The velocity hodograph for an arbitrary Keplerian motion

An interesting, useful, and simple, but not widely known property of Keplerian motion relating to the circular shape of the orbit in velocity space is discussed in this paper. The property is illustrated by a computer simulation program. A simple dynamical derivation of the circular shape of the velocity hodograph is suggested. 1. Circular hodograph of the velocity vector for closed and open orbits One of the most interesting aspects of Keplerian motion (the motion along conic sections governed solely by a central force whose magnitude is inversely proportional to the square of the distance from the force centre) concerns the shape of its trajectory in velocity space (or momentum space). The velocity vector of a moving body at any moment is directed tangentially to the spatial trajectory, so that in curvilinear motion the direction of the velocity vector changes continuously. We obtain the trajectory of motion in velocity space as follows. For each point on the spatial trajectory, we draw the corresponding velocity vector so that its tail lies at the origin of velocity space and its direction is parallel to the tangent to the spatial trajectory at the point in question. During the curvilinear non-uniform motion of the body, the direction and magnitude of this vector change. The tip of this varying velocity vector generates a curve in velocity space. The now-customary name of 'hodograph' was given to this curve by Hamilton in 1846. For a circular orbit, the magnitude of the velocity is constant and so the variation of the velocity vector is reduced to a uniform rotation about the origin of velocity space. It is evident that the hodograph of the velocity vector for the circular Keplerian motion is itself a circle whose centre is located at the origin of velocity space. The radius of this circle equals the constant magnitude of the circular velocity. As a planet or a satellite moves along a closed elliptical orbit or along an open parabolic or hyperbolic trajectory, rotation of the velocity vector is non-uniform, and both the direction and magnitude of the vector change. However, these variations occur in such a way that the end of the velocity vector in this case also generates a circle (or an arc of a circle) in velocity space but whose centre is not at the origin. In other words, the hodograph of the velocity vector for an arbitrary Keplerian motion is a circle. This interesting property is ignored in almost all the numerous textbooks on mechanics and general physics that treat the orbital motion. For closed orbits, the property is briefly discussed in an optional supplement in (1), and is used in (2) for a