Space Trajectories. Perturbed Motion

The movement of an artificial satellite around the Earth is the result of the action of numerous forces. As a first approximation, it is possible to consider only terrestrial attraction, assuming the Earth to be spherical and homogeneous. The solution to this two-body problem is then a Keplerian conic – a circle or ellipse – defined by its orbital parameters. The forces ignored by the Keplerian model are generally very weak compared to the central gravitational attraction. These forces are, in descending order of importance: gravitational potential terms due to the Earth's volumetric mass distribution, the attraction of other bodies (Moon, Sun), atmospheric friction, radiation pressure due to the solar wind, forces due to the magnetic field, tidal forces... The effect of these forces is slow, which means that the actual trajectory can be considered as a perturbation of the Keplerian trajectory.

The state of the satellite is represented by the orbital osculator parameters corresponding to the Keplerian orbit that would be followed from a given instant if the perturbations were cancelled. Gaussian or Lagrangian differential equations describe the evolution of these osculating parameters under the effect of perturbations. These differential equations can be solved by numerical methods, known as special perturbation methods, or with simplifications by analytical methods, known as general perturbation methods. The latter allow us to analyze the main effects of each type of disturbance. In particular, the modeling of gravitational potential by decomposition into spherical harmonics highlights the effects of nodal and apsidal precession due to terrestrial flattening.

This article deals with the treatment of orbital disturbances. It recalls notions and formulas useful to space engineers.