Data Bases • Custom Term Papers • Free Term Papers • Free Research Papers • Free Essays • Free Book Reports • Plagiarism? Links • Top 100 Term Paper Sites • Top 25 Essay Sites • Top 50 Essay Sites • Search 97,000 Papers @ DirectEssays.com • Search 101,000 Papers @ ExampleEssays.com • Search 90,000 Papers @ MegaEssays.com Free Essays • Term Paper Sites • Chuck III's Free Essays • Free College Essays • TermPaperSites.com • My Term Papers • Get Free Essays • Essay World • Planet Papers	Search Lots of Essays Back to Subjects - Mathematics Keplers Laws Keplers Laws In today’s world, we have very advanced technology. There have been many new technological and medical advancements as we entered the new century. The Internet allows us to shop, talk, and find valuable information on very scarce topics, and even check stocks with a simple click of a button. Medical advancements had recently been discovered on “The Human Genome Projects,” the first gene was mapped and within a short period of time we will have mapped out all the genes in a human chromosome. This is absolutely amazing because we will now be able to reveal the many causes of serious deadly diseases. Throughout the years, we have gained the technology to send astronauts into space to gather new information about our universe. However, without all of this technology that we have today, a man was able to discover a great deal of information about our universe. This man’s name was Johannes Kepler. Johannes Kepler was born on December 27, 1571 in the village of Leonberg outside the small town of Weil der Stadt, in Swabia. His father was a mercenary soldier and his mother the daughter of an innkeeper. Johannes was their first child out of seven children. His father left home for the last time when Johannes was five, and is believed to have died in the war of the Netherlands. As a child, Johannes lived with his mother in his grandfather’s inn. When Kepler was a child he went to a local school and then at a nearby seminary. Later on in his education, he enrolled at the University of Tubingen, which is now a bastion of Lutheran orthodoxy. During his lifetime he formulated three laws of planetary motion. However, he did not set out at first to formulate these three laws, instead he was originally working on understanding the orbit of the planet mars. Kepler was an assistant to Tycho Brahe, another astronomer / mathematician. Brahe did not trust Kepler and worried that Kepler would surpass him and become more well known than him. So Brahe assigned him the job of understanding the orbit of Mars because it gave Brahe much difficulty, and would keep Kepler occupied while Brahe worked on his theory of the solar system. Brahe also kept other data from Kepler hoping that he would be the one to discover the orbits of the planets before Kepler did. In 1601, when Brahe died Kepler received all of Brahe’s data. Whether he obtained this data legally or not is still in debate today, however, it is fortunate that he obtained this data. (Silverberg, 160) When Kepler first began his work on the orbit of Mars he was under the assumption, as many scientists were, that the planetary orbits were circular, and that the Sun was at the center of the orbits. This type of system is called a heliocentric system. Also at this time only six planets were known. When Kepler obtained Brahe’s data he discovered that the orbits were not perfect circles, but instead were ellipses that were only slightly flattened. The reason nobody else realized this was because the orbits were so slightly elliptical that extensive investigation and data would be needed to show this. It also turned out that the reason the orbit of Mars was very difficult to understand was because its orbit was more eccentric than the other planets that Kepler and Brahe had data about. To understand a lot of Kepler’s work you must first understand some simple ideas about an ellipse. An ellipse is defined, as the locus of all points, whose sum of distances from two fixed points, also called the foci, is a constant. Below is an example: (http://csep10.phys.utk.edu/astr161/lect/history/kepler.html) In the example the Xs represent the foci. No matter where the point is on the path of the ellipse, the sum of the distances from the foci will be the same. The distance between the foci is what gives the ellipse its shape, or eccentricity. As the foci move closer together the ellipse becomes more like a circle, and as they foci move farther apart the ellipse becomes flatter. The eccentricity of an ellipse is measured on a scale from zero to one, where an ellipse with an eccentricity of zero has the foci occupying the same region, also known as a circle. An ellipse with an eccentricity of one would be a straight line. Therefore, all ellipses have an eccentricity between zero and one. An ellipse also has two axises, the longer one which is called the major axis and the shorter one which is called the minor axis. The semi-major axis is half the size of the major axis and is called the radius of the ellipse or the size of the ellipse. (http://csep10.phys.utk.edu/astr161/lect/history/kepler.html) Once Kepler obtained Brahe’s data and knew that the orbits were not perfect circles, but instead were ellipses that were only slightly flattened, he was able to formulate his three laws of planetary motion. In order they are: 1. The orbits of the planets are ellipses with the Sun at one focus of the ellipse 2. The line joining the planet to the Sun sweeps out in equal areas in equal times as the planet travels around the ellipse 3. The ratio of the squares of the revolutionary periods for the planets is equal to the ratio of the cubes of their semi-major axis The first law was probably most simplest and easiest to understand, however, it was also the most important because up until that time all theories and models had the Sun at the center with the orbits of the planets being perfect circles. Kepler instead showed that the orbits were ellipses and the Sun was not at the center, but instead it was one of the foci of the ellipse while the other foci was usually just an empty region. Knowing that the Sun is one of the foci also meant that the distance between the planets and Sun is constantly changing. This law is demonstrated the following diagram: (http://csep10.phys.utk.edu/astr161/lect/history/kepler.html) Kepler’s first law showed the path that a planet would take during its orbit around the Sun. The second law shows that a planet goes through its elliptical orbit with constantly changing angular speed. Thus, when a planet is closest to the Sun it moves faster and when it is farther away from the Sun it moves slower. The point where the planet is closest to the Sun is called the perihelion, and the point where the planet is farthest away from the Sun is called the aphelion. This law is demonstrated in the diagram below: (http://csep10.phys.utk.edu/astr161/lect/history/kepler.html) The shaded region between the perihelion and the Sun, and the shaded region between the aphelion and the Sun is equal in area. This shows that in the same amount of time, the amount of area between the Sun and two different points on the orbit is the same. Lets infer that we could mark two thirty-day periods, such as May 1 to May 31 and October 1 to October 31, the area from the Sun to each of the points would be equal. The third law solved the problem of calculating the distance between the Sun and a given planet. The reason this was a problem is because the distance is always changing. The law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. For example Mercury is the innermost planet and takes 88 days to orbit the Sun, and Pluto the outermost planet, whose orbit has a longer radius, takes 248 years to orbit the Sun. The following diagram represents Kepler’s third law: (http://csep10.phys.utk.edu/astr161/lect/history/kepler.html) In the equation above P is the period of revolution for a planet and R is the distance between that planet and the Sun, or the semi-major axis. The period of revolution is measured in earth years and the distance from the Sun is measured in astronomical units. The equation can be simplified to: The equation can either be solved for the revolutionary period when given the semi-major axis: P (years) = R(A.U.)^3/2 or can be solved for the distance of the planet from the Sun, when given the revolutionary period: R (A.U.) = P (years)^ 2/3 Here is an example of using this formula. Mars takes 1.88 earth years to orbit the Sun. Therefore, by Kepler’s third law: R = 1.52 Which is the measured average distance from Mars to the Sun. Given Pluto’s observed average separation from the Sun is 39.44 astronomical units then: P = 248 years Which is the observed orbital period for Pluto. These two examples show that Kepler’s third law is correct and can accurately calculate either the revolutionary period of a planet or the planet’s distance from the Sun. (http://csep10.phys.utk.edu/astr161/lect/history/kepler.html) I have made two of my own make believe planets to show how Kepler's third law works. Planet Tara takes 1.65 earth years to orbit the sun. So we use the equation: R = 1.40 This calculates to be the measured average distance from Tara to the Sun. My next planet, Miss Healy, has an observed average separation of 23.68 astronomical units from the Sun. So we use the equation: P = 115 years This is the observed orbital period for Miss Healy. On my solar system diagram, I have written down the time it takes for each planet to revolve around the sun and I have also converted kilometers into astronomical units. I was given the distance of each planet from the sun in kilometers. Mr. Hayes told me that 1 astronomical unit is equal to 150 million kilometers. So I then converted the kilometers into astronomical units by dividing the distance from the sun into 150 million kilometers. Then following with Kepler’s third law P = R^3/2 to find the orbital period for each planet. For example, the distance from Jupiter to the sun is 778 million kilometers. 7780000 / 1500000 = 5.1866 I rounded 5.1866 to 5.2 astronomical units. I then used Kepler’s third law to figure out the orbital period of Jupiter. Jupiter is 5.2 astronomical units away from the sun and its orbital period is 11.8578 years. Kepler’s laws were important for many reasons. They allowed astronomers to track and predict the motions of the planets more accurately. They also showed that not only objects on earth follow laws, but also objects in the universe do to. This showed scientists that it was possible to study and understand the universe. (Fleisher, 20) After Kepler passed away Sir Issac Newton an astronomer / mathematician / physicist, improved upon much of Kepler’s work and also made some of his own discoveries. All of Kepler’s laws can also be used dealing with satellites because they are in orbit just like the planets. Much of the work done in modern astronomy is still based on Kepler’s three laws of planetary motion. He was very advanced for his time, and if he had not made such great discoveries the human race would have been far behind in the exploration of outer space. Bibliography: Works Cited 1) http://csep10.phys.utk.edu/astr161/lect/history/kepler.html. Johannes Kepler: The Laws of Planetary Motion 2) http://chabut.cosc.org/~arf/Kepler.Newton/intro.html. Johannes Kepler 3) http://es.rice.edu/Es/humsoc/Galileo/People/Kepler.html. Johannes Kepler 4) Fleisher, Paul. Secrets of the Universe. McMillan : New York, 1987. 5) Silverberg, Robert. Four Men Who Changed the Universe. GP Putnam’s Sons : New York, 1968. 6) Baker, Robert and Fredrick, Laurence. Astronomy Ninth Edition. Litton Educational Publishing: New York, 1971. Word Count: 1885
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