Abstract
The leading source of uncertainty to predict the orbital motion of asteroid (99942) Apophis is a nongravitational acceleration arising from the anisotropic thermal reemission of absorbed radiation, known as the Yarkovsky effect. Previous attempts to obtain this parameter from astrometry for this object have only yielded marginally small values, without ruling out a pure gravitational interaction. Here we present an independent estimation of the Yarkovsky effect based on optical and radar astrometry which includes observations obtained during 2021. Our numerical approach exploits automatic differentiation techniques. We find a nonzero Yarkovsky parameter, A_{2} = (−2.899 ± 0.025) × 10^{−14} au d^{−2}, with induced semimajor axis drift of (−199.0 ± 1.5) m yr^{−1} for Apophis. Our results provide definite collision probability predictions for the close approaches in 2029, 2036, and 2068.
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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 EarthMoon distance^{1,2}. 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 postNewtonian gravitational corrections, tidal effects and nongravitational forces^{3,4}. The leading source of orbital uncertainty for Apophis is the Yarkovsky effect^{2,5,6,7}, a dissipating nongravitational interaction which induces a semimajor axis acceleration due to reemission of the incident solar radiation^{8,9,10,11}. Several attempts to obtain this acceleration for Apophis have reported marginal detections, essentially indistinguishable from a pure gravitational model^{7,12,13,14}.
Here we report independent results for a nonzero 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 highquality optical observations^{15}. 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 highprecision ephemeris of the major Solar System objects^{16}, which includes the Sun, all planets, Pluto, the Moon, and 343 mainbelt asteroids, and is consistent with JPL’s DE430 model^{17}; see Supplementary Note 1 for details and Supplementary Figure 1 for some comparisons.
For the treatment of the NEA, we consider Apophis as a massless testparticle subject to the Newtonian gravitational attraction from the Sun, the eight planets, the Moon, Pluto and the 16 most massive mainbelt asteroids (see Supplementary Table 1), including postNewtonian corrections (except for the asteroids) and Earth’s J_{2} zonal harmonic; see Supplementary Note 2. The choice of 16 asteroids is to have a fair comparison with other models^{6,13,17}, 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 r_{0} = 1 au, r_{Sun} 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 A_{2} which fits the optical and radar astrometry, based on precise numerical integrations of the equations of motion^{18}. This coefficient allows one to estimate the associated semimajor axis drift^{6}, \(\langle \dot{a}\rangle =2{A}_{2}(1{e}^{2}){r}_{0}^{2}/(n{p}^{2})\), where n is the mean motion, e is the eccentricity, p = a(1 − e^{2}) is the semilatus rectum and a is the semimajor axis of Apophis. All parameters (planetary masses, zonal harmonics, etc.) used in our integrations (cf. Methods sections) are taken from the JPL DE430 ephemeris documentation^{17}. We select initial conditions for our integrations from JPL’s #197 solution for Apophis^{12}, 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 2013^{5,12}, as well as three timedelays and a Doppler shift from observations of the recent Apophis flyby, obtained at Goldstone in early March 2021. In addition, we included 7,902 rightascension/declination astrometry observation pairs, which span from March 15th, 2004 through May 12th, 2021; these data are available through the Minor Planet Center observational database^{19}. 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 publiclyavailable groundbased optical astrometry for Apophis. Systematic errors due to timing, observational technique, reference star catalogs, etc., induce correlations in the orbital solution^{20,21}, which in turn bias the results^{22}. To account for these biases, our optical astrometry error model considers appropriate star catalog debiasing techniques^{23}, and a suitable weighting scheme^{24} 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 degreesoffreedom (DOF) gravityonly solution which uses a fixed value zero for the Yarkovsky parameter A_{2}, the OR6 solution, and a 7DOF nongravitational 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 gravityonly solution display a clear bimodal (two mode) distribution, which is not present for the corresponding residuals of the nongravitational solution. As illustrated by the frequency histograms of rightascension 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 gravityonly solution is not compatible with the observational dataset. Moreover, as illustrated in Fig. 2 the radar astrometry residuals for the gravityonly solution (purple circles) display statistically significant offsets, while the corresponding radar astrometry residuals offsets from the nongravitational solution are smaller and all consistent with zero. These results can be cast in quantitative terms using the postfit normalized RMS, mean weighted residuals and χ^{2}, provided in Table 1. Clearly, the nongravitational solution OR7 provides a more consistent fit to the radar and optical astrometry data.
Our estimate of the Yarkovsky transverse parameter obtained from the nongravitational solution is A_{2} = (−2.899 ± 0.025) × 10^{−14} au d^{−2}, which yields a Yarkovskyinduced semimajor 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 nongravitational 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 SmallBody Database for Apophis^{1}, 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 highquality optical astrometry^{15}. Notice that the value reported^{15} 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 bplane, which is related to the arrival times of the NEA^{25}. The figure displays the nongravitational solution computed with the data available before the 2021 flyby (blue curve, which yields^{26} A_{2} = (−4.97 ± 2.75) × 10^{−14} au d^{−2}), and the corresponding solution which also includes the data gathered during the 2021 flyby (orange curve). The histogram obtained with the 2021 data corresponds to a very narrow distribution around ζ_{2029} = 47, 363 km. This rules out a collision event on April 13, 2029, and allows us to estimate a closestapproach distance of (38, 011.8 ± 1.6) km (1σ 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 bplane 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.
We also considered the dependency of A_{2} in terms of the number of mainbelt 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 mainbelt asteroids, the change in A_{2} corresponds to 0.002σ with respect to the orbital fit obtained with 16 asteroids; σ denotes the uncertainty in A_{2}. In terms of \(\langle \dot{a}\rangle\), the difference is 0.003 m yr^{−1}. Öpik’s ζ coordinate on the 2029 bplane, ζ_{2029}, 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 8DOF orbital fit which, in addition to the initial conditions and the Yarkovksy effect, estimates the nongravitational acceleration due to the direct solar radiation pressure^{13} in the force model of Apophis. We model this acceleration as^{13} \({{{{{{{{\bf{a}}}}}}}}}_{{{{{{{{\rm{r}}}}}}}}}={A}_{1}{({r}_{0}/{r}_{{{{{{{{\rm{Sun}}}}}}}}})}^{2}\hat{{{{{{{{\bf{r}}}}}}}}}\), where \(\hat{{{{{{{{\bf{r}}}}}}}}}\) is the radial unit vector, which is characterized by the radial nongravitational parameter A_{1}. The uncertainty associated with the value we obtain is large and comparable to the magnitude of A_{1}, and therefore it only represents a marginal detection. The lack of a proper constrain of the uncertainty of A_{1} can be attributed to the dataset itself, which may require more highquality observations^{15}. We have not pursued this any further. We note that recent computations by JPL^{1} 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 nonzero Yarkovsky transverse parameter for Apophis, including its uncertainty. Highorder automatic differentiation is exploited to obtain the actual orbits, the fit to data of the nominal A_{2} value and initial conditions, the corresponding semimajor axis drift, and for the uncertainty calculations. The solution obtained provides qualitatively and quantitatively better OC 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 semimajor axis drift of \(\langle \dot{a}\rangle =(199.0\pm 1.5)\) m yr^{−1}. Moreover, this nongravitational solution allows us to predict a closestapproach distance of (38, 011.8 ± 1.6) km (1σ formal uncertainty) on April 13, 2029. We have also analyzed the orbital uncertainty on the 2029 bplane 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 2021^{1,15} serves as an independent validation, which is important for potentially hazardous asteroids.
Methods
The integration of the differential equations uses a variable stepsize 25order 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 A_{2}, which initially is set to zero. The numerical integration of Apophis^{18}, besides considering the nominal initial conditions and the nominal A_{2}, 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 observedminuscomputed (OC) 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 A_{2} 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 nonlinear terms which are ignored when the covariance matrix is constructed from the transitionstate matrix. The numerical stability of our results was addressed by contrasting them against JPL’s #197 solution^{12}, a gravityonly solution.
Data availability
The astrometry used to obtain the results of this paper can be found at https://github.com/PerezHz/NEOs.jl, under the data directory. Optical astrometry was retrieved from https://minorplanetcenter.net^{19} and radar astrometry from https://ssd.jpl.nasa.gov/sb/radar.html.
Code availability
All code used to obtain the results presented in this paper is in the public domain and was developed by the authors. The (Julia) package PlanetaryEphemeris.jl^{16} is available in https://github.com/PerezHz/PlanetaryEphemeris.jl and NEOs.jl^{18} in https://github.com/PerezHz/NEOs.jl.
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Acknowledgements
The authors are thankful to Davide Farnocchia (JPL) for correspondence and valuable insights, W. Folkner (JPL) for his assistance with the fine details related to the JPL DE430/431 ephemeris and Lázaro Alonso for his help in producing the figures. We are also thankful to the referees, whose comments and suggestions greatly improved the manuscript. This research has made use of data and/or services provided by the International Astronomical Union’s Minor Planet Center. We acknowledge financial support from the PAPIITUNAM project IG100819, and computer time provided through the project LANCADUNAMDGTIC284.
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J.A.P.H. wrote most of the code, performed the numerical simulations and their subsequent reduction. L.B. designed the original project. Both authors contributed to the analysis of the results and writing the paper.
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PérezHernández, J.A., Benet, L. Nonzero Yarkovsky acceleration for nearEarth asteroid (99942) Apophis. Commun Earth Environ 3, 10 (2022). https://doi.org/10.1038/s4324702100337x
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DOI: https://doi.org/10.1038/s4324702100337x
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