Closest Approach in Universal Variables

In this paper, universal formulations of the closest approach problem are established and solved by two methods. The first method uses the technique of one-dimensional unconstraint minimization and needs the solution of the universal Kepler's equation twice, while for the second method, a constraint minimization technique is developed and needs the solution of two nonlinear simultaneous equations. Flexible iterative schemes of quadratic up to any positive integer order are developed for the solution of the universal Kepler's equation. The two methods of the minimization process are applied for the closest approach of Hyakutake and Hale–Bopp comets, while the first method is applied to obtain the minimum angular separation of ADS 9159, ADS 2959 and ADS 11632 visual binaries as typical examples of elliptic, parabolic and hyperbolic orbits.