Four thousand years ago, soothsayers, watching the skies for omens from which they might predict the future, observed that the stars circled the heavens every night from east to west. With the exception of a mere handful, however, the stars always seemed to remain in fixed positions with respect to one another. As an aid to their identification, these so-called fixed stars were later divided into groups now known as constellations (fig. 2.1). Within each of the constellations the stars formed an apparently unchanging pattern, and every constellation always retained the same position in the sky in relation to the others.
In addition to the fixed stars, the ancient observers noted that, apart from the Sun and the Moon, there were five celestial bodies which looked like stars but behaved in a different manner. In the first place, these apparent stars did not belong to any of the constellations. Although they traversed the sky from east to west each night like the fixed stars, they were found to move, in the course of time, from one constellation to another, generally from west to east.
At certain intervals, however, these wandering stars, which were called planets (from the Greek word planetes, meaning wanderer) , seemed to reverse their direction of motion through the constellations. Instead of traveling from west to east through the fixed stars, they would appear to move from east to west for a few weeks before resuming their normal motions. Another observed peculiarity of the planets was that their brightness varied quite considerably during the course of a year and from year to year. The brightness of the fixed stars, on the other hand, remained the same.
The seemingly erratic behavior of the planets aroused considerable interest. First, there were the astrologers, those early students of the stars who claimed the ability to foretell human events from the locations of the planets in relation to one another and to fixed stars. Then came the astronomers who attempted to develop a scientific explanation for the motions of the planets.
For many hundreds of years, in fact until
the middle of the 16th century, seven heavenly bodies were recognized as planets. They were the Sun, the Moon, Mars, Mercury, Jupiter, Venus, and Saturn (fig. 2.2). The Sun and the Moon were included among the planets because they appeared, like the others, to move from west to east among the constellations. Earth was not thought of as being a planet, as it is now, because it was supposed to be the stationary center of the universe.
A reminder of the seven planets of the ancients is to be found in the names of the days of the week: Sunday (Sun); Monday (Moon) ; Tuesday (French, mardi, Mars) ; Wednesday (mercredi, Mercury) ; Thursday (jeudi, Jove or Jupiter); Friday (vendredi, Venus); and Saturday (Saturn).
Of the planets, one in particular attracted special attention because of its reddish color, suggestive of blood and fire. To the Sumer-ians and other early civilizations, this planet became a symbol of the carnage and destruction of war, and the Chaldeans, about 3000 years ago, named it Nergal for the master of battles. Later, the Greeks referred to the red planet as Ares, for their god of war, and the Romans adopted the equivalent name Mars, who was identified with Ares. In the course of time, the red planet came to be called Mars (or its equivalent) in most languages. Nevertheless, the Greek Ares still persists in such words as areocentric and areography. It may be noted that the symbol for Mars, $ , represents a shield and spear, the ancient implements of war.
The earliest attempts to explain the apparent motions of the stars and the planets were made by the Greeks around 600 B.C. They thought that Earth was stationary, as indeed it seemed to be, and that all the heavenly bodies, including the Sun and the Moon, revolved about Earth once every day. This
FIGURE 2.2. The seven planets of the ancients. (From 17th Century engraving; courtesy Time, Inc.)
geocentric (or Earth-centered) picture of the universe seemed to agree with what the eyes saw when an observer looked at the sky, and there was no obvious reason for rejecting it. The Greek scholar Aristarchus of Samos, who lived in the third century B.C., is said to have suggested that Earth and the planets might be revolving about the Sun. But this view was not regarded with favor because of the strongly entrenched opinion among the Greeks that Earth was the fixed center of the universe.
A more detailed development of the geocentric concept was made in Greece during the second century B.C. by Hipparchus, the first systematic astronomer of whom there are any records. Hipparchus is said to have taken his ideas from an earlier philosopher, Apollonius of Perga (in Asia Minor), and they were adopted in turn by the Greco-Egyptian Claudius Ptolemaeus (Ptolemy), who lived in Alexandria in the early decades of the second century A.D. In his famous work that became known as the Almagest, an Arabic distortion of a word meaning "the greatest [book]," the Ptolemaic system was expounded. So convincing was Ptolemy, that his system of planetary motion remained virtually unchallenged for more than 1400 years.
According to the Ptolemaic system, Earth was stationary and around it revolved the seven planets in the following order: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn (fig. 2.3). The planets were supposed to move in a series of circles, known as deferents, with Earth at the center. In order to account for the reversal in their direction of travel through the constellations, which occurs at regular intervals, an additional motion was introduced. The planets, other than the Sun and Moon, were assumed to move on small circles, called epicycles; the centers of these epicycles were supposed to travel along the deferents. By combining the two types of motion, a qualitative interpretation could be provided for the periodic reverse (or retrograde) motion of the planets. The Sun and the Moon do not exhibit the retrograde behavior of the other planets and so they did not need epicycles.
The Greek astronomers were aware that, as seen from Earth, the planets Mercury and Venus always appeared to be close, in angular distance, to the Sun, whereas the angular dis
tances of the other planets—Mars, Jupiter, and Saturn—varied considerably. Consequently, the Ptolemaic system required that the centers of the epicycles of Mercury and Venus should always be on the line joining Earth and the Sun, as shown in figure 2.3.
Although the combinations of epicycles and deferents were able to provide a general picture of the apparent motions of the planets, it was not adequate to account for their actual positions over the years. Consequently, smaller, secondary epicycles were introduced with their centers moving on the primary epicycles. Eventually, more than 50 epicycles were required to represent the paths of the five starlike planets in the sky.
In 1543, after more than 30 years of preparation, the Polish astronomer Nicolaus Copernicus published his great work "On the Revolutions of the Celestial Bodies," in which he revived the heliocentric (Sun-centered) concept of Aristarchus. In the Copernican system, the Sun v/as stationary and the planets, including Earth but not the Moon, revolved around it (fig. 2.4). Instead of assuming that the heavenly bodies circle about Earth, Copernicus showed that the nightly motion of the stars from east to west could be explained equally well by postulating that Earth rotated about its north-south axis once every day in a counterclockwise direction; that is, from west to east. The main orbit (deferent) of each planet was still taken to be circular in the Copernican system, and epicycles were necessary to account for the apparent motion of the planets.
The ideas of Copernicus appeared to offer no striking advantage or simplification over the geocentric system of Ptolemy. Furthermore, in the 16th century, belief in Earth as the center of the universe had become a religious dogma that few dared to oppose. Consequently, the Copernican system received little support. One of the arguments against it was that, if Earth did indeed move, then the directions of the nearer fixed stars, at least, should appear to change in the course of time (fig. 2.5). But no such changes could be detected. Actually, the argument was sound
but the observations were incorrect because of the limitations of the available instruments. It is now known that the apparent directions of the nearer fixed stars do in fact change against the background of more distant stars.
Tycho Brake and Johannes Kepler
One of those who was interested in the work of Copernicus but could not accept his
Motion of Earth r
Apparent change in Fixed star direction
FIGURE 2.5. Change in direction of fixed star due to Earth's motion.
theory, largely because the directions of the stars seemed to remain unchanged, was the Danish astronomer Tycho Brahe. He died in 1601, before the telescope was invented, but during his lifetime Brahe developed several astronomical instruments. With these he was able to make more accurate observations than had been previously possible of the locations of the stars and planets in the sky. In particular, he made detailed records of the positions of the planet Mars during its nine apparitions from 1580 to 1600. These data, together with those obtained in the course of the next two opportunities, were destined to play an important role in the understanding of planetary motion.
Two years before his death, Tycho Brahe had moved to a location near Prague at the invitation of the German emperor, Rudolph II. There he was joined a year or so later, in 1600, by the young German astronomer Johannes Kepler who became Brahe's assistant and subsequently his successor as the imperial astronomer. In Kepler's first important work on cosmography, published in 1596, he stated his objections to the Ptolemaic system and gave reasons for favoring that of Copernicus in spite of its apparent defects. In this respect he was at variance with the man he was to serve a few years later. Tycho Brahe, in his Tychonic system, suggested that the five starlike planets revolved about the Sun, as in the Copernican theory, but he thought that the Sun and Moon moved around Earth, as in Ptolemy's system.
It is reported that, on his deathbed, Tycho Brahe asked Kepler to utilize the extensive observations that he, Brahe, had made on Mars to provide support for the Tychonic system. As an objective scientist, however, Kepler proceeded to use the data to test the three existing theories of planetary motion, namely those of Ptolemy, Copernicus, and Brahe. But none of these systems was able to account in a satisfactory manner for the recorded observations. After several years of tedious calculations on the motion of Mars, often made to six significant figures and without the benefit of logarithms, Kepler came to a revolutionary conclusion; this conclusion was to lead to the replacement of the earlier systems, with their circular deferents and epicycles, by one that was both much simpler and at the same time more accurate.
Kepler's Elliptical Planetary Orbits
For nearly 2000 years, the thinking of astronomers had been dominated by the Greek philosophical concept that the heavenly bodies could move only in perfect circles or in com binations of such circles. In the words of Aristotle, "the shape of the heavens is of necessity spherical . . . the circle is primary among figures, and the sphere occupies the same position among solids." Kepler, however, concluded from his study of Brahe's observations on Mars that the orbits of the planets are not circles but ellipses. By making this simple change, the complicated Ptolemaic system of deferents and epicycles was shown to be unnecessary and it was soon abandoned.
In his effort to understand the motion of Mars, Kepler started with two assumptions. His first was that, as in the Copernican system, Earth moves in a known circular orbit about the Sun. In such an orbit the distance from Earth to the Sun would always be the same. Furthermore, the time required for Earth to complete an orbit is exactly a year; that is, 365 days. Hence, if its rate of motion is uniform, as it should be in a circular orbit, the position of Earth in its orbit on any date can be determined.
The second assumption made by Kepler was that Mars also revolves about the Sun but in an unknown orbit. By noting the position of Mars relative to the Sun and to the fixed stars, it had been calculated that Mars requires 687 days to make a complete orbit around the Sun (p. 37). Consequently, for a certain location of Mars in the sky on any date, it was known that 687 days later it would return to exactly the same position. Because Earth has an orbital period of 365 days, it will be in different locations on these two dates, but the positions in its orbit will be known. From the quadrangle formed by the Sun, the two known positions of Earth in its orbit, and the particular location of Mars, as shown in figure 2.6, it is possible to calculate the distance of this planet from the Sun in terms of the constant distance of Earth from the Sun.
In this way, by utilizing Brahe's accumulated data and his own obtained in 1602 and
FIGURE 2.7. Determination of the orbit of Mars.
1604, Kepler was able to calculate the distance of Mars from the Sun for any specified position of the planet in the sky. Repetition of the same calculation for many different positions permitted Kepler to determine the distance of Mars from the Sun at each location and thus to define the orbit of the planet (fig. 2.7). Because the distances were found to vary with the position of Mars in its orbit, it was clear that the orbit could not be a circle with the Sun at its center. It might, however, have been a circle with the Sun displaced from the center. After much trial and error, Kepler was eventually forced to conclude that the orbit of Mars, and presumably of the other planets, is an ellipse.
Kepler's Laws of Planetary Motion
A circle is characterized by having a central point (or center) such that the distance from it to any point on the circle, that is, the radius, is a constant. An ellipse, on the other hand, has two points within it, called foci (plural of focus), so that the sum of the distances, known as the radius vectors, from them to any point on the ellipse is always the same. Thus, in figure 2.8, F± and F2 are the two foci of the ellipse, and the sum of the distances PF1 and PF2 is the same no matter
where the point P lies on the ellipse. According to Kepler's first law of planetary motion, published in 1609, the orbit of each planet is an ellipse with the Sun at one focus. Because the Sun is presumed to be stationary, all the planetary orbits must have one common focus, the location of the Sun.
Kepler's second law, enunciated at the same time as the first, was also based on a study of the orbit of Mars. This law, which has an important bearing on the length of the Martian seasons, as will be seen in chapter
III, states that a straight line joining the Sun and a planet in its orbit, that is, a radius vector of the ellipse, sweeps out equal areas in equal intervals of time. Thus, if the shaded portions of figure 2.9 have equal areas, it takes a planet the same time to cover the distances AB and CD. It is evident that when a planet is farther from the Sun, at AB, it covers a shorter distance in a given time, and hence travels more slowly, than when nearer to the Sun at CD. Kepler thought that the speed of a planet at any point in its orbit is inversely proportional to the length of the radius vector at that point. This is not true in general, however, but only at the two points when the planet is farthest and nearest, respectively, from the Sun.
The first two laws of Kepler did not, by themselves, account for the apparent occasional reversed or retrograde motion of the planets. But by taking into consideration the different orbital speeds of Earth and Mars, Kepler showed how the retrograde movement arises. A more detailed treatment of this subject is given in the next chapter. Another point that emerged was that the retrograde behavior should occur at regular intervals, once in each orbital period of a planet, as, in fact, it does.
In 1618, Kepler completed his work with the third law of planetary motion; namely, that the square of the orbital period of a planet is proportional to the cube of its mean distance from the Sun. Although Kepler's three laws, based on a Sun-centered system, gave a much more accurate representation of planetary motion than did the Earth-centered system of Ptolemy, they still lacked a unifying principle. This came in 1687 when Isaac Newton, the English mathematician, published his theory of universal gravitation. By postulating that the Sun attracted each planet by a force that was inversely proportional to the square of its distance, the three laws of Kepler followed as a natural consequence. Thus the already dying Ptolemaic system was finally laid to rest, and the heliocentric system of Copernicus gained general acceptance. Earth was no longer the center of the Universe but merely one of the members of the Sun's system of planets.
It should be no detraction from Kepler's brilliant and painstaking work to mention that he was fortunate in several respects. The orbits of the planets around the Sun do not differ very greatly from circles. It happens that Earth's orbit is very close to being circular, as Kepler postulated, but the orbit of Mars deviates appreciably from a circle. If the reverse had been true, the determination of the Martian orbit might well have proved beyond Kepler's capability. Furthermore, the availability of the observational data on Mars, and Brahe's recommendation that Kepler make use of them to study planetary motion, was another fortunate circumstance. The orbit of the planet Venus is even closer to a circle than is that of Earth. Had this planet been selected for study instead of Mars, the discovery of Kepler's first and second laws would undoubtedly have been delayed.
Another interesting point has been noted by the Yale University astronomer Dirk
Brouwer. "An ellipse," he writes, "is only a first approximation to a planetary orbit. If an attempt were made to represent modern . . . observations with the aid of elliptic orbits, the inadequacy of the representation would soon be apparent. The good fortune of Kepler was that . . . [Brahe's] observations were just accurate enough to reveal the elliptical character of the [Martian] orbit. If they had been too accurate, the perturbations would have shown up, and Kepler's effort might well have bogged down in endless experimentations that would have led nowhere." The perturbations (or deviations) mentioned are attributable mainly to the gravitational attractions of Earth and Jupiter on Mars. If these (and other planetary) attractions were absent, the Martian orbit would be very close to an ellipse. For many purposes, however, such as those to be considered in this book, it is adequate to assume that Mars has an essentially elliptical orbit.
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