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Astronomy for Amateurs

Chapter 3

Ptolemy and the Dark Ages of Astronomy

It's about 130 AD, and many scholars happily argue about the merits of a heliocentric universe, as promoted by Aristarchus, or a geocentric universe, as proposed by Aristotle. History tells us that the geocentric model was by far more widely accepted. However, I believe the heliocentric model was more widely accepted than surviving history suggests. Remember, the victors write history, so ancient history may be clouded by 1,500 years dominated by the geocentric model.

Now, along comes Claudius Ptolemy.

Ptolemy could not swallow the Earth rotating on its axis or revolving around the Sun. First, if the Earth were spinning, you would feel it. If you drop something, it would fall at an angle instead of straight down. There should also be tremendous winds. If the Earth revolved around the Sun, we should see stellar parallax or other distortions of star patterns. Aristarchus had already addressed the lack of parallax and distortion of star patterns by assuming the stars were too distant for the effects to be significant. I don't know how Aristarchus explained the lack of wind, etc. However, the most logical (and correct) explanation is that the air is dragged around with the surface of the Earth and is thus spinning with the Earth; therefore, there is no relative wind. His explanation for objects falling straight down could have been tantamount to the principle of inertia, but there is no record of him addressing that.

Unable to accept a heliocentric universe, Ptolemy had to invent an alternate explanation for the variable speeds, retrograde motion, etc., of the Sun, Moon, stars, and planets. Let's recall what he had to deal with.

The Sun goes around the earth once each day (that is the definition of a day).
The Moon revolves slower than the Sun, falling behind by about 12 degrees per day. It takes 30 days for the Moon to go full circle to realign with the Sun. The Moon's day-to-day motion against the stars varies. Its distance also varies, manifested by changes in size and brightness.
The Stars take about five minutes less than the Sun to complete one revolution. As the Sun and stars move relative to each other, the speed varies slightly throughout the year, being faster in the winter and slower in the summer. The Sun thus spends about seven more days north of the equator than to the south.
Mars, Jupiter, and Saturn revolve slightly slower than the fixed stars, appearing to move eastward day by day compared to the stars, each at its own speed. However, when they near opposition, they speed up, going faster than the stars, moving west among the stars instead of east (retrograde motion), then slowing again to resume their regular eastward motion.
Venus and Mercury swing back and forth from the evening to the morning skies, never venturing far from the sun. Therefore, they sometimes go faster than the Sun and sometimes slower.
The Sun follows a fixed path among the stars. This path, called the ecliptic, is inclined to the equator by 23.5 degrees. Thus, in the winter, the Sun reaches 23.5 degrees south of the equator, and in the summer, it reaches 23.5 degrees north. The ecliptic is the same year after year, but by Ptolemy's time, Hipparchus had discovered that it creeps to the west by about 1/100 of a degree each year.
The Moon and planets follow paths within about seven degrees of the ecliptic.

Ptolemy professed to believe what is currently called Occam's Razor (don't multiply entities unnecessarily), believing the simplest solution to a problem is the most likely solution. However, unable to accept a heliocentric universe, he developed his own mathematical model to explain and predict the motions of the heavenly bodies from a geocentric perspective.

We have already seen that Aristarchus proposed a universe with the Sun at the center and the planets orbiting at uniform speeds in concentric circles. This model needed refinement (Kepler discovered that the orbits are elliptical), but it worked very well. Ptolemy also believed that celestial objects must move in perfect circles at a uniform speed. However, the observed motions of the heavens were not uniform. It was impossible to model these motions with uniform speeds in perfect circular motion from a geocentric perspective—or was it?

Before going into Ptolemy's model, I have to point out that I am handicapped here. Even if I were able to read ancient Greek, only fragments of Ptolemy's written work remain. What we have are synopses of synopses that are fragmented and contradictory. I will do my best to consolidate what I have found into a simplified but consistent explanation.

As I describe Ptolemy's system, think about what causes each phenomenon from the heliocentric perspective that Aristarchus proposed.

The Earth

The Earth is a fixed sphere at the center of the universe.

The Stars

Ptolemy believed all the stars were embedded in a sphere (the firmament) at a uniform distance from the Earth, putting the Earth at the center of the universe as a whole. This sphere rotated around the earth, from east to west, in what Ptolemy called �the period of the fixed stars,� (what modern astronomers call the sidereal day). I cannot find a source that quotes Ptolemy, so I cannot say how he expressed the length of the sidereal day. Some sources say he determined the sidereal day to be 23 hours, 56 minutes, and four seconds, which is only about 1/100 of a second short. It is conceivable that Ptolemy reached such precision by averaging many observations. However, seeing that the second hadn�t yet been introduced to horology (the science of timekeeping), it is likely that Ptolemy stated the length of the sidereal day in hours and fractions of hours. Therefore, he probably gave the length of the sidereal day as about 23.93 hours, perhaps with one or two more decimal places of precision.

The Sun

The Sun also resides on a sphere (called a deferent) within the stellar sphere that rotates along with the stellar sphere. However, the Sun's sphere also rotates on a secondary axis that is inclined to that of the stellar sphere by 23.5 degrees. On this secondary axis, the Sun's sphere rotates within the stellar sphere, taking approximately 365 days per revolution compared to the stellar sphere. Therefore, the sun falls behind the stars by about one degree per day. This causes the sun to make one revolution around the Earth a bit slower than the stars, taking an average of 24 hours. Modern astronomers call this time a solar day or, more technically, a synodic day.

The Sun, residing on the equator of its heavenly sphere, doesn't follow the Earth's equator. Instead, it follows a path among the stars inclined to the equator by 23.5 degrees. This path, called the ecliptic, carries the Sun to 23.5 degrees south of the Earth's equator in the winter and to 23.5 degrees north of the Earth's equator in the summer.

Unlike the stellar sphere, the Sun's sphere is not centered on the Earth. It is offset (eccentric) such that the Sun's velocity among the stars appears to vary as seen from the Earth. This causes the Sun to spend seven more days north of the equator than it does south of the equator.

Figure 21: Here is an exaggerated diagram of Ptolemy's universe showing just the stars, the earth, and the Sun. The whole system rotates clockwise around the Earth (the green dot in the center). As everything rotates once per day, the sun's sphere (the blue circle) rotates a bit slower, such that it offsets the Sun to the east by about one degree per day. Note that the Sun's sphere is centered on the X instead of the Earth.

Figure 22: In addition to being off-center (eccentric), the Sun's sphere is inclined to the stellar sphere by 23.5 degrees. This diagram shows the inclination of the equator of the stellar sphere (which is exactly aligned with the Earth's equator) to the equator of the Sun's sphere. As the whole system rotates once per day, the Sun's sphere slowly rotates compared to the stars such that the Sun slowly offsets one degree to the east compared to the stars each day. The Sun takes 365 days to complete one revolution compared to the stars.

Mars

According to Ptolemy, Mars was embedded in a sphere next in line outside the Sun's sphere. Like the Sun's sphere, Mars's sphere is eccentric, with its own central point to account for its individual motion (the illustrations show only one central point to reduce complexity). Mars's sphere also has its own axis inclined to the stellar sphere's axis, which differs from the Sun's sphere's inclination by about two degrees. Therefore, Mars's path across the sky is inclined to the ecliptic by about two degrees.

Mars gets an added feature, thanks to its brightness variation and retrograde motion. Mars not only rides on its main sphere but also rides on an attached sphere called an epicycle.

Keep in mind when viewing the following illustrations that the whole system rotates clockwise around the stationary earth once each day (we are looking from outside the system from the north). However, the celestial objects, riding on their celestial spheres, revolve a bit slower than the system as a whole and thus appear to fall behind the stars day by day. Each illustration is at the same sidereal time. This means we are looking at the system with the stars at the same position they were 23 hours, 56 minutes, and four seconds earlier. It takes about another three minutes and 56 seconds on average for the Sun to catch up to where it was the day before. From this perspective, the stars are always in the same place, and we see the Sun appearing to revolve counterclockwise by about one degree per day. The other celestial objects also appear to creep counterclockwise compared to the stars but slower than the Sun (except the Moon, as we will see later).

Let's begin with Mars in conjunction with the Sun (behind the Sun as seen from the Earth). We see that Mars, riding on its epicycle, is farther from the Earth than the sphere to which Mars's epicycle is attached.

Figure 23: Mars in conjunction (behind the Sun as seen from the Earth). Note that Mars resides on its epicycle, which carries it farther from the Earth as Mars approaches conjunction with the sun. This accounts for Mars's variation in brightness.

As Mars's sphere carries Mars counterclockwise compared to the stars, the epicycle rotates counterclockwise every 13 months. This coincides with the rotation of the Sun's sphere such that Mars is always on the outermost portion of its epicycle when Mars is in conjunction with the Sun.

About 13 months after conjunction, Mars's sphere has made about 6/10 of a revolution counterclockwise among the stars. In that time, the Sun has made one revolution plus an additional 1/12 revolution among the stars. Meanwhile, Mars's epicycle has rotated counterclockwise such that, like the Sun, it has also rotated about one and 1/12 revolutions, putting Mars at its closest point to the Earth. Additionally, the counterclockwise rotation of Mars's epicycle is such that, even though Mars's sphere is generally moving eastward among the stars, the net motion of Mars is westward. Therefore, normally, Mars moves a bit eastward among the stars day by day, but when near opposition, it appears to move a bit westward among the stars day by day.

Figure 24: Mars at opposition. Residing on its epicycle, Mars is now closer to the Earth than when at conjunction. Also, with Mars revolving counterclockwise on its epicycle, it is now moving westward among the stars.

Mars's epicycle is synchronized with the Sun's sphere's rotation, not with Mars's sphere's rotation. Mars takes about 23 months to complete one trip around the stars, but it takes another three months for the Sun to lap Mars for a second time. In those 26 months, Mars's epicycle has made two rotations; the epicycle completes its second rotation when the Sun laps Mars, not when Mars completes one trip around the stars. Therefore, it takes approximately 26 months for Mars to go from one opposition to the next (hoo boy!).

Jupiter and Saturn

Jupiter and Saturn are the next planets in line outside the Sun's sphere. Like Mars, Jupiter and Saturn have the axes of their celestial spheres offset to the axis of the stellar sphere by roughly 23 to 24 degrees. This aligns their sphere's axes near that of the Sun, putting the paths of Jupiter and Saturn among the stars near the ecliptic. The equator of Jupiter's sphere is inclined to the ecliptic by about 1.3 degrees, and Saturn's is inclined by about 2.5 degrees.

Jupiter's and Saturn's epicycles are about the same size as Mars's epicycle. However, being more distant, their retrograde motions don't traverse as much distance as seen from the Earth.

Figure 25: Ptolemy's universe with Mars, Jupiter, and Saturn, but minus the moving bodies between the Earth and the Sun. Like Mars, Jupiter, and Saturn move independently of the Sun, but also like Mars, their epicycles are synchronized to the Sun such that Jupiter and Saturn are always farthest from the Earth when they pass conjunction and closest to the Earth when they pass opposition.

Like Mars's epicycle, Jupiter's and Saturn's epicycles are synchronized to the Sun such that the planets are always farthest from the Earth when passing conjunction and nearest the Earth when passing opposition.

Jupiter takes about 12 years to complete one trip around the stars, but the Sun laps Jupiter about every 13 months. Jupiter's epicycle is synchronized with the Sun such that as the Sun passes Jupiter (conjunction), Jupiter is at its farthest point from the Earth. About six and a half months later, when Jupiter approaches opposition, Jupiter's epicycle carries it to its closest point to the Earth.

Saturn takes about 30 years to complete one trip around the stars, with the Sun passing by about every 12 and a half months. Saturn's epicycle is also synchronized with the Sun to put Saturn at its farthest point at conjunction and its nearest point at opposition.

Mercury and Venus

Mercury's and Venus's spheres are inside the Sun's sphere. Like Mars, Jupiter, and Saturn, Venus and Mercury have epicycles, but these epicycles are independent of the Sun's movement. However, they are synchronized such that the axes of their epicycles always roughly line up between the Earth and the Sun.

Figure 26: The axes of Venus's and Mercury's epicycles always line up between the Earth and the Sun. Unlike Mars, Jupiter, and Saturn, the rotation of Venus's and Mercury's epicycles are independent of the Sun.

Venus revolves on its epicycle about every 19 months, whereas Mercury revolves on its epicycle about every four months. With the epicycle axes locked to the Sun, Venus and Mercury never get far from the Sun in the evening and morning skies.

Venus and Mercury are never in opposition to the Sun but instead have two conjunctions per revolution. Ptolemy apparently had no explanation for Venus's dimming as it approached its conjunction closer to the Earth.

The Moon

Ptolemy put the Moon closest to the Earth. The Moon's celestial sphere is the slowest, falling behind the stars by about 12 degrees each day. Like the other celestial bodies, the Moon revolves on an epicycle. However, the Moon's epicycle is not synchronized to the Sun but rotates about once each time the Moon circles the stars, which occurs about once every 29.5 days.

Figure 27: Ptolemy's Moon.

The Moon never travels retrograde but only varies its speed as it traverses the stars. Like the other bodies, the Moon's epicycle also causes the Moon to come closer to the Earth and then recede from the Earth once per cycle. The Moon appears about 14 percent larger and about 30 percent brighter at its closer point than at its farther point.

Figure 28: The Universe according to Ptolemy. The entire model rotates clockwise once per day, centered on the stationary Earth. The planetary spheres are offset (eccentric). Each has its own individual center of rotation, but they are shown offset equally for simplicity. Each planetary sphere rotates independently (except Mercury and Venus), slightly slower than the stars. Each body resides on an epicycle that rotates counterclockwise. The epicycles of bodies outside the Sun are synchronized to the Sun such that the body is always on the far side (away from the Earth) of its epicycle when in solar conjunction and on the near side when at opposition. The axes of Venus's and Mercury's epicycles always line up between the Earth and the Sun. Venus revolves around its epicyclic axis about every 19 months, and Mercury revolves around its epicyclic axis about every four months.

So there we have the complete universe, according to Ptolemy. Except it is oversimplified—yes, oversimplified. His system's Early versions didn't match observations, so Ptolemy had to add epicycles upon epicycles and eccentricities upon eccentricities to make it work. This complexity in the face of the simpler model proposed by Aristarchus is why I don't believe Ptolemy's was so universally believed as history tells us.

However, Ptolemy's system had one significant advantage over Aristarchus's heliocentric system: Ptolemy had taken the time and effort to develop the complex math to make it work. With Kepler's and Newton's laws yet undiscovered, a heliocentric model had no mathematical advantage over a geocentric model; Copernicus had to use eccentrics and epicycles to make his heliocentric model work in lieu of applying the laws of gravity, momentem, etc. Ptolemy's math predicted when and where astronomical events would occur as seen against the celestial sphere.

Ptolemy's model did not come close to making a visual model as depicted in virtually all drawings of the system; the math does not fit such drawings. Nevertheless, the model did the job it was required to do: accurately predict astronomical events. Therefore, even astronomers who may have questioned Ptolemy's model used Ptolemy's math to do their jobs (money talks).

Figure 29: Here is what the orbits and epicycles of the Sun, Venus, and Mars would have looked like if Ptolemy had drawn them according to his mathematical model. This violates Ptolemy's claim that each planet is confined to an area between the spheres containing other planets—oops.

As mentioned above, Ptolemy apparently faked some measurements to fit his model, but astronomers would have abandoned his model if it weren't accurate enough (their jobs depended on it). It wasn't until Brahe's precise measurements, followed by Kepler's analysis and Newton's laws of gravity and motion, that the heliocentric model surpassed Ptolemy's geocentric model in predictive accuracy.

In the end, Ptolemy's model was used for about 1,500 years, and Aristarchus's model was nearly lost to history. Even as measurement became more accurate, Ptolemy's model made the necessary predictions. The length of the Year was refined. Equinoxes and solstices were accurately marked and predicted. The calendar was improved, and life went on, all using Ptolemy's flawed model. So, there you go.

Terms learned in this chapter

Conjunction

Deferent

Eccentric

Ecliptic

Epicycle

Firmament

Occam's Razor

Opposition

Sidereal day

Solar day

Synodic day

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