Clones” of an Impact Orbit for the Earth-Asteroid Collision

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 appeared 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 after 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, representing 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 equations 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 method 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 observations 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). Orbital elements of the asteroids are presented in Table 1. Although the impact orbit differs from the nominal one (what is visible in the cases of 2004 VD17 and of Apophis), both orbits represent the observations with the same value of the mean residual. Using the impact orbits from Table 1 we generated a number of “clones” to determine for each asteroid a path of impact locations on the Earth surface (Fig. 1). The impact parameters for ﬁve chosen collision dates are presented in Table 2.

3.1. Asteroid 2004 VD17 (Collision in 2102, fs = 0.015)

Minor planet 2004 VD17 was discovered in November 2004, afterwords four observations made in 2002 were found. The asteroid was also observed in the next years and we used 933 observations from 2002.02.16–2006.05.24 for the orbit correction. The minor planet has a possibility to collide with the Earth just in May 2102, and until this date there are no approaches of the asteroid to the Earth closer than to within 0.013 a.u. Therefore, it was rather a simple case for applying our method to ﬁnd an arbitrary number of the impact orbit “clones” without the necessity of Vol. 56 287

Table1 Orbital elements of four asteroids: 0 – nominal orbit, 1 – impact orbit; the impact orbit was used for generating of its “clones” to ﬁnd a path of the impact places on the Earth surface. The last column shows the year of the expected hypothetical Earth-asteroid collision.

M a e ωJ2000 ΩJ2000 iJ2000 coll. [deg] [a.u.] [deg] [deg] [deg] year

2004 VD17; Epoch: 2006 May 25.0 TT 0 329.57001 1.50819228 0.58866861 90.68726 224.24216 4.22303 1 329.57012 1.50819195 0.58866880 90.68728 224.24210 4.22302 2102 (29075) 1950 DA; Epoch: 2004 Dec. 21.0 TT 0 240.10133 1.69880439 0.50762976 224.49692 356.80420 12.18451 1 240.10133 1.69880439 0.50762969 224.49681 356.80429 12.18449 2880 (99942) Apophis; Epoch: 2006 Sept. 22.0 TT 0 84.78634 0.92226343 0.19105782 126.39508 204.46043 3.33131 1 84.78611 0.92226364 0.19105725 126.39535 204.46021 3.33131 2053 (2340) Hathor; Epoch: 2005 Jan. 30.0 TT 0 260.82417 0.84382282 0.44991401 39.92896 211.53182 5.85429 1 260.82417 0.84382282 0.44991401 39.92896 211.53182 5.85429 2307

repeating the numerical integration many times. Indeed, we easily found the impact orbit for 2102 as well as its 130 “clones”. In this case we found 0.015 as an appropriate value of a scale factor fs .

3.2. Asteroid (29075) 1950 DA (Collision in 2880, fs = 0.025)

Minor planet (29075) 1950 DA has one of the best determined Near Earth Ob- ject’s orbit due to a 54-year astrometric data arc. Motion of the asteroid has been carefully investigated at NASA by Giorgini et al. (2002), and it was found that the impact probability with the Earth in 2880 amounts to 0.33%. The authors randomly selected 10000 statistically possible orbits and integrated each one to 2880. They found the orbit no. 37 on which 1950 DA would approach the Earth to within 0.001954 a.u. in March 2880. We determined the asteroid’s orbit based on 231 observations from the period 1950.02.22–2004.12.18 and applied our cracovian methods to ﬁnd an impact orbit for 2880 (Table 1). Then we could “clone” the impact orbit accepting 0.025 for a scale factor fs . It is evident that integration of thousands orbits until 2880 requires a long time of computer work. Our method of “cloning” the impact orbit allows to perform only one integration until 2880, and then an arbitrary number of impact “clones” can be obtained in several minutes. 288 A. A.

Fig. 1. Paths of the hypothetical impacts on the Earth surface for four Earth-crossing asteroids.

The paper by Giorgini et al. is very important since the authors published values of minimum distances to the planets for their orbit no. 37 (p. 133, Table 1) which might be compared with our results of integration. In Table 3 we present our values of planetary approaches of 1950 DA smaller than 0.1 a.u. If we compared our Table 3 with NASA Table 1 we could see a very good agreement of both results what means that the Warsaw ephemeris DE405/WAW is compatible with the JPL one used at NASA. Hence we may regard results of our investigations as fully comparable with those obtained by other authors.

3.3. Asteroid (99942) Apophis (= 2004 MN4 ) (Collision in 2053, −6 fs = 3.5 × 10 )

Minor planet Apophis was discovered as 2004 MN 4 in December 2004, and we soon have been alerted that the asteroid would closely approach the Earth and there Vol. 56 289

Table2 Impact parameters for hypothetical collisions of four asteroids; λ and ϕ denote, respectively, the geographical longitude and latitude of the impact place on the Earth surface, ψ is the angle between the geocentric velocity vector V and normal to the Earth surface at the impact point. For each asteroid the ﬁve impact dates have been chosen according to the ψ value.

Date [U.T.] λ [deg] ϕ [deg] ψ [deg] V [km/s]

2004 VD17 in May 2102 21020504.89135 148.02750 –40.69597 81.00 21.325 21020504.89006 99.69287 –55.17087 50.51 21.543 21020504.89070 26.78929 –49.14630 26.69 21.631 21020504.89348 339.05866 –27.15109 50.62 21.465 21020504.89612 307.48886 –09.71502 80.13 21.224 (29075) 1950 DA in March 2880 28800316.98912 89.90827 –21.82506 82.18 18.329 28800316.98736 60.88206 –01.26821 51.53 18.191 28800316.98861 8.15147 +29.17304 2.59 17.844 28800316.99450 295.87423 +29.64417 51.43 17.707 28800316.99766 264.80908 +15.65345 77.12 17.801 (99942) Apophis in April 2053 20530413.30831 139.77525 +32.77735 77.79 12.254 20530413.29825 200.41546 +10.79282 41.29 12.387 20530413.29057 253.51603 –15.87113 3.44 12.854 20530413.29104 311.68440 –24.74081 42.54 13.002 20530413.29654 1.17132 –10.17800 78.58 12.673 (2340) Hathor in October 2307 23071025.98495 48.68447 +45.61384 86.07 17.085 23071025.98190 112.79294 +54.61781 54.47 17.178 23071025.98033 182.93251 +41.66474 32.07 17.527 23071025.98198 228.59669 +24.06413 54.42 17.763 23071025.98496 266.49630 +12.37786 86.85 17.706

was a high impact probability in April 2029. Today we know that further observations used for the orbit improvement averted that threat of the Earth-asteroid collision, but in 2029 the asteroid will approach the Earth to within 0.00033 a.u. what makes an exceptional small “keyhole” for predicting the next impacts. Apophis is permanently observed and its motion is carefully investigated since it really is a very dangerous object. One may expect its close approaches to the Earth with a possibility of collision (we found the impact orbits!) in the years: 2036, 2037, 2044, 2046, 2048, 2053, and 2067. The alarm on a cosmic catastrophe in 2029 was retracted when old observations of Apophis were found: six observations made in June and six ones made in March 290 A. A.

Table3 1950 DA planetary close approaches smaller than 0.1 a.u. from the discovery up to 2880

Date Body Dist[a.u.] Date Body Dist[a.u.]

19500312.983 Earth 0.059287 25390901.736 Mars 0.082053 20010305.058 Earth 0.052073 26390511.520 Mars 0.070470 20320302.281 Earth 0.075751 26410314.330 Earth 0.015634 20740319.930 Earth 0.095462 27360320.928 Earth 0.048454 21050310.069 Earth 0.036316 28090319.957 Earth 0.033390 21360311.864 Earth 0.042600 28400815.065 Mars 0.077101 21870308.967 Earth 0.035214 28600320.280 Earth 0.036892 22180320.452 Earth 0.084994 28800316.579 Moon 0.002318 23730318.008 Earth 0.059031 28800316.847 Earth 0.001826 24550306.304 Earth 0.087684

2004. We improved the Apophis’ orbit using 1000 observations from 2004.03.15– 2006.08.16. The impact orbit for 2053 has been found earlier, but successively coming new observations always conﬁrmed an existence of that impact possibility. A hypothetical collision of Apophis with the Earth in April 2053 allows us to show how our method of “cloning” the impact orbit works in a difﬁcult case. Numerical problems appeared because the asteroid has to pass through the small “keyhole” of 0.00033 a.u. in 2029, and moreover, there is a second “keyhole” of 0.0074 a.u. in April 2040 which Apophis has to overcome if it “intends” to hit the Earth in 2053. Therefore, we had to accept such small value of the scale factor −6 fs = 3.5 × 10 when generating a number of “clones” of the Apophis’ impact orbit for 2053.

−9 3.4. Asteroid (2340) Hathor (Collision in 2307, fs = 6.5 × 10 )

Hathor was discovered in 1976 and observed until 2005 in the aphelion part of its orbit. The asteroid has rather an unusual orbit since its orbital period amounts to 283 days, the perihelion is placed near the orbit of Mercury, and aphelion between the orbits of Mars and the Earth. Using 153 astrometric observations we determined very exact orbital elements which served as starting data for numerical integration of the equations of motion. And then a quite improbable accident has happened: according to our results of integration, in October 2307 the asteroid should approach the Earth to within 0.000031 a.u. what means that the nominal orbit was just the impact orbit for the Earth-Hathor collision. Moreover, it is impossible to discover this impact orbit in another way than by happy chance since there are three “keyholes” smaller than 0.01 a.u. which Hathor has to pass before the collision in 2307, namely: 0.0066 Vol. 56 291 in 2069, 0.0059 in 2086, and 0.0089 in 2113. This is why there is no other way than using our method to “clone” the Hathor’s impact orbit since in this case it is −9 necessary to accept an extremely small value of the scale factor fs = 6.5 × 10 .

4. Discussion

There are two scientific centers where the Near-Earth Asteroids (NEAs) are carefully investigated, incoming observations are stored, and orbital elements of NEAs are successively improved. These are: University of Pisa in Italy, and Jet Propulsion Laboratory of NASA in Pasadena, USA. Both institutions cooperate and results of their researches are available on the web sites where one may find tables with dates of the NEA expected close encounters with the Earth, and – what is important – probability values of hypothetical collisions computed for those dates. A complicated procedure of estimating the probability was in details described by Milani et al. (2002). However, we have never encountered a case that impact orbital elements would be published even for the future collision with an alarming high probability. To search impact orbits of NEAs, we make use of data taken from the above web sites. We do not estimate probability of the potential collision, but an existence of the impact orbit certainly confirms the non-zero collision probability. Therefore, it seems that our investigations, resulting in impact orbits found for the NEA expected collisions, complete information connected with the earlier suggested dates of a potentially possible cosmic catastrophe. There is, however, a problem connected with the predicted dates of expected close encounters of NEAs and estimated probabilities of the possible collisions with the Earth. We have to stress that in our opinion an existence of the impact orbit implies the non-zero probability of the expected future collision. On the NASA web portal we may find dates of expected close approaches of NEAs to the Earth. The minimum Earth-asteroid distances are expressed in the Earth-radius units (rEarth) what means that if rEarth < 1 the collision should be expected. However, sometimes NASA provided the dates of possible collisions for which with our methods we were not able to find impact orbits, on the other hand, by applying our procedures of random orbit selection we discovered the collision dates which never appeared on the NASA web sites. For example, in early investigations of Apophis orbit, NASA suggested 2035, 2054, and 2056 as expected collision years for which we never found the impact orbits, and actually they pointed out only 2036 and 2037 as dangerous dates whereas we found the impact orbits for 2046, 2048, 2053, and 2067 as well. Applying the cracovian random orbit selection as well as searching for the impact orbits by the least squares correction with the “forced” equality constraints we perform all the computations in rectangular coordinates. Our methods provide the impact orbits representing the observations with the same mean residual like the 292 A. A. nominal one. To examine whether our results depend on the accepted coordinate system, we adopted the random orbit selection for acting in orbital elements. It appeared that the new (more complicated) procedure yielded values of the randomly selected orbital elements in the same ranges like the previous one in rectangular coordinates. To generate a set of different orbits representing well the observations, we also applied the method elaborated by Marsden (2005) for computing the ephemerides of asteroids not just from the nominal orbit but from a number of similar orbits to create the uncertainty map on the sky. In this method starting from the nominal orbit we changed step by step a value of the mean anomaly, each time improving the remaining five orbital elements by the least squares correction. Although we did not obtain a set of randomly selected orbits, the Marsden’s method appeared to be a very good way to detect the Apophis impact orbit for 2067. If the observation arc of a new discovered asteroid is very short its orbit is poorly determinable. In such a case our random orbit selection fails because the covariance matrix elements are very uncertain, and the selected orbits may sometimes represent the observations with a much greater mean residual than the nominal one. However, here we can solve the problem by generating the “clones” of the nominal orbit if we accept an appropriate value of the scale-factor fs . The method of “cloning” the orbit can also be applied in an iterative process if we start from the orbit of the minimum Earth-asteroid distance, and adequately changing the scale-factor fs we are trying to reach finally the impact orbit.

Acknowledgements. We are indebted to Dr. Sławomira Szutowicz, Space Re- search Centre, Warsaw, for preparation of maps with hypothetical impact points on the Earth surface, and to Dr. Ireneusz Włodarczyk, Astronomical Observatory of the Chorzów Planetarium, for many useful e-mail discussions. This work was supported by MSRIT grant 4 T12E 039 28.

REFERENCES

Bielicki, M., and Sitarski, G. 1991, Acta Astron., 41, 309. Giorgini, J.D., et al. 2002, Science, 296, 132. Marsden, B.G. 2005, The Minor Planet Ephemeris Service, http://cfa-www.harvard.edu/iau/info/MPES.pdf. Milani, A., Chesley, S.R., Chodas, P.W., and Valsecchi, G.B. 2002, in: “Asteroids III”, Eds: Bottke, W.F., Paolicchi, P., Binzel, R.P., Cellino, A., The University of Arizona Press, Tucson, Arizona, USA., p. 55. Sitarski, G. 1998, Acta Astron., 48, 401. Sitarski, G. 1999, Acta Astron., 49, 471. Sitarski, G. 2002, Acta Astron., 52, 295.