, we determine the dependence
of the detections on isolated tracklets, and only those that passed the check
are accepted. Among known NEAs, the algorithm identified 348 detections. The
procedure described here is now adopted by the ESA NEO Coordination Centre,
and data are automatically uploaded on the web portal.

The paper is organized as follows. In Sec. 2 we describe the model for the
Yarkovsky effect and the model of the physical parameters of the NEAs that we
used to predict the semi-major axis drift, the method we used to determine the
orbit, and the procedure we used to automatically detect the Yarkovsky effect.
In Sec. 3 we provide the results we obtained by running the procedure on the
known NEA population, we compare them with the results reported by the JPL SSD
group, and we discuss the effectiveness and limitations of our algorithm.
Finally, we summarize our conclusions in Sec. 4.

##  2 Methods

###  2.1 Orbit determination

The dynamical model we used to determine the orbit of a NEA includes the
gravitational perturbations of the Sun, the eight planets, the Moon, the 16
most massive main-belt asteroids, and Pluto. The masses and positions of all
these bodies were all retrieved from the JPL ephemeris DE441 (Park et al.,
2021). The masses of the 16 main-belt asteroids included in the dynamical
model and the mass of Pluto are given in Table 5. Relativistic effects of the
Sun, the planets, and the Moon were also added to the dynamical model as a
first-order Newtonian expansion (Will, 1993). The Yarkovsky effect is
expressed as an acceleration along the transverse direction $\hat{\mathbf{t}}$
of the form

$\mathbf{a}_{t}=A_{2}\bigg{(}\frac{1\textrm{
au}}{r}\bigg{)}^{2}\hat{\mathbf{t}},$ (1)

where $r$ is the distance from the Sun (Farnocchia et al., 2013). The
corresponding semi-major axis drift is obtained through the Gauss planetary
equations by

$\frac{\text{d}a}{\text{d}t}=\frac{2A_{2}(1-e^{2})}{np^{2}},$ (2)

where $n$ is the mean motion, and $p=a(1-e^{2})$ is the semilatus rectum.
Additionally, it is possible to optionally add solar radiation pressure (SRP)
to the dynamical model. The SRP is modeled as a radial force $\mathbf{a}_{r}$
of the form

$\mathbf{a}_{r}=A_{1}\bigg{(}\frac{1\textrm{
au}}{r}\bigg{)}^{2}\hat{\mathbf{r}},$ (3)

where $\hat{\mathbf{r}}=\mathbf{r}/r$ is the radial unit vector, and
$\mathbf{r}$ is the heliocentric position of the asteroid. The coefficient
$A_{1}$ is computed as in Montenbruck & Gill (2000),

$A_{1}=(1-\Theta)\times\frac{\phi}{c}\times\frac{A}{m},$ (4)

where $\phi=1.361$ kW m-2 is the solar radiation energy flux at 1 au, $c$ is
the speed of light, $A/m$ is the area-to-mass parameter, and $\Theta$ is the
occultation function, which determines whether the object is occulted by a
major planet.

The parameter $A_{2}$ (and possibly the $A/m$ parameter, when SRP is included
in the model) is determined together with the orbital elements by fitting the
model to the observations through a least-squares procedure (Milani & Gronchi,
2009). Observational outliers were identified and discarded with an automatic
rejection algorithm developed by Carpino et al. (2003). In all the
computations, we used the weighting scheme based on observatory statistics
developed by Vereš et al. (2017), and the debiasing scheme based on star
catalogs developed by Farnocchia et al. (2015). The orbital elements estimated
from the fit refer to the weighted mean of the observation times, as indicated
in Milani & Gronchi (2009). To perform the orbital fit, we used the ESA Aegis
orbit determination and impact monitoring software (Faggioli et al., 2023),
developed and maintained by SpaceDyS s.r.l.444https://www.spacedys.com/ under
ESA contracts.

Table 1: Gravitational parameters $GM$ of the 16 most massive main-belt
asteroids included in the dynamical model, and that of Pluto.

Asteroid | $GM/GM_{\odot}$
---|---
(1) Ceres | 4.7191422767$\times 10^{-10}$
(2) Pallas | 1.0297360324$\times 10^{-10}$
(3) Juno | 1.4471670475$\times 10^{-11}$
(4) Vesta | 1.3026836726$\times 10^{-10}$
(7) Hebe | 8.5890389994$\times 10^{-12}$
(10) Hygea | 4.2385986149$\times 10^{-11}$
(15) Eunomia | 1.5243642468$\times 10^{-11}$
(16) Psyche | 1.1978215785$\times 10^{-11}$
(31) Euphrosyne | 8.1331596146$\times 10^{-12}$
(52) Europa | 2.0216913527$\times 10^{-11}$
(65) Cybele | 7.0687100328$\times 10^{-12}$
(87) Sylvia | 1.6337820878$\times 10^{-11}$
(88) Thisbe | 8.9653065564$\times 10^{-12}$
(107) Camilla | 1.0878696848$\times 10^{-11}$
(511) Davida | 2.9345275748$\times 10^{-11}$
(704) Interamnia | 2.1327387534$\times 10^{-11}$
(134340) Pluto | 7.3504789732$\times 10^{-9}$
555The gravitational parameters of the 16 main-belt asteroids are extracted
from the header of the JPL ephemerides DE441
(https://ssd.jpl.nasa.gov/ftp/eph/planets/ascii/de441/). The gravitational
parameter of Pluto is taken from Brozović et al. (2015). The values are
reported in the units of the gravitational parameter of the Sun, corresponding
to $GM_{\odot}=1.32712440040944\times 10^{11}$ km3 s-2.

###  2.2 Expected Yarkovsky drift

####  2.2.1 Semi-analytical formula for the semi-major axis drift

The instantaneous semi-major axis drift produced by the Yarkovsky effect can
be expressed by analytical formulas (Vokrouhlický et al., 2017), assuming a
spherical body and a linearization of the boundary conditions. It is given by

$\frac{\text{d}a}{\text{d}t}=\frac{2}{n^{2}a}\mathbf{f}_{\text{Y}}\cdot\mathbf{v},$
(5)

where $a$ is the semi-major axis of the asteroid orbit, $\mathbf{v}$ is the
heliocentric orbital velocity, and $\mathbf{f}_{\text{Y}}$ is the
instantaneous value of the Yarkovsky acceleration. The term
$\mathbf{f}_{\text{Y}}$ is the result of the sum of the diurnal component
$\mathbf{f}_{\text{Y,d}}$ and the seasonal component
$\mathbf{f}_{\text{Y,s}}$. The diurnal component is expressed as

$\mathbf{f}_{\text{Y,
d}}=\kappa[(\hat{\mathbf{r}}\cdot\hat{\mathbf{s}})\hat{\mathbf{s}}+\gamma_{1}(\hat{\mathbf{r}}\times\hat{\mathbf{s}})+\gamma_{2}\hat{\mathbf{s}}\times(\hat{\mathbf{r}}\times\hat{\mathbf{s}})].$
(6)

In Eq. (6), $\hat{\mathbf{r}}=\mathbf{r}/r$ is the heliocentric unit position
vector, and $\hat{\mathbf{s}}$ is the unit vector of the asteroid spin axis.
In addition,

$\kappa=\frac{4\alpha}{9}\frac{SF}{mc},$ (7)

where $S=\pi R^{2}$ is the cross section of the asteroid, $R$ is the radius,
$F$ is the solar radiation flux at a heliocentric distance $r$, $\alpha$ is
the absorption coefficient, $m$ is the asteroid mass, and $c$ is the speed of
light. The coefficients $\gamma_{1},\gamma_{2}$ are expressed as

$\begin{split}\gamma_{1}&=-\frac{k_{1}(R^{\prime}_{\textrm{d}})\Theta_{\textrm{d}}}{1+2k_{2}(R^{\prime})\Theta_{\textrm{d}}+k_{3}(R^{\prime}_{\textrm{d}})\Theta_{\textrm{d}}^{2}},\\\
\gamma_{2}&=-\frac{1+k_{2}(R^{\prime}_{\textrm{d}})\Theta_{\textrm{d}}}{1+2k_{2}(R^{\prime})\Theta_{\textrm{d}}+k_{3}(R^{\prime}_{\textrm{d}})\Theta_{\textrm{d}}^{2}}.\end{split}$
(8)

In Eq. (8), $R^{\prime}=R/l_{\textrm{d}}$ is the scaled radius, where
$l_{\textrm{d}}=\sqrt{K/(\rho C\omega)}$ is the penetration depth of the
diurnal thermal wave, obtained from the thermal conductivity $K$, the heat
capacity $C$, the density $\rho$, and the rotation frequency $\omega$. The
functions $k_{1},k_{2}$, and $k_{3}$ are positive analytic functions of the
scaled radius (Vokrouhlický, 1998, 1999). In addition,
$\Theta_{\textrm{d}}=\sqrt{\rho KC\omega}/(\varepsilon\sigma T_{\star}^{3})$,
where $\sigma$ is the Stefan-Boltzmann constant, and $T_{\star}$ is the
subsolar temperature, which is given by $4\varepsilon\sigma
T_{\star}^{4}=\alpha F$.

The seasonal component is expressed as

$\mathbf{f}_{\text{Y,
s}}=\kappa[\bar{\gamma}_{1}(\hat{\mathbf{r}}\cdot\hat{\mathbf{s}})+\bar{\gamma}_{2}(\hat{\mathbf{n}}\times\hat{\mathbf{r}})\cdot\hat{\mathbf{s}}]\hat{\mathbf{s}},$
(9)

where $\hat{\mathbf{n}}$ is the unit vector normal to the orbital plane. The
coefficients $\bar{\gamma}_{1},\bar{\gamma}_{2}$ have the same expressions as
Eq. (8), but evaluated with $\overline{\Theta}_{\textrm{s}}=\sqrt{\rho
KCn}/(\varepsilon\sigma T_{\star}^{3})$, and with a scaled radius
$R^{\prime}_{\textrm{s}}=R/l_{\textrm{s}}$, where
$l_{\textrm{s}}=\sqrt{K/(\rho Cn)}$ is the penetration depth of the seasonal
thermal wave. The average Yarkovsky drift $\text{d}a/\text{d}t$ is finally
obtained by numerically averaging the instantaneous Yarkovsky drift of Eq. (5)
over an orbital period.

####  2.2.2 Modeling the orbital and physical parameters

The orbital parameters in the analytical formulas of Eq. (5) are the semi-
major axis $a$ and the eccentricity $e$. The uncertainties on these parameters
are usually small when the orbit is well determined, and they typically do not
affect the results given by Eq. (5). We therefore fixed them to their nominal
value.

The other parameters in Eq. (5) are the diameter $D$, the density $\rho$, the
obliquity $\gamma$, the rotation period $P$, the heat capacity $C$, the
thermal conductivity $K$, the emissivity $\varepsilon$, and the absorption
coefficient $\alpha$. The emissivity was fixed to $\varepsilon=0.984$, which
corresponds to the average value measured on meteorites (Ostrowski & Bryson,
2019). The heat capacity was fixed to a value of $C=800$ J kg-1 K-1, which is
appropriate for NEAs (Delbo’ et al., 2015). However, changing this value in
the range $500-1200$ J kg-1 K-1, which is thought to be valid for most of
materials present on asteroids, produces very small changes in the predicted
semi-major axis drift (Fenucci et al., 2021, 2023). Because we are interested
in estimating the maximum semi-major axis drift that can be reached by a NEA,
we assumed that the obliquity $\gamma$ can be either $0^{\circ}$ or
$180^{\circ}$. For the same reason, we also fixed $\alpha=1$.

To obtain the values of the other physical parameters, we used the SsODNet
service666https://ssp.imcce.fr/webservices/ssodnet/ (Berthier et al., 2023)
through the Python rocks API777https://github.com/maxmahlke/rocks. The
diameter $D$, the rotation period $P$, and the taxonomic class were retrieved
through ssoCard, which provides a best estimate by averaging all the available
measurements of a given parameter. When the values of $D$ or $P$ were
provided, we assumed that they are Gaussian distributed with a mean value
equal to the nominal value and a standard deviation equal to the error. On the
other hand, the taxonomic complex gives only indirect information about the
density. Table 19 reports all the taxonomic complexes found for NEAs in the
SsODNet database. The grouping of the different taxonomic types (Tholen, 1984;
Bus & Binzel, 2002; DeMeo et al., 2009) and complexes can be found in Berthier
et al. (2023). The density was assumed to be distributed according to a log-
normal distribution, with values of $\mu$ and $\sigma$ provided by Table 19.
These values were obtained by a statistics on the density values extracted
from the SsODNet database (see Appendix A).

Whenever the diameter $D$, the taxonomic complex, or the rotation period $P$
were lacking in the SsODNet database, we modeled them according to the
properties and available models of the whole NEA population. The distribution
of the measured rotation periods shows a dependency on the diameter (Pravec &
Harris, 2000), where objects rotating in less than about 2.2 h are found only
at sizes with diameters smaller than roughly 150$-$200 m. Here we assumed that
the rotation period varied with the diameter as

$P(D)=\cases{5}.6\text{ h}&\text{if }D\geq 200\text{ m},\\\ \displaystyle
5.6\text{ h}\times\bigg{(}\frac{D}{0.2\text{ km}}\bigg{)}^{2}\text{if
}D<200\text{ m}.{}$ (10)

Figure 1 shows the trend of the rotation period of Eq. (10) we adopted,
superimposed on the distribution of rotation period measurements of NEAs
extracted from the Asteroid Lightcurve Database (LCDB) (Warner et al., 2009).
Above the threshold of 200 meters of diameter, we used a constant rotation
period of 5.6 h, corresponding to the median value of the measured rotation
periods of asteroids larger than the threshold. At sizes smaller than 200
meters, asteroids may rotate very fast without being disrupted, and therefore,
we assumed a trend that scales as $D^{-2}$, which is a better fit to the data
than the $D^{-1}$ trend that was previously suggested by Farinella et al.
(1998). The biases in the rotation period measurements for asteroids with
diameters smaller than 200 m are unclear, and it is difficult to extrapolate a
general trend for the whole population. However, we found this solution to be
the best that can be done based upon the currently available observations.

Figure 1: Distribution of the rotation period measurements of NEAs extracted
from LCDB, together with the rotational disruption limit of 2.2 h and the
dependence of $P(D)$ on the diameter.

For the diameter $D$ and the density $\rho$, we used the model by Fenucci et
al. (2021, 2023), which combines the orbital distribution model for NEAs by
Granvik et al. (2018) and the albedo distribution model for NEAs by Morbidelli
et al. (2020). The model first produces a distribution of the visual albedo
$p_{V}$ that is later converted into a joined distribution $(D,\rho)$ of the
diameter and density. For this purpose, we also assumed that the absolute
magnitude $H$ is Gaussian distributed, with a standard deviation given by the
root mean square (RMS) error obtained by converting the visual magnitude
measurements into absolute magnitude. When $D$ is given in the SsODNet
database, but not the taxonomic complex (or when the taxonomic complex is
given in the SsODNet database, but not the diameter $D$), we used the marginal
distribution of $\rho$ (the marginal distribution of $D$) obtained by the same
model.

The thermal conductivity $K$ is related to the thermal inertia $\Gamma$ by the
relation

$\Gamma=\sqrt{\rho KC}.$ (11)

We assumed $\Gamma$ to be Gaussian distributed with a mean value of 250 J m-2
K-1 s-1/2 (Delbo’ et al., 2007) and a standard deviation of 100 J m-2 K-1
s-1/2, and the value of $K$ was then computed by inverting Eq. (11).

Table 8 summarizes the physical parameters for the model of the Yarkovsky
effect in Eq. (5) and the corresponding value or model we used to compute the
predicted drift of the semi-major axis.

Table 2: Physical parameters for the Yarkovsky effect model and the
corresponding value or model.

Parameter | Value or model
---|---
$D$ | From SsODNet or from [1] model
$\rho$ | From SsODNet taxonomy or from [1] model
$P$ | From SsODNet or Eq. (10)
$\Gamma$ |  $250\pm 100$ J m-2 K-1 s-1/2
$C$ |  $800$ J kg-1 K-1
$\gamma$ | 0∘ or 180∘
$\alpha$ | 1
$\varepsilon$ | 0.984
888Reference [1] above refers to Fenucci et al. (2021).

###  2.3 Treatment of poor astrometry

The fact that a high signal-to-noise ratio (S/N) detection of $A_{2}$ is found
from orbit determination does not imply that it is due to a true dynamical
effect. In fact, previous works (Farnocchia et al., 2013; Del Vigna et al.,
2018) found that old and poorly measured astrometry or isolated tracklets that
significantly increase the observational arc (see also Hu et al., 2023) may
lead to erroneous detections of the Yarkovsky effect. To treat old astrometry,
we adopted data weights at 10 arcsec for measurements taken before 1890 and at
5 arcsec for those taken before 1950, so that very old observations were less
significant in the orbit determination process.

To ensure that a detection with an S/N higher than 3 is reliable and does not
depend on short isolated tracklets, we adopted the following strategy. We
defined an isolated tracklet as a set of observations such that 1) they were
all taken from the same observatory, 2) the length of the arc was shorter than
15 days, and 3) they were separated by at least five years from the previous
and the following observation sets. When an isolated tracklet was found, we
removed it from the dataset of observations, and we repeated the process until
a dataset without isolated tracklets was obtained. Then, we performed the 7-D
OD with the dataset that did not contain isolated tracklets. When the S/N in
$A_{2}$ was lower than 3, the detection strongly depends on the old isolated
observations and may therefore not be reliable or possibly spurious. As a
conservative choice, the detection was therefore discarded. On the contrary,
if a signal with a large S/N in $A_{2}$ is found also with the smaller set of
observations and with a possibly shorter observational arc, the signal is most
likely true and due to dynamical effects rather than astrometric issues. In
this case the detection is considered valid.

Note that detections are more sensitive to isolated tracklets appearing at the
beginning of the observational arc, as they extend the arc length, however
they could be also affected by intra-arc isolated tracklets. In addition to
this, some isolated tracklets may be known to be reliable. For example, some
old observations may have been carefully remeasured from an experienced
observer who examined the detection and confirmed its authenticity and the
correctness of the astrometry (see Vokrouhlický et al., 2008, for an example),
or they may be produced by modern surveys using more advanced technologies. To
handle these cases, we introduce a list of exceptions to the rule described
above, where the isolated tracklet is always included in the orbital fit.
Regarding modern surveys, we assume that only tracklets from the Pan-STARRS
survey (IAU codes F51 and F52) are always included in the orbital fit, as a
conservative choice.

###  2.4 Automated Yarkovsky effect detection

To select candidates for Yarkovsky effect detection, we first discard all the
NEAs with uncertainty in semi-major axis $\sigma_{a}$ larger than $10^{-5}$
au, since their orbit is not known with a sufficient accuracy to attempt for
the detection. Here $\sigma_{a}$ refers to the uncertainty of the orbit
obtained from 6-dimensional orbit determination (6-D OD), estimated at the
weighted mean observational epoch. For the asteroids passing the first
criterion, we compute the distribution of the predicted maximum Yarkovsky
drift999Here we refer to the maximum because the obliquity is fixed at extreme
values of $0^{\circ}$ or $180^{\circ}$, that maximize the diurnal Yarkovsky
effect. $(\text{d}a/\text{d}t)_{\max}$ that each of them can be subjected to.
This is done with a Monte Carlo method by computing the expected Yarkovsky
drift with the model of Sec. 2.2.1, using 500 000 combinations of input
parameters chosen from the parameter distributions described in Sec. 2.2.2.
Then, we determine the 95-th percentile of the distribution of
$|(\text{d}a/\text{d}t)_{\max}|$, i.e. the value $Y_{M}$ such that

$P(|(\text{d}a/\text{d}t)_{\max}|<Y_{M})=0.95,$ (12)

where $P(\,\cdot\,)$ denotes the probability. Values of semi-major axis drifts
larger than $Y_{M}$ in absolute values are attained only in very unlikely
combinations of the input parameters. By denoting with $\Delta T$ the total
length of the observational arc of the asteroid, if the condition

$\sigma_{a}<\Delta T\times Y_{M}$ (13)

is fulfilled, the asteroid is considered a good candidate for Yarkovsky effect
detection, hence the value of $A_{2}$ is fitted through a 7-dimensional orbit
determination (7-D OD). If the signal-to-noise ratio
$\text{S}/\text{N}=|A_{2}/\sigma_{A_{2}}|$ is smaller than 3, then the
detection is considered not statistically significant, and therefore
discarded.

To understand whether a detection is spurious or if the value of $A_{2}$
obtained from orbit determination could be actually caused by the Yarkovsky
effect, we determine whether $A_{2}$ is statistically compatible with the
physical properties of the given object or not. To this purpose, we convert
$A_{2}$ and its uncertainty $\sigma_{A_{2}}$ to semi-major axis drift values
$(\text{d}a/\text{d}t)_{\text{fit}}$ and
$\sigma_{(\text{d}a/\text{d}t)_{\text{fit}}}$ respectively, by using Eq. (2).
Then, we check if the value $(\text{d}a/\text{d}t)_{\text{fit}}$ obtained from
orbit determination fulfills the condition

$|(\text{d}a/\text{d}t)_{\text{fit}}|-k\sigma_{(\text{d}a/\text{d}t)_{\text{fit}}}<Y_{M},$
(14)

where $k=1.645$ gives the 5-th percentile of the measured Yarkovsky drift. If
inequality of Eq. (14) is fulfilled, it means that semi-major axis drift
values down to
$|(\text{d}a/\text{d}t)_{\text{fit}}|-k\sigma_{(\text{d}a/\text{d}t)_{\text{fit}}}$
are compatible with the predictions obtained with the Yarkovsky model. On the
other hand, if inequality of Eq. (14) is not valid, the value detected from
astrometry is not compatible with the Yarkovsky physical model, the detection
is considered spurious and therefore discarded. Note that the criterion of Eq.
(14) is adopted because the distribution of the expected maximum Yarkovsky
drift is not Gaussian in general. Finally, for those detections fulfilling Eq.
(14) we check the dependence on isolated tracklets by using the method
described in Sec. 2.3, and keep only those detections passing the test.

##  3 Results and discussion

###  3.1 Accepted detections

We ran the above procedure on the NEOCC list of known NEAs on 7 August 2023,
soon after the last monthly orbital update issued by the Minor Planet
Center101010https://www.minorplanetcenter.net/ (MPC). Optical observations are
downloaded from the MPC, while radar observations are taken from the JPL Solar
System Dynamics website111111https://ssd.jpl.nasa.gov/sb/radar.html. Out of 32
396 NEAs, 5 308 were selected as candidates for a possible Yarkovsky effect
detection. After the orbit determination process, we found 461 NEAs with
$\text{S}/\text{N}\geq 3$, and 373 of them fulfilled also the condition of Eq.
(14). Among them, 69 objects presented isolated tracklets, and 25 were found
to be spurious detections. In conclusion, we found 348 positive Yarkovsky
effect detections. A short list of detections with high S/N is shown in Table
20, while the complete list of accepted detections in extended precision can
be found at the CDS. Figure 2 shows the distribution of the RMS error of the
normalized observational residuals of the accepted detections, obtained with
7-D OD (blue histogram). As a comparison, the red histogram shows the
distribution of the RMS of normalized residuals of the same NEAs obtained with
the 6-D OD. The median value decreases from 0.616 for the 6-D OD, to 0.548 for
the 7-D OD. The distribution obtained with 7-D OD is skewed towards smaller
values if compared to that obtained with 6-D OD, meaning that including the
Yarkovsky effect in the orbital fit of these NEAs generally improves the
quality of the fit of the accepted detections.

Figure 2: Distribution of the RMS of the normalized residuals for the NEAs
with a positive Yarkovsy effect detection. The blue (red) histogram represents
the RMS of the normalized residuals obtained with (without) the Yarkovsky
effect included in the dynamical model. The dashed vertical lines correspond
to the median of the two distributions, where the numerical value is reported
in the legend.

Figure 3 shows the distribution of the S/N of the accepted Yarkovsky effect
detections. About 110 objects have S/N between 3 and 4, and the presence of
radar observations generally improves the S/N, even though they are not
necessarily needed to obtain good detections. The asteroids with
$\text{S}/\text{N}>40$ are (101955) Bennu, 1998 SD9, (99942) Apophis, (480883)
2001 YE4, (524522) 2002 VE68, (2340) Hathor, and 2002 BF25, which are not
shown in Fig. 3.

Figure 3: Distribution of the S/N of the accepted Yarkovsky effect detections,
obtained with equi-spaced bins of width equal to 1. NEAs with only optical
observations are represented in the blue histogram, while the red histogram
refers to NEAs observed also with radar.

The Yarkovsky effect determinations can be used to extrapolate the ratio of
retrograde-to-prograde rotators (R/P), because negative (positive) semi-major
axis drift values are associated with retrograde (prograde) rotators. To this
purpose, we randomly generate 10 000 values of $A_{2}$, for each positive
detections. In this process, we assumed $A_{2}$ to be a Gaussian random
variable, with mean value equal to the nominal value and standard deviation
equal to the 1-$\sigma$ uncertainty of the detection. With this method, we
obtained $\text{R}/\text{P}=2.976$. Spin pole determinations of NEAs show that
$\text{R}/\text{P}=2^{+1}_{-0.7}$ (La Spina et al., 2004), in good agreement
from what NEA distribution models predict. Although the ratio R/P we found is
still consistent with spin pole determinations, the discrepancy with the
distribution models can possibly be explained by the biases in the orbital
distribution of NEAs with a measured Yarkovsky effect (Farnocchia et al.,
2013). Another possible cause contributing to this discrepancy might be that
the Yarkovsky detections are driven by cases with extreme obliquity, which may
have a larger R/P ratio than the general NEA population. However, it is worth
noting that the percentage of retrograde rotators found here is about 75%,
which is lower than the 81% found in (Farnocchia et al., 2013), and closer to
the 69% value expected in the NEA population. This is an indication that the
observation capabilities improved in the last 10 years, and the new generation
surveys may help in making the sample of NEAs with measured Yarkovsky effect
representative of the whole debiased NEA population. Note that a value of
$\text{R}/\text{P}=2^{+0.8}_{-0.7}$ consistent with observations was
previously found in Tardioli et al. (2017), by fitting the NEA obliquity
distribution to the Yarkovsky effect determinations.

Figure 4 shows the measured semi-major axis drifts
$|(\text{d}a/\text{d}t)_{\text{fit}}|$ as a function of the diameter $D$.
Error bars in the diameter were obtained by computing the 15-th and the 85-th
percentile of the diameter distribution, using the model described in Sec.
2.2.2. Analytical models of the Yarkovsky effect predict that
$\text{d}a/\text{d}t$ scales with the diameter as $D^{-1}$. To verify that
spurious detections are correctly discarded by our procedure, we fit the data
about accepted detections with a function of the form $\alpha\times D^{\beta}$
using the Orthogonal Distance Regression (ODR; Boggs et al., 1987) method. The
best fit values computed by the ODR algorithm are $\alpha=(31.65\pm
2.42)\times 10^{-5}$ and $\beta=-0.97\pm 0.03$. Note that the exponent $\beta$
is statistically compatible with the value of $-1$ predicted by Yarkovsky
analytical models within 1-$\sigma$. The power law function computed with the
nominal parameters is also shown in Fig. 4. Detections with
$\text{S}/\text{N}\geq 3$ that are considered spurious are also shown in Fig.
4 with red points, and most of them are far off the best-fit power law
computed from the accepted detections. This already shows that our method is
able to appropriately find spurious detections.

Figure 4: Distribution of the accepted Yarkovsky effect detections in the
plane $(D,|(\text{d}a/\text{d}t)_{\text{fit}}|)$. Green points are NEAs for
which an estimate of the diameter is available, while black points are those
for which the physical model is used. The error bars in the diameter are
represented by using the 15-th and 85-th percentile of the diameter
distribution (see Sec. 2.2.2), which roughly correspond to the 1-$\sigma$
uncertainty in the case the diameter is estimated. The error bars in the semi-
major axis drift represent the 1-$\sigma$ uncertainty obtained by the
conversion of $A_{2}$, that has been estimated with orbit determination. The
dashed blue line represents the fit with a function of the form $\alpha\times
D^{\beta}$. Red points are detections with $\text{S}/\text{N}\geq 3$ that are
considered spurious, hence rejected by our procedure.

The dispersion from the best fit power law of Fig. 4 is explained by the
differences in the physical properties of asteroids, that are able to slightly
modify the magnitude of the semi-major axis drift. To verify this we retrieved
taxonomic data from the SsODNet database about the accepted Yarkovsky effect
detections, finding 116 NEAs with such information. A histogram showing the
number of NEAs with measured Yarkovsky effect for each taxonomic complex is
shown in Fig. 5. As expected, the vast majority of the detections belong to
the S-complex, however almost all the complexes (see Table 19) are represented
in the Yarkovsky measurement sample. These facts indicate that our procedure
is able to discriminate between positive and negative Yarkovsky effect
detections from a statistical point of view.

Figure 5: Taxonomy distribution of the accepted Yarkovsky effect detections.
Data about the taxonomic complex were retrieved from the SsODNet database
Berthier et al. (2023).

###  3.2 Notable rejections

To further show the effectiveness of the algorithm introduced here, we
highlight some notable cases among the identified spurious detections. In
addition to the Yarkovsky effect, other nongravitational forces such as SRP
can influence the dynamics of very small bodies. So far, SRP has been detected
through orbit determination only on a short list of NEAs: 2009 BD (Micheli et
al., 2012), 2011 MD (Micheli et al., 2014), 2012 LA (Micheli et al., 2013),
2012 TC4, and 2015 TC25 (Del Vigna et al., 2018). All these objects do not
appear in our list of accepted Yarkovsky effect detections, because the semi-
major axis drift obtained by the 7-D OD is not compatible with what expected
from the physical model. This issue is actually fixed when SRP is added to the
dynamical model, and the area-to-mass ratio $A/m$ of the object is added to
the list of parameters to be estimated in the orbital fit.

A similar result has been found for 2021 GM1, an NEA that is currently in the
NEOCC Risk List121212https://neo.ssa.esa.int/risk-list. The Yarkovsky effect
detection of this object is rejected by our procedure, but a nongravitational
effect is needed to better fit the observations. Adding SRP to the dynamical
model, we fitted an area-to-mass ratio of $0.2564\pm 0.00257$ kg2 t-1 (see
Table 14 for the complete orbital solution), providing a new SRP detection
that was not available in the literature131313We note that this detection was
however already present in the JPL SBDB since May 2021.. This shows that the
procedure may also be helpful in finding asteroid candidates for detection of
other nongravitational effects.

Table 3: Orbital elements of 2021 GM1.

Six-parameter solution |  | Unit
---|---|---
Semi-major axis $(a)$ | $0.968020575\pm 1.43\times 10^{-8}$ | au
Eccentricity $(e)$ | $0.041997691\pm 1.58\times 10^{-8}$ | /
Inclination $(i)$ | $1.598736270\pm 4.09\times 10^{-7}$ | deg
Long. of Asc. Node $(\Omega)$ | $193.321099298\pm 1.95\times 10^{-6}$ | deg
Argument of Perihelion $(\omega)$ | $207.383938594\pm 8.52\times 10^{-6}$ | deg
Mean anomaly $(M)$ | $162.303893236\pm 1.00\times 10^{-5}$ | deg
Normalized residuals RMS | $0.776$ | /
Seven-parameter solution |  |
Semi-major axis $(a)$ | $0.968020505\pm 1.38\times 10^{-8}$ | au
Eccentricity $(e)$ | $0.041997755\pm 1.49\times 10^{-8}$ | /
Inclination $(i)$ | $1.598740133\pm 5.18\times 10^{-7}$ | deg
Long. of Asc. Node $(\Omega)$ | $193.321107921\pm 2.15\times 10^{-6}$ | deg
Argument of Perihelion $(\omega)$ | $207.383858669\pm 1.14\times 10^{-5}$ | deg
Mean anomaly $(M)$ | $162.303967843\pm 1.23\times 10^{-5}$ | deg
Area-to-mass $(A/m)$ | $0.2564\pm 0.0257$ |  m2 t-1
Normalized residuals RMS | $0.622$ | /
141414Orbital elements refer to date 2021$-$04$-$14.660191 (MJD 59318.660191
TDT). Two solutions are shown: one determined with 6-D OD, and another one
with 7-OD OD using SRP as additional force.

Other interesting cases are 2006 RH120 (Seligman et al., 2023) and (523599)
2003 RM (Farnocchia et al., 2023). In addition to the radial component $A_{1}$
and the tangential component $A_{2}$, it is possible to also fit a
nongravitational out-of-plane force component $A_{3}$ for these two NEAs,
showing a dynamical behavior similar to that of comets. In these two cases,
the value of $A_{2}$ obtained by a 7-D OD only is not compatible with the
Yarkovsky physical model (see also Table 22), while it is likely due to
outgassing (Chesley et al., 2016; Taylor et al., 2024). Still, these
detections are recognized as spurious by our procedure.

Another remarkable detection that was flagged as spurious is (433) Eros. This
NEA was selected as a candidate for Yarkovsky effect detection because of its
long observational arc of more than 100 years, and $A_{2}$ was fitted with a
S/N of about 5. However, the detection was not statistically compatible with
the prediction made by the Yarkovsky physical model, and therefore the
detection is spurious. This detection needs to be necessarily discarded,
because bad quality astrometric measurements performed before the 1950 cause
an artificial nongravitational acceleration that is actually not expected. In
fact, (433) Eros is known to have an obliquity of almost $90^{\circ}$ (Yeomans
et al., 2000), that minimizes the Yarvkosky effect, but the procedure is able
to discard the detection even by assuming an obliquity of $0^{\circ}$ or
$180^{\circ}$ (see Sec. 2.2.2). All these considerations provide evidences of
the effectiveness of the algorithm introduced here, in both the selection of
possible candidates for the Yarkovsky effect determination, and in the
identification of spurious detections.

The algorithm found 69 $A_{2}$ detections with S/N larger than 3 and
fulfilling Eq. (14) that were affected by the presence of isolated tracklets
in their dataset. Among them, 58 presented isolated tracklets at the beginning
of the observational arc, and the complete list of these asteroids is reported
in Table 23. The table reports also the S/N of $A_{2}$ obtained without
including the isolated tracklet in the orbital fit, following the procedure
described in Sec. 2.3, and 30 of them still present a signal in $A_{2}$ even
with the shorter observational arc. These objects were therefore accepted as
positive Yarkovsky effect detections. Regarding the list of exceptions for
which we know that astrometry is reliable, up to now it includes: 2012 BB124
and 2014 RO17, for which accurate measurements were performed by MM; 2015 LA2,
2018 KR, and 2016 UW80, that present isolated tracklets from F51; (152563)
1992 BF for which it is known that the tracklet has been accurately remeasured
(Del Vigna et al., 2018). While this list may increase in the future, it is
not expected to grow to a point that can not be maintained manually. Other 11
objects presented intra-arc isolated tracklets, but the $A_{2}$ detection was
affected only for 2005 TG50 and 2011 XC, that were therefore considered as
spurious. In addition, the S/N drops down also for 2009 OS5 when observations
from 2014 from Mauna Kea Observatory (IAU code 568) are excluded in the fit,
however we considered them as reliable151515These observations have program
code equal to 2, corresponding to D. Tholen. and we therefore accepted this
detection. Finally, among the detections that were not accepted, we note that
the algorithm flagged (469219) Kamo’oalewa as spurious, in agreement with Hu
et al. (2023).

###  3.3 Publication on the NEOCC portal

The availability of a large number of reliable Yarkovsky effect detections
opens possible future developments in other fields of asteroid research, such
as physical properties determination (Rozitis & Green, 2014; Fenucci et al.,
2021, 2023), long term dynamical studies (Fenucci & Novaković, 2021), or
population studies on NEAs (Tardioli et al., 2017). To this end, it is
important to continuously provide up-to-date data to the scientific community.
The procedure described here has been now adopted by the ESA NEOCC to
periodically update the Yarkovsky effect detections on NEAs, that are
automatically posted on the NEOCC web portal161616https://neo.ssa.esa.int/ and
available through the HTTPS APIs171717https://neo.ssa.esa.int/computer-access.
The catalogues of observations and of known NEAs are continuously growing, and
the algorithm is scheduled to run after every monthly orbital update issued by
the MPC, so that old determinations are updated and possible new detections
are added to the database.

###  3.4 Comparison with the JPL SBDB

We downloaded the list of determinations of $A_{2}$ from the JPL Small Body
Database181818https://ssd.jpl.nasa.gov/ (SBDB) maintained by the JPL SSD group
on 8 August 2023. A total of 333 measurements, 308 of which with
$\text{S}/\text{N}\geq 3$, were found. Differences in the number of detections
with a good S/N are possibly due to different debiasing and weighting schemes
of observations, different observational datasets and rejections, and timing
errors handling. These differences become particularly significant when the
S/N is close to the threshold of 3. Among the JPL detections, 13 objects have
also a determination of the component $A_{1}$, and 6 objects have a
determination of both $A_{1}$ and $A_{3}$. We did not make the comparison for
these objects, because the results are obtained with different dynamical
models. Of the remaining objects, 270 were found in our list of accepted
Yarkovsky effect measurements. Figure 6 shows the scatter plot of the common
$A_{2}$ determinations, together with their 1-$\sigma$ uncertainty. We fitted
the data with a linear function of the form $\alpha|A_{2}|+\beta$ with the ODR
algorithm, that gave an estimation of the parameters of $\alpha=1.00\pm
0.0001$ and $\beta=(-1.90\pm 0.97)\times 10^{-16}$, meaning that our
determinations and those of the JPL SBDB are statistically compatible, and no
clear outlier can be recognized. This is certainly due to the fact that the
orbit determination algorithms and the datasets used by the JPL SSD group are
similar to those used here for these cases.

Figure 6: Distribution of the common $A_{2}$ measurements found in this work
and at the JPL SBDB. The blue dashed line is the diagonal.

To make a quantitative comparison of the results for each asteroid, we
computed the relative errors

$\varepsilon_{1}=\frac{|A_{2}-A_{2}^{\text{JPL}}|}{\sigma_{A_{2}}},\quad\varepsilon_{2}=\frac{|A_{2}-A_{2}^{\text{JPL}}|}{\sigma_{A^{\text{JPL}}_{2}}},$
(15)

where the superscript JPL denotes the value provided by the JPL SBDB. For the
vast majority of the asteroids in common, the relative errors
$\varepsilon_{1},\varepsilon_{2}$ were both smaller than 3, indicating that
the results provided by us and by the JPL SBDB are all statistically
comparable, even case by case. The only cases in which the relative errors are
larger than 3 are (65717) 1993 BX3, 2003 AF23, (1685) Toro. However, also in
these cases the values of $\varepsilon_{1}$ and $\varepsilon_{2}$ are only
slightly larger than 3. We also performed a test by checking that

$\chi=\frac{|A_{2}-A_{2}^{\text{JPL}}|}{\sqrt{(\sigma_{A_{2}})^{2}+(\sigma_{A^{\text{JPL}}_{2}})^{2}}}<2,$
(16)

meaning that the hypothesis that the two distributions are not the same is
rejected at 95% confidence level. For all the common asteroids with a
Yarkovsky effect measurement we found $\chi<2$, with the only exceptions of
2003 AF23 and (65717) 1993 BX3, that have $\chi=2.23$ and $\chi=2.18$,
respectively. However, the two measurements are still considered the same if
the test is verified with a more relaxed confidence level of 90%. In addition,
we found that the vast majority of the measurements have $\chi<1$, implying an
excellent agreement between our results are those reported in the JPL SBDB.
Table 21 reports the numerical values of the relative errors $\varepsilon_{1}$
and $\varepsilon_{2}$, and of $\chi$ for all the NEAs with $\chi>1$. The full
table in extended precision, used to generate Fig. 6, is available through the
CDS.

Asteroids with a determination of $A_{2}$ that are reported in the JPL SBDB,
but are either discarded or not found by our procedure, are 54. Among them, 44
have $\textrm{S}/\textrm{N}\geq 3$ and their list is shown in Table 22, while
the complete list in extended precision is available through the CDS. Notable
cases are those of (65803) Didymos, (3200) Phaethon, and (25143) Itokawa.

For asteroid (65803) Didymos the JPL SBDB reports an orbital solution with S/N
$=16$ in the $A_{2}$ parameter, while the value we obtained here is only of
2.57. The reasons for this discrepancy are probably due to the availability of
some occultation observations took during the September 2022 campaign (Dunham
et al., 2023), the availability of data from the Double Asteroid Redirection
Test (DART, Cheng et al., 2018) mission, and the inclusion of Gaia astrometry
in the orbit computation. These high accuracy observations are not reported by
the MPC, and therefore they were not considered in our data set.

For (3200) Phaethon the JPL SBDB reports an orbital fit with a S/N in $A_{2}$
of 8.3. The Yarkovsky effect measurement for this object has been reported by
Hanuš et al. (2018), where the authors included also very precise Gaia
astrometric observations taken between 2014 and 2016 in the dataset. After the
publication of that work, occultation observations taken on 25 October 2019
permitted to further improve the orbit of (3200) Phaethon (Dunham et al.,
2021). However, both the Gaia astrometry and the occultations are currently
not reported by the MPC, and we believe this being the reason we are not able
to fit the $A_{2}$ parameter with such high accuracy.

The last case is that of (25143) Itokawa, for which the JPL SBDB gives an
orbital solution with S/N of 3.2 in the $A_{2}$ parameter. In this case, it is
not easy to identify what may cause this discrepancy in the results, and the
only noticeable difference that we could find is in the number of observations
used in the orbital fit. For other objects, we also obtain a S/N larger than 3
as for the JPL SBDB, but they are discarded because the drift is too large to
be explained by the Yarkovsky effect, and therefore Eq. (14) does not hold.

###  3.5 Limitations of the procedure

A possible limitation of the procedure may be due to the fact that the
Yarkovsky effect modeling takes into account the orbital elements of the NEA
at a certain epoch to compute the expected drift $\text{d}a/\text{d}t$.
However, this may have consequences if an object has a close encounter with a
planet that is able to change its orbital elements. While this is certainly
true, the changes induced in the 95-th percentile $Y_{M}$ are negligible
because the distribution of the expected semi-major axis drift is mostly
controlled by the uncertainties in the physical properties. To show this, we
took into account the case of (99942) Apophis, for which we have an estimate
of the diameter, a measurement of the rotation period, and we know that the
spectral type is S. Therefore, the physical model of (99942) Apophis has
certainly more constraints than the generic NEA. To show that changes in the
orbital elements have a little influence on the results, we computed $Y_{M}$
with the orbital elements computed at the mean epoch, and with those obtained
by propagating the nominal orbit to an epoch right after the close approach of
13 April 2029, that changes the semi-major axis by about 0.18 au. The value of
$Y_{M}$ went from $22.45\times 10^{-4}$ au My-1 obtained before the close
approach, to $22.90\times 10^{-4}$ au My-1 computed after the close approach.
For the generic NEA we expect the change to be even smaller, because the
distribution of the expected maximum semi-major axis drift is dominated by the
uncertainties in the physical properties, rather than by the uncertainties in
the orbital elements.

From the orbit determination point of view, an effect that is not taken into
account in the model described in Sec. 2.1 is that of timing errors in the
astrometric observations. It has been shown that the astrometric errors are
affected by time calibration errors made by observers. These calibration
errors can be either random or systematic, and they are especially important
when an object has a fast apparent motion (Farnocchia et al., 2022). In turn,
this may result in an erroneous estimation of the $A_{2}$ parameter, thus
affecting the Yarkovsky effect detection. In this respect, the Aegis software
used at the NEOCC is already able to treat timing errors by correcting the
weights of the observations using the correlation with the right ascension and
declination, with the method introduced in Farnocchia et al. (2022). However,
the information about the timing errors is communicated only in the new
Astrometry Data Exchange Standard (ADES, Chesley et al., 2017), which is not
currently used to automatically compute asteroid orbits at the NEOCC (Faggioli
et al., 2023). The switch of the Aegis system to automatically work with
observations in ADES format is planned for the future, and the procedure
described here will be able to automatically take into account also timing
errors in the orbit determination.

Finally, the detection of a nongravitational force $A_{2}$ along the
transverse direction does not necessarily mean that the Yarkovsky effect is
responsible for it. In fact, other dynamical effects, such as outgassing or
mass-shedding, may produce a transversal acceleration (Farnocchia et al.,
2023). The usage of a statistical model for the physical properties of NEAs
seems to help identifying these outliers, as confirmed by the cases of
(523599) 2003 RM and 2006 RH120 that were flagged as spurious. Still, there
may be cases in which the acceleration produced by such phenomena is small
enough to result compatible with the acceleration produced by the Yarkovsky
effect. In these cases, the detection of such acceleration is real, and it is
not possible to make a clear distinction between the Yarkovsky effect and
activity events, unless there are proofs of such phenomena obtained from
detailed studies of the single object.

##  4 Conclusions

In this paper we introduced a procedure for the automated Yarkovsky effect
detection on NEAs and for the identification of spurious detections. The
procedure makes use of a statistical model to estimate the physical properties
of an NEA, based on the most recent data and models about the NEA population.
Physical properties are then used to estimate the expected maximum Yarkovsky
semi-major axis drift. Candidates for detection are selected according to the
current precision in the knowledge of the orbit, the observational arc, and
the expected semi-major axis drift. Then, a 7-dimensional orbit determination
of the candidates, that includes the estimation of a transversal
nongravitational component, is performed. If the S/N of the $A_{2}$ detection
is larger than 3, a statistical test to check the compatibility with the
Yarkovsky effect physical model is carried out. Finally, the dependence of the
detection on isolated tracklets is analyzed.

Among the known NEAs, we identified 348 positive Yarkovsky effect detections.
This procedure is now adopted by the ESA NEO Coordination Centre, and data are
automatically uploaded on the web portal and updated at every monthly orbital
update issued by the MPC. A comparison with the results obtained by the JPL
SSD is also shown, and the measurements are generally in a very good
agreement, being based on similar orbit determination methods and datasets.

###### Acknowledgements.

We thank Davide Farnocchia for the constructive comments on an early version
of the manuscript, that permitted us to improve the quality of the work.

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##  Appendix A Densities of asteroid complexes

To determine values of the densities of asteroid complexes, we extracted data
from the SsODNet database using the best value provided by the ssoCard. We
selected only asteroids for which the taxonomic complex and the density were
both determined, and we found a total of 386 objects. To determine the density
distribution of a complex, we used the following procedure: 1) the density
distribution of each asteroid is assumed to be log-normal; 2) a density sample
for all the asteroids belonging to a given complex was randomly generated,
obtaining a sample for the whole complex; 3) the sample for the whole complex
was fitted with a log-normal distribution.

Table 4: Taxonomic complexes included in the SsODNet database.

Tax. | Tot. | # S/N$>$3 | #S/N$>$2 | #S/N$>$1.5 |  $\mu$ (kg m-3) |  $\sigma$ (kg m-3)
---|---|---|---|---|---|---
A | 2 | 2 | 2 | 2 | N/A | N/A
B | 5 | 3 | 4 | 4 | 1290 | 181
C | 78 | 30 | 44 | 72 | 1757 | 1088
Ch | 106 | 33 | 53 | 88 | 1739 | 1054
D | 4 | 1 | 1 | 1 | N/A | N/A
E | 5 | 3 | 3 | 3 | 2560 | 270
K | 3 | 1 | 2 | 2 | N/A | N/A
L | 3 | 0 | 0 | 2 | 2674 | 1551
M | 27 | 10 | 12 | 19 | 3614 | 1113
P | 54 | 14 | 20 | 40 | N/A | N/A
Q | 3 | 3 | 3 | 3 | 1915 | 932
R | 1 | 0 | 0 | 1 | N/A | N/A
S | 80 | 37 | 44 | 68 | 2721 | 1273
U | 1 | 1 | 1 | 1 | N/A | N/A
V | 5 | 4 | 4 | 5 | 1930 | 1000
X | 5 | 1 | 1 | 4 | 1427 | 1494
Z | 4 | 0 | 1 | 3 | 2225 | 1804
191919Columns report the name of the complex, the number of asteroids in
SsODNet for which the density was determined, the number of determinations
with S/N larger than 3, 2, and 1.5, and the values of the density assumed in
this work. The values of $\mu$ and $\sigma$ for the log-normal distribution
were obtained by fitting the available data. The flag N/A indicates the
complexes for which the data were not enough to obtain a reasonable fit.

Figure 7: Fit of the density distribution for the B$-$, C$-$, Ch$-$, E$-$,
L$-$, M$-$, Q$-$, S$-$, X$-$, and Z$-$complexes. The histogram is obtained by
generating log-normal distributions of asteroid density determinations of the
corresponding complex, while the red curve is the best-fit log-normal
distribution. Numerical values of the parameters $\mu$ and $\sigma$ of the
best-fit distribution are shown in the title of each panel.

Table 19 shows the list of asteroid complexes found in the SsODNet, together
with the total number of density determinations, and the number of
determinations with signal-to-noise ratio (S/N) larger than 3, 2, and 1.5. The
sixth and seventh columns contain the value adopted in this work for each
complex. The values for the C$-$, Ch$-$, M$-$, Q$-$, and S$-$complexes were
fitted by using only the data with $\text{S}/\text{N}>3$. The values for the
L$-$, X$-$, and Z$-$complexes were obtained from the measurements with
$\text{S}/\text{N}>1.5$. The B$-$complex showed two outliers with relatively
high density (i.e. (2) Pallas and (426) Hippo), that we removed from the
sample used for the density fit. The E$-$complex also has a similar issue,
because asteroids (44) Nysa and (3671) Dionysus show a density that is
considered too small for these asteroid types, and therefore we removed them
from the final fit. For the A$-$, P$-$, R$-$, K$-$, and U$-$complexes it was
not possible to reasonably fit the sample with a log-normal distribution,
therefore we used the population model mentioned in Sec. 2.2.2. Finally, the
densities determined for asteroids in the V$-$complex show a large variety of
values, therefore we adopted the value reported by Carry (2012). The log-
normal fits obtained for the with the method described above for the B$-$,
C$-$, Ch$-$, E$-$, L$-$, M$-$, Q$-$, S$-$, X$-$, and Z$-$complexes are all
shown in Fig. 7

##  Appendix B Additional tables

Table 5: Results of the automated Yarkovsky effect determination procedure.

Ast. | $H$ | RMS |  RMS6 | $A_{2}$ | $\sigma(A_{2})$ | $\text{d}a/d\text{t}$ | $\sigma(\text{d}a/d\text{t})$ | $Y_{M}$ | S/N | Res. | #OO | #ReO |  #ReO6 | #RaO | #ReRa |  #ReRa6 | $D_{15}$ | $D_{50}$ | $D_{85}$ | $P$ | Tax | $\Delta T$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
101955 | 20.6 | 0.40 | 0.45 |  $-$4.52 | 0.023 |  $-$18.89 | 0.009 | 23.20 | 191 | Acc. | 580 | 8 | 8 | 29 | 0 | 3 | 469 | 488 | 507 | 4.29 | B | 18.6
1998 SD9 | 23.9 | 0.78 | 6.84 |  $-$28.98 | 0.156 |  $-$197.06 | 1.063 | 317.9 | 185 | Acc. | 149 | 4 | 5 | 1 | 0 | 0 | 25 | 35 | 75 | / | / | 19.9
99942 | 18.9 | 0.30 | 0.33 |  $-$2.88 | 0.023 |  $-$13.25 | 0.107 | 28.10 | 122 | Acc. | 9523 | 18 | 20 | 50 | 0 | 5 | 300 | 340 | 380 | 27.38 | S | 17.1
480883 | 20.9 | 0.47 | 2.48 |  $-$6.91 | 0.073 |  $-$50.44 | 0.533 | 116.11 | 94 | Acc. | 430 | 1 | 5 | 7 | 0 | 2 | 103 | 172 | 287 | / | V | 20.0
2340 | 20.4 | 0.72 | 0.75 |  $-$2.99 | 0.052 |  $-$17.38 | 0.304 | 80.28 | 57 | Acc. | 1369 | 35 | 78 | 7 | 0 | 0 | 134 | 193 | 312 | 3.35 | Q | 45.1
524522 | 20.5 | 0.46 | 1.09 |  $-$5.98 | 0.105 |  $-$35.91 | 0.630 | 42.57 | 56 | Acc. | 606 | 8 | 21 | 10 | 0 | 7 | 112 | 176 | 337 | 13.43 | / | 16.0
2002 BF25 | 22.4 | 0.45 | 0.88 | 16.67 | 0.295 | 71.90 | 0.0001 | 75.61 | 56 | Acc. | 143 | 0 | 48 | 4 | 0 | 0 | 149 | 152 | 155 | / | / | 20.0
2012 UR158 | 20.6 | 0.57 | 0.63 |  $-$771.30 | 17.221 |  $-$8147.57 | 181.918 | 63.39 | 44 | Rej. | 161 | 1 | 15 | 0 | 0 | 0 | 320 | 629 | 938 | / | / | 10.0
523599 | 19.7 | 0.46 | 1.12 | 212.06 | 5.103 | 829.24 | 19.957 | 21.83 | 41 | Rej. | 329 | 4 | 51 | 0 | 0 | 0 | 66 | 200 | 602 | / | / | 15.2
2000 EB14 | 23.2 | 0.58 | 1.04 |  $-$27.00 | 0.685 |  $-$160.60 | 4.0749 | 214.11 | 39 | Acc. | 83 | 2 | 27 | 0 | 0 | 0 | 38 | 55 | 97 | 1.79 | / | 22.0
483656 | 23.7 | 0.60 | 1.25 |  $-$14.01 | 0.383 |  $-$80.064 | 2.1925 | 267.66 | 36 | Acc. | 207 | 5 | 41 | 1 | 0 | 0 | 28 | 40 | 84 | 1.69 | / | 18.0
2009 BD | 28.2 | 0.44 | 0.96 |  $-$157.65 | 4.955 |  $-$667.46 | 20.979 | 285.39 | 31 | Rej. | 197 | 0 | 1 | 0 | 0 | 0 | 2 | 4 | 8 | / | / | 2.4
202020 Columns are: Ast. = Designation of the asteroid; $H$ = Absolute
magnitude; RMS = RMS of normalized residuals for 7-D OD; RMS6 = RMS of
normalized residuals for 6-D OD; $A_{2}$ = Transversal nongravitational
component obtained with 7-D OD; $\sigma(A_{2})$ = Error in $A_{2}$;
$\text{d}a/d\text{t}$ = Semi-major axis drift obtained by converting $A_{2}$;
$\sigma(\text{d}a/d\text{t})$ = Error in the semi-major axis drift; $Y_{M}$ =
95-th percentile of the maximum expected Yarkovsky drift; S/N = Signal-to-
noise ratio of the detection; Res. = Acceptance flag, can be only accepted
(Acc.) or rejected (Rej.); #OO = Number of Optical Observations; #ReO = Number
of rejected Optical Observation in the 7-D OD; #ReO6 = Number of rejected
Optical Observation in the 6-D OD; #RaO = Number of Radar Observations; #ReRa
= Number of rejected Radar Observations in the 7-D OD; #ReRa6 = Number of
rejected Radar Observations in the 7-D OD; $D_{15}$ = 15-th percentile of the
diameter distribution; $D_{50}$ = 50-th percentile of the diameter
distribution; $D_{85}$ = 85-th percentile of the diameter distribution; $P$ =
Rotation period; Tax = Taxonomic class; $\Delta T$ = Length of the
observational arc. The full list of results in extended precision is available
at the CDS. The $A_{2}$ and $\sigma(A_{2})$ values are given in $10^{-14}$ au
d-2, while the values of the semi-major axis drifts
$\text{d}a/d\text{t},\sigma(\text{d}a/d\text{t})$ and $Y_{M}$ are given in
$10^{-4}$ au My-1. The diameter percentiles $D_{15},D_{50},$ and $D_{85}$ are
given in meters, the rotation period $P$ in hours, and the arc length $\Delta
T$ in years.

Table 6: Values of $\varepsilon_{1},\varepsilon_{2}$, and $\chi$ for the
asteroids with $\chi\geq 1$.

Ast. | $A_{2}$ | $\sigma_{A_{2}}$ | S/N | $A_{2}^{\text{JPL}}$ | $\sigma_{A_{2}}^{\text{JPL}}$ |  S/N${}^{\text{JPL}}$ | JPL Model | $\varepsilon_{1}$ | $\varepsilon_{2}$ | $\chi$
---|---|---|---|---|---|---|---|---|---|---
65717 | 4.33 | 0.99 | 4.3 | 7.74 | 1.04 | 7.4 | 1 | 3.42 | 3.26 | 2.36
2003 AF23 |  $-$4.44 | 0.32 | 13.7 |  $-$5.48 | 0.34 | 15.6 | 1 | 3.21 | 2.96 | 2.18
2011 PU1 |  $-$44.44 | 6.68 | 6.6 |  $-$30.30 | 4.99 | 6.0 | 1 | 2.11 | 2.83 | 1.69
2062 |  $-$1.40 | 0.16 | 8.3 |  $-$1.06 | 0.11 | 9.2 | 1 | 2.04 | 3.00 | 1.69
1998 KY26 |  $-$22.99 | 3.96 | 5.7 |  $-$13.77 | 5.66 | 2.4 | 3 | 2.32 | 1.62 | 1.33
4179 |  $-$0.61 | 0.06 | 8.8 |  $-$0.77 | 0.09 | 7.9 | 2 | 2.24 | 1.60 | 1.30
2018 CW2 |  $-$49.22 | 2.39 | 20.5 |  $-$55.01 | 3.85 | 14.2 | 1 | 2.41 | 1.50 | 1.27
1685 |  $-$0.44 | 0.10 | 4.3 |  $-$0.30 | 0.04 | 6.5 | 1 | 1.39 | 3.08 | 1.27
163000 |  $-$3.48 | 0.55 | 6.2 |  $-$2.46 | 0.60 | 4.0 | 1 | 1.84 | 1.70 | 1.25
85990 |  $-$3.15 | 0.21 | 14.4 |  $-$3.56 | 0.26 | 13.4 | 1 | 1.87 | 1.54 | 1.19
2018 GY |  $-$30.67 | 4.18 | 7.3 |  $-$39.49 | 6.32 | 6.2 | 1 | 2.10 | 1.39 | 1.16
306383 |  $-$2.50 | 0.44 | 5.6 |  $-$1.71 | 0.54 | 3.1 | 1 | 1.79 | 1.47 | 1.13
2010 VK139 |  $-$14.71 | 2.12 | 6.9 |  $-$9.81 | 3.80 | 2.5 | 1 | 2.30 | 1.28 | 1.12
363599 |  $-$6.37 | 0.49 | 12.8 |  $-$5.51 | 0.60 | 9.1 | 1 | 1.72 | 1.42 | 1.09
2005 VL1 |  $-$21.68 | 2.33 | 9.2 |  $-$83.20 | 56.12 | 1.4 | 3 | 26.36 | 1.09 | 1.09
524522 |  $-$5.98 | 0.10 | 56.9 |  $-$5.82 | 1.12 | 51.8 | 1 | 1.49 | 1.40 | 1.02
212121Columns are: Ast = Designation of the asteroid; $A_{2}$ = Transversal
nongravitational component estimated by NEOCC; $\sigma_{A_{2}}$ = Error in
$A_{2}$ estimated by NEOCC; S/N = Signal-to-noise ratio of the NEOCC
detection; $A_{2}^{\text{JPL}}$ = Transversal nongravitational component
estimated by JPL; $\sigma_{A_{2}}^{\text{JPL}}$ = Error in $A_{2}$ estimated
by JPL; S/N${}^{\text{JPL}}$ = Signal-to-noise ratio of the JPL detection; JPL
Model = Dynamical model used by JPL (1 = $A_{2}$; 2 = $A_{1}$ and $A_{2}$; 3 =
$A_{1},A_{2}$ and $A_{3}$); $\varepsilon_{1}$ = Relative error w.r.t. NEOCC;
$\varepsilon_{2}$ = Relative error w.r.t. JPL; $\chi$ = Chi value. The full
list in extended precision is available at the CDS. The $A_{2}$ and
$\sigma(A_{2})$ values are given in $10^{-14}$ au d-2.

Table 7: Asteroids with measured Yarkovsky effect for the JPL SBDB, not
identified or rejected by our procedure.

Ast. | $\text{d}a/d\text{t}$ | $\sigma(\text{d}a/d\text{t})$ | S/N | $Y_{M}$ | $\text{d}a/d\text{t}^{\text{JPL}}$ | $\sigma(\text{d}a/d\text{t}^{\text{JPL}})$ | S/N${}^{\text{JPL}}$ | JPL Model
---|---|---|---|---|---|---|---|---
523599 | 829.24 | 19.95 | 41.5 | 21.83 | 1292.37 | 22.09 | 58.4 | 1
66391 |  $-$3.12 | 2.25 | 1.384 | 13.53 |  $-$5.47 | 0.29 | 18.7 | 1
2006 CT |  $-$38.88 | 2.08 | 18.67 | 94.22 |  $-$38.51 | 2.09 | 18.39 | 1
65803 | 16.41 | 6.39 | 2.57 | 12.46 |  $-$4.04 | 0.25 | 16.0 | 1
2011 CL50 | 1949.59 | 138.59 | 14.0 | 340.71 | 2023.44 | 135.40 | 14.9 | 1
2009 BD |  $-$667.46 | 20.97 | 31.8 | 285.39 |  $-$473.66 | 34.58 | 13.6 | 2
2004 JN1 |  $-$164.31 | 11.51 | 14.2 | 123.80 |  $-$159.28 | 11.80 | 13.4 | 1
2001 BB16 | 153.56 | 11.37 | 13.4 | 133.76 | 150.48 | 12.25 | 12.2 | 1
613995 |  $-$95.23 | 8.76 | 10.8 | 63.79 |  $-$94.35 | 9.00 | 10.4 | 1
3200 |  $-$6.90 | 2.48 | 2.78 | 4.19 |  $-$12.11 | 1.44 | 8.3 | 1
2006 RH120 |  $-$54628.00 | 1438.9 | 37.96 | 1406.22 |  $-$21205.10 | 2664.47 | 7.9 | 3
2013 GZ7 |  $-$24.48 | 2.82 | 8.67 | 103.88 |  $-$23.11 | 2.97 | 7.78 | 1
2000 AA6 | 71.40 | 9.46 | 7.5 | 40.78 | 76.07 | 10.00 | 7.6 | 1
87024 | 9.41 | 5.83 | 1.61 | 27.04 | 9.33 | 1.25 | 7.4 | 1
2008 DB |  $-$588.54 | 65.53 | 8.9 | 222.81 |  $-$455.24 | 62.16 | 7.3 | 1
2012 UU136 | 40.45 | 6.00 | 6.73 | 60.55 | 40.58 | 6.09 | 6.65 | 1
2018 KC2 | 127.86 | 18.24 | 7.0 | 97.81 | 122.98 | 18.69 | 6.5 | 1
2009 EM1 |  $-$118.36 | 14.56 | 8.1 | 62.99 |  $-$122.27 | 18.62 | 6.5 | 1
162004 | 20.48 | 2.29 | 8.9 | 16.18 | 18.31 | 2.78 | 6.5 | 1
2018 HG2 |  $-$186.27 | 18.76 | 9.9 | 152.22 |  $-$186.85 | 29.04 | 6.4 | 1
2015 XC352 |  $-$323.04 | 49.74 | 6.4 | 218.19 |  $-$332.22 | 55.54 | 5.9 | 1
337248 |  $-$24.04 | 4.13 | 5.8 | 11.33 |  $-$23.73 | 4.64 | 5.1 | 1
2013 BA74 | 307.24 | 0.71 | 5.0 | 145.09 | 619.07 | 124.18 | 4.9 | 1
267759 |  $-$11.43 | 2.31 | 4.94 | 19.04 |  $-$10.99 | 2.31 | 4.74 | 1
491007 |  $-$42.92 | 9.04 | 4.74 | 38.79 |  $-$40.38 | 9.16 | 4.40 | 1
68950 |  $-$2.02 | 1.04 | 1.93 | 8.97 |  $-$2.99 | 0.68 | 4.3 | 1
397326 | 9.76 | 2.05 | 4.75 | 18.84 | 9.86 | 2.26 | 4.35 | 1
2012 TC4 |  $-$16.48 | 16.32 | 1.01 | 1165.75 |  $-$106.88 | 26.05 | 4.1 | 2
2004 FH | 50.16 | 31.56 | 1.58 | 374.62 | 85.37 | 21.15 | 4.0 | 1
65679 |  $-$18.50 | 4.38 | 4.22 | 17.96 |  $-$19.71 | 4.89 | 4.02 | 1
153201 |  $-$23.52 | 4.22 | 5.5 | 15.80 |  $-$18.59 | 4.69 | 3.9 | 1
5381 |  $-$0.18 | 2.94 | 0.060 | 12.23 | 2.70 | 0.69 | 3.9 | 1
7482 | 1.35 | 3.96 | 0.34 | 8.15 |  $-$2.22 | 0.61 | 3.6 | 1
526826 |  $-$56.99 | 19.35 | 2.94 | 74.76 |  $-$66.69 | 18.68 | 3.5 | 1
163899 |  $-$1.42 | 2.51 | 0.56 | 4.93 |  $-$6.94 | 2.01 | 3.4 | 1
5604 |  $-$12.27 | 3.90 | 3.14 | 16.03 |  $-$14.00 | 4.20 | 3.33 | 1
2003 MS2 |  $-$22.11 | 7.50 | 2.94 | 61.85 |  $-$27.63 | 8.36 | 3.3 | 1
2021 QH2 | 407.83 | 95.41 | 4.2 | 219.79 | 354.44 | 107.61 | 3.2 | 1
2016 GE1 |  $-$394.54 | 112.11 | 3.5 | 123.95 |  $-$582.52 | 177.24 | 3.2 | 1
25143 |  $-$7.84 | 4.86 | 1.61 | 24.69 |  $-$11.09 | 3.38 | 3.2 | 1
2014 QL390 |  $-$45.92 | 16.27 | 2.82 | 52.65 |  $-$51.76 | 15.99 | 3.2 | 1
138947 |  $-$13.20 | 4.61 | 2.86 | 21.90 |  $-$18.10 | 5.67 | 3.1 | 1
369984 |  $-$52.58 | 26.60 | 1.97 | 23.11 |  $-$11.65 | 3.68 | 3.1 | 1
22753 |  $-$8.69 | 3.345 | 2.59 | 19.28 |  $-$10.35 | 3.28 | 3.1 | 1
483423 |  $-$51.54 | 18.22 | 2.82 | 88.76 |  $-$58.96 | 18.97 | 3.1 | 1
396593 | 12.29 | 4.56 | 2.69 | 29.31 | 14.89 | 4.83 | 3.0 | 1
417210 |  $-$15.02 | 5.23 | 2.87 | 54.46 |  $-$16.46 | 5.34 | 3.0 | 1
22099 | 3.19 | 2.27 | 1.40 | 16.45 | 5.02 | 1.65 | 3.0 | 1

222222Columns are: Ast = Designation of the asteroid; $\text{d}a/d\text{t}$ =
Semi-major axis drift computed by the NEOCC; $\sigma(\text{d}a/d\text{t})$ =
Uncertainty in the semi-major axis drift computed by the NEOCC; S/N = Signal-
to-noise ratio of semi-major axis drift detection of NEOCC measurement;
$Y_{M}$ = 95-th percentile of the expected maximum Yarkovsky semi-major axis
drift; $\text{d}a/d\text{t}^{\text{JPL}}$ = Semi-major axis drift computed by
JPL; $\sigma(\text{d}a/d\text{t}^{\text{JPL}})$ = Uncertainty in the semi-
major axis drift computed by JPL; S/N${}^{\text{JPL}}$ = Signal-to-noise ratio
of semi-major axis drift detection of JPL measurement; JPL Model = Dynamical
model used by JPL (1 = $A_{2}$; 2 = $A_{1}$ and $A_{2}$; 3 = $A_{1},A_{2}$ and
$A_{3}$). The full list in extended precision is available at the CDS. The
values of $\text{d}a/d\text{t},\sigma(\text{d}a/d\text{t})$ and $Y_{M}$ are
given in $10^{-4}$ au My-1.

Table 8: Asteroids with measured Yarkovsky drift with isolated tracklets at
the beginning of the observational arc.

Ast. | Tracklet Initial Date | Obs. Code | Arc length | $\Delta t$ | $A_{2}$ | $\sigma(A_{2})$ | $\text{d}a/\text{d}t$ | $\sigma(\text{d}a/\text{d}t)$ | S/N | Except
---|---|---|---|---|---|---|---|---|---|---
2007 TF15 | 2007-10-09 | G96 | 1 | 6.9 |  $-$22.55 | 2.11 |  $-$91.13 | 8.54 | 10.67 | N
326290 | 1993-04-13 | 691 | ¡1 | 5.0 |  $-$3.17 | 0.32 |  $-$17.85 | 1.83 | 9.72 | N
4179 | 1934-02-14 | 012 | 4 | 42.2 |  $-$0.61 | 0.06 |  $-$2.72 | 0.30 | 8.81 | N
175706 | 1985-12-18 | 675 | 1 | 10.2 |  $-$1.20 | 0.14 |  $-$5.69 | 0.67 | 8.47 | N
2062 | 1955-12-17 | 675 | ¡1 | 20.0 |  $-$1.43 | 0.17 |  $-$6.42 | 0.77 | 8.27 | N
481442 | 2001-11-12 | 645 | ¡1 | 5.0 |  $-$5.64 | 0.70 |  $-$33.50 | 4.19 | 7.98 | N
385186 | 1986-12-29 | 381 | ¡1 | 7.0 | 1.18 | 0.14 | 4.80 | 6.05 | 7.93 | N
350751 | 1991-03-09 | 261 | ¡1 | 10.7 |  $-$5.20 | 0.65 |  $-$22.87 | 2.89 | 7.91 | N
85953 | 1971-03-26 | 675 | ¡1 | 27.9 |  $-$1.51 | 0.19 |  $-$14.82 | 1.87 | 7.89 | N
152563 | 1953-01-12 | 261 | 2 | 39.0 |  $-$2.96 | 0.39 |  $-$14.25 | 1.92 | 7.42 | Y
66400 | 1987-04-21 | 675 | ¡1 | 12.1 |  $-$2.33 | 0.36 |  $-$15.96 | 2.47 | 6.44 | N
1994 GL | 1994-04-10 | 691 | 3 | 25.9 | 45.31 | 7.16 | 11.96 | 9.35 | 6.32 | N
162361 | 1991-01-15 | 675 | ¡1 | 8.9 | 2.89 | 0.47 | 15.80 | 2.60 | 6.06 | N
2019 QU2 | 2004-08-12 | 807 | ¡1 | 6.0 |  $-$6.80 | 1.25 |  $-$63.26 | 11.68 | 5.41 | N
446924 | 1992-10-02 | 261 | ¡1 | 10.0 |  $-$3.37 | 0.72 |  $-$19.32 | 4.15 | 4.65 | N
138404 | 1982-07-20 | 260 | ¡1 | 6.6 | 2.09 | 0.46 | 9.26 | 2.05 | 4.50 | N
234341 | 1991-04-18 | 260 | ¡1 | 8.8 |  $-$3.28 | 0.72 |  $-$22.59 | 5.01 | 4.50 | N
511684 | 2005-12-03 | 568 | ¡1 | 9.1 | 10.02 | 2.34 | 62.68 | 14.63 | 4.28 | N
283457 | 1951-11-05 | 261 | ¡1 | 39.7 |  $-$7.05 | 1.66 |  $-$25.34 | 5.99 | 4.23 | N
203471 | 1994-01-08 | 675 | 1 | 8.0 |  $-$7.77 | 1.86 |  $-$41.49 | 9.97 | 4.16 | N
85774 | 1989-04-30 | 260 | ¡1 | 9.4 |  $-$0.53 | 0.12 |  $-$2.15 | 0.52 | 4.12 | N
7350 | 1963-10-15 | 675 | ¡1 | 22.5 |  $-$1.86 | 0.49 |  $-$8.03 | 2.12 | 3.77 | N
592968 | 2003-03-24 | 644 | ¡1 | 12.0 | 9.28 | 2.47 | 64.55 | 17.23 | 3.74 | N
85770 | 1990-10-12 | 261 | ¡1 | 8.0 |  $-$5.03 | 1.36 |  $-$24.26 | 6.60 | 3.67 | N
162181 | 1979-12-20 | 413 | ¡1 | 19.4 |  $-$1.90 | 0.57 |  $-$7.40 | 2.24 | 3.29 | N
398188 | 2001-07-02 | 644 | ¡1 | 8.9 |  $-$3.01 | 0.91 |  $-$14.87 | 4.51 | 3.29 | N
2017 DN109 | 2010-05-02 | C51 | ¡1 | 6.7 | 11.87 | 3.66 | 145.86 | 45.02 | 3.23 | N
7341 | 1981-10-23 | 095 | ¡1 | 9.8 |  $-$0.56 | 0.17 |  $-$2.38 | 0.73 | 3.23 | N
138911 | 1984-08-02 | 260 | ¡1 | 8.1 |  $-$2.49 | 0.80 |  $-$9.19 | 2.96 | 3.10 | N
162080 | 1981-03-03 | 413 | 1 | 16.9 |  $-$2.28 | 0.74 |  $-$11.76 | 3.81 | 3.08 | N
474158 | 1978-10-10 | 260 | ¡1 | 19.8 |  $-$10.39 | 3.50 |  $-$43.28 | 14.60 | 2.96 | N
65679 | 1954-11-20 | 675 | ¡1 | 34.9 |  $-$3.10 | 1.04 |  $-$14.84 | 5.00 | 2.96 | N
2006CT | 1991-01-15 | 691 | ¡1 | 13.6 |  $-$7.58 | 2.56 |  $-$32.48 | 10.99 | 2.95 | N
469219 | 2004-03-17 | 645 | ¡1 | 7.0 |  $-$11.73 | 4.50 |  $-$50.33 | 19.31 | 2.60 | N
2012 BB124 | 1976-08-27 | 260 | ¡1 | 35.4 | 6.64 | 2.909 | 27.48 | 12.03 | 2.28 | Y
2012 GA5 | 2001-03-20 | 645 | ¡1 | 10.9 | 42.33 | 19.68 | 168.96 | 78.59 | 2.14 | N
491007 | 2000-04-28 | 699 | ¡1 | 10.9 |  $-$11.77 | 5.89 |  $-$93.69 | 46.90 | 1.99 | N
267759 | 1953-04-13 | 261 | ¡1 | 44.7 |  $-$3.03 | 2.01 |  $-$11.37 | 7.55 | 1.50 | N
26310 | 1951-07-08 | 261 | ¡1 | 47.2 |  $-$11.79 | 7.99 |  $-$43.89 | 29.75 | 1.47 | N
358453 | 1994-03-06 | 691 | ¡1 | 13.0 |  $-$6.09 | 4.91 |  $-$30.33 | 24.44 | 1.24 | N
510190 | 2002-01-09 | 645 | ¡1 | 9.0 | 7.242 | 6.45 | 30.12 | 26.86 | 1.12 | N
5604 | 1976-04-23 | 413 | ¡1 | 8.9 |  $-$0.85 | 0.84 |  $-$4.49 | 4.44 | 1.01 | N
2018 KR | 2013-04-19 | F51 | ¡1 | 5.0 |  $-$53.82 | 52.94 |  $-$254.06 | 249.88 | 1.01 | Y
2013 GZ7 | 2003-03-31 | 645 | ¡1 | 10.0 |  $-$2.96 | 3.43 |  $-$11.03 | 12.77 | 0.86 | N
2015 XW261 | 2009-01-16 | 645 | ¡1 | 6.8 | 32.78 | 39.13 | 159.35 | 190.21 | 0.83 | N
2014 RQ17 | 1953-10-08 | 261 | ¡1 | 60.9 | 65.74 | 80.30 | 261.30 | 319.14 | 0.81 | Y
403775 | 2002-06-02 | 644 | 1 | 7.9 |  $-$2.77 | 4.37 |  $-$22.61 | 35.67 | 0.63 | N
2012 DY32 | 2003-03-22 | 644 | ¡1 | 8.1 | 37.01 | 76.56 | 155.24 | 321.12 | 0.48 | N
339492 | 1983-10-26 | 260 | ¡1 | 16.5 | 1.83 | 3.943 | 6.85 | 14.70 | 0.46 | N
2021 SP4 | 2005-09-28 | 645 | ¡1 | 6.9 |  $-$8.47 | 24.1 |  $-$35.54 | 101.45 | 0.35 | N
2012 UU136 | 2002-11-08 | 645 | ¡1 | 9.9 |  $-$2.47 | 7.10 |  $-$10.20 | 29.32 | 0.34 | N
2015 LA2 | 2010-05-16 | F51 | 5 | 5.0 | 54.38 | 171.79 | 266.77 | 842.74 | 0.31 | Y
2016 GL2 | 2009-04-02 | G96 | 1 | 7.0 | 17.64 | 94.08 | 66.90 | 356.78 | 0.18 | N
397326 | 1953-03-09 | 675 | ¡1 | 53.5 | 9.77 | 5.27 | 3.68 | 19.87 | 0.18 | N
375505 | 1998-11-17 | 645 | ¡1 | 9.9 |  $-$4.98 | 2.69 |  $-$1.81 | 9.80 | 0.18 | N
2016 UW80 | 2011-10-25 | F51 | ¡1 | 5.0 |  $-$6.43 | 579.31 |  $-$260.58 | 2347.70 | 0.11 | Y
465617 | 2001-12-18 | 645 | ¡1 | 7.2 |  $-$6.65 | 6.43 |  $-$2.67 | 25.89 | 0.10 | N
2005 TF49 | 1989-10-21 | 260 | ¡1 | 15.9 | 2.67 | 3.77 | 1.11 | 15.70 | 0.07 | N

232323Columns are: Ast. = Designation of the asteroid; Tracklet Initial Date =
Date of the first observation in the tracklet; Obs. Code = Observatory code;
Arc length: Arc length of the isolate tracklet; $\Delta t$: Time passed
between the isolated tracklet and the following available observation; $A_{2}$
= Value of $A_{2}$ obtained by excluding the isolated tracklet;
$\sigma(A_{2})$ = Error in $A_{2}$ obtained by excluding the isolated
tracklets; $\text{d}a/\text{d}t$ = Value of $A_{2}$ obtained by excluding the
isolated tracklet; $\sigma(\text{d}a/\text{d}t)$ = Error in $A_{2}$ obtained
by excluding the isolated tracklets; S/N = Signal-to-noise obtained by
excluding the isolated tracklets; Except = Flag to indicate that the tracklet
is reliable. The arc length is given in days, the time $\Delta t$ in years,
the values of $A_{2},\sigma(A_{2})$ in $10^{-14}$ au d-2, and
$\text{d}a/\text{d}t,\sigma(\text{d}a/\text{d}t)$ in $10^{-4}$ au My-1.