Theories of the World from Antiquity to the Copernican Revolution: Second Revised Editionby Michael J. Crowe
Newly revised edition of Professor Crowe's accessible, enlightening book re-creates the change from an earth-centered to a sun-centered conception of the solar system. The work is organized around a hypothetical debate: Given the evidence available in 1615, which system (Ptolemaic, Copernican, Tychonic, etc.) was most deserving of support?See more details below
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Newly revised edition of Professor Crowe's accessible, enlightening book re-creates the change from an earth-centered to a sun-centered conception of the solar system. The work is organized around a hypothetical debate: Given the evidence available in 1615, which system (Ptolemaic, Copernican, Tychonic, etc.) was most deserving of support?
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Theories of the World from Antiquity to the Copernican Revolution
By Michael J. Crowe
Dover Publications, Inc.Copyright © 2001 Michael J. Crowe
All rights reserved.
The Celestial Motions
The Motion of the Stars
This chapter doubly challenges the imagination. First, it draws on the geometrical or spatial imagination needed to conceive of the motions of the stars, sun, moon, and planets. Second, this chapter, nearly all of which could have been written by an ancient Greek astronomer, invokes the historical imagination by presenting these motions in the way that the Greeks envisioned them, that is, from a geocentric (earth-centered) perspective, which is, of course, how we see them. This approach will not only assist in understanding some of the ancient astronomies, but also facilitate a comprehension of these motions as conceptualized from the modern heliocentric (sun-centered) point of view.
Persons watching the stars over a number of nights see that nearly all of them appear to move in a counterclockwise direction along circles varying in size. The sole stationary stars are Polaris and the southern polar stars. The motions of the stars are identical to what they would be if they were all located on a huge sphere, the starry vault, rotating once approximately every twenty-four hours, and having as its center the earth, which is assumed to be motionless. The sense of rotation of the starry vault is such that a star on its right side is moving out of the page. Typically, a given star will appear to rise on the eastern horizon and set on the western horizon.
How do persons living on the earth's equator see the stars move? Polaris and the southern polar stars appear fixed in position. The remaining stars rise perpendicularly to the eastern horizon and set perpendicularly to the western horizon. These motions are represented in the accompanying diagram in which the ellipse represents the horizon plane.
At the north pole, Polaris is seen fixed in position at the zenith (the point in the heavens directly above a person's location). The other stars appear to move in circles parallel to the horizon plane and centered on Polaris.
Persons living in Chicago or Boston are located at 42° north terrestrial latitude, i.e., 42° up from the equator. As the next diagram indicates, for such persons, Polaris appears fixed in position, whereas the stars near it move in circles that are always visible. Stars farther down the starry vault move in circles that are cut by the horizon plane.
Problem 1: Polaris is the last star in the handle of the little dipper. Draw the motion of the stars in the Little Dipper as seen from 42° north latitude over a period of three hours.
Problem 2: Suppose that there exist only two bodies in the universe: one is identical to the starry vault and the other is a very small spherical planet located at the center of the starry vault. Let us assume that the starry vault rotates once every 24 hours, whereas the planet remains fixed in position, i.e., it neither rotates nor revolves. What motion would an inhabitant of the starry vault, convinced that the starry vault is at rest, attribute to the planet at the center? Represent this by means of a diagram. Are there any conclusive arguments that an inhabitant of the planet could formulate to prove that the starry vault is rotating? Or are there arguments that the inhabitant of the starry vault could use to show that the planet is rotating?
The Motion of the Sun
First, some definitions:
Celestial Equator: The celestial equator is the line on the starry vault that lies directly above the earth's equator. Some point on the celestial equator is always at the zenith of a person living on the earth's equator.
Ecliptic: The ecliptic is a line on the starry vault on which the sun always appears to be located. It is the apparent yearly path of the sun or the projection of the path of the sun on the starry vault during one year. The ecliptic is inclined at 23 1/2 degrees to the celestial equator. The sun completes one circuit of the ecliptic every 365.24220 days. From this it is evident that the sun moves approximately one degree each day on the ecliptic. Note that whereas the stars move from east to west, the motion of the sun on the ecliptic is from west to east. Let us now combine the motion of the starry vault with that of the sun. It is important to remember that the ecliptic is simply a line among the stars; it is like a seam on a basketball. If a basketball is rotated, its seam rotates with it. Correspondingly, the ecliptic rotates with the starry vault. Consequently, each day the sun makes one revolution around the earth along with the starry vault; however, the sun also moves about one degree per day along the ecliptic, moving in the opposite direction. As an aid to visualizing this, imagine an ant walking slowly down the side of a rapidly rotating basketball. The path of the ant from the point of view of the basketball is a straight line, but if seen from a fixed observer at a distance, the ant will appear to be moving along a helix. The next diagram shows the motion of the sun for one day.
Additional definitions are now needed.
Vernal Equinox: The point on the starry vault where the ecliptic crosses the celestial equator with the sun moving toward the northern half of the heavens. The sun is at the vernal equinox around March 21. When the sun is at an equinoctial point (vernal or autumnal equinox), people on earth in most cases experience days and nights of equal length.
Summer Solstice: The most northerly point on the ecliptic. The sun is at the summer solstice around June 22.
Autumnal Equinox: The point on the starry vault where the ecliptic crosses the celestial equator with the sun moving toward the southerly half of the heavens. The sun is at the autumnal equinox around September 23.
Winter Solstice: The most southerly point on the ecliptic. The sun is at the winter solstice around December 22.
These definitions can be presented diagrammatically.
The discussion of the sun's motion presented up to this point can be used to explain the seasons. This will also provide practice in applying these ideas. Many inhabitants of the northern hemisphere believe that summer is hot because the sun is closer at that time than in winter. In fact, the sun is farther from the earth in summer than in winter. The chief reason why summer is warm and winter cold is the difference between the angles at which the sun's rays reach us during those seasons. Let us imagine the sun on the celestial equator and determine the angle at which the sun's rays strike such cities as Chicago or Boston, which are located at 42° N. terrestrial latitude. As the diagram shows, when the sun is at an equinoctial point, it is directly above the equator. Hence its rays strike a point at 42° N. latitude at an angle of 42°. Thus at the vernal and autumnal equinoxes, solar rays arrive at Chicago at an angle of 42° off the vertical.
When the sun is at the summer solstice, it is 23 1/2° up from the equator, which causes its rays to strike Chicago at 42° - 23 1/2° = 18 1/2° from the vertical. Consequently, in summer the sun's rays come closer to the perpendicular than in fall or spring. Many people in the northern United States believe they have seen the sun directly overhead; in fact, the sun never gets closer to being directly overhead than at the time of the summer solstice and even then it is 18 1/2° from the zenith at 42° N. latitude. The above diagram shows the relative positions of the sun and earth at the time of the summer solstice. Note the difference between this and the previous diagram.
When the sun is at the winter solstice, it is located 23 1/2° below the equator. Consequently, its rays strike Chicago at 42° + 23 1/2° = 65 1/2° from the vertical. This is to say that when the sun is at the winter solstice, it is at most 24 1/2° (90° - 65 1/2°) above the horizon.
From this one sees that the greatest heating of a region occurs when the sun's rays strike it most directly, whereas when the sun's rays hit it more obliquely less heating occurs. This is the chief factor influencing our seasons.
The following problems illustrate the materials presented up to this point.
Problem 3: Determine the length of the night that began on September 23, 1845, in Clausville, which is located at 134° east terrestrial longitude and 90° north terrestrial latitude.
Problem 4: Given below is a plot of temperature versus month for the northern continental United States.
Draw a comparable plot of temperature versus month for (a) persons living in Sydney, Australia (about 35° south latitude), and for (b) persons residing in Quito, Ecuador, which is located on the equator.
Problem 5: The tropical or seasonal year, i.e., the average period between the beginning of spring in one year and that in the next, is 365.242200 days. This is the basic factor that must be kept in mind in devising a calendar. For each of the calendars proposed below, calculate the amount of error that will be introduced by that calendar in a period of two thousand years.
Calendar I: The year is 365 days long.
Calendar II: The year is 365 days long, unless the year number is divisible by 4, in which case the year is 366 days long.
Calendar III: The year is 365 days long, unless the year number is divisible by 4, in which case the year is 366 days long. If, however, the year number is divisible by 100, but not by 400, the year remains 365 days long.
Problem 6: Suppose only two bodies, A and B, exist in the universe. Suppose that the inhabitants of B see A move around B in a circle in one year. Let us suppose that inhabitants of A believe that A is motionless. Specify the period and the shape of the orbit that they will attribute to B. Let us assume that the body A always retains the same orientation; that is, were there actually a letter "A" on top of it, that letter would continue to remain right side up. Put another way, assume that body A does not rotate. Are there any arguments that inhabitants of B could use to prove to inhabitants of A that it is their body that is actually revolving?
The motions of the stars and sun are more complicated than presented so far. Another important aspect of the heavens, the precession of the equinoxes, was known to the Greeks who found that the equinoctial points, the points where the ecliptic crosses the celestial equator, change slightly; specifically, each equinoctial point makes a full circuit around the ecliptic in 26,000 years (the Greeks thought 36,000 years). To visualize this, imagine two lines running through the earth, one perpendicular to the plane of the celestial equator, the other perpendicular to the plane of the ecliptic. These lines extend respectively to the pole of the celestial equator (currently Polaris) and to the pole of the ecliptic. As the diagram shows, the polar line for the celestial equator turns around the ecliptic's polar line in 26,000 years. Thus the celestial equator slowly changes position over this period.
A major result of precession is that Polaris is ceasing to be our north polar star. In 13,000 years, a star 47° (twice 23 1/2°) from Polaris will be our north star.
This phenomenon can also be visualized by imagining that the earth rotates around an axis that turns through a circle of radius 23 1/2° in a period of 26,000 years. The detection of the precession of the equinoxes was one of the most important observational results achieved by ancient astronomers.
The Motions of the Moon
The analysis and prediction of the moon's motions presented a major challenge to ancient civilizations. One complexity in this regard derives from the fact that the nearness of the moon makes it possible to observe its motions with far greater precision than for most other celestial bodies. This challenge was also felt very strongly because of the importance of the moon's motions. An understanding of those motions was crucial not only for knowing when the moon would provide nocturnal illumination, but also for predicting eclipses, which were viewed as spectacular and very significant phenomena, and for constructing calendars, which in ancient times were frequently lunar. In what follows, the main motions of the moon will be examined in detail, not only because they are important in their own right, but also because an understanding of these motions is essential in any assessment of the claims made in recent decades that Stonehenge and other megalithic sites were designed as astronomical observatories.
First, some definitions:
Sidereal Period of the Moon: This is the time it takes the moon to orbit the earth and to realign itself with a star. It is equal to 27.32166 days.
It is very important to distinguish the sidereal period of the moon from its synodic period.
Synodic Period of the Moon: The moon's synodic period is the time it takes the moon to complete one orbit and to continue on to the point that it is realigned with the sun. It equals 29.53059 days, i.e., the time between successive new or full moons.
Zodiac: This is the region reaching 8° on either side of the 8° ecliptic. The motions of the sun, moon, and the classical planets are confined to the zodiac. It is divided into twelve constellations or houses. Were the zodiac to be unwrapped like a belt and laid on a page, the moon's orbit would appear as shown in the next diagram, from which it is evident that the moon's orbit is inclined at 5° to the ecliptic.
The Phases of the Moon
As is well known, the moon shows various phases.
The Greeks realized that the moon's phases do not result from the earth blocking the sun's light from the moon. Rather, as the next diagram shows, the phases are a result of how the moon is seen from earth. We always see half the moon (the side nearer to us) and half the moon is always illuminated by the sun (the side toward the sun). Depending on the time in the lunar month, we see all the moon's illuminated half or only a portion of it.
When the moon is in positions 4 or 6, it is seen as a crescent; when in positions 3 or 7, it appears as a half moon. At positions 2 or 8, it is in its gibbous phase. When at 1, it is seen full. When at position 5, the moon cannot be seen, both because of the sun's glare and because the moon's dark side faces us.
Eclipses are among the most dramatic celestial events. Two types of eclipses concern us: solar and lunar.
Solar Eclipse: Occurs when the moon comes between the earth and sun, the moon blocking the sun's rays from reaching the earth.
Lunar Eclipse: Occurs when the earth comes between the sun and moon, blocking the sun's rays from reaching the moon.
Because of the geometry of the two types of eclipses, solar eclipses are seen over only a very limited region of the earth, whereas lunar eclipses (which entail an actual darkening of the moon) are seen identically from all portions of the earth from which the moon can be seen at the time of the lunar eclipse. Let us now investigate the conditions necessary for eclipses of each type to occur.
One might think that a solar eclipse would occur once for each synodic period (29.53059 days), i.e., every time the sun and moon reach the same portion of the zodiac. This does not, however, occur, the reason being that the moon's orbit is inclined at 5° 9' (or 5.14°) to the plane of the sun's orbit. Because of this, when the moon passes the sun, it is usually above or below the sun, and although it blocks out the rays of the sun, the resulting shadow cone misses the earth. The point where the moon, moving upward, crosses the ecliptic is called the "ascending node"; the point where the moon, moving downward, crosses the ecliptic is known as the "descending node."
As this diagram suggests, for a solar eclipse to occur, the moon must be on or very near the ecliptic and in the same region of the zodiac as the sun. For a lunar eclipse to occur, the moon must be on or near the ecliptic and 180° distant on the zodiac from the sun. An understanding of this relationship makes clear why, if a solar eclipse occurs, it is probable that a lunar eclipse will follow it, occurring about 14 days later.
It is important to note that the time between the moon's passing through successive descending nodes is slightly shorter than its sidereal period. This shorter period, known as the moon's draconitic period, is illustrated in the diagrams by the fact that the moon makes one sweep up and down—and a fraction more—as it moves through the zodiac. The moon's draconitic period is 27.2122 days, whereas the sidereal period of the moon is 27.32166 days.
Hint: Finding solutions to most of these problems will be facilitated by beginning with a diagram that incorporates the information provided in the problem.
Excerpted from Theories of the World from Antiquity to the Copernican Revolution by Michael J. Crowe. Copyright © 2001 Michael J. Crowe. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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