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1 Artificial Satellites : Journal of Planetary Geodesy

1 2021

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Distance between two Keplerian orbits

Mohamed Ayman Homda, Dwidar Hany Ramdan, Adham Inal, Bakry Abd-Elazaz, El-Raffie Ahmed

Artificial Satellites : Journal of Planetary Geodesy

2021

Vol. 56, No. 3

57--77

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), (paraboli-elliptic), (paraboli-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 ≤ε1G||solution ≤ε2, where ε1,2 < 10 -10. For the cases of (paraboli-parabolic), (paraboli-hyperbolic), and (hyperbolic-hyperbolic), we studied thousands of comets, but the condition of the closest approach was not met.