Close Approaches of Asteroid 1999 An10: Resonant and Non-resonant Returns

The Earth passes very close to the orbit of the asteroid 1999 AN10 twice per year, but whether or not this asteroid can have a close approach depends upon the timing of its passage across the ecliptic plane. Among the possible orbits there are some with a close approach in 2027. The period of the asteroid may be perturbed in such a way that it returns to an approach to the Earth at either of the possible encounter points. We have developed a theory which successfully predicts the 25 possible such returns up to 2040. We have also identi ed 6 more close approaches resulting from the cascade of successive returns. Because of this extremely chaotic behaviour there is no way to predict all possible approaches for more than a few decades after any close encounter, but the orbit will remain dangerously close to the orbit of the Earth for about 600 years. 1 The 2027 encounter with 1999 AN10 The asteroid 1999 AN10 was discovered by the LINEAR telescope on 13 January 1999 [MPEC 1999-B03]. The discovery was somewhat unusual in that the declination was +70 . The nominal orbit computed by our online information service [NEODyS], that is the solution of the least squares t to 94 observations (with one outlier removed, RMS of the residuals 0:59 arc-sec), is as follows (J2000): a = 1:458432 AU, e = 0:562093, I = 39 :932, = 314 :556, ! = 268 :255, M = 321 :958, for epoch JD 2 451 206:805. The absolute magnitude is estimated at 17:9 0:5; given that the albedo is unknown, this object could be between 0:5 and 2 km in diameter. The ascending node is only 0:000 25 AU closer to the Sun than a point where the Earth is in early August; the descending node is 0:004 78 AU inward from a position of the Earth in early February. This means that, whenever the asteroid and the Earth are in phase at each node, close approaches are possible. Indeed a close approach is possible in August 2027. To analyse the 2027 encounter, we need to consider not only the nominal solution, but also all the solutions compatible with the observations, that is resulting in residuals which are not much larger than the ones of the nominal orbit [Milani 1999]. The region of these compatible solutions can be approximated by an ellipsoid of con dence, which, for a value up to 3, contains solutions with RMS of the residuals up to 0:63 arc-sec. The nominal solution undergoes a close approach in August 2027 with a minimumdistance from Earth's center of 0:025 7 AU. The plane normal to the geocentric velocity at closest approach is the Modi ed Target Plane [Milani & Valsecchi 1999]. The hypothetical objects lling the con dence region evolve along a bundle of orbits; their intersections with the MTP de ne the con dence region of the encounter. If the Earth is not touched by this con dence region, then a collision can be ruled out. The con dence regions are very thin, the width being only 2 600 km, and 0:42 AU long. Thus the occurrence of a very close approach is not very likely: the true orbit could be anywhere along a very long line, including long stretches corresponding to very shallow encounters. In conclusion, the 2027 encounter could be a shallow approach, or could be, with a low, nonnegligible probability, very close. In any case it cannot result in an impact. But the case for a possible dangerous encounter is not closed after 2027; indeed, it is just opened. 2 Resonant and non-resonant returns Resonant returns after a close approach have been discussed in di erent contexts, e.g., the close approaches of comet Lexell to Jupiter [Leverrier 1844] and the repeated visits to Mercury of the Mariner 10 spacecraft. B. G. Marsden recently applied this idea to the asteroid 1997 XF11 in the assumption that the 1990 precovery observations had not been discovered [Marsden 1999]. We can formulate the basic theory of resonant returns as follows. When an asteroid undergoes a close approach in the future, decades after the last available observation, the con dence region on the MTP is thin, with a width much less than the diameter of the Earth and very long; thus it is enough to perform the analysis on the long axis of the con dence region, which we call the Line Of Variation (LOV) [Milani 1999]. The alternate solutions along this line undergo di erent degrees of perturbations, as a result of the close approach. The elements after the encounter describe a curve in the orbital elements space, e.g., in the (a; e) plane; the shape of such curves can be understood by using  Opik's piecewise two body approximation [Greenberg et al. 1988]. These curves are almost closed, they go back to nearly the unperturbed values when the encounter is 1 1.68 1.7 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 −10 −5 0 5 10 Period (years) D el ta t (d ay s) 34 34 34 36 36 39 39 40 40 32 38 38 Figure 1: Resonant returns of 1999 AN10 after the 2027 encounter, taking place until August 2040. The circles represent 1 001 alternate orbits along the LOV. The solid line represents other solutions in the region of highest stretching. The resonant returns are labeled with the year (after 2000) in which the return takes place. shallow, on both extremes of the LOV [Valsecchi & Manara 1997]. Let Pmin and Pmax be the corresponding minimum and maximum orbital periods; every rational number in the interval between them corresponds to at least two resonant returns. If the period P = h=k years with h; k integers, then after h years the asteroid has completed k orbits, the Earth has completed h orbits, and both return to nearly the same position. As an example, 1999 AN10 can have several di erent 7=4 resonant returns in 2034, resulting in an approach potentially closer than the one in 2027, down to 0:000 10 AU. However, two re nements must be taken into account. First, the amount of time by which the rst encounter has been missed needs to be recovered to make the second encounter a close approach. If t is the amount of time by which the asteroid is early for an encounter, the condition to be satis ed for a resonant return at the minimum distance is h + t = k P , where t and P are in years. Thus the resonant returns are described, in the ( t; P ) plane, by lines which are somewhat slanted with respect the P = h=k lines. Figure 1 depicts these resonant lines for the returns of 1999 AN10 after 2027 and for h 13. Where these resonance lines intersect the LOV, one nds a resonant return leading to a close approach. In the gure the LOV has been traced by using the multiple solutions algorithm of [Milani 1999, Sec. 5]. We have used 1 001 solutions equally spaced along the axis between 3 and +3; we have added a denser sampling of solutions along the axis in the region near the 2027 closest approach. The intersections with the resonant lines can be counted from the gure; the resonances not touching the LOV, e.g., the P = 5=3 resonance, cannot result in deep encounters. The other re nement is to consider the Minimum Orbital Intersection Distance (MOID), the minimum distance between the two osculating ellipses representing the orbit of the Earth and 2 1.68 1.7 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 −10 −5 0 5 10 Period (years) D el ta t (d ay s) 36 36 36 29 29 29 34 34 38 38 39 39 40 40 Figure 2: Non-resonant returns of 1999 AN10 taking place until February 2040, in the same style as the previous gure. of the asteroid. Even if the asteroid were exactly on time at the rendezvous with the Earth, the unperturbed close approach distance cannot be less than the MOID. For 1999 AN10, there are in fact two local MOIDs, one per node; each is ' 0:74 times the minimum of distance at the respective node. If the MOID were to remain small forever, since every real number P is approximated arbitrarily well by a rational number h=k a resonant return after h years would be always possible. What is the evolution in time of the local MOIDs of 1999 AN10? It is not enough to compute the evolution of the MOIDs along the nominal solution, because the close approaches can change them: in particular, an encounter near the ascending node (in August) can reduce the distance at the descending node, and make possible a closer approach at the descending node (in February). We have asked G.F. Gronchi to compute the evolution of the mean orbital elements, `averaged' in the sense of [Gronchi & Milani 1998], [Gronchi & Milani 1999], in a way accounting also for the secular e ects of the close approaches. The answer is that 1999 AN10 will continue to have a very low distance at both nodes for about 600 years. Thus it is simply not possible to perform close approach analyses in the sense of [Milani & Valsecchi 1999] for all possible resonant returns: there are hundreds of them. Because of the low nodal distance also at the descending node, there is the possibility of a non-resonant return. This can occur if the Earth completes h + 1=2 revolutions while the asteroid completes k+ 1=2 revolutions, so that they are both at the descending node at the same time. Taking into account the eccentricities of both orbits, the time required to go from the ascending node to the descending node is tE for the Earth (not exactly half a year), and tA for the asteroid (much more than half a period). Again allowing for the timing of the 2027 encounter, the condition to be satis ed for an encounter at the descending node is h+ tE + t = k P + tA. If we 3 add the condition that the distance is zero at both nodes, we have 4 conditions on the 5 variables (a; e; !; u1; u2), where u1; u2 are the eccentric anomalies at the nodes, and we can explicitly compute tA as a function of a. Thus the above condition de nes a curve in the (P; t) plane, as in the resonant case. Note that this analysis would equally apply even if the rst encounter were with another planet. Figure 2 shows all the possible non-resonant returns to the Earth after the 2027 encounter with h 12. 3 Global return analysis Combining the 11 resonant returns of Fig. 1 and the 14 non-resonant ones of Fig. 2, our theory predicts 25 close approach solutions; for each of these we could perform a detailed close approach analysis to determine the minimum distance possible. However, this is not necessary because the minimum distance is essentially the local MOID near the relevant node; this is also not su cient to identify all the possible returns, because a secondary return from a previous one is possible, and so on. For this reason we have devised a global method to nd returns. We started from the same catalog of 1 001 alternate orbital solutions used for the gures. Each solution was propagated forward from the 2027 encounter, recording the position and the nodal distances every time the Earth is passing at the nodes. We determine if there was a crossing of the relevant node (near that time) by the changes of sign of the z coordinate in an ecliptic reference frame. We interpolate between these adjacent solutions to nd the value corresponding to the node crossing at the time when the Earth is there. We similarly obtain the minimum distance between the orbit of the Earth and of the asteroid around the relevant node. By a continuity argument, if z changes sign between two solutions at 1 and 2, there is an intermediate value of for which z is zero, that is at least one solution along the LOV always exists that passes at a distance from the Earth as low as the local MOID (even slightly less, due to gravitational focusing). This argument cannot be applied for values of jzj too large, otherwise the two consecutive solutions could be out of phase by more than one period. Thus the limit of the method is the stretching , which is the ratio between the distance in physical space of two orbit solutions at some time and the distance of the corresponding values of , which parametrises the LOV. For = 0:006, as in our 1 001 solutions catalog, ' 1 500 AU would result in two consecutive orbits being out of phase by 1 revolution; we can reliably detect a close approach only up to ' 200. (P. Chodas, private communication, has found another return in August 2039 which has escaped our search because it has ' 400.) After a very close approach such values of do occur, and even more after a sequence of close approaches. For this reason we have densi ed our sampling of the LOV in the region of high stretching around the solution with the closest approach in 2027, namely for 0:46 0:26, by computing another 2 001 alternate orbits. With = 0:000 1, even returns with > 10 000 AU can be detected. Table I presents all the returns up to August 2040 that we have found with this method, using both the = 0:006 catalog and the denser = 0:000 1 catalog. The stretching MTP in the Table is not , computed with distances in the 3-D space, but its projection upon the MTP, which is in a xed ratio to . That is, we use the product of the time di erence in the node crossing and the relative encounter velocity divided by . MTP allows one to compute the size of the interval 4 Table 1: Earth close approaches possible through 2040. Date MOID MTP % prob. (10 3 AU) (AU/1 ) w/in L.D. Aug. 2027 0.26 0.071 1.2 0:358 Feb. 2029 3.0 0.065 1:801 6.5 4.5 0:388 42.8 240. 0:362 Feb. 2034 21.4 110. 0:352 2.0 650. 0:000 08 0:360 Aug. 2034 0.12 0.077 1.1 +2:734 0.10 3.8 0.023 0:299 0.16 1100. 0:000 08 0:361 Feb. 2036 2.8 0.074 +1:317 4.5 0.91 0:226 4.5 4.8 +2:695a 6.8 760. 0:299a 29.6 1400. 0:361 Aug. 2036 0.15 52. 0:001 7 0:379 0.17 1300. 0:000 07 0:362 0.54 10000. 0:000 009 0:299b Feb. 2038 10.9 52. 0:380 42.9 1600. 0:362 8.1 4600. 0:379c 7.2 5600. 0:379c Aug. 2038 0.09 195. 0:000 4 0:370 0.16 1100. 0:000 08 0:363 Feb. 2039 23. 220. 0:355 6.0 710. 0:359 Aug. 2039 0.073 110. 0:000 8 0:348 0.15 1500. 0:000 06 0:360 0.011 15000. 0:000 006 0:299d Feb. 2040 24.1 200. 0:371 49.8 1300. 0:363 Aug. 2040 0.080 220. 0:000 4 0:366 0.051 450. 0:000 2 0:364 aSecondary non-resonant return from Aug. 2034 bSecondary resonant return from Aug. 2034 cSecondary non-resonant return from Aug. 2036 dSecondary resonant return from Aug. 2034 5 along the LOV, in units, where approaches within a given distance occur. Given a probabilitydensity function on the LOV, the probability of such an event can be determined. But, there is nosuch thing as a unique probability of an event involving an orbit obtained by a least squares t: itdepends upon assumptions on the statistical distribution of the observational errors. In the Tablewe have used a uniform probability density along the LOV for3, to estimate the probabilityof an encounter within the mean distance of the Moon. Note that the lower the stretching, thehigher the probability of an encounter within a given distance; thus shallow encounters can bemore e ective in generating likely returns than the deep ones.Each of the 25 returns predicted by our theory appear in the Table, with MTP 1 500. 6solutions not predicted by the gures appear; they can all be interpreted as secondary returns.Both the 5=3 and 2=1 returns become possible after the 2034 encounter. Among these secondaryreturns there is one in August 2039 for which the interpolated MOID is less than the radius of theEarth. Since the stretching is extreme, we have checked by performing close approach analysis:a collision solution does exist. But MTP appears as divisor in the formula for the probability,so the probability for this impact is of the order of 10 9. If the probability of an impact byan undiscovered 1 km asteroid is of the order of 10 5 per year [Chapman & Morrison 1994], theprobability of impact in 2039 is less than the probability of being hit by an unknown asteroid ofthis size within the next few hours. In any case the asteroid orbit will soon be re ned by furtherobservations and this possible solution may be ruled out.The stretching coe cients used here are related to the dimensionless stretching used in thecomputations of the Lyapounov characteristic exponents: they di er only by a constant factor.Thus the data in the Table indicate the level of chaos of each return orbit. The cascade of successivereturns could be described by a symbolic dynamics, as in other chaotic celestial mechanics problems[Zare & Chesley 1998].4 In the long runAsteroid 1999 AN10 was observed until the angular distance from the sun became < 70 . It willbe again at > 60 from the Sun after the beginning of June; by that time the uncertainty ofits position on the sky will grow to 1:5 arc minutes (corresponding to = 3), that is any newobservation will signi cantly contribute to an improvement of the orbit. It is very likely that theobservations made in the second half of this year will constrain the orbit well enough to predictaccurately the 2027 encounter. This implies that some of the returns of the Table will be discardedas incompatible with the observations; in fact, most of them if the 2027 encounter is not very deep.The problem, however, will not go away, because all along the LOV there are possible encoun-ters occurring later, almost every six months. We have analysed with our global method the same1 001 multiple solutions over a time span of 50 years after 2027, and found 165 possible returns,out of which 117 in the moderate to low stretching region. This situation is qualitatively stable:whatever the actual orbit is, it will not be possible to predict with certainty the returns after thenext close approach for a time span longer than a few decades.Since at least one node of 1999 AN10 will remain close to the orbit of the Earth for centuries,this asteroid shall have to be monitored, by observations and computations, for a very long time.It is conceivable that at some time in the future a decision could be made to de ect it; but, ade ection decreasing the depth of some speci c close approach could increase the impact risk at6 a later date. Thus before such a decision can be contemplated we need to better understand thetheory of resonant and non-resonant returns, which has only been outlined in this paper.References[Chapman & Morrison 1994] Chapman, C.R., Morrison, D., 1994, Nature 367, 33[Greenberg et al. 1988] Greeenberg, R., Carusi, A., Valsecchi G.B., 1988, Icarus 75, 1[Gronchi & Milani 1998] Gronchi, G.F., Milani, A., 1998, Cel. Mech. 71, 109[Gronchi & Milani 1999] Gronchi, G.F., Milani, A., 1999, A&A; 341, 928[Leverrier 1844] LeVerrier, U.-J., 1844, C. R. Acad. Sci. Paris 19, 982[Marsden 1999] Marsden, B.G., 1999, JBAA 109, 39[Milani 1999] Milani, A., 1999, Icarus 137, 269[Milani & Valsecchi 1999] Milani, A., Valsecchi, G.B., 1999, Icarus, in press.[MPEC 1999-B03] MPEC 1999-B03, Minor Planet Center, 16 January 1999.[NEODyS] NEODyS, Univ. Pisa, http://newton.dm.unipi.it/neodys/[Valsecchi & Manara 1997] Valsecchi, G.B., Manara, A., 1997, A&A; 323, 986[Zare & Chesley 1998] Zare, K., Chesley, S., 1998, Chaos 8, 475