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1 The 2027 encounter with 1999 AN tex2html_wrap_inline340

The asteroid 1999 AN tex2html_wrap_inline340 was discovered by the LINEAR telescope on 13 January 1999. The discovery was somewhat unusual in that the declination was tex2html_wrap_inline368 . We checked for possible prediscovery observations in the archives made available by the Minor Planet Center, and found none; this is not surprising, given that this asteroid is typically visible in a portion of the sky which has been very little surveyed in the past. The asteroid was observed until 20 February: afterwards the angular distance from the sun became tex2html_wrap_inline370 .

Figure 1:  The orbit of 1999 AN tex2html_wrap_inline340 and the orbit of the Earth, in a reference system with the z axis normal to ecliptic plane, in which the Earth's orbit lies, and the x axis towards the tex2html_wrap_inline348 point. The line of intersection of the two orbital planes is also drawn. The nodes are both very close to the orbit of the Earth. This results in two periods of the year when close approaches are possible, in August and in February.

The nominal orbit published by the online information service NEODyS[1], that is the solution of the least squares fit to 94 observations (with one outlier removed, RMS of the residuals 0.59 arc-sec), is as follows: a=1.458432 AU, e=0.562093, tex2html_wrap_inline386 , tex2html_wrap_inline388 , tex2html_wrap_inline390 , tex2html_wrap_inline392 , for epoch tex2html_wrap_inline394 . The unique feature of this orbit is shown in Figure 1: both its intersections with the ecliptic plane, the nodes, are very close to the Earth's orbit. The ascending node is only 0.00025 AU closer to the Sun than a point where the Earth is in early August; the descending node is 0.00478 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 those resulting in residuals which are not much larger than the ones of the nominal orbit. The region of these compatible solutions can be approximated by an ellipsoid of confidence, which can be computed according to the standard theory of normal and covariance matrices. In the notation of [2], the confidence ellipsoid corresponding to a tex2html_wrap_inline400 value up to 3 contains solutions with RMS of the residuals up to 0.63 arc-sec.

Figure 2:  The close approach of August 2027, with its uncertainty as projected on the Modified Target Plane. The prediction according to the linear theory would allow for an approach closer than the one allowed by the more accurate semi-linear theory. In any case, the Earth (shown to scale) is not at risk. The crosses mark the alternate orbits computed at a separation of tex2html_wrap_inline350 along the variation axis.

The nominal solution undergoes a close approach in August 2027 with a minimum distance from Earth's center of 0.0257 AU. The plane normal to the geocentric velocity at closest approach is the Modified Target Plane (MTP) [3]. The hypothetical objects filling the confidence region evolve along a bundle of orbits; their intersections with the MTP define the confidence region of the encounter. If the Earth is not touched by this confidence region, then a collision cannot occur at that encounter unless the RMS of the observational errors is significantly larger than the minimum. The boundary of the confidence region on the MTP can be approximated by the linear confidence ellipse [4]; however, in cases where the nominal solution is far from the closest possible approach, the linear approximation may prove unreliable and the semi-linear method described in [3] should be used. Figure 2 shows the MTP of the 2027 encounter, with the Earth and the nearby portions of the linear and semi-linear confidence boundaries. The confidence regions are very thin, the width being only tex2html_wrap_inline410 km: they can be represented as lines in the plot. The confidence region is much longer than the portion shown, indeed it extends up to 0.21 AU in both directions (for tex2html_wrap_inline414 ). 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.

There is no such thing as a unique probability of an event involving an orbit obtained by a least squares fit; the probability depends upon assumptions on the statistical distribution of the observational errors. There are, however, mathematically rigorous methods to deduce a probability for an event given a hypothesis on the error statistics. As an example, if the errors were normally distributed with zero mean and variance of tex2html_wrap_inline416 arc-sec tex2html_wrap_inline418 , then the probability of a 2027 close approach within the distance of the Moon would be tex2html_wrap_inline420 . Such a model for observational errors is questionable, because it underestimates the effect of correlations, and it assumes an unrealistically low occurrence of large errors. To compensate for this, we prefer to use a uniform probability density along the line of variation for tex2html_wrap_inline414 ; in the example above, this would result in a probability of tex2html_wrap_inline424 . As this example shows, the values obtained by different statistical error models are different, and the selection of the appropriate model is not a simple issue; in the following we shall use the uniform distribution assumption.

In conclusion, the August 2027 encounter could be a very shallow approach, or could be, with a low, non-negligible probability, very close, but in any case cannot result in an impact. The case for a possible dangerous encounter, however, is not closed after 2027; indeed, it is just opened.

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Andrea Milani Comparetti
Tue Apr 6 19:21:11 MET DST 1999