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  • State Kepler’s laws of planetary motion.
  • Derive the third Kepler’s law for circular orbits.
  • Discuss the Ptolemaic model of the universe.

Examples of gravitational orbits abound. Hundreds of artificial satellites orbit Earth together with thousands of pieces of debris. The Moon’s orbit about Earth has intrigued humans from time immemorial. The orbits of planets, asteroids, meteors, and comets about the Sun are no less interesting. If we look further, we see almost unimaginable numbers of stars, galaxies, and other celestial objects orbiting one another and interacting through gravity.

All these motions are governed by gravitational force, and it is possible to describe them to various degrees of precision. Precise descriptions of complex systems must be made with large computers. However, we can describe an important class of orbits without the use of computers, and we shall find it instructive to study them. These orbits have the following characteristics:

  1. A small mass m size 12{M} {} orbits a much larger mass M size 12{M} {} . This allows us to view the motion as if M size 12{M} {} were stationary—in fact, as if from an inertial frame of reference placed on M size 12{M} {} —without significant error. Mass m size 12{m} {} is the satellite of M size 12{M} {} , if the orbit is gravitationally bound.
  2. The system is isolated from other masses . This allows us to neglect any small effects due to outside masses.

The conditions are satisfied, to good approximation, by Earth’s satellites (including the Moon), by objects orbiting the Sun, and by the satellites of other planets. Historically, planets were studied first, and there is a classical set of three laws, called Kepler’s laws of planetary motion, that describe the orbits of all bodies satisfying the two previous conditions (not just planets in our solar system). These descriptive laws are named for the German astronomer Johannes Kepler (1571–1630), who devised them after careful study (over some 20 years) of a large amount of meticulously recorded observations of planetary motion done by Tycho Brahe (1546–1601). Such careful collection and detailed recording of methods and data are hallmarks of good science. Data constitute the evidence from which new interpretations and meanings can be constructed.

Kepler’s laws of planetary motion

Kepler’s First Law

The orbit of each planet about the Sun is an ellipse with the Sun at one focus.

In figure a, an ellipse is shown on the coordinate axes. Two foci of the ellipse are joined to a point m on the ellipse. A pencil is shown at the point m. In figure b the elliptical path of a planet is shown. At the left focus f-one of the path the Sun is shown. The planet is shown just above the Sun on the elliptical path.
(a) An ellipse is a closed curve such that the sum of the distances from a point on the curve to the two foci ( f 1 size 12{f rSub { size 8{1} } } {} and f 2 size 12{f rSub { size 8{2} } } {} ) is a constant. You can draw an ellipse as shown by putting a pin at each focus, and then placing a string around a pencil and the pins and tracing a line on paper. A circle is a special case of an ellipse in which the two foci coincide (thus any point on the circle is the same distance from the center). (b) For any closed gravitational orbit, m size 12{m} {} follows an elliptical path with M size 12{M} {} at one focus. Kepler’s first law states this fact for planets orbiting the Sun.

Kepler’s Second Law

Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times (see [link] ).

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