In this chapter, we present the theory that is used in the analysis and design of ballistic guidance systems, with an emphasis on the principles rather than on the implementation. The trajectory of a ballistic missile typically consists of three phases: a powered lift-off, a free flight (this typically lasts about 80% of the engagement), and an aerodynamic reentry (see Figure 6.1). The purpose of ballistic guidance is then to determine the boundary conditions between the first and second phases that lead to a hit, and to analyze the miss due to navigation errors. Note that the optimization of the first phase leads to interesting “rocket staging” problems that can be treated with methods introduced in Chapter 8.

Section 6.1 describes the restricted two-body problem. Section 6.2 deals with the two-dimensional hit equation, whereas Section 6.3 contains the in-plane error analysis. Section 6.4 deals with three-dimensional error analysis, Section 6.5 with accounting for effects of the Earth's rotation. Section 6.6 considers the effect of the Earth's oblateness and geophysical uncertainties. Section 6.7 presents a general framework for numerical solution of general ballistic guidance problems. Sections 6.8, 6.9, and 6.10 present a summary of the key results in the chapter, bibliographic notes for further reading, and homework problems, respectively.

The Restricted Two-Body Problem

Assume that a ballistic missile, modeled as a particle, moves above the sensible atmosphere (i.e., above an altitude of 80 km). We assume that the Earth is nonrotating and perfectly spherical.