Gravitational Keyhole - Background

Background

Due to observational inaccuracies, bias in the frame of reference stars, and largely unknown non-gravitational forces on the asteroid, mainly the Yarkovsky effect, its position at the predicted time of encounter is uncertain in three dimensions. Typically, the region of probable positions is formed like a hair, thin and elongated, because visual observations yield 2-dimensional positions at the sky but no distances. If the region is not too extended, less than about one percent of the orbital radius, it may be represented as a 3-dimensional uncertainty ellipsoid and the orbits (ignoring the interaction) approximated as straight lines.

Now imagine a plane comoving with the planet and perpendicular to the incoming velocity of the asteroid (unperturbed by the interaction). This target plane is named b-plane after the collision parameter b, which is the distance of a point in the plane to the planet at its coordinate origin. Depending on a trajectory's position in the b-plane its post-encounter direction and kinetic energy is affected. The orbital energy is directly related to the length of the semi-major axis and also to the orbital period. If the post-encounter orbital period of the asteroid is a fractional multiple of the orbital period of the planet, there will be a close encounter at the same orbital position after the given numbers of orbits.

According to Ernst Öpik's theory of close encounters, the set of points in the b-plane leading to a given resonance ratio forms a circle. Lying on this circle are the planet and two gravitational keyholes, which are images of the planet in the b-plane of the future encounter (or rather of the slightly larger catchment area due to gravitational focusing). The form of the keyholes is a small circle elongated and bent along the circle for the given resonance ratio. The keyhole closest to the planet is smaller than the other, because the variation of deflection becomes steeper with decreasing collision parameter b.