Kepler's Laws

Type of physical science: Kepler's Laws, Kepler, Johannes, Planetary motion, Solar system, Astronomy and astrophysics

Field of study: Planetary systems

Johannes Kepler's three laws of planetary motion simplified astronomical theory, supported the Sun-centered cosmology of Nicolaus Copernicus, and for the first time based astronomy on physical causality rather than geometry and metaphysics.

Overview

The three laws of planetary motion formulated by the German mathematician Johannes Kepler (1571-1630) describe the conditions of a planet's orbit around the Sun and its relation to the orbits of the other planets. Because of these laws, the first of their kind in science, historians consider Kepler the first modern astronomer. In fact, however, Kepler proposed his laws in part because he was an astrologer who cast horoscopes and believed that the planets exert a physical influence on human lives, although he rejected much of the mystical and magical aspects of astrology. Nevertheless, Kepler was the first astronomer to derive ideas completely by empirical methods: He founded his theories of motion on physical causality rather than theology, philosophy, or pure abstractions of mathematics, and he thereby initiated scientific astronomy. His influence on later astronomers and cosmologists, among them Isaac Newton, has been fundamental.

Kepler published the first two laws in Astronomia nova, aitiologetos seu physica coelestis (1609; new astronomy, causally explained, or celestial physics) as part of a thoroughgoing review and overhaul of the astronomical systems of Ptolemy, Tycho Brahe, and Nicolaus Copernicus. All three were elaborate geometries of the heavens based on the assumption that the planets move in perfect circles. Ptolemy placed Earth at the center of these circles; Copernicus put the Sun at the center; Tycho had the planets circling the Sun, which with them circled Earth. Kepler was a Copernican, but an atypical one. Many adherents thought of the Copernican model as a method to simplify the calculation of orbits, not as a physical reality. Kepler believed that the Sun was in fact the physical center of planetary orbits. Furthermore, he insisted that the Sun exerts a force to keep the planets on their proper paths. He did not know the true nature of that force, nor would anyone else until Newton devised his theory of gravitation nearly eighty years later, but Kepler supposed it was magnetism or some force like it. These assumptions, correct in spirit if not wholly accurate, were the physics upon which Kepler based the mathematical relationships of his laws.

According to the first law, a planet's orbit is an ellipse, and the Sun lies at one of the two foci. Geometers define an ellipse as the intersection of a plane and a cone, and ellipses are sometimes called "conical sections." One can also define the ellipse in the manner of analytic geometry as the closed sequence of points around two foci such that lines connecting both foci to any of the points are together equal in length to the major axis of the ellipse. Kepler discerned the law from the data that Tycho, a Danish astronomer and Kepler's employer, had compiled on the orbit of Mars during thirty years of observations. Kepler found that an ellipse explained the complicated data most simply, assuming that Mars revolved around the Sun.

Like most of the inner planets, the orbit of Mars is nearly circular--Mercury's orbit is the most elliptical--and so the variations from a circle are not large. For Kepler, though, they were too large to dismiss as errors in Tycho's observations, which were by far the most accurate of the era. With its basis in the ellipse, the first law was a daring departure from the ideas of earlier theorists, who for metaphysical reasons believed nature required the planets to move in perfect circles. To Kepler, a circle was only an ellipse in which the foci coincide, a rare special case. Every planet in the solar system, he argued, swings through a near point to the Sun, the perihelion, and a far point, the aphelion, at the other end of the long diameter of the orbital ellipse.

Like the first law, Kepler's second law concerns geometrical area, and it is a consequence of the ellipse: As a planet orbits, a line drawn between it and the Sun passes through equal areas during equal time intervals. Whereas the first law describes a static shape, the second law concerns relative velocity, that is, dynamics. The most important insight of the law is that a planet's velocity changes throughout its orbit. It accelerates toward perihelion as the distance from the Sun decreases and decelerates toward aphelion as the distance increases. As physics, the second law expresses the conservation of angular momentum, a principle not known as such to Kepler. Most simply put, angular momentum is the tendency of a body to continue orbiting, and it can be expressed as the product of its mass, velocity, and average distance from the center of the orbit.

The best way to understand this speed-distance relation, as the second law states it, is to imagine a wedge the apex of which is the Sun; the sides are two lines drawn from the planet to the Sun, one from the planet's position in the orbit at the beginning of a time interval and one from its position at the end of the interval. Near the Sun, the sides of a wedge are short, and the arc made during the interval is long; therefore, the wedge is broad. Far away from the Sun, the sides of a wedge are long, and the arc connecting them during the same time interval is short; therefore, the wedge is narrow. All such wedges defined during equal time intervals, according to Kepler, have the same area. As the first law did, the second law departed from the established principles of astronomy. Both Ptolemaists and Copernicans had assumed that planets orbit at uniform velocities.

As Kepler continued to study Tycho's data from planetary observations, he tried to find some proportional relation among the planets that hinted at the fundamental structure of the universe. Only Mercury, Venus, Mars, Jupiter, and Saturn (and the Moon, which he studied separately) were then known to move against the background of fixed stars and so could reveal such a proportion. He succeeded just in time to include the result, his third law, in his next major astronomical work, Harmonices mundi (1618; harmonies of the world); this principle is sometimes called the "harmonic law."

The third law states that a planet's orbital period squared is directly proportional to its average distance from the Sun cubed. Astronomers measure the orbital period of planets in Earth years and the distances in astronomical units (AUs). One AU equals Earth's average distance from the Sun. Accordingly, for any planet, the third law gives the orbit-to-distance relation as years squared proportional to AUs cubed. The law thus enables one to calculate either a planet's orbital period or its distance from the Sun if the other quantity is known. For example, Venus takes 0.615 years to complete an orbit. That number squared is 0.378, the cube root of which gives Venus' average distance from the Sun, 0.723 AU. Uranus is 19.18 AUs from the Sun. That number cubed is 7,055.8; the square root yields Uranus' orbital period, 84 years. Actually, the results for Venus' distance and Uranus' orbital period are not quite accurate, because other forces than the Sun's gravity affect orbits slightly. For this reason, Kepler's third law gives only a good first approximation.

When Kepler derived the law, however, it was a triumph of precision and simplicity. Because astronomers had measured the orbital periods of the first six planets accurately, for the first time they could calculate the planets' distance from the Sun in comparison to Earth's distance with similar accuracy. Once Earth's distance was measured in standard units, such as kilometers (approximately 150 million) or miles (approximately 93 million), the other planets' distances in AUs could be converted as well.

Unfortunately, the third law, as well as the first and second, went largely unnoticed by Kepler's contemporaries. In his Principia mathematica philosophiae naturalis (1687; mathematical principles of natural philosophy), however, Newton recognized the importance of Kepler's laws and generalized them as part of his own theory of motion. Every celestial body--comet, moon, meteor, asteroid, planet, or star--obeys the three laws when orbiting another body, unless an external force, such as gravity from yet a third body or a collision, perturbs the orbit. Furthermore, Newton showed, as Kepler could not, why the laws are true: The mutual attraction of gravitation, the force of which weakens in proportion to the square of the distance between two bodies, is the physical cause. From Newton's extension of Kepler's laws came the science of celestial mechanics.

Applications

In 1627, Kepler published the Tabulae Rudolphinae (Rudolphine tables), a compilation and analysis of the data from Tycho's decades of planetary observations. Together, the tables and Kepler's three laws increased the possible accuracy of astronomical calculations by a factor of one hundred. The increased accuracy improved the most common applications of astronomy in Kepler's day, astrology and navigation, but Kepler's work has since had practical uses in astronomical research and discovery itself, as well as setting astronomy on a foundation of physics.

To chart their progress at sea when they cannot see land, sailors need to find their locations north or south of the Equator--the latitude--and east or west from a meridian running through Greenwich, England--the longitude. Although the methods differed, to find the latitude or longitude, navigators measured the angle of a heavenly body, usually the Sun or Moon but sometimes the planets, from the horizon or from certain stars and compared the result with published tables indicating where on Earth the sighted angle was possible at a given time of year. Kepler's work, especially his observation that Earth and other planets orbit the Sun in ellipses, made it possible to compile more accurate tables.

Kepler's contemporary Galileo Galilei introduced the telescope as an astronomical instrument early in the seventeenth century. Efficient use of the instrument for planetary observations required that an observer be able to aim it quickly and accurately. Kepler's laws helped astronomers do so. Yet far more important, Newton's generalized versions of the laws were used to discover new planets. The square of the orbital period times the total mass of an object equals the cube of the mean radius of its orbit. Astronomers located Uranus in 1781, Neptune in 1843, and Pluto in 1930 by noting deviations of a known planet from its expected elliptical orbit and calculating the likely mass, distance, and orbital path of a body the gravity of which could cause the deviation. The calculations narrowed the search areas for observers, who eventually spotted the new planets. Astronomers continue to search for a hypothetical tenth planet and other suspected members of the solar system with the same basic methods, although with vastly improved equipment.

Kepler's laws enabled astronomers to calculate the orbits and masses of the moons of the outer planets as well as of asteroids and comets. The orbits of asteroids and comets became a public concern after the discovery in the early 1980's that large bodies occasionally strike Earth and cause mass extinctions of living species and that smaller bodies hit regularly, sometimes causing extensive destruction as well. For example, a small comet exploded over Siberia in 1908 and leveled trees for thousands of square kilometers. Naturally, looking for comets and asteroids that cross Earth's orbit and calculating their paths has become an important job for both professional and amateur astronomers in order to provide early warning of danger. The orbits of comets are so elliptical that some swing past the Sun only once on a hyperbolic orbit or return only after many years or centuries on a parabolic orbit. These orbital characteristics prompted astronomers to theorize that comets come from two regions in the distant solar system: the spherical Oort cloud, between 20,000 and 100,000 AUs from the Sun, and the Kuiper belt, just beyond Neptune's orbit. In 1992, observers confirmed the existence of the Kuiper belt; the Oort cloud is beyond the range of telescopes, but astronomers generally accept its existence.

Kepler's third law has also revealed the dynamics of stellar systems beyond the solar system. Astronomers use it to calculate the masses of binary stars after they have determined the orbital period observationally. During the early 1990's, astronomers were able to detect planets orbiting other stars than the Sun, even though they could not see these planets directly. The observed rhythmic wobbling in a star's course through interstellar space can be a sign that a planet is orbiting it. From the size and period of this wobbling motion, astronomers can estimate the mass of the planet and its distance from the star.

It is space exploration, however, that has made the laws of motion most vividly familiar to nonscientists. The manned orbital missions of space capsules, the space shuttle, and military, scientific, and commercial satellites depend upon precise use of Kepler's laws by ground controllers and pilots to maintain them at the proper altitude in their orbits and to avoid collisions, an increasing possibility as the space near Earth grows ever more crowded. Likewise, trips to the Moon and voyages of space probes to the planets require controllers to apply the laws of motion, but with an important difference. Kepler's third law, as modified to accommodate gravitation, gives accurate results for only two bodies, and then only if their orbits are not perturbed. During interplanetary travel, however, at least three bodies are part of plotting a course: Earth in its motion around the Sun, the target planet or moon, and the space vessel. Calculations for this "three-body problem" very seldom give definite solutions, and so controllers, using sophisticated computers, must recalculate frequently and fire a vessel's engines to adjust its course.

Context

In the late sixteenth century, when Kepler began his career, astronomers were having increasing difficulties fitting contemporary observations of the planets to either of the two prevailing theories. In his Almagest (circa 150 c.e.), Ptolemy, an Alexandrian mathematician, depicted the heavens as a series of nested spheres, with Earth at the center; each sphere carried a planet (from the Greek word for "wanderer") against the backdrop of the fixed stars. His model accorded with Aristotle's dictum that since the heavens are perfect and the circle is the perfect geometric form, the planets must circle Earth. The Catholic church embraced this doctrine, as did most philosophers and thinkers. For example, the nine celestial levels of Neoplatonist philosophy and the circles of heaven in Dante Alighieri's Paradiso (1321) draw from Ptolemy's model.

Yet the planets did not appear to obey Ptolemy's model strictly. Deviations from their proper circular courses were noticed by Ptolemy himself. To account for the observed motion of the planets, Ptolemaists, who made up the majority of scientists throughout antiquity and the Middle Ages, had to resort to complicated geometric modifications, essentially theorizing that the planets swung in secondary circles on the faces of their spheres as the spheres themselves wheeled around Earth. Copernicus tried to simplify the system in De revolutionibus orbium caelestium (1543; on the revolutions of the celestial spheres) by replacing Earth with the Sun as the center of the planetary orbits. He too assumed that the orbits of planets were perfect circles, however, and to accommodate observations, he also had to admit circles-within-circles into his model, although fewer than in the Ptolemaic system.

Kepler's greatest conceptual step was to reject the metaphysical tenet that the heavens consisted of perfect circles; in so doing, he freed astronomy from the traditional authority of Aristotle in the sciences. Instead, Kepler worked empirically, drawing conclusions that accounted for observations most comprehensively and simply. He thereby became part of the slow change of emphasis that distinguishes the Renaissance from the Middle Ages: Investigators such as Kepler and his Italian contemporary Galileo Galilei looked to direct experience, in the form of observations and experimentation, to settle questions about nature rather than relying on the pronouncements of ancient authors and religious authorities. Thus, Kepler fostered the modern scientific method, which holds that if verifiable data fails to support basic assumptions and theories, those assumptions and theories, and not the data, are to be abandoned.

Yet Kepler was not wholly modern. His long search for the laws of planetary motion came from a deep conviction that basic harmonies, expressible mathematically, organize the universe and that these harmonies reveal not only the mind of God but also the direct physical influence of the heavens on human affairs. Although the aspiration to decipher God's design also motivated Newton, his incorporation of Kepler's laws into the theory of gravitation was another large step away from astrology toward a rationalist astronomy. Albert Einstein's theories of relativity extended Newtonian laws of motion into four dimensions by treating gravitation as distortions in the space-time continuum of the universe. Mathematically and conceptually, relativity is far removed from Kepler's laws, it is but their descendent nevertheless.

Principal terms

ACCELERATION: Any change in an object's velocity

APHELION: The point in a planet, comet, or asteroid's orbit when it is farthest from the Sun

ELLIPSE: An oval; a closed figure formed by the intersection of a plane and a cone, having a major axis (long diameter) and minor axis (short diameter)

INERTIA: The property whereby an object maintains its motion unless some force alters it; resistance to acceleration

MASS: The quantity of matter comprising a body

ORBITAL PERIOD: One revolution of a planet around the Sun or a satellite around a planet, usually measured in years (Earth's orbital period)

PERIHELION: The point in a planet, comet, or asteroid's orbit when it is closest to the Sun

VELOCITY: An object's speed and direction of motion

Bibliography

Christianson, Gale E. This Wild Abyss. New York: Free Press, 1978. Part biography and part history, this book chronicles astronomy from the earliest conceptions of the heavens to Isaac Newton's theory of gravitation. Christianson devotes a chapter to Kepler. A lucid work for general readers.

Gingerich, Owen. The Eye of Heaven: Ptolemy, Copernicus, Kepler. New York: American Institute of Physics, 1993. An astronomer and historian of science, Gingerich offers a thorough review of the change from Ptolemy's geometric universe through Copernicus' heliocentric model to Kepler's physical astronomy. Requires a high-school level understanding of algebra and physics.

Goodstein, David L., and Judith R. Goodstein. Feynman's Lost Lecture: The Motion of Planets Around the Sun. New York: W. W. Norton, 1996. The Goodsteins provide a delightfully written history of the Copernican revolution and then re-create one of Nobel laureate Richard Feynman's<$IFeynman, Richard> legendary lectures, which offers an original proof of Kepler's second law and requires only high-school geometry to understand.

Kepler, Johannes. New Astronomy. Translated by William H. Donahue. New York: Cambridge University Press, 1992. A translation of Astronomia nova, in which Kepler introduces the first two of his laws of planetary motion and systematically argues how to overcome the failings of geometric astronomy with his new physics-based principles.

Kozhamthadam, Job. The Discovery of Kepler's Laws: The Interaction of Science, Philosophy, and Religion. Notre Dame, Ind.: University of Notre Dame Press, 1994. A painstaking study of Kepler's development of the first two laws of planetary motion, emphasizing his role as a transitional thinker from the theologically centered views of the Middle Ages to modern science.

Motz, Lloyd, and Jefferson Hane Weaver. The Story of Astronomy. New York: Plenum Press, 1995. Two distinguished science historians present a comprehensive review of astronomy from ancient times through the twentieth century, well suited for readers who want detailed information on the genesis and effect of Kepler's laws.

North, John. Astronomy and Cosmology. New York: W. W. Norton, 1995. A brilliantly written history of astronomy that presents dominant concepts and theories nontechnically but thoroughly. In chapter 12, North explains the context and substance of Kepler's ideas and their influence on later natural philosophers.

Peterson, Ivars. Newton's Clock: Chaos in the Solar System. New York: W. H. Freeman, 1993. Peterson explains the development of celestial mechanics, modern computational techniques, and problems involved in understanding the long-term stability of the solar system. Many helpful photographs and diagrams.

Stephenson, Bruce. Kepler's Physical Astronomy. Princeton, N.J.: Princeton University Press, 1994. Stephenson closely studies the physical principles, often mistaken, upon which Kepler based his laws of planetary motion. Capably written, the book explains the genesis, meaning, and purpose of the laws in detail. Stephenson works out the mathematics for readers and provides graphs to clarify Kepler's ideas.