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Fig. 3-13. Lagrange Points of the Earth-Moon System

'Named after the 18th-Century French mathematician and astronomer Joseph Lagrange.

Orbit Maneuvers. A significant portion of spaceflight analysis is concerned with orbit maneuvers and optimal methods of changing orbits. Normally we wish to minimize the energy which must be supplied or, equivalently, the velocity change, A V, required to go from one orbit to another. There are two basic types of maneuvers: in-plane maneuvers, which do not affect the plane of the spacecraft orbit and out-of-plane or plane change maneuvers, which change the orbital plane. Because the spacecraft velocity vector is in the orbit plane, any component of the rocket thrust normal to the plane of the orbit will have only a small effect on the magnitude of the velocity and, therefore, on the total energy and semimajor axis. If we wish to obtain a specified semimajor axis with a minimum expenditure of energy, then plane changes should be minimized.

The principal orbit characteristic relevant for in-plane maneuvers is that the semimajor axis depends only on the total energy. Therefore, the most efficient way to change the semimajor axis and raise or lower the spacecraft is to change only the magnitude of the velocity by firing the rocket either parallel or antiparallel to the velocity vector. If we wish to transfer between two circular orbits (for example, to travel from the Earth to Mars or from low Earth to synchronous orbit), then we start at the lower orbit and fire the rocket so that the propellent is expelled in the direction opposite the velocity vector, as illustrated in Fig. 3-14. For a minimum energy expenditure, the semimajor axis of the transfer orbit should be such that its apofocus is at the radius of the larger orbit. Such an elliptical orbit with perifocus at the smaller orbit and apofocus at the larger orbit is called a Hohmann transfer ellipse and is the minimum energy path between the orbits, either from the smaller to the larger or the larger to the smaller. The Hohmann transfer ellipse is tangent to both the inner and outer orbits.

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