The possibility of a resonant return of an asteroid involved in a close approach was proposed by B. Marsden . This idea did not receive the attention it deserved. Marsden applied this idea only to a hypothetical case, namely the asteroid 1997 XF in the assumption that the 1990 precovery observations had not been discovered. In that hypothetical case there are indeed more than 20 different solutions leading to a return after injection by the 2028 close approach into a resonance with the Earth. The real 1997 XF , with the orbit determined by using all the available observations, is only mildly perturbed by the 2028 encounter , and cannot have a resonant return.
We can formulate the basic theory of resonant returns as follows. When an asteroid undergoes a close approach in the future, several decades after the last available observation, the confidence region on the MTP is thin (with a width much less than the diameter of the Earth) and very long, as in Figure 2; thus it is enough to perform the analysis on the line of variations, which is essentially the long axis of the confidence region (for a precise definition, see ). The alternate solutions along this line undergo different degrees of perturbations, as a result of the close approach. The values of the elements after the encounter describe a curve in the orbital elements space, e.g. in the (a,e) plane; an idea of the shape of such curves can be obtained by using Öpik's approximation, which is essentially a piecewise two body solution, and is reasonably accurate for encounters with a high relative velocity . It turns out that these curves are almost closed, that is they go back to nearly the unperturbed values when the encounter is shallow, on both extremes of the variations line. In particular a minimum value of the post encounter semimajor axis a results from a close approach such that the asteroid is slightly early for the encounter with the Earth, a maximum from an orbit which is late . Let and be the corresponding minimum and maximum orbital periods. Then every rational number in the interval between and 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 they were at the previous encounter. Because the curve in the plane (P,e) is nearly closed, the lines P=h/k in general cross it an even number of times. As an example, 1999 AN can have several different 7/4 resonant returns in 2034, resulting in an approach potentially closer than the one in 2027, down to AU.
Figure 3: Resonant returns of 1999 AN after the 2027 encounter, taking place until August 2040. The circles represent alternate solutions along the variations line for the asteroid orbit. The solid line represents other solutions in the region of highest stretching. The resonant returns are the intersections between the straight lines and the line of variations, all labeled with the year (after 2000) in which the return takes place.
The above theory is still too simple, however, because two further elements have to be taken into account. The first one is that the amount of time by which the first encounter has been missed needs to be recovered to make the second encounter a close approach. As an example, let us take the 2027 encounter of 1999 AN . If is the amount of time by which the asteroid is early for the encounter with the Earth in August 2027, the condition to be satisfied for a resonant return at the minimum distance is , where and P are in years. Thus the resonant returns are described, in the plane, by lines which are somewhat slanted with respect the P=h/k lines. Figure 3 depicts these resonant lines for , that is for returns taking place until August 2040. Where these resonance lines intersect the line of variations, one finds a resonant return leading to a close approach. In the figure the line of variation has been traced by using the multiple solutions algorithm of [2, Sec. 5,]. We have used 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 easily counted from the figure; the resonances not touching the line of variations cannot result in deep encounters, e.g. the P=5/3 resonance can only result in an approach at 0.07 AU in 2032.
The other element to be taken into account is the so called Minimum Orbital Intersection Distance (MOID), the minimum distance between the two osculating ellipses representing the orbit of the Earth and 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. In the case of 1997 XF , within a few decades after the 2028 encounter the MOID becomes large enough to make the most dangerous encounters impossible. Thus in the hypothetical case without 1990 precovery only 11 resonances have to be taken into account to assess the risk of resonant return; for the real case, there is no resonance with low enough h. For 1999 AN , there are in fact two local MOID's, one per node; each is times the minimum of distance at the respective node. If the MOID was 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 MOID's of 1999 AN ? It is not enough to compute the evolution of the MOID's 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 , ,  in a way accounting also for the secular effects of the close approaches. The answer is that 1999 AN will continue to have a very low distance at both nodes, until the crossing at the descending node, which should take place `on average' in 2633, and for a few decades later. Thus it is simply not possible to perform close approach analysis, in the sense of , for all possible resonant returns: there are hundreds of them.