2. Resonant and non-resonant returns

Resonant returns after a close approach have been discussed in different 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 confidence 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 confidence region, which we call the Line Of Variation (LOV) [Milani 1999]. The alternate solutions along this line undergo different 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 Öpik'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 shallow, on both extremes of the LOV [Valsecchi & Manara 1997]. Let P_min and P_max 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/kyears 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 different 7/4 resonant returns in 2034, resulting in an approach potentially closer than the one in 2027, down to $0.000\,10$AU.

  

Figure: 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.

However, two refinements must be taken into account. First, the amount of time by which the first encounter has been missed needs to be recovered to make the second encounter a close approach. If $\Delta t$is the amount of time by which the asteroid is early for an encounter, the condition to be satisfied for a resonant return at the minimum distance is $h+\Delta t= k\, P$, where $\Delta t$ and P are in years. Thus the resonant returns are described, in the $(\Delta t, P)$plane, by lines which are somewhat slanted with respect the P=h/klines. Figure 1 depicts these resonant lines for the returns of 1999 AN₁₀ after 2027 and for $h\leq 13$. Where these resonance lines intersect the LOV, one finds a resonant return leading to a close approach. In the figure 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 $\sigma$ axis between -3 and +3; we have added a denser sampling of solutions along the $\sigma$ axis in the region near the 2027 closest approach. The intersections with the resonant lines can be counted from the figure; the resonances not touching the LOV, e.g., the P=5/3 resonance, cannot result in deep encounters.

The other refinement is to consider the 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. For 1999 AN₁₀, there are in fact two local MOIDs, one per node; each is $\simeq 0.74$ 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/ka 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 effects 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.

  

Figure: Non-resonant returns of 1999 AN₁₀ taking place until February 2040, in the same style as the previous figure.

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 t_E for the Earth (not exactly half a year), and t_A for the asteroid (much more than half a period). Again allowing for the timing of the 2027 encounter, the condition to be satisfied for an encounter at the descending node is $h+t_{E}+ \Delta t= k\,P + t_{A}$. If we add the condition that the distance is zero at both nodes, we have 4 conditions on the 5 variables $(a,e,\omega,u_1, u_2)$, where u₁, u₂ are the eccentric anomalies at the nodes, and we can explicitly compute t_A as a function of a. Thus the above condition defines a curve in the $(P,\Delta t)$ plane, as in the resonant case. Note that this analysis would equally apply even if the first encounter were with another planet. Figure 2 shows all the possible non-resonant returns to the Earth after the 2027 encounter with $h\leq 12$.