Mathematics and Society - OS X Lion Server

(Kepler was not content with his firsttwo discoveries. He looked for a rulethat would relate the average distanceof a planet from the Sun to the time theplanet takes to complete one orbit.From Tycho Brahe's observations,Kepler was able to find the time for acomplete orbit (orbital period) of eachplanet. From a scale model (It's A GreatSystem) he was able to determine theaverage distances of the planets from theSun in astronomical units. The table tothe right shows the orbital periods andaverage distances of the planets that wereknown in the 1600's.Suppose a planet has an average distance fromthe Sun of 4 a.u. Use the graph to make a guessabout its orbital period.It took Kepler seventeen years before he discoveredthe relationship between the orbitalperiod of a planet and its distance from the Sun.Cube the average distance (4 a.u.) of the imaginaryplanet above. Now square its orbital period.The two numbers are the same!Pick another point on the curve. Suppose aplanet has an average distance of 5 a.u. Thegraph suggests that its period is between 11 and12 years. Let's try Kepler's discovery on thesenumbers:Cube the average distance: 5 x 5 x 5 = 125PLAt-JE.T PERIOD (YEARS) DISTANCE (A.U.)ME.RCURY 0.2."+\ 0.38'1VENUS 0.40\5 0.723EARTH 1.00 1.00MARS 1.88 \.52JUPITE.R 11.9 5.2.0SATURtJ 2.9.5 9.55Do you see any patterns in the table? Is there a relationship between the distancesof the planets from the Sun and their orbital periods?The facts in the table were used to plot apoint on the graph for each planet.Notice the regularity about them. They seemto lie on a smooth curve. It appears that if aplanet has an orbit of a certain size, then itsperiod will be a certain definite number of years.Square 11: 11 x 11 = 121A number a little larger thanwith the graph?Square 12:11 will give&),'>;) 131--1--1--I--HH-I­>- l~ I-+-+~I-+--t.l-+-!­:2- "I-+-+I-I-+-+-+-!­2::> ,. I-+-+I-I-+-H--+-!­(jJ91-+--+--+-I+-+-+--+-~C>281-+--+--+-I1-+-+--+-­::>~7f--j-+-+-fI;-t--j-+-~.I-+-+-t;,,+-+---j-1-­8 51--+-+--'1-++++_t:!'+I--+--f-14-+-+-+-I1­0. 31--+-+-++-+-+-11­...J 21--+-f-l-++-+-+-I1-~~\o "'--'-~"-'3'-,>t--:5~..I.G~~'--:'812 x 12 = 144AVERAGE DISTAJJCE F'ROM5UI\J llll A.U.us 125 when squared.Does this agree(Kepler's third discovery--the square of the period in years equals the cube of theaverage distance in a.u.--seems to work! Unlike "Bode's Law," Kepler's discovery does (work in all cases; Isaac Newton proved this.110IDEA FROM:The Universe in Motion, Book 2, The University of Illinois Astronomy Prograrr.