Asteroid Orbit Determination Using Gaia FPR: Statistical Analysis

The Gaia mission has provided astrometric observations of unprecedented accuracy for more than 156,000 asteroids. The reported astrometric uncertainties are of the order of milliarcseconds, about 2 orders of magnitude smaller than that of traditional ground-based observations. The accuracy of Gaia data requires a high-fidelity orbit determination process, especially in the observation modeling. We present a statistical analysis of Gaia Focused Product Release to test the accuracy of the reported positions and associated uncertainties. We find that center-of-light offsets due to phase variations need to be modeled to properly fit the observational data. Prediction tests show that the uncertainty in the fitted orbits can be optimistic unless the observational uncertainty is inflated to account for errors in finding the center-of-mass of the body. Moreover, errors in the masses of small-body perturbers can cause differences in the orbital solution that exceed formal uncertainties of the best constrained orbits. As an example, we provide an update of the impact hazard analysis of 1950 DA, one of the asteroids observed by Gaia, and find that the impact probability in the year 2880 increases to 3.8 × 10−4.


Introduction
The astrometric observations of solar system objects obtained by the Gaia spacecraft provide a revolutionary opportunity to more accurately estimate the orbits of small bodies.Even though the main aim of the Gaia mission is to generate stellar astrometry (Gaia Collaboration et al. 2016b), solar system objects pass through its field of view frequently and are detected with much higher precision than ground-based observations (Spoto et al. 2018).The latest release at the time of writing this paper is Gaia Focused Product Release (FPR; Gaia Collaboration et al. 2023), which extends from 34 to 66 months the available observations of 156,792 objects that were included in Gaia Data Release 3 (DR3; Tanga et al. 2023).
Gaia Data Releases (from DR1; Gaia Collaboration et al. 2016a; to Gaia FPR) have already been used in multiple solar system studies and applications.Hanuš et al. (2018) used Gaia astrometry to perform a detailed analysis of asteroid (3200) Phaethon.Eggl et al. (2020) constructed a star catalog debiasing table to improve the accuracy of ground-based astrometry using the Gaia DR2 star catalog (Gaia Collaboration et al. 2018) as a reference.The use of Gaia's stellar catalogs improves stellar occultations, a technique to obtain highprecision astrometry (Spoto et al. 2017;Ferreira et al. 2022).Occultation data impose constraints on planetary motion.For example, Park et al. (2021) uses occultations of Pluto reduced against Gaia stellar astrometry in the development of the planetary ephemerides DE440 and DE441.Recent studies also point to the possibility to characterize small, yet relevant dynamical effects such as the Yarkovsky effect (Dziadura et al. 2023;Hung et al. 2023) and to estimate the mass of asteroids after flybys (Baer & Chesley 2017;Siltala & Granvik 2020).
In this paper we aim to provide a statistical treatment of Gaia astrometric measurements for the orbit determination of solar system objects.At the level of precision of Gaia FPR, we need high-fidelity observation and dynamical models.In particular, Tanga et al. (2023) show that the center-of-light offset relative to the center-of-figure as a function of the phase angle is significant and can lead to systematic errors.Kaasalainen & Tanga (2004) proposes models for this correction, including analytical expressions assuming a spherical shape.A higherfidelity formulation would need to take into account the shape and spin state of the asteroid, which are generally unknown (Muinonen & Lumme 2015).Gaia Collaboration et al. (2023) compute orbit solutions solely based on Gaia data and compare them to JPL's orbital solutions,1 which are mostly based on ground-based astrometry.They find a general agreement with a small yet significant bias in the semimajor axis for large asteroids.This bias could be due to the fact that Gaia Collaboration et al. (2023) do not model the phase-dependent photocenter offset, which is also more significant for larger objects.We derive a model for a photocenter correction and also increase the observation uncertainty to account for this systematic effect.
High fidelity in asteroid dynamics may require detailed physical information about the asteroid, as illustrated by the trajectory analysis of (101955) Bennu (Farnocchia et al. 2021) after the visit of NASA's OSIRIS-REx spacecraft (Lauretta et al. 2019).The masses of small-body perturbers are another important consideration, which can occasionally be estimated as their gravity affect the motion of other asteroids (Baer & Chesley 2017;Siltala & Granvik 2022;Li et al. 2023).The error in the perturber masses is generally not accounted for in the uncertainties of orbital solutions.Using a Monte Carlo experiment, we quantify the contribution of the uncertainty of perturber masses to the uncertainty in the orbit solutions.
We start in Section 2 by describing Gaia FPR data set, including the steps to preprocess the observations.Section 3 describes the orbit determination process, where we derive the photocenter correction.Section 4 shows the results of the orbit determination analysis using Gaia FPR.Next, in Section 5, we fit a shortened arc and compare the solutions to the full arc, with the aim of studying the prediction errors and their associated uncertainties.In Section 6, we assess the sensitivity of the uncertainty in the semimajor axis to the unknown masses of the perturbers.Using Gaia FPR observations we update the impact hazard of 1950 DA in Section 7. We conclude with the discussion of our results in Section 8.

Gaia FPR Solar System Astrometry
The full description of Gaia FPR Solar System Survey is discussed in Gaia Collaboration et al. (2023).In this section, we provide a brief description of the properties of the data set that are relevant to our study.
Gaia FPR includes observations taken from 2014 July 26 to 2020 January 20, or about 66 months.Therefore, Gaia FPR significantly extends the 34 months data arc from Gaia DR3.
When an asteroid enters the field of view of Gaia, it can be detected multiple times (up to 9) in a transit.Figure 1 shows the distribution of the absolute magnitude H of the asteroids in Gaia FPR, as well as the arc length and the number of transits as function of the absolute magnitude.The absolute magnitudes are obtained from the JPL Small-Body database, as estimated from photometric data reported to the MPC.Most objects brighter than H = 14 are observed consistently, with an average observational arc of 61 ± 3 months.Objects fainter than H = 14 are observed for an average of 55 ± 10 months.
Gaia's plane-of-sky observations of solar system objects are best described in a frame tied to the scanning direction: the along-scan direction is measured with great accuracy (order of mas) while the orthogonal direction (across scan) is orders of magnitude less accurate.(See postfit residuals in Figure 11 of Tanga et al. 2023.)As a result, the uncertainties in R.A. and decl.are expressed by a highly correlated covariance matrix.The ingestion of Gaia FPR requires careful steps (Gaia Collaboration et al. 2023).First, it is important to use the geocentric position of the spacecraft instead of the barycentric one, both of which are included in the data release.In fact, the barycentric position of the spacecraft depends on the specific planetary ephemeris adopted.A different set of bodies and different masses included in the solar system model can lead to significant differences in the state of the solar system barycenter.For example, between JPL Planetary and Lunar Ephemerides DE430 and DE440 the barycenter shifted by ∼100 km (Park et al. 2021).Instead, the geocentric position of the Gaia spacecraft is substantially independent of the specific planetary ephemeris model.
A second aspect to be considered is that the Gaia pipeline uses the Barycentric Coordinate Time as timescale, while the JPL Planetary and Lunar Ephemerides and our code use the Barycentric Dynamical Time.As a result, the Gaia position provided in the Focused Data Released needs to be multiplied by a factor (1 − L B ), where L B = 1.550519768 × 10 −8 (Klioner et al. 2010).Using the geocentric position of Gaia instead of the barycentric one would prove beneficial by greatly diminishing the consequences of neglecting this scaling factor.In fact, the Gaia spacecraft is roughly 1 au away from the solar system barycenter, and so the resulting error would be about 2 km.If fully projected on the plane of sky, the resulting astrometric error for an object observed when at 1 au from the spacecraft could be as large as 3 mas, which exceeds the typical quality of Gaia astrometry (Gaia Collaboration et al. 2023).On the other hand, the Gaia spacecraft is about 0.01 au from the Earth and so the resulting error in its position due to neglecting the proper scaling would be about 20 m.
A final consideration is about ensuring that the fitting process accurately models the observable.The Gaia astrometric positions correspond to the apparent position of the target, i.e., corrected for one-way light time.Gravitational light bending (Section 7.2.4 of Urban & Seidelmann 2012) must be included in the observable model and can be significant at the level of quality of Gaia data.Note that for Gaia data this is an absolute effect, while it is a differential effect for conventional astrometry, which is measured relative to background stars.On the other hand, the astrometric positions in Gaia FPR have already been corrected for annual aberration, due to the motion of the observer relative to the solar system barycenter.

Orbit Determination Methodology
The algorithm used to solve the orbit determination problem is based on a batch least-squares method, as described in more detail in Chodas & Yeomans (1999), Farnocchia et al. (2015).The force model used for the positions of the objects consists in an n-body formulation.The motion of the planets is modeled using the ephemeris model DE441 (Park et al. 2021) and SB441-N373 for the small-body perturbers (Farnocchia 2021).2These perturbers include 343 small-body pertubers and 30 Kuiper Belt objects.We also included the Yarkovsky effect as described in Farnocchia et al. (2013) in objects for which it was previously detected.Finally, we include relativity effects via a parameterized post-Newtonian formulation (Moyer 2003).As detailed in Tanga et al. (2023), Gaia FPR astrometric uncertainties have two terms: a random, measurement-specific component and a systematic component common to the whole transit.To account for the systematic nature of part of the error, we weight each observation using a covariance matrix C that represents the error in R.A. and decl.: where C rnd is the covariance of the random error component, C sys is the covariance of the systematic error component, and N the number of observations in the transit.The factor N is included to counteract the N decrease in the uncertainty of the mean.We reject outliers in the fit using the automatic rejection algorithm in Carpino et al. (2003).In some of those cases there is a full transit that is removed or a large fraction of it.These events do not necessarily mean that the full transit should be removed, instead, it may point to the fact that there are mismodelled dynamics or photocentric offsets, as we discuss in the following section.
The coordinates of the sources detected in astrometric observations correspond to the center-of-light (or photocenter).The photocenter can be offset from the center-of-figure, the outline of the body as seen from the observer.This offset in general increases with the solar phase angle.Also, the centerof-figure can be offset from the center-of-mass, the position of which is estimated in the orbit determination process.In the Gaia catalog, there are objects large enough that the offset between photocenter and center-of-figure is significant at the level of precision of the measurements.The observing geometries are limited by Gaia's scanning constraints, limited to solar elongations between 45°and 135° (Spoto et al. 2018).We consider this phenomenon at a systematic level in two ways: a phase-dependent correction to the astrometric position of the object center and an increase in the uncertainty of the measurements.

Photocenter Correction
The precise computation of the position of the photocenter would require knowledge of the shape of the object and photometric models of the reflectivity of their surfaces, all of which are generally unknown.Thus, we need to make some assumptions to simplify the computation and apply these corrections to represent systematic offsets in the photocenter.Kaasalainen & Tanga (2004) provide a number of expressions to compute the photocenter offset of a spherical object.The next step in fidelity to model this offset would be using an ellipsoidal shape model, which was considered in Muinonen & Lumme (2015).Here we assume the shape of the objects to be spherical due to aforementioned generalized lack of knowledge.
Using the Lommel-Seeliger scattering law, the expression for the offset in units of the object radius is (Batrakov et al. 1999): where α is the solar phase angle and R is the object radius.To speed up the evaluation of this correction at each observation, we approximate the d L−S /R function by the following polynomial: which removes the singularity at exactly α = 0 and differs by less than 0.4% of the body radius, which can be ignored.The maximum error occurs at a phase angle of 180 deg, while observations are typically taken at least tens of degrees aways from the Sun.Moreover, this difference is far smaller than uncertainties in size, shape or photometric properties.In Figure 3 we show the magnitude of the corrections in km as well as an equivalent angular offset if observed at 1 au.We estimate the radius of the object using Equation (4) from Harris & Harris (1997): where we use the absolute magnitude and assume an albedo of p V = 0.125, a typical value for C-type asteroids.
We obtain the plane-of-sky correction t considering the observation geometry in Figure 2. The correction is in this direction under the assumptions of spherical shape and isotropic reflectivity surface.The angular correction to the observation model is: where we use a bold font to denote vectors and the hat ( ˆ) operator to indicate a unit vector.Here, r ˆis the line-of-sight direction and rˆis in the direction of the heliocentric position of the object.The direction of the correction t ˆis opposite to the projection of the heliocentric position of the object into the plane of sky: Note that even if the direction becomes singular at α = 0, the photocenter offset is also 0 under the spherical object assumption.The corrections in R.A. and decl.become: • ˆ( ) where i R.A. ˆand i Decl ˆare the directions in the plane of sky corresponding to increasing R.A. and decl., respectively.

Photocenter Offset Uncertainty Term
The photocenter offset correction uses assumptions for the shape and light reflection properties.The use of an object's shape model can lead to a significantly different photocenter offset compared to the spherical correction (Kaasalainen & Tanga 2004;Muinonen & Lumme 2015).This additional complexity in the modeling of the photocenter requires information of the rotational state, which is frequently unknown and usually uncertain.In addition, there is also uncertainty in the object's radius as derived from H and p V .
Besides applying a photocenter offset, we also augment the astrometric uncertainty by adding a diagonal covariance with the nonzero elements (λR/ρ) 2 , where λ > 0 is a photocenter offset uncertainty factor that parameterizes the fraction of the asteroid's size to be assumed for the offset correction uncertainty.We adopt λ = 0.3 based on prediction tests discussed in Section 5. Figure 3 includes the magnitude of the photocenter offset uncertainty factor in contrast to the photocenter offset correction for asteroids of different sizes.A larger λ results in a larger uncertainty in the astrometric position and, in turn, in the estimated orbit.We note that this contribution is most significant for large objects since the photocenter offset uncertainty factor is proportional to R.Moreover, larger objects are observed more often and their orbits are better constrained, which means that the errors in the photocenter offset models are more statistically significant.
This increase in the uncertainty also captures the fact that the center-of-figure in general does not coincide with the center-ofmass of the body.The difference between the centers can be even larger if the asteroids are binaries.This effect can be significant, as modeled in the examples of Section 5.2 of Tanga et al. (2023).Considering that ∼15% of NEAs are binaries (Pravec et al. 2006), assuming even a small fraction of binaries in the main belt would mean a large number of cases in Gaia FPR data set.

Orbit Determination Results
After estimating the orbit of the 156,792 objects in Gaia FPR using the methodology described above, we proceed to show general results of the fit.There were only 63 instances of the data arc being too short to compute an orbit solution.For the other objects, we focus on the residuals of the along-scan direction (ΔAL), which is the high-accuracy Gaia observable.
Figure 4 shows the histograms of the residuals of the fit.About 93.3% of the residuals along-track residuals are below 10 mas, which shows the great overall precision of Gaia FPR data set.The mean of the along-track residuals is −0.004 mas with a standard deviation of 5.1 mas. Figure 5 shows the same residuals normalized by the uncertainty in the measurements.The result is an almost exactly normal distribution, which indicates that the reported uncertainty and data treatment are realistic.The standard deviation of the distribution is 0.997, which suggests that the uncertainties σ AL are slightly overestimated.This is partly a consequence of the use of the photocenter offset uncertainty factor, which is conservative but beneficial to the orbit solutions, as discussed in Section 5.
The duration of a typical nine-observation transit is roughly 39 s, so systematic effects will in general not average out in such a small window.Thus, it is worth looking at the statistical distribution of the per-transit mean residuals.Figure 6 shows the weighted mean of the residuals of each transit.We obtain the distribution mean and standard deviation after removing transits with weighted mean beyond ±10 mas, which we   consider as outliers.The standard deviation of the per-transit weighted mean of the residuals is mas.The mean of 0.006 mas is statistically compatible with 0. We note that we find a significant amount of transits with absolute mean residual larger than 10 mas, roughly 0.9% of all transits, even after the photocenter offset correction.These transits can be partially or totally removed from the fit when their uncertainty is considered.When λ > 0, more transits are accepted in the fit thanks to the more conservative uncertainty.
The residuals in the across-scan direction, shown in the bottom panel of Figure 4, are as inaccurate as expected from the observation system and display strong non-Gaussian features, as previously reported for Gaia DR2, DR3 and FPR (Spoto et al. 2018;Gaia Collaboration et al. 2023;Tanga et al. 2023).The standard deviation of the distribution is roughly 50 times wider with a nonzero mean and a non-Gaussian negative tail.For this reason, we considered the possibility of heavily deweighting the across-scan direction, effectively making Gaia observations one-dimensional.However, this approach yielded no material difference in the orbital solutions, thus showing that the along-scan component controls the orbital solutions anyway.Therefore, we decided that deweighting in the acrossscan direction was an unnecessary complication and use the highly correlated covariance previously described.
One key metric of the orbit uncertainty is the uncertainty in the semimajor axis, as it feeds into the mean motion and the position along the orbit.We use the symbol σ a for the uncertainty in semimajor axis, and σ a,FPR when important to specify that the full observational arc in Gaia FPR was used.The resulting uncertainties are shown in Figure 7, which also includes the uncertainties using shorter arc through 2017 June, corresponding to the Gaia DR3 arc.The median uncertainty is 2 × 10 −9 au, roughly 350 m.
The effect of the photocenter offset uncertainty factor is more significant for brighter objects.Figure 8 shows the cdf of the uncertainty in the semimajor axis for the group of 20,536 asteroids with H < 14 and σ a,FPR < 10 −8 au obtained with the described approach and corrections.If λ = 0 (neglecting the photocenter offset uncertainty factor) the resulting uncertainties are smaller, but as we describe in the following section, the predictions are less statistically consistent.
Because the corrections are more significant in larger asteroids, we show two examples of the residuals of the fit.Asteroid 779 Nina is close to the 0.1% percentile of brightest objects in Gaia FPR with an absolute magnitude of H = 8.02.
Asteroid 3557 Sokolsky is close to be at the 1% percentile of brightest objects included in Gaia FPR with an absolute magnitude of H = 11.06.We show the absolute residuals of 779 Nina in Figure 9, demonstrating the effect of the photocenter correction.Next, we show the normalized residuals of 3557 Sokolsky in Figure 10, demonstrating the effect of the photocenter offset uncertainty factor.The general photocenter offset correction effectively removes the bias in many of the transits that otherwise would have been rejected from the orbital fit of 779 Nina (Figure 9).The offset of the transits is corrected throughout the observation arc except around the end of 2016.Overall, the correction improved the fit but the magnitude of the correction might have been overestimated in these transits.Without correction, the transits become increasingly further from being zero mean as the solar phase angle increases, as seen in the bottom panel of Figure 9.In the case of 3557 Sokolsky (Figure 10) there is only one transit fully removed from the fit after applying the photocenter offset correction.However, there are a few with multiple observations excluded, as well as transits that are not zero mean even if accepted in the fit.The example of 3557 Sokolsky shows the effect of including observations in the fit that were excluded without the photocenter offset uncertainty factor.

Prediction Tests
In this section we test the uncertainties in the orbital solutions using a series of prediction tests.We compute short-  We include the uncertainties using the full observational arc (full FPR) and for short-arc solutions based on data through 2017 June, i.e., the data cutoff for Gaia DR3.The bin frequency is the number of objects in the bin divided by the total number of objects.arc solutions with data through 2017 June 1 and compare to the full-arc solution, assumed as the truth.The goal is to obtain unbiased predictions with errors statistically consistent with uncertainties.We map the orbit elements and uncertainties to the same reference epoch: 2017 June 1, which is shortly after the end of Gaia DR3 observations and close to the center of the arc.
Figure 11 shows the prediction test results using all the objects of the data set.The mean prediction error is −0.013σ, with a standard deviation of 1.18 indicating that the uncertainties are underestimated roughly by 18%.This result indicates that the uncertainties of the short arc are optimistic.However, the predictions are more conservative when setting λ > 0, as discussed next.In Figure 12 we show the significance of the photocenter correction and uncertainty factor in the prediction.We choose the group of 20,915 asteroids with H < 14 and σ a,FPR < 10 −8 au to study this effect.We select the brightest asteroids, as this effect increases with the size of the object and is magnified by tighter constraints on the orbit (see Figure 1).Because of that, their orbits are often well constrained.Moreover, a better constrained orbit ensures that the residuals reflect the astrometric errors.
Table 1 reports the mean and standard deviations of the different methodologies.The application of the correction is successful in reducing the mean prediction error from −0.61 ± 0.01 to −0.005 ± 0.01.The distribution of predicted semimajor axes still has a larger standard deviation than a Gaussian distribution would imply, indicating that the uncertainties in the predictions are optimistic.Once we include the photocenter offset uncertainty factor (λ > 0), the standard deviation of the predicted semimajor axes is reduced from 1.58 to 1.22.The difference in semimajor axis is normalized by the predictor uncertainty (i.e., (a long − a short )/σ a,short ).A Gaussian distribution is included for reference.The pdf is estimated by the bin count divided by the number of objects times the bin width (i.e., count/(N • w i )).
Figure 12.Semimajor axis prediction tests of the 20,915 asteroids with H < 14 and σ a,FPR < 10 −8 au truncating the observational arc until 2017 June 1, shown as in Figure 11.The predictions are made with or without photocenter offset correction and setting λ to the following values: with correction, λ = 0.3 (blue); with correction, λ = 0.0 (orange); and without correction, λ = 0.0 (yellow).The mean and standard deviations of the prediction errors using the three approaches are in Table 1.The photocenter correction could partially explain the bias in semimajor axis that Gaia Collaboration et al. (2023) found in large objects when comparing their solutions with the ones from the JPL's Small-Body Database.Those two solutions use different sets of observations and long observational arcs.The effect of the photocenter correction in semimajor axis as function of size is shown in Figure 13 for the same group of asteroids with H < 14 and σ a,FPR < 10 −8 au.We use the full observational arc of the Gaia FPR, either with and photocenter correction/uncertainty factor.The magnitude of the bias in semimajor axis continuously increases with size, starting from −0.10 ± 0.01 at H = 14 to −4 ± 1 at H = 10.This size-dependent bias is mitigated by the photocenter correction, which brings the median to −0.001 ± 0.007 at H = 14 and 0.4 ± 0.5 at H = 10.Note that the differences in semimajor axis due to the photocenter correction can easily exceed the formal semimajor axis uncertainty when λ = 0.

Sensitivity to Unknown Perturber Masses
Previous studies in the context of high-fidelity asteroid trajectory modeling (Farnocchia et al. 2021) and planetary ephemeris (Kuchynka & Folkner 2013) highlight the uncertainty in the masses of the main belt perturbers as a potentially significant source of uncertainty.In addition, in recent years the precise Gaia astrometry of asteroids that flyby these perturbers has been used to estimate the masses of a few of them (Baer & Chesley 2017;Siltala & Granvik 2020, 2021, 2022;Li et al. 2023).These results lead us to the following question: how often are the errors in the perturber masses significant at the level of the orbital uncertainty for the sample of asteroids in Gaia FPR?
We inspected this question with a Monte Carlo experiment consisting of 1000 asteroids randomly selected among our group of asteroids with H < 14 and σ a,FPR < 10 −8 au.Then, we varied the masses of the 40 most massive main belt perturbers for 100 test runs, according to the list in Farnocchia (2021).We modified the masses according to a normal distribution with mean and standard deviation as the estimated values reported in Appendix A of Farnocchia et al. (2021).The measured statistical dispersion in the semimajor axis obtained is the sensitivity to perturber masses, and we report it in Figure 14.
We find that in 2.8% of the cases the sensitivity of semimajor axis to perturber masses is larger than the formal uncertainty.If we do not include the photocenter offset uncertainty factor, the percentage goes up to 7.0%, indicating that the formal uncertainty of the orbital solution can be optimistic and not capture the orbital element variations due to the errors in perturber masses.Note that this experiment was of 1000 objects with with σ a < 10 −8 au, which is true for 84.6% of the objects when fitting the full Gaia FPR data set.
7. Updated Impact Hazard for 1950 DA Among the objects observed by Gaia is (29075) 1950, a kmsized near-Earth object discovered by C.A. Wirtanen at Lick Observatory in 1950 (Wirtanen & Vasilevskis 1950).Thanks to the orbital constrains from radar observations collected in 2001, Giorgini et al. (2002) was able to analyze the long-term trajectory of 1950 DA through 2880, when the minimum asteroid intersection distance between the Earth and the asteroid is near-zero (Fuentes-Muñoz et al. 2023) and there is a nonnegligible chance of reaching the Earth.On such long timescales, small terms in the force model become significant and a rigorous assessment was challenging because the Yarkovsky effect was not yet detected.Farnocchia & Chesley (2014) combined constraints from orbital fit and physical properties, both specific to 1950 DA and general of the asteroid population, to model the Yarkovsky effect and derived a 2.5 × 10 −4 impact probability.This result was eventually updated to 2.9 × 10 −5 as additional observational data were reported, new astrometric data weighting and debiasing schemes developed, and planetary ephemeris versions released. 3aia FPR contains 58 observations of 1950 DA from 2014 August to December.Gaia astrometry provides additional constraints on the 2880 encounter, which can be parametrized using Öpik's ζ coordinate (Valsecchi et al. 2003;Farnocchia et al. 2019).The ζ parameter reflects how early or late compared to Earth the asteroid is at the encounter.Therefore, a ζ value of 0 in 2880 would lead to an Earth impact (Farnocchia & Chesley 2014).Before Gaia data were included in the orbital solution, the estimate was ζ = 23.0 ± 11.6 million km, while the updated estimate obtained combining the data available from the MPC, radar astrometry, and Gaia FPR astrometry is ζ = 9.8 ± 8.0 million km (1σ).Therefore, the nominal trajectory moves closer to Earth and the uncertainty shrinks by 30%.As a result the impact probability is now 3.8 × 10 −4 , which corresponds to a Palermo Scale (Chesley et al. 2002) of −0.9.

Discussion
Using high-precision astrometry such as Gaia FPR for orbit determination of small bodies requires detailed observational and dynamical models, as described in Section 3. We validated Gaia FPR astrometric positions and uncertainties.By accounting for both random and systematic uncertainties, we find near Gaussian statistics for the along-scan direction, which is the one tightly constrained by Gaia astrometric measurements.
To properly model Gaia data, we implemented a phasedependent photocenter correction that significantly improves the accuracy of the orbit solutions, as demonstrated by prediction tests.We find that the distribution of prediction errors is wider than expected from the formal prediction uncertainties.This effect is partially mitigated by the photocenter offset uncertainty term, which is proportional to the object's size and deweights the observation.Possible improvements to reduce the statistical spread of prediction errors include a better photocenter offset model (e.g., ellipsoidal shape rather than spherical) and improved set of perturbers and corresponding masses.
At an individual level, the real photocenter offset due to the nonspherical shapes or binary systems is still present and causes systematics in the residuals of the orbital fit.The upside is that this sensitivity could enable/provide constraints on the object's size or even binary system orbit, as demonstrated in Tanga et al. (2023).Kaasalainen & Tanga (2004) describes the challenges of inverting physical properties solely from astrometry, as the photocenter position has limited information about the object's shape.Gaia DR3 included photometry, which they use to invert some physical properties (Tanga et al. 2023).The photometric inversion genetic algorithm includes an estimation of the spin pole and axial ratios of a triaxial ellipsoid Cellino et al. (2019).These shape models could potentially be used in the astrometric model to improve the fit.
The high-precision of Gaia FPR data set also calls for high fidelity in dynamical models.We inspected the sensitivity of the uncertainty in the semimajor axis to the masses of the main perturbers in the solar system.Acknowledging that asteroids with direct close encounters serve as probes to estimate the perturber's masses (Baer & Chesley 2017;Siltala & Granvik 2020), we performed an experiment at the systematic level.We find that errors in perturber masses, which are generally neglected, can lead to orbit solution differences exceeding their formal uncertainties.While this is not a widespread effect, perturber masses are bound to become an ever more important factor in modeling asteroid orbits.These results highlight the need to estimate perturber masses, which Gaia FPR and future releases facilitate (Siltala & Granvik 2022;Li et al. 2023).It is clear that the inclusion of Gaia astrometry can significantly reduce the uncertainty in orbit solutions.For example, we find a reduction in the uncertainty of the trajectory of 1950 DA during its Earth encounter of 2880, which leads to an increased impact probability.We leave outside of the scope of this work the evaluation of the accuracy of the solutions obtained solely with Gaia FPR compared to historically available observations.

Figure 1 .
Figure 1.The top panel shows the length of the observation arc as function of the absolute magnitude H, for solar system objects with H > 6.The color of the dots represents the number of observation transits.The bottom panel shows the distribution of absolute magnitudes of the objects in Gaia FPR data set, as histogram of object number count and estimated cdf from sorting the objects by H.

Figure 2 .
Figure 2. The asteroid position relative to the observer ρ and the heliocentric position of the asteroid r define the photocenter offset correction direction t ˆand are related by the solar phase angle α.The correction t is the projection into the plane of sky directions i RA ˆand i Decl ˆafter being scaled using Equation (5).

Figure 3 .
Figure 3. Photocenter offset as function of the solar phase angle for objects of 0.1, 1 and 10 km of radius.The equivalent angular correction if observed at 1 au (mas) is included for reference.The dashed lines correspond to the photocenter offset uncertainty term as observed at 1 au (λ = 0.3).

Figure 4 .
Figure 4.The top panel shows the along-scan residuals (mas) of the fit of Gaia FPR data set, bottom panel shows the across-scan residuals (mas).The orange bins account for all along-scan outliers beyond 30| | mas, some of which may be excluded from the fit.The bin frequency is the number of residuals in the bin divided by the total number of residuals.

Figure 5 .
Figure 5. Normalized along-scan residuals (mas) with respect to the uncertainty in the measurements shown as estimated pdf.The pdf is estimated by the bin count divided by the number of residuals times the bin width (i.e., count/(N Δ • w i )).A Gaussian distribution is overlapped for reference.

Figure 6 .
Figure 6.Transit weighted mean of along-scan residuals (mas) from Gaia FPR data set.The orange bins account for all outliers beyond 10 | | mas in the distribution of the mean.The gray histogram shows the residuals of Figure 4 for reference.The bin frequency is the number of transits in the bin divided by the total number of transits.

Figure 7 .
Figure 7. Uncertainty in the semimajor axis normalized by the semimajor axis.We include the uncertainties using the full observational arc (full FPR) and for short-arc solutions based on data through 2017 June, i.e., the data cutoff for Gaia DR3.The bin frequency is the number of objects in the bin divided by the total number of objects.

Figure 8 .
Figure 8. Uncertainty in the semimajor axis of the 20,536 asteroids with H < 14 and σ a,FPR < 10 −8 au used in the detailed prediction test.The uncertainties are shown for full arc Gaia FPR observational arc (Long arc), observations until 2017 June (Short arc), with and without the photocenter offset uncertainty factor (λ = 0, dotted, λ = 0.3 otherwise).The estimated cdf corresponds to the objects sorted by σ a .

Figure 9 .
Figure 9. Orbital fit residuals of asteroid 779 Nina (H = 8.02) using the photocenter correction (blue) and without photocenter correction (orange).Neither solution includes the photocenter offset uncertainty factor (λ = 0).The top panel shows the residuals over time, whereas the bottom panel shows the residuals as function of the solar phase angle.Residuals removed from the fit are shown with a cross symbol, whereas circles are included in the fit.

Figure 10 .
Figure10.Normalized orbital fit residuals of asteroid 3557 Sokolsky (H = 11.06), using the photocenter offset uncertainty factor (blue, λ = 0.3) and without photocenter offset uncertainty factor (orange, λ = 0).Both solutions include the photocenter offset correction.Residuals removed from the fit are shown with a cross symbol, whereas circles are included in the fit.The dashed lines show the ±3σ bounds for reference.The automatic rejection algorithm is fromCarpino et al. (2003).

Figure 11 .
Figure 11.Semimajor axis prediction test: truncating the observational arc until 2017 June 1 (subscript short ) with respect to the full arc (subscript long ).The difference in semimajor axis is normalized by the predictor uncertainty (i.e., (a long − a short )/σ a,short ).A Gaussian distribution is included for reference.The pdf is estimated by the bin count divided by the number of objects times the bin width (i.e., count/(N • w i )).

Figure 13 .
Figure13.Semimajor axis difference between long-arc solutions of the 20,915 asteroids with H < 14 and σ a,FPR < 10 −8 au.The nominal long-arc solution is compared to the long-arc solution without photocenter correction or photocenter offset uncertainty factor.The differences are normalized to the uncertainties in the latter (λ = 0).Cyan line indicates the moving median of the distribution, with 3σ standard deviation of the median bounds.

Figure 14 .
Figure14.Sensitivity of the semimajor axis to perturber masses via Monte Carlo as a function of the formal uncertainty in the semimajor axis.The results are of 1000 asteroids randomly picked from the H < 14 and σ a,FPR < 10 −8 au group.

Table 1
Mean and Standard Deviation of Prediction Tests Test with group of 20,915 filtered by H < 14 and σ a,FPR < 10 −8 au (using nominal).The number of solutions excludes outliers (±6σ). a