Introduction

Apophis (99942), a Near Earth Asteroid (NEA) discovered in 2004, will have a close approach to the Earth on April 13, 2029, with a proximity close to 1/10 of the Earth-Moon distance1,2. Computing the orbital dynamics of a NEA is a difficult problem, especially for assessing possible hazardous events where high precision is crucial: the NEA orbit is sensitive to several interactions, including post-Newtonian gravitational corrections, tidal effects and non-gravitational forces3,4. The leading source of orbital uncertainty for Apophis is the Yarkovsky effect2,5,6,7, a dissipating non-gravitational interaction which induces a semi-major axis acceleration due to re-emission of the incident solar radiation8,9,10,11. Several attempts to obtain this acceleration for Apophis have reported marginal detections, essentially indistinguishable from a pure gravitational model7,12,13,14.

Here we report independent results for a non-zero value of the unknown transverse Yarkovsky acceleration of Apophis, as determined from optical and radar astrometry data that include 2021 observations. Our results provide definite predictions for the close approach in 2029 of Apophis, as well as in 2036 and 2068. The results use an independent implementation for the planetary ephemeris and for the orbital dynamics of Apophis, as well as a different numerical integration technique. The value we obtain is essentially the same as the current one reported by the Jet Propulsion Laboratory (JPL)1 with minor differences in the uncertainties, and is also consistent with preliminary estimates which use high-quality optical observations15. While the underlying ephemeris and NEA models differ, the agreement of the values is a reproducible validation for the calculations, which is particularly important and desirable specially for potentially hazardous objects.

Results and discussion

Very precise ephemerides are required to compute possible hazardous events of NEAs. This imposes having a realistic model for the Solar System dynamics and for the asteroid. We have implemented a planetary model to compute high-precision ephemeris of the major Solar System objects16, which includes the Sun, all planets, Pluto, the Moon, and 343 main-belt asteroids, and is consistent with JPL’s DE430 model17; see Supplementary Note 1 for details and Supplementary Figure 1 for some comparisons.

For the treatment of the NEA, we consider Apophis as a mass-less test-particle subject to the Newtonian gravitational attraction from the Sun, the eight planets, the Moon, Pluto and the 16 most massive main-belt asteroids (see Supplementary Table 1), including post-Newtonian corrections (except for the asteroids) and Earth’s J2 zonal harmonic; see Supplementary Note 2. The choice of 16 asteroids is to have a fair comparison with other models6,13,17, though not all these models use the most massive asteroids. More importantly, the model also comprises the Yarkovsky effect on Apophis as an acceleration along the transverse direction of motion, \({{{{{{{{\bf{a}}}}}}}}}_{{{{{{{{\rm{t}}}}}}}}}={A}_{2}{\left({r}_{0}/{r}_{{{{{{{{\rm{Sun}}}}}}}}}\right)}^{2}\hat{{{{{{{{\bf{t}}}}}}}}}\), where r0 = 1 au, rSun is the heliocentric distance expressed in au, and \(\hat{{{{{{{{\bf{t}}}}}}}}}\) is the heliocentric unit transverse vector, which depends upon the velocity. Our goal is to determine the unknown proportionality constant A2 which fits the optical and radar astrometry, based on precise numerical integrations of the equations of motion18. This coefficient allows one to estimate the associated semi-major axis drift6, \(\langle \dot{a}\rangle =2{A}_{2}(1-{e}^{2}){r}_{0}^{2}/(n{p}^{2})\), where n is the mean motion, e is the eccentricity, p = a(1 − e2) is the semi-latus rectum and a is the semi-major axis of Apophis. All parameters (planetary masses, zonal harmonics, etc.) used in our integrations (cf. Methods sections) are taken from the JPL DE430 ephemeris documentation17. We select initial conditions for our integrations from JPL’s #197 solution for Apophis12, at the epoch December 17, 2020 00:00:00.0 (TDB).

The results discussed below include the whole radar astrometry measurement data sets spanning from 2005 through 20135,12, as well as three time-delays and a Doppler shift from observations of the recent Apophis flyby, obtained at Goldstone in early March 2021. In addition, we included 7,902 right-ascension/declination astrometry observation pairs, which span from March 15th, 2004 through May 12th, 2021; these data are available through the Minor Planet Center observational database19. Apart from a set of observations performed on January 28, 2021, which we excluded due to clear systematic bias, we have included essentially all the publicly-available ground-based optical astrometry for Apophis. Systematic errors due to timing, observational technique, reference star catalogs, etc., induce correlations in the orbital solution20,21, which in turn bias the results22. To account for these biases, our optical astrometry error model considers appropriate star catalog debiasing techniques23, and a suitable weighting scheme24 which accounts for sources of systematic errors other than star catalog biases. See Supplementary Note 4 for further details.

We have produced two orbital fits to the astrometry data described above: a 6 degrees-of-freedom (DOF) gravity-only solution which uses a fixed value zero for the Yarkovsky parameter A2, the OR6 solution, and a 7-DOF non-gravitational solution, which fully takes into account the Yarkovsky effect, the OR7 solution. The results are presented in Figs. 1 and 2; see also Supplementary Note 5. Optical astrometry residuals for the gravity-only solution display a clear bimodal (two mode) distribution, which is not present for the corresponding residuals of the non-gravitational solution. As illustrated by the frequency histograms of right-ascension and declination residuals, upper and right panels of Fig. 1, the purple distributions have two bumps. These follow from the two clusters of (purple) points illustrated in the main panel. Therefore, since we expect the residuals to follow a Gaussian distribution, we interpret these bimodal distributions as an indication that the gravity-only solution is not compatible with the observational dataset. Moreover, as illustrated in Fig. 2 the radar astrometry residuals for the gravity-only solution (purple circles) display statistically significant offsets, while the corresponding radar astrometry residuals offsets from the non-gravitational solution are smaller and all consistent with zero. These results can be cast in quantitative terms using the post-fit normalized RMS, mean weighted residuals and χ2, provided in Table 1. Clearly, the non-gravitational solution OR7 provides a more consistent fit to the radar and optical astrometry data.

Fig. 1: Optical astrometry observed-minus-computed residuals.
figure 1

a The horizontal axis corresponds to right-ascension residuals and the vertical to declination residuals. Purple circles correspond to the residuals obtained from the gravity-only solution (A2 = 0), and the average residual is illustrated as a blue square. Green circles correspond to the non-gravitational solution residuals; their average is marked with an orange circle. The ellipses centered at each average residual value represent the 3-σ level confidence. b Frequency histogram of right-ascension residuals. c Frequency histogram of declination residuals. Notice that the purple histograms display bimodal (two mode) distributions.

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Fig. 2: Goldstone and Arecibo radar astrometry residuals in time.
figure 2

a Time-delay and b Doppler-shift residuals. Purple circles correspond to the gravity-only solution, while green squares correspond to the non-gravitational solution. The horizontal axis corresponds to the observation date; notice that it has a different scale in each segment of the plot. Error bars represent observational uncertainties. The residuals corresponding to the non-gravitational solution, together with their uncertainties, are compatible with zero; this is not the case for all gravity-only residuals.

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Table 1 Fit quality statistics for the OR6 gravity-only and OR7 non-gravitational orbital solutions.
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Our estimate of the Yarkovsky transverse parameter obtained from the non-gravitational solution is A2 = (−2.899 ± 0.025) × 10−14 au d−2, which yields a Yarkovsky-induced semi-major axis drift of \(\langle \dot{a}\rangle =(-199.0\pm 1.5)\) m yr−1; the specific initial conditions and orbital elements for the solutions obtained are provided in Supplementary Tables 2 and 3. Therefore, our results show clearly a non-gravitational acceleration acting upon Apophis, due to the Yarkovsky effect. The value obtained for the Yarkovsky transverse parameter practically coincides with the current value reported at the JPL Small-Body Database for Apophis1, namely \({A}_{2}^{{{{{{{{\rm{JPL}}}}}}}}}=(-2.901\pm 0.019)\times 1{0}^{-14}\) au d−2. In addition, it is consistent with preliminary results which incorporate high-quality optical astrometry15. Notice that the value reported15 for \(\langle \dot{a}\rangle\) does not quite match the value we obtain; it is difficult to comment on this due to the lack of details on this estimate.

Based on these results, we turn now to the assessment of the collision probability for the close approach that will take place in April 13th, 2029. In Fig. 3 we plot the probability density of Öpik’s ζ coordinate on the 2029 b-plane, which is related to the arrival times of the NEA25. The figure displays the non-gravitational solution computed with the data available before the 2021 fly-by (blue curve, which yields26 A2 = (−4.97 ± 2.75) × 10−14 au d−2), and the corresponding solution which also includes the data gathered during the 2021 fly-by (orange curve). The histogram obtained with the 2021 data corresponds to a very narrow distribution around ζ2029 = 47, 363 km. This rules out a collision event on April 13, 2029, and allows us to estimate a closest-approach distance of (38, 011.8 ± 1.6) km (1-σ formal uncertainty). In Fig. 3 we have also included the estimated ζ values which correspond to possible future impacts during the close approaches in 2036 and 2068; these values were obtained from the resonant circles on the 2029 b-plane using jet transport techniques; details are included in Supplementary Note 6. The values corresponding to these events are located, respectively, at −756 σ and 1163 σ, with respect to the nominal prediction and therefore their probability of occurrence is negligible. Yet, these results should be revisited in the future to incorporate new observations which will refine the quality of the fit that defines the nominal orbit and hence the collision assessment.

Fig. 3: Probability distribution of ζ2029.
figure 3

Probability distribution on the 2029 b-plane due to the uncertainty in the non-gravitational orbital solution, which includes the Yarkovsky effect, with (orange line) and without (blue line) including the 2021 optical and radar astrometry. The vertical dashed and dotted lines represent, respectively, the ζ coordinate value on the 2029 b-plane corresponding to a resonant return in 2036 and 2068.

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We also considered the dependency of A2 in terms of the number of main-belt asteroids used in the dynamical model. The results presented use perturbations from the 16 most massive asteroids in the main belt. If we instead consider the gravitational perturbations on Apophis from the 32 most massive main-belt asteroids, the change in A2 corresponds to 0.002σ with respect to the orbital fit obtained with 16 asteroids; σ denotes the uncertainty in A2. In terms of \(\langle \dot{a}\rangle\), the difference is 0.003 m yr−1. Öpik’s ζ coordinate on the 2029 b-plane, ζ2029, is shifted by 2 m, while the ζ values corresponding to resonant returns in 2036 and 2068 change by −0.6 km and 2.5 km, respectively.

We also considered an 8-DOF orbital fit which, in addition to the initial conditions and the Yarkovksy effect, estimates the non-gravitational acceleration due to the direct solar radiation pressure13 in the force model of Apophis. We model this acceleration as13 \({{{{{{{{\bf{a}}}}}}}}}_{{{{{{{{\rm{r}}}}}}}}}={A}_{1}{({r}_{0}/{r}_{{{{{{{{\rm{Sun}}}}}}}}})}^{2}\hat{{{{{{{{\bf{r}}}}}}}}}\), where \(\hat{{{{{{{{\bf{r}}}}}}}}}\) is the radial unit vector, which is characterized by the radial non-gravitational parameter A1. The uncertainty associated with the value we obtain is large and comparable to the magnitude of A1, and therefore it only represents a marginal detection. The lack of a proper constrain of the uncertainty of A1 can be attributed to the dataset itself, which may require more high-quality observations15. We have not pursued this any further. We note that recent computations by JPL1 report the value \({A}_{1}^{{{{{{{{\rm{JPL}}}}}}}}}=(5\pm 4.903)\times 1{0}^{-13}\) au d−2.

Summarizing, using automatic differentiation techniques together with optical and radar astrometry we have obtained an orbital solution with a non-zero Yarkovsky transverse parameter for Apophis, including its uncertainty. High-order automatic differentiation is exploited to obtain the actual orbits, the fit to data of the nominal A2 value and initial conditions, the corresponding semimajor axis drift, and for the uncertainty calculations. The solution obtained provides qualitatively and quantitatively better O-C residuals than the one which includes only a gravitational interaction, i.e., which assumes a zero value for the Yarkovsky parameter. The acceleration associated with the Yarkovsky effect induces a semi-major axis drift of \(\langle \dot{a}\rangle =(-199.0\pm 1.5)\) m yr−1. Moreover, this non-gravitational solution allows us to predict a closest-approach distance of (38, 011.8 ± 1.6) km (1-σ formal uncertainty) on April 13, 2029. We have also analyzed the orbital uncertainty on the 2029 b-plane to conclude that this close approach will not lead to impacts on neither 2036 nor 2068, though a more refined analysis is required to discard the possibility of impacts in the next 200 years. The consistency of these results with other estimations using also data from 20211,15 serves as an independent validation, which is important for potentially hazardous asteroids.

Methods

The integration of the differential equations uses a variable step-size 25-order Taylor integration method for the planetary ephemeris and for Apophis orbit, with a small absolute tolerance (10−20). An additional differential equation is added for the Yarkovsky parameter, \(\dot{{A}_{2}}=0\), which assumes the constancy of Yarkovsky parameter as a first approximation, and allows one to consider small variations around an a priori nominal value of A2, which initially is set to zero. The numerical integration of Apophis18, besides considering the nominal initial conditions and the nominal A2, also integrates small deviations from these nominal values. These deviations are included as truncated polynomials in the small deviations, up to order 5, and are transported along the integration in time; see Supplementary Note 3, for details. At each step of the numerical integration we compute polynomials in time for the positions and velocities of all the Solar System objects considered. For Apophis, the coefficients of those polynomials are polynomials in the small deviations from the nominal values of the initial conditions and Yarkovsky parameter. These series expansions, combined with radar and optical astrometry data, allow us to obtain the optimal deviations from the nominal values by minimizing the observed-minus-computed (O-C) residuals. The optimal deviations in the unknown parameters are obtained through a Newton iterative method, which exploits the derivatives of the polynomials that the integration produce. We further exploit automatic differentiation when constructing the covariance matrix used to compute the uncertainty of A2 and the initial conditions. The covariance matrix is the inverse of the Hessian, which is obtained directly from the explicit time series produced by our integrator. In this sense, we include non-linear terms which are ignored when the covariance matrix is constructed from the transition-state matrix. The numerical stability of our results was addressed by contrasting them against JPL’s #197 solution12, a gravity-only solution.