<abstract xmlns="http://www.w3.org/1999/xhtml">

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 <italic>L</italic>|<italic>solution</italic> ≤ ɛ1; <italic>G</italic>|<italic>solution</italic> ≤ ɛ2, where ɛ1,2 &lt; 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.
</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.

Distance Between Two Keplerian Orbits

Department of Astronomy, Space Science, and Meteorology, Faculty of Science

Astronomy and Meteorology Department, Faculty of Science

Artificial Satellites

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"Distance Between Two Keplerian Orbits"}

In this paper, constrained minimization for the point of closest approach of two conic sections is developed. For this development, we considered the nine...

Distance Between Two Keplerian Orbits

Publié en ligne: 08 oct. 2021

Pages: 57 - 77

Reçu: 27 oct. 2020

Accepté: 28 juin 2021

DOI: https://doi.org/10.2478/arsa-2021-0006

Mots clés
distance function, Lagrange multiplier, objective function, Hessian matrix, minimum distance

© 2021 Ayman Homda Mohamed et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Distance Between Two Keplerian Orbits

Publié en ligne: 08 oct. 2021

Pages: 57 - 77

Reçu: 27 oct. 2020

Accepté: 28 juin 2021

DOI: https://doi.org/10.2478/arsa-2021-0006

Mots clésdistance function, Lagrange multiplier, objective function, Hessian matrix, minimum distance

© 2021 Ayman Homda Mohamed et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Mots clés
distance function, Lagrange multiplier, objective function, Hessian matrix, minimum distance