Kepler's laws of planetary motion

• 1. The Mars Data Crisis

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  In 1609, Johannes Kepler published a book titled Astronomia nova in Prague. This work contained the first two laws of planetary motion derived from the precise astronomical observations collected by Tycho Brahe. Kepler had spent years trying to fit Brahe's data for the planet Mars into a circular orbit model. The data simply would not align with any circle, creating a crisis that forced him to abandon centuries of accepted geometry. Mars possessed the highest orbital eccentricity among all planets except Mercury, making it the perfect test case for this failure. The Sun emitted magnetic fibrils according to Kepler's physical theory at the time, which pulled planets into their paths. These fibrils were somewhat elastic, allowing for non-circular motion driven by the inertia of the planets themselves. Kepler realized that the distance between the Sun and Mars varied significantly enough to prove the orbit was an ellipse.

• 2. Replacing Circular Orbits

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  Johannes Kepler improved upon the heliocentric model proposed by Nicolaus Copernicus decades earlier. Copernicus claimed that planetary orbits were circles with epicycles and that the Sun sat approximately at the center of these orbits. He also argued that the speed of a planet in its main orbit remained constant throughout its journey. Kepler proved these assumptions incorrect through his analysis of Mars. He established that the planetary orbit is actually an ellipse with the Sun located at one of the two foci rather than the geometric center. This change meant that neither linear nor angular speed remains constant as a planet travels. Instead, the area swept out by a line joining the planet and the Sun remains equal during equal intervals of time. The Earth provides a clear example of this variation. The time from the March equinox to the September equinox spans around 186 days. In contrast, the period from the September equinox back to the March equinox takes only about 179 days. This difference arises because the plane through the Sun parallel to the Earth's equator cuts the elliptical orbit into two parts with areas in a 186 to 179 ratio.

• 3. The Harmonic Law Discovery

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  Kepler published his third law in 1619 within a work titled Harmonice Mundi. He described this relationship as the harmonic law or music of the spheres expressed through precise mathematical notation. The law states that the square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit. Kepler became aware of John Napier's recent invention of logarithms before he discovered this pattern. He used log-log graphs to visualize the relationship between distance and time for various planets. A table of data from 1618 shows Mercury at 0.389 AU with a period of 87.77 days. Venus follows at 0.724 AU taking 224.70 days. Earth sits at exactly 1 AU with a period of 365.25 days. Mars measures 1.524 AU over 686.95 days. Jupiter extends to 5.204 AU with a period of 4,332.62 days. Saturn completes its cycle at 9.510 AU in 10,759 days. Kepler noted that the ratio of the square of the period to the cube of the mean distance remained nearly constant across all these bodies.

• 4. Newton's Physical Cause

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  Isaac Newton demonstrated in 1687 that relationships like Kepler's would apply in the Solar System as a consequence of his own laws of motion and law of universal gravitation. His work appeared in Philosophiæ Naturalis Principia Mathematica. Newton showed that the acceleration of a planet moving according to Kepler's first two laws points directly toward the Sun. The magnitude of this acceleration is inversely proportional to the square of the planet's distance from the Sun. This inverse square law implies that the Sun may be the physical cause of planetary acceleration. Newton defined force acting on a planet as the product of its mass and acceleration. He stated that every planet is attracted towards the Sun by a force directly proportional to the planet's mass. The force also varies inversely with the square of the distance between them. Newton assumed that all bodies in the Solar System attract one another symmetrically. Since planets have small masses compared to the Sun, their orbits conform approximately to Kepler's laws while Newton's model fits actual observations more accurately.

• 5. Standardizing Scientific Laws

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  It took nearly two centuries for the current formulation of Kepler's work to take on its settled form. Voltaire published Eléments de la philosophie de Newton in 1738, which was the first publication to use the terminology of laws for these discoveries. Voltaire wrote that each planet describes equal areas in equal times under one of the great laws of Kepler. Joseph de Lalande later confirmed that the terminology of scientific laws for these discoveries was current at least from his time. Robert Small standardized these three specific discoveries into a formal set in 1804 through An account of the astronomical discoveries of Kepler. Small claimed against history that these were empirical laws based on inductive reasoning rather than derived from physical causes. Godefroy Wendelin provided a detailed account of the third law in 1652 regarding Jupiter's moons. He communicated a letter showing that the periods and distances of Jovian satellites conformed to Kepler's third law. The four brightest moons of Jupiter obey this relationship approximately as described by Kepler in 1621.