Fundamentals of astrodynamics and applications 4th Edition (2013)

12 EQUATIONS OF MOTION 1.2
1.1.5 Other Early Astrodynamic Contributions
Although this chapter deals mainly with the two-body problem, the multi-body problem
has great importance in celestial mechanics, as well as a rich history. Joseph Louis
Lagrange (1736–1813) showed the first solutions for the three-body problem in Essai
sur le problème des trois corps in 1772. In the same year, Leonhard Euler (1707–1783)
arrived at a similar conclusion for equilibrium points. Carl Gustav Jacob Jacobi (1804–
1851) found an exact integral in 1836 (Comptes rendus de l’Académie des Sciences de
Paris), and Henri Poincaré (1854–1912) proved this was the only exact integral for the
three-body problem in 1899. This work even won a prize from the King of Sweden. The
technique was only of mathematical interest until 1906 when the Trojan planets were
discovered near Jupiter. These small planetoids, named after heroes in the Trojan war,
remain near equilibrium positions described by the restricted three-body problem. History
has favored Lagrange’s contributions, so the equilibrium positions bear his name.
The achievements of the people discussed in this section may seem small, but they’re
actually momentous in the history of modern astrodynamics. Together with many other
scientists of the time, these pioneers broke a 1500-year logjam and created the basic
tools needed to advance astrodynamic thought and study. The mathematical techniques,
instruments, and means of distribution had caught up to the scientific thought of the
time. Indeed, even with the advent of the computer, much of our work in astrodynamics
uses the results these men formed long before the modern era!
1.2 Geometry of Conic Sections
Kepler’s first law states that the planets travel in conic sections, namely, ellipses. We
must consider the types of conic sections because they represent all the possibilities
allowed by Newton’s law of gravity for satellite orbits. A conic section is the intersection
of a plane and a right circular cone.
Figure 1-3 illustrates generally how to find each conic section. The first four conic
sections (circle, ellipse, parabola, and hyperbola) are considered to be orbits. The point
at the intersection of the two cones is a special case that doesn’t represent realistic satellite
motion. Rectilinear orbits are special cases of elliptical, parabolic, and hyperbolic
motion. * Later chapters treat both standard and special orbits.
1.2.1 Basic Parameters
A few geometrical concepts, particularly conic sections, underlie many derivations
throughout this book. Every conic section has two foci. From a geometrical standpoint,
the foci provide the locations from which to construct a conic section, as illustrated in
* Rectilinear orbits result from the plane lying on an outer surface of the cone, a plane parallel to
and infinitely far from the surface, and a plane perpendicular to and infinitely far from the center
of the base, respectively. Although these aren’t really closed orbits, they represent ideal limiting
cases that we may approximate by sections of parabolic and hyperbolic orbits. We use them to
represent sections of maneuvers, comets, or flights of missiles. Because we may encounter these
orbits, computer software should account for them to avoid numerical problems.