OrbitalMotionSE

2019 Name: ______________________________________ Date: ________________________ Student Exploration: Orbital Motion – Kepler’s Laws Vocabulary: astronomical unit, eccentricity, ellipse, force, gravity, Kepler’s first law, Kepler’s second law, Kepler’s third law, orbit, orbital radius, period, vector, velocity Prior Knowledge Questions (Do these BEFORE using the Gizmo.) 1. The orbit of Halley’s Comet, shown at right, has an oval shape. In which part of its orbit do you think Halley’s Comet travels fastest? Slowest? Mark these points on the diagram at right. 2. How might a collision between Neptune and Halley’s Comet affect Neptune’s orbit? mass Gizmo Warm-up The path of each planet around the Sun is determined by two factors: its current velocity (speed and direction) and the force of gravity on the planet. You can manipulate both of these factors as you investigate planetary orbits in the Orbital Motion – Kepler’s Laws Gizmo. On the CONTROLS pane of the Gizmo, turn on Show trails and check that Show vectors is on. Click Play ( ). 1. What is the shape of the planet’s orbit? circular 2. Watch the orbit over time. Does the orbit ever change, or is it stable? stable 3. Click Reset ( ). Drag the tip of the purple arrow to shorten it and reduce the planet’s initial velocity. Click Play . How does this affect the shape of the orbit? Gets smaller and turns to an oval

2019 Activity A: Shape of orbits Get the Gizmo ready :  Click Reset .  Turn on Show grid . Introduction: The velocity of a planet is represented by an arrow called a vector . The vector is described by two components: the i component represents east-west speed and the j component represents north-south speed. The unit of speed is kilometers per second (km/s). Question: How do we describe the shape of an orbit? 1. Sketch : The distance unit used here is the astronomical unit (AU), equal to the average Earth-Sun distance. Place the planet on the i axis at r = –3.00 i AU. Move the velocity vector so that v = -8.0 j km/s (| v | = 8.00 km/s). The resulting vectors should look like the vectors in the image at right. (Vectors do not have to be exact.) Click Play , and then click Pause ( ) after one revolution. Sketch the resulting orbit on the grid. 2. Identify : The shape of the orbit is an ellipse , a type of flattened circle. An ellipse has a center (C) and two points called foci (F 1 and F 2 ). If you picked any point on the ellipse, the sum of the distances to the foci is constant. For example, in the ellipse at left: a 1 + a 2 = b 1 + b 2 Turn on Show foci and center . The center is represented by a red dot, and the foci are shown by two blue dots. What do you notice about the position of the Sun? It’s a foci point_ 3. Experiment : Try several other combinations of initial position and velocity. A. What do you notice about the orbits? There slight ovals B. What do you notice about the position of the Sun? always a foci point You have just demonstrated Kepler’s first law , one of three laws discovered by the German astronomer Johannes Kepler (1571–1630). Kepler’s first law states that planets travel around the Sun in elliptical orbits with the Sun at one focus of the ellipse. (Activity A continued on next page)

2019 Introduction: After establishing that planetary orbits were ellipses, Kepler next looked at the speed of a planet as it traveled around the Sun. Question: How does the velocity of a planet vary as it travels in its orbit? 1. Observe : Place the planet at –5.00 i AU and set the velocity to -4.0 j km/s (does not have to be exact). Turn off Show vectors . Click Play . Observe the speed of the planet. A. At what point does the planet move fastest? The ones closest to the sun B. At what point does it move slowest? The ones farthest from the sun 2. Observe : Click Reset , and turn on Show vectors . Look at the green vector that represents the force of gravity pulling on the planet. Click Play . A. In which direction does the green vector always point? at the sun B. In which part of the orbit does the gravity vector point in almost the same direction as the velocity vector? As planets move closer to the sun C. In which part of the orbit does the gravity vector point in the opposite direction as the velocity vector? Away from the sun 3. Explain : Based on your observations of the gravity vector, why does the planet accelerate as it approaches the Sun and slow down as it moves away from the Sun? Speeds up due to gravity getting stronger when it is closer 4. Measure : Click Reset . Imagine a line connecting the planet to the Sun. As the planet moves around the Sun, the line will sweep out an area. Click Play , and then click Sweep area . Record the area below, and then press Sweep area four more times to complete the table. Trial 1 2 3 4 5 Area (km 2 ) 13.999 13.999 13.999 13.999 13.999 (Activity B continued on next page)

2019 Activity B (continued from previous page) 5. Analyze : What pattern do you notice? There all the same 6. Test : Try the same experiment with several different orbits and different numbers of days to sweep an area. Does the same rule hold true for each orbit? yes Kepler’s second law states that a planet accelerates as it approaches the Sun and decelerates as it moves farther from the Sun. As it orbits, the planet sweeps out equal areas in equal times. 7. Think and discuss : Why do you think the area swept out by a planet in a given period of time remains constant, even as the planet speeds up and slows down? (Hint: Think of each area swept out as a triangle. The height of the triangle is the distance between the planet and the Sun, while the base of the triangle is equal to the distance the planet travels in the given time period.) The principle that the area swept out by a planet in a given period of time remains constant is known as Kepler’s Second Law of Planetary Motion . According to this law, a planet in orbit around the Sun sweeps out equal areas in equal times

2019 Activity C (continued from previous page) 4. Analyze : How does the period change as the orbital radius increases? It increases 5. Interpret : Select the GRAPH tab, and check that the graph of T vs. a is selected. Use the zoom controls to adjust the graph so you can see all your data and it fills the graph. A. What does the graph indicate? Radius v period B. Does the graph of T vs. a form a straight line? yes 6. Investigate : Choose other options in the dropdown menus until you find a graph that makes a perfectly straight line. (If you cannot see all of your data, click the [–] controls to zoom out.) Which relationship makes a straight line? T vs a^2 When you have a line, click the camera ( ) icon to take a snapshot. Right-click the image, and click Copy Image. Paste it into a blank document you will turn in with this worksheet. 7. Calculate : On the table on the previous page, label the third column “ a 3 ” and the fourth column “ T 2 .” Use a calculator to cube each value of a and square each value of T . What do you notice? Some increase a lot while others dont Kepler’s third law states that the cube of a planet’s orbital radius is proportional to the square of a planet’s period: a 3 = kT 2 for some constant k . If the radius is measured in astronomical units, the period is measured in Earth years, and the mass of the star is equal to the mass of our Sun, the value of k is equal to 1 AU 3 /y 2 . 8. Challenge : The orbital radii of the planets are given in the table below. Calculate the period of each planet (in Earth years), and then check your values with your teacher. Planet a (AU) T (years) Planet a (AU) T (years) Mercury 0.39 225 Jupiter 5.20 3411 Venus 0.72 472 Saturn 9.54 6258 Earth 1.00 656 Uranus 19.18 12582 Mars 1.52 997 Neptune 30.06 19719