On the integration of Cid’s radial intermediary (original) (raw)
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Acta Astronautica, 2020
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Taking advantage of the radial intermediaries and the regularization and linearization methods, the zonal Earth satellite theory is studied in the polar nodal canonical set of variables (r, 0, v, R, | N). The variable 0 is eliminated in the first order of the Hamiltonian by applying Deprit's method. Then, the elimination of the perigee is carried out by another canonical transformation. As a consequence, a new radial intermediary, which contains all the J2,(n >~ 1) harmonics, is given. A comparison with the previous radial intermediaries of Cid and Lahulla, Deprit and Alfriend and Coffey is made. Finally, a regularizing transformation which allows us to linearize part of the radial intermediary is proposed, and an analytical study of this process is presented.
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The orbital motion around a central body is an interesting problem that involves the theory of artificial satellites and the planetary theories in the solar system. Nevertheless some difficult situations appear while studying this apparently simple problem, depending on each particular case. The real problem consists of searching the perturbed solution from a basic two-body motion problem. In addition, the perturbed problem must be solved using a numerical method and its efficiency depends on the selected coordinate system and the corresponding time. In fact, local and global errors are not necessarily homogeneously distributed over the orbit. In other words, there is a strong relationship between the spatial distribution of the selected points and the temporal independent variable. This is particularly dramatic in specially difficult cases. This issue leads us to consider different anomalies as temporal variables, searching for both precision and efficiency. Therefore, we are inter...
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Acta Astronautica, 2012
The purpose of this paper is to obtain a third-order expression, for the in-plane and outof-plane amplitudes, of the solutions of the elliptic Hill-Clohessy-Wiltshire non-linear equations. The resulting third-order solution is explicit in terms of true anomaly. The coefficients of the expansions are given as functions of the eccentricity e of the orbit of the leader (i.e., are valid for all values of e). For e ¼0 we recover the solution given by Richardson and Mitchell for the circular case; for ea0 the linear terms of the solution recover the solution found by Lawden for the linearised elliptic HCW equations, also known as the Tschauner-Hempel equations. In the last part of the paper we explain how a formal series solution of the elliptic HCW non-linear equations (in powers of the two amplitudes and the eccentricity) can be obtained, using the Lindstedt-Poincaré procedure.