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A&A
Volume 667, November 2022
Article Number A150
Number of page(s) 10
Section Celestial mechanics and astrometry
DOI https://doi.org/10.1051/0004-6361/202244488
Published online 22 November 2022

© L. Liu et al. 2022

Licence Creative CommonsOpen 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.

This article is published in open access under the Subscribe-to-Open model. Subscribe to A&A to support open access publication.

1 Introduction

The asteroid (469219) Kamo’oalewa, hereafter ‘HO3’ for brevity, is an Apollo-type near-Earth 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 quasi-satellites, and could have been a companion of our planet for more than 1 Myr (De la Fuente Marcos & De la Fuente Marcos 2016).

Quasi-satellites are objects in a solar co-orbital configuration (1:1 orbital resonance) with a planet and are always sufficiently ‘close’ to it. Nevertheless, quasi-satellite orbits always lie outside the planet’s Hill sphere (hence the prefix quasi-), and have unstable orbits, with the possibility of going back-and-forth 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 quasi-satellite 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 Earth-Moon distance). Therefore, several space-exploration 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 Tianwen-2 small body exploration mission, which is a sample-return visit to HO3 planned for sometime around 2024 (CNSA 2019; Jin et al. 2020).

The Yarkovsky effect is a subtle non-gravitational 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 sub-kilometre NEA is 10−15–10−13 au day−2 (Del Vigna et al. 2018). Though small, the Yarkovsky effect may influence the long-term orbital evolution of asteroids (Desmars 2015).

Two teams of investigators have comprehensively studied the short- and long-term 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’ quasi-satellite configuration and a more distant horseshoe configuration. The current quasi-satellite 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 co-orbital 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 semi-major 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 Yarkovsky-related drift using only ground-based astrometric observations of the asteroid (152563) 1992BF. Subsequently, the Yarkovsky-related drifts of a large number of NEAs were detected using ground-based 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 non-gravitational accelerations (Yarkovsky effect and solar radiation pressure) for HO3 based on the Tianwen-2 mission are discussed in Sect. 5. We draw conclusions in Sect. 6.

Table 1

Known physical properties of HO3 (up to date).

2 Data and method

2.1 Observational data and treatment

We used ground-based optical astrometry (right-ascension (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 Center1. 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 χrej = 3.

2.2 Dynamical model

The point-mass gravitational accelerations acting on HO3 from the Sun, the Moon, the eight planets, Pluto, and 16 massive main-belt 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 solve-for parameter in the dynamical model.

More specifically, the Yarkovsky perturbation can be modelled as a transverse acceleration of the form at = A2(r0/r)d, where A2 is an estimated parameter, r is the heliocentric distance in astronomical units (au), and r0 = 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 out-of-plane 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 A2 value, the corresponding semi-major 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 A2 corresponds to a negative orbital drift, while a positive value of A2 corresponds to a positive orbital drift. Therefore, the sign of an estimated A2 allows us to distinguish whether an asteroid is a retrograde rotator (A2 < 0) or a prograde rotator (A2 > 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 ar = A1(r0/r)2 (Farnocchia et al. 2015), where A1 is a solve-for parameter. The A1 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 area-to-mass 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.

thumbnail 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 area-to-mass ratio of 1.965 × 10−5 m2 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 well-known OrbFit 5.0.7 software2. 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 least-squares 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 A2 estimate in close agreement with the A2 estimates already published (see the Appendix A for details).

3 Yarkovsky effect detection

3.1 Least-squares regression

To try to detect both the SRP and the Yarkovsky effect, we performed an eight-parameter solution which included gravity and relativety, the Yarkovsky effect, and SRP. There were no prior constraints introduced in the least-squares fit. The estimate of A1 is a negative, non-physical value with a large uncertainty. This is due to the fact that for the near-circular 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 six-parameter solution which only included gravity/relativity without the Yarkovsky effect, and a seven-parameter solution with the inclusion of the Yarkovsky effect. For the six-parameter solution, we removed six outliers from 310 available observations, leaving 304 positions in the orbital fit. For the seven-parameter solution, we removed four outliers from the 310 available observations, leaving 306 positions in the orbital fit. Figure 2 depicts the post-fit plane of the sky astrometric residuals.

We also present the orbital solutions and root mean square (RMS) of the normalised post-fit residuals (χ2) of the six- and seven-parameter 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 χ2 for the six- and seven-parameter solutions were 0.415 and 0.403, respectively. This indicates that the seven-parameter solution provides a better fit to ground-based optical astrometry, although this decrease in the χ2 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 semi-major axis increases most significantly because of the strong correlation between A2 and the semi-major axis (Del Vigna et al. 2018). A correlation matrix of the seven-parameter solution (see Fig. 3) clearly indicates that high correlations are present between the A2 parameter and some other orbital elements, and especially the semi-major axis. This problem was already noted by Del Vigna et al. (2018), and indicates that including the A2 parameter in the least-squares fit results in an almost ill-posed problem.

In order to be sure that our determination of the A2 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 p-value 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 p-value can be computed as (3)

where the f(mY − mo, N − mY, x) is the F-distribution probability density function with mY − m0 and N − mY degrees of freedom. F is defined as (4)

where and . The value Oi is the ith observation and δi is the associated uncertainty. The values C0,i and CY,i are the ith computed value from the six-parameter solution and seven-parameter solution, respectively. The value N is the number of observations. The value m0 is the number of estimated parameters in the six-parameter solution and mY is the number of estimated parameters in the seven-parameter solution. In our case, we obtained a p-value 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 signal-to-noise ratio defined as , where is from an a posteriori covariance matrix of the seven-parameter 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 A2 from the non-gravitational solution is (−1.434 ± 0.410) × 10−13 au day−2, which yields a Yarkovsky-induced semi-major axis drift of da/dt = (-6.155 ± 1.758) × 10−3 au Myr−1. The negative value of A2 suggests that HO3 is a retrograde rotator.

thumbnail Fig. 2

Ground-based 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 gravity-only solution. Red dots correspond to the non-gravitational solution residuals.
Table 2

Orbital elements at epoch 2016 December 30 11:22:22.2765 TDT.

thumbnail Fig. 3

Posteriori correlation matrices corresponding to the six-parameter solution (panel a) and the seven-parameter solution (panel b).

3.2 The external assessment of the Yarkovsky detection

Although we detected the Yarkovsky effect of asteroid HO3 by fitting long-term ground-based 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 long-term behaviour of the HO3 orbit, by looking at the Yarkovsky drift (da/dt) of the semi-major 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 pv 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 (aB, eB) of asteroid Bennu were from JPL Small-Body Database3. The adopted orbital elements (a, e) of HO3 were values from our seven-parameter solution, as in Table 2. We obtained from Eq. (5) an expected semi-major 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 S-type 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 Yarkovsky-related value of semi-major 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 Yarkovsky-induced semi-major axis drift (see Fig. 3 in their work) for HO3 corresponding to different values of thermal conductivity K using statistical methods. The Yarkovsky-induced semi-major 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.

Table 3

Values of the physical properties for the asteroid (101955) Bennu and HO3 used in Eq. (5).

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 long-term (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 Yarkovsky-induced drifts, resulting in several thousand cases. We reran their simulation, but only with respect to our estimated Yarkovsky parameter A2 value corresponding to a semi-major 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 Short-term 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 short-term 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 λT librating around 180°, with amplitude of almost 180°) and quasi-satellite (i.e. resonance angle λT librating around 0°) dynamic states. The present-day HO3 dynamic state is that of a quasi-satellite 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 Long-term orbital evolution

We investigated the effect of the Yarkovsky effect on the long-term dynamics of HO3 by computing the distribution of the time spent in the Earth co-orbital region for the clonal orbits over the time interval (−5, 5) Myr. We used a semi-major 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 short-term orbital evolution of this object (see the third panel in Fig. 4). The distributions of the time spent in the Earth co-orbital 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 co-orbital 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 long-term 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 co-orbital 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 co-orbital 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).

thumbnail 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 (dEarth), resonance angle (λT), semi-major axis (a), eccentricity (e), and inclination (i) over time (order from top to bottom). The resonance angle (λT) 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 Tianwen-2 small-body 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 sample-return to Earth (Yan et al. 2022). In this section, we discuss the potential improvement of the detection of non-gravitational accelerations (Yarkovsky effect and SRP) acting on HO3 that can be provided by radio-tracking 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 area-to-mass ratio S/M of a spherical asteroid of 36 m in diameter and a density of 2120 kg/m3 (see Table 3), that is, 1.965 × 10−5 m2 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 A1 was derived from Eq. (2). In the real case, these physical parameters will be inferred from in situ measurements made by the Tianwen-2 spacecraft.

We simulated pseudo-range measurements (roundtrip delay of a two-way 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 non-gravitational accelerations (Chesley et al. 2014; Farnocchia et al. 2021a,b; Konopliv et al. 2002, 2014, 2018). We simulated eight potential post-rendezvous 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 one-night ground-based optical astrometry by a large telescope in mid-April 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 ground-based optical observations were simulated. We did not consider the ground-based radar observations because the size of HO3 is so small that it is almost impossible to observe it with ground-based radar. We performed an additional eight-parameter solution (including the Yarkovsky effect and SRP) with these additional simulated data, and if the eight-parameter solution failed, an additional seven-parameter solution (including Yarkovsky effect only) was performed. Table 4 lists the uncertainties obtained before and after the inclusion of these simulated Tianwen-2 pseudo range data and typical ground-based 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 ground-based 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 Tianwen-2 pseudo range data, the uncertainty of A2 decreased again by a factor of 6.8. This indicates that the Tianwen-2 radio science observations could improve the precision of the detection of the Yarkovsky effect down to 1%. Such a high-precision 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 Tianwen-2 pseudo range data.

Additionally, the Tianwen-2 radio science observations could significantly improve the predictions of future ephemeris. Potential ground-based 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 Tianwen-2 pseudo range data.

Table 4

Formal uncertainties with and without simulated Tianwen-2 pseudo range data and typical ground-based optical observation data.

thumbnail Fig. 5

Distributions of the time spent in the Earth co-orbital region T1:1 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 quasi-satellites, 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 ground-based optical astrometry. We detected a reliable Yarkovsky-induced semimajor axis drift of da/dt = (−6.155 ± 1.758) × 10−3 au Myr−1 for this object.

We used this semi-major 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 short-term orbital evolution. However, the Yarkovsky effect can delay the entry of the object into the Earth co-orbital region and accelerate its exit from this region in the case of past and future long-term 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 co-orbital 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 Tianwen-2 small-body exploration mission will improve both the detection of non-gravitational 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 Tianwen-2 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 (SKL-LPS(MUST)-2021-2023). 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 ground-based 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 outlier-removal 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 seven-parameter 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 A2 = (−2.135 ± 0.300) × 10−13 au/day2, in very good agreement with the value of (−2.121 ± 0.301) × 10−13 au/day2 as reported by the JPL Small-Body Database4 and the value of (−2.241 ± 0.298) × 10−13 au/day2 as given by the NEODyS5.

Table A.1

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

Table B.1

A posteriori covariance matrix of the six-parameter solution.

Table B.2

A posteriori covariance matrix of the seven-parameter solution.

Appendix C Orbital numerical integration

Numerical integrations were performed with a modified N-body code mercury 6by 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 point-mass 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 sb-N16 (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 N-body problem, small uncertainties in the initial orbit can cause large differences in the long-term 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 six-parameter solution and a gravitational plus Yarkovsky effect model for seven-parameter 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 Bulirsch-Stoer 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

Table 1

Known physical properties of HO3 (up to date).

In the text
Table 2

Orbital elements at epoch 2016 December 30 11:22:22.2765 TDT.

In the text
Table 3

Values of the physical properties for the asteroid (101955) Bennu and HO3 used in Eq. (5).

In the text
Table 4

Formal uncertainties with and without simulated Tianwen-2 pseudo range data and typical ground-based optical observation data.

In the text
Table A.1

Orbital elements at epoch 2015 July 3, 12:49:34.597424 TDT.

In the text
Table B.1

A posteriori covariance matrix of the six-parameter solution.

In the text
Table B.2

A posteriori covariance matrix of the seven-parameter solution.

In the text

All Figures

thumbnail 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 area-to-mass ratio of 1.965 × 10−5 m2 kg−1 and a Bond albedo of 0.083. See Sect. 5 for details.
In the text
thumbnail Fig. 2

Ground-based 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 gravity-only solution. Red dots correspond to the non-gravitational solution residuals.
In the text
thumbnail Fig. 3

Posteriori correlation matrices corresponding to the six-parameter solution (panel a) and the seven-parameter solution (panel b).
In the text
thumbnail 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 (dEarth), resonance angle (λT), semi-major axis (a), eccentricity (e), and inclination (i) over time (order from top to bottom). The resonance angle (λT) 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
thumbnail Fig. 5

Distributions of the time spent in the Earth co-orbital region T1:1 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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