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Astronomy for Amateurs
Chapter 1
Aristarchus
The sun, moon, stars, and planets were essential to the very lives of ancient people. These divided time into days, months, and years. The heavens told when to plant crops in the spring and when to celebrate festivals. The Egyptians expected the Nile to flood shortly after the star Sirius first rose from the sun's glare in the morning sky. Less scientifically, Mediterranean astrologers predicted war and other calamities about every two years when Mars returned to an appointed part of the sky.
From an Earthly point of view, the heavens appear to revolve around the Earth. Each day, the celestial bodies circle the Earth from east to west. But they appear to revolve at different speeds. Scholars wanted to understand how the heavens worked and made observations to help them discern the workings of the celestial mechanisms.
Here are some of the celestial phenomena early scholars would have observed concerning the revolutions of the heavens:
- Everything in the sky revolves around the earth from east to west once per day.
- The moon revolves slower than the sun, falling about 12 degrees on average behind the sun each day.
- The stars revolve a little faster than the Sun, moving ahead of the Sun by about one degree per day.
- Five stars are brighter than most others and, although still revolving faster than the Sun, revolve slightly slower than the rest of the stars (the fixed stars). The ancient Greeks called these planetae (wanderers). The planetae appear to move a little east each day compared to the stars.
- Two planetae, Mercury and Venus, never venture far from the Sun. Over the months, they swing back and forth from the evening sky to the morning sky. Mercury stays closest to the Sun and makes its oscillation in only 120 days. Venus swings much farther from the Sun, taking over a year and a half to complete its cycle.
- Three planetae, Mars, Jupiter, and Saturn, circle the entire sky compared to the Sun. Usually, they move a little east each day compared to the stars, but when they near the opposite side of the sky to the sun (opposition), they appear to stop and then move a little west each day. This retrograde motion continues for a time, and then they resume their daily eastward trek. Mars is especially dramatic, moving east among the fixed stars about 1/5 the diameter of the Moon each day, taking about 26 months to go from one opposition to the next. At opposition it is the fourth-brightest object in the sky after the Sun, Moon and Venus. Months later, when it has drawn closer to the Sun, it becomes an inconspicuous red “star.” Jupiter and Saturn take a more leisurely stroll among the stars, moving east compared to the stars only a few degrees per year with their retrograde motions smaller and shorter-lived.
These are just a few of the general observations. There were many questions about the nature of the universe. For example, what are the Sun, Moon, stars, and planets? How far away are they? What is the nature of their motions? Without modern tools to measure the heavens, they had to use what they had available. One tool was observing eclipses. Let’s see some of the things ancient scholars learned from them, as documented in ancient Indian and Greek texts.
Observations from solar eclipses
- A solar eclipse occurs when the Sun, Moon, and Earth line up. Therefore, when the Sun darkens, it is because the moon moves in front of the Sun.
- The moon moves in front of the sun during a solar eclipse. Therefore, the moon is closer to the Earth than the sun.
- From our point of view, the Sun and the Moon have the same apparent diameter (the same angular diameter). Since the Sun is farther away than the Moon, the Sun must be larger than the Moon.
Observations from lunar eclipses
- A lunar eclipse occurs when the Sun, Earth, and Moon line up in that order. Therefore, the Moon darkens during a lunar eclipse because the Earth’s shadow falls on the Moon, demonstrating that the Moon’s light comes from the Sun.
- From our perspective, the Earth’s shadow is circular regardless of the angle at which the sun is shining around the Earth. When an eclipse occurs around sunset or sunrise, sunlight passes across the earth, casting a circular shadow on the moon. When an eclipse occurs around midnight, with the sun coming from behind the Earth, the Earth still casts a circular shadow. The only shape that casts a circular shadow no matter what direction it is lit from is a sphere. Therefore, the earth is spherical.
- No shadow of a supporting structure appears during a lunar eclipse. Therefore, the Earth hangs or floats free in space.
- The earth's shadow is relatively small at the moon's distance, showing that it gets smaller at greater distances. Therefore, the Sun is larger than the Earth. If the Sun were smaller than the Earth, the Earth’s shadow would get larger with distance.
The above observations were widely known by about 400 BC (BCE) and certainly much earlier. Scholars rarely doubted the conclusions, but that which could not be demonstrated with the technology of the time was regularly disputed. The foremost question was, do the heavens revolve around the Earth, or does the Earth rotate at or near the center of the heavens?
Contrary to popular belief, a geocentric universe, with the Earth at the center and the heavens revolving around the Earth, was not universally believed by ancient scholars. Despite the obvious appearance of the heavens revolving around the Earth, many scholars thought it made more sense that the vast heavens were stationary and that the smaller Earth rotated in the opposite direction to the apparent motion of the heavens. The first scholar credited with making measurements of the heavens was Aristarchus, around 280 BC. Others certainly made similar measurements, but Aristarchus got his name in print, so the credit goes to him.
Calculating the distance to the Sun
One of the first notable things Aristarchus did was measure the distance of the Sun compared to the distance to the Moon. At the quarter moon (when the side of the moon facing the Earth is half-lit), a line drawn from the Sun to the Moon will form a right angle with a line drawn from the Moon to the Earth.
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Knowing the geometry of the quarter Moon, the angle between the Sun and the Moon, as seen from the Earth, can be used to calculate the relative distance to the Sun. Aristarchus determined the angle between the Sun and the Moon at the quarter Moon to be 87 degrees. This put the Sun’s distance at 18 to 20 times the distance to the Moon. It also made the Sun 18 to 20 times larger than the Moon (recall that from our perspective, the Sun appears to be the same size as the Moon). Aristarchus’ error was most likely in determining when the quarter moon occurs, when the light/dark line on the moon (the terminator) is straight, as seen from the Earth. This was difficult to determine with the tools of his day, essentially a ruler and his naked eye. The small size of the moon in the sky—only ½ degree—only added to the difficulty. Being off by less than three degrees, Aristarchus made his Sun-Moon angle measurement only five or six hours too soon. Try matching his accuracy without a telescope; he did pretty well. The actual angle between the Sun and the Moon at the time of the quarter Moon is only 1/6 of a degree short of 90 degrees. This gives an actual distance to the Sun of 400 times the distance to the Moon. Some scholars think that Aristarchus was being conservative. Aristarchus may have said that the Sun-Moon angle is at least 87 degrees but could be greater. Perhaps he thought it was unlikely that the Sun was hundreds of times more distant than the Moon.
Calculating the size and distance of the Moon
Aristarchus determined the relative size and distance of the Moon by observing lunar eclipses. He had no way of determining the size of the Moon except as a ratio to the size of the Earth, as the size of the Earth had not yet been determined (Eratosthenes would take care of that about 100 years later).
Aristarchus noted that the sun is about ½ degree in diameter. Regardless of the Sun’s actual size, the angular size in the sky determines the geometry of the Earth’s shadow. Whether the Sun is very large and far away or smaller and closer, the geometry is the same,
Through observations of lunar eclipses, it was already known that the Sun is larger than the Earth. This means that the umbra, the dark part of the Earth’s shadow where no sunlight reaches, will get smaller with distance in the shape of a cone. With the Sun having an angular diameter of ½ degree, the Earth’s shadow, as seen from the side, is a triangle with the two major sides separated by an angle of ½ degree.
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We can easily calculate the length of the Earth’s shadow cone in Earth radii by taking the reciprocal of the tangent of ½ of the cone angle (1/tan 0.25 degrees). Using modern measurements, that distance is 215 Earth radii. With that information, we can calculate the diameter of the shadow cone at any relative distance. For example, the diameter of the Earth’s shadow cone at a distance of 107.5 Earth radii (half the distance to the end of the shadow) is ½ of the Earth’s radius. When the Moon moves through the shadow during an eclipse, we can measure the size of the shadow in the sky and, in turn, calculate the distance to the Moon (where the shadow falls). Knowing the distance to the moon and the angular size of the moon, we can then calculate the actual size of the moon as a percentage of the size of the Earth.
Unfortunately, Aristarchus didn’t have trigonometric functions and had to again use Euclidean methods. He observed that the diameter of the Earth’s shadow at the distance of the Moon is about 1½ degrees and about 2.5 times the diameter of the Moon. With that, he calculated the distance to the Moon to be about 60 Earth radii, which agrees with modern measurements. He calculated the Moon’s diameter to be about 1/3 of the Earth's diameter (the actual size is about 27 percent of the Earth's diameter). He also calculated the length of the Earth’s shadow to be about 144 Earth radii.
Having previously calculated the relative distance to the Sun compared to the distance to the moon, Aristarchus could now calculate the relative size of the Sun. Knowing that the Moon is about 1/3 of the Earth’s diameter and the Sun is about 20 times the distance to the Moon, he could calculate the size of the Sun as 20 times 1/3 of the Earth’s diameter or about seven times the size of the Earth.
Aristarchus thought it illogical for the larger Sun to revolve around the smaller Earth, so it must be the Earth that rotates. This notion was not without its problems. Some scholars (including Ptolemy) thought a rotating Earth would result in tremendous winds. However, it appears that scholars accepted this heliocentric model until Ptolemy came along around 100 AD.
With a rotating Earth and stationary Sun, we still have a problem if we assume that the Earth is also stationary. All the heavenly bodies other than the Sun still appear to revolve around something. Each day, we find the stars have moved about a degree to the west compared to the Sun, making a complete circuit in about 365 days. Aristarchus solved this by having the Earth revolve around the Sun and making the stars stationary. As our perspective changes as we revolve around the Sun, we see the Sun appear to move about one degree east compared to the stationary stars each day. Aristarchus gave us a heliocentric system with the stationary sun at the center, the fixed stars around the edge, and the Earth revolving around the Sun—only one moving object (discounting the moon and planets to be dealt with later). There was no proof that this was correct. However, it was simpler than a system with the Sun, stars, and planets, each moving at different speeds around the Earth. Therefore, it was the more logical system.
Critics, however, pointed out that the heliocentric model should produce an annual distortion of the positions of the stars. Let’s assume all the stars are the same distance from the Sun. As the Earth revolves some distance from the Sun and closer to the stars, perspective should distort the shapes of the constellations, like parallel lines converging in the distance. If the stars are at different distances, the movement of the Earth should produce parallax, a shift in the apparent positions of nearer stars compared to the more distant background stars.
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Aristarchus countered this argument by assuming that the fixed stars are so distant that neither perspective nor parallax produces distortions large enough to see or measure with existing tools. He also assumed that the stars are bodies like the Sun, but many times farther away.
Concerning the planets, Aristarchus proposed that they orbited the Sun like the Earth, in circular orbits inside and outside the Earth’s orbit. Mercury and Venus, never venturing far from the sun, must orbit inside the Earth’s orbit. Mercury and Venus are, therefore, called the inferior planets. Mercury, having the smallest swing from the morning to the evening sky and the shortest period, must be the closest to the Sun, with Venus following. Mars must orbit outside the Earth's orbit as it ventures to the opposite side of the sky from the sun. Of the superior planets (those orbiting outside the Earth’s orbit), Mars moves fastest, taking just under two years to complete one cycle around the fixed stars (although it takes the Earth about another two months to catch up to come between Mars and the Sun, putting the opposition dates about 26 months apart). Mars also has the greatest variation in brightness. Aristarchus thought this was consistent with Mars being the innermost of the superior planets.
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Jupiter comes next, completing the celestial circuit in about 12 years. With Jupiter going slower (from our perspective), it takes less time for the Earth to catch up at each cycle, so the Earth passes Jupiter, putting Jupiter at opposition, about every 13 months. Saturn is the farthest of the planets known in Aristarchus’ time. It is the faintest and has the least variation in brightness. Saturn takes about 30 years to complete one circuit among the fixed stars, with the Earth passing it about every 12½ months.
Returning to the inferior planets, Venus also varies considerably in brightness, but not so much as Mars. When Venus first becomes visible out of the Sun’s glare (beginning apparition) after passing the far side of the sun (superior conjunction), it is already the third brightest object in the sky. However, it is nearly five times brighter when it reaches its greatest brilliancy about 7½ months later.
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Venus never ventures farther than about 47 degrees from the sun in the evening sky (greatest eastern elongation).
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After its greatest eastern elongation, Venus continues to approach the Earth and appears to get closer to the Sun in the sky as it prepares to pass between the Earth and the Sun (inferior conjunction). As Venus approaches the Earth, it becomes brighter. However, it is also turning its lit face away from the Earth. Venus’ apparent brightness is a compromise between its distance and how much of its lit face is presented to us. It reaches the sweet spot of greatest brilliancy about five weeks after its greatest eastern elongation.
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Venus then rapidly draws closer to the Sun, disappearing into the Sun’s glare, passing roughly between the Earth and the Sun to emerge in the morning sky shortly thereafter. Venus repeats the above phenomena in reverse in the Morning sky. Shortly after the inferior conjunction, Venus again brightens to its greatest brilliancy. About five weeks after that it reaches its greatest western elongation. Over the next six months it draws closer to the sun from our perspective, fading to about 1/5 of its greatest brilliancy, and then to again be lost in the Sun’s glare.
Aristarchus thought the above observations were consistent with Venus revolving around the Sun in an orbit inside the Earth’s orbit. Venus occasionally passes directly between the Earth and the Sun, thus passing in front of the Sun (a transit). However, this is impossible to observe without a solar telescope, so Aristarchus had no proof that Venus passes between the Earth and the Sun. Nevertheless, the above observations are consistent with Venus circling the Sun inside the Earth’s orbit but not with Venus revolving around the Earth at a uniform distance.
Aristarchus placed the Earth and the planets in orbit around the Sun in their proper order, creating a simple model of the Universe with the Sun at the center, fixed stars at a tremendous distance, and six planets, including the Earth, in their concentric circular orbits. As for the Moon, it gets to be special. Going both between the Earth and the Sun and to the opposite side of the Earth to the Sun, the Moon clearly revolves around the Earth. What about the retrograde motion of the superior planets? Aristarchus’ answer to that is simple. For a modern example, as a slower car appears to go backward as you pass it, Mars appears to go backward as we pass it on the celestial highway. Jupiter and Saturn do likewise. This explanation requires no complex motion of the planets; each continues in its uniform circular motion around the Sun.
This simple model describes and predicts the motions of the heavens with such precision that it took the meticulous measurements of Tycho Brahe, about 1,800 years later, to expose its errors. This led to Keppler revising the Aristarchus/Copernicus model to become today’s model with elliptical orbits. History tells us that Aristarchus’ model of the universe was lost and rediscovered independently about 1,700 years later by Copernicus. However, some historians believe Copernicus said that he was aware of Aristarchus’ model. Other scholars don’t believe this, seeing that there is no record of any references to Aristarchus’ work being available during Copernicus’ lifetime.
Aristarchus gets credit for correctly describing the universe, but there is ample evidence that he was not alone. However, about 400 years later, Ptolemy rejected a heliocentric model because of the parallax and wind problems. This was despite his own postulation that the simplest model that makes accurate predictions is most likely the correct model. Ptolemy developed a geocentric model that predicted the movements of the celestial bodies about as accurately as Aristarchus’ model. However, Pholemy’s model was complex to absurdity. Nevertheless, the Catholic Church, the Byzantine Empire, and, later, Islam endorsed Ptolemy’s geocentric model, so Western and Near Eastern scholars mostly accepted it based on that sponsorship. There were undoubtedly scholars who questioned Ptolemy’s geocentric model, but their voices have been lost in history. It wasn’t until Nicholas Copernicus successfully revived the heliocentric model in the mid-1500s that Ptolemy’s model was finally retired. As a final nail in the coffin, many modern scholars believe Ptolemy altered facts to fit his model. Tycho Brahe cast doubts on the legitimacy of Ptolemy’s measurements as early as the 1500s. In 1977, Physicist Robert Newton, in his book The Crimes of Claudius Ptolemy, accused Ptolemy of inventing data and altering data recorded by earlier scholars. In 1978, Herbert A. G. Lewis attempted to prove Robert Newton wrong but, ultimately labeled Ptolemy an outrageous fraud. In the end, Aristarchus wins the race for no other reason than Ptolemy is disqualified for cheating.
Next time you look at the full moon, focus on an area near the western (left) edge. Even with the naked eye, you can see a bright spot in the darker lunar basin. This is Aristarchus, a crater named after our Greek Astronomer who figured out the heavens so long ago.
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Terms learned in this chapter:
Angular Diameter
Apparition
Eclipse
Fixed stars
Greatest brilliancy
Greatest eastern elongation
Greatest western elongation
Inferior conjunction
Inferior planet
Opposition
Parallax
Quarter Moon
Retrograde
Revolve
Rotate
Superior conjunction
Superior planet
Terminator
Transit
Umbra
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