The aim of this work is to study the evolution of the high-altitude frozen orbits using the eccentricity-inclination diagram. Furthermore, the evolution of the frozen eccentricity and inclination with time is studied. The disturbing function is expressed in terms of non-singular elements (a,λ,ξ,η,P,Q) under the Perturbation due to the third body. The short periodic term are omitted by single averaged on the disturbing function. Lagrange planetary equations are obtained. A comparison between the solutions obtained from ordinary and non-singular Lagrange planetary equations is presented.

In this paper, the constrained minimization for the point of closest approach of two conic sections is developed. We considered the nine cases of the possible conics, which are: 1) Elliptic-Elliptic, 2) Elliptic-Parabolic, 3) Elliptic-Hyperbolic, 4) Parabolic-Elliptic, 5) Parabolic- Parabolic, 6) Parabolic-Hyperbolic, 7) Hyperbolic-Elliptic, 8) Hyperbolic-Parabolic, 9) Hyperbolic-Hyperbolic orbits. These developments are determined from the two methods (analytic and computation). For the analytical developments, the literal expression of the minimum distance equation (S) and the constrained equation (G) and there first and second derivative for each case are established. While for the computational developments, we construct an efficient algorithm to calculate the minimum distance by using the Lagrange multiplier method under the constraint on time. Finally, we compute the closest distance (S) between two conic orbits.

This paper is concerned to evaluate the perturbations in orbital elements of a low Earth-orbiting satellite by using a semi-analytical method. We have obtained the solution of the equations of motion of near-Earth satellites under the combined perturbing effect due to Earth's figure (J2-gravity) and drag force. We have been studied, for a fixed initial position and initial velocities and initial six elements, till satellite collapses on Earth. The atmospheric model takes into account a linear variation of the density scale height with altitude its rotation. The perturbation theory is based upon Lie transforms. A precise calculation of the perturbations is possible only if the orbit is sufficiently well known. The results can be obtained by using tow methods which are Cowell's Method and Average Method. This paper discusses the development of Mathematical Modeling and Analysis Software on Perturbation Effects and with the help of the Mathematica Program to visualize the Effect by a numerical example.