LAB 2

1 EXPERIMENT# 2 KEPLER’s LAWS By reproducing ellipse via the “string -and- pencil method,” the students will draw ellipses and determine the eccentricities; by measuring the orbits of five of Jupiter’s moons, and the students will test Kepler’s third law; and by using characteristics of Mercury’s orbit, the students will confirm Kepler’s second law. APPARATUS String, pin, paper, ruler, and a computer THEORY: 1. Kepler’s First Law: Orbiting objects travel in elliptical paths with the central mass at one focus. 2. Kepler’s Second Law: Objects in elliptical orbits sweep out equal areas in equal times. This implies that orbital speed of the a planet around the sun is uniform – it moves fastest at the point closest to the Sun (known as the PERIHELION) and slowest at the point farthest away (known as APHELION) a. To do this, we need to find the area swept out by planet’s orbit. This can be approximately described as a triangle with: i. Area = ½ x (Distance to the Sun) x (Current Velocity) x (Time) b. The law states that planets sweep out equal areas in equal times: this means that the area swept out in a fixed time interval (say a week) is the same at the perihelion as it at the aphelion. Therefore we can say: i. ½ x (D perihelion ) x (V perihelion ) x (time) = ½ x (D aphelion ) x (V aphelion ) x (time) c. And finally the ratio between distances and orbital speed can be found as: i. (D perihelion ) x (V perihelion ) = (D aphelion ) x (V aphelion ) 3. Kepler’s Third Law : The periods and semi-major axes of bodies orbiting a common object are related by a. 𝑃 𝑏𝑜𝑑𝑦1 2 ? 𝑏𝑜𝑑𝑦 1 3 = 𝑃 𝑏𝑜𝑑𝑦2 2 ? 𝑏𝑜𝑑𝑦 3

2 PROCEDURE 1. Kepler’s First Law: In this section you will get acquainted with ellipses by sketching one yourself a. Get two thumb tacks and a piece of string. On your paper, place the two tacks a small distance apart, pinning down the ends of the string. Be sure to leave some slack in the string b. Using the string as a guide (i.e., place the pencil inside the string loop and pull the loop taut), draw an ellipse. As shown in image 1. Image 1: Image of drawing the ellipse. Source: https://www.grasshopper3d.com/forum/topics/gardener-ellipse-with-grasshopper-and-galapagos c. Now measure with a ruler and write down the distance between the foci AND the length of the major axis of the ellipse. i. Distance between foci = ____10.6_______ cm ii. Major axis = ___17.1______ cm

4 2. Kepler’s Second Law: In this section, you will calculate the difference in the orbital speed using Pluto as an example. Mercury’s orbit has an eccentricity e=0 .206. Its semi-major axis is 5.79 x 10 7 km. a. Determine the distance (D aphelion ) between the Sun and Mercury at aphelion. You should be able to determine this using just the semi-major axis and the eccentricity. D aphelion = a+ea = a(1+e) where a is the semi-major axis and e is the eccentricity i. D aphelion = (5.79 x 10^7) x (1+ 0.206) = 69827400 km b. Determine the distance (D perihelion ) between Mercury and the sun at perihelion. Again, you should be able to determine this using just the semi-major axis and the eccentricity. D perihelion = a-ea = a(1-e) i. D perihelion = (5.79 x 10^7) x (1- 0.206) = 45972600 km ii. c. Kelper’s Second Law allows use to determine the ratio between Mercury’s velocity at aphelion and perihelion: v aphelion /v perihelion . To do this, you will need to use the equation: i. (D perihelion ) x (V perihelion ) = (D aphelion ) x (V aphelion ) 1. v aphelion /v perihelion = 45972600 / 69827400 = 0.658 Hint: You have found D aphelion and D perihelion in parts a and b, now you want to find the ratio between the velocities. d. Mercury’s minimum orbital velocity is 47.9 km/sec. Determine the values for v aphelion and v perihelion i. V aphelion = x / y = 0.658 with x= V aphelion and y = V perihelon = 47.9 km/s so 0.658 x 47.9 = 31.51 km/s ii. V perihelon = 47.9 km/s since it corresponds to the minimum velocity Hint: first you need to decide that the given velocity of 47.9 km/sec belongs to which one of v aphelion and v perihelion .

5 3. Kepler’s Third Law: In this section you will verify this law for the five largest moons of Jupier: Almathea, Io, Europa, Callisto, and Ganymede. a. Create a table (Table 1) to hold the values of orbital period (P), semi-major axis (a), and P 2 /a 3 for all five moons. Use the appendix of your textbook to find the values. Look at your values of P 2 /a 3 : i. Does Kepler’s Third Law hold? YES Table 1: P 2 /a 3 rations by using actual periods and semi-major axis values Moons of Jupiter Orbital Period, P (days) Semi-major axis, a (km) P 2 /a 3 Almathea 0.498 1.828x10 5 0.498^2 / (1.828x10^5) ^3 = 4.06003812 ^-17 Io 1.769 4.23x10 5 4.13461012 ^-17 Europa 3.551 6.77x10 5 4.06382819 ^-17 Ganymede 7.155 1.07x10 6 4.17895739 ^-17 Callisto 16.689 1.88x10 6 7.70452251 ^-18 b. Go to the website: http://www1.phys.vt.edu/~jhs/phys1155/orbits_of_jovian_moons.gif c. Measure the semi-major axes of the moons on the screen with a ruler and record in Table 2. d. Measure the orbital periods either by noting the times in the movie or by timing with a watch and record in Table 2. e. Look at your new values of P2/a3: i. Does Kepler’s Third Law hold? YES Table 2: P 2 /a 3 rations by measuring periods and semi-major axis values from the simulation.

7 4. Calculate the percent difference of each P2/a3 from the average P2/a3 found in Step 3 and record in Table 4. Table 4: Percent Difference from Table 2: Moons of Jupiter P 2 /a 3 Percent Difference (%) Almathea 4.0401 (4.0401 - 1002952.467) / 1002952.467 = -100,00% Io 799.899806 -99,92% Europa 9862.55613 -99,02% Ganymede 193916.651 -80,67% Callisto 4810179.19 379,60% 5. Your numbers are probably not exactly as you expected. Comment on sources of error in your measurements. For me, we need to add absolute value in the calculus for them to be positive and for them to make some kind of sense. It is also hard to not make any mistake with this type of calculus so I checked it with excel but i possibly made a mistake with the first measurements because of the screen.