PH112Man19 Lab06.pdf

PH-112 SPACE, ASTRONOMY and OUR UNIVERSE LABORATORY LAB # 6 Testing Kepler's First Law: The Orbit of Mercury OVERVIEW About four centuries ago, the mathematician, Johannes Kepler, discovered three laws that describe the motions of the planets accurately. Using the observations of a Danish astronomer, Tycho Brahe, Kepler found that the true path of every planet was an ellipse. He stated this in his first law: Law #1: Planets travel in elliptical orbits with the Sun at one focus. Most astronomers before Kepler thought that the paths were circular or were a composite of several motions all based on the circle. Kepler's discovery brought about a genuine revolution in astronomical thinking. After examining the features of the ellipse, we will use some of Tycho Brahe's observations of Mercury to verify Kepler's first law. Part 1: The Ellipse The figure of an ellipse is made as follows: « Take a length of string and tie the ends together to make a loop. » Place the loop around two fixed points on a surface. Each of these points is called a focus. « Hold a pencil against the string inside the loop and pull it to form a triangle. While keeping the pencil point on the surface, slide the pencil along the string. The figure you draw is an ellipse. focus ““‘ ‘.Illlll.llll.ll.lll.lllllglll.l.lllla ellipse (unfinished) Rewinder: All astronomy students are required to attend one evening telescope observing session. Have you attended a session?

Lab #6: Testing Kepler's First Law (continued) Page 2 of 6 The important features of an ellipse are shown in the diagram below. The shape of an ellipse is given by the ratio c¢:a, which is called the eccentricity (e). Note that the eccentricity of a circle is zero. Tear off the paper millimeter ruler on page five of this lab and use it to find ¢ and a. Compute e. 2 e=_28 mm : 34 4 a=___ mm Jocus e=c/a= 82 semi-major axis () Place the bottom of this page on a cork board and insert a pin at each of the two foci. Place a loop of string around the two pins and draw the ellipse in the manner shown on page one. Measure ¢ and a with the millimeter ruler and compute e. c=_ 26 mm a=__ 2’ mm e=c/a= 26/57=.45

Lab #6: Testing Kepler's First Law (continued) Page 3 of 6 Part 2: The Orbit of Mercury Because its orbit lies entirely within the Earth's orbit, Mercury is never seen far from the Sun. In the illustration below, the Sun is setting in the western sky as seen from New York and Mercury is at its Greatest Eastern Elongation, which occurs when the angle Sun-observer-Mercury reaches its greatest extent. When Mercury is at its Greatest Western Elongation it can be seen in the morning sky. Tycho Brahe and other astronomers measured the greatest elongations of Mercury in order to better define the planet’s orbit. A list of some of these observations are printed on the next page. The first thing to note is that the angles are not the same. Why this is so will become clear once you have plotted Mercury’s orbit. Mercury On the next page a circle has been drawn with the months of the year along its outer edge. It is an approximate representation of the Earth's orbit around the Sun. If we know the date of an observation of a greatest elongation as well as the elongation in degrees, the actual orbit of Mercury can be plotted. « Tear off the protractor at the bottom of the page. Fold over the edges as indicated. « Use the protractor to plot the greatest elongations in the table. To do this, place the point representing the Earth on the protractor on the date of an elongation. (The first date has been plotted.) Adjust the protractor so that the zero degree line points toward the Sun. Make a mark at the elongation listed in the table and draw a line from the Earth through the mark almost to the other side of the Earth's orbit. Do this for all the dates in the table.

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Lab #6: Testing Kepler's First Law (Continued) Page 4 of 6 (represented as a circle here) 4z Tycho's Observations of 7 m a’ZCI'l Mercury at Greatest Elongation Apr 18 1588 20° East Aug 17 1588 27° East Dec 11 1588 21° East Apr 011589 19° East Jul 29 1589 27° East Nov 23 1589 22° East Mar 14 1590 18° East Jul 10 1590 26° East Nov 05 1590 23° East Feb 06 1588 26° West Jun 05 1588 24° West Sep 28 1588 18° West Jan 18 1589 24° West May 17 1589 26° West Sep 12 1589 18° West Jan 01 1590 23° West Apr 28 1590 27° West Aug 251590 18° West Dec 14 1590 21° West NOILVINOT3 NY3ISIM / 09 0§ OF 0 Oc SN\ NOILVONOT3 & % 08 NH31SVv3 0c o0e OoF 05 09 0/ NN ol b3 23+ Q104 3343+ Q104

Lab #6: Testing Kepler's First Law (continued) Page 5 of6 After all the elongation lines have been drawn, you should notice an oval region around the Sun where no lines appear. The edge of this region is the rough outline of Mercury's orbit. « Draw a smooth elliptical figure that best represents the edge of the region described above. Don't try to center the Sun in the figure you draw. If you find it at the center, check the accuracy of the elongation lines you drew. » Draw the major and minor axes by finding the largest and smallest widths of the ellipse. The Sun should lie along or close to the major axis since it is at one focus. « Label the points of perihelion, where the planet is closest to the Sun, and aphelion, where the planet is farthest from the Sun. (Do you now understand why all of Mercury’s greatest elongations are not the same?) « Find the following: 25 c=_95 mm a= mm Measured value of e = c/a = 0.2 « Look up the accepted value of the eccentricity of Mercury's orbit on the Internet or in a textbook and find the difference between your value and the accepted value in percent. 95% Accepted value of e for Mercury's orbit: TEAR ALONG BROKEN LINE, THEN FOLD THIS EDGE ¥ OVER 100 110 120 130 140 150 MILLIMETERS

Lab # 6: Testing Kepler’s First Law (continued) Part 3: Elliptical orbits in the Solar System In addition to planets, millions of smaller objects, such as comets, asteroids and spacecraft, orbit the Sun in elliptical paths. To see what the orbits of these objects look like, consult the internet to find the information needed to fill in the table below. Then sketch the orbits in the diagram below the table. Note that the scale in the diagram is in Astronomical Units or AU. Page 6 of 6 Object Orbitqli Semi-major axis |Distance f{om Sun eccentricity (AU) at Perihelion (AU) Asteroid Pallas | 0.23 2.8 2.1 Comet Tempel 1 0.52 3.12 0.34 CometEncke |85 2.91 0.34 Eovthtooupitonieg) 089 &0 Lo

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