© The Authors 2024

1 Introduction

The Gaia mission is a revolutionary mission of the European Space Agency that delivers exceptionally precise astrometric measurements of celestial bodies throughout the entire sky (Prusti et al. 2016). Although the primary objective is to generate a precise 3D map of stars in the Milky Way, the mission also measures accurate positions of asteroids, including near-Earth objects (NEOs). The latest data release, known as the Focused Product Data Release (FPR), contains astrometric data for 446 NEOs, with an average astrometric uncertainty of ~50 mas (David et al. 2023; Tanga et al. 2023; Spoto et al. 2018).

These precise observations can increase our capability of determining the Yarkovsky effect. This phenomenon is associated with the anisotropic thermal emission of solar radiation absorbed by asteroid surfaces. It leads to secular changes in the orbital semi-major axis (Farinella et al. 1998; Bottke et al. 2006; Vokrouhlický et al. 2015). The effect plays a critical role in the delivery of NEOs from the main asteroid belt, long-term orbit propagation, calculation of Earth-impact probabilities, impact hazard assessment, and development of asteroid deflection strategies (Bottke et al. 2006; Farnocchia et al. 2015b).

The Yarkovsky effect has seasonal and diurnal components (Vokrouhlický et al. 2015). The seasonal component is related to the seasonal heating and cooling of an object’s hemispheres in its orbital motion. The diurnal effect relates to the heating and emission of thermal photons during the object’s rotation. The seasonal effect was first detected for the LAGEOS satellite (Rubincam 1988) and the diurnal effect for the asteroid (6489) Golevka (Chesley et al. 2003). The overall Yarkovsky effect depends on the surface conductivity, density, shape, and heliocentric distance of the object (Bottke et al. 2006; Vokrouhlický et al. 2015). For a 1 km sized NEO, the effect is aboutf 10⁻⁴ au/Myr (Greenberg et al. 2020).

The significance of the Yarkovsky effect was recently highlighted by the case of the asteroid (99942) Apophis, which is considered one of the most hazardous objects for nearly two decades prior to its 2021 observations. Precise radar measurements in 2021 allowed an accurate determination of the semi-major axis drift due to the Yarkovsky effect of da/dt = −199.0 ± 1.5 m/year¹ (Pérez-Hernández & Benet 2022). This resulted in the reduction of its Earth-impact probability during its 2029, 2036, and 2068 close approaches, which ultimately led to the removal of Apophis from impact risk tables². On 13 April 2029, during its next approach, Apophis will pass Earth at a distance smaller than the radius of the geostationary orbit. Currently, more than 1500 asteroids with non-zero impact probabilities are listed for the next 100 years. The Yarkovsky effect has recently been included in routine Earth-impact probability computations at the impact-monitoring system Sentry-II at the JPL (Roa et al. 2021). Previously, the Yarkovsky effect was factored into impact monitoring for only four asteroids: (99942) Apophis, (410777) 2009 FD, (101955) Bennu, and (29075) 1950 DA because the effect was sufficiently well determined (NEODys³). Through the NASA OSIRIS-REx mission, the Yarkovsky drift of (101955) Bennu was determined at da/dt = −284.6 ± 0.2 m/year, with an impressive signal-to-noise ratio of ~1400 (Farnocchia et al. 2021). The Yarkovsky effect for asteroids (410777) 2009 FD and (29075) 1950 DA was established through radar and optical observations (Del Vigna et al. 2019; Farnocchia & Chesley 2014). The estimation of the Yarkovsky drift for the 2009 FD asteroid played a crucial role in excluding its potential Earth impact in 2185 (Del Vigna et al. 2019), and for 1950 DA, it allowed the estimate of the impact probability in the year 2880, while before this, only an upper limit for its impact probability was derived (Giorgini et al. 2002). The Yarkovsky effect has already been detected for numerous other NEOs (e.g. Farnocchia et al. 2013a; Chesley et al. 2015; Nugent et al. 2012; Liu et al. 2023; Del Vigna et al. 2018; Fenucci et al. 2024), but with a lower statistical significance. The precise astrometry of the ESA Gaia mission is proving to be a game changer for many of these asteroids (Delbo et al. 2008).

In this work, we present the determination of the Yarkovsky effect for 52 NEOs contained in the Gaia FPR. Additionally, we conduct an analysis of close approaches for selected NEOs. Section 2 provides a description of the key features of the Gaia FPR astrometry and other data used in this investigation. The results are presented in Section 4, and Sections 3 and 5 cover the methods and summary, respectively.

2 Data

We used optical astrometry from the Gaia FPR and the Minor Planet Center (MPC) as well as radar data from the Jet Propulsion Laboratory (JPL). The Gaia FPR, was produced by the European Space Agency’s Gaia mission launched in 2013 (Prusti et al. 2016). Through a systematic scanning of the sky, the mission continues to provide milliarcsecond astrometry for a variety of celestial objects, including near-Earth asteroids. The newest Gaia FPR published on 10 October 2023, offers a substantial enhancement over previous releases, particularly Gaia Data Release 3 (DR3) (David et al. 2023; Vallenari et al. 2023; Tanga et al. 2023). The FPR includes precise astrometric data for more than 150000 asteroids over observational arcs that span up to 66 months. The observing arcs are about twice as long as those recorded by DR3. These extended orbital arcs provide a significant improvement in the determination of orbital parameters. They reduce the uncertainties of the orbital elements (David et al. 2023). Gaia asteroid observations have a different accuracy in the along-scan (AL) and across-scan (AC) directions. The AL direction offers a superior precision at the submilliarcsecond level, in contrast to the AC direction, where the accuracy is considerably lower, up to approximately one arcsecond. This disparity arises because AL data are derived from the Gaia astrometric field, whereas the AC information is solely sourced from the sky-mapper field of the CCD. This leads to a significant correlation between right ascension (RA) and declination (DEC) in Gaia astrometry, and it highlights the need to use the full observation covariance matrix for the orbit determination. This approach is essential to fully exploit the precision of Gaia astrometric measurements (David et al. 2023; Tanga et al. 2023; Spoto et al. 2018). In Gaia FPR, a marked advance over Gaia DR3 is not only the extended observational arc, but also the comprehensive reprocessing of the data. The independent processing method of the FPR, which used distinct selection criteria and acceptance thresholds for residuals, has resulted in occasional differences in the datasets. FPR has improved the orbital determination uncertainty significantly compared to DR3 (David et al. 2023).

In addition to Gaia FPR astrometry, we used the complete dataset available from the Minor Planet Center (MPC) on 7 November 2023⁴. The MPC database provides a comprehensive catalogue of positional information for minor planets, comets, and irregular outer natural satellites with an accuracy ranging from a few to several hundred milliarcseconds (mas). This dataset complements the Gaia data by providing a much longer observational arc that is essential for the long-term study of the orbital evolution of non-gravitational transverse acceleration. In the realm of astrometric observations of small Solar System bodies reported to the Minor Planet Center (MPC), the accuracy can vary significantly. It often ranges from a fraction of to over 1 arcsecond (Vereš et al. 2017). This variability in precision is influenced by factors such as the observation equipment, the observing conditions, and the data-processing methods employed. Consequently, the MPC astrometric data, which encompass a wide range of sources, exhibit a broad spectrum of accuracy levels that reflect the diverse nature of the contributing observations. Furthermore, we incorporated all available radar astrometry data from the Jet Propulsion Laboratory⁵. Radar astrometry is particularly valuable for determining precise orbits of NEOs, as it provides highly accurate range and velocity measurements that are not subject to the same limitations as optical observations (Ostro et al. 2002). The integration of the JPL radar data with optical astrometry from Gaia and the MPC allows a preciser and more reliable orbit determination. For this study, we used a comprehensive dataset comprising 625 531 observations from the MPC and 66 071 from Gaia, and 794 radar measurements. The summary of the observations is provided in Table A.1.

The combination of these diverse datasets, that is, the Gaia optical astrometry, the MPC minor planet catalogue, and the JPL radar measurements, enables a multifaceted approach to understanding the dynamical properties of asteroids. This is crucial for a precise orbit determination, especially for potentially hazardous asteroids (PHAs), and for supporting future space missions.

3 Methods

3.1 Orbit computation

The orbits of the investigated Near-Earth Objects (NEOs) were determined through the well-established least-squares differential correction procedure that was described extensively (Milani & Gronchi 2010; Milani et al. 2005; Farnocchia et al. 2015b). In each iteration, adjustments were made to the six orbital elements, along with an additional parameter A₂ (which parametrises the Yarkovsky drift), until the algorithm converged to the minimum of the linearised target function (Milani & Gronchi 2010; Milani et al. 2005; Farnocchia et al. 2015b), resulting in a nominal orbit. The transverse acceleration ${\mathbf{a}}_{\mathbf{t}}=A_{2}g(r)\hat{\mathbf{t}}$ was assumed to be directly related to the secular drift in the semi-major axis caused by the Yarkovsky effect, and g(r) is a function of the heliocentric distance r of the asteroid. Following Farnocchia et al. (2013a), we used g(r) = r⁻²au⁻². The unit vector $\hat{t}$ indicates the direction tangent to the orbit. The secular drift in semi-major axis da/dt can be estimated from Gauss variational equations (Farnocchia et al. 2013a), $$\frac{\mathrm{d}a}{\mathrm{d}t}=\frac{2A_2(1-e^2)}{n{p}^2},$$(1)

where e is the eccentricity, n is the mean motion, and p is the semilatus rectum.

The force model incorporated gravitational accelerations from the Sun, the planets, the Moon, the 16 largest asteroids, and the non-gravitational Yarkovsky effect. We used a modified version of the OrbFit software, which was developed at the Minor Planet Center⁶. The orbital solution was computed at an epoch near the midpoint of the observational arc. The OrbFit fitting program fully accounts for the gravitational deflection of light, refining the approximation used in Gaia FPR by accurately modeling the finite distances of observed objects. We selected geocentric positions from the Gaia data release, avoiding potential errors linked to the use of barycentric positions and discrepancies in planetary ephemerides (David et al. 2023). Additionally, the difference in timescales between the Gaia pipeline (TCB) and OrbFit (TDB) was addressed by converting into TDB within OrbFit. Because our study focused on smaller NEAs, the offset between the centre-of-light and centre-of-figure was not incorporated into our calculations. This decision reflects the limited impact of this factor on the precision required for our investigation of the Yarkovsky effect.

Before we computed the orbit, we corrected the astrometry for stellar biases originating from various catalogues (Farnocchia et al. 2015a). Additionally, an observation weighting scheme was incorporated that considered factors such as the year of observation, the observatory code, the apparent magnitude, and the type of observations. High-precision Gaia astrometry was incorporated, which encompassed considering the complete error covariance matrix, which accounts for both random and systematic errors (Spoto et al., in prep.). For a more detailed description of the fitting procedure, we refer to our earlier works (Dziadura et al. 2022 and Dziadura et al. 2023). Close approaches were computed through the propagation of nominal orbits (Milani & Gronchi 2010).

3.2 Acceptance criteria

To decide whether the empirically detected Yarkovsky effect was reliable, we compared the derived value with the theoretically expected value. The expected value was computed on the basis of the orbital and physical properties of the NEAs as (Spoto et al. 2015; Del Vigna et al. 2018) $$⟨\mathrm{d}a/\mathrm{d}t{⟩}_{\exp }=(\mathrm{d}a/\mathrm{d}t{)}_{\mathcal{A}}\times \left(\frac{\sqrt{a_{\mathcal{A}}}\left(1-e_{\mathcal{A}}^2\right)}{\sqrt{a}\left(1-e^2\right)}\right)\left(\frac{D_{\mathcal{A}}}{D}\right)\left(\frac{{\rho }_{\mathcal{A}}}{\rho }\right)\left(\frac{\cos \varphi }{\cos {\varphi }_{\mathcal{A}}}\right)\left(\frac{1-A}{1-A_{\mathcal{A}}}\right).$$(2)

Here, a is the orbit semi-major axis, e is the eccentricity, D is the diameter, ρ is the density, ϕ is the obliquity, A is the Bond albedo, and the values denoted with 𝒜 are the properties of (99942) Apophis. The parameter S was then calculated as S = A_{2 empirical}/A_{2 expected}, where A_{2 empirical} is the empirical value derived from orbital fitting and A_{2 expected} is the expected value (Eq. (2)). We defined accepted values as results that showed S > 2 and a signal-to-noise ratio (S/N) > 3 and marginal with S/N > 2.5. These thresholds are depicted as vertical and horizontal lines in Figure 1. Our method agrees with the approaches documented in Del Vigna et al. (2018); Spoto et al. (2015); Dziadura et al. (2022, 2023) and is compatible with the method recently adopted by the ESA NEO Coordination Center (Fenucci et al. 2024). Objects with S/N above the threshold and S above two were considered spurious detections. These detections should not occur, except in cases caused by different mechanisms for non-gravitational accelerations (A2). High A2 values may result from physical mechanisms such as thermal effects and solar radiation pressure (Farnocchia et al. 2023). Furthermore, the biases from older data might contribute to the erroneous detection of non-gravitational accelerations (Farnocchia et al. 2013a). The S value was estimated based on well-known orbital parameters and poorly known physical parameters of asteroids. However, the criterion is very strict, which ensured that we do not claim values that might not be true. We also analysed the results of A2 using different observational arcs to validate our findings.

4 Results

We computed orbits for 446 NEAs that incorporated the additional A₂ parameter. Of these, 43 can be considered credible, and 7 are probable. The detailed results for these 43 accepted A₂ values can be found in Table A.1. The orbital solutions for seven asteroids deemed probable are significantly influenced by a few older, separated, or precovery observations. The comparison of results with and without these earlier data is presented in Table A.2. These precovery detections provided additional data points that improved the orbital fits and reduced the uncertainties. However, the question remains whether we fully trust these few observations. Asteroid (4179) Toutatis, with a result of A₂ = −6.26±0.60, a tumbling asteroid, was excluded from our analysis because it iss sensitive to the selected number of perturbers and errors in their masses (Farnocchia et al. 2013a; Fenucci et al. 2024). Our acceptance and validation criteria involved comparing all results with the theoretically expected values of the A₂ parameter, which were estimated based on the orbital and physical parameters as described in Section 3.2. We required that the Yarkovsky drift derived from the orbit fitting be no more than twice as large as the expected drift based on the physical and orbital properties of the object. In addition, we required that the accepted objects have a signal-to-noise ratio S/N greater than 3 and marginal cases greater than 2.5. These thresholds are represented by vertical and horizontal lines in Figure 1. We identified 50 accepted objects (green) and 14 marginal objects (blue). Further details can be found in Table A.1.

We highlight the impact of the new Gaia FPR catalogue on orbit determination processes compared to Gaia DR3. Figure 2 illustrates the change in the accepted Yarkovsky drift objects, where the start of each grey line represents the value determined by DR3, and the end, marked by a coloured point, indicates the value determined by FPR. The other parameters and observations remained unchanged. The shift between using the two catalogues is mostly visible along the S/N axis. Due to the extended observational arc in the FPR for most objects, we observed an increase in the S/N ratio. Three asteroids moved to marginal cases as a result of a small decrease in S/N. In particular, asteroid (66391) Moshup shifted into the rejected category. This change was attributed to a minor decrease in A₂ in the FRP data A₂ = −2.7 10⁻¹⁵ au d⁻² compared to the DR3 data A₂ = −3.7 10⁻¹⁵ au d⁻², while the uncertainty levels remained similar. However, a larger number of objects benefitted from Gaia FPR: Nine objects moved to the accepted category, and 23 objects significantly increased the S/N ratio of their detections.

The further comparison is shown in Figure 3, which displays σ_A2 as a function of the arc in orbital periods covered exclusively by Gaia data for the orbit determination, accompanied by the corresponding distribution plots. The difference between the distributions comes from the additional data in the FPR as opposed to the earlier DR3. For a few objects with extensive radar observations and a well-established orbit, this addition does not significantly alter the uncertainty. However, even these slight variances in orbital uncertainties have led to the identification of new detections. In particular, the overall distribution of the results differs between the datasets; in the case of FPR, this distribution is moved towards lower values of σ_A2. A direct comparison of the accepted values between the two catalogues is presented in Figure A.1, where we plot all accepted objects (top panel) and marginal objects (bottom panel) using the FPR (purple) and juxtapose them with the DR3 values (green). In the upper panel, any DR3 value that does not meet the acceptance criteria is indicated by a diamond. We indicate an overall consistency between DR3 and FPR, accompanied by an improvement in the uncertainty of A₂. In particular, the application of FPR data reduced the uncertainty in the orbital parameters in most cases and resulted in nine additional accepted detections.

In addition, we compared our results with the available A₂ values from the JPL database Figure 4, which show minor differences from the values reported in other studies. These differences arise primarily from the use of alternative observational datasets and the instability of the orbital solutions, which depends on the specific datasets. The final result appears to be sensitive to the chosen weighting model, and other observational arcs are potentially included in the calculations. Most of our solutions agree within the 1σ confidence level. However, as shown in the figure, a few asteroids deviate significantly, with a value S > 2, and we therefore rejected them. In the corresponding figure, we compare all the asteroids in our sample with the available A₂ values from the JPL database. Of the 72 asteroids, some were rejected from our analysis, and these are also shown in the comparison.

Furthermore, we computed the orbits using only Gaia and radar data. Various studies (Spoto et al. 2018, 2019; Mouret & Mignard 2011) previously proposed that it may be possible to detect the Yarkovsky effect using Gaia data alone. Our analysis resulted in the accepted value for only one object with Gaia DR3 and four objects with Gaia FPR. This success was largely due to the significant number of available radar observations. However, it was not sufficient to rely on Gaia or radar data alone to detect the Yarcovsky effect. This highlights the importance of incorporating all available data sources for a precise orbit determination. The potential for detection using only Gaia data may increase with the anticipated publication of the complete Gaia catalogue in the future (Spoto et al. 2018, 2019; Mouret & Mignard 2011).

The impact of the Yarkovsky effect on orbit predictions has been highlighted in previous studies, such as Farnocchia et al. (2013b), whose analysis of the close approach of Apophis in 2029 shows significant differences in the uncertainty estimates when the Yarkovsky effect is considered versus a gravity-only model. To extend this analysis, we included the Yarkovsky effect in our computations of future close-approach distances of the 50 studied NEOs for 165 years forward, starting in November 2023 and ending in November 2188. We then compared these distances to close-approach distances computed based on purely gravitational orbits using orbits computed based on orbits determined using Gaia DR3 and FPR. For each asteroid, we computed the close-approach distance based on a purely gravitational orbit and based on an orbit including the non-gravitational Yarkovsky effect. The differences between these distances for the 50 asteroids are shown in Figure 5 for orbits computed on the basis of the two Gaia catalogues. Generally, the differences are consistent between the two catalogues. The differences between the distance based on a purely gravitational orbit and on an orbit with the Yarkovsky effect reach up to 0.1 lunar distance for most asteroids. For a single asteroid ((164121) 2003 YT1), we found a difference of up to 10 lunar distances. The results appear to be highly sensitive to the Gaia data and the weighting scheme applied. We now consider this result to be accurate, as it agrees with the solution provided by JPL database. Close approaches to Earth amplify the effect on an asteroid orbit. Even a small difference in the orbital parameters can cause this effect to accumulate over time, which explains the significant differences in the close-approach distances shown in Figure 5 for asteroid 164121. This asteroid passes relatively close to Earth (0.011 au in 2073), and as a result, the gravitational interactions during these approaches play a significant role in shaping its orbital evolution. However, additional observations are necessary to confirm the stability of the result in the current observational framework.

Fig. 1 Parameter S in relation to S/NA2 for all NEAs, as derived using Gaia FPR data. The green circles denote accepted values, characterised by S/NA2 greater than 3 and S values below 2. This agrees with the scaled expected A2 parameter for (99942) Apophis. The blue diamond indicates borderline cases for which the S/NA2 ranges between 2.5 and 3 and S remains under 2. The red stars highlight the instances in which S/NA2 falls below 3, and the thin orange diamonds represent objects fr which the S/NA2 exceeds 3 and S surpasses 2. The plot features a horizontal demarcation at S = 2 and a vertical threshold at S/NA2 = 3.

Fig. 2 Grey lines originate from the positions corresponding to Gaia DR3 data, leading to their new locations as determined by the application of Gaia FPR data. The axes and colours are the same as in Figure 1.

Fig. 3 Relative formal uncertainty of the Gaia A2 orbital parameter as a function of the arc-length coverage expressed in orbital periods. The green, blue, and red objects are accepted, marginal, and rejected objects computed using Gaia FPR (the density plot is shown in green). The grey dots show the solution based on Gaia DR3 (the density plot is shown in blue).

Fig. 4 Comparison of A2 values between our dataset and the JPL database for 72 asteroids. The rejected objects (in this study) with S > 2 are plotted in yellow.

Fig. 5 Difference in distance during close approaches between the distances determined from orbits with and without the Yarkovsky effect based on two different Gaia catalogues.

5 Summary

We searched for the signal of the Yarkovsky effect among 446 NEAs that were observed by Gaia. Of these, 50 successfully passed our validation criteria. However, 7 NEAs depended on old separated data. Our study confirmed the crucial role of combined data sources in enhancing the precision in a determination of the Yarkovsky effect. In particular, the integration of the Gaia FPR catalogue was shown to be useful in the refinement of orbital parameters. This is also evidenced by the identification of additional detections of the Yarkovsky effect. Furthermore, the limited detection of the Yarkovsky effect based on Gaia data alone highlights the limitations of the current catalogue and emphasises the need for including other astrometry and/or an extension of the observational arcs.

Our estimation of the close-approach distances of NEOs demonstrated the importance of including the Yarkovsky effect for a close-approach analysis. Several detections we presented have a very high S/N (>10). However, the Yarkovsky effect should be included in the long-term orbital evolution close-approach and Earth-impact analysis for all NEOs with reliable detections and S/N > 3. The final Gaia catalogue may provide a much higher number of high-S/N detections of the Yarkovsky effect. This would allow us to include the effect in routine close-approach and impact analyses for more NEOs.

Acknowledgements

The research leading to these results has received funding from the National Science Center, Poland, grant number 2022/45/N/ST9/01403 in the years 2023/2024. DO was supported by the National Science Center, Poland, grant number 2022/45/ST9/00267, P.B. was supported by Grant nr 2022/45/ST9/00267 and through the Spanish Government retraining plan Mara Zambrano 2021-2023 at the University of Alicante (ZAMBRANO22-04). This work has used data from the European Space Agency (ESA) Gaia mission https://www.cosmos.esa.int/gaia. This work presents results from the European Space Agency (ESA) Gaia space mission. Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC). The funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement (MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia. The Gaia archive website is https://archives.esac.esa.int/gaia and was possible thanks to the open source OrbFit software http://adams.dm.unipi.it/orbfit/. This research has used data and/or services provided by the International Astronomical Union Minor Planet Center, The JPL Database On-Line Ephemeris System, and the Gaia Collaboration. The authors thank all observers who submitted data to the Minor Planet Center. During the preparation of this work, the authors used Writefull for Overleaf and chatGPT for language editing and to improve readability. After using this tool/service, the authors reviewed and edited the content as needed and assume full responsibility for the content of the publication. The authors declare no competing interests.

Appendix A Estimated Yarkovsky effect

Fig. A.1 Comparison of the A2 with 1-σ uncertainty values computed using Gaia FPR and DR3. The top panel - accepted objects and the bottom panel - marginal objects. Diamond shaped purple points were rejected in Gaia DR3.

References

All Tables

All Figures

Fig. 1 Parameter S in relation to S/NA2 for all NEAs, as derived using Gaia FPR data. The green circles denote accepted values, characterised by S/NA2 greater than 3 and S values below 2. This agrees with the scaled expected A2 parameter for (99942) Apophis. The blue diamond indicates borderline cases for which the S/NA2 ranges between 2.5 and 3 and S remains under 2. The red stars highlight the instances in which S/NA2 falls below 3, and the thin orange diamonds represent objects fr which the S/NA2 exceeds 3 and S surpasses 2. The plot features a horizontal demarcation at S = 2 and a vertical threshold at S/NA2 = 3.

In the text

	Fig. 2 Grey lines originate from the positions corresponding to Gaia DR3 data, leading to their new locations as determined by the application of Gaia FPR data. The axes and colours are the same as in Figure 1.
In the text

	Fig. 3 Relative formal uncertainty of the Gaia A2 orbital parameter as a function of the arc-length coverage expressed in orbital periods. The green, blue, and red objects are accepted, marginal, and rejected objects computed using Gaia FPR (the density plot is shown in green). The grey dots show the solution based on Gaia DR3 (the density plot is shown in blue).
In the text

	Fig. 4 Comparison of A2 values between our dataset and the JPL database for 72 asteroids. The rejected objects (in this study) with S > 2 are plotted in yellow.
In the text

	Fig. 5 Difference in distance during close approaches between the distances determined from orbits with and without the Yarkovsky effect based on two different Gaia catalogues.
In the text

	Fig. A.1 Comparison of the A2 with 1-σ uncertainty values computed using Gaia FPR and DR3. The top panel - accepted objects and the bottom panel - marginal objects. Diamond shaped purple points were rejected in Gaia DR3.
In the text