Page 49 - Satellite Communications, Fourth Edition

Chapter
2

Orbits and Launching Methods

2.1 Introduction
Satellites (spacecraft) orbiting the earth follow the same laws that
govern the motion of the planets around the sun. From early times
much has been learned about planetary motion through careful obser-
vations. Johannes Kepler (1571–1630) was able to derive empirically
three laws describing planetary motion. Later, in 1665, Sir Isaac Newton
(1642–1727) derived Kepler’s laws from his own laws of mechanics and
developed the theory of gravitation [for very readable accounts of much
of the work of these two great men, see Arons (1965) and Bate et al.
(1971)].
Kepler’s laws apply quite generally to any two bodies in space which
interact through gravitation. The more massive of the two bodies is
referred to as the primary, the other, the secondary or satellite.

2.2 Kepler’s First Law
Kepler’s first law states that the path followed by a satellite around the
primary will be an ellipse. An ellipse has two focal points shown as F 1
and F in Fig. 2.1. The center of mass of the two-body system, termed
2
the barycenter, is always centered on one of the foci. In our specific case,
because of the enormous difference between the masses of the earth and
the satellite, the center of mass coincides with the center of the earth,
which is therefore always at one of the foci.
The semimajor axis of the ellipse is denoted by a, and the semiminor
axis, by b. The eccentricity e is given by

2
2a b 2
e (2.1)
a
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