Kepler’s vanishing circles hidden in Hamilton’s Hodograph

The Greeks regarded the heavens as the epitome of perfection. All flaws and blemishes were confined to the terrestrial domain. Since the circle is perfect in its infinite symmetry, it was concluded by Aristotle that the Sun and planets move in circles around the Earth. Later, the astronomer Ptolemy accounted for deviations by means of additional circles, or epicycles. He stuck with the circular model.

The Platonic ideal of circular motions continued to dominate our view of the heavens for centuries. In the 16th century, Copernicus moved the centre to the Sun, but the planetary orbits remained resolutely circular. Later, more precise observations of planetary motions became available through the work of the Danish astronomer Tycho Brahe.

Johannes Kepler struggled to find orbits that were consistent with these measurements. For Mars, he could not reconcile the circular model with the observed patterns. After much mental anguish, he had to abandon the effort and accept that the orbit of Mars was in the form of an ellipse – a flattened circle. In 1609 he published his law of the ellipse. This was a courageous leap away from received wisdom.

Where did the circles go?

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We can illustrate the orbit of a planet about the Sun by showing its position at different times. This gives the elliptical curve, closing on itself as the motion recurs periodically, as shown in the illustration. But there is another way to describe the motion: we can represent the planetary velocity by a vector, an arrow with a definite length and direction.

When the planet moves slowly, the arrow is short; when the planet moves rapidly, it is long. The arrows can be attached to the relevant points on the ellipse. But they can also be plotted all together from a common point. This gives a diagram called a hodograph, also shown in the illustration.

The great Irish scientist William Rowan Hamilton devised the hodograph, and used it in his analysis of the motions of the planets. In December 1846, Hamilton presented his ideas at a meeting of the Royal Irish Academy. His work was later published in the Proceedings of the Academy with the title "The hodograph, or a new method of expressing in symbolical language the Newtonian law of attraction".

The Law of the Circular Hodograph

Hamilton showed that if the force of gravity varies inversely with the square of the distance, as shown by Newton, the tips of the arrows trace out - almost miraculously - a circular curve. Thus, “the angular motion of a body in its orbit is exactly represented, with all its variations, by the circular motion on the hodograph”.

Since the speed of the planet varies as it travels around the ellipse, the common origin of the vectors or arrows is not at the centre of the circle, but is off-set by an amount that depends on the eccentricity, or flattening, of the orbit. Hamilton remarked that “the Newtonian law may be characterised as being the Law of the Circular Hodograph.”

We may conclude that the circles that were banished by Kepler in 1609 did not vanish entirely, but were hiding in the hodographic representation, to be revealed 235 years later by Hamilton. Perfect.

Peter Lynch is emeritus professor at UCD School of Mathematics & Statistics, University College Dublin – ge blogs at thatsmaths.com