(Neolithic record of Moon Phases) Introduction
However, very few formed mathematical or physical models based on their observations.
At the same time, that focus should not cause us to forget the impressive accomplishments of other cultures. Other Examples
The story we will follow in more detail begins with the Babylonians / Mesopotamians / Sumerians, the cultures that inhabited the “fertile crescent.” Babylonian Astronomy
Their observations and mathematics was instrumental to the development of Greek astronomy and continues to influence science today. They were the first to provide a mathematical description of astronomical events, recognize that astronomical events were periodic, and to devise a theory of the planets. Babylonian Astronomy
• The accurate prediction of solar and lunar eclipses. • They developed mathematical models to predict the motions of the planets. • Accurate star charts. • Recognized the changing apparent speed of the Sun’s motion. • Developed models to account for the changing speed of the Sun and Moon. • Gave us the idea of 360◦ in a circle, 600 in a degree, 6000 in a minute. Alas, only very fragmentary records of their work survives. Early Greek
The conquests of Alexander the Great are oen credited with bringing knowl- edge of the Babylonians science and mathematics to the Greeks. Alexander ordered many translations of their work.
The Greeks did benefit from these observations, particularly their records of eclipses, observations of the motion of the Sun, and their star charts. The Greeks also learned many techniques of time measurement from the Babylonians. Unfortunately, for most of them, only scraps of their work remains, oen as mentions by other authors. This is partly due to the decline of Alexandria. Greek Astronomers Solar System Models
Starting with the Greeks, we will see a tension between models for the solar system, a geocentric or Earth-centered model, and a heliocentric or Sun-centered model. Although we know today that the heliocentric model is correct and the Earth really does move, remember that for most of recorded history the geocentric model was accepted as “obviously” correct. Solar System Models Parallax
Some of the resistance to the heliocentric model was due to Aristotle’s influence, some because they did not observe stellar parallax. Astronomers realized it might mean the stars were very far away, but that much empty space was inconceivable. Anaximander of Miletus
Anaximander was a student of Thales, who is oen called the first scientist. Anaximander drew the first known map of the world. In addition, he believed life began in weer environments and simpler forms gave rise to more advanced forms. He even traced human ancestors to fish! Anaximander of Miletus
Anaximander’s astronomical ideas were radical for his time. He was the first to propose that Earth floated freely in space. He described the Sun, Moon, and stars as coming from rings of fire with openings in them that could change size or close. Note that this puts them at dierent distances from the Earth. Euxodus of Cnidus
Euxodus developed many of the foundational ideas in Greek mathematics and astronomy. • He was the first to give a mathematical framework for a model of the “Universe” (solar system), based on spherical geometry. • It consisted of nested celestial spheres to explain celestial motions, including retrograde motions. The Earth was at the center. • There were 3 spheres each for the Sun and Moon, 1 for the stars. • There were 4 spheres each for the 5 known planets. • Each sphere provided one motion, such as diurnal motion and the eastward motion relative to the stars. • The major flaw was an inability to explain brightness changes. Euxodus’ Model Aristotle
Aristotle was the most revered scholar in science, mathematics, and philosophy for almost two millennia.
Over that time, his writings dominated Western thought. To break with Aristotle was tantamount to heresy.
At the same time, many of his ideas in physics and astronomy were incorrect and actually held back progress in some areas. Aristotle’s Physics
Aristotle believed that the logic and reason were the best way to figure out how the Universe worked. Experiments or observations were not as trustworthy, because the senses could be fooled.
• Aristotle extended the four elements established by Empedocles, earth, air, fire, and water, to include a fih one found only in the heavens: aether. • He believed that there were two basic types of motion. Forced motion occurred when something was physically disturbed, like throwing a rock. Natural motion occured as an object sought its “proper” place in the Universe. • The forced motion stopped as soon as the cause of it was removed. Aristotle’s Physics
• Earth and water moved down. Fire and air moved up. Aether, which made up objects in the heavens, underwent circular motion. • Aristotle thought that since everything in the heavens must be perfect, and the circle was the most perfect shape, everything in the heavens must undergo uniform, circular motion. • This idea became so entrenched that it was not seriously questioned until aer Copernicus. • Aristotle thought that it was impossible to have a vacuum. One argument was that an object’s motion could increase to infinity with nothing to slow it down. Aristotle’s Astronomy
Aristotle refined the models of the celestial spheres by adding counter-rotation spheres in-between to decouple the motions. The mechanics of the celestial spheres concerned him. Contrary to popular belief, it was known since ancient times that the Earth is round. Aristotle was one of many who provided proofs of this.
• There were stars that could be seen at some latitudes, but not others. • The Earth always cast a round shadow during a lunar eclipse. • Wherever you are, objects always fall straight down. However, he thought it impossible that Earth itself might be moving, or spinning. Aristarchus of Samos
Aristarchus of Samos was one of the first to make an estimate of the relative sizes of the Earth, Sun, and Moon, and the distances to the Sun and Moon.
He measured the angle between the Moon and Sun at first and last quarter, when they formed a right triangle with the Earth, to get the distance to the Moon and Sun. Since the Sun and Moon are about the same angular size, their distances and sizes must be proportional. He used the size of the Earth’s shadow at the Moon’s location during a lunar eclipse to get the size of the Moon in Earth diameters. This also gave him the size of the Sun. Aristarchus’s Measurements Aristarchus’s Measurements
His technique was sound, but very diicult to carry out. The real angle is 89◦ 500. His measurement was about 87◦. So he estimated the distance to the Sun was 19x the distance to the Moon. The real value is closer to 390X. However, he did find that the Sun was much larger than the Earth. So, he thought, the smaller object should orbit the larger. Aristarchus, not Copernicus, was the first astronomer, to advocate for a heliocentric model of the solar system. He further wrote that the Earth spun on its axis and that the stars were distant suns that were much farther away than anyone had imagined. No one took these ideas seriously until aer Copernicus. Apollonius of Perga
Apollonius was best known for his work on conic sections. His mathematical descriptions of the parabola, ellipse, and hyperbola are still basically what we use today. Many historians give him credit for the idea that solved the biggest problem with geocentric models - the retrograde motion of the planets. Apollonius reputedly showed that by combining two circular motions he could replicate retrograde motion. He moved the planet onto a smaller, the epicycle the center of which moved around the Earth on an orbit called the deferent. Apollonius of Perga
Some kind of epicycle would be used in all models of the solar system for the next 1500 years, until Kepler devised his laws of planetary motion. Eratosthenes of Cyrene
Eratosthenes excelled in many areas, particularly mathematics. This won him the position of Director of the Library of Alexandria.
Like Aristarchus, Eratosthenes also determined the distance to the Sun and Moon, and the size of the Sun. His value was 27 times the diameter of the Earth. The real answer is 109 times. He may have been o, but he got even closer and verified that the Sun was much larger than the Earth. Eratosthenes of Cyrene
One of his many other accomplishments was the invention of the armillary sphere. In time the armillary became the pre- eminent astronomical instrument for de- termining positions, until the invention of the telescope. It should be noted that Chinese astronomers independently in- vented this device. Size of the Earth
Eratosthenes was most well known for making the first accurate estimation of the size of the Earth. He did this with a lile simple geometry. (Since we can’t be sure of his units, it’s hard to say how close he got.) It was known that at a certain day of the year the Sun was directly overhead at Syene (modern Aswan, Egypt). It could be seen at noon at the boom of a well and objects there cast no shadow. Eratosthenes showed that if on that day he measured the angle from the vertical for the Sun in Alexandria, the ratio of that angle to 360◦ is equal to the ratio of the distance between the two cities to the circumference of the Earth. Eratosthenes’ Method Size of the Earth
In Alexandria Eratosthenes measured a shadow angle of 7◦ 140. Now all he needed was the distance between the two cities. That was diicult in those times. Also, we do not know the definition of the units he used, stadia. He was said to have hired someone to essentially pace it o. The distance estimate was 5,000 stadia which is thought be about 800 kilometers. Try it! With those numbers, what do you get for the size of the Earth? Hipparchus of Nicaea
Hipparchus was the greatest astronom- ical observer of the ancient world. His measurements were more accurate than any that came before him.
In his work, Hipparchus made good use of the earlier work of the Babylonians in mathematics and astronomy. Thus he was able to compare observations made over a longer span of time. He even worked on a heliocentric model but abandoned it when he could not reconcile it with circular motions. Note Aristotle’s influence again. Some of Hipparchus’ Accomplishments
• Invented trigonometry. • May have invented the astrolabe. • Devised new techniques for determining the size of the Sun and Moon. • Very accurately measured the distance to the Moon using parallax and solar eclipses. • Developed accurate methods to predict solar eclipses. • Developed a more accurate model for the Sun’s motion. • Discovered precession of the equinoxes. • Perfected Eratosthenes grid system and in doing so was the first to accurately measure positions (latitude and longitude) on the Earth. • Made the most complete and extensive star charts yet to be produced, partially in hope future astronomers might detect relative motion of the stars. Claudius Ptolemy
Much of what we know of some ear- lier Greek astronomers comes from Ptolemy’s great tome, The Almagest. Earlier writings were lost but some of the results were described in the Almagest.
The Almagest is indisputably one of the most influential scientific works of all time. It became THE authoritative reference for almost 1200 years. However, there is evidence that key parts were taken from earlier astronomers. For instance, the star catalog seems to have been copied from Hipparchus’ catalog, with the coordinates precessed for a later time. Claudius Ptolemy
Although he did draw from earlier works, he updated and refined many of the models and techniques, presented the work in tables that could be used for predictions and presented the most sophisticated geocentric model ever devised. Ptolemy’s geocentric model was the most accurate ever devised in the ancient world. Deferents and epicycles had been used in earlier models, but not even Hipparchus could get them to match all observed features of retrograde motion. Ptolemy’s innovation was the introduction of the equant. He displaced the Earth from the center of the deferent (this was called the eccentric) and defined a point, the equant, from which the planet at a uniform speed in angle. This mimics the changing distance in an elliptical orbit. Ptolemy’s Model
This idea worked well, even though it violated the Aristotelian ethic of uniform circular motion centered on the Earth. Because of this, many astronomers objected to the idea, even while using it for calculations. Ptolemy’s Model Islamic Astronomy
Aer Greek culture fell into decline, Arabic and Muslim scholars became the pre-eminent astronomers in the world. Their influence extended as far as China and Korea. It was around the 12th century when their work was starting to be translated into Latin and became more widely known in Europe. This reached its peak in the 14th century. Even today, not all of their surviving works have been well studied. Islamic Astronomy
Just a few of their accomplishments include: • The design of a much improved armillary sphere capable of unprecedented accuracy. • Greatly designs for other astronomical instruments, especially sundials, quadrants, and astrolabes. • Updated observations and many improvements on the observation and mathematical techniques of the Greek astronomers. • They began to accept the idea and write about the possibility that the Earth was in motion. • Alternatives to Ptolemy’s equants using other geometric constructions. Most importantly, they kept the preceding astronomical knowledge alive by copying the Almagest and continuing the work of the Greeks and Babylonians. Nicolaus Copernicus
As noted earlier, Copernicus did NOT come up with the very first heliocentric model, but he was the first to have such a model be accepted.
He was motivated by a philosophical belief in the perfection of the heavens (Aristotle again), rather than any scientific evidence. He wanted to restore uniform, circular motion instead of relying on the eccentrics and equants of Ptolemy. Because of this, even though it was heliocentric, his model still needed epicycles to explain changes in apparent speed and brightness. So it may not have been less complicated than Ptolemy’s but it was more accurate. Copernicus’ Model
Copernicus’ model: • naturally explained retrograde motion, • accurately models the relative positions of the planets, • explains the cause of the seasons, • correctly puts the stars vastly farther away than the Sun. Copernicus’ Work
An early version of his work was distributed to his friends long before he published his De revolutionibus orbium coelestium, “On the Revolutions of the Heavenly Spheres.” So rumors of his work spread, as well as positive and negative commentary. For espousing a moving Earth, his works were placed on the Index of Forbidden Books. This is why, when he did publish, he proposed it as only as a mathematical convenience, not necessarily a model of reality. His delay in publishing the work may not only have been because of religious objections. There is evidence he was also wary of criticisms from other astronomers and philosophers. It was only near the end of the life that it was finally published. Legend has it that he was presented with the final published work on his deathbed. Galileo
Galileo Galilei could arguably be called the first modern scientist. He put exper- iments and observations at the forefront of his work, breaking from the Aristolean principle that thought and logic should take precedence.
His work in astronomy, physics, and optics marked the beginnings of these as modern scientific fields. He made major contributions in many other areas as well. We’ll get back to his physics in just a bit. Galileo
It should be noted that Galileo did NOT invent the telescope. (It is unclear who did.) But he understood the principle, built his own, and was the first to make systematic astronomical observations.
Many of his observations directly contradicted the ideas of heliocentric cosmology and supported Copernicus’ geocentric model. Galileo’s Observations
Some of his discoveries: • The Milky Way is made up of innumerable stars. • The Sun has “sunspots”. By tracking them he showed that the Sun rotated. So if the much larger Sun could rotate, why not the Earth? Also, the Sun was not a “perfect” sphere. • Jupiter had 4 moons. So the Earth was NOT the center of all rotations in the Universe. • The Moon has craters. It was not perfect either. • He saw things like ears around Saturn, but could not tell that they were rings. • Venus exhibits all phases. This would be completely impossible in a geocentric Universe, so this was convincing proof for heliocentrism and a moving Earth. Venus’ Phases
His advocacy for Copernicus’ model got him in trouble with the Catholic church and led to him being convicted of heresy and put under house arrest for the rest of his life. He was not oicially “forgiven” until 1992. Tycho Brahe
Tycho Brahe was the most prolific and accurate observer in the pre-telescopic era. He was member of the nobility and so got the Danish King, Frederick II to fund his eorts.
He originally became famous for his observations of the supernova of 1572, now called Tycho’s Supernova. The stars were supposed to be immutable, and therefore the “new star” or nova was thought to be a terrestrial phenomena. Tycho showed it had no parallax and stayed in the same position relative to the stars, so belonged in that sphere. Tycho Brahe
Later he observed the comet of 1577-1578 and showed again that it was not terrestrial in nature. It was beyond the Moon and moved through the solar system, shaering once and for all the notion of solid celestial spheres. He was driven by the desire to improve the accuracy of all his observations, partially in hopes of providing proof for his model of the solar system. His measurements were routinely accurate to 10-20, and even beer for many stars. To do so, he originally built the observatories of Uraniborg and Stjerneborg on the island of Hven. These were abandoned when he fell out of favor with the Fredrick’s successor, Christian IV. Tycho eventually resumed his work in Prague. Tycho Brahe’s Model
Tycho developed a hybrid geo-heliocentric model for the solar system. The Moon and Sun revolved around the Earth, but all the planets revolved around the Sun. Tycho in Prague
Yes, religious considerations caused him to reject the Copernican model and a moving Earth. But he also objected because he could detect no parallax, while at the same time he thought he could see the sizes of stars. If they were so far away that he could detect no parallax, they would also be vast compared to the Sun. This he thought absurd. In Prague he began working closely with Johannes Kepler. Even though Kepler was unconvinced of the accuracy of Tycho’s model he defended Tycho against Tycho’s detractors. Kepler knew who paid his salary! Tycho’s Innovations
Tycho innovations were novel for his time. He: • made large instruments, permanently fixed in place. • made them of metal instead of wood, held in masonry. • had multiple observers take measurements to average the results. • made measurements on a regular basis, not just special times, such as the start of a retrograde loop. • used equatorial coordinates. • was able to compensate for atmospheric refraction. Kepler
Tycho died in 1601 as a result of a burst bladder. At this point Kepler gained full access to Tycho’s data to begin his work on the laws of planetary motion. Kepler was appointed as his successor and tasked with completing his work.
Like Tycho he observed the supernova of 1604, now known as Kepler’s Supernova, and showed it belonged in the sphere of the stars, further weakening the argument for immutable heavens. By the end of that year he had formulated his first two laws of planetary motion, based primarily on his aempt to fit a theory to the excellent observations of Tycho. Astronomia nova or New Astronomy was finally published in 1609. Kepler
In the intervening years, corresponding with Galileo on his observations of the moons of Jupiter, he turned his aention to optics and not only extended the theory, but also came up with an improved design for the telescope. He published his third law a decade later, in 1619. It took a few decades aer that for Kepler’s work to really be accepted. The successful prediction of transits of Venus and Mercury were very sensitive tests of his theory. The 1631 transit of Mercury was the first to ever be observed. Kepler’s Laws
It is important to recognize that Kepler’s Laws also provided a foundation for Newton’s Law of Universal Gravitation. Newton’s derivation of Kepler’s Laws from his theory of gravity was essentially the end for geocentric theories. Kepler’s laws apply to anything in orbit. We use them for stars, galaxies, and even within clusters of galaxies. They are important tools in many areas of astronomy. Kepler’s 1st Law
Planets move in ellipses with the Sun at one of the foci. Kepler’s 2nd Law
The planet moves such that a line joining the planet to the Sun sweeps out equal areas in equal times. Kepler’s 3rd Law
If the planet’s period, P, is measured in years and its average orbital distance in AUs: P2 = a3 An AU is the average distance of the Earth from the Sun, a handy unit for solar system sized measurements. This gives us a way to tell where an object is, just by measuring its orbital period. Much more importantly, by the time Newton is done we can use orbits to measure the mass of the object being orbited. Again, it doesn’t maer what it is, planet, star, galaxy, whole clusters of galaxies. Segue
We will pick up the story again in the next section, where we discuss gravity in detail. Although Newton, of course, developed and proved the theories, Galileo, Robert Hook, and Rene` Descartes all made important contributions as well. Answer - Size of Earth
7◦ 140 = 7.233◦ 7.233◦ 800 km ◦ = 360 CircumferenceEarth
CircumferenceEarth = 39, 816 km
The “modern” value for the circumference is 40,075 km at the equator, 39,941 km pole-to-pole. (Rotation makes the Earth a bit squished.) Eratosthenes’ value is astonishingly good. But remember, we can’t be sure what the size of the “stadia” he used was.