Rowan Introduction to Astronomy Lab 4 / An Introduction to Kepler’s 3 Laws Name: A&P‘é/‘ %d’vo loS Score: AObjectives 1 i A4 tring-and-pencil method,” students will draw ellipses and determine their eccentricities. > Reproducing ellipses via the s f Jupiter's moons, students will test Keple's third law. > By measuring the orbits of five 0 [> By using characteristics of Pluto’s orbit, students will confirm Kepler’s second law. _AProcedure Kepler's three laws are simply a mathematical way of describing motions of objects that orbit a large central mass, such as the planets which orbit around the Sun or the moons which orbit around Jupiter. This lab explores each of Kepler's three laws. [45 pts: you must draw and submit two ellipses to receive credit for this part] AKepler's First Law hat orbiting objects travel in elliptical paths with the central mass at one focus. In this section, you will get Kepler’s first law states t acquainted with ellipses by sketching one yourself. Note that the string loops around the pins. Do NOT stick the pins into the string. Make sure there is enough slack in your loop of string. C e Major {perihelion 2ol - 2 . ] 5 rt . = = B ¢ Semi-major axis X - = i ‘E\"~ D o - o -2 R To draw an elipse loop srng around thumb facks at i the distance between the center of the 2 4 each focus and stretch string fight with a pencil while ellipse (orbit) and one of the foci, F, which N 4 the penci around the tacks. The Sun is at one typically is the Sun or a planet (like Earth) e el focus. that is being orbited. Steps to draw an ellipse (a) Su:tntwlo mak;soor su;)h pi.ns and a piece of string. On your paper, place the two tacks or pins a small distance apart. Place your g loop around the pins. Be sure to leave some slack in the string. Use a piece of cardboard to help secure the pins or iacks (b) Using the string as a guide (i.c., place the pencil inside the stri B vour sketoh {0 this lab weport, g loop and pull the loop taut), draw an ellipse. [10 pg] i d) MUST come from your drawing; guesses are NOT accept: Now i R 7 (€) Now measure and write down the distance between the foci and the length of the major axis of the ellipse. [§ pts] distance between foci = L mm length of major axis = L mm (d) Divide the di i e the distance between the foci by the length of the major axis. This quantity is known as the eccentricity, “e.” [S pts] (distance between foci) W E ) g 2 e= hat is the eccentricity of the elli 2 2 pse youdrew? e= [i 5] [ to 2 places) (length of major axis) (¢) What famili e . AT amiliar shape (it is a conic section) is an ellipse with an eccentricity e = 0,07 C t ( C(‘-L

Lab 4/ An Introduction to Kepler's Three Laws (f) Sketch Pluto’s orbit, which has e = 0.245. (Note that cccentricity, e, is a quantity without units.) [10 pts for df.fl"fingl For a string loop that is 254 mm (10 in) long, the tacks or pins must be 99 mm (3.9 in) apart. (Note that the 10-inch 13“_8‘1‘ refers to the length of the loop. The length of the string /il be doubl 20 inches long ) Attach your S‘ii’i‘l‘lé&““s ez report. [H at: This is a large cllipse. You s séther 4 shieets of copy paper in order ! o fit this ellipse.] o From your ellipse of Pluto’s orbit, determine the cccentricity: epiuo = 2 q ?/ an aphelion distance () Eris, the largest Kuiper Belt Object found to date, has a perihelion distance (closest approach) of 38 AU and “Calculator Two.” (farthest approach) of 97 AU. Go to Eclipse Calculator at http:/www.1728.com/ellipse.htm; scroll down to “C 0 Click on the radio button for “Perihelion Distance and Aphelion Distance.” Input the perihelion and aphelion distances for Eris And press “calculate.” |5 pts] eccentricity = _’_[1}’?’__-— {round o 3 places] foci distance = __Sj__”— AU [state AKepler's Third Law [15 pts] Kepler's third law states that the periods (P) and semi-major axes (a) of bodies orbiting a common object are related by 2 2 Pbmm i, PhndyZ 3 T Bpossr Vo2 In this section, you will verify this law for the five largest moons of Jupiter: Almathea, lo, Europa, Callisto, and Ganymede. (a) Create a table to hold the values of orbital period (P), semi-major axis (a), and P2/a? for all five moons. [Table is worth 15 pts] Jupiter’'s Moon Orbital Period (P) Semi-Major Axis (a) Pa® Aimathea A c 1S i 061667, J lo 66 s Z_ZS mm Své(gbsb Europa \S o 2 L ()O%({Zg Ganymede q/b s 97 ,5 e .Ob 3556 Callisto b S s \(7 O mm | 00 %Z’LS J (b) Mea.tswe the semi-major axes of the moons on the screen with a ruler, (Use millimeters.) (Hint: Measure the entire width of each orbit—the major axis—and divide by 2 to get the semi-major axis a.) (c) Measure the orbital periods either by noting the times in the movie or by timing with a watch (in seconds). (d) I:ook at your valu::s or P%/a*: Considering the limitations of the animation and your ability to measure accurately distance and time, does Kepler’s third law hold? Your numbers are probably not exactly as you expected. Comment on sources of error. Batth o0 M gumid e dn Qraph, He 31y loWw J digifly polds ab ¥ Awmbacs vire hot ot g much aJ 5 exfe<. PrAm Te he The instructor will supply the animated GIF of Jupiter’s moons. Save the .GIF file to a local drive and use QuickTime to view it so you can stop and start the movie at will, For Callisto and Gany i i i i ) i 510 4 ymede, you will have to time half an orbital period and th i two to get the complete period. Note that the one moon whose label is very hard to see is To. % B AdKepler's Second Law [40 pts] Kepler’ j i ipti i pl:;:‘ eetr; :;clgr::elasvlvl :l;':\su;sol‘hal.(;bjcclsT 1: cllllpllcal orbits sweep out equal areas in equal times. This implies that the orbital speed of a uniform. The planet moves fastest at the point closest to the sun (kn iheli ! h i ! S S own as the perihel ;)hlet;x.)mt fgnhest away (kn(.nt/n as aphelion). In this section, you will calculate the difference in th il R g uto's orbit has an eccentricity e = 0.25. Tts semi-major axis, a, is 5.9 x 10? km. (Note: € sure to. speed using Pluto as an example. @ scientific not )

Lab 4/ An Introduction to Kepler's Three Laws cis the distance between the center of the ellipse (orbit) and one of the foci, F, which typically is the Sun or a planet (like Earth) that is being orbited. C a = semimajor axis