An Imaginary Tale by Nahin Paul J.;

Author:Nahin, Paul J.; [Nahin, Paul J.;]
Language: eng
Format: epub
ISBN: 9780691169248
Publisher: PrincetonUP
Published: 2017-07-15T05:00:00+00:00

In chapter 7 I will show you how to evaluate this integral using Cauchy’s theory of contour integration in the complex plane. If, however, you simply can’t wait until then to verify Kepler’s third law for E ≠ 0, then here’s the value of the integral for you to play with: .

5.4 WHY AND WHEN THE OTHER PLANETS SOMETIMES APPEAR TO GO BACKWARD

When observed from the Earth, the other planets (ancient Greek astronomers knew of five: Mercury, Venus, Mars, Jupiter, and Saturn) periodically seem, at various times, to reverse direction temporarily in their motions across the night sky. The ancient astronomers could “explain” these mysterious retrograde motions with Ptolemy’s crystal spheres, but in fact these motions are simply illusions caused by watching one moving thing (a planet) from another moving thing (the Earth). Kepler knew this, and was the first to explain the illusion using diagrams to illustrate his qualitative reasoning. Complex exponentials, however, make the mathematics of what is happening easy to understand as well. Here’s how.5

By definition, the Earth has an orbital period of one year. Also by definition, the Earth’s average distance from the sun of 93,000,000 miles is one astronomical unit (A.U.). So, with the Sun as the center of our coordinate system, and making the good approximation of a circular orbit, we can write the position vector of the Earth as

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