The Wanderers (Planetary Motion)

The Inner (Minor) and Outer (Major) Planets

Among the thousands of visible stars there are five objects which pursue very strange courses among the stars. The brightest of these is the planet Venus, which is far brighter than Sirius, and which appears alternately as an evening and a morning star, never setting more than six hours after the Sun, or rising more than six hours before the Sun. It's "greatest elongation," the largest angle it ever makes with the Sun is 47 degrees. In general the planet which is most difficult to find is Mercury, for its greatest elongation varies between 18 degrees and 28 degrees, so you must view it in twilight or at dawn, depending upon whether it is an evening or morning star.

The planets Mars, Jupiter and Saturn pursue increasingly leisurely courses about the sky. All five planets are never found very far from the ecliptic, i.e., the path of the Sun across the celestial sphere. While Mars, Jupiter and Saturn can be as much as 180 degrees from the Sun, i.e., in opposition, their motion across the sky is quite complicated. From time to time they stop moving eastward across the sky, and for a spell they move westward in retrograde motion. Later the eastward motion resumes. This is illustrated by the apparent paths of the planets Uranus and Neptune, which cannot be seen without optical aid.

Periodically each planet lies (approximately) on a line through the earth and the sun. In the case of Venus and Mars, this occurs with the planet between the Sun and the Earth, but in the case of Mars, Jupiter and Saturn, it occurs with the earth between the Sun and the planet. In any event, the time between such events is easily measured. For example, in the case of Mars opposition reoccurs every 2 years 50 days (2.137 years). Hence the difference of angular speeds of Earth and Mars in their orbits is 0.468 revolutions per year. Since the Earth's angular speed is 1 revolution per year, it follows that the angular speed of Mars is 0.532 revolutions per year. Thus, it requires 1.88 years for one revolution of Mars in its orbit.

Astronomical Knowledge of the Ancients

It was originally Eudoxus (409-353 B.C.) who suggested that the motion of the planets was similar to that which would be pursued by an object constrained to move in a circle about a center which itself moves in a circle. The theory of such epicycles reached its culmination in the hands of Ptolemy (2nd cent. A.D.), who developed a geocentric model involving intricate epicyclic motions. In this way Ptolemy was able to predict many of the features of planetary motions. Without doubt, however, Ptolemy would have been astounded to learn that no improvement over his model would be made for nearly 1400 years.

The Dark Ages

The rise of Christianity very nearly spelled disaster for physical science, for the ideas of classical astronomy were vigorously suppressed whenever the naive views expressed in Genesis conflicted with the inferences of direct observation. The spirit of scientific inquiry, still young and fragile, was so stifled that the earth once again became flat, and remained so for over a thousand years. One of the most spectacular events in recent astronomical history, the birth of the Crab Nebula, was recorded in 1054 by people all over the world, except in Western Europe, where the authorities must have been busy planning the First Crusade. Nevertheless, this astronomical event was supposed to have been so bright that it was visible for several weeks during broad daylight.

Fortunately, when it became too dangerous to think about Science in the west, other people, specifically the Hindus and the Arabs, preserved classical Greek knowledge. Arab astronomers, in particular, gave to the stars many of the names we use today, and they continued to compare the Ptolemaic model with their own observations. By the close of the fifteenth century most of the Greek ideas concerning astronomy had made their way back in to the consciousness of western Man, although it was still very dangerous to espouse these views publicly.

The Renaissance (Rebirth)

The introduction of the compass from China about 1300 as a result of the voyage of Marco Polo led to a new era in navigation. The compass was indispensable to Columbus at the end of the 15th century, when he sailed westward in quest of an alternative route to the orient.

Practical engineering began to flourish. Especially important for the science of the Renaissance was the existence of accurate tables of trigonometric functions, compiled because of their utilitarian value. Here was an early example of the beneficial interplay between Science and technology.

In 1500 the Polish astronomer Copernicus went to Italy where among other things he studied the Greek classics, and resurrected the heliocentric model of Aristarchus. This model appeared to offer a natural explanation for Ptolemy's observation that the epicycles of Mercury and Venus are centered very nearly on the Sun, as well as his observation that the epicycles of Mars, Jupiter and Saturn seem to have the same period as the Sun.

Copernicus went far beyond Aristarchus in developing a heliocentric model. He appreciated that it would not suffice to place the planets in circular orbits about the Sun. He, therefore, went into retreat in Germany, where he developed an elaborate epicyclic picture of the planetary orbits.

Determining the Relative Sizes of the Planetary Orbits

In addition, Copernicus determined the approximate radii of the orbits of the six known planets in terms of the earth's orbital radius. It is easy to see how this may be done for Mercury and Venus. Remember that the observed greatest elongation of Venus is 47 degrees. That is the greatest angle from the Sun's direction we ever see Venus. At this time Venus through a telescope appears like the moon at quadrature, i.e., first or last quarter. It can be seen from this diagram that the sine of this angle 47 degrees is equal to the radius of Venus's orbit expressed in "astronomical units." The astronomical unit of distance is the radius of the earth's almost circular orbit. Thus, we find that the radius of Venus's orbit is about 0.7 A.U., while a similar calculation for Mercury yields about 0.4 A.U. I shall leave it as a suggested exercise to figure out how Copernicus may have deduced the radii of the orbits of the outer planets, Mars, Jupiter and Saturn.

The actual size of the earth's orbit was not known to Copernicus, although from the work of the Greeks it was apparent that 1 A.U., the distance from the earth to the Sun, must be at least 5 million miles. An accurate determination was not made until the end of the 18th century. How would you suggest measuring the distance to the Sun, given the technological status of the world of 1500?

Tycho Brahe

While the model of Copernicus, refined with numerous epicycles, was fairly accurate, the development of more accurate observations revealed apparent discrepancies. Most noteworthy are the painstaking efforts of the Danish astronomer Tycho Brahe (1546-1601). (H p. 106) In several elaborate observatories, Tycho had devices such as the great equatorial armillary, with which he was able accurately to plot the positions of stars and planets upon the celestial sphere.

Tycho's unbearable temperament eventually caused much trouble in Denmark, so, fearing that his equipment might be confiscated, he fled to Bohemia. In Prague he spent the last year of his life with Johannes Kepler (1571-1630), who had fled religious persecution in Graz, Austria.

Kepler

Kepler suffered Tycho's insults, and from his teacher acquired knowlege which would prove indispensable for the emergence of his own genius. It might be pointed out that Tycho never accepted the Copernican idea that the Sun was fixed with the earth moving about it. He had his own model, and he rather expected Kepler to adopt the Tychonic model too.

Kepler was always coming up with weird ideas, which he would pursue with avid dedication. The thing which saved him from folly was his insistence upon absolute accuracy. In this quest for accuracy Kepler took upon himself the formidable problem of determining more accurately than ever before the orbits of the planets, beginning with the planet earth.

Kepler's analysis of the earth's orbit proceeded from a simple supposition, which it turns out was very nearly true. He supposed that after the elapse of one Martian year (1.88 earth years) Mars returns to the same point in space; i.e., the vector SM is of the same length and direction every 1.88 years. Now, if one looks at Mars and the Sun at intervals of 1.88 years, the earth will be in a different position each time. Thus, over a period of many years the earth's orbit will be plotted out in our diagram. The angle SEM may be measured directly, and it is unnecessary to have someone stationed on Mars to determine the angle SME, for the angle SME may be inferred from the apparent position of Mars against the background of the stars. Knowing these two angles the position of the earth may be calculated using simple trigonometry. In this way the shape of the earth's orbit can be determined, given observations over a sufficiently long time span.

Fortunately, Kepler inherited Tycho's data, accumulated over a lifetime. What Kepler concluded is that the earth moves in an ellipse with the Sun at one focus and that SE swings over equal areas during equal periods of time. These comprise the first two of Kepler's laws of planetary motion. The epicycles of the Copernican model were seen for the first time to be an approximate way of describing elliptical motion.

Kepler proceeded then to the study of the orbit of Mars. This he did by repeating the calculations which determine the earth's orbit, using another sequence of times 1.88 years apart, but with Mars at a different point M in space. The point was that the deduced orbit of the earth had to agree with that of the earlier calculation, and hence two points on Mars's orbit could be determined. By repeating such calculations many times Kepler was eventually able to establish that Mars also moves in accordance with his first two laws of planetary motion. Furthermore, he observed that the square of the period of orbital motion was proportional to the cube of the semi-major axis of the ellipse. Subsequent laborious studies of the other planets revealed that they also move in accordance with these three laws. In fact, not only do the ellipses play an important role, but the other conic sections also describe orbits, those of bodies that have sufficient energy to escape the gravitational field of the massive central body.

In the relation between the observer Tycho and the theoretician Kepler we see the necessity for cross-fertilization of ideas and skills. While the ideas which are born of Science are often simple, the acquisition of these ideas is usually far beyond the patience and skill of most people.