The Circular Kinematic and Dynamic Equation of a Satellite Orbit

Abstract

Here we discuss the kinematic equation of a satellite orbit based on a circular representation of the velocity vectors of a Kepler orbit, otherwise known as the two-body problem in celestial mechanics. The velocity vector for Keplerian orbit describes a circle, i.e., we show that the velocity vector of the satellite in the presence of any point-like mass will rotate about that object along a circle with a constant radius. Thus, an interesting advantage of using circular perturbations is that this method preserves the orthonormality of the rotational transformation, i.e., the geometrical properties of the orbit. We show that the proposed circular model could be applied to kinematic as well as dynamic modeling of the orbit and rotation of a rigid body (satellite, Earth, etc.). In the case of circular perturbations, the radius of rotation is preserved, as is also the case with rotation of a rigid body (satellite, planet, etc.). At the end of this section, we discuss the proposed model in the light of geometrical integration, a special kind of integration that preserves the properties of the orbit, i.e., the exact flow of differential equations or Hamiltonian systems that govern satellite motion and rotation. In the light of circular perturbations we extend Newton’s theorem of revolving orbits that defines a special central force as one that is changing the angular speed of the orbit by some constant factor, while the radial motion remains unaffected.