| Vocademy |
As mentioned above, I personally doubt the universal acceptance of Ptolemy’s geocentric system. Even though Aristarchus’s writings had long been lost, they were referenced in later works, including those of Ptolemy and Copernicus. Astronomers and philosophers debated conflicts between the systems of Aristotle and Ptolemy. Arab astronomers debated the validity of Ptolemy’s system as early as the 10th century AD. It was also not a leap of genius to see that the cosmos could be heliocentric. Of course, we have 20/20 hindsight today. But the motions of the celestial bodies scream heliocentricity to the meticulous observer. Aristarchus demonstrated that long before Ptolemy. The concept of a heliocentric system was not lost during the reign of Ptolemy’s system. However, Ptolemy’s mathematics made sufficiently accurate predictions, which was what medieval astronomers needed. Anyone promoting a heliocentric system would have to develop the predictive mathematics to accompany it. It is likely this was never done simply because no one was inclined to devote a decade or two to the task.
Nicolaus Copernicus was born in Toruń, Royal Prussia (modern Poland). His father was Niklas Koppernigk. Over his lifetime, the junior Niklas signed his name as Niklas or Niclas Kopperlingk, Nicolaus Copernik, and Nicolaus Coppernicus, finally settling on Nicolaus Copernicus. He lived in Toruń and studied liberal arts at the University of Kraków. He also studied canon law, medicine, and astronomy in Bologna, Padua, and Ferrara. His day job was that of canon (an administrator) at the Cathedral of Frauenberg, though he was not an ordained priest. He was also a medical doctor, a diplomat, and an economic advisor to the Royal Prussian Parliament. He did his astronomical work unpaid in his spare time. Therefore, by strict definition, he was an amateur astronomer.
Sometime before 1514, Copernicus wrote his first outline proposing a heliocentric celestial system, but at that time, he had no mathematical model to support it. This work, made available only to his friends, was not intended for publication but was merely a rough draft of his ideas. Starting around 1515, he recorded astronomical measurements using the standard instruments of the day. In the following text, we will skip Copernicus’s highly detailed discussions of the geometry he used to determine the relevant celestial coordinates, as they are probably not of interest except to the most dedicated astronomical history nerds.
One of the measurements made by Copernicus was the location of the Sun’s apparent apogee (greatest distance from the Earth). In Ptolemy’s system, the Sun’s orbit was not perfectly centered on the Earth. Therefore, the distance to the Sun varied over the year. Ptolemy introduced this eccentricity to account for the Sun’s varying speed as it moved among the fixed stars each day. The location of the Sun’s apogee would be where this speed is slowest. Comparing his observations with existing ones made over the centuries, Copernicus discovered that the Sun’s apogee drifted by about one degree every 60 to 70 years. However, the ancient records he relied upon contained errors that led Copernicus to erroneously conclude that the Earth’s aphelion oscillated, or librated back and forth.
Copernicus used recently developed mathematical and geometrical techniques originating with Arab scientists, including Mo'ayyeduddin al-Urdi, Nasir al-Din al-Tusi, and Ibn al-Shatir. He not only revived the heliocentric model but also developed the mathematics to make it more accurate, enabling more precise predictions than Ptolemy’s geocentric model (although his greater accuracy is disputed by some).
What is lost to modern commentaries on Copernicus’s achievements is the main thing contemporary astronomers were interested in. It was not the heliocentricity of his model but the elimination of Ptolemy’s equants. Recall that Ptolemy created what he called deferents as the primary circles on which an object’s orbit is based. The centers of Ptolemy’s deferents were not aligned with the earth; they were offset or eccentric. This was to partially account for the variable speed at which the planets move across the sky. However, this is insufficient to accurately match the observations. Therefore, Ptolemy created a point on the opposite side of the center of the deferent to the Earth, which he called the equant. He then adjusted the speed of the planet's revolution around the deferent so that an observer at the equant would see the planet moving at a uniform speed (not accounting for the motion around the epicycle). Copernicus, like many others, balked at Ptolemy’s use of a fictitious uniform motion to account for an observed non-uniform motion.
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Copernicus retained the notion that celestial motions must be perfect circles. This adherence was not due to any religious or philosophical prejudice. He argued that any other motion would require an impetus to alter the body’s motion from a circle. With celestial objects attached to rotating spheres, the natural motion would be circular at a uniform speed. The observed non-uniform speeds were nevertheless periodic and regular, so these must be accounted for by one uniform circular motion superimposed upon another (epicycles on deferents), or by placing the observer eccentric to the object’s orbital center. Copernicus noted that the apparent motions of the planets could be derived from the use of eccentrics, epicycles, or both. Ultimately, he employed both. We will see how this worked shortly. Also, Copernicus used epicycles in a completely different way than Ptolemy. Instead of using epicycles to account for the retrograde motion of planets, Copernicus used them to counter natural rotation to keep certain angles consistently oriented in space
By 1532, Copernicus had essentially completed the manuscript on his proposed heliocentric system. Nevertheless, he resisted publication to avoid controversy. In 1533, Johann Albrecht Widmannstetter delivered a series of lectures on Copernicus’s theories, attended by Pope Clement VII and several Cardinals. In 1536, Cardinal Nikolaus von Schönberg wrote to Copernicus, urging him to send back copies of his writings and to publish his work to scholars.
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Copernicus described his final system in six books, published as a single volume titled De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres, hereafter referred to as On the Revolutions).
In the first book, On the Principles of Heavenly Motion, he respectfully and cautiously refutes Ptolemy’s model, introducing the heliocentric model. He establishes that the universe and the Earth are spherical, citing, among other things, the facts discussed in previous chapters in this text. He then establishes that the motions of the heavenly bodies are uniform and circular or compound circular motions (epicycles on deferents). He describes the Earth as having three motions: its daily rotation, its annual revolution around the Sun, and an annual revolution of its axial tilt, which is opposite to the direction of its orbital revolution.
The latter is often misunderstood. The knowledge of physics in Copernicus’s time did not include the idea that a spinning body would maintain its orientation in space. He believed that as the Earth revolved around the Sun, being attached to a transparent sphere, the direction of the Earth’s rotational axis should maintain its orientation relative to the Sun; he thought that if the axis pointed toward the Sun in June, it should pivot in space and still point toward the Sun in December, thus pointing to a different location among the fixed stars. He correctly argued that such a motion would eliminate the seasons. To account for the fact that the Earth’s axis maintains its orientation to the fixed stars, Copernicus introduced an annual rotation in the opposite direction to cancel the natural rotation of the Earth’s axis.
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Copernicus’s counter rotation, his third motion of the Earth, is often confused with the Earth’s 26,000-year axial precession. However, Copernicus discusses this separately and in greater detail in his third book. To account for the Earth’s axial precession, Copernicus made the above rotation and counterrotation of the Earth’s axis an imperfect match. This causes the counterrotation to lag the rotation by ever so slightly, resulting in slow precession of the axis (Previous astronomers accounted for axial precession by a slow rotation of the stellar sphere).
Hold on to this thought. We will see later that Copernicus employs the same counterrotation with his epicycles to produce planetary orbits. Ptolemy used co-rotating epicycles—epicycles that rotate in the same direction as the deferent—to account for retrograde motion. When we plot Ptolemy’s epicycles over time, we generate compound circular motions that produce intricate curves reminiscent of those created by geared drawing devices like the Spirograph. By removing the Earth from the center of the system, Copernicus eliminated the need for epicycles to account for retrograde motion. Instead, his model produces retrograde motion as a natural consequence of an inner planet passing a slower outer planet.
Unlike Ptolemy’s epicycles, Copernicus’s epicycles rotate in the opposite direction to the deferents and at the same speed. This causes the orientation of a planet with respect to the center of its epicycle to remain fixed in space, much like the Earth’s rotational axis in the above turntable illustration. The end result is that the planet’s orbit is circular but offset from the center of the deferent. Therefore, to reproduce the non-uniform motion of a celestial body as seen from the Earth, Copernicus uses a circular deferent rotating at a uniform speed centered on the Sun. Riding on this deferent is an epicycle on which the planet rides. This epicycle counterrotates such that the final orbit of the planet is a circle, identical to the deferent, but with its center offset from the Sun. This closely approximates the cyclic speed changes observed from Earth as the planet orbits the Sun.
He also discusses the ordering of the celestial bodies, placing the Sun near the center, with the planets, including the Earth, in their correct order, with the Moon orbiting the Earth. As proof, he cites numerous logical conclusions drawn from the established observations we have already discussed. One of these is that the planets maintain a cyclic motion, coming relatively near the Earth, then moving to a much greater distance. He cites Mars in particular, which is brighter than Jupiter at opposition, but “…is found barely among the stars of the second magnitude, being recognized [only] by those who track it…,” when near the Sun in the sky.
He places the stars on a sphere as the outermost part of his universe. He contends that this sphere is so large that the Earth’s size is infinitesimally small compared to it. He also refers to the distance between the Earth and the Sun as equally small compared to the stellar sphere. As such, no parallax or distortion of the constellations is observable as the Earth orbits the sun. As empirical evidence of this tremendous distance, he cites how the horizon splits the stellar sphere into two equal halves (he and other astronomers had measured this to the highest accuracy possible at the time). If the stellar sphere were smaller, the Earth below the horizon would obscure more than half of it.
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It’s worth noting that an observer on a flat plane will actually see slightly more than half of the stellar sphere due to atmospheric refraction; stars below the horizon are refracted such that they appear above the horizon. There appears to be no record that Medieval astronomers corrected for this phenomenon. However, they were aware of it and made celestial measurements well above the horizon to avoid the errors it would introduce.
It is in the first book that Copernicus makes his case for a rotating Earth. Ptolemy argued that if the Earth rotated, due to its size, the surface would be traveling at a tremendous speed, producing an outward force that would tear the Earth apart. Copernicus argued that if this were true, the stellar sphere would be subject to the same forces and would be torn apart by its tremendous speed, or would be inflated by the outward force. Thus, he further argued, “Were this reasoning sound, surely the size of the heavens would likewise grow to infinity. For the higher they are driven by the power of their motion, the faster that motion will be, since the circumference of which it must make the circuit in the period of twenty-four hours is constantly expanding, and, in turn, as the velocity of the motion mounts, the vastness of the heavens is enlarged. In this way, the speed will increase the size, and the size the speed, to infinity. …the heavens will therefore necessarily remain stationary.” In short, rotating should make it larger, and becoming larger would force it to rotate faster (to maintain its 24-hour cycle), and so on, until it becomes infinitely distant, rotating at infinite speed. This being impossible demonstrates that the stellar sphere, and thus the stars, must be stationary.
Copernicus also argued that a person on the rotating and revolving earth would no more feel the motion than a sailor drifting on a ship in a calm sea. The sailor would see distant landmarks drift by and would not, by his own sensations, be able to tell if he or the land were drifting. Copernicus quotes Vergil's poem, the Aeneid, “Forth from the harbor we sail, and the land and cities slip backward.”
Copernicus believed that objects close to the Earth, such as clouds, birds, and falling items, would be carried around with the Earth’s daily rotation. He also believed that a falling object would fall toward the Earth’s center of gravity. It is clear from his writings that the concepts of conservation of momentum and gravity were not unknown in Copernicus’s time. However, these concepts were not yet mature.
He also argued that comets, which appear to rise and set with the stars and planets, are far from the Earth. He claimed they were, “[in] that part of the air, which we can maintain, is unaffected by the Earth’s motion on account of its great distance from the Earth. The air closest to the Earth will accordingly seem to be still.”
In the last three chapters of the first book, he lays the groundwork for the rest of the volume by explaining the mathematics and geometry used to derive his system. This includes plane and spherical geometry (geometry on the surface of a sphere). He includes extensive tables to facilitate calculations with the described techniques
The second book, On the Sphere, or the Sky, begins with detailed definitions of the geometric entities of the universe, such as the celestial sphere, the celestial equator, the ecliptic (the apparent path of the Sun among the fixed stars), and the horizon with key angular relationships. He explains the ecliptic as a consequence of the Earth’s orbit around the Sun, where the Earth, “…traces the ecliptic around the Sun. Its direction is likewise from west to east…” He also describes methods of determining the latitude and longitude (referenced to the ecliptic and vernal equinox) of celestial bodies. He also includes extensive star tables listing stellar positions by longitude and latitude (also referenced to the ecliptic and vernal equinox). Note that medieval astronomers referenced celestial positions relative to the ecliptic and the vernal equinox, whereas modern astronomers use the celestial equator and the vernal equinox.
Copernicus begins his third book, On the Motion of the Earth, by discussing the 26,000-year precession of the Earth’s rotational axis (aka the precession of the equinoxes). He notes the difference between the tropical year, measured from equinox to equinox (also called the solar year), and the Olympic year, which ancient astronomers measured by the Heliacal rising of Procyon. He noted that Hipparchus of Rhodes was the first to report this discrepancy (around 130 BC), finding that the year measured according to the fixed stars was ever so slightly longer than the year measured from equinox to equinox. Hipparchus was the first to presume there was a slow rotation of the stellar sphere compared to the equinoxes. Copernicus notes that the zodiacal signs, which are measured according to the equinoxes, had shifted more than 24 degrees westward from the constellations that originally named them. (The vernal equinox, called the first point of Aries by astrologers, has precessed westward through the constellations Aries and Pisces and is about to enter Aquarius [“This is the dawning of the age of Aquarius,” as the song goes]. Did you know your astrological sign doesn’t line up with the constellation it’s named after?)
Unfortunately, the measurements Copernicus had to rely upon, made over more than a millennium, were inaccurate. As a result, he agreed with previous astronomers that the precession was not uniform. He describes the variation as ranging from 1 degree every 65 years to 1 degree every 100 years. Copernicus then devotes considerable space to proving the non-uniform precession and describing a mechanism of epicycles to account for the non-uniformity. We will mostly pass over this here for obvious reasons. By eliminating these mechanisms, we return to Copernicus’s explanation for the existence of this precession in the first place: a slight mismatch in the rotations that account for the Earth’s rotational axis maintaining its orientation in space, as discussed above.
The actual precession rate of the equinoxes is 1 degree every 71.6 years. The only significant non-uniformity is a tiny wobble (nutation) of the Earth’s axis over 18.6 years, discovered by James Bradley in 1748. This is primarily caused by the Moon's gravitational attraction and traces tiny circles along the Earth’s axial precession path, with a diameter of approximately 42 arc seconds (0.012 degrees). This translates to about 15 millimeters at the Earth’s surface.
One thing we will discuss, because it’s relevant to the motion of other objects, is his demonstration of how oscillating motion (libration) can be constructed out of circular motions. Here, he uses Tusi’s couple (without crediting Tusi).
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After several chapters devoted to the Earth’s supposed axial libration—which we now know doesn’t exist—we finally arrive at the central question: how Copernicus determined the orbit of the Earth and, by extension, the orbits of the other planets. Beginning with Chapter 13, The length and nonuniformity of the solar year, Copernicus introduces his model to explain the Sun’s non-uniform apparent motion, which, of course, reflects the true orbital motion of the Earth.
He begins by discussing the length of the year and how it is determined. Copernicus measured this length according to a fixed location among the stars—a sidereal year. This is in contrast to measuring from one equinox to the next—a tropical year, or solar year. Since the equinox moves slightly westward each year, these measurements will differ. Copernicus determined the sidereal year to be 365 days, 6 hours, 9 minutes, and 40 seconds. This is only 30.24 seconds greater than the actual value (the Arab astronomer, Thabit ibn Qurra, was closer, being only 3.24 seconds too long). Such precision was not derived from accurate clocks, which medieval astronomers lacked, but from averaging many measurements from various astronomers over the centuries. They essentially used the Earth’s rotational period as a clock, averaging the number of days and fractions of a day it took for the Sun to return to a designated point in the sky.
Copernicus also discusses that the interval from the vernal equinox to the autumnal equinox is about six days longer than the interval from the autumnal equinox to the vernal equinox, proving that the Earth’s motion around the sun is not uniform. The discussion includes many descriptions of proofs and explanatory geometry that we will not go into here. Most importantly, Copernicus argues that it is the Earth that moves, not the Sun.
The following illustration demonstrates Copernicus’s explanation of the Earth’s motion, which is the basis for the motion of every other body.
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. In Figure 113 the Earth’s epicycle has revolved counterclockwise about 45 degrees along the deferent. However, since the Earth has also revolved around the epicycle clockwise by the same amount, it has maintained its original orientation at the 12 o’clock position in space relative to the epicycle's center. This is the same motion described above, where the globe on the turntable has a mechanism to maintain the orientation of the globe’s axis in space. As this continues in a full circle, the Earth will trace an orbit with its center at point K, eccentric to the Sun at point D.
To understand how this accounts for the Earth’s apparent non-uniform motion, we can imagine observing the Earth's motion as viewed from the Sun. The Earth moves at a uniform speed around its orbit. However, when viewed from the location of the Sun, it will appear faster when close to the Sun, in the direction of point C, and slower when farther away, near point G. The apparent motion of the Sun mirrors the motion of the Earth. Therefore, from the Earth’s point of view, the Sun will appear to move faster when the Earth is closer to the Sun and will appear to move more slowly when farther from the Sun. This is identical to how a fast-moving airplane appears to move slowly when viewed from a great distance, but faster when viewed from up close.
Note again that Copernicus uses epicycles to maintain a planet’s orientation relative to space, not to account for retrograde motion. With the epicycle revolving on the deferent in one direction but the planet revolving on the epicycle in the opposite direction, the planet will remain at the 12 o’clock position in space relative to the center of its epicycle, regardless of where it is on the deferent.
Contrary to many modern commentaries on Copernicus, he places the Sun at the center of the deferent, not eccentric to it. In fact, when describing the accompanying illustration, he writes, “Let E, the center of the Universe, where the Sun is situated…” This becomes evident when one carefully follows his construction: placing the deferent off-center from the Sun would merely require a smaller epicycle to produce the same apparent orbit. The net result, in either case, is an eccentric circular path for the Earth. This is how Copernicus demonstrates that a planet’s orbit can be determined by either eccentrics or epicycles, and neither way is preferred over the other. In other words, Copernicus could have drawn the Earth’s orbit by simply drawing the green circle around point K. Copernicus said, “…the same apparent nonuniformity always occurs either through an epicycle on a concentric or through an eccentric equal to the concentric. There is no difference between them provided that the distance between their centers is equal to the epicycle’s radius.”
Notice also that Copernicus didn’t actually rid his model of the equant. Recall that Ptolemy placed each planet on an epicycle, the center of which rode the deferent around the earth at a non-uniform speed. The equant was a point in space where an observer would see the center of the epicycle circle the sky at a uniform speed. Copernicus had each planet orbit in a circle revolving at uniform speed. Thus, an observer at the center of the orbit would see the planet move at a uniform speed. The difference is that Copernicus’s planet revolved around its actual orbit at a uniform speed, unlike Ptolemy’s model, where the uniform speed was fictitious.
What Copernicus doesn’t address is the discrepancy between the planet’s observed motion and the uniform motion around his derived orbit. The model provides only a close approximation of the Earth’s non-uniform motion as observed from the Sun's location. Copernicus makes no attempt to improve the match.
Now we are going to step into another realm where Copernicus got it wrong. We are only doing this because modern commentaries tend to misinterpret the following illustration as depicting the 26,000-year precession of the Earth’s axis. It doesn’t.
Copernicus described the Earth as having three motions. He discusses axial precession extensively, but doesn’t include it with the other motions, which would have made it a fourth motion. He also omits a fifth motion: apsidal precession. The line that goes through the Earth’s perihelion (closest approach to the Sun) and aphelion (farthest point) slowly rotates in space. This occurs over a period of 112,000 years. Thus, the Earth’s perihelion currently occurs in early January. In the year 6430, the Earth’s perihelion will occur in early July.
Thanks again to the inaccurate measurements of his predecessors, Copernicus determined that the Earth’s apsidal precession was an oscillating motion (libration) rather than a complete circle around the Sun. The following diagram shows how he accounted for this motion.
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Doesn’t that look like fun?
In short, Copernicus added the second epicycle or epicyclet (light green) to account for the Earth’s apsidal libration. Copernicus’s second epicycle rotates over a period of 8,333 1/3 years, causing point P, the center of the Earth’s orbit, to revolve around point L. The Earth’s apsidal axis extends along the line that intersects point C and point P. As point P revolves around point L, the line through points C and P—the Earth’s apsidal axis—librates back and forth 12 degrees to either side of the vertical line. In the illustration, the green line represents part of the Earth’s orbit, with the aphelion positioned approximately 15 degrees to the right of the vertical line. Copernicus could have achieved the same result by eliminating the two epicycles and simply having the center of the Earth’s orbit, point P, revolve around point L on the circle shown. Nevertheless, this illustration solves a nonexistent problem and has nothing to do with the precession of the Earth’s rotational axis.
Again, we discussed the above diagram only because many modern commentaries confuse it with Earth’s axial precession. Unfortunately, such mistakes are common and are picked up by generative large language model artificial intelligence systems. So, don’t trust them, or, at least, trust but verify.
In the fourth book, On the Motion of the Moon and Its Eccentricities, Copernicus first expresses the importance of accurately measuring the Moon’s motion. The position of the Moon was a primary reference to measure the positions of the stars and planets, “…[it is] principally through the moon, which takes part in the day and the night, that the positions of any asters whatever are found and verified.”
Copernicus devotes the first two chapters to the inadequacies of the Ptolemeic model before he presents his determination of the Moon’s orbit. Copernicus used the same techniques to describe the Moon’s orbit as he did for the Earth and other planets. Here, we will concentrate on the characteristics of the Moon that Copernicus had to model.
Copernicus begins by modeling the Moon’s orbit around the Earth using the same method as he modeled the Earth’s orbit around the Sun. However, he adds a second, smaller epicyclet. This shifts the Moon’s circular orbit up and to the right in his diagram. The result is that the eccentric circular orbit of the Moon better matches observation when the Moon is new or full, but less of a match when the Moon is at quadrature (when a line between the Sun and the Earth forms a right angle with a line from the Earth to the Moon)
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The motion of the Moon
Now is a good time to examine the orbits of the Moon and the planets. Copernicus spends the bulk of his books describing how to model the observed motion using deferents and epicycles, resulting in eccentric circular orbits. We won’t delve into this further because it's largely a rehash of what we have already discussed. We will instead examine the characteristics he had to imitate in his model.
Let’s start with some other aspects of the Moon’s orbit that Copernicus had to account for. To start with, the Moon’s orbit is inclined about five degrees to the ecliptic. The points where the Moon crosses the ecliptic are called the nodes (or ecliptics by medieval astronomers). The node where the moon crosses the ecliptic from south to north is the ascending node, and the opposite is the descending node. The Moon’s orbital inclination, and thus the celestial location of the orbital nodes, precesses westward over a period of 18.6 years ( “…completing a revolution in the nineteenth year.”). This causes the nodes to move west around the sky, traversing the full circle over the same period.
Eclipses
Eclipses occur when the Sun happens to be near one of the Moon’s orbital nodes when the Moon crosses either node (the Sun crosses each node twice per year as it traverses the ecliptic). Depending on how closely the Sun is aligned with a node when the Moon crosses that node or the opposite node, the resulting eclipse will be a partial or total eclipse. When the alignment is nearly perfect, with the Moon positioned between the Earth and the Sun, the resulting solar eclipse will be either a total or an annular eclipse. An annular eclipse occurs when the Moon is near its apogee, thus so distant that it appears smaller than the Sun. In that case, the Moon does not completely block the whole solar disk, leaving a “ring of fire” around the Moon.
There are between four and seven eclipses, lunar and Solar, each year. These occur during eclipse seasons, which take place at opposite times of the year. The eclipse seasons gradually occur earlier in the year as the Moon’s orbit precesses. An eclipse season lasts up to six weeks or one and a half lunar orbits. There are typically two eclipses per season (solar, followed two weeks later by lunar or lunar followed two weeks later by solar), but there are occasionally three (solar, lunar, solar or lunar, solar, lunar). Of course, eclipses seem rare because each eclipse is visible from only a portion of the Earth. With a solar eclipse, you must be in the Moon’s shadow to see the eclipse. However, with a lunar eclipse, because we are observing the Earth’s shadow on the Moon, if you can see the Moon, you can see the eclipse.
While the tilted plane of the Moon’s orbit precesses to the west, the axis of the Moon’s apogee and perigee (the Moon’s apsidal axis) precesses eastward with a period of 8.85 years. This, coupled with the nodal precession, governs the degree of total and annular solar eclipses. A supermoon occurs when the full moon coincides with the Moon passing near its perigee, making the full Moon conspicuously large. The full Moon appears about 14 percent larger and 30% brighter at its perigee than at its apogee.
The Moon rotates on its axis at the same rate that it orbits the Earth. Therefore, the Moon acts like the globe on the turntable in the above illustration before the mechanism is added to keep the globe oriented in the same direction in space. Consequently, the same side of the Moon constantly faces the Earth. However, although the Moon's rotational rate is constant, its orbital speed isn’t. Thus, as the Moon is near its perigee and is moving faster than at its apogee. When near its parigee, the Moon’s rotation lags slightly behind its orbit, and we see the Moon rotated just a bit to our left. Likewise, when near its apogee, the Moon’s rotation gets ahead of its orbit, and we see the Moon rotated a bit to our right. This apparent rocking back and forth is called its libration.
Like the Earth, the Moon’s rotational axis is tilted to its orbit. Therefore, the north pole of the Moon is sometimes tilted slightly toward the Earth, and about two weeks later, it is tilted away from the Earth. The result of these motions is that as the Moon traverses its orbit, it appears to undergo a slight circular wobble while it approaches and recedes.
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Copernicus finishes the fourth book by discussing various anomalies of the Moon’s motion and how he modeled them. Particularly, he discusses the problems of parallax. Measuring the Moon’s location is complicated by the fact that the size of the Earth is significant compared to the size of the Moon’s orbit; the exact position of the Moon against the fixed stars is complicated by the location on Earth from which the measurement is taken. He also discusses how the solar parallax is significant enough that it also has to be taken into account.
To estimate the Moon’s parallax, Copernicus measured the angular separation between the Sun and the Moon over a period of several hours. He then calculated Moon’s parallax after accounting for the apparent movements of the Sun, Moon, and the Earth’s surface over that time (which were known with remarkable precision considering his era). This method allowed him to approximate the Moon’s distance using geometric reasoning in lieu of direct triangulation (modern trigonomotry was unavailabe at Copernicus’s time). It should be noted that Copernicus measures the distances in his orbital models from the center of the Sun to the Center of the Earth or from the center of the Earth to the center of the Moon, etc. This eliminates angular differences between the surface of the Earth and the diameter of the object in question (parallax).
He also discusses the timings of eclipses and how they are used to, in effect, calibrate his measurements. He finishes by discussing the relative sizes and distances of the Sun, Moon, and Earth. He allowed these to remain at the known quantities of his day, which have been discussed previously.
In his fifth book, On the Motions of the Five Planets, Copernicus discusses the motions of Mercury, Venus, Mars, Jupiter, and Saturn. As with the Moon, there is nothing new revealed in his methods, so we will not focus on the finer details of them. Copernicus begins by once again demonstrating the errors of the ancient astronomers.
Next, he discusses the non-uniformity of the planet’s orbits caused by the Earth’s motion. We have already discussed how a planet’s retrograde motion is the result of the Earth passing that planet on the celestial speedway. However, we are always viewing the other planets from a moving Earth. So, astronomers were trying to measure the unknown location of a moving planet from the unknown location of another moving planet.
Since there is little more to discuss about Copernicus’s methods, let’s list the modern measurements of the orbital properties of the remaining planets. We will start with Saturn, being the outermost known planet in Copernicus’s time. Copernicus also describes Saturn first.
Saturn
About two to three weeks after Saturn’s superior conjunction with the Sun, it begins to emerge from the solar glare, appearing low in the morning twilight. Each day, shortly before sunrise, it climbs roughly one degree higher in the eastern sky. After about three months, Saturn stands high near the meridian as morning twilight begins, rising around midnight. From this point onward, it rises earlier each night. Six months after conjunction, Saturn reaches opposition, rising in the east at sunset and remaining visible all night. This is its brightest and most favorable viewing period. In the weeks following opposition, Saturn shifts westward in the evening sky, rising earlier each evening. About three months later, it appears high near the meridian shortly after sunset. Another three months on, Saturn sinks deeper into the western twilight, eventually vanishing into the Sun’s glare. After a brief invisibility period of four to six weeks, the cycle begins anew.
Saturn’s sidereal period, the time it takes to complete one orbit compared to the fixed stars, is 29.45 years. Its synodic period, essentially the time between two successive oppositions, is 378.1 days—about one year and 12 days. Thus, one year after opposition, Saturn has moved eastward in its orbit, so that it takes about 12 more days for the Earth to catch up and pass it again.
Being so distant, its retrograde loop or zigzag is the smallest at only about 7 degrees. Saturn’s inclination to the ecliptic is 2.49 degrees, so it may be found as far as that distance above or below the ecliptic. Saturn’s visual magnitude at opposition is about -0.55.
Saturn is of special interest because of its rings. However, they hadn’t yet been discovered in Copernicus’s time, so we will discuss them later.
Jupiter
Jupiter follows the same pattern as described for Saturn. However, Jupiter orbits the Sun faster than Saturn, so it takes a little longer for each phase of the cycle.
Jupiter’s sidereal period is 11.86 years. Its synodic period is 398.9 days. Therefore, after one year, it takes about another month to overtake and lap Jupiter. Its orbit’s inclination to the ecliptic is 1.03 degrees, and its visual magnitude at opposition is -2.94.
Mars
Mars is the closest of the superior planets, and accordingly, it has the shortest orbital period—about 686.9 days. Because of this, Earth catches up to Mars less frequently than it does with Jupiter or Saturn. if you observe it night after night, you’ll notice it drifts eastward among the stars by roughly 0.5 degrees per day. That’s half the rate of Earth’s daily motion through its orbit, making Mars a worthy rival in the celestial race.
One year after opposition—when Earth has completed a full orbit—Mars has advanced through about half of its own orbit. At that point, it appears nearly on the opposite side of the Sun, and it has become much less conspicuous among the other stars. After two Earth years, Mars has completed just over one orbit, meaning Earth hasn’t quite lapped it. In fact, it takes about two years and seven additional weeks for Earth to overtake Mars and bring about the next opposition.
Mars also exhibits the greatest variation in brightness of any superior planet. This is due in part to the proximity of its orbit to the Earth’s orbit, and also to the eccentricity of its path around the Sun. At a favorable opposition—when Mars is near perihelion—it can approach as close as 56 million kilometers (35 million miles), shining brilliantly at magnitude –2.94 and rivaling Sirius. But about a year later, Mars will be near aphelion and positioned almost directly across the Sun from Earth, at a distance of roughly 400 million kilometers (250 million miles). At that time, its brightness fades to around magnitude +1.7.
Mars’s orbit is notably eccentric, ranging from a perihelion of about 207 million kilometers (129 million miles) to an aphelion of about 249 million kilometers (155 million miles). This variation in distance contributes significantly to its changing brightness. If opposition occurs near aphelion, Mars reaches only about magnitude -1.0—still bright, but far less dazzling than its perihelion performance.
Finally, Mars’s orbit is tilted about 1.85 degrees relative to the ecliptic, adding subtle complexity to its path across the sky during each apparition.
One thing that ancient and medieval astronomers missed is a subtle but visible change in Mars’s color every five and a half years or so. If you are practiced in observing Mars with the naked eye, this color shift is quite visible, especially if it happens near a favorable opposition. We now know that this yellowing of Mars’s color is due to periodic golbal dust storms which occur roughly every three Mars orbits during the martian southern hemiphere summer.
Venus
Venus, being an inferior planet, orbits the Sun faster than Earth, with a sidereal period of 224.7 days. Its synodic period is 583.9 days, so Venus laps us about every year and seven months. Like Mars, Venus is visible in the evening sky about every other year. The orbit of Venus is inclined to the ecliptic by 3.39 degrees. It also has the most circular orbit of all the planets, with an eccentricity of only 0.007.
Because it orbits inside the Earth’s orbit, as opposed to Mars, which orbits outside the Earth’s orbit, the distance between the Earth and Venus varies much less than the distance between the Earth and Mars. Therefore, the brightness of Venus varies much less than the brightness of Mars; Venus is conspicuously bright throughout most of its apparition.
When Venus makes an evening apparition, shortly after its superior conjunction, it first appears low on the horizon as twilight darkens. Even though Venus is then nearly on the other side of the Sun, it is already very bright. Night by night, it will appear farther from the sun. If Venus makes its apparition in the winter, when the ecliptic lies at a low angle to the southwestern horizon, it will track south each night. As spring arrives and the ecliptic rotates to a higher angle, Venus’s track will curve upward, having its maximum elongation from the sun high in the western sky. However, if it starts its apparition in the summer, it will first track higher each night, then move south, falling toward the horizon, then track north until it disappears, approaching its inferior conjunction with the sun.
While the above is happening, Venus gets brighter by the night. After it passes greatest eastern elongation (the greatest angular distance from the Sun in the evening), it continues to brighten until it reaches its maximum brilliancy 10 to 15 days later. At that point, Venus is the third brightest object in the sky after the Sun and the Moon, becoming as bright as magnitude -4.92. After that, it dims rapidly as it approaches inferior conjunction, disappearing into the Sun’s glare after only about 3 to four weeks.
About seven weeks after Venus disappears from the evening sky, all the above happens in reverse in the morning sky. Venus appears, then rapidly brightens to its greatest brilliancy, dims as it reaches its greatest western elongation, and fades slightly as it moves to the opposite side of the Sun to pass its superior conjunction. All the time, it makes a nightly curving track in the morning sky as it stays near the ecliptic in its trek among the stars.
The track described above, as is everything in this book, is discussed from the perspective of the Mediterranean area. In the southern hemisphere, some things get reversed. For example, when Venus is fairly high in the sky as viewed from the northern hemisphere, because of the angle of the ecliptic to the horizon, it will be low on the horizon viewed from the southern hemisphere.
Occasionally, Venus goes directly between the Sun and the Earth. When this occurs, we have a transit of Venus. Venusian transits occur in pairs, eight years apart, followed by a 121.5-year gap. This last occurred in 2004 and 2012. The next Venusian transit occurs in 2117.
Mercury
Mercury has a sidereal period of 87.97 days and a synodic period of 115.88 days. Its orbital inclination to the ecliptic is 7.01 degrees. Its name comes from the swift, wing-footed messenger of the gods. With such a short synodic period, it appears in the evening sky about four times a year. It follows a track and brightness profile similar to Venus, but is not as bright, reaching a maximum brightness of about magnitude -2.48. Mercury doesn’t stray far from the Sun and is only visible for five or six weeks between conjunctions, usually inconspicuous in the Sun’s glare. However, when its greatest elongation coincides with the ecliptic being angled to the west, as in the spring, Mercury can get fairly high in the sky.
As mentioned above, Copernicus’s theories were not at first rejected by the Catholic Church. At least one cardinal urged Copernicus to distribute his writings in scientific circles. Immediately after publication, which was shortly before his death, his theories were controversial but not universally rejected. Those who rejected his heliocentric model often dismissed it as a usable mathematical model, not grounded in reality. Further fueling the controversy, Andreas Osiander, a Lutheran theologian, added an unauthorized preface to the first edition of On the Revolutions, stating that the model was purely hypothetical. The model began to gain traction after the invention of the telescope and Galileo’s subsequent discovery of Jupiter’s four largest satellites (looking like a miniature solar system).
Before the early 1600s, the Catholic Church took little dogmatic interest in Copernicus’s theories. Many Catholic scholars, including Jesuit priests, used and taught Copernicus’s mathematics as a computational tool without endorsing its physical accuracy. However, by 1616, the Church determined that Copernicus’s model challenged biblical references to an unmovable Earth and a moving Sun. The Church declared heliocentrism official heresy and placed On the Revolutions on the Index of Forbidden Books. In 1620, his books were allowed with edits, presenting the model as hypothetical. It wasn’t until the early 1800s that the Church began to embrace a non-literal interpretation of biblical scripture and partially lifted the ban. The books were finally fully removed from the index in 1835.
At least one historian has claimed that Copernicus’s treatise was a failure, calling it the book that nobody read. However, Owen Gingerich, an eminent astronomer and historian, spent 35 years tracking down every surviving copy of the first two editions and found that virtually all the leading mathematicians and astronomers of the time owned and read the book. However, marginal notes made by the readers indicate that they mostly ignored the cosmology and were only interested in the equant-free mathematics of planetary motion.
Copernicus didn’t invent anything new concerning a basic heliocentric system. Aristarchus already presented a fairly accurate model 1,800 years earlier. However, Copernicus developed the mathematics and geometry to make a heliocentric system work at least as good as Ptolemy’s geocentric system for predicting celestial motions. He also went far beyond simply describing a heliocentric system, detailing many nuances of celestial architecture that are beyond the scope of this text. Lacking tools such as Newton’s laws, he used the geometry of the day to create something that worked. Nevertheless, with his many eccentrics and epicycles, he didn’t have an elegant model such as we have today. That took the work of Tycho Brahe, Johannes Kepler, Galileo Galilei, and Issac Newton to develop in the future.
Copernicus had a religious or philosophical prejudice toward perfect circular motion.
Copernicus preferred circular motion because it required no impetus to change the speed of a planet. This left all speed changes as artifacts of viewing circular motion from a point outside the orbit’s center.
Copernicus did not mention Aristarchus in On the Revolutions.
Copernicus casually references Aristarchus in On the Revolutions as if Aristarchus’s work were common knowledge.
Copernicus’s deferents had their centers eccentric to the Sun.
Copernicus’s deferents were centered on the Sun. His epicycles cause the planets to trace circles identical to the deferents but offset from the Sun. He notes that he could have eliminated the deferents and epicycles and simply used circular orbits with their centers eccentric to the Sun, but also noted that one method worked as well as the other.
Copernicus used multiple epicycles with each planet to closely match the observed motion of a planet.
I did not find this in On the Revolutions. Each orbit used one epicycle to create a circular orbit with its center offset from the Sun. He occasionally added a second epicycle to account for what we now know were errors in the available measurements. The eccentric orbits closely approximated the non-uniform speeds as seen from the Earth. Copernicus made no effort to refine his model to improve the approximations.
Annular eclipse
Apoapsis
Aphelion
Ascending node
Descending node
Libration
Nadir
Nutation
Oblateness
Olympic year
Periapsis
Perihelion
Quadrature
Solar disk
Tropical year
Zenith
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