Space Trajectories. Keplerian Motion

The two-body problem is the fundamental model of orbital mechanics. It describes the motion of two point bodies interacting purely gravitationally, to the exclusion of all other forces. These simplifying assumptions lead to an analytical expression of trajectories in the form of conics, verifying the properties of conservation of energy and angular momentum. The nature of the conic (circle, ellipse, parabola or hyperbola) depends solely on the initial conditions of position and velocity. The Keplerian orbit of a satellite around the Earth is represented geometrically by its orbital parameters, which are analytically related to position and velocity as a function of time. The right choice of orbital parameters enables the satellite to be synchronized with the movements of the Earth (geosynchronism) or the Sun (heliosynchronism). These properties are particularly beneficial for telecommunication or observation space applications.

The two-body problem naturally obeys the equations of mechanics, but analytical solutions are only obtained under certain assumptions. In particular, the two bodies are assumed to be spherical and homogeneous, and there is no relativistic effect (remember that the displacement of Mercury's perihelion is one of the first proofs of the validity of the theory of general relativity).

Although simplified, the Keplerian model gives a very good approximation of the real motion of an artificial satellite. To study motion more precisely, we need to take into account disturbing forces, mainly due to the Earth's gravitational potential, atmospheric friction, the attraction of the Moon and Sun, and solar radiation pressure. The effect of these forces on a Keplerian orbit can be studied using analytical perturbation methods or numerical simulations. The subject of this article is Keplerian motion. It recalls the notions and formulas useful to engineers working on space applications.