Kepler's Laws of Planetary Motion


 Archibald Morris Clarke
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1 Writing Assignment Essay n Kepler s Laws. Yu have been prvided tw shrt articles n Kepler s Three Laws f Planetary Mtin. Yu are t first read the articles t better understand what these laws are, what they mean, and why they are imprtant t scientists. Then, yu are t fill in the graphic rganizer t help yu t cmpse and rganize yur thughts fr the essay. Finally, yu will cmpse yur essay using the graphic rganizer as yur guide. Yur best writing skills are required and yu are t use the infrmatin frm bth articles fr yur essay. Yur essay will be five paragraphs in length and must be wrdprcessed. Yu will als turn in yur graphic rganizer with this assignment. This assignment is due Friday, Nvember 9 th. Kepler's Laws f Planetary Mtin In the sixteenth century, mst peple believed in the ideas f the ancient astrnmer Ptlemy, that the planets, Mn, and Sun all rbited arund the Earth. Then in 1543, Niclaus Cpernicus prpsed the idea that the planets and the Earth rbited arund the Sun. Hwever, Cpernicus' new thery was n better at predicting the psitins f the planets in the sky than the lder, Earthcentered thery. There was still smething missing... Half a century later, Jhannes Kepler sught t refine the Cpernican system and truly understand hw the planets mve arund the Sun. He studied bservatins f Mars recrded by his mentr, Tych Brahe. Rather than trying t frce the data t supprt a predetermined view f the Universe, Kepler used Tych's bservatins t guide the creatin f his theries. This was a radical departure frm the thught prcesses f his era, and it is a signal f the beginning f ur mdern scientific age. In 1609, Kepler published his first and secnd laws f planetary mtin, The Law f Ellipses and The EqualAreas Law. Ten years later he published a third law, The Harmnic Law. He had succeeded in using a scientific methd t create a simple, elegant, and accurate mdel t describe the mtin f planets arund the Sun Kepler's First Law: The Law f Ellipses
2 Previus theries f the Slar System, including thse f Ptlemy and Cpernicus, believed that the rbits f the planets were perfect circles. Kepler was unable, hwever, t fit Tych's bservatins with circular rbits. He rejected the ancient idea f circular rbits had discvered that: The rbits f the planets are ellipses with the Sun at ne f the fci. This is nw called Kepler's First Law r The Law f Ellipses. What is an ellipse? Glad yu asked. An ellipse is a clsed, curved shape that is defined by tw fci. An ellipse is a like a flattened circle. In fact, if bth f the fci f an ellipse are at the same pint, an ellipse becmes a circle! If yu think abut it, the relatinship between an ellipse and a circle is similar t the relatinship between a rectangle and a square. An ellipse has tw axes. The lng ne is called the majr axis, and the shrt ne is called the minr axis. Astrnmers will ften use the term "semimajr axis". That's just half the length f the majr axis! The shape f an ellipse is measured by its eccentricity. The "flatter" the ellipse, the greate the eccentricity. A circle, fr example, has an eccentricity f zer since bth fci are at the center. As the ellipse becmes flatter and flatter, the fci get farther frm the center, and the eccentricity will apprach, but never equal, ne. Kepler's Secnd Law: The EqualAreas Law In additin t determining that the rbits f the planets arund the Sun were ellipses, Kepler als nticed that their speeds varied thrughut their jurnies Kepler nted that the planets seemed t mve fastest when they were at their clsest pint t the Sun (called perihelin) and slwest when they were at their farthest pint frm the Sun (called aphelin). Using sme rather brilliant insights f gemetry, Kepler discvered that: The line that cnnects the planet t the Sun sweeps ut equal areas in equal times. This is nw knwn as Kepler's Secnd Law r The Equal Areas Law. The mtin this law describes als tells us that the average distance frm a planet t the Sun is equal t the length f the semimajr axis. That's why astrnmers lve the term s much! Kepler's Third Law: The Harmnic Law After determining his first tw Laws f Planetary Mtin, Kepler cntinued t study the rbits f the planets. Ten years later, he discvered a relatin between the time f a planet's rbit nad its distance frm the Sun: The squares f the rbital perids f the planets arund the Sun are prprtinal t the cubes f the rbital semimajr axes. What des this mean? This means that if yu knw either hw much time a planet's rbit arund the Sun takes yu can easily knw it's average distance frm the Sun, r viceversa! Nw yu will ften see Kepler's Third Law written like this: P 2 =a 3 Where P is the rbital perid in Earth years and a is the length f the semimajr axis (average distance frm the Sun) in Astrnmical Units.
3 The Physics Classrm Kepler's Three Laws In the early 1600s, Jhannes Kepler prpsed three laws f planetary mtin. Kepler was able t summarize the carefully cllected data f his mentr  Tych Brahe  with three statements that described the mtin f planets in a suncentered slar system. Kepler's effrts t explain the underlying reasns fr such mtins are n lnger accepted; nnetheless, the actual laws themselves are still cnsidered an accurate descriptin f the mtin f any planet and any satellite. Kepler's three laws f planetary mtin can be described as fllws: The path f the planets abut the sun is elliptical in shape, with the center f the sun being lcated at ne fcus. (The Law f Ellipses) An imaginary line drawn frm the center f the sun t the center f the planet will sweep ut equal areas in equal intervals f time. (The Law f Equal Areas) The rati f the squares f the perids f any tw planets is equal t the rati f the cubes f their average distances frm the sun. (The Law f Harmnies) Kepler's first law  smetimes referred t as the law f ellipses  explains that planets are rbiting the sun in a path described as an ellipse. An ellipse can easily be cnstructed using a pencil, tw tacks, a string, a sheet f paper and a piece f cardbard. Tack the sheet f paper t the cardbard using the tw tacks. Then tie the string int a lp and wrap the lp arund the tw tacks. Take yur pencil and pull the string until the pencil and tw tacks make a triangle (see diagram at the right). Then begin t trace ut a path with the pencil, keeping the string wrapped tightly arund the tacks. The resulting shape will be an ellipse. An ellipse is a special curve in which the sum f the distances frm every pint n the curve t tw ther pints is a cnstant. The tw ther pints (represented here by the tack lcatins) are knwn as the fci f the ellipse. The clser tgether that these pints are, the mre clsely that the ellipse resembles the shape f a circle. In fact, a circle is the special case f an ellipse in which the tw fci are at the same lcatin. Kepler's first law is rather simple  all planets rbit the sun in a path that resembles an ellipse, with the sun being lcated at ne f the fci f that ellipse. Kepler's secnd law  smetimes referred t as the law f equal areas  describes the speed at which any given planet will mve while rbiting the sun. The speed at which any planet mves thrugh space is cnstantly changing. A planet mves fastest when it is clsest t the sun and slwest when it is furthest frm the sun. Yet, if an imaginary line were drawn frm the center f the planet t the center f the sun, that line wuld sweep ut the same area in equal perids f time. Fr instance, if an imaginary line were drawn frm the earth t the sun, then the area swept ut by the line in every 31day mnth wuld be the same. This is depicted in the diagram belw. As can be bserved in the diagram, the areas frmed when the earth is clsest t the sun can be apprximated as a wide but shrt triangle; whereas the areas frmed when the earth is farthest frm the sun can be apprximated as a narrw but lng triangle. These areas are the same size. Since the base f these triangles are shrtest when the earth is farthest frm the sun, the earth wuld have t be mving mre slwly in rder fr this imaginary area t be the same size as when the earth is clsest t the sun.
4 Kepler's third law  smetimes referred t as the law f harmnies  cmpares the rbital perid and radius f rbit f a planet t thse f ther planets. Unlike Kepler's first and secnd laws that describe the mtin characteristics f a single planet, the third law makes a cmparisn between the mtin characteristics f different planets. The cmparisn being made is that the rati f the squares f the perids t the cubes f their average distances frm the sun is the same fr every ne f the planets. As an illustratin, cnsider the rbital perid and average distance frm sun (rbital radius) fr Earth and mars as given in the table belw. Planet Perid Average T 2 /R 3 (s) Dist. (m) (s 2 /m 3 ) Earth x 10 7 s x x Mars 5.93 x 10 7 s x x Observe that the T 2 /R 3 rati is the same fr Earth as it is fr mars. In fact, if the same T 2 /R 3 rati is cmputed fr the ther planets, it can be fund that this rati is nearly the same value fr all the planets (see table belw). Amazingly, every planet has the same T 2 /R 3 rati. Planet Perid Ave. T 2 /R 3 (yr) Dist. (au) (yr 2 /au 3 ) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Plut (NOTE: The average distance value is given in astrnmical units where 1 a.u. is equal t the distance frm the earth t the sun x m. The rbital perid is given in units f earthyears where 1 earth year is the time required fr the earth t rbit the sun x 10 7 secnds. ) Kepler's third law prvides an accurate descriptin f the perid and distance fr a planet's rbits abut the sun. Additinally, the same law that describes the T 2 /R 3 rati fr the planets' rbits abut the sun als accurately describes the T 2 /R 3 rati fr any satellite (whether a mn r a manmade satellite) abut any planet.
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7 Written cmmunicatin (50 pints) Grading Rubric Kepler Essay Organizatin Inadequate (10 pints): There appears t be n rganizatin f the essay s cntents. N graphic rganizer used. Needs Imprvement (15 pints): Organizatin f the essay is difficult t fllw, due t inadequate transitins and/r rambling frmat. Graphic rganizer used sparingly. Adequate (20 pints): The essay can be easily fllwed. A cmbinatin f the fllwing is apparent: Basic transitins are used; a structured frmat is used. Graphic rganizer is used fully. Prfessinal quality (25 pints): The essay can be easily fllwed. A cmbinatin f the fllwing is apparent: Effective transitins are used; a prfessinal frmat is used. Graphic rganizer is used fully. Mechanics and grammar Inadequate (10 pints): Sentences and paragraphs are difficult t read and understand due t pr grammar r mechanics Cntent (50 pints) Needs imprvement (15 pints): The essay cntains numerus grammatical and mechanical errrs. Adequate (20 pints): The essay cntains minimal grammatical r mechanical errrs. Prfessinal quality (25 pints): The essay is clear and cncise and cntains n grammatical r mechanical errrs. Crrectness f facts Inadequate (10 pints): Mst facts are wrng. Needs imprvement (15 pints): Sme facts are wrng. Adequate (20 pints): Technical details are generally crrect. Prfessinal quality (25 pints): All facts are crrect, and the technical explanatin is bth cncise and cmplete. Cmpleteness Inadequate (10 pints): Did nt address sme f the questins. Needs imprvement (15 pints): Addressed the questins, but prvided few details. Adequate (20 pints): Address the questins, but left ut sme details. Prfessinal quality (25 pints): Addressed all questins cmpletely.
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