# A Concept for Satellite Orbit Sensitivity Analysis

## Abstract

A satellite’s trajectory around a celestial body, e.g. the Earth, is mainly influenced by the gravitational field of the body. This causes perturbations of the orbit which would be a Keple-rian ellipse in the ideal case. Measuring these perturbations allows an analysis of the gravitational field itself. It is quite obvious that different trajectories are perturbed in different ways depending on the direction, position and velocity of the satellite’s movement in the gravitational field. For analysis purposes it is vital to know which components of the gravitational field have the strongest effect on the satellite’s orbit, with other words, which component of the gravitational field’s description can be obtained from an analysis of the measured orbit perturbations. This task is called satellite orbit sensitivity analysis. It is mainly performed during the planning process for a new geodetic satellite mission to get an idea of the benefits the mission may yield. Today there are two main methods to perform a sensitivity analysis. The classical way is based on the solution of the Lagrange planetary equations for the satellite using Kaula’s representation of the perturbing potential by use of the six Ke-plerian elements. The second method is some how a try-and-error method: A complete orbit integration (orbit synthesis) is performed using a reference gravitational field model in it’s original state first. After that single representation coefficients are varied, the integration performed again and the perturbations caused with respect to the reference orbit are a measure for the sensitivity of the satellite with respect to the varied coefficient. Especially the second method yields high computational time expenses.

The idea of the method presented in the following is to gain some sort of pre-information on the coefficients of a gravitational field’s representation a satellite is most sensitive to by means of a Fourier analysis of both the perturbing potential and the satellite’s orbit data. To be more precise the pre-information will, as we will see, consist of information on the so-called lumped coefficients, linear combinations of gravitational coefficients being the Fourier coefficients of the development of the gravitational field. This information will be introduced to a parameter adjustment process to determine the accuracy with which the preselected coefficients are possibly estimated from satellite orbit data. This adjustment procedure will be based solely on an existing gravitational model as used for the reference trajectory before and the approximate orbit positions computed from it. Thus it will yield no improvement of the unknown gravitational coefficients but their variance-covariance-matrix. By using this two-step method for orbit analysis we want to reduce the dimension of the equation systems to be evaluated during the adjustment process and with that the computation time expenses by omitting irrelevant gravitational coefficients introducing pre-computed information from the Fourier representations.