Issue 
A&A
Volume 667, November 2022



Article Number  A150  
Number of page(s)  10  
Section  Celestial mechanics and astrometry  
DOI  https://doi.org/10.1051/00046361/202244488  
Published online  22 November 2022 
Yarkovsky effect detection from groundbased astrometric data for nearEarth asteroid (469219) Kamo’oalewa
^{1}
State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University,
129 Luoyu Road,
Wuhan
430070, PR China
email: jgyan@whu.edu.cn; mye@whu.edu.cn
^{2}
State Key Laboratory of Lunar and Planetary Science, Macau University of Science and Technology,
Macau, PR China
^{3}
Geodesy Observatory of Tahiti, University of French Polynesia,
BP 6570,
98702
Faa’a, Tahiti, French Polynesia, France
Received:
13
July
2022
Accepted:
23
September
2022
Context. The Yarkovsky effect is a weak nongravitational force but may significantly affect subkilometresized nearEarth asteroids. Yarkovskyrelated drift may be detected, in principle, from astrometric or radar datasets of sufficient duration. To date, the asteroid Kamo’oalewa, the most stable of Earth’s quasisatellites, has an ~18 yrlong arc of groundbased optical astrometry. These data provide an opportunity to detect the Yarkovsky effect acting on the asteroid Kamo’oalewa.
Aims. We determined the Yarkovskyrelated drift of asteroid Kamo’oalewa from ~18 yr of groundbased optical astrometry. Furthermore, we investigated the influence of the Yarkovsky effect on the orbital evolution of asteroid Kamo’oalewa based on this estimated value, and evaluated the potential improvements in the detection of nongravitational accelerations (Yarkovsky effect and solar radiation pressure) for the asteroid Kamo’oalewa that could be provided by the future Chinese smallbody exploration mission, Tianwen2.
Methods. The Yarkovskyrelated drift of asteroid Kamo’oalewa was detected from the orbital fitting of the astrometry measurements. We checked the Yarkovsky effect detection based on both the orbit fitting results and the physical mechanisms of the Yarkovsky effect.
Results. We report for the first time the detection of the Yarkovsky effect acting on asteroid Kamo’oalewa based on ~18 yr of groundbased optical astrometry data. The estimated semimajor axis drift is (−6.155 ± 1.758) × 10^{−3} au Myr^{−1}. In addition, our numerical simulation shows that the Yarkovsky effect has almost no influence on the shortterm orbital evolution of the asteroid Kamo’oalewa, but does have a longterm influence, by delaying the entry of the object into the Earth coorbital region and accelerating its exit from this region, with a more significant signature on the exit than on the entry. In the context of spacecraft tracking data, the Tianwen2 mission will improve both nongravitational accelerations (Yarkovsky effect and solar radiation pressure) and predictions of its future ephemeris.
Key words: minor planets, asteroids: general / minor planets, asteroids: individual: (469219) Kamo’oalewa / astrometry / celestial mechanics
© L. Liu et al. 2022
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1 Introduction
The asteroid (469219) Kamo’oalewa, hereafter ‘HO3’ for brevity, is an Apollotype nearEarth asteroid (NEA) that is believed to be composed of lunar ejecta (Sharkey et al. 2021). The currently known physical properties of this object are limited to its absolute magnitude, rotation period, and possible taxonomic class (see Table 1). This asteroid is the smallest and most stable of Earth’s five known quasisatellites, and could have been a companion of our planet for more than 1 Myr (De la Fuente Marcos & De la Fuente Marcos 2016).
Quasisatellites are objects in a solar coorbital configuration (1:1 orbital resonance) with a planet and are always sufficiently ‘close’ to it. Nevertheless, quasisatellite orbits always lie outside the planet’s Hill sphere (hence the prefix quasi), and have unstable orbits, with the possibility of going backandforth in a resonance mode in a complicated way (Christou & Asher 2011; De la Fuente Marcos & De la Fuente Marcos 2016). HO3 is a perfect example of such a quasisatellite and is also currently the closest to Earth, being always between 38 and 100 times the distance of the Moon (The Earth’s Hill sphere is about four times the EarthMoon distance). Therefore, several spaceexploration missions to the asteroid HO3 have been proposed (Wiegmann 2018; Heiligers et al. 2019; Venigalla et al. 2019; CNSA 2019). Among these is the future Chinese Tianwen2 small body exploration mission, which is a samplereturn visit to HO3 planned for sometime around 2024 (CNSA 2019; Jin et al. 2020).
The Yarkovsky effect is a subtle nongravitational acceleration caused by the anisotropic emission of thermal radiation, and associated with solar heating on a rotating body (Bottke et al. 2006; Vokrouhlický et al. 2000). This effect depends on several physical quantities of asteroids such as spin state, size, mass, shape, thermal properties, and surface roughness (Chesley et al. 2014; Vokrouhlický & Farinella 1999). The Yarkovsky effect includes both seasonal and diurnal components. The magnitude of the diurnal effect is generally larger than that of the seasonal effect (Vokrouhlický et al. 2000). The typical value of the Yarkovsky acceleration for a subkilometre NEA is 10^{−15}–10^{−13} au day^{−2} (Del Vigna et al. 2018). Though small, the Yarkovsky effect may influence the longterm orbital evolution of asteroids (Desmars 2015).
Two teams of investigators have comprehensively studied the short and longterm dynamics of HO3: De la Fuente Marcos & De la Fuente Marcos (2016) and Fenucci & Novaković (2021). The former found that the orbital geometry of the asteroid, computed without considering the Yarkovsky effect, switches repeatedly between a ‘close’ quasisatellite configuration and a more distant horseshoe configuration. The current quasisatellite orbit started about 100 yr ago, and will remain stable for about 300 more years. The inclusion of a possible range of amplitude for the Yarkovsky effect was investigated by Fenucci & Novakovic (2021), who found that the Yarkovsky effect may cause HO3 to exit from the Earth’s coorbital region slightly faster than with respect to a purely gravitational model. However, it can still maintain its Earth companion identity for at least 0.5 Myr in the future instead of 1 Myr in the previous model.
The Yarkovsky effect primarily acts to promote secular variation on the semimajor axis, which causes a runoff in orbital anomalies that accumulate quadratically with time (Bottke et al. 2006). Therefore, the orbital drift due to the Yarkovsky effect may be detected, in principle, from astrometric or radar datasets of sufficient duration (Chesley et al. 2014; Del Vigna et al. 2018). For example, asteroid (6489) Golevka became the first asteroid for which the Yarkovsky effect was detected through radar ranging (Chesley et al. 2003). Vokrouhlický et al. (2008) detected the Yarkovskyrelated drift using only groundbased astrometric observations of the asteroid (152563) 1992BF. Subsequently, the Yarkovskyrelated drifts of a large number of NEAs were detected using groundbased optical astrometry (Del Vigna et al. 2018; Desmars 2015; Farnocchia et al. 2013; Greenberg et al. 2020; Nugent et al. 2012).
The present paper is organised as follows: our data and method are detailed in Sect. 2. Our detection of the Yarkovsky effect is discussed in Sect. 3. The impact of the Yarkovsky effect on the orbital evolution of HO3 is presented in Sect. 4, and potential improvements in nongravitational accelerations (Yarkovsky effect and solar radiation pressure) for HO3 based on the Tianwen2 mission are discussed in Sect. 5. We draw conclusions in Sect. 6.
Known physical properties of HO3 (up to date).
2 Data and method
2.1 Observational data and treatment
We used groundbased optical astrometry (rightascension (RA) and declination (Dec)) from 2004 March 17 to 2021 May 14, for a total of 310 observations. These optical astrometry data can be downloaded from the Minor Planet Center^{1}. We corrected the astrometry for the star catalogue systematic errors using the Eggl et al. (2020) debiasing scheme and weighted the data according to Vereš et al. (2017). Outliers were rejected based on the Carpino et al. (2003) method with a rejection threshold of χ_{rej} = 3.
2.2 Dynamical model
The pointmass gravitational accelerations acting on HO3 from the Sun, the Moon, the eight planets, Pluto, and 16 massive mainbelt asteroids were computed from the Jet Propulsion Laboratory (JPL) planetary ephemerides DE441 (Farnocchia 2021; Park et al. 2021). Relativistic perturbations were applied for the Sun, the Moon, and the eight planets based on the Einstein–Infeld–Hoffman formulation (Moyer 2005). Because HO3 is always within 1 au of the Earth, we also accounted for the oblateness term in the geopotential of Earth (Kaula 1966). The Yarkovsky effect was considered as a solvefor parameter in the dynamical model.
More specifically, the Yarkovsky perturbation can be modelled as a transverse acceleration of the form a_{t} = A_{2}(r_{0}/r)^{d}, where A_{2} is an estimated parameter, r is the heliocentric distance in astronomical units (au), and r_{0} = 1 au is used as the normalisation factor. The exponent d is related to the thermophysical properties of the asteroid, and is always between 0.5 and 3.5 (Farnocchia et al. 2013). d = 2 is typically used for all asteroids without complete thermophysical data, and therefore we also applied this convention in this work. We note that this model ignores radial and outofplane accelerations, as well as some of the finer details in the transverse acceleration such as hysteresis, but captures the salient aspects of the Yarkovsky effect and is computationally fast (Chesley et al. 2014, 2021).
After obtaining an estimated A_{2} value, the corresponding semimajor axis drift can be computed as (1)
where e is the eccentricity, n is the mean motion, and p is the semilatus rectum (Farnocchia et al. 2013). From Eq. (1), we can see that a negative value of A_{2} corresponds to a negative orbital drift, while a positive value of A_{2} corresponds to a positive orbital drift. Therefore, the sign of an estimated A_{2} allows us to distinguish whether an asteroid is a retrograde rotator (A_{2} < 0) or a prograde rotator (A_{2} > 0) (Vokrouhlický et al. 2000).
Given the very small size (diameter of about 36 m) of HO3, the solar radiation pressure (SRP) imposed on HO3 may also be significant. Therefore, we also considered a case where we tried to detect SRP along with the Yarkovsky effect. In this case, the SRP is modelled as a radial acceleration a_{r} = A_{1}(r_{0}/r)^{2} (Farnocchia et al. 2015), where A1 is a solvefor parameter. The A_{1} can be related to physical quantities of asteroids as follows (Vokrouhlický & Milani 2000): (2)
where A is the Bond albedo, ϕ = 1.371 kW m^{−2} is the solar radiation energy flux at 1 au, c is the speed of light, and S/M is the areatomass ratio of the asteroid. It is noteworthy that, although the intensity of the Yarkovsky effect and SRP scale down as the square of the Sun–HO3 distance, they behave with different directional signatures (transverse for the Yarkovsky effect and radial for the SRP). Typical amplitudes of the accelerations acting on HO3 during the observation period are given in Fig. 1.
Figure 1 indicates that the dynamical model used in the orbital fit is reliable, as the smallest acceleration amplitude that we imposed on HO3 – inferred from the oblateness term of Earth – is much smaller than the acceleration amplitude imposed on HO3 by the Yarkovsky effect. As a consequence, the acceleration from the oblateness term of Earth can be neglected.
Fig. 1 Typical amplitudes of the accelerations acting on the HO3 in heliocentric J2000 ecliptic reference system. The SRP acceleration amplitude is derived from Eq. (2) based on an assumed areatomass ratio of 1.965 × 10^{−5} m^{2} kg^{−1} and a Bond albedo of 0.083. See Sect. 5 for details. 
2.3 Software
Taking into account the force model, the orbital solutions were computed in this work using the wellknown OrbFit 5.0.7 software^{2}. The latest online version (2020) includes a weighting and outlier rejection scheme for optical astrometry described in Sect. 2.1. We also incorporated the Eggl et al. (2020) astrometric debiasing scheme into the software. The software uses an iterative weighted leastsquares batch estimator to find an initial orbital solution with the minimum residual variance for a given set of astrometry data. In order to check our implementation and correct use of this software, we used it to model the Yarkovsky effect for the asteroid NEA 2011PT. We obtained an A_{2} estimate in close agreement with the A_{2} estimates already published (see the Appendix A for details).
3 Yarkovsky effect detection
3.1 Leastsquares regression
To try to detect both the SRP and the Yarkovsky effect, we performed an eightparameter solution which included gravity and relativety, the Yarkovsky effect, and SRP. There were no prior constraints introduced in the leastsquares fit. The estimate of A_{1} is a negative, nonphysical value with a large uncertainty. This is due to the fact that for the nearcircular orbit of HO3, the SRP force is perfectly aliased with the solar gravity (Chesley et al. 2014) and is absorbed in the fit of the initial value of the semimajor axis. We therefore considered only the Yarkovsky effect in our force model.
To model the Yarkovsky effect, we considered two solutions, in sequence: a sixparameter solution which only included gravity/relativity without the Yarkovsky effect, and a sevenparameter solution with the inclusion of the Yarkovsky effect. For the sixparameter solution, we removed six outliers from 310 available observations, leaving 304 positions in the orbital fit. For the sevenparameter solution, we removed four outliers from the 310 available observations, leaving 306 positions in the orbital fit. Figure 2 depicts the postfit plane of the sky astrometric residuals.
We also present the orbital solutions and root mean square (RMS) of the normalised postfit residuals (χ^{2}) of the six and sevenparameter solutions in Table 2. The covariance matrices corresponding to the two solutions are shown in Tables B.1 and B.2 in the Appendix B. The χ^{2} for the six and sevenparameter solutions were 0.415 and 0.403, respectively. This indicates that the sevenparameter solution provides a better fit to groundbased optical astrometry, although this decrease in the χ^{2} can be qualified as marginal.
From Table 2, we can see that after estimating the Yarkovsky parameter, the uncertainty on all the orbital elements increases (see also Tables B.1 and B.2). In particular, the uncertainty on semimajor axis increases most significantly because of the strong correlation between A_{2} and the semimajor axis (Del Vigna et al. 2018). A correlation matrix of the sevenparameter solution (see Fig. 3) clearly indicates that high correlations are present between the A_{2} parameter and some other orbital elements, and especially the semimajor axis. This problem was already noted by Del Vigna et al. (2018), and indicates that including the A_{2} parameter in the leastsquares fit results in an almost illposed problem.
In order to be sure that our determination of the A_{2} parameter is statistically significant, we performed the same check as in Greenberg et al. (2020). Their survey of the detection of the Yarkovsky effect for a large set of asteroids included an analysis of variance to verify whether the data warrant the inclusion of the Yarkovsky parameter. This analysis using the pvalue formalism from the life sciences was based on the null hypothesis that the additional degrees of freedom introduced by the Yarkovsky force model are redundant. A detection result is rejected if p > 0.05 and vice versa. The pvalue can be computed as (3)
where the f(m_{Y} − m_{o}, N − m_{Y}, x) is the Fdistribution probability density function with m_{Y} − m_{0} and N − m_{Y} degrees of freedom. F is defined as (4)
where and . The value O_{i} is the ith observation and δ_{i} is the associated uncertainty. The values C_{0,i} and C_{Y,i} are the ith computed value from the sixparameter solution and sevenparameter solution, respectively. The value N is the number of observations. The value m_{0} is the number of estimated parameters in the sixparameter solution and m_{Y} is the number of estimated parameters in the sevenparameter solution. In our case, we obtained a pvalue of 1.2 × 10^{−8}, indicating that our detection of the Yarkovsky effect for HO3 was at an acceptable level of confidence. The second check considered the signaltonoise ratio defined as , where is from an a posteriori covariance matrix of the sevenparameter solution. Farnocchia et al. (2013) used as a threshold to define reliable detection. In this work, we obtained of 3.5, that is, a ~3.5 σ detection of the Yarkovsky effect for HO3.
In summary, the estimated A_{2} from the nongravitational solution is (−1.434 ± 0.410) × 10^{−13} au day^{−2}, which yields a Yarkovskyinduced semimajor axis drift of da/dt = (6.155 ± 1.758) × 10^{−3} au Myr^{−1}. The negative value of A_{2} suggests that HO3 is a retrograde rotator.
Fig. 2 Groundbased RA and Dec residuals of the modelling of the orbit of HO3. The RA residuals include the cos(Dec) scaling factor. Blue dots represent residuals obtained from the gravityonly solution. Red dots correspond to the nongravitational solution residuals. 
Orbital elements at epoch 2016 December 30 11:22:22.2765 TDT.
Fig. 3 Posteriori correlation matrices corresponding to the sixparameter solution (panel a) and the sevenparameter solution (panel b). 
3.2 The external assessment of the Yarkovsky detection
Although we detected the Yarkovsky effect of asteroid HO3 by fitting longterm groundbased optical astrometry with a reasonable level of confidence, an external check is necessary (Del Vigna et al. 2018; Desmars 2015; Farnocchia et al. 2013). This external check verified the consistency of our estimated value of the Yarkovsky effect with the expected longterm behaviour of the HO3 orbit, by looking at the Yarkovsky drift (da/dt) of the semimajor axis of the orbit with respect to an ‘expected’ value (da/dt)_{exp}. This expected value was computed by scaling the value obtained for Bennu as follows (Del Vigna et al. 2018). (5)
where a is the semimajor axis of the asteroid, e is the eccentricity, ρ is the density, ϕ is the obliquity, and A is the Bond albedo. The Bond albedo was computed from geometric albedo p_{v} according to the relation (Muinonen et al. 2010). The of asteroid (101955) Bennu is considered as the most reliable detection of the Yarkovsky effect. We used physical parameters of asteroid Bennu and HO3 listed in Table 3. The adopted orbital elements (a_{B}, e_{B}) of asteroid Bennu were from JPL SmallBody Database^{3}. The adopted orbital elements (a, e) of HO3 were values from our sevenparameter solution, as in Table 2. We obtained from Eq. (5) an expected semimajor drift of 1.380 × 10^{−2} au Myr^{−1}.
Unlike the physical properties of asteroid Bennu presented in Table 3, the physical properties of HO3 in Table 3 were obtained on the basis of reasonable assumptions given that these properties are currently unknown. The geometric albedo of 0.250 was assumed for HO3 (Reddy et al. 2017). The diameter of 36 m was estimated from the absolute magnitude (see Table 1) following the relationship: (Pravec & Harris 2007). A density of 2.120 g cm^{−3} was assumed (Del Vigna et al. 2018), with reference to Stype asteroids (Fohring et al. 2018; Sharkey et al. 2021). The obliquity was assumed as 0 or 180 deg to ensure that the expected maximum absolute Yarkovskyrelated value of semimajor axis drift for HO3 was obtained.
The ratio is the key parameter to understand whether the estimated orbital drift (da/dt) is physically consistent with the Yarkovsky effect. If the estimated orbital drift (da/dt) is significantly larger than the expected maximum absolute value (da/dt)_{exp}, then the result is inconsistent with the mechanism of the Yarkovsky effect.
Del Vigna et al. (2018) classified Yarkovsky effect detections into three categories  namely accepted detection, marginal significance, and rejected detection – by combining the value of the with the value of the filtering parameter R. The accepted detection must satisfy both and R ≤ 2. The marginal significance is defined as and R ≤ 2. A detection is considered rejected if and R > 2. In this work, we obtained of 3.5 and R of 0.446 for HO3, which indicates this Yarkovsky detection was an acceptable detection based on these two criteria. Fenucci & Novaković (2021) derived the ranges of possible Yarkovskyinduced semimajor axis drift (see Fig. 3 in their work) for HO3 corresponding to different values of thermal conductivity K using statistical methods. The Yarkovskyinduced semimajor axis drift of da/dt = (−6.155 ± 1.758) × 10^{−3} au Myr^{−1} is still within the ranges of possible semimajor axis drift as they provided, which also indicates that our Yarkovsky detection is acceptable.
4 Effect of the Yarkovsky effect on orbital evolution
Fenucci & Novakovic (2021) investigated the role played by the Yarkovsky effect in the short (less than 1 Myr) to longterm (several Myr) orbital evolution of HO3 in a statistical framework. This method considers clone orbits, that is, orbits differing by small amounts in their initial states to a nominal orbit, with each clone orbit being affected by a set of the Yarkovskyinduced drifts, resulting in several thousand cases. We reran their simulation, but only with respect to our estimated Yarkovsky parameter A_{2} value corresponding to a semimajor drift of da/dt = (−6.155 ± 1.758) × 10^{−3} au Myr^{−1}. For the convenience of the reader, the approach of Fenucci & Novakovic (2021) is summarised in Appendix C.
4.1 Shortterm orbital evolution
Figure 4 corresponds to a numerical integration of the clone orbits in the interval (−5, 5) kyr, and shows that the Yarkovsky effect has almost no impact on the shortterm evolution of the orbit of HO3. This is clearly illustrated by the small differences in the parameters when considering or not the Yarkovsky effect (see right panels). Within this 10 kyr interval, HO3 switches repeatedly between the horseshoe (i.e. resonance angle λ_{T} librating around 180°, with amplitude of almost 180°) and quasisatellite (i.e. resonance angle λ_{T} librating around 0°) dynamic states. The presentday HO3 dynamic state is that of a quasisatellite of the Earth, a state that started about 100 yr ago and will remain for about 300 yr into the future, before transitioning to a horseshoe state.
4.2 Longterm orbital evolution
We investigated the effect of the Yarkovsky effect on the longterm dynamics of HO3 by computing the distribution of the time spent in the Earth coorbital region for the clonal orbits over the time interval (−5, 5) Myr. We used a semimajor axis range of 0.994 au ≤ a ≤ 1.006 au as given by De la Fuente Marcos & De la Fuente Marcos (2016), which is also in agreement with our results for the shortterm orbital evolution of this object (see the third panel in Fig. 4). The distributions of the time spent in the Earth coorbital region for clonal orbits over the time interval (−5, 5) Myr are shown in Fig. 5.
From Fig. 5, we can clearly see that either in the past or in the future, the Yarkovsky effect reduces the time that clonal orbits spend in this Earth coorbital region. This implies that the Yarkovsky effect has an impact on the reconstruction of the dynamic past of HO3 and any predictions made regarding its future orbital evolution. Moreover, the Yarkovsky effect will have a more significant impact on the longterm orbital evolution of HO3 in the future, which can be clearly observed in the first bin of the histogram shown in Fig. 5. This is also further illustrated by the magnitude of the reduction in medians of distributions that include the Yarkovsky effect relative to those medians of distributions that do not include the Yarkovsky effect.
We found that when including the Yarkovsky effect, 81% of the clonal orbits leave this Earth coorbital region in less than 1 Myr from the present time (see top panel in Fig. 5), implying a low probability that HO3 will remain in the Earth coorbital region for more than 1 Myr. Further, our results also show that the probability that HO3 reached this region as early as 1 Myr ago seems unlikely, given that 77% of the clonal orbits reached this region later than 1 Myr ago (see bottom panel in Fig. 5).
Fig. 4 Time evolution of relevant parameters for the clonal orbits of HO3 during the time interval (−5, 5) kyr. The left panels correspond to the case of purely gravitational perturbation plus Yarkovsky effect. The right panels correspond to the differences in parameters seen with and without considering the Yarkovsky effect. Panels show the evolution of the distance between HO3 and Earth (d_{Earth}), resonance angle (λ_{T}), semimajor axis (a), eccentricity (e), and inclination (i) over time (order from top to bottom). The resonance angle (λ_{T}) is the difference between the mean longitude of the object and that of the Earth. The mean longitude of an object is given by λ = M + Ω + ω. The radius of the Earth’s Hill sphere, 0.0098 au, is plotted as a magenta line. The black thick curve displays the average evolution of 500 clonal orbits, and the red thin curves show the standard deviations in the values of these parameters at a given time. 
5 The Chinese small body exploration mission
The future Chinese Tianwen2 smallbody exploration mission is planned to visit the HO3 around 2024 (CNSA 2019). One of the scientific objectives of this mission includes the measurement of the thermal properties of the asteroid and a samplereturn to Earth (Yan et al. 2022). In this section, we discuss the potential improvement of the detection of nongravitational accelerations (Yarkovsky effect and SRP) acting on HO3 that can be provided by radiotracking from Earth.
For this purpose, an a priori value of A1 is needed to model the SRP of HO3 in this simulation. We assumed that HO3 has the areatomass ratio S/M of a spherical asteroid of 36 m in diameter and a density of 2120 kg/m^{3} (see Table 3), that is, 1.965 × 10^{−5} m^{2} kg^{−1}. We also assumed a Bond albedo (A) value of 0.083 for HO3. Finally, an a priori value (4.646 × 10^{−12}) au day^{−2} of A_{1} was derived from Eq. (2). In the real case, these physical parameters will be inferred from in situ measurements made by the Tianwen2 spacecraft.
We simulated pseudorange measurements (roundtrip delay of a twoway radio signal from the Earth geocentre to the asteroid centre of mass) in a fashion similar to Farnocchia et al. (2021a) for asteroid Bennu. Range measurements can improve the estimation of spacecraft and asteroid positions and nongravitational accelerations (Chesley et al. 2014; Farnocchia et al. 2021a,b; Konopliv et al. 2002, 2014, 2018). We simulated eight potential postrendezvous synthetic pseudo range points, assuming an a priori uncertainty of 0.01 µs in time delay, which corresponds to about 1.5 m in range (Farnocchia et al. 2019). The simulated range measurements were performed at monthly intervals from 2025 December 01 to 2026 July 01. Asteroid HO3 is also favourably placed for observations once a year around April when it becomes sufficiently bright (with visual magnitude V < 23.0 mag) to be observed by large telescopes on Earth (Sharkey et al. 2021). Therefore, we also simulated onenight groundbased optical astrometry by a large telescope in midApril each year from 2022 to 2026. We simulated three optical observations (RA/Dec) per night with an uncertainty of 0.15″, which is consistent with the typical and preferred contribution with three to five observations per night (Chesley et al. 2014). In the end, a total of 15 groundbased optical observations were simulated. We did not consider the groundbased radar observations because the size of HO3 is so small that it is almost impossible to observe it with groundbased radar. We performed an additional eightparameter solution (including the Yarkovsky effect and SRP) with these additional simulated data, and if the eightparameter solution failed, an additional sevenparameter solution (including Yarkovsky effect only) was performed. Table 4 lists the uncertainties obtained before and after the inclusion of these simulated Tianwen2 pseudo range data and typical groundbased optical observation data.
From Table 4, we can see that the uncertainty of the Yarkovsky parameter A2 drops by a factor of 3.3 after the inclusion of groundbased optical observation data. We conclude that potential optical observation data could resolve the Yarkovsky effect acting on HO3 with a higher 9% precision in 2026. However, the SRP acting on HO3 still cannot be detected. After including simulated Tianwen2 pseudo range data, the uncertainty of A_{2} decreased again by a factor of 6.8. This indicates that the Tianwen2 radio science observations could improve the precision of the detection of the Yarkovsky effect down to 1%. Such a highprecision measurement of the Yarkovsky effect can not only further constrain the orbital evolution, but also potentially provides strong constraints on the physical properties of this object, such as bulk density (Chesley et al. 2014) and thermal inertial (Fenucci et al. 2021), if combined with the physical parameters of the body. Moreover, the SRP can also be detected and its estimated precision is better than 20% with the contribution of Tianwen2 pseudo range data.
Additionally, the Tianwen2 radio science observations could significantly improve the predictions of future ephemeris. Potential groundbased optical observation data could be used to constrain the position uncertainty of HO3 to the order of tens of kilometres on 2026 October 1, which could be reduced to under 1.0 km with the simulated Tianwen2 pseudo range data.
Formal uncertainties with and without simulated Tianwen2 pseudo range data and typical groundbased optical observation data.
Fig. 5 Distributions of the time spent in the Earth coorbital region T_{1:1} for clonal orbits over the time interval (−5, 5) Myr. The top panel corresponds to the distribution of the time spent over the time interval (0, 5) Myr. The bottom panel corresponds to the distribution of the time spent over the time interval (−5, 0) Myr. The blue colour corresponds to the case of including the Yarkovsky effect. The red colour corresponds to the case of purely gravitational perturbation. Vertical dashed lines denote the medians of the distributions, and the corresponding numerical values are written in each panel. Error bars in the histograms are relative to each entire bin, which are computed using Bernoulli statistics. Each bin width is set to 0.5 Myr. 
6 Conclusions
Asteroid (469219) Kamo’oalewa is the smallest and closest of Earth’s five known quasisatellites, and has the most stable orbit. It is therefore considered as a potential target for in situ exploration in the future. We performed Yarkovsky effect detection for this object using ~18 yr (up to present) of groundbased optical astrometry. We detected a reliable Yarkovskyinduced semimajor axis drift of da/dt = (−6.155 ± 1.758) × 10^{−3} au Myr^{−1} for this object.
We used this semimajor axis drift to further constrain its dynamic orbital evolution. We found that the Yarkovsky effect does not change the dynamical state of the asteroid Kamo’oalewa in the case of shortterm orbital evolution. However, the Yarkovsky effect can delay the entry of the object into the Earth coorbital region and accelerate its exit from this region in the case of past and future longterm orbital evolution. The Yarkovsky effect is more significant in accelerating the departure of this asteroid from this region in the future. Specifically, it seems unlikely that this object will remain in the Earth’s coorbital region for more than 1 Myr in the future, and it also seems unlikely that it reached this region as early as 1 Myr ago.
The future Chinese Tianwen2 smallbody exploration mission will improve both the detection of nongravitational accelerations (Yarkovsky effect and SRP) of the asteroid Kamo’oalewa and predictions of its future ephemeris with the contribution of the spacecraft tracking data. Our simulations show that with the contribution of range data from the Tianwen2 mission, the Yarkovsky effect can be estimated with 1% accuracy, that the SRP estimated precision is better than 20%, and that the ephemeris position uncertainty can be reduced to under 1.0 km.
Acknowledgements
We thank the referee Prof. David Vokrouhlický for his useful comments that have improved the quality of the paper. This research is supported by the National Natural Science Foundation of China (U1831132, 42030110) and open project of State Key Laboratory of Lunar and Planetary Science, Macau University of Science and Technology (SKLLPS(MUST)20212023). JPB was funded by a DAR grant in planetology from the French Space Agency (CNES). The numerical calculations of this paper have been done in the Supercomputing Center of Wuhan University.
Appendix A Detection of the Yarkovsky effect for NEA 2011 PT
In order to verify our implementation of the OrbFit5.0.7 software, we estimated the Yarkovsky effect for asteroid NEA 2011PT, an estimation already done by JPL and Near Earth Objects Dynamic Site (NEODyS) group. We used a total of 199 groundbased optical astrometry observations (from 2011 August 01 to 2017 November 17) downloaded from the Minor Planet Center. The same optical astrometry processing schemes (debiasing, weighting, and outlierremoval schemes) as in Sect. 2.1 and the same force model as discussed in Sect. 2.2 – but excluding the SRP – were applied to the orbit fit. A sevenparameter solution (see Table A.1) was computed using the OrbFit5.0.7 software. For this orbital solution, we removed five outliers out of 199 available observations in the orbital fitting. We obtained an estimated A_{2} = (−2.135 ± 0.300) × 10^{−13} au/day^{2}, in very good agreement with the value of (−2.121 ± 0.301) × 10^{−13} au/day^{2} as reported by the JPL SmallBody Database^{4} and the value of (−2.241 ± 0.298) × 10^{−13} au/day^{2} as given by the NEODyS^{5}.
Orbital elements at epoch 2015 July 3, 12:49:34.597424 TDT.
Appendix B Additional tables for the detection of the Yarkovsky effect for HO3
A posteriori covariance matrix of the sixparameter solution.
A posteriori covariance matrix of the sevenparameter solution.
Appendix C Orbital numerical integration
Numerical integrations were performed with a modified Nbody code mercury ^{6}by Fenucci & Novaković (2021). The modified code can include the Yarkovsky effect in the dynamical model by modelling it as a secular force along the orbital velocity of the asteroid. The orbital drift is treated as an input, which can be derived using Eq. 1. In addition to the Yarkovsky effect, our dynamical model also included the pointmass gravitational perturbations of the Sun, the Moon, the eight planets, Pluto, and the three large asteroids (Ceres, Pallas, and Vesta). These initial conditions for both planets and minor bodies at epoch 2016 December 30, 11:22:22.2765 TDT, were taken from DE441 and sbN16 (Park et al. 2021), respectively. The nominal orbit of HO3 was taken from Table 2. The covariance matrix corresponding to the nominal orbit can be found in the Appendix B.
The nominal orbit of HO3 has a certain uncertainty. Given the chaotic nature of the Nbody problem, small uncertainties in the initial orbit can cause large differences in the longterm numerical integration. We applied a Monte Carlo method using the Covariance Matrix (MCCM) approach to produce clones of the initial orbit (Avdyushev & Banschikova 2007; Bordovitsyna et al. 2001; De La Fuente Marcos & De La Fuente Marcos 2015; Fenucci & Novakovic 2021). The MCCM approach can consider how these elements affect each other and their associated uncertainties when generating clonal orbits. By integrating these cloned orbits, the effect of the uncertainty of the nominal orbit on the results of the orbit integration can be evaluated.
We considered two cases: a purely gravitational model for sixparameter solution and a gravitational plus Yarkovsky effect model for sevenparameter solution. For each case, we produced 500 orbital clones using the MCCM approach. Each clone orbit was propagated forward and backward 5 Myr using the BulirschStoer integration algorithm (Bulirsch et al. 2002). The integrated step is set to 4 h and the orbital elements were outputted for every 10 yr.
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All Tables
Formal uncertainties with and without simulated Tianwen2 pseudo range data and typical groundbased optical observation data.
All Figures
Fig. 1 Typical amplitudes of the accelerations acting on the HO3 in heliocentric J2000 ecliptic reference system. The SRP acceleration amplitude is derived from Eq. (2) based on an assumed areatomass ratio of 1.965 × 10^{−5} m^{2} kg^{−1} and a Bond albedo of 0.083. See Sect. 5 for details. 

In the text 
Fig. 2 Groundbased RA and Dec residuals of the modelling of the orbit of HO3. The RA residuals include the cos(Dec) scaling factor. Blue dots represent residuals obtained from the gravityonly solution. Red dots correspond to the nongravitational solution residuals. 

In the text 
Fig. 3 Posteriori correlation matrices corresponding to the sixparameter solution (panel a) and the sevenparameter solution (panel b). 

In the text 
Fig. 4 Time evolution of relevant parameters for the clonal orbits of HO3 during the time interval (−5, 5) kyr. The left panels correspond to the case of purely gravitational perturbation plus Yarkovsky effect. The right panels correspond to the differences in parameters seen with and without considering the Yarkovsky effect. Panels show the evolution of the distance between HO3 and Earth (d_{Earth}), resonance angle (λ_{T}), semimajor axis (a), eccentricity (e), and inclination (i) over time (order from top to bottom). The resonance angle (λ_{T}) is the difference between the mean longitude of the object and that of the Earth. The mean longitude of an object is given by λ = M + Ω + ω. The radius of the Earth’s Hill sphere, 0.0098 au, is plotted as a magenta line. The black thick curve displays the average evolution of 500 clonal orbits, and the red thin curves show the standard deviations in the values of these parameters at a given time. 

In the text 
Fig. 5 Distributions of the time spent in the Earth coorbital region T_{1:1} for clonal orbits over the time interval (−5, 5) Myr. The top panel corresponds to the distribution of the time spent over the time interval (0, 5) Myr. The bottom panel corresponds to the distribution of the time spent over the time interval (−5, 0) Myr. The blue colour corresponds to the case of including the Yarkovsky effect. The red colour corresponds to the case of purely gravitational perturbation. Vertical dashed lines denote the medians of the distributions, and the corresponding numerical values are written in each panel. Error bars in the histograms are relative to each entire bin, which are computed using Bernoulli statistics. Each bin width is set to 0.5 Myr. 

In the text 
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