Abstract
Abstract In this paper, constrained minimization for the point of closest approach of two conic sections is developed. For this development, we considered the nine cases of possible conics, namely, (elliptic–elliptic), (elliptic–parabolic), (elliptic–hyperbolic), (parabolic–elliptic), (parabolic–parabolic), (parabolic–hyperbolic), (hyperbolic–elliptic), (hyperbolic–parabolic), and (hyperbolic–hyperbolic). The developments are considered from two points of view, namely, analytical and computational. For the analytical developments, the literal expression of the minimum distance equation (S) and the constraint equation (G), including the first and second derivatives for each case, are established. For the computational developments, we construct an efficient algorithm for calculating the minimum distance by using the Lagrange multiplier method under the constraint on time. Finally, we compute the closest distance S between two conics for some orbits. The accuracy of the solutions was checked under the conditions that L| solution ≤ ɛ1; G| solution ≤ ɛ2, where ɛ1,2 < 10−10. For the cases of (parabolic–parabolic), (parabolic–hyperbolic), and (hyperbolic–hyperbolic), we studied thousands of comets, but the condition of the closest approach was not met.
Highlights
The problem of determining the point of closest approach of two orbits has important applications
The close approach analysis can be applied on the orbital elements and for geometrical analysis of the orbital elements
The numerical methods are based on the orbital ephemeris of objects either at certain time steps during a certain interval or the position and velocity information obtained from the orbital model
Summary
The problem of determining the point of closest approach of two orbits has important applications. The close approach analysis can be applied on the orbital elements and for geometrical analysis of the orbital elements In these cases, the information of the closest approach events could be obtained by applying the differential method. The numerical methods are based on the orbital ephemeris of objects either at certain time steps during a certain interval or the position and velocity information obtained from the orbital model. The relative position and closest approach information are obtained by numerical processing methods, such as difference, interpolation, fitting, and polynomial root-finding to the orbital ephemerides or positions. The method is valid for all values of eccentricities
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