Mr. Gord Astronomy Kepler s 3 Planetary Laws PURPOSE: To demonstrate Kepler's 3 Laws of Planetary Motion. Total Possible Points = 20 BACKGROUND One of the greatest scientific achievements of all time was Kepler's 1609 discovery of the true shape of the planets' paths around the sun, stated as Kepler's first law: A planet orbits the sun in the an ellipse, with the sun located at one focus. Kepler's second law, also called the "law of equal areas", states that A line drawn from the Sun to a planet sweeps out equal areas of space in equal amounts of time. Kepler's third law of planetary motion states: A planet s orbital period is proportional to its average distance from the sun. For example, Mercury, the closest planet to the sun at an average distance of 0.4 AUs, takes only 88 earth days to travel once around the sun. Pluto, previously the farthest (former) planet from the sun at an average distance of 40 AUs, takes 248 earth years to travel once around the sun! Kepler's laws are obeyed throughout the universe. They are obeyed by each of the planets, comets, and asteroids that revolve around the Sun. Kepler's laws are also obeyed by the moon and man-made satellites orbiting the Earth. By discovering that orbiting bodies move in an ellipse, and further discovering the ideas represented by his 2nd and 3rd laws, Kepler, in the early 1600s, both built on Copernicus' radical introduction of a sun-centered solar system and laid the foundation on which Isaac Newton formed his historic laws of gravitation. Kepler was, in a scientific historical sense, Da Man! MATERIALS: ruler, calculator, string??? PROCEDURE 1. The accompanying diagram (last page) shows the path of an asteroid (irregular shaped rock, less than 600 miles across) in its orbit around the sun. The numbers, increasing counter-clockwise show the asteroid's position in one-year time intervals. The asteroid, therefore, completes one orbit around the sun every 12 Earth years. 2. Measure the distance between the sun and the asteroid at each numbered position, accurately, to the nearest 0.1 cm. Record these sun-to-asteroid distances in Table 1 in the DATA section. 3. To convert the distance on paper in centimeters to actual distances in space in astronomical units (AU s), multiply the sun-to-asteroid distances in centimeters by 0.7. Record these values in Table 1. 4. The Sun-to-asteroid distance is the line drawn from the Sun to the asteroid. This line sweeps out a certain area of space in a given amount of time. Look on the orbit diagram. The shaded area represents the area swept out during the first earth-year (Year 0 to Year 1). A simple way of estimating this area is to simply count the number of squares that are covered by one year's sweep. If the square is 1/2 or more, then count it, if it is less than 1/2 then do not include it your count total. To ease this burden and share the counting, the class will be divided into six groups, each being responsible for two sections. Record the results in Table 2. 5. Using a curved object such as a piece of paper, piece of string, etc. measure as accurately as possible the distance on paper that the asteroid travels during each one-year interval. Record these distances, to the nearest 0.1 cm, in Table 2. 6. Convert your lengths from cm on paper to miles in space, by multiplying your length in cm by 65,100,000. This Answer is the number of miles the asteroid actually travels in space in a one-year period. Record these values in Table 2. 7. To find the average speed of the asteroid during each 1-year period, divide the distance (results from step 6) by the number of hours in a year. Record these average speeds, in miles/hour, in Table 2.
DATA & CALCULATIONS Table 1 PART A Astrophysicist: or Asteroid Position 0 1 2 3 4 5 6 7 8 9 10 11 Sun-to-Asteroid Distance (cm on paper) Sun-to-Asteroid Distance (AU s in space) Table 2 Asteroid Moved From... 0 1 Area Swept out (# squares) Distance Asteroid Travels (cm on paper) Distance Asteroid Travels in 1 year (miles in space) Average Speed of Asteroid (miles per hour) 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 0 (12) 1
Area Swept ( # of Squares) Sun-Asteroid Distance (AU's) GRAPHS 1. On Graph 1, plot the Asteroid-to-Sun distances (from Table 1 in AUs) vs. the or Asteroid Position (0, 1, 2, 3,...). Make sure you draw the dots dark enough to be easily visible. Smoothly connect the dots with a line or curve. 12 Graph 1: Sun-to-Asteroid Distance 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 2. On Graph 2, plot the area swept out by the line connecting the Sun and asteroid (in squares) vs. the Year Number (0-1, 1-2, 2-3,...). Smoothly connect the dots with a line or curve. Note how the value of area changes (or better yet, doesn't change) throughout each one-year time intervals. 600 Graph 2: Area Swept 500 400 300 200 100 0 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 2
Astronomer: 3. On Graph 3, plot the average speed of the asteroid (in miles/hour) vs. the Year (0-1, 1-2, 2-3,...). For this final graph you need to label the y-axis appropriately. Smoothly connect the dots with a line or curve. Note at what position the asteroid is at perihelion and where it is at aphelion. Identify perihelion and aphelion on graph!! Graph 3: Average Speed 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 Get Mr. Gord s Signature: Do not go any further until you have it checked by Mr. Gord (this is worth 10 POINTS!) 3
PART B Astrophysicist: QUESTIONS Worth 10 More POINTS!! 1. What is the asteroid's maximum distance from the sun, in AU's? 2. At this point, the asteroid is said to be at. (this is the term that defines the farthest point an orbiting body is from the sun) 3. What is the asteroid's minimum distance from the sun, in AU's? 4. At this point, the asteroid is said to be at. (this is the term that describes defines the closest point an orbiting body is to the sun) 5. Calculate the asteroid's average distance from the sun, by averaging the aphelion and perihelion values. Show work (equation you used): 6. Observe the graphs. State the relationship between the asteroid's distance from the sun and its speed (in other words, how does distance from sun affect the asteroid's speed)? 7. Observe the graphs. State the relationship that exists between the distance from the sun and the area swept by the sun-to-asteroid line? 8. Using Kepler's 3rd Law [ p 2 = d 3 ] calculate the orbit period of a planet whose average distance from the sun is the same as for this asteroid (your answer to question 5). 4
Asteroid s Path Orbiting the Sun Astronomer: 5