Satellite in Keplerian Orbit

Abstract

Consider a satellite in periodic motion around the Earth. Let us define the frame (O; x, y, z). The origin O is the centre of the Earth, which is taken to be a sphere Σ. The axis Oz is the axis joining the poles, oriented from the south to the north. The plane xOy is the equatorial plane of the Earth, denoted E, which cuts the terrestrial sphere at the equator. The axis Ox is chosen arbitrarily to point towards a distant star. The axis Oy is deduced from the other two axes in such a way as to obtain a right-handed orthonormal frame. The frame associated with this coordinate system is considered to be Galilean and will be denoted by. The motion of the satellite is Keplerian, i.e., it occurs on a Keplerian orbit. In , the trajectory is a conic section, in this case an ellipse, with one focus at the centre of attraction O, and lying in a plane P, the orbital plane. In this Galilean frame, the orbital plane P is fixed. Let OZ denote the straight line perpendicular to P at O. The intersection of the planes P and E is a straight line through O, called the line of nodes. 2.3.2 Specifying a Point on an Orbit In order to specify a point in Keplerian motion in space, the first step is to identify the orbit, and then the point on the orbit. We thus define successively: (a) the location of the orbital plane in this frame, (b) the position of the elliptical orbit in this plane, (c) the characteristics of the ellipse, (d) the position of the moving point (i.e., the satellite) on the orbit. We shall find that six parameters are necessary and sufficient to determine the position of the satellite in. Let us now go through each of these points. Points (a) and (b): Locating the orbital plane in the coordinate system, and the orbit in the plane. The orbit is considered as a solid. When a solid has a fixed point in a frame , its position relative to is determined by three parameters corresponding to the three degrees of freedom. We choose the Euler angles, defined classically as the angle of precession ψ, the angle of nutation θ, and the angle of proper rotation 5 χ. We shall return to the details of the decomposition of a rotation into elementary rotations when we come to study the ground track in Chap. 5. In the present context, to specify the plane P of the ellipse, we consider the intersection of P with the sphere Σ, which gives the circle T , as shown in Fig. 2.1. T is called the ground track of the orbit. The projection of P , the perigee, on the ground track is P 0 (the intersection of OP with Σ). The two points of intersection of the ground track T 5 The three Euler angles are traditionally denoted by φ, θ and ψ, respectively, in the literature. We have broken with this tradition by using χ, to avoid confusion with the latitude, which already uses the symbol φ.