Great Astronomers: Kepler
Great Astronomers in Modern English
by Sir Robert S. Ball, 1895 (paraphrased by Leslie Noelani Laurio)
To view the table of contents for the rest of this book, click here.
Johannes Kepler, 1571-1630
Discovered three laws of planetary motion, starting with his observation that planets orbit in ellipses.
Tycho [pronounced TEE ko] Brahe spent his life accumulating vast stores of careful observations of the positions of the heavenly bodies. He didn't live long enough to make any interpretations about them. As he lay on his death-bed, he summoned a brilliant young philosopher to come to his bed-side, and he requested that the young man spare no effort to calculate his observations in order to reveal the secrets of the movements of the heavens and the great truths of nature. This solemn trust was accepted. The young man to whom this sacred trust was passed down was Johannes Kepler.
Johannes Kepler was born December 27, 1571 in Weir in the Duchy of Wurtemberg [now part of the Stuttgart Region in the German state of Baden-Wurttemberg]. He did not have a happy childhood. His father was born into a well-connected family, but was a shiftless and idle adventurer. His mother was ignorant and ill-tempered. His father abandoned the family when Johannes, their oldest son, was 18. At the age of four, Johannes had a severe attack of smallpox that permanently damaged his eyesight and weakened his constitution.
The physical limitations of young Johannes Kepler are what started his interest in the pursuit of knowledge. If he had been able to do ordinary physical work like other boys, his life would most likely have been devoted to manual labour. But although his body was weakened, he soon showed a talent for learning. It was thought that he might be able to work in some religious clerical position, since the Church was the only professional option available for an intellectual career. By the time Johannes Kepler was seventeen years old, he had attained enough education to be admitted to the University at Tubingen.
During his time at the university, Kepler seems to have divided his attention between astronomy and religion. He did equally well at both and couldn't decide which course to follow as a profession. He leaned more towards a career in the Church, but his friends, who knew him better than he knew himself, convinced him to pursue astronomy. So in 1594 he accepted the prestigious position as Professor of Astronomy at the University of Gratz [in present day Austria].
The duties expected from an astronomy professor in the 1500's were much different than they are today. He was expected to use his expertise to predict eclipses, and the movements of the heavenly bodies in general. That doesn't seem unreasonable -- but astronomers were also expected to predict the fates of countries and the destinies of individuals!
In those days, it was almost universally believed that all the heavenly spheres revolved in some mysterious way around the earth, which appeared to be the most central and important body in the universe. It was believed that the changing positions of the sun, the moon, and the stars indicated the course of nations and people. This was the generally accepted notion, so it seemed reasonable to expect a professor of astronomy to decipher celestial clues regarding the fate of man.
Kepler eagerly took on the job, including this part of his task. He diligently studied the rules of astrology that had been handed down over the ages. He sincerely believed in the connection between the position of the stars and the state of human affairs. He even thought he could see confirmation of astrology in the events of his own life.
But astrology wasn't the only delusion that people believed in at the time. It's amazing to us that the most intelligent men only a few centuries ago could have believed such nonsense. For example, geometricians had known for a long time that there were only five regular, solid figures. [Platonic solids: 4-sided pyramid-looking Tetrahedron, 6-sided cube, 8-sided octahedron, 12-sided icosahedron, 20-sided dodecahedron; they make dice in these shapes.]
Coincidentally, there were five planets known at the time: Mercury, Venus, Mars, Jupiter, and Saturn. It seemed obvious to Kepler that the five solids corresponded to the five planets, and in his book 'Mysterium Cosmographicum,' he created a model of the solar system in which the five shapes were set one inside the other to signify the planetary orbits. [Below is an image of his model of the solar system.]
Today we know there are more than five planets, so this theory doesn't hold together. But in Kepler's day, his theory was praised as an intellectual triumph. It brought him esteem and respect -- and may be what brought him to the attention of Tycho Brahe and Galileo.
At the time Kepler lived, science professors were more affected by compelling changes than they are in modern universities. Johannes Kepler was a Protestant, and had been appointed to Gratz University by a Protestant board of directors. But a change had taken place at Gratz and the Protestant professors were expelled. An exception was made for Kepler because of his distinction and prestige, and he was allowed to come back and resume his position. But now his students were gone, so when Tycho Brahe offered him a job at his observatory near Prague, he accepted.
When Tycho Brahe died a couple years later (in 1601), Kepler was appointed to succeed him as imperial mathematician, and gained exclusive access to Tycho's recorded observations. It was Kepler's work with Tycho's observations that brought him to his most important discoveries.
Johannes Kepler was among the first astronomers to view the heavenly bodies through a telescope; in 1610, he was able to use the kind of telescope that led Galileo to his discoveries. But his world-changing achievements didn't come from looking through a telescope; they were deduced from Tycho Brahe's records and measurements of planets made with the instruments he had at that time -- and that didn't include a telescope.
All astronomers working before Kepler assumed that planets must revolve in a perfect circle. If their observations didn't seem to show a perfect circle, they explained it away with Ptolemy's theory that planets were moving in circles within larger circles.
When Kepler looked at Tycho Brahe's records of his observations of Mars, he couldn't work out any mathematical formula that corresponded to circular movements, or circles within circles. No matter how many possibilities he tried, no mathematical calculation lined up with Tycho's actual records. This led Kepler to discover that planets orbit in ellipses, not perfect circles. An ellipse has two focus points rather than one, and he determined that the sun was one of the focus points in the orbit of Mars.
It's easy to draw an ellipse -- stick two pins in a piece of cardboard. Those are the two focus points. Tie a string into a loop that will fit loosely over both focus points. Now use a pencil inside the loop of string to draw a circular shape. [Watch this method of drawing an ellipse on YouTube.] Try this with the pins closer, and then farther apart. The closer the pins are to each other, the more like a circle the shape will be. The farther apart the focus points, the more elongated the ellipse will be. Planets generally orbit in ellipses that are more rounded, like a circle [meaning that the focus points are relatively close together -- which is probably why it looked to astronomers like planets orbited in perfect circles]. But Mars orbits in a more elongated ellipse, and that helped make its ellipse more obvious to Kepler. If Kepler had been working with Tycho Brahe's measurements of a planet like Venus whose orbit is almost a circle, he might never have noticed that the measurements were off.
Today we know that a planet can't possibly orbit in any shape other than an ellipse. But Johannes Kepler didn't know that. Even up to the end of his life, he had no knowledge of any natural cause why planets should follow ellipses. He merely took Tycho Brahe's observations at face value and made a brilliant guess, and his mathematical calculations confirmed his guess.
Johannes Kepler also identified the law that helps to determine the speed a planet is orbiting at different points on its path. [The second of Kepler's three laws of planetary motion has a mathematical equation showing that a planet moves fastest when it's closest to the sun.] Here again, Kepler knew of no reason to think this had to be true; he merely made a brilliant guess based on what he saw. He did know something about gravitation, and he had written that the ebb and flow of tides must be attributed to the moon's attraction to the waters on the earth. But he knew nothing about the discoveries that Isaac Newton would make almost fifty years later -- in which Newton showed that universal gravitation meant that the laws discovered by Kepler's insight were not only accurate, but they were the only possible way planets could move.
In order to fully appreciate the accomplishments of Tycho Brahe and Johannes Kepler, we must reflect on the different ways these two men viewed the system of the heavens. Copernicus [from Poland] had already explained the true system, which is that the sun, and not the earth, is at the center of our planetary system. But in the days of Tycho Brahe, this theory wasn't widely accepted. Tycho himself didn't accept Copernicus's theory. His own observations seemed to show that the earth was at the center [and the religious teaching of the day said that man, as the apex of God's creation, must be at the center]. We may find it surprising that such an accurate observer of the heavens as Tycho Brahe could have rejected Copernicus's theory, which seems so logical to our minds, in favor of a system that seems ridiculous to us. But he carefully observed the positions of the sun, the moon, and the planets, and what he saw and recorded seemed to confirm that they all revolved around the earth.
But Johannes Kepler had the advantage of belonging to the new school. He used Tycho Brahe's observations to develop the Copernican theory that Tycho persistently resisted.
We can make a modern comparison by looking at a more contemporary example. The intellectual revolution produced by Copernicus's new theory about the heavens can be compared to the intellectual revolution produced by Darwin's theories. It radically changed the views held by biologists about life on the earth. It was not universally accepted even by naturalists who had spent their entire lives and built successful careers studying organisms. Professor Richard Owen greatly expanded our knowledge of animals through their fossils, and Darwin even used Owen's observations to confirm his own generalizations. But Owen never accepted Darwin's conclusions. Like Tycho Brahe, he continued compiling his own observations while sticking rigidly to ideas that are considered erroneous by most modern scientists. In this analogy, Kepler is like the modern scientists who interpret the same data upon more currently sanctioned principles.
If we in our modern era read Kepler's works, we are sometimes surprised that the sublime truths he found are mixed with wild errors and absurdities. We must remind ourselves that he wrote in an age when even the foundations of science as we understand them were almost entirely unknown.
Is there any joy more genuine than the bliss that rewards someone who successfully searches after truths of nature? Even the most humble observer of nature will be able to understand Kepler's enthusiastic delight when, after years of toil, the glorious light of truth dawned on him. He had correctly deduced that the number of days a planet takes to travel around the sun must somehow be related to its distance from the sun -- the radius of its orbit, if the orbit were a circle.
Here again, Kepler was in the dark about accurate scientific principles. He had a few hazy notions, but they were fairly bizarre compared with what we know now. His notions were mixed up with vague ideas about the five Platonic solids corresponding to planets, and the five elements -- earth, air, fire, water, and 'cosmos.' Even more lamentable, his reasoning was mixed up with astrology. But the influence of these faulty impressions didn't prevent him from skillfully calculating the speed and distance of orbits around the sun.
It was easy to demonstrate that, the farther away from the sun a planet was, the longer it would take to travel around it. It would seem obvious that the time would be directly proportional to the distance. But there's more to the calculation than just time and distance. Something was still missing. After many attempts, Kepler was overjoyed to discover the solution: the square of the time in which a planet revolves around the sun was proportional to the cube of the average distance of the planet from that body. [This is Kepler's third law of planetary motion.] This equation made it possible to determine how far a planet was from the sun by measuring how long it took to revolve around the sun.
An example of how Kepler's brilliance is intermingled with some wild speculations is illustrated in the book that included the law just referred to. The explanation and equation of this law connecting distances of planets with their orbit times was mixed with comments about the musical properties of the planets. There was an old Greek theory still believed at the time that the planets hummed with tones that were inaudible to human ears, making profound and harmonic "music of the spheres" to be enjoyed by their Creator.
Kepler was the first astronomer to predict the transit of a planet in front of the sun. In 1629, he announced that Mercury would make a transit across the sun on November 7, 1631, and Venus on December 6, 1631. He died before this happened, but the French scientist Pierre Gassendi witnessed the transit of Mercury as predicted. The Venus transit happened on cue, but no one recorded it.
In addition to the discoveries already mentioned, Johannes Kepler also published a catalogue of stars and planetary tables that he called the 'Rudolphine tables' in memory of Rudolf II, Holy Roman Emperor. This catalogue enabled star gazers to find planets in the sky with greater accuracy than before.
Kepler was not primarily an astronomical observer. His role was dealing with Tycho Brahe's observations. By studying them and comparing results, he was able to work out movements of heavenly bodies. Thus, Tycho supplied the raw data, and Kepler's genius turned that data into a usable form. For more than a century, the 'Rudolphine tables' were regarded as a standard among astronomy books. In our day (1895), we have access to exact tables in the 'Nautical Almanack' and similar government publications, but it was Kepler who started the practice of publishing star tables.
In 1595, at the age of 26, Johannes Kepler married Barbara Muller, a 23-year old widow with a young daughter. Her first husband had died and left her a fortune, and her second husband had divorced her. This does not seem to have been a happy marriage for Kepler, and Barbara died in 1611, leaving him with 3 young children. Two years later, he decided to re-marry -- but he determined not to make a mistake this time. He kept notes about the methodical manner in which he made his choice. [His method is referred to as the proverbial "marriage problem."] He notes that there were no fewer than eleven options that he was considering. He carefully estimated and recorded the merits and flaws of each candidate, and chose a young woman named Susanna Reuttinger with no fortune who 'won me over with love, humble loyalty, economy of household, diligence, and the love she gave the stepchildren.' This second marriage seems to have been much more successful than the first one. They had seven children [but only three survived into adulthood].
In his middle age, Johannes Kepler's life was distracted by a circumstance that seems unusual to us, but wasn't so unusual in the era of the Inquisition. His mother, Katharina, was arrested and imprisoned for witchcraft in 1620. It was a year before Kepler was able to get her acquitted and released (and she died a year after that).
Around this time, the English ambassador in Venice invited Kepler to come to England, where his work was esteemed and he would do better financially. But Kepler refused to leave his native country. He was 49 years old, and even the inducement of money couldn't persuade him to leave his homeland and go to a strange country where he didn't speak the language.
It had taken a good bit of money to publish the Rudolphine tables. The government of Venice helped with some of this expense, but Kepler was responsible for much of it himself, and the financial burden caused some stress for him. The strain of that anxiety, and of the traveling involved, took a toll on his health. He had never been strong to begin with, and he finally fell ill and died in November, 1630, at the age of 59.
Johannes Kepler didn't have the colorful, eccentric characteristics that made Tycho such a romantic, adventurous figure, but he did have his own striking characteristics. His distinction, though, was of an intellectual kind. His reasoning abilities and his imagination always worked in sync. He was motivated by the most fascinating reflections. Most of his speculations were preposterous and absurd, but every now and then, one of his fancies was actually spot on, and an eternal truth was discovered.
One time I visited the observatory where one of our modern astronomers works, and he showed me a large desk full of photographs he had taken. Most of them were attempts that hadn't been successful. Then he showed me the rare few pictures that had succeeded and revealed important truths. With a jubilant expression I'll never forget, he told me that the contents of the desk were his "chips" -- useless but necessary efforts in the truly successful work. That's the way it is with all great and useful work. Even the most gifted scientist pursues many wrong trails. The greater the man's genius and intelligence, the more numerous these ventures will be, and the vast majority of them will be fruitless. They are the "chips." Kepler had many "chips," and they were fascinating and varied. But every now and then, he made such a sublime discovery, that we marvel at his mind and regard even his "chips" with the greatest respect.
At this point, the natural laws that Kepler had discovered became very important. He had shown how the planets revolve in ellipses around the sun, and the sun is only one of two focus points. He had come to this conclusion by interpreting the observations that Tycho Brahe had recorded. He didn't offer any possible reasons why planets should revolve in ellipses rather than perfect circles, he just measured what he saw and took it at face value. When he compared the movements of two planets with each other, he concluded that the squares of the periodic times in which each planet revolved were proportional to the cubes of their mean distances from the sun. He knew it was true, but he didn't know why. He also discovered the law that explained how the velocity [speed] of a planet changes at different points on its path and demonstrated it by showing how the line drawn from the sun to the planet described equal areas around the sun in equal times. Isaac Newton looked at all of this and set to work. He thought there might be a relationship between these laws and the attraction force from the sun that guides the planets as they orbit. Here is where his sublime mathematical genius came in handy as he worked step by step until he accounted for all of the phenomena.
First, he [Newton] showed that as the planet's orbit makes equal areas in equal times as it goes around the sun, the attraction force that the sun exerts on the planet must be pulling the planet towards itself in a straight line, drawing the planet straight in. Conversely, that attraction force, whatever it was, if it was coming from the very center of the sun, meant that anything revolving around it would have to describe equal areas in equal times. It couldn't do anything else! He still wasn't sure what it was that made the attraction force pull on the planet at different intensities at different points in its orbit, though. Still, this was a tremendous advance in understanding how the universe works.
The next step was to figure out an equation to explain the force within the sun that seemed to draw the planet with a stronger pull at some points. It seemed to depend on the planet's distance from the sun. With his superb mathematical reasoning, Isaac Newton showed that with an elliptical orbit that had the sun as one of the focus points, the intensity of the attraction force has to vary inversely as the square of the planet's distance. With anything other than an inverse square of the distance, there would be no ellipse -- or at least not an ellipse with the sun as one focus point. By using nothing but Kepler's laws, he was able to show that the power which guided the planets was an attraction force coming from the sun, and that the intensity of this attraction force varied with the inverse square of the distance between the two bodies. [View a 4 minute explanation of Newton's Inverse Square Law on YouTube.] Once this was worked out, it was easy to show that Kepler's third law must necessarily follow. [Kepler's third law: the square of the time in which a planet revolves around the sun is proportional to the cube of the average distance of the planet from that body.] If a few planets were revolving around the sun, and each of those planets were equally affected by gravity, then the square of the periodic time that it takes each planet to complete one orbit is proportional to the cube of the greatest diameter in that orbit. [View a 6 minute YouTube video about Kepler's third law, called The Law of Periods, here.]
But these marvelous discoveries were just the starting point for Isaac Newton. From here, he embarked on a whole series of studies which revealed many of the most profound secrets in the mechanical details of astronomy. He realized that it wasn't only large bodies like the sun and the moon that had this attraction force; everything down to the smallest particle must also have it, and be attracting everything else towards it with the same force whose mathematical equation varies inversely as the square of the distance between them. If any two particles -- not just the sun and moon, but anything -- were placed twice as far apart as they were to begin with, then the force trying to pull them towards each other would be reduced by one-fourth. If two masses were attracting each other from ten miles apart, and then they were moved closer so that they were now one mile apart, the attraction between them would increase a hundred times. And this is true throughout all of the universe. But, in some cases, putting the inverse-square equation into use wouldn't be possible without another discovery Newton made. When the particles are two spheres, such as the earth and the moon, we aren't dealing with separate particles but huge masses of matter composed of millions of tiny particles. Every particle in the earth is attracting every particle of the moon with a force that varies inversely as the square of their distance. How is it possible to calculate the different forces of millions of particles? Isaac Newton figured it out. If we think of the earth as consisting of materials that are arranged in layers with different densities, like a shell around a shell, we can imagine that the whole mass of the sphere is concentrated at a single point in the center. If we do the same thing with the moon, we can work our equation from those two points -- it's much simpler to work out a calculation from two single points than millions of different points that represent different particles.
This led to an avalanche of other revelations. It explained what causes the ebb and flow of the tides on our shores. Even the ancients knew that the tides were somehow connected with the moon. It had been recognized that tides were especially high during full moons or new moons, which seemed to mean there was some kind of relationship between them. But beyond noticing that there was a connection, it was a mystery -- until Isaac Newton explained the law of gravitation. Newton showed that the rise and fall of the water was simply a result of the attraction force of the moon pulling on our oceans. He also showed that the sun, too, has an effect on tides. In fact, when both the sun and the moon exerted their pulling force on the earth at the same time, the result was extra-high tides -- what we call "spring tides." Related to this, when the sun causes lower tides at the same time the moon is causing high tides, they cancel each other out and we end up with "neap" tides.
But the most remarkable application of Newton's law of gravitation was connected with discrepancies in the moon's movements. When it revolves around the earth like a satellite, its orbit is mostly guided by the earth's gravitational attraction force. If there were no other celestial bodies in the universe, the center of the moon would travel in an ellipse with the earth's center as a focus point. But nothing in nature is that simple. The sun's presence also exerts an attractive force of its own, complicating the moon's perfect orbit around the earth. The sun pulls on the moon, but it also pulls on the earth. However, it attracts the earth with a stronger force because the earth is so much bigger than the moon. This, of course, adds its own complication to the moon's simple journey around the earth. It can't make a straightforward ellipse around the earth, nor is the earth exactly in the focus. But Isaac Newton wasn't able to mathematically calculate this complex orbit until he created the mathematical tools to work with by inventing calculus.
by Sir Robert S. Ball, 1895 (paraphrased by Leslie Noelani Laurio)
To view the table of contents for the rest of this book, click here.
Johannes Kepler, 1571-1630
Discovered three laws of planetary motion, starting with his observation that planets orbit in ellipses.
Tycho [pronounced TEE ko] Brahe spent his life accumulating vast stores of careful observations of the positions of the heavenly bodies. He didn't live long enough to make any interpretations about them. As he lay on his death-bed, he summoned a brilliant young philosopher to come to his bed-side, and he requested that the young man spare no effort to calculate his observations in order to reveal the secrets of the movements of the heavens and the great truths of nature. This solemn trust was accepted. The young man to whom this sacred trust was passed down was Johannes Kepler.
Johannes Kepler was born December 27, 1571 in Weir in the Duchy of Wurtemberg [now part of the Stuttgart Region in the German state of Baden-Wurttemberg]. He did not have a happy childhood. His father was born into a well-connected family, but was a shiftless and idle adventurer. His mother was ignorant and ill-tempered. His father abandoned the family when Johannes, their oldest son, was 18. At the age of four, Johannes had a severe attack of smallpox that permanently damaged his eyesight and weakened his constitution.
The physical limitations of young Johannes Kepler are what started his interest in the pursuit of knowledge. If he had been able to do ordinary physical work like other boys, his life would most likely have been devoted to manual labour. But although his body was weakened, he soon showed a talent for learning. It was thought that he might be able to work in some religious clerical position, since the Church was the only professional option available for an intellectual career. By the time Johannes Kepler was seventeen years old, he had attained enough education to be admitted to the University at Tubingen.
During his time at the university, Kepler seems to have divided his attention between astronomy and religion. He did equally well at both and couldn't decide which course to follow as a profession. He leaned more towards a career in the Church, but his friends, who knew him better than he knew himself, convinced him to pursue astronomy. So in 1594 he accepted the prestigious position as Professor of Astronomy at the University of Gratz [in present day Austria].
The duties expected from an astronomy professor in the 1500's were much different than they are today. He was expected to use his expertise to predict eclipses, and the movements of the heavenly bodies in general. That doesn't seem unreasonable -- but astronomers were also expected to predict the fates of countries and the destinies of individuals!
In those days, it was almost universally believed that all the heavenly spheres revolved in some mysterious way around the earth, which appeared to be the most central and important body in the universe. It was believed that the changing positions of the sun, the moon, and the stars indicated the course of nations and people. This was the generally accepted notion, so it seemed reasonable to expect a professor of astronomy to decipher celestial clues regarding the fate of man.
Kepler eagerly took on the job, including this part of his task. He diligently studied the rules of astrology that had been handed down over the ages. He sincerely believed in the connection between the position of the stars and the state of human affairs. He even thought he could see confirmation of astrology in the events of his own life.
But astrology wasn't the only delusion that people believed in at the time. It's amazing to us that the most intelligent men only a few centuries ago could have believed such nonsense. For example, geometricians had known for a long time that there were only five regular, solid figures. [Platonic solids: 4-sided pyramid-looking Tetrahedron, 6-sided cube, 8-sided octahedron, 12-sided icosahedron, 20-sided dodecahedron; they make dice in these shapes.]
Coincidentally, there were five planets known at the time: Mercury, Venus, Mars, Jupiter, and Saturn. It seemed obvious to Kepler that the five solids corresponded to the five planets, and in his book 'Mysterium Cosmographicum,' he created a model of the solar system in which the five shapes were set one inside the other to signify the planetary orbits. [Below is an image of his model of the solar system.]
Today we know there are more than five planets, so this theory doesn't hold together. But in Kepler's day, his theory was praised as an intellectual triumph. It brought him esteem and respect -- and may be what brought him to the attention of Tycho Brahe and Galileo.
At the time Kepler lived, science professors were more affected by compelling changes than they are in modern universities. Johannes Kepler was a Protestant, and had been appointed to Gratz University by a Protestant board of directors. But a change had taken place at Gratz and the Protestant professors were expelled. An exception was made for Kepler because of his distinction and prestige, and he was allowed to come back and resume his position. But now his students were gone, so when Tycho Brahe offered him a job at his observatory near Prague, he accepted.
When Tycho Brahe died a couple years later (in 1601), Kepler was appointed to succeed him as imperial mathematician, and gained exclusive access to Tycho's recorded observations. It was Kepler's work with Tycho's observations that brought him to his most important discoveries.
Johannes Kepler was among the first astronomers to view the heavenly bodies through a telescope; in 1610, he was able to use the kind of telescope that led Galileo to his discoveries. But his world-changing achievements didn't come from looking through a telescope; they were deduced from Tycho Brahe's records and measurements of planets made with the instruments he had at that time -- and that didn't include a telescope.
All astronomers working before Kepler assumed that planets must revolve in a perfect circle. If their observations didn't seem to show a perfect circle, they explained it away with Ptolemy's theory that planets were moving in circles within larger circles.
When Kepler looked at Tycho Brahe's records of his observations of Mars, he couldn't work out any mathematical formula that corresponded to circular movements, or circles within circles. No matter how many possibilities he tried, no mathematical calculation lined up with Tycho's actual records. This led Kepler to discover that planets orbit in ellipses, not perfect circles. An ellipse has two focus points rather than one, and he determined that the sun was one of the focus points in the orbit of Mars.
It's easy to draw an ellipse -- stick two pins in a piece of cardboard. Those are the two focus points. Tie a string into a loop that will fit loosely over both focus points. Now use a pencil inside the loop of string to draw a circular shape. [Watch this method of drawing an ellipse on YouTube.] Try this with the pins closer, and then farther apart. The closer the pins are to each other, the more like a circle the shape will be. The farther apart the focus points, the more elongated the ellipse will be. Planets generally orbit in ellipses that are more rounded, like a circle [meaning that the focus points are relatively close together -- which is probably why it looked to astronomers like planets orbited in perfect circles]. But Mars orbits in a more elongated ellipse, and that helped make its ellipse more obvious to Kepler. If Kepler had been working with Tycho Brahe's measurements of a planet like Venus whose orbit is almost a circle, he might never have noticed that the measurements were off.
Today we know that a planet can't possibly orbit in any shape other than an ellipse. But Johannes Kepler didn't know that. Even up to the end of his life, he had no knowledge of any natural cause why planets should follow ellipses. He merely took Tycho Brahe's observations at face value and made a brilliant guess, and his mathematical calculations confirmed his guess.
Johannes Kepler also identified the law that helps to determine the speed a planet is orbiting at different points on its path. [The second of Kepler's three laws of planetary motion has a mathematical equation showing that a planet moves fastest when it's closest to the sun.] Here again, Kepler knew of no reason to think this had to be true; he merely made a brilliant guess based on what he saw. He did know something about gravitation, and he had written that the ebb and flow of tides must be attributed to the moon's attraction to the waters on the earth. But he knew nothing about the discoveries that Isaac Newton would make almost fifty years later -- in which Newton showed that universal gravitation meant that the laws discovered by Kepler's insight were not only accurate, but they were the only possible way planets could move.
In order to fully appreciate the accomplishments of Tycho Brahe and Johannes Kepler, we must reflect on the different ways these two men viewed the system of the heavens. Copernicus [from Poland] had already explained the true system, which is that the sun, and not the earth, is at the center of our planetary system. But in the days of Tycho Brahe, this theory wasn't widely accepted. Tycho himself didn't accept Copernicus's theory. His own observations seemed to show that the earth was at the center [and the religious teaching of the day said that man, as the apex of God's creation, must be at the center]. We may find it surprising that such an accurate observer of the heavens as Tycho Brahe could have rejected Copernicus's theory, which seems so logical to our minds, in favor of a system that seems ridiculous to us. But he carefully observed the positions of the sun, the moon, and the planets, and what he saw and recorded seemed to confirm that they all revolved around the earth.
But Johannes Kepler had the advantage of belonging to the new school. He used Tycho Brahe's observations to develop the Copernican theory that Tycho persistently resisted.
We can make a modern comparison by looking at a more contemporary example. The intellectual revolution produced by Copernicus's new theory about the heavens can be compared to the intellectual revolution produced by Darwin's theories. It radically changed the views held by biologists about life on the earth. It was not universally accepted even by naturalists who had spent their entire lives and built successful careers studying organisms. Professor Richard Owen greatly expanded our knowledge of animals through their fossils, and Darwin even used Owen's observations to confirm his own generalizations. But Owen never accepted Darwin's conclusions. Like Tycho Brahe, he continued compiling his own observations while sticking rigidly to ideas that are considered erroneous by most modern scientists. In this analogy, Kepler is like the modern scientists who interpret the same data upon more currently sanctioned principles.
If we in our modern era read Kepler's works, we are sometimes surprised that the sublime truths he found are mixed with wild errors and absurdities. We must remind ourselves that he wrote in an age when even the foundations of science as we understand them were almost entirely unknown.
Is there any joy more genuine than the bliss that rewards someone who successfully searches after truths of nature? Even the most humble observer of nature will be able to understand Kepler's enthusiastic delight when, after years of toil, the glorious light of truth dawned on him. He had correctly deduced that the number of days a planet takes to travel around the sun must somehow be related to its distance from the sun -- the radius of its orbit, if the orbit were a circle.
Here again, Kepler was in the dark about accurate scientific principles. He had a few hazy notions, but they were fairly bizarre compared with what we know now. His notions were mixed up with vague ideas about the five Platonic solids corresponding to planets, and the five elements -- earth, air, fire, water, and 'cosmos.' Even more lamentable, his reasoning was mixed up with astrology. But the influence of these faulty impressions didn't prevent him from skillfully calculating the speed and distance of orbits around the sun.
It was easy to demonstrate that, the farther away from the sun a planet was, the longer it would take to travel around it. It would seem obvious that the time would be directly proportional to the distance. But there's more to the calculation than just time and distance. Something was still missing. After many attempts, Kepler was overjoyed to discover the solution: the square of the time in which a planet revolves around the sun was proportional to the cube of the average distance of the planet from that body. [This is Kepler's third law of planetary motion.] This equation made it possible to determine how far a planet was from the sun by measuring how long it took to revolve around the sun.
An example of how Kepler's brilliance is intermingled with some wild speculations is illustrated in the book that included the law just referred to. The explanation and equation of this law connecting distances of planets with their orbit times was mixed with comments about the musical properties of the planets. There was an old Greek theory still believed at the time that the planets hummed with tones that were inaudible to human ears, making profound and harmonic "music of the spheres" to be enjoyed by their Creator.
Kepler was the first astronomer to predict the transit of a planet in front of the sun. In 1629, he announced that Mercury would make a transit across the sun on November 7, 1631, and Venus on December 6, 1631. He died before this happened, but the French scientist Pierre Gassendi witnessed the transit of Mercury as predicted. The Venus transit happened on cue, but no one recorded it.
In addition to the discoveries already mentioned, Johannes Kepler also published a catalogue of stars and planetary tables that he called the 'Rudolphine tables' in memory of Rudolf II, Holy Roman Emperor. This catalogue enabled star gazers to find planets in the sky with greater accuracy than before.
Kepler was not primarily an astronomical observer. His role was dealing with Tycho Brahe's observations. By studying them and comparing results, he was able to work out movements of heavenly bodies. Thus, Tycho supplied the raw data, and Kepler's genius turned that data into a usable form. For more than a century, the 'Rudolphine tables' were regarded as a standard among astronomy books. In our day (1895), we have access to exact tables in the 'Nautical Almanack' and similar government publications, but it was Kepler who started the practice of publishing star tables.
In 1595, at the age of 26, Johannes Kepler married Barbara Muller, a 23-year old widow with a young daughter. Her first husband had died and left her a fortune, and her second husband had divorced her. This does not seem to have been a happy marriage for Kepler, and Barbara died in 1611, leaving him with 3 young children. Two years later, he decided to re-marry -- but he determined not to make a mistake this time. He kept notes about the methodical manner in which he made his choice. [His method is referred to as the proverbial "marriage problem."] He notes that there were no fewer than eleven options that he was considering. He carefully estimated and recorded the merits and flaws of each candidate, and chose a young woman named Susanna Reuttinger with no fortune who 'won me over with love, humble loyalty, economy of household, diligence, and the love she gave the stepchildren.' This second marriage seems to have been much more successful than the first one. They had seven children [but only three survived into adulthood].
In his middle age, Johannes Kepler's life was distracted by a circumstance that seems unusual to us, but wasn't so unusual in the era of the Inquisition. His mother, Katharina, was arrested and imprisoned for witchcraft in 1620. It was a year before Kepler was able to get her acquitted and released (and she died a year after that).
Around this time, the English ambassador in Venice invited Kepler to come to England, where his work was esteemed and he would do better financially. But Kepler refused to leave his native country. He was 49 years old, and even the inducement of money couldn't persuade him to leave his homeland and go to a strange country where he didn't speak the language.
It had taken a good bit of money to publish the Rudolphine tables. The government of Venice helped with some of this expense, but Kepler was responsible for much of it himself, and the financial burden caused some stress for him. The strain of that anxiety, and of the traveling involved, took a toll on his health. He had never been strong to begin with, and he finally fell ill and died in November, 1630, at the age of 59.
Johannes Kepler didn't have the colorful, eccentric characteristics that made Tycho such a romantic, adventurous figure, but he did have his own striking characteristics. His distinction, though, was of an intellectual kind. His reasoning abilities and his imagination always worked in sync. He was motivated by the most fascinating reflections. Most of his speculations were preposterous and absurd, but every now and then, one of his fancies was actually spot on, and an eternal truth was discovered.
One time I visited the observatory where one of our modern astronomers works, and he showed me a large desk full of photographs he had taken. Most of them were attempts that hadn't been successful. Then he showed me the rare few pictures that had succeeded and revealed important truths. With a jubilant expression I'll never forget, he told me that the contents of the desk were his "chips" -- useless but necessary efforts in the truly successful work. That's the way it is with all great and useful work. Even the most gifted scientist pursues many wrong trails. The greater the man's genius and intelligence, the more numerous these ventures will be, and the vast majority of them will be fruitless. They are the "chips." Kepler had many "chips," and they were fascinating and varied. But every now and then, he made such a sublime discovery, that we marvel at his mind and regard even his "chips" with the greatest respect.
At this point, the natural laws that Kepler had discovered became very important. He had shown how the planets revolve in ellipses around the sun, and the sun is only one of two focus points. He had come to this conclusion by interpreting the observations that Tycho Brahe had recorded. He didn't offer any possible reasons why planets should revolve in ellipses rather than perfect circles, he just measured what he saw and took it at face value. When he compared the movements of two planets with each other, he concluded that the squares of the periodic times in which each planet revolved were proportional to the cubes of their mean distances from the sun. He knew it was true, but he didn't know why. He also discovered the law that explained how the velocity [speed] of a planet changes at different points on its path and demonstrated it by showing how the line drawn from the sun to the planet described equal areas around the sun in equal times. Isaac Newton looked at all of this and set to work. He thought there might be a relationship between these laws and the attraction force from the sun that guides the planets as they orbit. Here is where his sublime mathematical genius came in handy as he worked step by step until he accounted for all of the phenomena.
First, he [Newton] showed that as the planet's orbit makes equal areas in equal times as it goes around the sun, the attraction force that the sun exerts on the planet must be pulling the planet towards itself in a straight line, drawing the planet straight in. Conversely, that attraction force, whatever it was, if it was coming from the very center of the sun, meant that anything revolving around it would have to describe equal areas in equal times. It couldn't do anything else! He still wasn't sure what it was that made the attraction force pull on the planet at different intensities at different points in its orbit, though. Still, this was a tremendous advance in understanding how the universe works.
The next step was to figure out an equation to explain the force within the sun that seemed to draw the planet with a stronger pull at some points. It seemed to depend on the planet's distance from the sun. With his superb mathematical reasoning, Isaac Newton showed that with an elliptical orbit that had the sun as one of the focus points, the intensity of the attraction force has to vary inversely as the square of the planet's distance. With anything other than an inverse square of the distance, there would be no ellipse -- or at least not an ellipse with the sun as one focus point. By using nothing but Kepler's laws, he was able to show that the power which guided the planets was an attraction force coming from the sun, and that the intensity of this attraction force varied with the inverse square of the distance between the two bodies. [View a 4 minute explanation of Newton's Inverse Square Law on YouTube.] Once this was worked out, it was easy to show that Kepler's third law must necessarily follow. [Kepler's third law: the square of the time in which a planet revolves around the sun is proportional to the cube of the average distance of the planet from that body.] If a few planets were revolving around the sun, and each of those planets were equally affected by gravity, then the square of the periodic time that it takes each planet to complete one orbit is proportional to the cube of the greatest diameter in that orbit. [View a 6 minute YouTube video about Kepler's third law, called The Law of Periods, here.]
But these marvelous discoveries were just the starting point for Isaac Newton. From here, he embarked on a whole series of studies which revealed many of the most profound secrets in the mechanical details of astronomy. He realized that it wasn't only large bodies like the sun and the moon that had this attraction force; everything down to the smallest particle must also have it, and be attracting everything else towards it with the same force whose mathematical equation varies inversely as the square of the distance between them. If any two particles -- not just the sun and moon, but anything -- were placed twice as far apart as they were to begin with, then the force trying to pull them towards each other would be reduced by one-fourth. If two masses were attracting each other from ten miles apart, and then they were moved closer so that they were now one mile apart, the attraction between them would increase a hundred times. And this is true throughout all of the universe. But, in some cases, putting the inverse-square equation into use wouldn't be possible without another discovery Newton made. When the particles are two spheres, such as the earth and the moon, we aren't dealing with separate particles but huge masses of matter composed of millions of tiny particles. Every particle in the earth is attracting every particle of the moon with a force that varies inversely as the square of their distance. How is it possible to calculate the different forces of millions of particles? Isaac Newton figured it out. If we think of the earth as consisting of materials that are arranged in layers with different densities, like a shell around a shell, we can imagine that the whole mass of the sphere is concentrated at a single point in the center. If we do the same thing with the moon, we can work our equation from those two points -- it's much simpler to work out a calculation from two single points than millions of different points that represent different particles.
This led to an avalanche of other revelations. It explained what causes the ebb and flow of the tides on our shores. Even the ancients knew that the tides were somehow connected with the moon. It had been recognized that tides were especially high during full moons or new moons, which seemed to mean there was some kind of relationship between them. But beyond noticing that there was a connection, it was a mystery -- until Isaac Newton explained the law of gravitation. Newton showed that the rise and fall of the water was simply a result of the attraction force of the moon pulling on our oceans. He also showed that the sun, too, has an effect on tides. In fact, when both the sun and the moon exerted their pulling force on the earth at the same time, the result was extra-high tides -- what we call "spring tides." Related to this, when the sun causes lower tides at the same time the moon is causing high tides, they cancel each other out and we end up with "neap" tides.
But the most remarkable application of Newton's law of gravitation was connected with discrepancies in the moon's movements. When it revolves around the earth like a satellite, its orbit is mostly guided by the earth's gravitational attraction force. If there were no other celestial bodies in the universe, the center of the moon would travel in an ellipse with the earth's center as a focus point. But nothing in nature is that simple. The sun's presence also exerts an attractive force of its own, complicating the moon's perfect orbit around the earth. The sun pulls on the moon, but it also pulls on the earth. However, it attracts the earth with a stronger force because the earth is so much bigger than the moon. This, of course, adds its own complication to the moon's simple journey around the earth. It can't make a straightforward ellipse around the earth, nor is the earth exactly in the focus. But Isaac Newton wasn't able to mathematically calculate this complex orbit until he created the mathematical tools to work with by inventing calculus.
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