Kepler's Laws of Planetary Motion
Kepler solved the problem over which so many astronomers had labored for centuries. The orbits of the planets are ellipses, not circles, and their variable motion is due to their varying distances from the sun. These are his first two laws, briefly stated. The third contains the relationship between the planets' orbital periods and their distance from the sun.
The First Law: Elliptical Orbits
Kepler's first law of planetary motion can be stated as follows:
KEPLER'S FIRST LAW (LAW OF ELLIPTIC ORBITS): Each planet moves in an elliptic orbit around the sun, with the sun occupying one of the two foci of the ellipse.
The ellipse, a "family" of mathematical curves, is important in discussing the orbits of any bodies about each other, not just planets. Roughly peaking, an ellipse is a circle with the opposite ends of a diameter pulled outward, thus distorted into an oval figure. The long axis of the ellipse is called the major axis, and perpendicular to it through the center of the figure is the minor axis. There are two points on the major axis, called the foci (the singular form is focus), about which the figure is roughly symmetrical. The sum of the distances from each of the foci to every point of the ellipse is a constant. This immediately suggests a means of drawing an ellipse: Loop a piece of string around two tacks (the foci), and wield a pencil. In a planet's orbit the sun occupies one focus; the other one is empty.
The farther the foci are from each other, the more elongated the ellipse; the closer together they are, the more nearly circular the ellipse. Thus the ratio of the distance of a focus from the center to the length of the semimajor axis (half the major axis), known as the eccentricity, determines the shape of the elliptic orbit. When the ratio is zero, the foci and center coincide, and the elliptic orbit degenerates into a circular orbit. The more elongated the elliptical orbit is, the nearer the eccentricity is to 1. They vary from near zero for Venus to around 0.2 for Mercury and Pluto. Thus the planetary orbits are not very elongated but very nearly circular, which is why it was not obvio us to Kepler or to his predecessors that the orbits of planets were not indeed perfect circles as Plato had envisioned. Finally the size of the elliptic orbit is the length of the major axis.
The Second Law: Speed in Orbits
If the planets were orbiting the earth in circular orbits, they would traverse equal angles on the celestial sphere in equal intervals of time anywhere in their orbits. The fact that they do not actually do this but traverse variable angles in equal intervals of time (depending on where they are located in their orbits) was well known to the Greeks. This irregular motion was what necessitated epicycles in Ptolemy's geocentric system and in Copernicus's heliocentric system. In the elliptical orbits determined by Kepler the planets' distance from the sun, which occupies one focus, varposition in the orbit. In addition the speeds in its vary from one position to another such that planets move fastest when closest to the sun and slowen farthest away. This is a consequence of Kepler's second law of planetary motion, which may be stated as follows:
Kepler's Second Law (LAW OF AREAS): The imaginary line connecting any planet to the sun sweeps over aJeas of the ellipse in equal intervals of time.
The mean distance of a planet from the sun is the ge of the distance between the point of closest approach, called perihelion, which is located at one end of the major axis, and the most distant point of the orbit, aphelion, which is
located at the other end of the major axis. The average is one-half the length of the major axis, or the semimajor axis, as shown in the figure. The alternate gray and colored sectors in the figure are of equal area. Therefore, according to Kepler"s second law, the planet passes th rough the numbered positions in equal intervals of time.
The Third Law: Yardstick for the Solar System
Kepler's third law is extremely important in the sense that it provides a means of determining the relative size of the solar system in units of the mean earth-sun distance, the astronomical unit (AU). The law may be stated as follows:
KEPLER'S THIRD LAW (HARMONIC LAW): The square of any planet's period of orbital revolution is proportional to the cube of its mean distance from the sun.
The period of orbital revolution is theplanet's sidereal period; that is, the time to move through 3600 in its orbit. The sidereal period cannot be measured directly since there is no marker along the orbit to tell us when the planet has come back to its starting point 3600 later. But astronomers can measure the synodic period directly, say, the time from opposition until the planet returns to opposition again. And from a simple mathematical relation the sidereal period can be computed from the synodic period. If we know the sidereal period, from Kepler's third law in units of years and astronomical units we can compute the mean distance (semimajor axis) of a planet's orbit.
Newton later modified Kepler's third law to show that the mass of the sun and the planet under consideration enter into the equation. This modification is all important; for it allows astronomers to determine masses not only for the planets of the solar system but also for many stars and even galaxies.
Universal Nature of Kepler's Laws
Kepler's laws are universal. They apply to any two bodies gravitationally bound to each other, whether in the solar system or elsewhere in the universe. Kepler's work put to rest any notion that the planets moved in perfectly circular orbits because nature had decreed that the heavenly bodies must show perfection in their movements. Of course, bodies may move in circular orbits when the two focal points coincide with the center, in which instance a circle is a special case of an ellipse.
Kepler did not know why the planets move by these empirical relationships, which he had established from Brahe's observations. He sought a cause of which his three laws were the effect. As he stated, "I am much occupied with the investigation of physical causes. My aim in this is to show that the celestial machine is to be likened not to a divine organism, but rather a clockwork .... " Kepler vaguely sensed that bodies have a natural "magnetic" affinity for each other and guessed that the sun has an attractive force. It remained for Newton, half a century later, to formulate a unified theory of motion, which includes planetary motion with gravity as its cause.
The substitution of a kinematic planetary model for a purely geometric one was Kepler's primary achievement. He prepared the way for the modern theory of force and the mathematical analysis of motion in terms of forces. A chief proponent of this new idea was the Italian physicist-astronomer Galileo Galilei. From his experiments with bodies in motion and the forces controlling them came the foundations of modern mechanics.
