Correlation of Optical Observations of Objects in Earth Orbit
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A sequence of optical measurements of an Earth orbiting object over one track has sufficient information to determine the angles and angular rates with some degree of precision, but cannot measure the range or range rate. Despite the lack of complete state information, constraints on range and range rate can be determined by taking into account certain physical considerations, which ultimately restrict the object's state to lie within a two-dimensional submanifold of phase space. Such a region can be mapped into orbital element space and propagated in time. As the regions in question are two dimensional in nature it is possible to model them with high precision without excessive computational burden. A second observation of a space object can similarly be mapped into a similar submanifold of orbit element space and intersected with a previous observation mapped to the same epoch. If the object is the same, this intersection process yields a nonzero set which may be unique, depending on observational geometries. If the object is different, the intersection is null in general. The addition of uncertainty in the angle and angle-rate measurements yields finite regions of intersection, sufficient to localize an initial estimate for a connecting orbit if the two mapped observation manifolds have regions of nonzero intersection. If the submanifolds are mapped into a Hamiltonian canonical set of elements, such as the Delaunay or Hamiltonian elements, the projection of this submanifold into the conjugate pairs of coordinates and moments must sum to a constant, due to the integral invariants of Poincar-Cartan. This provides additional structure to these regions as this integral invariance is conserved when mapping in time and thus the area of these projections remains constant.
