Unified parameterization for three types of ballistic trajectories

Аuthors

Denisov M. M., Denisova I. P.^*

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: kaf.pmitet.mai@yandex.ru

Abstract

Solving various tasks in celestial mechanics and space dynamic, as unperturbed motion the movement on three types of trajectories such as elliptic, parabolic, and hyperbolic which are characterized by the value of eccentricity, have been used. It is convenient to specify the laws of motion along these trajectories in a parametric form. However, the existing parametric formulas for setting motion have one serious drawback as they have a different functional form for each type of trajectory. This circumstance had not caused a particular concern before the beginning of the space age, since the transition from one trajectory to another for very large space objects in move was virtually impossible. With the advent of spacecraft equipped with thrusters, such transitions have become possible as the engine operation change the value of eccentricity. The existing parameterization could not be used for these transitions theoretical studies, since one parameterization will not enter another while a continuous variation of the eccentricity. So, performing these calculations, it seems inconvenient to describe the transition of the spacecraft from one orbit to another, and it is not clear at what point you need to start using another parameterization.

To eliminate this non-analytic behaviour of the formulae in the transition of the eccentricity using the value of one, in this paper a unified parametrization that enables such a transition has been we construct. This enables to describe the motion of the spacecraft under the action of the thrusters with a continuous transition from one orbit type to another one, using the unified parameterization for the organization of iterative scheme of the method of successive approximations ( a method of osculating elements in a more general scheme). In this scheme, in the initial approximation the eccentricity, the argument and the orientation angles of the orbit in space are considered constant. The following approximation is taken into account a small variation of these orbital elements under the action of a thruster. Consistent application of this iterative scheme with unified parameterization allows to obtain both analytical and numerical solution of the problem on motion of a spacecraft moving from one orbit type to another.

Keywords:

parametrization of the trajectories, the eccentricity, the Kepler problem

References

1. Duboshin G.N. Nebesnaya mekhanika (Celestial mechanics), Moscow, Nauka, 1968, 799 p.

2. Shatina A.V., Sherstnyov E.V. Satellite motion in the gravitational field of a viscoelastic planet with a core, Cosmic Research, 2015, vol. 53, no. 2, pp. 163-170.

3. Pavlenko Y.G. Lektsii po teoreticheskoi mekhanike (Lectures on theoretical mechanics), Moscow, Nauka, 2002, 392 p.

4. Arnold V.I. Matematicheskie metody klassicheskoi mekhaniki (Mathematical methods of classical mechanics), Moscow, URSS, 2000, 408 p.

5. Petkevich V.V. Teoreticheskaya mekhanika (Theoretical mechanics), Moscow, Nauka, 1981, 496 p.

6. Brumberg V.A. Relyativistskaya nebesnaya mekhanika (Relativistic celestial mechanics), Moscow, Nauka, 1972, 382 p.

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