So apparently I'm in a models-of-the-solar-system mood recently, so tonight I am going to tell you about Tycho Brahe's geocentric model, generally referred to as the Tychonic model of the solar system.
Tycho Brahe lived around the same time as Galileo. During this period, the Catholic Church was busy with the Counter Reformation, an attempt to strengthen its authority which was under assault from the Protestant Reformation throughout Europe. This involved an increased sensitivity to threats to Church authority, and among the perceived threats was heliocentrism, which implied that humans did not hold a special place at the center of creation and was in apparent contradiction with several Bible passages.
Geocentrism, on the other hand, placed Earth at the very center of creation but did not provide very accurate predictions of the planets' positions in the long term. By the fifteenth century, the Ptolemaic model (the geocentric model in use, developed by Claudius Ptolemy in the first or second century) had become incredibly complicated. In its original form, it had everything moving around the earth in perfectly circular orbits. However, this failed to account for some aspects of the planets' movement, especially retrograde motion--the phenomenon when a planet appears to stop moving, travel in the opposite direction for a bit, and then resume its course. People compensated by adding epicycles: circular deviations from the circular orbit, so that the planets would literally travel backward occasionally, but would do so because they were following a path determined by a system of circles. The end result looks something like this:
Over time, as the Ptolemaic system's predictions deviated from the observed paths of the planets, more and more epicycles were added on top of each other. Copernicus (known for developing what was essentially the heliocentric system we use today) was driven to look for alternatives because he felt that God would not have made his system so messy. The heliocentric system solved the problem of retrograde motion by showing that it was an illusion created by our own motion relative to the other planets. This video demonstrates how that works:
However, as I mentioned above, the Church was not ready to adopt a heliocentric model. Tycho Brahe developed his model as a compromise between the two systems. According to the Tychonic system, the Earth is at the center of the solar system, with the sun and moon orbiting it, while all the other planets orbit the sun. This still allows for retrograde motion without resorting to epicycles and provided more accurate predictions than the Ptolemaic system.
In fact, the Tychonic system is mathematically indistinguishable from the Copernican system Galileo was teaching (and in fact it is the system he was fighting, rather than the Ptolemaic system as is commonly believed), in which the moon orbited the Earth while the Earth and other planets orbited the sun. The difference is simply where you place your reference point.
To explain: We know today that nothing in our universe is truly stationary because everything is in motion relative to everything else. However, picking some stationary reference point simplifies our diagrams and calculations. Any point will do, provided we make the appropriate adjustments around it. When you are sitting in a car traveling at sixty miles per hour and tossing a baseball up and down, you don't need to throw it forward at sixty miles per hour to avoid breaking your own nose; as far as you are concerned, the car isn't moving, and those trees outside are simply whizzing by. It is the same idea here. If I wanted to, I could propose a model of the solar system with Titan at its center, Jupiter orbiting Titan, the sun and Jupiter's other moons orbiting Jupiter, and all other planets orbiting the sun. If you were in a high school physics class solving a word problem set on Titan, you might even find it convenient to do this.
The main scientific reason for using a heliocentric system today is that we conventionally say that lighter objects orbit heavier objects, as the heavier objects are less influenced by the mutual gravitational attraction. But if you know a fifth grader who likes to stump the teacher in class, you should tell him about Tycho Brahe's geocentric model of the solar system.
Monday, October 24, 2011
Sunday, October 23, 2011
Kepler's Model of the Solar System
Disclaimer: I know that this model is not terribly accurate, but I still think it is a pretty neat idea and so I am going to tell you about it. If you get annoyed by people getting excited about outdated attempts at science, you probably should not read this.
Before we begin talking about Kepler, we are going to have to take a trip back to ancient Greece and talk about geometry. You may remember discussing regular polygons in your high school geometry class: those two-dimensional figures which had all of their sides the same length and all of their angles equal (the math word for same size is "congruent"). There are infinitely many such figures, as you can make one for any number of sides greater than or equal to three. What if we move up a dimension?
In three dimensions, a regular solid is a figure which has congruent regular polygons as its faces (all the same shape and the same size) and all of the vertices the same (meaning the same number of polygons meeting at any vertex) and convex (meaning they all poke out rather than in). You might be tempted to think that since there are infinitely many regular polygons to choose from, there are also infinitely many regular solids to be made, but in fact there are only five.
To see why, we must start with the fact that the sum of the angles meeting at a vertex must be less than 360 degrees, because 360 degrees would make a flat plane (so that you don't get a point and thus don't really have a vertex) and any more than that would have to have some part poking in the wrong way in order to make everything fit, so that the vertex is not convex. Moreover, in order to form a vertex you must have at least three faces meeting, because if you only have two they are lying flat on top of each other and you no longer have a three-dimensional object.
Armed with this information, we can start trying to construct some regular solids.
First suppose that all of our faces are equilateral triangles. Every angle of an equilateral triangle measures sixty degrees. If you have three meeting, the sum is 180 degrees, which is less than 360, so we can get a solid this way. This gives a regular tetrahedron (solid with four faces).
If we have four equilateral triangles meet, the angles sum to 240 degrees, which is still less than 360, so we get another regular solid. This time it is the octahedron (solid with eight faces).
With five, the angles will sum to 300 degrees, which is still less than 360, so we get yet another
regular solid: the icosahedron (solid with twenty faces).
If we try to have a vertex with six equilateral triangles, the angles sum to 360 degrees, which doesn't work, so we now move on to regular solids with square faces. Each angle of a square measures ninety degrees, so if we put three together we get 270 degrees. This is less than 360, so it yields a regular solid: the cube.
If we put four squares together, the angles at the vertex sum to 360, which again means we have run out of solids. We now move on to solids with regular pentagons for faces. Each angle of a regular pentagon measures 108 degrees, so three of them together makes 324 degrees. Once again, this is less than 360 so we get a regular solid. This time it is the dodecahedron (solid with 12 faces).
Moving up to four pentagons, the angles sum to 432 degrees, which is greater than 360, so we won't get any more regular solids from the pentagon. Let's move on to hexagons. A regular hexagon has angles measuring 120 degrees each. Three of them together will measure 360 degrees, so this doesn't work. Since a regular polygon with more sides will have even larger angles, none of them will allow us to construct another regular solid. We have found them all.
A quick review: there are exactly five regular solids. They are the regular tetrahedron, octahedron, icosahedron, cube, and dodecahedron, having four, eight, twenty, six, and twelve faces, respectively.
These solids are often referred to as the platonic solids after the Greek philosopher Plato. This is because Plato believed that the classical elements were composed of these solid figures. According to Plato, water was composed or icosahedra (this is the plural of icosahedron) because this is the regular solid closest to the sphere, and water flows smoothly as though its base particles are rolling smoothly over each other. Earth was made from cubes, since these shapes are solid and strong and would thus make things which are difficult to break, such as rocks. Air was made from octahedra because these shapes are light and will float easily. Fire was made from tetrahedra because they are the sharpest and fire hurts if you get to close. At this point we've exhausted the classical elements found on earth, but we haven't done anything with the dodecahedron. Plato postulated that everything in the heavens was made from a mysterious fifth element or "quintessence" whose base particles were shaped like dodecahedra. This fifth element is more perfect than anything found on earth, which is why celestial bodies glow and are shaped (as far as Plato could tell) like perfect spheres.
During the Renaissance in Europe, the old Greek classics were rediscovered and held in high esteem as relics of a past golden age of science and learning in Europe. The Catholic Church adopted Plato's classification of matter and fit it into Christian theology, using it as evidence of Earth's uniqueness in the universe, an explanation of human imperfection, and so on.
In the fifteenth and sixteenth centuries, with Europeans sailing all over the world, it became increasingly evident that the existing geocentric model of the solar system was not good enough. Over time, the deviation between the predicted and actual positions of the planets grew, which presented problems for sailors who needed accurate sky charts for navigation. In response, people began positing heliocentric models of the solar system, which were found to be simpler and more accurate for several reasons which I won't go into here.
Generally when you hear the phrase "heliocentric model of the solar system" the two names that come to mind are Copernicus and Galileo, because their model comes closest to the one we use today. But there were many people developing models, most of which have fallen out of fashion. Among them were the German Johannes Kepler.
Kepler set out to answer two questions: why are there six planets in our solar system instead of some other number (six was the amount they were aware of at that time, since nothing beyond Saturn can be seen by the naked eye), and what determines the size of their orbits. He began by assuming a heliocentric model with circular orbits for the planets. He then found that if you let the radius of Saturn's orbit be the radius of a sphere and inscribe a cube in that sphere then another sphere in that cube, the radius of the inner sphere is the radius of Jupiter's orbit. If you then inscribe a tetrahedron and another sphere, you will get the radius of Mars's orbit. Continuing down with a dodecahedron, sphere, icosahedron, sphere, octahedron, and sphere, you will get the radius of the orbit for Earth, Venus, and Mercury to within reasonable experimental error based on his data (before you get too excited, we now know that some of them are off by as much as ten percent, which is quite a lot on the scale of planetary orbits). If that's a lot to take in, the picture looks something like this:
If you're concerned that this is a bit random, I should mention he did come up with detailed reasons for this configuration as opposed to some other ordering of the solids, based upon Plato's classification of matter. For example, Earth's place inside the dodecahedron is significant because the dodecahedron is supposed to be the shape of the fifth element from which the heavens are composed, and Earth is the only planet in the solar system with life and the one on which we are found. Thus, our special place in God's creation is marked off by the one shape that is used in building the perfection of the rest of the rest of the universe.
So, with one model Kepler had answered both his questions: there are six planets because there are five platonic solids, leaving room for exactly six spheres in this nested system, and the sizes of their orbits are determined by the nesting. His results have not withstood modern scrutiny, but he did succeed in building a single model that elegantly answered his questions and offered further information about the system he was studying (through the perceived properties of the platonic solids and their order in the nesting), which is still one of the main goals of science today.
Before we begin talking about Kepler, we are going to have to take a trip back to ancient Greece and talk about geometry. You may remember discussing regular polygons in your high school geometry class: those two-dimensional figures which had all of their sides the same length and all of their angles equal (the math word for same size is "congruent"). There are infinitely many such figures, as you can make one for any number of sides greater than or equal to three. What if we move up a dimension?
In three dimensions, a regular solid is a figure which has congruent regular polygons as its faces (all the same shape and the same size) and all of the vertices the same (meaning the same number of polygons meeting at any vertex) and convex (meaning they all poke out rather than in). You might be tempted to think that since there are infinitely many regular polygons to choose from, there are also infinitely many regular solids to be made, but in fact there are only five.
To see why, we must start with the fact that the sum of the angles meeting at a vertex must be less than 360 degrees, because 360 degrees would make a flat plane (so that you don't get a point and thus don't really have a vertex) and any more than that would have to have some part poking in the wrong way in order to make everything fit, so that the vertex is not convex. Moreover, in order to form a vertex you must have at least three faces meeting, because if you only have two they are lying flat on top of each other and you no longer have a three-dimensional object.
Armed with this information, we can start trying to construct some regular solids.
First suppose that all of our faces are equilateral triangles. Every angle of an equilateral triangle measures sixty degrees. If you have three meeting, the sum is 180 degrees, which is less than 360, so we can get a solid this way. This gives a regular tetrahedron (solid with four faces).
regular solid: the icosahedron (solid with twenty faces).
A quick review: there are exactly five regular solids. They are the regular tetrahedron, octahedron, icosahedron, cube, and dodecahedron, having four, eight, twenty, six, and twelve faces, respectively.
These solids are often referred to as the platonic solids after the Greek philosopher Plato. This is because Plato believed that the classical elements were composed of these solid figures. According to Plato, water was composed or icosahedra (this is the plural of icosahedron) because this is the regular solid closest to the sphere, and water flows smoothly as though its base particles are rolling smoothly over each other. Earth was made from cubes, since these shapes are solid and strong and would thus make things which are difficult to break, such as rocks. Air was made from octahedra because these shapes are light and will float easily. Fire was made from tetrahedra because they are the sharpest and fire hurts if you get to close. At this point we've exhausted the classical elements found on earth, but we haven't done anything with the dodecahedron. Plato postulated that everything in the heavens was made from a mysterious fifth element or "quintessence" whose base particles were shaped like dodecahedra. This fifth element is more perfect than anything found on earth, which is why celestial bodies glow and are shaped (as far as Plato could tell) like perfect spheres.
During the Renaissance in Europe, the old Greek classics were rediscovered and held in high esteem as relics of a past golden age of science and learning in Europe. The Catholic Church adopted Plato's classification of matter and fit it into Christian theology, using it as evidence of Earth's uniqueness in the universe, an explanation of human imperfection, and so on.
In the fifteenth and sixteenth centuries, with Europeans sailing all over the world, it became increasingly evident that the existing geocentric model of the solar system was not good enough. Over time, the deviation between the predicted and actual positions of the planets grew, which presented problems for sailors who needed accurate sky charts for navigation. In response, people began positing heliocentric models of the solar system, which were found to be simpler and more accurate for several reasons which I won't go into here.
Generally when you hear the phrase "heliocentric model of the solar system" the two names that come to mind are Copernicus and Galileo, because their model comes closest to the one we use today. But there were many people developing models, most of which have fallen out of fashion. Among them were the German Johannes Kepler.
Kepler set out to answer two questions: why are there six planets in our solar system instead of some other number (six was the amount they were aware of at that time, since nothing beyond Saturn can be seen by the naked eye), and what determines the size of their orbits. He began by assuming a heliocentric model with circular orbits for the planets. He then found that if you let the radius of Saturn's orbit be the radius of a sphere and inscribe a cube in that sphere then another sphere in that cube, the radius of the inner sphere is the radius of Jupiter's orbit. If you then inscribe a tetrahedron and another sphere, you will get the radius of Mars's orbit. Continuing down with a dodecahedron, sphere, icosahedron, sphere, octahedron, and sphere, you will get the radius of the orbit for Earth, Venus, and Mercury to within reasonable experimental error based on his data (before you get too excited, we now know that some of them are off by as much as ten percent, which is quite a lot on the scale of planetary orbits). If that's a lot to take in, the picture looks something like this:
So, with one model Kepler had answered both his questions: there are six planets because there are five platonic solids, leaving room for exactly six spheres in this nested system, and the sizes of their orbits are determined by the nesting. His results have not withstood modern scrutiny, but he did succeed in building a single model that elegantly answered his questions and offered further information about the system he was studying (through the perceived properties of the platonic solids and their order in the nesting), which is still one of the main goals of science today.
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