Number 20: 24/05/2003 A scientific publication by SGF and NEODyS
 Resonant returns, keyholes and all that... by Giovanni B. Valscchi, Director of the SCN

A rather popular word in NEO jargon is "keyhole". It was introduced by Paul Chodas to denote the following: suppose that a certain small body (a comet, an asteroid or, far more often, a small meteoroid) encounters the Earth in a certain year. It can happen that the perturbation imparted by the Earth to the small body puts the latter in a resonant orbit such that, on the same day of some later year, when the Earth is again at the same place, also the small body comes back there, and a collision takes place.
The "coming back" of the small body after a certain number of years is called a "resonant return", and the collision takes place only if the small body passes through a certain small region of the "target plane" in the vicinity of the Earth; this small region is the keyhole associated to the given impact.

Examples of keyholes can be seen on the JPL website (http://neo.jpl.nasa.gov/news/news018.html); they were computed by Paul Chodas in May 1999, when the impact possibilities of 1999 AN10 in the years following the 2027 encounter with the Earth were being analyzed.
To obtain such results one needs a state-of-the-art orbit computation program, able to take into account all the perturbations one can possibly think of, so as to compute with the highest accuracy the precise positions of the collision solutions. One wonders whether it is possible to have an idea of the location, size and shape of impact keyholes, leaving aside the achievement of great precision, in exchange of some geometric understanding. In fact, with the help of Opik's theory of close encounters, we can say a lot about location, size and shape of impact keyholes.

Let us start by discussing the location. As we said before, for a resonant return to be possible, the orbital period of the small body after the first encounter must be in a certain resonant relationship with the orbital period of the Earth: its value in years must be equal to a fraction like p/q, with p and q integer and not too large (say, up to something like 50 or so).

It turns out that, if we compute, using Opik's theory, the target plane points where the small body has to pass in order to be put by the gravitational pull of the Earth in an orbit of prescribed period, we find that these points lie on a circle, whose radius and center are simple functions of the initial orbit of the small body, and of the given period of the post-encounter orbit.
On the other hand, we know that, on the target plane, the region of uncertainty is essentially very narrow, so as to resemble a line segment. Its intersections with the "resonant" circle thus give us the locations of the keyholes; in fact, there are generally two keyholes associated to the same resonant return, and one of them is much smaller than the other, for reasons that will be clear in a moment.
We now discuss the size of keyholes, and especially why they are in most cases very small. Let us see what happens to two imaginary particles that cross the target plane of the first encounter, and let us further suppose that they are very close to each other at the beginning. As a consequence of their different positions on the target plane, between their orbital parameters after the encounter there will be slight differences; the most important of these is the one in semimajor axis, i.e., in orbital period, since it is the one that forces the two particles to become more and more spatially separated over time.

 The dotted line shows the circle, in the b-plane of the 7 August 2027 encounter of 1999 AN10 with Earth, leading to a resonant return in 2044. Units are Earth radii and the Earth is shown as a circle in the center. The continuous lines on the circle enclose the keyholes resulting in impacts on the Earth in 2044. The vertical line at about 5.8 Earth radii from the centre corresponds to the region of uncertainty.
Let us consider the target planet of the second encounter, the one taking place at the resonant return. The two particles will be much more separated on this second plane than they were at the time of crossing of the first target plane, and this separation is all in one specific direction, the so-called "along-track" direction; actually, this direction corresponds to a time difference, along the same orbital path.
If we just look at the two target planes, forgetting the details of what happens in between, it is as if the distance between the two points had been magnified by a lens in passing from the first to the second plane.
Conversely, passing from the second to the first plane, we have a sort of "compressing" lens. Thus, since the colliding trajectories hit the SECOND plane on a disk of about the size of the Earth (actually, somewhat larger, due to the "gravitational focussing"), the size of the keyholes on the FIRST plane must be smaller, due to the "compressing" lens by which we characterize the correspondence between the two planes. The compression factor can be rather large, up to tens of thousands, and thus the keyholes are smaller than the diameter of the Earth by the same factor.
Moreover, of the two keyholes associated with every possible resonant return, one is always closer, and often much closer, than the other to the Earth; since the compression factor becomes greater the closer we go to the Earth, we have an easy explanation of why the two keyholes differ in size.
Now that we understand location and size, what about shape? Here we need some geometric intuition; we have already all the elements of the puzzle, we just have to assemble them in our mind. The main issues are just two: the keyholes must be close to resonant circle, and they correspond to a disk (the image of the Earth on the second target plane) compressed, in its corresponding image on the first target plane, by a large amount along a specific direction. Since this direction is the one corresponding to changes in semimajor axis, it turns out to be nearly perpendicular, in a given point on the resonant circle, to the circle itself.

Thus, the recipe is the following: first, take the disk of the Earth of the second target plane and put it on the first target plane, at one of the intersections between the appropriate resonant circle and the region of uncertainty; second, squeeze the disk by the appropriate compression factor, in the direction normal to the resonant circle (i.e., radially). Voila', it's done: the keyhole resembles an extremely thin lunar crescent, lying on the resonant circle.