1. Evolving perspectives - a historical prologue

Moving points of light

The ancient wanderers

Fig. .. 

There are seven objects that move with varying speeds against the background of fixed stars. Ranked in order of greatest apparent brightness, they are the Sun, Moon, Venus, Jupiter, Saturn, Mercury and Mars. Our ancestors called them planets, the Greek word for "wanderers"; a designation we still use for all but the Sun and Moon. For much of recorded history, astronomy has been mainly occupied with describing and predicting the movements of these ancient wanderers.

Fig. .. 

The Sun does not rise at precisely the same point on the horizon each day. Instead, the location of sunrise drifts back and forth along the horizon in an annual cycle that is tied to the seasons. Ancient astronomers used monuments to line up the limits of these excursions. The length of the Sunís arc across the sky also changes with a yearly rhythm. The Sun rises highest in the sky every summer, with its longest trajectory and the most daylight hours.

Circles and spheres

Fig. .. 

The Moon repeats its motion around the Earth on a monthly cycle, periodically changing its appearance. Once each month, the Moon comes nearly in line with the Sun, vanishing into the bright daylight. On the next night the Moon has moved away from this position, and a thin lunar crescent is seen. The crescent thickens on successive nights, reaching the rotund magnificence of full Moon in two weeks. Then, in another two weeks, the Moon disappears into the glaring Sun, completing the cycle of the month and providing another natural measure of time.

Even the earliest sky-watchers must have noticed that the wanderers are confined to a narrow track around the sky, known as the zodiac from the Greek word for "animal". The Sunís annual path, called the ecliptic, runs along the middle of this celestial highway. The paths of the other wanderers also lie within the zodiac. Its narrowness is a sign that the planets move almost like marbles on a table because the planes of their orbits are closely aligned with each other.

Fig. .. 

It was obvious to astronomers from the earliest times that the wanderers do not move at uniform speeds or follow simple paths across the sky. When Mars, Jupiter and Saturn shine brightly in the midnight sky, each planet will gradually come to a stop in its eastward motion, move backward toward the west, and then turn around again and resume moving toward the east. Although these planets travel eastward in the "prograde" direction most of the time, they sometimes appear to move in the westward "retrograde" direction before continuing on in their eastward course. Ancient and modern explanations of this looping, retrograde motion differ in their perspective on the locations and motions of the planets.

Fig. .. 

In arguments used by the Pythagoreans, and subsequently recorded by Aristotle (384-322 BC), it was shown that the Earth is a sphere. During a lunar eclipse, when the Moonís motion carries it through the Earthís shadow, observers at different locations invariably saw a curved shadow on the Moon. Only a spherical body can cast a round shadow in all orientations. The curved surface of the ocean was also inferred by watching a ship disappear over the horizon; first the hull and then the mast disappear from view.

In the Greek geocentric model, the central spherical Earth was supposed to be bounded by a much greater sphere, the imaginary celestial sphere of fixed stars. It wheeled around the central Earth once every day, with uniform circular motion and perfect regularity, night after night and year after year. Such a celestial sphere would also explain why people located at different places on Earth invariably saw just half of all the stellar heavens.

Fig. .. 

Combinations of uniform circular motion were additionally required to account for the looping, or retrograde, paths of the planets. Each planet was supposed to move with constant speed on a small circle, or epicycle, while the center of the epicycle rotated uniformly on a large circle, or deferent. In this way Ptolemy, in his Mathematical Compilations, or Almagest, written about 145 AD, was able to predict the motions of every one of the seven wanderers, compounding them from circles upon circles.

In Ptolemyís model, the Earth was located to one side of the center of the deferent circle. An imaginary point, called the equant, was symmetrically positioned on the opposite side of the center, and each planet was supposed to move uniformly with respect to the equant point. By selecting suitable radii and speeds of motion, Ptolemy could use this system of uniform motion around two circles to reproduce the apparent motions of the planets with remarkable accuracy. He succeeded so well that his model was still being used to predict the locations of the planets in the sky more than a thousand years after his death.

The Earth moves

Fig. .. 

The Sun might not be moving across the bright blue sky each day, for the Earthís rotation could produce this motion. Every point on the surface of a spinning Earth can be carried across the line of sight to an unmoving Sun, from sunrise to sunset, producing night and day. Such a perspective involves a certain amount of detachment Ė the ability to separate yourself from the ground and use your mind's eye to look down on the spherical, rotating Earth, like a spinning ball suspended in space.

Fig. .. 

The Moonís nightly motion from horizon to horizon could also be neatly explained by the rotation of the Earth, and the Moonís monthly circuit against the background stars could be ascribed to its slower orbital motion around the Earth. This would also account for the Moonís varying appearance. The Moon borrows its light from the Sun, and the Sun illuminates first one part of the Moonís face and then another as the Moon orbits the Earth. On any given night, all observers on Earth will see the same phase of the Moon as our planet's rotation brings it into view.

The globe on which we live might not only spin on its axis; it could also be whirling endlessly around the Sun, completing one circuit each year. This notion was presented by the Polish cleric and astronomer Mikolaj Kopernik (1473-1543), better known as Nicolaus Copernicus. Since the orbits of Venus and Mars lie inside that of Earth and closer to the Sun, these planets are only seen around dawn or dusk. In contrast, the orbits of Mars, Jupiter and Saturn lie outside that of the Earth, so they are visible throughout the night.

Fig. .. 

The Sun-centered view also provides a simple explanation of the occasional backward, or retrograde, motions that were so hard to reproduce using an Earth-centered, or geocentric, model. Most of the time we see Mars, Jupiter and Saturn moving around the Sun in the same direction as the Earth, but during the relatively short time that the Earth overtakes one of these planets, that planet appears to be moving backward. Moreover, one could confidently predict when a planetís apparent motion would come to a halt and turn around, and for how long it would seem to move backwards.

Fig. .. 

We now realize that the seasons can be explained in the heliocentric model by the tilt of the Earthís rotational axis and the annual orbit of the Earth about the Sun. As the Earth orbits the Sun, its rotational axis points toward the same direction in the sky, at the star Polaris, but the northern and southern hemispheres are tilted toward or away from the Sun by up to 23.5 degrees. The greatest sunward tilt in a given hemisphere occurs in summer when the Sun is more nearly overhead and its rays strike the surface more directly. Winter occurs when that hemisphere is at its greatest tilt away from the Sun.

The harmony of the world

In the hope of developing a more precise description of planetary motions, the Danish astronomer Tycho Brahe (1546-1601) amassed a great number of observations that were more accurate and complete than any previous ones. This was before the days of telescopes, and he used ingenious measuring instruments that resembled large gun sights with graduated circles. These data were eventually interpreted by Tychoís assistant and successor, Johannes Kepler (1571-1630), who was able to determine precise mathematical laws from them.

Fig. .. 

Since circular motions could not describe Tychoís accurate observations of the planets, Kepler concluded that non-circular shapes were required. In 1605, after four years of computations, Kepler found that the observed planetary orbits could be described by ellipses with the Sun at one focus. This ultimately became known as Keplerís first law of planetary motion.

Fig. .. 

A planet speeds up when it approaches the Sun, and slows down when it moves away from the Sun, and that accounts for the varying planetary speeds observed from Earth. Kepler was able to state the relationship in a precise mathematical form that can be explained with the help of the Figure. Imagine a line drawn from the Sun to a planet. As the planet swings about its elliptical path, the line (which will increase and decrease in length) sweeps out a surface at a constant rate. This is also known as the "law of equal areas". During the three equal time intervals shown, the planet moves through different arcs because its orbital speed changes, but the areas swept out are equal.

Fig. .. 

Kepler investigated arithmetic patterns between the periods and sizes of the planetary orbits, discovering the harmonic relation that is now known as Keplerís third law. It states that the squares of the planetary periods are in proportion to the cubes of their average distances from the Sun. If Pp denotes the orbital period of a planet measured in Earth years, and ap describes its semi-major axis measured in AU, then Keplerís third law states that Pp3 = ap2, where the subscript "p" denotes the planet under consideration. This expression is illustrated in Figure 1.13, for the major planets and for the brighter moons of Jupiter. The mean orbital velocity of each planet is proportional to the ratio ap/Pp, so the velocity varies inversely with the square root of the distance or as (ap)-1/2.

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Copyright 2010, Professor Kenneth R. Lang, Tufts University