Understanding the distribution of the propagated angles-only position vector

All rights reserved. Consider a space object in an orbit about the earth with an initial Gaussian uncertainty for position and velocity in ECI coordinates. In this paper we explore the propagated uncertainty under Keplerian dynamics for the angles-only position vector. If the initial state defines an orbit in the equatorial plane, then the angles-only position vector can be described in terms of “latitude” and “longitude”. In some cases the joint distribution of latitude and longitude simplifies to an approximate bivariate Gaussian distribution . However, in other cases (especially in the case of a breakup event, where the initial position is known nearly exactly, but the initial velocity shows uncertainty) the joint density shows a pronounced “pinching” or “bow-tie” effect in a scatterplot of latitude vs. longitude whenever the propagation time is an integer multiple of the half-period for the initial state. This behavior is well-known in breakup events where the cloud of debris tends to coalesce every half-integer period. Implications of this new “pinched Gaussian distribution” for association problems will also be explored.