ACTA ASTRONOMICA Vol. 56 (2006) pp. 283–292 Generating of “Clones” of an Impact Orbit for the Earth-Asteroid Collision by G. Sitarski Space Research Center, Polish Academy of Sciences, Bartycka 18A, 00-716 Warsaw, Poland e-mail: [email protected] Received September 14, 2006 ABSTRACT If we find an impact orbit of the Earth-crossing asteroid we can determine the impact point location on the Earth surface. If we want to find other orbits, very similar to the impact one, we have to select randomly a number of such “clones” and to integrate equations of motion many times from the osculation epoch to the collision date. Then we can determine a path of hypothetical impact points on a map of the Earth. We elaborated a method allowing us to avoid the repeating of long-term integration. The method is based on a special feature of the cracovian least squares correction applied to the random orbit selection. After finding the impact orbit we randomly select an arbitrary number of “clones”, perform only one time-consuming integration, and find quickly many similar impact orbits for the collision date. We applied our method for four chosen asteroids: 2004 VD 17 , 1950 DA, Apophis (2004 MN 4 ), and Hathor. We show that we are able to “clone” the impact orbit in a very difficult case and when it is impossible to do this in another way. Key words: Minor planets, asteroids – Celestial mechanics 1. Introduction Problem of finding the impact orbit for an asteroid hazardous to the Earth ap- peared when the minor planet 1999 AN 10 was discovered in January 1999. In March 1999 it was announced that very close approaches of the asteroid to the Earth would be possible in August 2027 and 2039. Soon it turned out that in 2027 the asteroid would reach the minimum distance to the Earth equal to 0.0007 a.u., but the close approach in 2039 with a great probability of the Earth-asteroid collision could occur too. 284 A. A. During INTERNET discussions in 1999, specialists for orbital computations considered the question how to find an impact orbit of 1999 AN 10 for 2039 af- ter its close approach to the Earth in 2027, or as one has said: after the “keyhole” in 2027. Since the orbital elements determined from observations are uncertain, many similar orbits can be found by varying the elements in ranges of their mean errors. To find the impact orbit either the Monte Carlo or the “trials and erros” method was then applied, however, it could not ensure that the selected orbits would represent the observations quite well. The problem of random orbit selection for an arbitrary number of orbits, rep- resenting the observations as the nominal orbit, has been solved by using a special feature of the cracovian algorithm applied to the least squares orbit improvement (Sitarski 1998). The selected orbits are the initial data sets for integrating the equa- tions of motion until the expected date of collision. As a result we find an orbit when the asteroid reaches the Earth at the minimum distance which, as a rule, is not a collision distance. Then one should perform a new selection starting from this minimum distance orbit and repeat the numerical integration many times hoping to discover the impact orbit. We elaborated the method of finding an impact orbit by the cracovian least squares correction with the “forced” equality constraints (Sitarski 1999). The meth- od was successfully applied to searching the impact orbit of 1999 AN10 in 2039. Having the impact orbit we can locate the impact place on the Earth surface. How- ever, to determine different impact places for another orbits very similar to the first one, we have to make a new selection and to integrate equations of motion many times again. Using our earlier experience in searching the impact orbit (Sitarski 1999) we found a new way for determining a number of “clones” of the impact orbit. The method allows us to perform only one integration starting from the impact orbit, and then to find quickly an arbitrary number of the randomly selected “clones”. 2. Method of Computations We have to remind a procedure of computing an arbitrary number of randomly selected orbits well fitting to the given set of observations. The method is based on improving the orbit in rectangular coordinates applying the cracovian algorithm of the least squares correction (Sitarski 1998). Suppose we have n observational equations: ai∆x + bi∆y + ci∆z + di∆x˙+ ei∆y˙+ fi∆z˙ = li, (i = 1,...,n) with six unknown corrections ∆x, ∆y, ∆z, ∆x˙, ∆y˙, ∆z˙, and let us denote: n n n [aa] = ∑ aiai, [ba] = ∑ biai,... [la] = ∑ liai. i=1 i=1 i=1 Vol. 56 285 We can create six normal equations and then six elimination equations which can be schematically written as below: normalequations eliminationequations ∆x ∆y ∆z ∆x˙ ∆y˙ ∆z˙ ∆x ∆y ∆z ∆x˙ ∆y˙ ∆z˙ [aa] [ba] [ca] [da] [ea] [ fa] [la] ra,1 rb,1 rc,1 rd,1 re,1 r f ,1 ρ1 [ab] [bb] [cb] [db] [eb] [ fb] [lb] rb,2 rc,2 rd,2 re,2 r f ,2 ρ2 [ac] [bc] [cc] [dc] [ec] [ fc] [lc] rc,3 rd,3 re,3 r f ,3 ρ3 [ad] [bd] [cd] [dd] [ed] [ fd] [ld] rd,4 re,4 r f ,4 ρ4 [ae] [be] [ce] [de] [ee] [ fe] [le] re,5 r f ,5 ρ5 [af ] [bf ] [cf ] [df ] [ef ] [ ff ] [lf ] r f ,6 ρ6 what means that e.g.,: [aa]∆x +[ba]∆y +[ca]∆z +[da]∆x˙+[ea]∆y˙+[ fa]∆z˙ =[la], ra,1∆x + rb,1∆y + rc,1∆z + rd,1∆x˙+ re,1∆y˙+ r f ,1∆z˙ = ρ1. Values of ra,1 , rb,1 ,... ρ1 , and further rb,2 , rc,2 ,... ρ2 and so on, can be found using earlier computed values of [aa], [ba],... [la]. Connection between tables of normal and of elimination equations is as follows: multiplying the table of elimination equations by itself according to the cracovian rule “column by column” we obtain the table of normal equations. After iteratively improving the orbit by the least squares method we save the final values of coefficients in elimination equations ra,1 , rb,1 ,... r f ,1 , and so on, and also save the improved values of rectangular coordinates r0 =[x0,y0,z0] and velocity components r˙0 =[x˙0,y˙0,z˙0] and of the mean residual µ. Further we apply a random number generator with Gaussian distribution with dispersion σ = µ to select random values of the right sides of elimination equations, i.e.,values of ρ1 , ρ2 ,... ρ6 . Solving again the elimination equations we obtain the new values of corrections ∆r, ∆r˙ and adding them to the nominal r0 , r˙0 we obtain a new orbit which is, of course, different than the nominal one, but it represents the observa- tions almost with the same mean residual as the nominal residual µ. Let r0 , r˙0 be the parameters of an impact orbit of the asteroid for the osculation epoch t0 . To find “clones” of the impact orbit we have to select a set of random values of ∆ri , ∆r˙i for i = 1,...,N , however, using the random number generator we should put now σ = fsµ where fs < 1 is a scale-factor. If we choose an epoch T , say, a week before the impact date, then starting from the initial values of r0 , r˙0 we integrate equations of motion as well as the special differential equations for vectors Gk , k = 1,...,6 (Sitarski 1999), from the epoch t0 until T . Thus for the epoch T we obtain values of rectangular coordinates and velocity components RT , R˙ T , and we can compute ∆Ri , ∆R˙ i for i = 1,...,N as follows: ∆Xi = G1x∆xi + G2x∆yi + G3x∆zi + G4x∆x˙i + G5x∆y˙i + G6x∆z˙i 286 A. A. ∆Yi = G1y∆xi + G2y∆yi + G3y∆zi + G4y∆x˙i + G5y∆y˙i + G6y∆z˙i ∆Zi = G1z∆xi + G2z∆yi + G3z∆zi + G4z∆x˙i + G5z∆y˙i + G6z∆z˙i ∆X˙i = G˙1x∆xi + G˙2x∆yi + G˙3x∆zi + G˙4x∆x˙i + G˙5x∆y˙i + G˙6x∆z˙i ∆Y˙i = G˙1y∆xi + G˙2y∆yi + G˙3y∆zi + G˙4y∆xi + G˙5y∆y˙i + G˙6y∆z˙i ∆Z˙i = G˙1z∆xi + G˙2z∆yi + G˙3z∆zi + G˙4z∆x˙i + G˙5z∆y˙i + G˙6z∆z˙i Hence we have a set of parameters Ri = RT + ∆Ri , R˙ i = R˙ T + ∆R˙ i for i = 1,...,N, being the starting data for a quick integration of the equations of motion from the epoch T to the impact date. The number of “clones” of the impact orbit, as found among the N selected random orbits, will depend on the accepted value of the scale-factor fs . 3. Numerical Examples As examples of generating the “clones” of impact orbits we have chosen four asteroids which potentially threaten the Earth: 2004 VD 17 , (29075) 1950 DA, (99942) Apophis (discovered as 2004 MN 4 ) and (2340) Hathor. Orbital elements of the minor planets have been improved using their actual sets of observations which earlier have been selected according to Bielicki’s mathematically objective criteria (Bielicki and Sitarski 1991). Equations of motion have been integrated by the recurrent power series, the planetary coordinates from Mercury to Pluto (treating the Moon as a separate body), and also four biggest asteroids, were taken from our Warsaw numerical ephemeris of the Solar System DE405/WAW (Sitarski 2002).