Kepler's laws of planetary motion

• 1. Introduction

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  Johannes Kepler stared at his calculation and could barely accept what it told him. "I first believed I was dreaming," he wrote, before steadying himself. "But it is absolutely certain and exact that the ratio which exists between the period times of any two planets is precisely the ratio of the 3/2th power of the mean distance." That moment of doubt sits at the heart of three laws describing how planets orbit the Sun. Kepler published them between 1608 and 1621 across three works: Astronomia nova, Harmonice Mundi, and Epitome Astronomiae Copernicanae. He built them on the precise observations of one man, Tycho Brahe, and on a strange belief that the Sun emitted invisible threads. How does a planet's orbit bend into an ellipse, why does it speed up and slow down, and how did errors that cancelled each other out lead Kepler to truths that still hold today? The answers begin with a planet that refused to behave.

• 2. The Stubborn Orbit Of Mars

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  Mars had the highest eccentricity of any planet except Mercury, and that accident of nature changed astronomy. In Astronomia nova, published in 1609, Kepler tried to fit Brahe's highly precise observations of Mars to a circle and failed. He had believed in the Copernican model, which demanded circular orbits, yet the data would not bend to it. From that failure came his first law: the orbit of every planet is an ellipse with the Sun at one of the two foci. Kepler then reasoned outward from Mars. He inferred that other bodies in the Solar System, including those farther from the Sun, also traced ellipses. The shape itself can be written as a relationship between the eccentricity and the distance from the Sun, where the eccentricity falls between zero and one. At an eccentricity of exactly zero, the ellipse collapses into a circle with the Sun at its centre. The point of closest approach is called perihelion, and the farthest point, exactly opposite it, is aphelion. Mars, the planet that broke the circle, had handed Kepler a new geometry for the heavens.

• 3. Equal Areas, Unequal Speeds

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  A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. That single sentence became the second law, and it means a planet travels faster when it is closer to the Sun and slower when it is farther away. Kepler reached it before he found his first law, and he reached it through assumptions that were either only approximately true or outright false. He believed planets needed a constant push to keep moving, a notion drawn from incorrect Aristotelian physics. He thought the Sun's force fell off in simple inverse proportion to distance, reasoning that gravity spreading in three dimensions would be wasted on planets confined to a plane. He also assumed the orbits were circular, which contradicted the law he had not yet discovered. The errors cancelled. The result was exactly true because it is logically equivalent to the conservation of angular momentum, which holds for any body under a radially symmetric force. Kepler kept two versions of this idea, an early "distance law" correct only for nearly circular orbits, and the later "area law" correct for all ellipses. He did not present the second law in its modern form until his Epitome Astronomiae Copernicanae of 1621.

• 4. The Harmonic Law

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  The square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit. Kepler published this third law in 1619 in his Harmonice Mundi, and he called it the harmonic law because he found it while trying to express the "music of the spheres" in precise terms and musical notation. The pattern emerged from a table of his own data. Mercury sat at a mean distance of 0.389 astronomical units with a period of 87.77 days, while Saturn stretched to 9.510 units and 10,759.2 days, yet a derived ratio held steady near 7.5 across every planet. Kepler had learned of John Napier's recent invention of logarithms and log-log graphs before the pattern revealed itself. In 1621 he noticed the same law governed the four brightest moons of Jupiter. The original form, stated in terms of a "mean distance" rather than the semi-major axis, holds true only for planets with small eccentricities near zero. Godefroy Wendelin, the first well-known astronomer to adopt Kepler's laws, gave a detailed account of this third law in 1652.

• 5. Where Copernicus Went Wrong

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  Copernicus had been right that planets revolve around the Sun, but wrong about how. His model held that a planetary orbit was a circle with epicycles, that the Sun sat approximately at the centre, and that the planet's speed in its main orbit stayed constant. Kepler overturned all three claims. The orbit was an ellipse, not a circle with epicycles. The Sun stood at a focal point, not the centre. And neither the linear nor the angular speed was constant; only the area speed remained fixed, a quantity historically tied to angular momentum. The eccentricity of Earth's own orbit makes the case visible. The stretch from the March equinox to the September equinox runs about 186 days, while the return from September to March takes only about 179 days. A plane through the Sun parallel to Earth's equator divides the orbit into two parts whose areas stand in a 186 to 179 ratio. Working from this, one can estimate Earth's eccentricity, though the simple version lands a factor of two off the correct value of 0.016710218. The calculation assumes perihelion falls on a solstice. The current perihelion, near January 4, sits fairly close to the solstice of December 21 or 22, but that gap of 14 to 15 days exceeds the 7-day difference in inter-equinox times, which explains the substantial error.

• 6. Magnetic Fibrils And A Theory Of Force

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  Kepler did not see geometry alone; he saw a physical cause. His laws grew from a theory in which the Sun emitted magnetic fibrils that pulled the planets into their orbits. These fibrils were somewhat elastic, which allowed non-circular motion driven by the inertia of the planets. The idea sounds alien now, yet it pushed astronomy past mere description toward physical explanation. Kepler's contemporaries were slow to follow. His work was a strong defense of Copernicanism at a time when that view had fallen out of fashion, partly because of opposition from Tycho Brahe himself. The second law, in particular, had little impact at first, since calculations of planetary positions using it were approximate and time consuming. Nicolaus Mercator contested the area law in a book from 1664, yet by 1670 the Philosophical Transactions had turned in its favour, and acceptance widened as the century went on. The turning point came earlier, in 1627, when Kepler published the Rudolphine Tables. They held many accurate observations accumulated by Brahe, and their breadth let astronomers test Kepler's formulas against good data. The calculations were off-putting at first, but once undertaken, more astronomers were convinced.

• 7. From Three Insights To Newton's Gravity

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  Isaac Newton took Kepler's first and second laws and computed what they demanded of a planet's acceleration. He found the acceleration points toward the Sun and falls off as the inverse square of the planet's distance from it. Newton understood that the second law is not special to the inverse-square law; it follows from any force that is purely radial. The first and third laws, by contrast, depend on the inverse-square form itself. From this he built his law of universal gravitation, published in 1687, holding that all bodies in the Solar System attract one another in proportion to the product of their masses and in inverse proportion to the square of the distance between them. Because the planets carry small masses compared with the Sun, their orbits conform approximately to Kepler's laws, and Newton's model fits actual observations more accurately. Newton was cautious about meaning. He stated in his Principia that he considered forces from a mathematical point of view rather than a physical one, assigning no cause to gravity. Long afterward, Carl Runge and Wilhelm Lenz identified a symmetry principle in the phase space of planetary motion, the orthogonal group O(4), that accounts for the first and third laws under Newtonian gravity. The reception in Germany shifted noticeably between 1688, the year Newton's Principia appeared and was read as basically Copernican, and 1690, by which time Gottfried Leibniz's work on Kepler had been published.

• 8. The Making Of Three Laws

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  Kepler never spoke of three numbered laws. It took nearly two centuries for his work to settle into the form taught today. Voltaire's Eléments de la philosophie de Newton of 1738 was the first publication to use the terminology of "laws," and the language of scientific laws for these discoveries was current at least from the time of Joseph de Lalande. The set of three as we know it came from Robert Small, whose An account of the astronomical discoveries of Kepler, published in 1814, gathered them together by adding in the third. Small also claimed, against the history, that these were empirical laws based on inductive reasoning, when in truth Kepler had reasoned from his theory of solar force. The laws themselves outlasted that tidy story. They became the basis for computing a planet's position as a function of time, through a transcendental equation known as Kepler's equation, solved by way of an intermediate quantity called the eccentric anomaly. A deviation from uniform circular motion, once treated as the natural state, is still called by the name Kepler's predecessors gave it: an anomaly.