Development of Impact Probability Estimation System for Near-Earth Objects

Trans. JSASS Space Tech. Japan Vol. 7, No. ists26, pp. Pk_11-Pk_15, 2009

By Tomohiro Yamaguchi1) and Makoto Yoshikawa2)

1)Department of Space and Astronautical Science, The Graduate University for Advanced Studies, Sagamihara, Japan 2)Japan Aerospace Exploration Agency, Sagamihara, Japan

(Received April 25th, 2008)

In this paper, the estimation method of impact probability for Near-Earth Objects (NEOs) is investigated. The impact probability of NEOs has been calculated by a linear target plane analysis and a Monte-Carlo method. Since the collisions of NEOs with the Earth are quite sensitive problems, the calculations have to be conﬁrmed by several methods. A linear target plane analysis cannot be applied if the position uncertainty is too large since the uncertainty ellipsoid is not a good assumption in this case. A Monte-Carlo method can be used for a large position uncertainty, but the computational cost is high. The limitation of using a linear target plane analysis is investigated using the Monte-Carlo method for the close approach of 99942 Apophis (2004 MN4). The relation between impact probability and observation accuracy is investigated by analyzing the close approach of 2007WD5.

Key Words: Near•Earth Objects, Impact Probability, Orbit Estimation

1. Introduction 2. Linear mapping to the modiﬁed target plane

Asteroids may provide significant information on the solar 3. Impact probability calculation system formation process. Since sending a mission to Near- The details are described in following subsections. Earth Objects (NEOs) require less energy than for main-belt 2.1. Orbit determination of asteroid asteroids, spacecraft can rendezvous and orbit NEOs. For The orbit determination of asteroid is necessary for the example, Hayabusa is aimed to become the first sample return impact probability calculation. Asteroid orbits are calculated 1) spacecraft from the Near-Earth Asteroid Itokawa . by a weighted least square method. The orbit determination These small celestial bodies provide us not only with problem is to estimate the six orbital elements X 0 at epoch scientific knowledge but also catastrophic hazard, as some t0 and with covariance matrix CX from a set of m astrometric asteroids approach the Earth and have the possibility of measurements. The measurement set is given by right ascension impacting the Earth. A lot of ground Earth observation α and declination δ. The weighted least squares procedure efforts are made for searching NEOs, and several NEO survey minimizes the following weighted sum of the residuals Q, programs are running to discover such dangerous celestial 2) 1 bodies . Then the orbit of discovered asteroids are calculated Q = ρ TWρ (1) and the impact probabilities with respect to the Earth and m the other planets are estimated. For example, the close where ρ is a vector of residuals and W is symmetric matrix approach to the Earth of asteroid 1997XF11 and 1999AN10 which reflects the observation errors. The best-fitting solution 3,4) is investigated . Since, the impact probability calculation X 0 and covariance matrix CX are solved by the Newton- is quite a sensitive problem, confirming the results by several Raphson method. methods is important. The asteroid motion can be calculated as follows: In this paper, the impact probability calculation using the ( ) d2r µ n r − r r data of optical observation is investigated. The linear mapping = − r + ∑ µ k − k + ∆ (2) dt2 r3 k |r − r|3 r3 technique is used to calculate the uncertainty ellipse on the k=1 k k modified target plane. The analysis for close approach of where r is heliocentric position vector of the asteroid, rk and asteroid Apophis (2004MN4) and 2007WD5 is described as µk are heliocentric position vector and gravity constant of the examples. perturbing body k, µs is a gravity constant of the Sun, n is the number of perturbing bodies and ∆ is a vector of relativistic 2. Impact Probability Calculation for NEOs perturbative accelerations of the Sun. Perturbations due to eight planets, Pluto and the three largest asteroids, Ceres, Pallas, and The impact probability calculation for NEOs is calculated by Vesta are included. The positions of eight planets, Pluto, and the following three steps. Moon are taken from the JPL ephemeris DE405. The others 1. Orbit determination of the asteroid are taken from JPL Horizons. The differential equations of

4.0 V∞out RA ^ Modified 3.0 SMTP Target Plane 2.0

^ 1.0 SBP V T^ 0.0 CA MTP -1.0

Residuals[arcseconds] -2.0 rCA T^ -3.0 B BP Error ellipse R^ -4.0 MTP 4.0 ^ DEC RBP 3.0 B-plane 2.0 Hyperbolic path 1.0 of asteroid Incoming asymptote 0.0 V∞in -1.0

Residuals[arcseconds] -2.0

Fig. 1. Modiﬁed target plane and B-plane (Target plane) -3.0

-4.0 Jan 01 Jul 01 Jan 01 Jul 01 Jan 01 Jul 01 Jan 01 RMTP 2004 2004 2005 2005 2006 2006 2007 Date [UTC] Nominal trajectory

Fig. 3. Post-ﬁt residuals of Apophis optical observations

1σ 2σ 3σ 4σ 5σ MTP

TMTP T CCA = (RΦ)CX (RΦ) (3) Target planet disk where R describes the matrix representing the rotation from the inertial reference frame to the modified target plane frame. Fig. 2. Bivariative Gaussian distribution and target planet disk The closest approach time is calculated by monitoring the distance between the asteroid and the Earth during the asteroid orbit propagation. The propagation is calculated using Eq. (1). the asteroid are solved by the 8th order Runge-Kutta method 2.3. Impact probability calculation developed by Prince & Dormand5). The impact probability is calculated by integrating the 2.2. Linear mapping to the modified target plane confidence region over the target planet disk. The confidence We use the confidence region on the modified target plane4) region (ellipsoid) is described as follows: to calculate the impact probability. The modified target plane (MTP) is oriented normal to the relative velocity with respect ∆ T −1 ∆ ≤ σ 2 ( X) CCA ( X) (4) to the target planet at the point of closest approach. The MTP ∆ coordinate is described in Fig. 1, where SˆMTP axis is parallel where X is the vector of the error from the nominal (best- σ to the relative velocity at closest approach V CA, TˆMTP axis is fitting) solution on the MTP frame and is the sample standard parallel to the equatorial plane and normal to SˆMTP axis, and deviation. RˆMTP axis is the cross product of SˆMTP and TˆMTP. The B-plane Then the bivariate Gaussian probability density function coordinates (SˆBP,TˆBP,RˆBP) are also described in Fig. 1, and the on the MTP can be calculated from the confidence ellipsoid difference between the modified target plane and the B-plane (Fig. 2). The integration of the probability density over the cross arises from the gravitational force of the target planet. sectional area of the target plane is calculated by the algorithm 6) The covariance matrix at the time of close approach CCA is developed by Michel . necessary to calculate the confidence region. The subscript CA implies the close approach time. CCA is calculated by mapping 3. Results and Discussion the covariance matrix at epoch CX onto the MTP. The state transition matrix Φ and rotation matrix R map the CX on the In this paper, the Earth impact probability of asteroid 99942 Apophis and the Mars impact probability of 2007WD5 are calculated.

Pk_12 T. YAMAGUCHI et al.: Development of Impact Probability Estimation System for Near-Earth Objects

Table 1. Orbit solution of Apophis (May 14.0, 2008) 1.0e+10 2029 Earth σ Value Uncertainty (1 ) 1.0e+08 encounter a AU 0.9224082716 9.95803806e-08 e 0.1911777108 3.72952034e-07 1.0e+06 i deg 3.3314018464 5.78335595e-06 Ω deg 204.44800475 1.06188046e-04 ω deg 126.38626504 1.08841089e-04 1.0e+04 M deg 32.481408398 2.12724026e-04

Uncertaintyregion volume ratio 1.0e+02

10000 1.0e+00 Earth disk 2005 2015 2025 2035 Date

0 Fig. 5. Change of uncertainty region R R [km] -10000 37,870 km wide. The nominal miss distance, the minimum miss distance, the relative velocity and the impact probability at the closest

-20000 Error ellipse is 13,740 km long approach are collected in Table 2. The Monte-Carlo results are by 726 km wide also described in Fig. 4, and most points are in the 3-sigma

-40000 -30000 -20000 -10000 0 10000 uncertainty ellipse. The nominal miss distance is base on the T [km] best-fitting solution and the minimum miss distance is based on the 3-sigma uncertainty ellipse. The minimum miss distance is Fig. 4. Uncertainty ellipse of Apophis 2029 Earth encounter (3σ) larger than 30,000 km and the impact probability is predicted to be about zero. 3.1.1. Limitation of linear mapping 3.1. Example 1: 99942 Apophis Linear mapping can be available when the uncertainty of the Asteroid 99942 Apophis (2004MN4) was discovered on June solution is small enough. If the uncertainty becomes large, 19, 2004. The diameter is estimated as 27060 m7). the confidence region cannot be approximated as the ellipsoid We calculate the Apophis orbit based on 986 optical described in Eq. (4) and the true confidence region becomes a astrometric measurements between March 2004 and June 2005. curved ellipsoid. The following six Keplerian orbital parameters are used for the The Apophis 2036 Earth encounter is investigated as an orbit determination, which are semi-major axis a, eccentricity example of large uncertainty. The uncertainty is evaluated using e, inclination i, longitude of ascending node Ω, argument of the uncertainty region volume, which is calculated from the 3- perihelion ω and mean anomaly M. The best-fitting solution sigma uncertainty ellipsoid. The calculation is based on the and the standard deviation are described in Table 1. The solution described in section 3.1. The ratio is plotted between standard deviation of Table 1 is calculated by assuming the 2005 and 2036 in Fig. 5, where the Y axis indicates the ratio of observation error as 1.0 arc-second (1-sigma). Fig. 3 describes uncertainty region volume between the volume at the epoch and Apophis residuals of right ascension α and declination δ for the the volume of the X axis. It is found that the uncertainty grows solution. rapidly after the 2029 Earth encounter. Next, the uncertainty ellipse and the impact probability of The Monte-Carlo method is used to illustrate the true Apophis are calculated using the techniques outlined in section ellipsoid for the Apophis 2036 encounter. The close approach 2. The 3-sigma uncertainty ellipse on the modified target plane time is April 13.375, 2036. 1,500 statistically possible orbits at the 2029 earth encounter is described in Fig. 4. The ellipse are sampled from the solution in Table 1. Fig. 6 describes the is elongated, and its size is about 13,740 km long by 730 km curved ellipsoid and indicates that the linearized Eq. (3) has no longer enough accuracy. 3.2. Example 2: 2007WD5 Table 2. Apophis 2029 Earth encounter The Mars impact probability of 2007WD5 is described in Close approach time [TDB] 2029/04/13- this section. 2007WD5 was discovered on November 2007 and 21:46:10.589 its diameter is estimated to be 50 m. 2007WD5 has passed Nominal miss distance [km] 37870.65 within 7.5 million km of the Earth on November 1, 2007 and Minimum miss distance [km] 31098.64 approached within about 20,000 km of the martian surface on 8) Relative velocity [km/s] 7.427809 January 30, 2008 . The orbit diagram of 2007WD5 is described Impact probability [%] 0.000 in Fig. 7.

Pk_13 Trans. JSASS Space Tech. Japan Vol. 7, No. ists26 (2009)

2.0 J2000 ecliptic 2007-Nov-27.0 TDB

1.0 Mars close approach 1.0e+00 2008-Jan-30

0.0 0.0e+00 Y [AU] Y Y [AU] Y

-1.0 -1.0e+00

-2.0

-1.0e+00 0.0e+00 1.0e+00 X [AU] J2000 ecliptic -3.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 X [AU ] Earth -4.0e-01

Fig. 7. Orbit diagram of 2007WD5

-6.0e-01 Table 4. 2007WD5 Mars encounter Y [AU] Y

Close approach time [TDB] 2008/01/30- 12:00:51.090 -8.0e-01 Nominal miss distance [km] 27899.26 Minimum miss distance [km] 6111.18 -1.0e+00 -8.0e-01 -6.0e-01 -4.0e-01 -2.0e-01 Relative velocity [km/s] 12.6314 X [AU] Impact probability [%] 0.0346

Fig. 6. Monte-Carlo results for the Apophis 2036 Earth encounter

the uncertainty ellipse, it is strongly influenced by the We solved the 2007WD5 orbit based on 44 optical astromet- observation accuracy. In order to evaluate the relation between ric measurements between November 2007 and January 2008. the impact probability and the observation accuracy, the impact The solution is shown in Table 3, and the standard deviation of probability with respect to the observation accuracy is collected the observation error is assumed to be 0.6 arc-seconds in the fit. in Table 5, where SMAA indicates the semi-major axis of a 3- The residuals of right ascension and declination for the solution sigma uncertainty ellipse. The uncertainty ellipse with respect are described in Fig. 8. to the observation accuracy is illustrated in Fig. 10, and it The uncertainty ellipse of 2007WD5 is calculated using a indicates that the impact probability changes dramatically with linear target plane technique. The 3-sigma uncertainty ellipse the observation error. on the modified target plane is illustrated in Fig. 9. The uncertainty ellipse is also elongated like that of Apophis in 4. Conclusion Fig. 4, but the size is larger because the number of observations is small. The close approach time is January 30.51, 2008 and its This paper investigates the impact probability calculation of impact probability is predicted to be 0.035%. Other results are NEOs using a linear target plane analysis. The case of asteroid collected in Table 4. 99942 Apophis and 2007WD5 are described. The linear target Since the impact probability can be calculated by integrating plane analysis is confirmed by a Monte-Carlo method. The

Table 3. Orbit solution of 2007WD5 (January 9.0, 2008) Table 5. Observation accuracy and impact probability

Value Uncertainty (1 σ) Observation SMAA Impact accuracy (1 σ) probability a AU 2.5444907097 5.7376891847e-04 arcseconds km % e 0.6029941339 8.8682556536e-05 i deg 2.3772859384 2.6777181925e-04 0.5 18,196 0.0023 Ω deg 67.423553211 1.5235721832e-04 0.6 21,836 0.0346 ω deg 312.82365824 6.5504195001e-04 0.7 25,475 0.1771 M deg 19.998600515 6.7741342774e-03 0.8 29,115 0.5016

Pk_14 T. YAMAGUCHI et al.: Development of Impact Probability Estimation System for Near-Earth Objects

0.8 12000 RA 10000 σw = 0.5 arcseconds 0.6 8000 6000 4000

0.4 R[km] 2000 0 0.2 -2000 -4000 12000 0.0 10000 σw = 0.6 arcseconds 8000 -0.2 6000 4000

R[km] 2000 Residuals[arcsecconds] -0.4 0 -2000 -0.6 -4000 12000 10000 σw = 0.7 arcseconds -0.8 8000 0.8 6000 DEC 4000 R [km] R 0.6 2000 0 -2000 0.4 -4000 12000 10000 σw = 0.8 arcseconds 0.2 8000 6000 0.0 4000 R [km] R 2000 0 -0.2 -2000 -4000

Residuals[arcsecconds] -0.4 -50000 -40000 -30000 -20000 -10000 0 10000 T [km]

-0.6

-0.8 Fig. 10. Observation accuracy and uncertainty ellipse Nov 05 Nov 20 Dec 05 Dec 20 Jan 04 Jan 19 2007 2007 2007 2007 2008 2008 Date [UTC] 2) Morrison, D.: The Spaceguard Survey Report of the NASA International Near-Earth-Object Detection Workshop, NASA- Fig. 8. Post-ﬁt residuals of 2007WD5 optical observations TM-107979, 1992. 3) Chodas, P. W. and Yeomans, D. K.: Orbit determination and estimation of impact probability for Near Earth Objects, 12000 10000 Advances in the Astronautical Sciences, 101 (1999), pp. 21-40. 8000 27,900 km 6000 4) Milani, A., Chesley, S. R. and Valsecchi, G. B.: Close approaches 4000 Error ellipse is 43,600 km long Mars disk of asteroid 1999 AN10: resonant and non-resonant returns, A&A, R [km] R 2000 by 316 km wide 346 (1999), pp. L65-L68. 0 -2000 5) Montenbruck, O. and Gill, E.: Satellite Orbits, Springer, 2000, -4000 -50000 -40000 -30000 -20000 -10000 0 10000 pp. 117-156. T [km] 6) Michel, J. R.: A New Method for Accurately Calculating the Integral of the Bivariate Gaussian Distribution over an Offset Fig. 9. The uncertainty ellipse of the 2007WD5 Mars encounter Circle, JPL Eng. Memo. 312/77-34, 1977. 7) Giorgini, J.D., Benner, L.A.M., Ostro, S.J., Nolan, M.C. and Busch, M.W.: Predicting the Earth encounter of (99942) Apophis, Icarus, 193 (2008), pp. 1-19. limitation of linear mapping is shown in the case of the Apophis 8) Włodarczyk, I.: Impact solutions of Asteroid 2007 WD5 with 2036 Earth encounter. The strong inﬂuence between impact Mars, Icarus, 203 (2009), pp. 119-123. probability and observation accuracy is also investigated. Although radar measurements reduce the volume of the predicted uncertainty region7), these measurements aren’t included in our software. We are improving the software to include available radar measurements.

Acknowledgments

The authors would like to express their appreciation to Dr. H. Ikeda (Japan Aerospace Exploration Agency) for fruitful discussions and for validating the software.

References

1) Kawaguchi, J., Fujiwara, A. and Uesugi, T.: Hayabusa- Its technology and science accomplishment summary and Hayabusa-2, Acta Astronautica, 62 (2008), pp. 639-647.

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