The number of discoveries of Near Earth Asteroids (NEAs) has been growing significantly in the recent past, due to the increased effort dedicated to this task, and maybe even more significantly, to the increased efficiency of the detection techniques. Coupled to the scientific interest, there is obviously the urge of knowing whether there is any body of significant size that may come dangerously close to the Earth in the near future. The ability to make such predictions implies that the orbits of all known asteroids must be known well enough, and what this means from the quantitative point of view is the main problem discussed in this paper.
Whenever it happens that, after the discovery of a NEA, the orbit of the latter is determined with a precision sufficient to show that its Minimum Orbital Intersection Distance (MOID, defined as the minimum distance between the osculating ellipses) with respect to the Earth's orbit is small, but not sufficient to reliably predict the closest approach distance, we are put in the uncomfortable situation of knowing that there is a NEA in a potentially dangerous orbit, but of not being able to answer the simple question: is it going to hit? It is customary to represent the range of possible orbits by a line (in the orbital elements space), the line of variations, along which all the solutions are more or less equally compatible with the observations. If one of the alternate orbits along this line undergoes, at some time in the near future, a dangerously close approach to the Earth, the astronomical community can not evade the responsibility of assessing the risk of impact.
The problem can be solved by identification, that is by combining the information from observations taken at a different apparition; this is possible if the orbit is determined well enough to allow one of the possible forms of identification, such as either orbit identification with an object already in the catalogues, or recovery at a following apparition, or precovery in some image archive of observations performed in the past and not yet attributed. If the accuracy of the available orbit is poor, identification of any kind can be difficult; sometimes the situation improves when an accidental recovery is made, but this may not happen for a long time.
This is what has recently happened with 1997 XF11, an asteroid with an exceptionally small MOID; the orbit determined on the basis of the observations at the discovery apparition could have a very close approach to the Earth in the year 2028, within three Earth radii from the surface. The prospect of an actual impact forced the celestial mechanics community to perform exceptionally detailed computations, and to use several different mathematical techniques to assess the possibility of collision. Even though relief was brought by an identification, more exactly by a precovery, after just one day of worry, it is nevertheless relevant to ask the following question: with the data available before the precovery, and with the appropriate mathematical tools, was it possible to exclude the possibility of a collision, with computations that would take much less than a day on a standard workstation? It is important to be able to answer this kind of question because, with the increased rate of discovery, similar cases may occur more and more frequently.
This paper belongs to a series dedicated to the asteroid identification problem, begun with [Milani 1999], hereafter referred to as Paper I. The similarity of the close approach assessment problem to the asteroid identification problem is that in both problems one has to compute the future evolution of a bundle of orbits with initial conditions in the region of elements space allowed by the available observations. For a given level of confidence, the set of orbits consistent with the observation is a 6-dimensional confidence region. In the case of recovery, described in Paper I, one is interested in the projection of the confidence region onto the sky plane at a given epoch; in the case of collision assessment, one is interested in the projection of the same confidence region onto a suitably defined target plane containing the Earth. The two problems are mathematically equivalent and, provided the map from the initial orbital elements to the prediction plane is differentiable, the theory of semilinear confidence boundaries applies equally well to the collision assessment problem. The fact that the coordinates in two prediction planes correspond to physically different quantities does not matter, as far as the algorithm definition is concerned.
Whether making predictions on the sky plane or the target plane, it is important to approximate the full nonlinear confidence boundary as accuratelt as possible. The simpler linear analysis of [Yeomans and Chodas 1994] maybe accurate enough in some cases, but in other cases, where the initial orbit is poorly determined or the orbit is strogly perturbed by the close approach itself, the semilinear approximation will perform much better. To draw practically useful conclusions, one needs to examine the nonlinear effects which bend the linear ellipses; this can be done by careful examination of some examples.
Another important characteristic of an algorithm for assessing the possibility of impact is the time it takes to perform the analysis, since an answer may be needed quickly. The most time-consuming part of the computation, in all risk assessment methods, is the accurate propagation of the orbit from the time of the observations to the close approach time. A Monte Carlo method has been proposed by [Muinonen and Bowell 1993] and used for several applications, including risk assessment. However, the Monte Carlo method, if applied rigorously (without extrapolation), requires to compute a number of accurate orbits of the order of the inverse of the probability of the event being investigated. The problem is to find out whether a reliable risk assessment can be completed by computing only a few tens to a few hundreds of accurate orbital propagations.
We do not anticipate the conclusions, which can be read in Section 5. The rest of this paper is organised as follows: in Section 2 we give all the mathematical theory of the predictions on the Modified Target Plane (which is also defined there); the formulas in common with the sky plane problem are given without proof, because the proofs and derivations can be found in Paper I. In Section 3 we analyse the example of the close approach of 1997 XF11 to the Earth in 2028, which was of special interest for the reasons outlined above. In Section 4 we analyse the fictitious case of an asteroid which is indeed Earth impacting, to see how the algorithms we are proposing would perform in case they were critically needed.
In the Appendix we describe the software system we have used to perform all the computations used in the examples of this paper, and which is available as free software. Readers may therefore reproduce our results for themselves, and, since source code is provided, they may examine the details of our algorithms.