Astronomy Principles and Practice Fourth Edition.pdf

Chapter 13
Celestial mechanics: the two-body problem
13.1 Introduction
By his telescope observations, Galileo showed that the Ptolemaic theory failed as an adequate
description of planetary geocentric phenomena. The Copernican System, however, was able to
embrace the new discoveries regarding the phases of Venus. No-one, however, was able as yet to
give any satisfactory explanation of why the movements of the planets were as observed or why the
Moon revolved about the Earth. Half a century had to pass before the explanation was given by Sir
Isaac Newton (1642–1727 AD). His work was built on the foundations laid by Tycho Brahé (1546–
1601 AD), Johannes Kepler (1571–1630 AD) and Galileo Galilei (1564–1642 AD).
13.2 Planetary orbits
13.2.1 Kepler’s laws
Johannes Kepler, from the study of the mass of observational data on the planets’ positions collected
by Tycho Brahé, formulated the three laws of planetary motion forever associated with his name. They
are:
(1) The orbit of each planet is an ellipse with the Sun at one focus.
(2) For any planet the radius vector sweeps out equal areas in equal times.
(3) The cubes of the semi-major axes of the planetary orbits are proportional to the squares of
the planets’ periods of revolution.
Kepler’s first law tells us what the shapes of the planetary orbits are and gives the position of the
Sun within them.
Kepler’s second law states how the angular velocity of the planet in its orbit varies with its distance
from the Sun.
Kepler’s third law relates the different sizes of the orbits in a system to the periods of revolution
of the planets in these orbits.
At the time of their formulation these laws were based upon the most accurate observational
material available. The great Danish astronomer, Tycho Brahé, had worked in the pre-telescope era
of astronomy but the accuracy of his observations had been of a high standard, certainly high enough
in the case of the Martian orbit to convince Kepler that the age-old idea of circular orbits had to be
discarded in favour of elliptical ones.
Kepler’s laws are still very close approximations to the truth. They hold not only for the system of
planets moving about the Sun but also for the various systems of satellites moving about their primaries.
Only when the outermost retrograde satellites in the Solar System are considered, or close satellites of
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