This study is based on the definition of the admissible region introduced by Milani et al.(2004); in the search for potential Earth impactors, this theory allows to take into account the partial data of the TSA (Too Short Arcs) from which it is impossible to deduce a full orbit. Only a set of 4 variables (two angles and their instantaneous time derivatives), called an attributable, is known; a few suitable boundary conditions allow to restrict the motions to a specific bounded 2-dimensional region. In this work, a new inner boundary of this region is introduced, based on the geocentric hyperbolic motion of the immediate impactors; the nodal distances (crossings of the virtual asteroidal orbits with the Earth's orbit) are drawn for two different test attributables, associated with a determination of circular and linear orbits. This could reduce the search for impactors (by propagation of the orbits) to a one-dimensional set. A few comments about elongations and complementary curves complete this paper.

The Minimum Orbit Intersection Distance (MOID) between two confocal Keplerian orbits is a useful tool to know if two celestial bodies can collide or undergo a very close approach. We describe some results and open problems on the number of local minimum points of the distance between two points on the two orbits and the position of such points with respect to the mutual nodes. The errors affecting the observations of an asteroid result in uncertainty in its orbit determination and, consequently, uncertainty in the MOID. The latter is always positive and is not regular where it vanishes; this prevents us from considering it as a Gaussian random variable, and from computing its covariance by standard tools. In a recent work we have introduced a regularization of the maps giving the local minimum values of the distance between two orbits. It uses a signed value of the distance, with the sign given to the MOID according to a simple orientation property. The uncertainty of the regularized MOID has been computed for a large database of orbits. In this way we have searched for Virtual PHAs, i.e. asteroids which can belong to the category of PHAs (Potentially Hazardous Asteroids) if the errors in the orbit determination are taken into account. Among the Virtual PHAs we have found objects that are not even NEA, according to their nominal orbit.