Mutual geometry of confocal Keplerian orbits: uncertainty of the MOID and search for virtual PHAs
Abstract
The distance between two confocal Keplerian orbits, called MOID in the literature, is a useful tool to know if two celestial bodies can collide or undergo a very close approach. Two confocal orbits may get close at more than one pair of points, thus it is useful to compute not only the absolute minimum of the distance function d between two points along the orbits, but all its local minimum values (local MOID). Recently some new algorithms for the computation of the critical points of d^{2} have been proposed, based on an algebraic formulation of the problem. When a new celestial body is discovered the orbit of the body can be determined by means of a least squares fit. Moreover the uncertainty in the determination of the nominal orbit produced by the errors affecting the observations can be represented by a covariance matrix. The errors in the orbit determination also affect the computation of the MOID, and it is important to estimate the size of this effect. The covariance of the MOID can be computed, but the possibility of orbit crossings produces a singularity in this computation. Furthermore the uncertainty of a small orbit distance may allow negative values of the distance, that are meaningless. This is particularly unpleasant because we would like to know its confidence interval just when the MOID can be small or vanishing. We present the results of a recent work, in which we regularize the minima of d as functions of the orbit configurations according to an intuitive geometric rule. Then we can compute a meaningful confidence interval for the local MOID also when it vanishes. We have computed the covariance of this "local MOID with sign" for a large database of orbits and we have searched for the Virtual PHAs, i.e. asteroids which can belong to the category of PHAs 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.
 Publication:

Near Earth Objects, our Celestial Neighbors: Opportunity and Risk
 Pub Date:
 May 2007
 DOI:
 10.1017/S1743921307003018
 Bibcode:
 2007IAUS..236....3G