Kepler's three laws: ellipses with the Sun at one focus; equal areas in equal times; the harmonic law T squared proportional to a cubed. Their derivation and meaning.

What Are Kepler’s Three Laws of Planetary Motion?

The three laws of planetary motion that bear Johannes Kepler’s name are among the most famous results in this history. They are also among the most consequential. They replaced two thousand years of circular-orbit orthodoxy with a precise mathematical description of the planetary paths. They gave the heliocentric theory the empirical precision it had previously lacked. And they set the empirical challenge that Newton’s gravitational theory would, half a century later, be designed to meet. This page looks at the three laws in detail: their content, their derivation from Tycho Brahe’s data, and their physical meaning. The biographical and historical context is on the Kepler page. The way the laws clinched the heliocentric case is treated in How Did Kepler’s Laws Support Heliocentrism?. The way Newton derived them from the law of universal gravitation is described in What Is in Newton’s Principia Mathematica? and in The Laws of Motion and Gravity.

The First Law: The Ellipse

The first law states that the orbit of each planet is an ellipse, with the Sun at one focus of the ellipse. This was the breakthrough that opened the way for the rest of Kepler’s work. An ellipse is a curve defined as the set of points for which the sum of the distances to two fixed points, called the foci, is constant. The longest axis of the ellipse, passing through the foci, is called the major axis; the shortest, perpendicular to the major axis through the center, is the minor axis. Half the major axis is the semi-major axis, often denoted a; half the minor axis is the semi-minor axis, b. The distance from the center of the ellipse to each focus is c, and the eccentricity of the ellipse, denoted e, is c/a. When e = 0, the ellipse is a circle. As e approaches 1, the ellipse becomes more and more elongated. (A parabola is the limiting case e = 1, and a hyperbola is e > 1.) Kepler’s discovery of the elliptical orbit is one of the great set-pieces in this history. He had been trying, for years, to fit Tycho Brahe’s observations of Mars with a circular orbit. The fit was good — within about eight arcminutes — but not perfect. The eight arcminutes (the angular size of a U.S. quarter coin seen from a hundred meters) was the discrepancy that would not go away. Kepler tried a series of alternatives: an eccentric circle, an oval, a “vicarious” hypothesis involving an epicycle. None of them fit. In 1605, in a moment of frustration, he tried an ellipse with the Sun at one focus. The fit was exact, within the precision of Tycho’s observations. The result was published in 1609, in the Astronomia Nova, as Kepler’s first law. The law has several important features. It applies to each planet individually: each planet moves in its own ellipse, with the Sun at one focus. The Sun is not, in general, at the center of the orbit; the distance from the Sun to the planet varies through the orbit, with the planet closest to the Sun (at perihelion) and farthest from the Sun (at aphelion). For the Earth, the difference is about 3 percent, which is small but measurable. For Mercury, it is about 40 percent, which is large. For most of the planets, the orbits are close to circles, and the elliptical shape was difficult to detect with the naked eye. Mars, with an eccentricity of about 0.09, was the planet for which the deviation from a circle is most pronounced and most easily detected; this is one of the reasons Kepler was able to discover the law using Mars data.

The Second Law: Equal Areas in Equal Times

The second law states that a line joining a planet and the Sun sweeps out equal areas in equal times. This is the dynamical heart of the new astronomy. It is also the law that requires the most care to understand. The line joining a planet and the Sun is, in modern terms, the “radius vector” of the planet’s orbit. As the planet moves along its orbit, this line sweeps out area. The second law says that the rate at which this area is swept out is constant. Equivalently, the planet moves fastest when it is closest to the Sun (at perihelion) and slowest when it is farthest (at aphelion). The variation in speed is a consequence of the elliptical shape of the orbit. The second law was also published in the Astronomia Nova, in 1609. Kepler derived it from the data, though the derivation was not straightforward, and he had to make some auxiliary assumptions about the way the planet’s “natural” motion is affected by the Sun. The full physical interpretation of the law — that the area-sweep rate is constant because the force is directed toward the Sun — came later, with Newton, in the Principia. Newton’s second law of motion, combined with the law of universal gravitation, implies the equal-areas law. Kepler, who had no such laws, derived the result empirically, and he was forced to use somewhat obscure arguments about the way the Sun “drives” the planet around. The second law is sometimes called the “area law,” and it is one of the most useful tools in celestial mechanics. It allows the astronomer to compute the position of a planet in its orbit as a function of time, given the position at one particular time. The problem of computing the position of a planet from the area law is called “Kepler’s equation,” and it is one of the central problems in the older planetary astronomy.

The Third Law: The Harmonic Law

The third law states that the square of a planet’s orbital period is proportional to the cube of its semi-major axis. If T is the orbital period and a is the semi-major axis, then T² ∝ a³, or, in modern notation, T² = ka³ for some constant k that is the same for all planets. The third law is the simplest and most famous of the three. It is also the most profound, because it relates the periods of the planets to the sizes of their orbits, and it does so in a way that any gravitational theory would have to explain. The law was published in 1619, in Kepler’s Harmonices Mundi. Kepler’s derivation of the law was, by his own account, partly empirical and partly mystical. He had been collecting data on the periods and distances of the planets for years, and on 15 May 1618, he wrote in a letter, “the thing has come together in my head.” The law holds, with high precision, for the six planets known to Kepler (Mercury, Venus, Earth, Mars, Jupiter, Saturn), and it has been confirmed for the planets discovered since (Uranus, Neptune, the dwarf planets) and for the satellites of the planets. The physical meaning of the third law is the central clue to Newton’s gravitational theory. The law says that the period of a planet’s orbit grows with the size of the orbit in a specific way. From Newton’s second law and the law of universal gravitation, it follows that the period is related to the size of the orbit and the mass of the central body by T² = (4π²/GM) a³, where G is the gravitational constant and M is the mass of the central body. The constant of proportionality in Kepler’s law is, therefore, a measure of the mass of the Sun. The way Newton derived the law is treated in What Is in Newton’s Principia Mathematica? and in How Did Newton Discover Gravity?.

The Derivation from Data

Each of Kepler’s laws was derived from Tycho Brahe’s data, and each was the result of years of patient calculation. Kepler did not, in modern terms, have a computer. He used a method of approximation that involved, among other things, computing the area of a sector of an ellipse by numerical integration, dividing the area into many small triangles and adding up the areas. The method was laborious, and it required Kepler to repeat the same calculation hundreds of times. The first law, in particular, was the result of a long, frustrating process. Kepler had inherited Tycho’s data set, which included observations of Mars at many different points in its orbit. He had tried, in succession, to fit the data with a circular orbit, an eccentric circle, an oval, and a “vicarious” hypothesis involving an epicycle. None of them worked. The eight arcminutes of discrepancy was the breaking point. Kepler tried an ellipse, and it fit. The second law was derived from the data on the time it took Mars to traverse different parts of its orbit. Kepler had observed, and Tycho had confirmed, that Mars moved more slowly when it was far from the Sun and more quickly when it was near. The relation between the time and the position was not, at first, obvious. Kepler tried several different forms, including one that related the time to the distance from the aphelion and another that related it to the length of the orbit. The form that finally worked was the equal-areas law. The third law was the most empirical of the three. Kepler had the periods of the planets (the time for a complete orbit) and the semi-major axes (the sizes of the orbits, in astronomical units) for the six known planets. He tried several different relationships between them. The relationship that finally worked was T² ∝ a³.

The Physical Meaning

The three laws together describe the kinematics of planetary motion. They say, in effect, that each planet moves in an ellipse, with the Sun at one focus; that the rate of sweeping out area is constant; and that the period of the orbit is related to the size of the orbit in a specific way. The laws are kinematic: they describe how the planets move, not why they move that way. The “why” was the question Newton answered. Newton’s theory of universal gravitation, set out in the Principia of 1687, derived Kepler’s three laws from the combination of three more general principles: the law of inertia, the law of force equal to mass times acceleration, and the law that every mass attracts every other mass with a force proportional to the product of the masses and inversely proportional to the square of the distance between them. The first two are the first and second of Newton’s three laws of motion. The third is the law of universal gravitation. The derivation is described in What Is in Newton’s Principia Mathematica?. The relation between Kepler’s laws and Newton’s theory is one of the great examples in this history of a phenomenological law (a law that describes the phenomena) being explained by a more fundamental theory (a law that explains why the phenomena behave as they do). Kepler gave a precise, mathematical description of the planetary motions. Newton explained why the motions had to be that way. The story is the central narrative of the Newton page and of How Did Newton Discover Gravity?.

The Legacy of the Three Laws

The three laws of planetary motion are still in use today. They are the basis of the calculation of planetary positions, the design of space missions, the prediction of eclipses, and the analysis of double-star systems. The laws also play a central role in the analysis of any system in which a small body orbits a large one: artificial satellites, the moons of the planets, the moons of other planets discovered by recent spacecraft, and the orbits of binary stars. The three laws were, in their own time, a decisive break with the past. They said, against two thousand years of tradition, that the planetary orbits were not circles. They said, against the philosophical tradition that celestial motions were “natural” and needed no explanation, that the planetary motions had to be explained by some kind of physical cause. And they said, against the tradition that astronomy was the science of saving the appearances, that astronomy was the science of how the planets really moved. The combination of these three claims is the central achievement of Kepler’s work, and it is the foundation of the modern science of celestial mechanics.

What Are Kepler's Three Laws of Planetary Motion?