Article by Dave Scott
Kepler
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In November, the body of Tycho Brahe, nobleman, mathematician, alchemist and astronomer was exhumed for the second time. It is perhaps the mark of a man's legend, notoriety, influence, and enigma that his restful peace (and that of his wife) should be interrupted more than once by historians seeking to determine his cause of death. For whilst it could possibly be that he contracted a bladder infection for neglecting to relieve himself at a social function, it is also possible that he was murdered. Such disparate eventualities are these that it is certainly worth the investigation, and this is obviously part of his enduring enigma.
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If this were insufficient to demonstrate just what a colourful character he was, we could also consider the fact that in his younger days, Tycho lost part of his nose in a duel, and it was replaced, repaired, augmented with a 'realistic' metal prosthetic. Perhaps we should consider this to be rhino-metallurgy, but strangely, this false nose was not found following his original exhumation in the early 20th century, and thus far, nobody is sure from what metal the prosthetic was forged. At the time of writing, we are yet to be told any of the findings of this most recent exhibition of historical morbid curiosity, but I urge you to keep an eye out for whatever conclusions are drawn from the study of his remains, if by the time of publication, we have not already been informed. If we have, look it up.
Tycho's interest in astronomy developed over some time, but what initially brought him into contact with such academic interests was his kidnapping (my words) by his uncle when he was two years old (Tycho, not his uncle, for those who appreciate a dangling participle). His uncle (Jorgan) without the knowledge, nor strangely, the objection of Tycho's parents took the young Tycho (or if you will, young tyke) so that he could become a scholar, and that, he did. He became interested in astronomy, and was particularly impressed by the correct prediction of a solar eclipse. His appetite duly whetted, he began to seek an understanding of this most fascinating of sciences, and it is as well for us that he did, for he essentially led the way for modern astronomy, making at that time, the most accurate and comprehensive study of the heavens, and all before the invention of the telescope.
In 1572, Tycho noticed a bright star in the constellation of Cassiopeia. Though he was not the first person to observe this phenomenon, it would nonetheless become known as Tycho's Supernova, or Tycho's Star. Visible for some weeks in daylight, and visible for sixteen months in all, this supernova marked an important stage in our understanding of the universe, as it disproved the hitherto accepted notion that the heavens were unchanging. So surprised was Tycho by this sight that he began to 'doubt the faith of my own eyes'.
In 1577, Tycho studied, and as with all his observations, meticulously recorded, the appearance of one of the most spectacular comets ever witnessed by man. At the age of five, a young Johannes Kepler, together with his mother, viewed the same comet from atop the hill overlooking his village of Leonberg. Far from enraptured by the sight, and perhaps more taken with the rare instance of motherly love for having taken him to see the comet, it is notable here for juxtaposing in our story the two men, yet to meet, one still a boy, who together and apart, with reluctance, animosity, genius and curiosity, forever changed our understanding of the universe.
Unlike most modern scientists, who cannot accept the coexistence of science and god, and who believe the two subjects to be irreconcilable, Kepler was a man of god. He had originally intended to serve the church as a pastor, and considered the universe to be the image of god. However, following the death of a mathematics teacher from a school in the south of Austria, and a subsequent request to Kepler's university of Tubingen, the theological faculty offered him as a replacement.
In addition to his teaching duties, Kepler was also appointed the position of district mathematician, for which he was required to produce astrological predictions, and about which he later wrote that he disliked 'nourishing the superstition of fatheads'. Tycho too, had responsibility for astrological prediction in Prague. I suppose it is a sign of the times that an unenlightened populace, preoccupied with religion and superstition (no offence intended to anyone of faith reading this) would expect the foremost scholars of astronomy, with their superior knowledge of the stars and planets, to produce astrological predictions; for who better to read and interpret the heavens?
Kepler accepted his assignment as a mathematics teacher reluctantly, seeing it as a somewhat lowly position, removed from his greater interests of theology and philosophy, and personally feeling he did not possess any great aptitude for mathematics. For this reason, he decided to elevate his studies to a more philosophical level. He studied the heliocentric model of the solar system posited by Copernicus, noting that it was by no means comprehensive. In the Copernican system, the planets were located at specific distances from the sun, but Kepler felt that these distances were somewhat arbitrary, questioning why god would construct the solar system in this way and not another.
Whilst teaching before his class in 1595, and having drawn a diagram of an equilateral triangle within a circle, he realised that if he drew another circle inside the triangle, the ratio between the sizes of the large and small circle were similar to the ratio between the orbits of Saturn and Jupiter. This led Kepler to believe that the key to planetary orbits lay in geometry. Drawing a square inside the inner circle, and another circle inside the square, he surmised that this circle was representative of the orbit of Mars, relative to the larger circles and the orbits of Jupiter and Saturn. Kepler decided that god must have used geometry when constructing the universe.
He soon decided that solid geometry would be more appropriate than plain geometry, the universe being three-dimensional. Therefore, he would have to use spheres rather than circles to represent the orbits of the planets, and expanding upon his two-dimensional discovery, Kepler used regular solids to fit within and around the orbital spheres. As there are only five regular solids, this matched very conveniently, and therefore to him, somewhat conclusively with the fact that there were only six planets, as the five regular solids would fit within the orbital spheres of the planets. Of course, this hypothesis does hinge on the notion that there are only six planets, that no more planets would ever be discovered, that god had used geometry to stick them where they are, and subsequently, that god exists. Kepler, rather immodestly, regarded his discoveries as 'stupendous miracles of god'.
This model, and the Copernican system failed to address the issue of orbital duration, and this is something on which Kepler had pondered for many years. The outer planets have longer orbital durations than the inner planets, and whilst this is reasonable given that the further a planet is from the sun, the longer its path around the sun will be, Kepler believed that the sun provided the power to move the planets around it. Therefore, the closer a planet is to the sun, the more quickly it will move in its path around the sun. Based on these issues, he devised a formula to explain the orbital periods of the planets. Whilst this formula was ultimately incorrect, and indeed, Kepler realised this, the results were conducive to his three-dimensional geometric model.
Kepler published his three-dimensional system in a book called 'Mysterium Cosmographicum', or rather, this is a truncated name. The full name of this book is so verbose as to beggar belief (and that's saying something coming from me). The book was published in 1597, and he sent it to various astronomers for their opinions. Among those interested in his book was Nicolas Reimer, who goes by the name Ursus, and who sought to use Kepler in a dispute with Tycho Brahe, as both Ursus and Tycho claimed to have conceived a new system in which the planets orbit the sun, but the sun orbits the earth. Kepler found himself caught up in the dispute between Ursus and Tycho, as the former had printed a complimentary letter from Kepler in his own book 'On Astronomical Hypotheses'. In the same book, Ursus said some very unpleasant things about Tycho. Tycho praised Kepler's model, but suggested that Copernicus's distances for the planets were inaccurate, and invited Kepler to use his own, more accurate data.
Kepler arrived in Prague in January 1600, and he met Tycho for the first time on 4th February. It is possible, had they never met, or certainly, had Kepler never been granted access to Tycho's data, that Kepler's greatest accomplishments would never have been realised. The synergy of two men, one with the data from a lifetime's punctilious observations, with foresight but without the insight, and the other with the creativity, genius and inspiration but without the relevant data coming together, enabled the world finally to understand the nature of the solar system. However, things between the two men were uneasy to say the least. Tycho was distrusting of Kepler, not least because of the letter printed in the book by Ursus, who was essentially a sworn enemy of Tycho, but also because he was very protective of his data. He had spent years amassing it and if it were to yield any cosmological revelations, he would prefer to conceive them himself, rather than allowing it to benefit someone else.
Tycho allowed Kepler very little information, but primarily, allowed Kepler access to his data for Mars. Tycho knew that this data was particularly complicated from a Copernican perspective, and he choose to give Kepler the data on Mars specifically because he thought it would be too difficult to rationalise. He hoped that Kepler would fail to accomplish anything using this data.
Conversely, it is the data on Mars that enabled Kepler to form the first of his laws. Though Kepler had been convinced that planetary orbits would be circular, and as much as he tried to reconcile this theory with Tycho's data for Mars, he painfully, and somewhat reluctantly had to accept that this was not the case. Indeed, Kepler referred to this as his 'warfare with Mars'. It led to the formulation of his first two laws of planetary motion. Strangely though, the first law came second and the second law came first, but as I don't wish to cause any confusion, I shall deal with them in the order by which they are known, rather than in the order by which they were conceived.
Kepler's first law is simplicity itself, but it is important to remember that understanding the solution to a problem is substantially easier than conceiving the solution to a problem. This, together with sophisticated communication, is one of the things that has enabled the human race to progress in the manner that it has. We owe so much to the visionaries of history for figuring out how to do things that we are subsequently taught, and which we understand, so easily, but this should never detract from the effort and genius involved in the conception of such scientific discoveries. It is because of such things that we do not have to figure out for ourselves Pythagoras' Theorem, trigonometry, buoyancy, gravity, spectroscopy, the Doppler Effect, and every other item of human knowledge that we are taught, and that we are able to take for granted. Instead, those who came before have saved time for those who came after so that rather than wasting time either literally or metaphorically reinventing the wheel, we can instead pass on and use knowledge and understanding of such concepts to build upon them rather than rediscover them, and the progression of our species owes much to this ability. That one genius may spend their life making such discoveries subsequently affords another genius an advantageous starting point to use that knowledge to discover more. In this way, the human race has climbed upon the shoulders of its forebears for millennia, so that now we stand so tall we can reach the sky.
Kepler's First Law:
The planets move in elliptical orbits with the sun at one focus.
The OED defines the word 'ellipse' as follows:
A regular oval shape, traced by a point moving in a plane so that the sum of its distances from two other points (the foci) is constant, or resulting when a cone is cut by an oblique plane which does not intersect the base.
But for the sake of simplicity, it's a squashed circle. Had Tycho not tried to flummox him, Kepler might never have been in a position to conceive this law, but the first law (which remember, came to him second) owes much to his second law, which came to him first... Mars spent too long along the sides of its orbit, and it is this that led him to the conclusion that orbits are oval. However, it took him an entire year to determine that the specific type of oval was elliptical. The amount by which an ellipse is squashed, is referred to as its eccentricity, and the orbits of the planets are not very eccentric, but of those on which Tycho had data, Mars was the most eccentric. In an epiphany, it came to him, such that he wrote it was 'as if I were aroused from a dream and saw a new light'. The elliptical orbit solved an accuracy problem with his second law.
Kepler's Second Law:
The line connecting the planet and the sun sweeps out equal areas in equal times.
Now given that a planet's orbit is elliptical, and that the sun resides at one of the foci, and not at the centre, for the planet to sweep out a given area at the furthest distance from the sun, where the area would be narrow and long, but in the same length of time, when near the sun to sweep out an equal area that is broad and short, the distance travelled around that section of the elliptical orbit must be greater in the second instance than in the first for those areas to be equal. The result, or rather the cause, of such behaviour of course, is that the planet must travel more quickly the closer it is to the sun, which means that the sun's gravitational influence is stronger the closer an object is to it, which is of course relevant to Keper's Third Law. We also see this behaviour with comets, which move slowly when in the outer reaches of the solar system, but travel far more quickly when they come close to the sun. It is also the basis of the slingshot effect used by man to increase the velocity of the objects we launch to send them to wherever they are destined.
Kepler's Third Law:
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
This is a bit of a mouthful, and at first glance seems less simple than the previous two laws, but what it describes is the relationship between a planet's distance from the sun, and the length of time it takes to orbit the sun. The result is that the further a planet is from the sun, the longer it takes to orbit the sun, for two reasons. The first is that the further a planet is from the sun, the longer its orbit is, and the second is that the further a planet is from the sun, the more slowly it moves (hence the relevance to the second law).
Such was the profundity of Kepler's laws, so important were they of their time, and so famous did he become that the English poet John Donne wrote of him that 'no new thing should be done in heaven without his knowledge'.
Kepler's life was as colourful as Tycho's, and there is much of interest to read about him, but I have omitted here such details, lacking as they are in relevance to his work. However, I would be remiss were I not to pique your interest by telling you that as a schoolchild, he kept a list of his enemies, that his mother was tried for witchcraft, that on looking for a suitable second wife, he referred to all potential suitors (eleven of them) by number rather than name, and that he is also considered a potential suspect in the murder of Tycho (if it turns out he was murdered), as indeed is the Danish king.
His works did not begin or end with his three famous laws, and he published many other theses on various subjects, all of interest, but with varying success. Nonetheless, his prominence was assured, and it is hardly surprising therefore that NASA's mission to search for extra-solar planets with properties similar to earth was named for him. Launched in 2009, to date, Kepler has identified eight new planets.
Dave Scott -May 2011
Tycho's interest in astronomy developed over some time, but what initially brought him into contact with such academic interests was his kidnapping (my words) by his uncle when he was two years old (Tycho, not his uncle, for those who appreciate a dangling participle). His uncle (Jorgan) without the knowledge, nor strangely, the objection of Tycho's parents took the young Tycho (or if you will, young tyke) so that he could become a scholar, and that, he did. He became interested in astronomy, and was particularly impressed by the correct prediction of a solar eclipse. His appetite duly whetted, he began to seek an understanding of this most fascinating of sciences, and it is as well for us that he did, for he essentially led the way for modern astronomy, making at that time, the most accurate and comprehensive study of the heavens, and all before the invention of the telescope.
In 1572, Tycho noticed a bright star in the constellation of Cassiopeia. Though he was not the first person to observe this phenomenon, it would nonetheless become known as Tycho's Supernova, or Tycho's Star. Visible for some weeks in daylight, and visible for sixteen months in all, this supernova marked an important stage in our understanding of the universe, as it disproved the hitherto accepted notion that the heavens were unchanging. So surprised was Tycho by this sight that he began to 'doubt the faith of my own eyes'.
In 1577, Tycho studied, and as with all his observations, meticulously recorded, the appearance of one of the most spectacular comets ever witnessed by man. At the age of five, a young Johannes Kepler, together with his mother, viewed the same comet from atop the hill overlooking his village of Leonberg. Far from enraptured by the sight, and perhaps more taken with the rare instance of motherly love for having taken him to see the comet, it is notable here for juxtaposing in our story the two men, yet to meet, one still a boy, who together and apart, with reluctance, animosity, genius and curiosity, forever changed our understanding of the universe.
Unlike most modern scientists, who cannot accept the coexistence of science and god, and who believe the two subjects to be irreconcilable, Kepler was a man of god. He had originally intended to serve the church as a pastor, and considered the universe to be the image of god. However, following the death of a mathematics teacher from a school in the south of Austria, and a subsequent request to Kepler's university of Tubingen, the theological faculty offered him as a replacement.
In addition to his teaching duties, Kepler was also appointed the position of district mathematician, for which he was required to produce astrological predictions, and about which he later wrote that he disliked 'nourishing the superstition of fatheads'. Tycho too, had responsibility for astrological prediction in Prague. I suppose it is a sign of the times that an unenlightened populace, preoccupied with religion and superstition (no offence intended to anyone of faith reading this) would expect the foremost scholars of astronomy, with their superior knowledge of the stars and planets, to produce astrological predictions; for who better to read and interpret the heavens?
Kepler accepted his assignment as a mathematics teacher reluctantly, seeing it as a somewhat lowly position, removed from his greater interests of theology and philosophy, and personally feeling he did not possess any great aptitude for mathematics. For this reason, he decided to elevate his studies to a more philosophical level. He studied the heliocentric model of the solar system posited by Copernicus, noting that it was by no means comprehensive. In the Copernican system, the planets were located at specific distances from the sun, but Kepler felt that these distances were somewhat arbitrary, questioning why god would construct the solar system in this way and not another.
Whilst teaching before his class in 1595, and having drawn a diagram of an equilateral triangle within a circle, he realised that if he drew another circle inside the triangle, the ratio between the sizes of the large and small circle were similar to the ratio between the orbits of Saturn and Jupiter. This led Kepler to believe that the key to planetary orbits lay in geometry. Drawing a square inside the inner circle, and another circle inside the square, he surmised that this circle was representative of the orbit of Mars, relative to the larger circles and the orbits of Jupiter and Saturn. Kepler decided that god must have used geometry when constructing the universe.
He soon decided that solid geometry would be more appropriate than plain geometry, the universe being three-dimensional. Therefore, he would have to use spheres rather than circles to represent the orbits of the planets, and expanding upon his two-dimensional discovery, Kepler used regular solids to fit within and around the orbital spheres. As there are only five regular solids, this matched very conveniently, and therefore to him, somewhat conclusively with the fact that there were only six planets, as the five regular solids would fit within the orbital spheres of the planets. Of course, this hypothesis does hinge on the notion that there are only six planets, that no more planets would ever be discovered, that god had used geometry to stick them where they are, and subsequently, that god exists. Kepler, rather immodestly, regarded his discoveries as 'stupendous miracles of god'.
This model, and the Copernican system failed to address the issue of orbital duration, and this is something on which Kepler had pondered for many years. The outer planets have longer orbital durations than the inner planets, and whilst this is reasonable given that the further a planet is from the sun, the longer its path around the sun will be, Kepler believed that the sun provided the power to move the planets around it. Therefore, the closer a planet is to the sun, the more quickly it will move in its path around the sun. Based on these issues, he devised a formula to explain the orbital periods of the planets. Whilst this formula was ultimately incorrect, and indeed, Kepler realised this, the results were conducive to his three-dimensional geometric model.
Kepler published his three-dimensional system in a book called 'Mysterium Cosmographicum', or rather, this is a truncated name. The full name of this book is so verbose as to beggar belief (and that's saying something coming from me). The book was published in 1597, and he sent it to various astronomers for their opinions. Among those interested in his book was Nicolas Reimer, who goes by the name Ursus, and who sought to use Kepler in a dispute with Tycho Brahe, as both Ursus and Tycho claimed to have conceived a new system in which the planets orbit the sun, but the sun orbits the earth. Kepler found himself caught up in the dispute between Ursus and Tycho, as the former had printed a complimentary letter from Kepler in his own book 'On Astronomical Hypotheses'. In the same book, Ursus said some very unpleasant things about Tycho. Tycho praised Kepler's model, but suggested that Copernicus's distances for the planets were inaccurate, and invited Kepler to use his own, more accurate data.
Kepler arrived in Prague in January 1600, and he met Tycho for the first time on 4th February. It is possible, had they never met, or certainly, had Kepler never been granted access to Tycho's data, that Kepler's greatest accomplishments would never have been realised. The synergy of two men, one with the data from a lifetime's punctilious observations, with foresight but without the insight, and the other with the creativity, genius and inspiration but without the relevant data coming together, enabled the world finally to understand the nature of the solar system. However, things between the two men were uneasy to say the least. Tycho was distrusting of Kepler, not least because of the letter printed in the book by Ursus, who was essentially a sworn enemy of Tycho, but also because he was very protective of his data. He had spent years amassing it and if it were to yield any cosmological revelations, he would prefer to conceive them himself, rather than allowing it to benefit someone else.
Tycho allowed Kepler very little information, but primarily, allowed Kepler access to his data for Mars. Tycho knew that this data was particularly complicated from a Copernican perspective, and he choose to give Kepler the data on Mars specifically because he thought it would be too difficult to rationalise. He hoped that Kepler would fail to accomplish anything using this data.
Conversely, it is the data on Mars that enabled Kepler to form the first of his laws. Though Kepler had been convinced that planetary orbits would be circular, and as much as he tried to reconcile this theory with Tycho's data for Mars, he painfully, and somewhat reluctantly had to accept that this was not the case. Indeed, Kepler referred to this as his 'warfare with Mars'. It led to the formulation of his first two laws of planetary motion. Strangely though, the first law came second and the second law came first, but as I don't wish to cause any confusion, I shall deal with them in the order by which they are known, rather than in the order by which they were conceived.
Kepler's first law is simplicity itself, but it is important to remember that understanding the solution to a problem is substantially easier than conceiving the solution to a problem. This, together with sophisticated communication, is one of the things that has enabled the human race to progress in the manner that it has. We owe so much to the visionaries of history for figuring out how to do things that we are subsequently taught, and which we understand, so easily, but this should never detract from the effort and genius involved in the conception of such scientific discoveries. It is because of such things that we do not have to figure out for ourselves Pythagoras' Theorem, trigonometry, buoyancy, gravity, spectroscopy, the Doppler Effect, and every other item of human knowledge that we are taught, and that we are able to take for granted. Instead, those who came before have saved time for those who came after so that rather than wasting time either literally or metaphorically reinventing the wheel, we can instead pass on and use knowledge and understanding of such concepts to build upon them rather than rediscover them, and the progression of our species owes much to this ability. That one genius may spend their life making such discoveries subsequently affords another genius an advantageous starting point to use that knowledge to discover more. In this way, the human race has climbed upon the shoulders of its forebears for millennia, so that now we stand so tall we can reach the sky.
Kepler's First Law:
The planets move in elliptical orbits with the sun at one focus.
The OED defines the word 'ellipse' as follows:
A regular oval shape, traced by a point moving in a plane so that the sum of its distances from two other points (the foci) is constant, or resulting when a cone is cut by an oblique plane which does not intersect the base.
But for the sake of simplicity, it's a squashed circle. Had Tycho not tried to flummox him, Kepler might never have been in a position to conceive this law, but the first law (which remember, came to him second) owes much to his second law, which came to him first... Mars spent too long along the sides of its orbit, and it is this that led him to the conclusion that orbits are oval. However, it took him an entire year to determine that the specific type of oval was elliptical. The amount by which an ellipse is squashed, is referred to as its eccentricity, and the orbits of the planets are not very eccentric, but of those on which Tycho had data, Mars was the most eccentric. In an epiphany, it came to him, such that he wrote it was 'as if I were aroused from a dream and saw a new light'. The elliptical orbit solved an accuracy problem with his second law.
Kepler's Second Law:
The line connecting the planet and the sun sweeps out equal areas in equal times.
Now given that a planet's orbit is elliptical, and that the sun resides at one of the foci, and not at the centre, for the planet to sweep out a given area at the furthest distance from the sun, where the area would be narrow and long, but in the same length of time, when near the sun to sweep out an equal area that is broad and short, the distance travelled around that section of the elliptical orbit must be greater in the second instance than in the first for those areas to be equal. The result, or rather the cause, of such behaviour of course, is that the planet must travel more quickly the closer it is to the sun, which means that the sun's gravitational influence is stronger the closer an object is to it, which is of course relevant to Keper's Third Law. We also see this behaviour with comets, which move slowly when in the outer reaches of the solar system, but travel far more quickly when they come close to the sun. It is also the basis of the slingshot effect used by man to increase the velocity of the objects we launch to send them to wherever they are destined.
Kepler's Third Law:
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
This is a bit of a mouthful, and at first glance seems less simple than the previous two laws, but what it describes is the relationship between a planet's distance from the sun, and the length of time it takes to orbit the sun. The result is that the further a planet is from the sun, the longer it takes to orbit the sun, for two reasons. The first is that the further a planet is from the sun, the longer its orbit is, and the second is that the further a planet is from the sun, the more slowly it moves (hence the relevance to the second law).
Such was the profundity of Kepler's laws, so important were they of their time, and so famous did he become that the English poet John Donne wrote of him that 'no new thing should be done in heaven without his knowledge'.
Kepler's life was as colourful as Tycho's, and there is much of interest to read about him, but I have omitted here such details, lacking as they are in relevance to his work. However, I would be remiss were I not to pique your interest by telling you that as a schoolchild, he kept a list of his enemies, that his mother was tried for witchcraft, that on looking for a suitable second wife, he referred to all potential suitors (eleven of them) by number rather than name, and that he is also considered a potential suspect in the murder of Tycho (if it turns out he was murdered), as indeed is the Danish king.
His works did not begin or end with his three famous laws, and he published many other theses on various subjects, all of interest, but with varying success. Nonetheless, his prominence was assured, and it is hardly surprising therefore that NASA's mission to search for extra-solar planets with properties similar to earth was named for him. Launched in 2009, to date, Kepler has identified eight new planets.
Dave Scott -May 2011
