Chebyshev polynomials in space dynamics

Barrio R., Elipe A.

Grupo de Mecanica Espacial, University of Zaragoza, 50009 Zaragoza, Spain

The increase of requirements in precision for orbit determination needs new approaches in both, models and methods of integration. The use of Chebyshev polynomials in ephemeris generation for bodies of the Solar system has revealed to provide good approximations and a big compactness [5].

Several integration methods have been designed based on the use of schemes different from the Taylor series and taking into account the properties of the solution to be obtained. Thus, Fourier series have been employed to integrate not only polynomials, but periodic functions. It seems natural to use Chebyshev series too; it is known that this kind of approximation gives a good approximation and is not difficult to obtain it. This idea was already used by Lanczos, Clenshaw, Norton, etc...[4]. However, it was early forgotten for initial value problems but extensively used for boundary value problems. Recently, new attempts to use that idea are appearing in the litterature, trying to improve the efficiency.

We present another view of these methods for the problem of satellite orbit determination. Our approach follows the Clenshaw [4] and Belikov [2] ones but in such a way that it is possible the choice of the step size of integration, either small or big, for instance one orbital period.

In this way, we obtain the solution as a Chebyshev series that will be possible to evaluate at any time in order to obtain the position, velocity and acceleration [1]. That is, our output is dense (typical of all collocation methods) and is compressed in the sense that is expressed directly as a Chebyshev series. When a big stepsize of integration has been chosen, the initial step in the iteration must be very accurate to asses the convergence. Some differential systems based on Encke method [3] and special sets of variables on ideal frames are formulated.

We apply our method to the artificial satellite subject to the geopotential and drag models.

[1]
J.C.Agnese, R.Barrio and A.Elipe: "Orbit determination in Chebyshev series" in Spaceflight Dynamics, 207-213, Ed: Cepadues, Toulouse , France, (1995).
[2]
M.V.Belikov: ''Methods of numerical integration with uniform and mean square approximation for solving problems of ephemeris astronomy and satellite geodesy'', manuscripta geodaetica, 18, 182-200, (1993).
[3]
R.Broucke: ''Perturbations in rectangular coordinates by iteration'', Celestial Mechanics, 1, 110-126, (1969).
[4]
C.W.Clenshaw and H.J.Norton: ''The solution of nonlinear ordinary differential equations in Chebyshev series'', Computer J., 6, 88, (1963).
[5]
A.Deprit, H.Pickard and W.Poplarchek: ''Compression of Ephemerides dy discrete Chebyshev approximations'', {\sl Navigation}, 26, 1-11, (1979).