(All the conic sections problems presented in the printed version of "Space Mathematics" are not included in this adaptation. The printed document offers twenty-two problems while the adaptation presents the first three problems as typical examples.)
The conic sections play a fundamental role in space science. As shown in the Appendix, any body under the influence of an inverse square law force (i.e., where force is inversely proportional to the square of distance) must have a trajectory that is one of the conic sections. In celestial mechanics the forces are gravitational; however, it is also of interest that the forces of attraction or repulsion between electrically charged particles obey an inverse square law, and such particles also have paths that are conic sections.
Telescopes with mirrors that are conic sections are also important in space technology because of their reflective properties. We shall close this chapter by considering the design of an X-ray telescope that requires two reflections in sequence from surfaces whose cross sections are conics.
In the analysis of orbits, where a celestial body, such as a planet, comet, meteor, star, or artificial satellite moves under gravitational attraction to a primary celestial body, the center of mass of the primary body is at one focus of the conic section along which the satellite moves. Because the simplest nontrivial conic section is the circle, we shall begin with a consideration of circular orbits. (The word "nontrivial" is included because a conic section could be a point or a pair of intersecting straight lines, if the sectioning plane passes through the cone's vertex.) Most of us understand from experience Newton's first law of motion, which states that an object in motion continues in a straight line unless it is acted on by some force. If we wish to make an object move in a circular path rather than in a straight line, we must give it a constant push toward the center. Thus a central, or centripetal, force is required. For example, when we tie a string to an object and whirl it in a circle, the pull of the string is the force that keeps the object in the circular path. If we represent the centripetal force by F1, then F1 = (mv^2)/r, where m is the mass of the object, v is its speed or velocity, and r is the radius of the circle.
When a spacecraft is moving in a circular orbit about any primary body, the force toward the center is supplied by the force of gravity F2. According to Newton's law of universal gravitation, F2 = (GMm)/r^2. In this equation, G is the constant of universal gravitation, assumed to be constant throughout the universe; M and m are the masses of any two bodies; and r is the distance between their centers of gravity. The physical situation, if the forces F1 and F2 are equal, is represented in Fig. 9.1.
The arrow toward the center represents the force of gravity, the dashed arrow represents the tangential velocity of the spacecraft, and the curved arrow indicates the circular path. (In rigorous use, velocity is a "vector" quantity, because it has both magnitude and direction, whereas speed, having magnitude only, is a "scalar" quantity. We will be using the symbol v for speed, the magnitude of the velocity vector.) Thus the force of gravity holds the body in the circular orbit.