Trans. JSASS Space Tech. Japan Vol. 7, No. ists26, pp. Pk_11-Pk_15, 2009 Development of Impact Probability Estimation System for Near-Earth Objects 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 a and declination d. 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 = r TWr (1) and the impact probabilities with respect to the Earth and m the other planets are estimated. For example, the close where r 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 m n r − r r data of optical observation is investigated. The linear mapping = − r + ∑ m k − k + D (2) dt2 r3 k jr − rj3 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 mk are heliocentric position vector and gravity constant of the examples. perturbing body k, ms is a gravity constant of the Sun, n is the number of perturbing bodies and D 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 Copyright© 2009 by the Japan Society for Aeronautical and Space Sciences and ISTS. All rights reserved. Pk_11 Trans. JSASS Space Tech. Japan Vol. 7, No. ists26 (2009) 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 = (RF)CX (RF) (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 D T −1 D ≤ s 2 ( X) CCA ( X) (4) to the target planet at the point of closest approach. The MTP D coordinate is described in Fig. 1, where SˆMTP axis is parallel where X is the vector of the error from the nominal (best- s 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 F 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 s 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 W deg 204.44800475 1.06188046e-04 w 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 [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 (3s) 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.